Time-resolved infrared studies of superconducting molybdenum-germanium thin films

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Time-resolved infrared studies of superconducting molybdenum-germanium thin films
Tashiro, Hidenori ( Dissertant )
Tanner, David B. ( Thesis advisor )
Hebard, Arthur F. ( Reviewer )
Cheng, Hai-Ping ( Reviewer )
Hagen, Stephen J. ( Reviewer )
Pearton, Stephen J. ( Reviewer )
Place of Publication:
Gainesville, Fla.
University of Florida
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Subjects / Keywords:
Electrons ( jstor )
Lasers ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Phonons ( jstor )
Quasiparticles ( jstor )
Signals ( jstor )
Superconductors ( jstor )
Synchrotrons ( jstor )
Transmittance ( jstor )
Physics thesis, Ph. D.
Dissertations, Academic -- UF -- Physics


Superconducting amorphous molybdenum-germanium ([alpha]-MoGe) thin films show progressively reduced transition temperatures T sub c as the thickness is reduced. This suppression has been explained in terms of electron localization effects and reduced screening. This dissertation presents the results of both linear spectroscopy and time-resolved studies of a set of [alpha]-MoGe films to understand more fully this weakened superconducting state. The observed optical conductivity shows the presence of an energy gap. The effects of reduced thickness in these films are to depress T sub c and the superfluid density, while maintaining the normal-state resistivity. All of the results from our linear optical measurements appear to be consistent trivially with those expected for weak to intermediate coupling dirty limit superconductors. Our time-resolved study reveals the overall relaxation of the samples at a time scale on the order of 100 ps. The temperature dependence of the relaxation time seems to be consistent with the prediction based on weak-coupling BCS theory for all films we measured without changing any material parameters for different thickness. The application of magnetic field did not change the relaxation times, which was unexpected.
amorphous, BCS, conductivity, disorder, dynamics, films, infrared, lifetime, MoGe, NSLS, optical, pump, quasiparticles, recombiantion, relaxation, spectroscopy, superconductivity, superconductors, synchrotron, time
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Thesis (Ph. D.)--University of Florida, 2004.
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Copyright 2004


Hidenori Tashiro

To my mother, Kimiko Tashiro


Over the past few years, I have received full of support, encouragement, and valu-

able advice from many people. Here, I would like to acknowledge some individuals who

helped me in various v--v-. I would like to express my foremost and sincere gratitude

to Professor David B. Tanner, my research advisor. My work could not possibly have

been completed without his guidance and support. He not only continuously encour-

aged me to keep going but also supplied me just the right amount of pressure to get

things done. The many things I have learned from him will be my treasure. I am truly

grateful to Professor ('!Ci -l"1 .1!. r J. Stanton for introducing me to Dr. Tanner. I would

like equally to thank my supervisory committee members, Arthur F. Hebard, Hai-Ping

('C!, i- Stephen J. Hagen, and Stephen J. Pearton, for their guidance and reading this


I would also like to express my appreciation to Professor David H. Reitze for his

occasional advice and his step-by-step instruction for using and maintaining the laser

system used in my work, Johon M. Graybeal for supplying me a set of samples and

information about them, and ('! i l. Porter for providing me with algorithms which

helped the data analysis.

I would like to express my special thanks to G. Lawrence Carr, who was my daily

advisor at the National Synchrotron Light Source (NSLS). His guidance, input, and

patience made my project to go smoothly. I equally thank Ricardo Lobo, who is the

author of valuable programs used for data acquisition and analysis, for helping me to

start my project during his visit at the NSLS. Additional thanks go to Jiufeng J. Tu,

('!Ci -I 1.!i. r Homes, Laszlo Mihaly, Diyar Talbayev, Gregory D. Smith, Randy J. Smith,

and all of who helped me at the Brookhaven National Laboratory (BNL).

I am dearly grateful to all of my past and present colleagues in Dr. Tanner's group

for their cooperation, conversations, and mostly friendship. I am also thankful to Kevin

T. McCarthy, Stephen Arnason, Zhihong C'!li n Suzette A. Pabit, Amol Patel, Susumu

Takahashi, and Naoki Matsunaga.

I would also like to acknowledge the help of the members of the machine shop,

electronic shop, and cryogenics team of the University of Florida Physics Department,

as well as the members of technical staff at the NSLS. Acknowledgement also goes to

the Physics Department staff for their assistance, especially Jill Kirkpatrick and Darlene

Latimer for taking care of the bureaucratic details while I was at the BNL.

Finally, I would like to address my exceptional thanks to my parents, Hiroyuki and

Kimiko Tashiro, for their support over my entire life. Finally, my deepest appreciation is

due to my wife and two sons, Yasuko, Mitsuru, and Hikaru, for their constant support

and patience. They made my life so much fun.


ACKNOWLEDGMENTS ..................... .......... iv

LIST OF TABLES .................................... x

LIST OF FIGURES ...................... ........... xii

ABSTRACT ....................................... xv


1 INTRODUCTION ................................. 1

1.1 Introductory Remarks ........................... 1
1.2 M otivation . . . . . . . . 2
1.3 Organization ................... ............. 3

2 OPTICAL PROPERTIES ............................. 4

2.1 Introduction . . . . . . . . 4
2.2 Optical Phenomena ............................. 4
2.3 Interaction of Light with Matter .................. 7
2.4 Experimental Determination of Optical Constants . . ... 15
2.4.1 Reflection and Transmission at a Plane Interface . ... 15
2.4.2 Kramers-Kr6nig Dispersion Relations . . . 16
2.4.3 Reflection and Transmission at Two Parallel Interfaces ...... .18
2.4.4 Optics in Thin Film on a Substrate . . ..... 20
2.4.5 Photoinduced Absorption ................ .. 23
2.5 Microscopic Models ............... ......... .. 23
2.5.1 Lorentz M odel ....... . . . 24
2.5.2 Free Carrier Response and Drude Model . . 27
2.5.3 Drude-Lorentz Model ................ .. .. 31
2.5.4 Sum Rules ............... .......... .. 31

3 FOURIER SPECTROSCOPY ............... ....... .. 32

3.1 Introduction ............... ............ .. 32
3.2 Fourier Transform Interferometry ............... .. .. .. 33
3.2.1 General Principles .................. .. .... .. .. 33
3.2.2 Finite Retardation and Apodization . . 36
3.2.3 Sampling .............. . . ... 38
3.2.4 Phase Error and Correction ................... ... 41
3.2.5 Step-Scan and Rapid-Scan Interferometers . . ... 43
3.3 Polarization Modulation .................. ..... .. 45

4 SUPERCONDUCTIVITY .............................

4.1 Introduction . . . . . . . . .
4.2 Fundamentals of Superconductivity .. ................
4.2.1 Fundamental Phenomena .. ..................
4.2.2 Thermodynamic Properties .. .................
4.2.3 Types of Superconductor .. ..................
4.2.4 Length Scales . . . . . . .
4.2.5 B C S Theory . . . . . . . .
4.2.6 Eliashberg Formalism .. ....................


5.1 Synchrotron Radiation .. ......................
5.1.1 Introduction . . . . . . . .
5.1.2 Radiated Power from a Bending Magnet .. ............
5.1.3 Angular Collimation and Polarization .. ............
5.1.4 RF Cavity and Pulsed Nature .. ...............
5.1.5 Beam Lifetim e . . . . . . .
5.1.6 Infrared Synchrotron Radiation .. ..............
5.1.7 Source Comparison .. ....................
5.2 Principle of Pump-Probe Studies .. .................
5.2.1 Laser-Synchrotron Pump-Probe Measurement .. .........
5.2.2 Interferometry Using Pulsed Source .. .............
5.2.3 Advantage of Laser-Synchrotron Technique .. ..........

6 EX PERIM ENT . . . . . . . . .

6.1 Introduction . . . . .
6.2 National Synchrotron Light Source .. .....
6.2.1 G general . . . . .
6.2.2 Vacuum Ultraviolet Ring .. .......
6.2.3 Beamlines U12IR and U10A .......
6.3 Spectrometers .. ................
6.3.1 Bruker IFS 66v/S .. ...........
6.3.2 Bruker IFS 125 HR .. ..........
6.3.3 Sciencetech SPS-200 .. ..........
6.4 Pump Laser System .. .............
6.4.1 System Overview .. ..........
6.4.2 Mode-locked, Solid-State Ti:Sapphire Las'
6.4.3 Optics and Light Distribution ......
6.4.4 Laser-Synchrotron Synchronization .
6.5 Other Experimental Components .. ......
6.5.1 Oxford Optistat Bath Cryostat ......
6.5.2 Ox-Box Custom-made Sample C('i ihler .


Oxford Instruments Vertical-bore Supercoi
D etectors . . . . .
Ratio Box . . . . .
Fiber Optic Cable and Pulse Delivery .

r . . .

iducting Magnet


6.6 Experimental Techniques and Setups .................. 129
6.6.1 Photoinduced Measurements ............. .. 130
6.6.2 Laser Insertion ............. . . ...... 136


7.1 Introduction ................... . . ..... 140
7.2 Background ................ ............. ..141
7.2.1 Infrared Properties of Superconductors . . ..... 141
7.2.2 Effects of Disorder upon 2D Superconductivity . .... 145
7.2.3 2D Model Systems ............... .... .. 147
7.3 Experimental Details .................. ....... 148
7.3.1 Samples .................. . . ..... 148
7.3.2 M easurements .................. ....... 151
7.4 Analysis ............... ............. ..151
7.5 Discussion ................... . . ... 154
7.6 Conclusion . . . . . . . .... 156


8.1 Introduction ................. . . 159
8.2 Background .................. ............. .. 160
8.2.1 Nonequilibrium Superconductivity . . 160
8.2.2 Temperature Dependence of Lifetimes . . .... 165
8.3 Experimental Details ......... . . . ..... 169
8.3.1 Time-resolved Measurements: Quasiparticle Decay . ... 171
8.3.2 Photoinduced Gap Shift Measurements . . 174
8.3.3 Fluence Dependence ................. .... 176
8.3.4 Spectrally Averaged Far Infrared Transmission . .... 178
8.4 Analysis and Discussion ............... .... .. 179
8.4.1 Relaxation Times .................. .. ..... 179
8.4.2 Photoinduced Gap Shift . ........... . 185
8.5 Conclusion .................. . . 187


9.1 Introduction ............... . . . ..... 189
9.2 Transmittance Ratio in Magnetic Fields ..... . . 190
9.3 Relaxation Times in Magnetic Fields .................. .. 191

10 SUMMARY AND CONCLUSION .................. ..... 192


A VUV STORAGE RING PARAMETERS ............... .. 197

B INFRARED BEAMLINES .................. .......... 199

B.1 Infrared Programs at NSLS VUV ring .................. 199
B.2 Hight Resolution Far-infrared Spectra at U12IR . . 200


C.1 Hazardous Beam Control .......... ............... 201
C.2 Personal Protective Equipment ............ ... . 201
C.2.1 Eye Protection .................. ......... .. 201
C.2.2 Skin Protection .................. ........ .. 202
C.3 Laser Safety Training .................. ......... .. 202
C.4 Alignment .................. .............. 202
C.4.1 Gross Alignment .................. ...... .. .. 202
C.4.2 Fine Alignment .................. ...... .. .. 202
C.4.3 At Beamline Endstation. ................. .. 203
C.5 Daily Operation Procedure .................. ..... 203
C.6 Optimization of the Downstream Optics ................ .. 205
C.7 ANSI Laser Classifications .................. ..... 206

D USEFUL INFORMATION ........... ...... . . .. 209

D.1 Frequency Ranges .................. ......... .. .. 209
D.2 Energy and Pressure Units Conversion ................. .. 211
D.3 Gas-phase Contamination .... ............ ..... .. 211

REFERENCES ................... ..... .... ....... 213

BIOGRAPHICAL SKETCH ........... ....... . .... 219


Transition temperatures for several superconductors

6-1 Operation modes of the VUV ring .

6-2 Frequency ranges of various sources .

6-3 Frequency ranges of various beam-splitters

6-4 Frequency ranges of various detectors .

6-5 Specifications of the SPS-200 . ..

6-6 Specification of the Mira . ....

6-7 Properties of Oxford cryostat windows .

6-8 ('!CI i i:teristics of fiber optic cable .

7-1 c-MoGe film parameters . .

7-2 ,I / 4, fitting parameters . ...

7-3 Values of N, and A ....... ......

8-1 a-MoGe film used for timing experiment

8-2 Parameters for the timing experiment .

8-3 Fluence dependence data . ....

8-4 Teff and A at various temperatures .

8-5 Material parameters in TRo and TBo .

8-6 T R To, TBO, and -Ro/TBO . .

8-7 2Ao Bo/TRo ....... .........

8-8 Relaxation times and material parameters

8-9 Photoinduced gap shifts . ....

A-1 VUV storage ring parameters . .

A-2 VUV storage ring's arc source parameters

. ... . 88

. ... . 95

. . 96

. ... . 96

. . .. . 10 1

. . .. . 0 3

. ... . 14

. . ... . 2 7

. . .. . 4 9

. . .. . 5 2

. . . . . 5 5

.. . . 70

. ... . 17 1

. . .. . 7 7

. . ... . 8 0

. ... . 8 2

. . .. . 8 2

. . . . . 8 3

from Kaplan et al. . . 184

. . .. . 8 6

. . .. . 9 7

.. . . 98

A-3 VUV storage ring's insertion device parameters ......





NSLS linac parameters . .....

NSLS booster parameters . ...

NSLS booster magnetic elements . .

Infrared beamlines of the VUV ring .

Infrared spectral regions . ....

Frequency ranges of conventional light sou

Frequency ranges of detectors . .

Spectral ranges of beam-splitters . .

Transmission range of optical window and

Relations between energy units . .

Relations between pressure units . .

Absorption peaks due to air . ..

rces .

filter mater

. .. . 98

. .. . 98

. .. . 98

. .. . 99

. .. . 209

. . 209

. .. . 209

. .. . 210

rials . ... 210

. .. . 211

. .. . 211

. .. . 212

Figure page

2-1 Reflection and transmission at two parallel interfaces . ..... 18

2-2 Reflection and transmission with a thin film on a substrate . ... 21

3-1 Spectrometer classification .................. ........ .. 32

3-2 Schematic view of a Michelson interferometer . . . 34

3-3 Sine function convolved with a single spectral line ........... .37

3-4 Comparison of the Happ-Genzel and boxcar apodization . ... 39

3-5 Relation between spectrum replication and sampling rate . ... 40

3-6 Two sine waves drawn through the same sampling points . ... 41

3-7 Schematic view of a Martin-Puplett interferometer . . 46

3-8 Interferograms produced by a polarizing interferometer . .... 47

4-1 Density of states for a BCS superconductor ................. ..65

4-2 Variation of the gap with temperature in the BCS approximation . 66

5-1 Incoherent and coherent radiations .................. ..... 72

5-2 Angular distribution of the radiation .................. .. 72

5-3 Angular distribution of polarization components .............. ..74

5-4 RF and higher harmonic cavity voltages .................. 76

5-5 Natural opening angle of IRSR with the VUV ring parameters ...... ..78

5-6 Power comparison between blackbody and synchrotron . .... 79

5-7 Brightness comparison between blackbody and synchrotron . ... 80

5-8 Principle of the pump-probe experiment ............... .. 81

6-1 C('!i 1, of the pulse width during the detuned mode operation ...... ..89

6-2 Elevation view of U12IR beamline .................. ...... 91

6-3 Transmitted power with and without a light cone ............. ..92

6-4 Emission spectra of conventional internal sources .............. ..95

6-5 Bruker IFS 125HR . ........

6-6 Sciencetech SPS-200 . .

6-7 Optical schematic of the Mira laser head .

6-8 Optical Layout of the U6 laser system .

6-9 Effects of the pulse pickers . ....

6-10 Timing scheme . ..........

6-11 Synchronized laser and synchrotron pulses

6-12 Oxford Optistat bath cryostat . ..

6-13 Custom made sample compartment .

6-14 Off-axis paraboloidal reflector . ..

6-15 Oxford magnet setup . .......

6-16 Transfer function . .........

6-17 Classification of detectors . ....

6-18 Composite bolometer . ......

6-19 Structure and profile of typical optical fiber

6-20 Experimental setup for timing experiment

6-21 New dithering scheme . ......

6-22 Laser insertion setups with Optistat .

6-23 Laser insertion setup with Heli-tran .

6-24 Coupling of diode laser with optical fiber ca

7-1 Mattis-Bardeen relative conductivity and tr

7-2 R i vs. 1/d . . . . .

7-3 T,/To vs. R. . ...

7-4 Measured transmittance ratio . ..

7-5 Mattis-Bardeen fit to / . .

7-6 Measured reflectance ratio . ....

7-7 Mattis-Bardeen fit to J, / . .

7-8 Optical conductivities of a-MoGe . .

7-9 A, N vs. R ....... ........

. .. . 98

. .. . 99

.. . 102

... . 105

. .. . 0 7

. .. . 0 8

.. . . 10

. .. . 11 1

. ... . 17

. .. . 18

. .. . 19

. .. . 12 1

. .. . 2 2

. .. . 24

cable . . ..... 128

. . . 134

. .. . 135

. ... . 137

. ... . 138

ble . . .... 138

ansmittance . . 143

. . . . . 4 9

. . .. . 5 0

. . .. . 5 2

. . .. . 153

. . .. . 5 4

. . .. . 5 5

. . .. . 5 6

. . . . . 5 7

10 N vs. T . . . . .

1 Simplified relaxation processes . ............

2 Universal temperature dependence of lifetimes . ....

3 Differential, integrated signal vs. time . ........

4 Quasiparticle decay signal vs. time and model function .

5 Points on a decay curve for gap shift measurements . .

6 Fluence dependence of eff . .....

7 Spectrally averaged far-IR transmittion vs. T/T . ..

8 Quasiparticle decay signal for 16.5 nm film . .....

9 Teff vs. T/Tc ... . ...................

10 2AoBo/T o Vs. T ... ...........

11 TRO/TBO vs. b . . . . . . .

12 Photoinduced spectral changes . ............

1 Measured transmittance ratio in magnetic fields . ...

2 Quasiparticle decay signal in magnetic field for the 33 nm film

1 High resolution far-IR synchrotron spectra at U12IR . .

. . 158

. . 163

. . 168

. . 172

. . 73

. . 175

. . 176

. . 178

. . 180

. . 181

. . 183

. . 185

. . 186

. . 190

. . 191

. . 200

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Hidenori Tashiro

December 2004

C'!I i': David B. Tanner
Major Department: Physics

Superconducting amorphous molybdenum-germanium (a-MoGe) thin films show

progressively reduced transition temperatures T, as the thickness is reduced. This

suppression has been explained in terms of electron localization effects and reduced

screening. This dissertation presents the results of both linear spectroscopy and time-

resolved studies of a set of a-MoGe films to understand more fully this weakened

superconducting state. The observed optical conductivity shows the presence of an

energy gap. The effects of reduced thickness in these films are to depress T, and the

superfluid density, while maintaining the normal-state resistivity. All of the results from

our linear optical measurements appear to be consistent trivially with those expected

for weak to intermediate coupling dirty limit superconductors. Our time-resolved

study reveals the overall relaxation of the samples at a time scale on the order of 100

ps. The temperature dependence of the relaxation time seems to be consistent with

the prediction based on weak-coupling BCS theory for all films we measured without

changing any material parameters for different thickness. The application of magnetic

field did not change the relaxation times, which was unexpected.


1.1 Introductory Remarks

Spectroscopy is a very useful technique for investigating the properties of various

types of materials. Information on the material is encoded in the radiation spectrum

modified through interaction with the material. The extensive energy range covered by

electromagnetic radiation allows us to study many properties (e.g., electronic, magnetic,

lattice, and so on) depending on the frequency ranges. For example, the conductivity

peak for free carriers is centered at zero frequency. Lattice vibrations (i.e., phonons)

interact with electromagnetic radiation at far-infrared frequencies. Electronic transitions

across the energy gap of a semiconductor like Si happen in the near infrared. Transitions

from core levels require even higher energies. In superconductors, an energy gap

develops in the electronic density of states around Fermi energy. The typical energy

scale of this gap is in meV, which corresponds to the frequencies between microwave and

far infrared. Thus, optical studies in this frequency range provide an important tool for

investigating superconductors.

A synchrotron is a source of high brightness electromagnetic radiation emitted from

electrons orbiting around a closed path of a storage ring. It is a very broad-band source,

extending from the very far infrared to the hard x-ray. Because electrons are bunched

as they travel, the radiation emitted from the synchrotron source is pulsed. Beamlines

U10A and U12IR are two beamlines on the VUV ring at the National Synchrotron

Light Source (NSLS) of Brookhaven National Laboratory (BNL) dedicated to solid-state

physics experiments. The NSLS provides a powerful, tunable, near-infrared/visible

mode-locked Ti:Sapphire laser, which produces pulses of a few picoseconds duration

synchronized to the synchrotron pulses. A specimen can be excited (pumped) by laser

pulses, and probed by infrared subnanosecond-duration synchrotron pulses from the

VUV ring [1, 2]. Depending on the mode of synchrotron operation, this facility provides

a unique opportunity for measuring transient phenomena on time scales in a few 100

ps up to 170 ns range over a broad spectral region. With proper tuning of excitation

energy, the dynamics of various systems can be investigated: quasiparticle relaxation

in conventional BCS superconductors, recombination of the photogenerated electron-

hole plasma in semiconductors, dynamics of the photodoped polarons and solitons

in conducting polymers, and relaxation of photoinduced conductivity effects in the

insulating phase of high-T, superconductors, to name a few. Use of a superconducting

magnet also allows us to study materials in magnetic fields.

1.2 Motivation

This dissertation describes optical studies on a set of superconducting amorphous

molybdenum-germanium (a-MoGe) thin films deposited on thick insulating substrates.

All measurements were performed at the two beamlines on the NSLS VUV ring. It is

well known that increasing disorder leads to localization and the related enhancements

of the repulsive electron-electron Coulomb interaction [3, 4, 5]. It is also well known

that in superconductors, two electrons form a pair due to phonon mediated attractive

interaction. Thus, the enhanced Coulomb interaction inherently competes with super-

conductivity. a-MoGe is a well-studied disordered system used for studying the interplay

between superconductivity and disorder. Its transport properties were investigated by

Graybeal [6, 7, 8], and showed progressively reduced transition temperatures (i.e., a

weakened superconducting state) as the thickness is decreased. In this work, we have set

out to understand this system even further, and possibly find some sort of connections

between reduced transition temperature and the degree of disorder. For this purpose,

we first undertook a thorough study of linear spectroscopy in the far infrared. Then,

the dynamics of the system was studied by a pump-probe technique. Driving super-

conductors to nonequilibrium states corresponds to breaking Cooper pairs, producing

excitations called quasiparticles. In returning to equilibrium, these quasiparticles recom-

bine into pairs, releasing energy usually as phonons. The rate at which this relaxation

progresses involves the interaction between quasiparticles, which is of fundamental

interest for any theory of superconductivity.

1.3 Organization

In C'!i lpter 2 we review a basic theory of optical properties. Some of the common

techniques and models used for extracting optical parameters from experiments are

discussed. C'!i lpter 3 provides the concepts of Fourier spectroscopy. Two types of

interferometers are described: amplitude modulation and polarization modulation.

C'! Ilpter 4 summarizes a few fundamental properties of superconductors within the

framework of BCS theory. The first part of ('!i lpter 5 describes the properties of

synchrotron radiation. Comparison to conventional thermal sources reveals advantages

of using synchrotron radiation for spectroscopic study particularly in the far infrared

frequencies. The second part of the chapter is intended to introduce the basic idea of

the pump-probe technique using a laser as the pump source and synchrotron radiation

as the probe source. Reading this part of the chapter prior to reading the following

chapters is recommended. ('!i lpter 6 describes in detail all experimental components,

techniques, and setups used in this project. The specifics of the apparatus such as the

NSLS VUV storage ring, beamlines, spectrometers, laser system, and others are all

included in this chapter. ('!i lpters 7 and 8 discuss our experimental results of linear

and time-resolved study on a-MoGe thin films, respectively. Each chapter contains a

theoretical background specific to the analysis used in the chapter. Finally, Chapter 9

shows the results of our most recent experiment of magneto-optical measurements.

In optical studies, different energy units are quite common for the different tech-

niques, frequency ranges, and disciplines. In the infrared spectral region, the most

commonly used unit is the wavenumber given by cm-1. It is defined as the frequency in

Hz divided by the speed of light in cm/s or the reciprocal of wavelength in cm. In this

dissertation, the wavenumber is used interchangeably with frequency or energy since the

values for these units are simply related.


2.1 Introduction

In this chapter, we will provide a general background of the theory of optical

properties. The chapter begins with very introductory discussion of a number of

phenomena that can occur as light propagates through a medium and the coefficients

that are used to quantify them. The basic results of Maxwell's equations also will be

summarized. In the following section, we will show several techniques for extracting

optical parameters from experimental data measured on samples in common forms. At

the end several microscopic models will be developed to explain the optical phenomena

that are observed in the solid state. The main purpose of the chapter is to give a basic

theory and techniques in general terms, and also to be used as quick reference.

2.2 Optical Phenomena

In the simplest way, reflection, transmission, and propagation are the three sim-

plest groups of optical phenomena that are observed in solid state materials. When a

light wave propagating in one medium encounters another medium, some of the light is

reflected from the interface, while the rest transmits into the medium and propagates

through it. The light also experiences a v ii, I v of optical phenomena during the propa-

gation within each medium. The light rays are bent at the interface due to the change

in the velocity of the light wave in different media. This is known as refraction, and is

described by Snell's law. The light may be attenuated as it propagates caused by pro-

cesses such as absorption or scattering. Absorption occurs if the frequency (i.e., energy

of photon) of the light is resonant with the transition energy of the atoms and electrons

in the medium. Hence, absorption causes reduction of the number of photons in the

forward direction. In the event of scattering, the light beam is re-directed in other direc-

tions caused by the presence of impurities, defects, or inhomogeneities. This obviously

causes attenuation in the original direction in an analogous way to absorption even

though the number of photons is unchanged. Scattering can also accompany changes in

frequency of the light. If the frequency of the scattered light is changed, it is said to be

inelastic; if it is unchanged, it is said to be elastic. Luminescence is the phenomenon of

spontaneous emission of light by the excited atoms in a medium. The light is emitted in

all directions, and has a different frequency to the incoming light.

With conventional sources of light, optical properties are described by linear optics,

where it is assumed that quantities such as the refractive index, absorption coefficient,

and reflectivity are independent of light intensity. This is based on an approximation

that is only valid in the low intensity limit, and practically everything we will be

discussing in this dissertation falls into the realm of linear optics. When high intensity

light propagates through a medium, a number of nonlinear phenomena can occur.

Frequency doubling and tripling are examples of these nonlinear effects, and are realized

through the use of lasers. This is the subject of nonlinear optics, where it allows the

electric susceptibility as well as all the properties that follow from it to vary with the

strength of the electric field of the light beam. Even though nonlinear optics is an

interesting subject in its own right, it will not be discussed further.

Optical phenomena can be quantified by a number of optical constants (optical

parameters) that describe the macroscopic behavior of the medium. The reflection from

an interface of different media is described by the reflectance J, defined as the ratio

of the power of reflected light to that of incident light on the surface. Transmission

through the interface, on the other hand, is described by the transmittance 3, defined

as the ratio of the transmitted power to the incident power. At the every interface the

light wave encounters, conservation of energy requires that

+ 1. (2.1)

The refraction is described by the refractive index n, defined as

n (2.2)

where c and v are the speed of light in free space and in the medium, respectively.

The refractive index depends on the frequency of the light wave, which is known

as dispersion, and characterizes the propagation of the light through a transparent

(i.e., non-absorbing) medium.

The absorption of light by a medium is quantified by the absorption coefficient a,

defined as the fraction of the power absorbed in a unit length of the medium. In terms

of a differential equation, the effect of absorption is given by

dl -adx I(x) (2.3)

where I(x) is the intensity of light at position x. The solution to this equation is the

exponential decay of light intensity as it propagates through the medium:

I(x) = 1o e-C (2.4)

The frequency dependence of the absorption coefficient is responsible for the color of


Rayleigh scattering is caused by variations of the refractive index of the medium on

a length scale smaller than the wavelength of the light. As mentioned earlier, scattering

has similar attenuation effect as absorption, and the intensity of light as it propagates

can be expressed by

I(x) = Io e-Nez (2.5)

where N is the number of scattering center per unit volume, and o-, is the scattering

cross-section of the scattering center.

The frequency dependent refractive index n(w) and absorption coefficient a(w)

are the two important quantities that characterize the propagation of light wave in a

medium since they describe the dispersive and absorptive nature of a material in the

most direct way. The refraction and absorption of a medium can be described by a

single quantity called the complex refractive index:1

N n+ i (2.6)

where n (real part of N) is the refractive index defined in Eq. 2.2, and K (imaginary

part of N) is the extinction coefficient that is directly related to the absorption coeffi-

cient a as will be discussed shortly.

A few examples of optical quantities that have been discussed so far provide

descriptions of the optical phenomena only from the point of view of the reflection,

transmission, and propagation, and offer the most useful information to manufacturers

of optical elements. The frequency dependent reflectance J (w) and transmittance 7(w)

are the experimentally measurable quantities from which other parameters such as the

complex refractive index N can be derived. The microscopic models, however, usually

enable us to calculate other parameters such as the complex dielectric function E and

complex conductivity a rather than N. The relationship between E and N provides

direct connection between microscopic models of materials and propagation properties of

electromagnetic waves.

2.3 Interaction of Light with Matter

This section summarizes the principal results of electromagnetism that are sufficient

for the study of optical properties of solids. Details of the subject can be found in most

books on optics and electromagnetism [9, 10, 11, 12, 13, 14, 15, 16]. CGS units are used

throughout unless otherwise specified.

The response of a material to external electric fields E is characterized by a few

macroscopic vectors: polarization P, electric displacement D, and current density J.

Within the linear approximation, these three vectors are proportional to the fields E

1 The use of complex quantities such as the complex refractive index, dielectric con-
stant, and conductivity to describe properties of medium naturally arrives from the use
of complex solutions to the Maxwell's equations. This point will be clear after reading

(e.g., P oc E), and their proportionality constants are the linear response functions

which describe properties of the solid-state system itself and are independent of the

driving force. The linear response is formulated in time and space. Since the response is,

in general, frequency and wave vector dependent and it is convenient to handle harmonic

functions, a discussion in Fourier space, both with respect to time and coordinates, is

more appropriate. Thus, rather than studying the response function directly, the linear

relation between the Fourier transform of the driving force and the Fourier transform

of the system response is considered. Also, we will use the time-varying E in the

form of exp(-iat), and express the proportionality constant as a complex quantity to

account for the phase shift between two fields: a real (an imaginary) part represents the

response of a medium in (out of) phase with the applied electric field. In addition, the

assumption of isotropic, homogeneous medium was made to simplify the discussions.2

The polarization is defined as the net dipole moment per unit volume. Within the

assumption made above, the microscopic dipoles (both permanent and induced dipole

moments) tend to align in the direction of the external field. This allows us to define a

polarization as

P = XE (2.7)

where X) is the complex electric susceptibility of the medium, which is one of the most

fundamental response function.

The electric displacement of the medium is defined by

D E + 47rP (2.8)

The above two equations can be combined to give an alternative expression:

D =E, (2.9)

2 Anisotropic crystals have nonequivalent optical properties along different (< i 11iw'
axes. The phenomenon of birefringence is an example of optical anisotropy. In such ma-
terials, the proportionality constants must be represented by a tensor.


SC1 + i2= 1 + 47re (2.10)

The parameter E is a complex dielectric constant (or dielectric function).

When time varying E is applied, there is an associated motion of each element of

charge. This leads to a relationship between the current density J and polarization as

J -iwP (2.11)

In a similar way to P and D, the current density can also be written as

J = E, (2.12)


a i + ia2 (2.13)

The parameter a is the complex conductivity of the medium. Generally, the current

density J is the sum of two contributions: one arising from the motion of charges that

are free to move through the medium J free, and the other arising from charges that are

restricted to localized motion Jbound.

Optical constants such as Xe, e, and a represent the response of a medium (i.e., re-

sponse functions) to a perturbing field of frequency c.3 All of these parameters,

however, are not independent. They are all interrelated to one another. As can be

seen from Eq. 2.10, Xe and E provide the same information. Using Eqs 2.7, 2.10, 2.11,

and 2.12, we can find a useful relationship between E and a:

=1 + i- (2.14)

3 The response functions should be considered as a function of both frequency w
and wave vector k. However, the explicit dependence of the response functions on k
(i.e., wavelength), the so-called spatial dispersion, can be neglected in case the fields
could be averaged over a unit cell. Spatial dispersion arises whenever the relation be-
tween D and E is not exactly local with D at a particular point determined solely by E
at that point.

or explicitly

a-1 2 (2.15)
0(1 el)
wo2 (2.16)

Thus optical measurements of E(w) are equivalent to conductivity measurements of o(w).

In case our interest is in the optical responses due to the free carrier gas in materials

such as metals and doped semiconductors, optical data are frequently discussed in terms

of the conductivity rather than the dielectric constant.

Later, we will show the connection between the optical parameters described here

and the propagation constants of electromagnetic waves in a medium, namely the

complex refractive index N.

The response of a material to external magnetic fields is characterized in a similar

way. The magnetization M is defined as the net magnetic moment per unit volume, and

is proportional to magnetic field strength H:

M = H. (2.17)

The parameter X is the magnetic susceptibility.

The magnetic flux density B is defined by

B H + 47M (2.18)

The above two equations can be combined to give

B = H, (2.19)


I = 1 + 47m (2.20)

The parameter p is the permeability of the medium. At optical frequencies, any

paramagnetic or ferromagnetic moments can not follow the rapid oscillations of magnetic

field because of their long relaxation times. The remaining diamagnetic moments are

so small as to have no appreciable effect on optical behavior. Thus, unless we study

magneto-optical phenomena, we can set B = H.

The starting point for the treatment of interaction between electromagnetic fields

and matter is contained within the four Maxwell's equations for the average fields.4 In

the absence of external charges, these equations are given by

V eE = 0, (2.21)

VH 0, (2.22)
1 8H
V x E H (2.23)
c Bt '
Vx H -= (2.24)

where E is the complex dielectric constant defined in Eq. 2.14, which allows the current

density arising from free carriers (i.e., Ohm's law) to be included in Eq. 2.24.

We consider the solution corresponding to a plane wave of the angular frequency w:

(H) (E)exp[i(k -x t)] (2.25)

where a constant amplitude vector Eo is in general complex. The complex wave vector

k was used to describe energy dissipation of the wave. Substitution of Eq. 2.25 into the

Maxwell's equations yields

k E 0 (2.26)

k H 0, (2.27)

k x E H (2.28)

k x H --E (2.29)

4 The use of the so-called macroscopic Maxwell's equations can be justified as follows.
In optical measurements, features that can be probed are the size of the order of a wave-
length of light or larger. Since a solid contains numerous atoms within the length scale
of the wavelength of light, it can be treated as a continuous medium

Here we have assumed an isotropic, homogeneous, and non-magnetic medium so that

E has no spatial variation. Equations are separately correct for both the real and

imaginary parts. These equations are combined to yield a relation between the wave

vector and frequency known as the dispersion relationship:

k.k (2.30)

In a non-absorbing medium of refractive index n, the wavelength of the light is

reduced by a factor n compared to the free space wavelength Ao (= 2rc/.). Therefore,

wave vector k is given by
2k r wn
k (2.31)
Ao/n c
This leads to the phase velocity u/k =c/n in Eq. 2.2. The wave vector can be

generalized to the case of an absorbing medium by allowing n (and as a result k,

too) to be complex:

k= N -(n + i) (2.32)
c c
Eq. 2.30 and Eq. 2.32 allow us to relate the propagation properties of light through a

medium to the response of the medium in the electromagnetic fields as

N = (2.33)

or explicitly

1 n2 2 (2.34)

C2 2nK (2.35)


2 Li
S- [- + (C + C2)2] (2.37)


Eqs. 2.26-2.29 also show that E, H, and k are mutually perpendicular (i.e., trans-

verse waves), and the scalar relation between E and H is given by

H vE = NE (2.38)

The ratio (47/c)E/H is called the wave impedance Z:

Z E 1 1
Z H -t t (2.39)
Zo0 H c N'

where Zo is the wave impedance of free space, which has a value of Zo = 47/c (or 377 Q

in SI).

On substituting Eq. 2.32 into Eq. 2.25, we find that the fields attenuate as e- K/c.

The optical intensity of light is proportional to the absolute square of the electric

field5 (I oc E*E). Thus, from Eq. 2.4, we find that

2wJc 47Tr
a (2.40)
c Ao

where a and K are the absorption and extinction coefficients, respectively. The pen-

etration (or skin) depth 6 is the characteristic length of the fields' penetration into a

medium defined by
2 c
= (2.41)
a wtK

The average rate of dissipation of electromagnetic energy density is

1 1
W =(Re(E) Re(J)) Re(E* J) t|E|2 (2.42)
2 2

Thus only the current that is in phase with E contributes to an energy loss, and a1

represents the resistive response (i.e., absorption that accompanies the energy loss)

of the medium in the fields. The out of phase current, on the other hand, does not

accompany the energy loss, and a2 describes the reactive response. The a is called the

5 The time-averaged energy flow in the electromagnetic wave is calculated from the
real part of the complex Poynting vector S (E x H*). The magnitude of this vector
gives the intensity of the light wave proportional to the square of the field.

optical conductivity since the response concerned here arises from transitions as a result

of photon absorption.

We have so far considered only transverse waves (i.e., k I E). However, Eq. 2.26

can also be satisfied for longitudinal waves (i.e., k I| E) for any frequency Ui provided

that e(1) = 0. At this frequency, longitudinal waves can propagate through a medium

and contribute to the energy loss that is proportional to the so-called loss function

defined as
(1 62 2n
-Im n (2.43)
G)j +1 | (n +K2)2
The longitudinal waves can excite LO phonon modes at the LO frequencies.

In an only weakly absorbing medium (i.e., n > K), Eqs. 2.34 and 2.35 simplify to

n = V (2.44)

C2 =Ao (2.45)
2n en

These equations tell us that the refractive index n is approximately determined by C1,

while the absorption is mainly determined by C2 (or Ol).

The purpose of this section has been to provide some of the optical constants

that describe optical properties of a medium, as well as the relationship between these

constants. The relations such as Eq. 2.30 and 2.33 are the connections between the

macroscopic optical parameters such as N and quantities that can be calculated by

microscopic theory such as E. Note that all the optical constants described here are in

general frequency dependent providing information about how photons of particular

energy, hw, interact with electrons, phonons, and other excitations in the system.

Detailed analysis of optical constants allows us to understand various properties of

solids. For example, knowledge of the electronic properties of solids is the key to

understanding most of their physical and chemical properties.


2.4 Experimental Determination of Optical Constants

Information about solid materials is often obtained by studying the electromagnetic

waves reflected from and/or transmitted across interfaces between materials with dif-

ferent optical properties. This can be done by considering the boundary conditions of

the E and H fields and the energy conservation. In experiment, we usually measure the

fraction of energy reflected [i.e., reflectance, J (w)] from and/or transmitted [i.e., trans-

mittance, 9(u()] through a specimen. The form of a specimen usually determines which

measurement technique has to be employ, ,1 Our main goal is to deduce the dielectric

function as well as other functions directly related to it. In this section, we will discuss a

few examples of simple procedures used for determining optical constants.

2.4.1 Reflection and Transmission at a Plane Interface

We first consider the transmission and reflection of light at a plane interface

between two media with different refractive indices, 1N and N2. For simplicity, we will

assume the light is incident normal to the interface. Then, the boundary conditions

require that the tangential components of the electric and magnetic fields are conserved

such that

E, + E = Et and H, H = H, (2.46)

where i, r, and t refer to the components of the incident, reflected, and transmitted

fields, respectively. Using the relation between E and H from Eq. 2.38 the bound-

ary conditions yield the amplitude reflection coefficient and amplitude transmission

coefficient as6
E, N,- 1N2
Ei 1,+ 12 (2.47)

Et 2N1
t l+ f 1 (2.48)
E, 1,+ 1N2

6 For an arbitrary angle of incidence a more general treatment is required. The reflec-
tion and transmission coefficients are then given by formulae known as Fresnel's equa-
tions [9].

The reflectance (or reflectivity) is the intensity reflection coefficient. If the light is

incident on a medium from a vacuum side, the reflectance off the medium is simply

given by

S ( n + (2.49)
(1 + n)2 + K2
where we have used N = 1 and 2 = n + iM. This is the valid equation for the

single-bounce reflectance measured from a thick < i, ii (i.e., a bulk material) with its

thickness much greater than the penetration depth (d > 6).

From Eq. 2.49, it is obvious that reflectance data alone can not determine both n

and K. It is in general not possible to determine both components from the measure-

ment of just one optical parameter, such as reflectance. Therefore, we need separate

measurement of either n or K by some other means, or to do something else in con-

junction with reflectance measurement. The Kramers-Krinig relations offer practical

solution to this problem as discussed below.

2.4.2 Kramers-Kr6nig Dispersion Relations

The Kramers-Kr6nig relations (KK) are integral relationships between real and

imaginary parts of a complex function, such as the linear response functions e(w),

(cju), and N(w), as a result of invoking the law of causality and applying the complex
ai 1,v-i- 7 One of the requirements for the relationships to be valid is that the response

function vanishes for w -- oo. The KK relations for the complex refractive index and the

complex dielectric function may be stated as follows:

n^)- 1 j^j J dcJ, (2.50)
2 J (') 1

(w) 2 -J L2dwu (2.51)
7T Jo U 'LC)/2 LC)2

7 In a physical system, response functions must satisfy G(-w) = G*(w). For the di-
electric function, this condition leads to cl(-uw) = ci(w) and 2(-w) = -C2(o). In other
words, 1(w) is an even and c2(w) is an odd function of the frequency w.


61(W)- 1 2 2~- d-cJ (2.52)

C (w) 2= ') d (2.53)
7" J0 )2 a) 2

where 2 stands for the Cauchy principal value of the integral. Similar relations are

available for other linear response functions. From these relations we see that if the real

part of a response function is known over an entire frequency range (0 < w < oo), the

imaginary part can be determined, and vice versa. In real experimental situations, there

is a limit in frequency range that can be measured. Therefore, this technique requires

proper extrapolations for the frequencies outside the range covered by measurement.

Inappropriate extrapolations may result in severe errors in the calculated counterparts.

Returning to the discussion of the single-bounce reflectance at a plane interface,

the amplitude reflection coefficient given in Eq. 2.47 is a complex quantity which can be

expressed as r(w) = &(w) ei( or

lnr(w) In (w) + i(w) (2.54)

where J (w) is the reflectance given by Eq. 2.49 and 0(w) is the phase shift of the

reflected electric field, which is related to n and K by

Im[F] -2r
tan (2.55)
Re [r] 1 n2 K2

One of commonly used technique is to measure the reflectance over a wide frequency

range. Then, the KK-related phase shift is calculated by

w [ ln ()- In () d
O(w) = w2 2i (2.56)
1 + co d-ln (w' )
In ada' (2.57)
27r 0o 'c Lo dud'

Medium 1, N,

Medium 2, N2 -- d

Medium 3, N3

Figure 2-1: Reflection and transmission at two parallel interfaces. The thickness of the
second medium is d. We assume the case of normal incidence, but the beams are drawn
at an angle for a clarity.

From Eq. 2.49 and Eq. 2.55, we can determine n(w) and (aw), the dielectric function,

and all other related functions.8

2.4.3 Reflection and Transmission at Two Parallel Interfaces

If the light is incident on a plane interface (between medium 1 and medium 2),

and transmitted through the second parallel plane interface (between medium 2 and

medium 3), the expressions of transmittance and reflectance become more complicated

since now we have to consider the multiple reflection as well as absorption absorption

within the second medium. This situation is depicted in Figure 2-1. The first and third

media are assumed to be non-absorbing, and span the semi-infinite space with their

complex refractive index N1 and N3, respectively. The second medium has its thickness

d with the refractive index N2. We again assume normal incidence for simplicity. Then,

the general formulae for the resultant amplitude transmission and reflection coefficients

8 This technique is quite practical, yet the requirements of wide range measurement
can be inconvenient in some situation. One of the technique called ellipsometry can de-
termine simultaneously both real and imaginary parts of the dielectric function over a
limited frequency range, and may serve as an alternative method to consider [17].

including multiple reflections are

t d ti23 C6[1t ( 23 21 Ci26) + (r2321 ei2)2 +

1 r23r21 (2.58)


r = 12 + t12r21 i[ + (r2123 6i2e) + ( 23 6e12)2 + *
r12 + r23 i2 (2.59)
1 r21r23 6i2

where rij and tij are the amplitude reflection and transmission coefficients between

mediums i and j as given by Eqs. 2.47 and 2.48, and 6 is the complex phase depth of

the second medium which is defined by

S w a
6 -N2d -n2d + -d (2.60)
c c 2

where a is the absorption coefficient defined by Eq. 2.40. From Eqs. 2.58 and 2.59, the

resultant transmittance and reflectance are obtained:

n3 2 n3 1212 12 23 12 -ad
|t| 1+ 12C2 r233 2 Cud s (2.61)
nI ni 1 + lr23 121 rll -2 2 r3 r2l -a cos 0

SI 1 2 2 + 1 r23 2 e-2ad +21 r3 1112 e-ad Cos (
S- 1 + |r232 r21 2 e-2ad 2 23 21 e-adcos (


S2-n2d + 23 + 21 (2.63)

where ', is the phase shift upon reflection at either interface. The cosine term leads

to interference fringes in the spectrum due to multiple internal reflection in the second

medium. When the second medium is thick (d > A) or wedged, there is no coherence

among multiple reflections. In a low resolution measurement, those fringes are not

resolved, and the transmittance or reflectance is averaged over the phase angle for the

partial beams as
S -12 r23 11212 -ad (2.64)
g v 1 1 23 1 21 2 | 1

1 i22 2 -2ad 2 ^12 2 -2rd
Jaoe -12 2 23 2 923 12 (2.6)
S- 23 122112 -2d (2.5)

When a thick sample of thickness d with complex refractive index n is measured in

a vacuum, it is straight forward to find the averaged transmittance and reflectance:

(1 _- )2+ 2/n2) -ad
ave -2ad (2.66)


ave ( +Js(?1 0e ad) (2.67)

where Js is the single-bounce reflectance given by Eq. 2.49. Experiments of this type

are very important and are often applied to measure absorption coefficients of solids.

When wavelengths of incident light are comparable to the thickness d, the interfer-

ence fringes are resolved with sufficiently high resolution measurements. From Eq. 2.63,

it is apparent that the spectrum exhibits periodic fringes of the frequency spacing

between two successive fringes given by

AV =2 (2.68)

where Av is in cm-1 and d is in cm. This equation is sometimes useful to determine the

thickness of the sample from the fringe -I. ii.- and vice versa. For example, mylar films

used as a beam-splitter in far infrared have the refractive index between 1.64 and 1.67.

Then, we can expect that the first minimum for 6 pm mylar beam-splitter appears near

500 cm-

2.4.4 Optics in Thin Film on a Substrate

A structure of a thin film of thickness d (< wavelength or penetration depth) laid

on a thick but non-absorbing (or weakly absorbing) substrate with refractive index

n and thickness x is quite common in the optical experiment. Figure 2-2 shows the

schematic diagram of the situation. Again we only consider the normal incidence to a


t h in f ilm .........................................

substrate o ......... n x


Figure 2-2: Reflection and transmission with a thin film on a weakly absorbing sub-
strate. The thickness of the thin film and the substrate are d and x respectively. We
assume the case of normal incidence, but the beams are drawn at an angle for a clarity.

sample in free space. It is obviously more complicated since now we are dealing with a

four 1 1-, i. I structure. In such case, the KK technique is inapplicable. However, it is

possible to extract linear response functions from measurements of both reflectance and

transmittance over a finite frequency range.

At first we consider the case when multiple reflections inside the substrate may be

neglected. This simplifies the situation significantly since the thickness of the substrate

x becomes unimportant, and the system can be considered as a three 1 ,li. I structure

(vacuum-film-substrate) just like the one discussed above. Then, from Eqs. 2.58 and 2.59
with the following approximations:

I N2 1 ,

IN21 >N3 n3 n (3 < n3) (2.69)
d < wavelength or skin depth ,

it can be shown that the transmittance across the film into substrate and the reflectance

from the film are given by the Glover-Tinkham equations [18, 19]:

1 4n
*7 = 4 (2.70)
1 + 1 12 (y + +1)2+ (7
(y + n- 1)2 +y
1 +2 (2.71)
f (y +n+1)2 2

where n is the refractive index of the substrate, yl and Y2 are the real and complex

part of the dimensionless complex admittance of the film, y, respectively. y is related

to the complex conductivity a = r1 + ia2 of the film by y =Zoad where Zo is the

impedance of free space (4r/c in cgs; 377 Q in mks). Thus, the optical behavior of a

film is determined by its electrical properties of the film that is modified by the surface


The actual measured transmittance and reflectance are influenced by multiple

internal reflections within the substrate of the thickness x and the absorption coefficient

a. Then, the system is a four 1li,. structure with vacuum as the forth medium. If a

substrate is thick (x > A) or wedged, coherence among multiple reflections are lost, the

measured transmittance and reflectance are simplified to

(1 J- 3)p-ax
3= e-2-x (2.72)
1 4s4e-2ax


g 1 j e-2ax

(yi + n + 1)2 + y2

and 4, is the single-bounce reflectance of the substrate given by Eq. 2.49. For a weakly

absorbing substrate such that = ca/2C < n, J4s may be approximated as

1- n\ 2
t ~ (2.75)
t +n)2

This is usually satisfied for measurements at low temperature and low frequencies.

From measurements of the transmittance and reflectance of the bare substrate, we

can find the index of refraction n and the absorption coefficient a of substrate using

Eqs. 2.66 and 2.67. Note that the term 2 /n2 in Eq. 2.66 can be neglected for a weakly

absorbing substrate.

With the knowledge of substrate's optical properties, al and -2 and in turn all

other response functions can be extracted by inverting Eqs. 2.72-2.74 after measuring

both transmittance and reflectance of the film-on-substrate system. For a structure

with more V1 --iSr (e.g., vacuum-film-buffer-substrate-vacuum), the analysis becomes

progressively more complicated. A more general discussion of the optical response from

multi-l .,-,;r is given in [15,20].

2.4.5 Photoinduced Absorption

In the photoinduced measurements, we are interested in changes in the optical

behavior of a sample in photoexcited state with respect to non-excited state (ground

state). For a sample in the form of film with thickness d deposited on a substrate, it

would be ideal to have a substrate material that is insensitive to the photoexcitation.

In such case, the photoinduced change in the transmittance, A9, is due to the photoin-

duced absorption by the film itself. If the measurement is done in low resolution, the

transmittance through the film into substrate is given by Eq. 2.66, and it can be shown

that the normalized photoinduced transmittance is written as

-- (Aa)d (2.76)

where I and go are the transmittance of the film in excited state and ground state,

respectively. Note that the negative of the quantity Al/9 is customary used as the

photoinduced signal.

2.5 Microscopic Models

Up to this point, we have not described the optical phenomena from a microscopic

point of view. There are various microscopic models that try to explain the optical

behavior observed experimentally. These models may be classified as either classical,

semiclassical, or quantum mechanical, depending on how we treat interaction between

light and matter.

In the classical model, both light and matter are treated classically. The dipole

oscillator model (Drude-Lorentz model), which will be discussed shortly, is a example

of a classical treatment. This model has been proven to be very successful, and is often

used for understanding the general optical properties of medium.

In the semiclassical approach, the atoms in the medium are treated quantum

mechanically, while the light is still treated as a classical electromagnetic wave. The

absorption coefficient or oscillator strength due to transition between two states or two

bands can be calculated using Fermi's golden rule, which requires knowledge of the wave

functions of the states.

In the completely quantum approach, the light is also treated quantum mechani-

cally, namely as photons. Feynman diagrams can be drawn to represent the interaction

processes between photons and atoms.

In this section, we will discuss only a few of microscopic models that are commonly

used during analysis.

2.5.1 Lorentz Model

In a solid, there are various processes (or excitations) that contribute to the

dielectric function which, in turn, describes its optical behaviors. For example, free

carrier absorption and phonon (including multi-phonon) absorption are the typical

processes at far- and mid-infrared frequencies.9 In the spectral range of near-infrared

and ultraviolet, processes such as excitons and fundamental absorption across the

energy gap, interband transitions, and plasma absorption may be seen. In the vacuum-

ultraviolet and X-ray spectral region, the transitions of the core electrons can be

expected to dominate the dielectric function. At the very high energies beyond nuclear

excitations, nothing can respond to the driving field, and the dielectric function becomes

unity since the medium does not possess any polarization. Note that all transitions

require the conservation of energy and momentum.

9 In principle, magnetic excitations could exist at even lower energies.

The Lorentz model is a simple, yet very useful classical model dielectric function

that can be derived for a set of damped harmonic oscillators. When a harmonic oscilla-

tor with mass m, charge q, damping constant 7, and resonant frequency wo is excited by

a harmonic electric field of the form E(t) Eo e-it, the equation of motion is given by

mr + myr + mwor = qE(t) (2.77)

The second term models the energy loss mechanism of the oscillating dipole. Note that

the resonant frequency wo is a transverse oscillator frequency that is coupled to the

transverse electric field. Inserting a solution of the form r = ro e-it into Eq. 2.77 yields

q 1
r 2 E (2.78)
m LCQ LC) 17LCj

If there are N oscillators per unit volume, the resonant contribution to the macro-

scopic polarization is
Nq2 1
P Nqr Nq2 1 E. (2.79)
m LCo2 LC2 i

Note that the isotropic medium is assumed here. Then, the susceptibility arising from

the oscillator is
Nq2 1
S 2 (2.80)

The total polarization is given by

Ptotal = eE =(- Xo)E (2.81)

where X, is the background susceptibility that arises from the polarization due to all

the other oscillators at higher frequencies.

The dielectric function is determined from Eq. 2.10:

w(h) = d, + (2.82)

where we have defined the high frequency limit of E(cw) as

co = 1 + 47rX, ,


and the plasma frequency up as

2 4Nq (2.84)

Note that the subscript oo should be understood as contributions above a certain

resonance. Separating the real and imaginary parts, we obtain

2 2 _2
( 2)2 + (7U)2(

C2(Uw) ( > + c> 7 (2.86)

From these equations, it is straight forward to see that Cl gradually increases from the

value co + /wu0 as frequencies increase toward wo, and peaks at wo 7/2. It takes

sharp negative slope, passing through Co at w0, and bottoms at Uo + 7/2. As frequencies

increase further, it finally approaches the high frequency limit of C~. As mentioned

briefly in the earlier section of this chapter, the frequency for which Ci(w) = 0 is labelled

as ul at which electromagnetic waves are coupled to the longitudinal component of

the oscillator. Compared with el, C2 has a simple bell shape with a strong peak at Uo

and the full with at half maximum given by 7. Note that both ei and C2 vary on the

frequency scale of 7, and the damping of the oscillator has the effect of broadening. It

is in general that material is highly absorbing near the resonance, for obvious reason,

strongly reflecting between wo and wl, and transparent at frequencies further .li. from

the resonance where ei does not vary strongly.

Eq. 2.82 can be generalized to an arbitrary number of different oscillators as
~({) + Y3 (2.87)

where uj, 7j, and ,pj are the resonant frequency, damping constant, and plasma

frequency of the oscillator of type j, respectively. The plasma frequency is defined by

2 4 'j (2.88)

where Ny, qj, and mj are the number density, effective charge, and effective mass of the

oscillator of type j, respectively. These vales must be appropriately chosen to account


for the different oscillators. For example, in the case of a phonon, up is the ion plasma

frequency with q and m as the effective charge and the reduced mass of the particular

lattice vibration mode.

A corresponding quantum mechanical version of Eq. 2.87 can be written as

(w) =e + f (2.89)

where we have introduced a oscillator strength fj in order to account for the strength

of the response of different transitions to the perturbing electric field. In the quantum

picture, Lj is the transition frequency between two states which are separated in energy

by hyj, and 7j is the uncertainty (or width) in energy of the initial and final states

of transition. The oscillator strength fj is related to the probability of a quantum

mechanical transition, which can be calculated using Fermi's golden rule.10 It satisfies a

sum rule

1 (2.90)

The oscillator strength provides us an explanation for the different absorption strength

in different transitions.

2.5.2 Free Carrier Response and Drude Model

The equation of motion given in Eq. 2.77 can also be used to derive the dielectric

response of free carriers of charge q and effective mass m* by taking the restoring force

term zero (i.e., w O 0):
m*v+- = qE(t), (2.91)

10 Fermi's golden rule shows that the transition rate between two states is proportional
to the square of a matrix element and also to a density of states for both the initial and
final states. The oscillator strength and absorption coefficient are related to the quan-
tum mechanical transition rate.


where we have expressed the damping constant 7 as a reciprocal of the collision time r

that characterizes loss of momentum of carriers due to scattering.11 This is a simple

equation based on the Drude model. Inserting a solution of the form v = vo e-i into

Eq. 2.91 yields
qr 1
v q E (2.92)
m* 1 iur

For N free carriers per unit volume, the current density is then

Nq2r 1
j Nqv = E = aE (2.93)
m* 1 iUr

Thus, the ac conductivity based on the Drude model is

oD(0) (2.94)
1 iU-

where co is the dc conductivity defined as

-o = (2.95)

The real and imaginary parts are

JDo1 (2.96)
1 + W2r2

D2 o (2.97)
1 + U2,-T2

From Eq. 2.14, the dielectric function for the free carriers is given by

ED (W) =1 (2.98)
L) + iLL/'T

11 A typical value of r for a metal or doped semiconductor is in the range ~ 10-14
10-13 seconds, which correspond to ~3000 cm-1 and ~300 cm-.


where ucpD is the Drude plasma frequency defined by12

2 47rNq2
SPD (2.9

Eq. 2.98 is obviously the same expression as Eq. 2.82 with wo = 0, 7 = -1, and c 1.

Note that Eq. 2.98 assumes that only free carriers contribute to the dielectric function.

When other processes at higher frequencies give contributions, the unity should be

replaced by c,. The real and imaginary parts are
2 2
D1 1 (2.100)

D2 (2.101)
w(1 + Uw22)

In the limit of low frequency where w ;< T- is satisfied, we can obtain following


CD1 DT2 (2.102)

eD2 opDT/U 47 ro/w; >> CD1 (2.103)

n t (rc 2/2)1/2 (2.104)

S t 1 2/n 1 (2w/7ao)1/2 (2.105)

Eq. 2.105 is known as the Hagen-Rubens relation. From the second expression we can

find the absorption coefficient:

a =- (2.106)
c c

12 The Drude plasma frequency ucpD is related to the dc conductivity and carrier mo-
bility p as ao = ~Di-/4r Nqp. The mobility (a ratio of the carrier drift velocity to
the field) is given by p q-r/m*. For metals, ao is (nearly) independent of temperature
assuming that r does not (or only weakly) vary with temperature, and is used to char-
acterize metals. For semiconductors, on the other hand, their carrier densities can be
varied by changing the temperature or the dopant concentration. Therefore, the mobility
is more convenient quantity to characterize semiconductor since the carrier density is
taken out.

or the skin depth:
2 c
S- (2.107)
a y2w7a7o0
Therefore, the skin depth is inversely proportional to the square root of dc conductivity

and frequency. This implies that a material with higher dc conductivity allows shorter

penetration of ac fields for a given frequency.

As another limiting case, consider an undamped free carrier system like a perfect

conductor. In this special case, the Drude width -1 = 0 and the dielectric function is

real and given by

CD1 oo (2.108)

eD2 DI = 0 (w / 0) (2.109)

Here we have used c. just for the purpose of generality. This equation tells us that

CD < 0 for frequencies below the plasma edge (u < wUpD/ /-g). Then, the complex

refractive index N is purely imaginary and thus the reflectance J is 1 in this frequency

range and the system suddenly becomes transparent above the plasma edge.13 This,

so-called a plasma reflection, happens without loss of energy since there is no resistive

current (i.e., cUD = 0) associated with this free carrier response.

In real materials, the damping T-1 has a non-zero value which may be deduced

from dc conductivity or other measurements. The effect of the damping may be small

but results in slightly less reflectance as well as broadening of the plasma edge. The

reflectance may be even lower and have structures due to other absorption processes

such as interband transitions. If other processes occurs near the plasma edge, sharp

onset of transmission may not be observed.

13 For metals, c, t 1 and the reflectance is very high for frequencies up to UcpD. For
semiconductors, on the other hand, c. can be large and as a result the plasma edge at
wpD//-- is lower than UpD.

2.5.3 Drude-Lorentz Model

When both the Drude and the Lorentz types of dielectric response is observed in a

spectrum, the total dielectric function can be expressed as the sum of various different

processes that cause a polarization:
2 2
(w) 2 2 i 7 a)+3 w -2 +i (2.110)
SJ U1- tjU U J

This relation is called the Drude-Lorentz model, and can be used in fitting the experi-

mental reflectance data for extracting optical parameters. Unlike the KK-methods, the

fitting data with the model function can be employ,, 1 in a finite frequency range as long

as we have a well defined background contribution co beyond the measured frequency


2.5.4 Sum Rules

In 2.5.1, we introduced the notion of oscillator strength f. Using quantum

mechanics, it can be shown that the total absorption by all transitions for the whole

frequency range is constant, and can be expressed by the f-sum rule:

ja 1 (a)) dw y- f. (2.111)

This tells that the total are under the real part of the conductivity is independent of

temperature, phase transition, photo-excitations, and so on. There exist several other

sum rules, but we will not discuss them here.

The sum rule is often applied to a certain process. If only free carriers are con-

cerned, Eq. 2.111 is rewritten as

SUai( )dw ) (2.112)

This is an exceptionally useful equation to see how the spectral weight shifts into the

delta function at zero frequency as a superconductor experiences phase transition.


3.1 Introduction

A spectrometer is an instrument that is designed to yield spectral information

contained in the electromagnetic waves under study. There exist several types of

spectrometers used for a number of research fields. Figure 3-1 shows the classification

of spectrometers. Of all, the scanning two-beam interferometric types are probably the

mainstream instrument now owing to their various advantages which will be explained


The monochrometer spatially separates the individual frequency components

by means of a dispersive element such as prism or diffraction grating. An individual

frequency component is selected by a slit, and its intensity is sequentially sampled. A

power spectrum is produced after measuring over all frequencies of interest. Although

this type of instrument is still used commonly, especially for near infrared and visible

spectroscopy, they have met with several limitations. The main difficulty comes from

their slow scanning process. Because the monochrometer measures each frequency

individually, it takes a long time (typically 10 minutes or more depending on the

signal to noise as well as resolution) to complete a single scan. The interferometric

Dispersing Prism spectrometer
(monochrometer) Diffraction spectrometer
Michelson interferometer
(amplitude separation)
Twin-beam interferometer-- Lamellar grating interferometer
(wavefront separation)
Interference Martin-puplett interferometer
spectrometer -(polarization separation)
Fabry-Perot interferometer
Multi-beam interferometer--
-- Etalon

Figure 3-1: Classification of spectrometers.

techniques were developed to overcome some of the limitations encountered with

dispersive instruments.

The interferometer is an instrument that can divide the incoming beam of light

into two paths and then recombine the two beams after a (optical) path difference (or

retardation) has been introduced. These recombined beams produce interference and

the resulting signal is detected. The measured signal as a function of path difference,

called an interferogram, is the Fourier transform of the power spectrum of the incident

light. Thus it can be inverse-Fourier transformed to yield the power spectrum. However,

because of the fact that the detected signal must be treated mathematically before

obtaining meaningful spectrum, certain care must be taken to avoid introducing errors

into the spectrum.

Interferometric technique has two basic advantages over dispersive methods. The

fact that nearly alv-l- the total intensity hits the detector during the whole period of

measurement improves the signal-to-noise (S/N) ratio, particularly for weak radiation

sources. This is known as the throughput (or Jacquinot) advantage. The interferometer

measures all frequency components simultaneously. This leads to considerable multiplex

(or Fellgett) advantage allowing quick data acquisition and higher S/N.

3.2 Fourier Transform Interferometry

3.2.1 General Principles

The general principle of interferometry can be understood by considering a simpli-

fied Michelson interferometer [21, 22, 23], which is shown schematically in Figure 3-2.

Consider that a monochromatic plane wave of the form

Es = Eo Ci(27vx-t)


Source C

Beam splitter
(r(wo), t(wo))

1 M1 (r,(o), #(o))

. x/2 (M
SM2 (ro(o), 2(o))

V Detector

Figure 3-2: A schematic view of a simplified Michelson interferometer. The light travels
to the beam-splitter with its amplitude reflectivity r and transmissivity t. The partially
reflected beam travels toward the fixed mirror (\! I) that has the reflectivity ri and
introduces phase shift 1. The partially transmitted beam travels a variable distance
toward the movable mirror (\ 2) with r2 and 2. The beams are recombined at the
beam-splitter after a optical path difference x has been introduced, and half of the total
beam returns to the source, and the other half proceeds to a detector. A sample can be
place between interferometer and the detector.

is incident on the beam-splitter from the source. Here, v is the wavenumber.1 This

beam-splitter partially reflects the beam toward the fixed mirror M1, and transmits

the rest toward the movable mirror M2.2 After travelling their respective paths, the

two beams are recombined at the beam-splitter, and the resultant beam proceeds

to a detector. If the beam-splitter has the amplitude reflectance r and amplitude

transmittance t,3 the resulting field emerging from the interferometer toward the

1 The wavenumber is defined as v = 1/A = 2rk = w/2rc.
2 Here for simplicity, we will assume that both M1 and M2 are equivalent perfect
reflectors. The amplitude reflectivity is a complex number which can be expressed as
r = r/E, = r(ow)|ei K), where E, is the reflected field, Ei is the incident field, and Q(w)
is the phase shift. For a perfect mirror, r| = 1 and Q = r.

3 For an ideal beam-splitter, r = t = 2-1/2. They are both frequency dependent.

2 -7r

detector is a superposition of fields from two beams which is given by

ED ri .,,, L- t) + Ci(2V 2- t)] (3.2)

where xi and x2 are the total distances of respective beam's optical path (see Fig-

ure 3-2). Since the energy reaching the detector is proportional to EDES, the time

averaged detector signal can be written as

S(x) = -lo(v)[1 + cos(2vx)] (3.3)

where we have defined the optical path difference, x = x2 x, the beam-splitter

efficiency, c = 4|rt|2, and the source intensity, lo(v). This expression may be simplified


S(x) = f(v)[1 + cos(27vx)] (3.4)

where f(v) is an arbitrary spectral input that depends only on v. S(x) is the detector

signal for a monochromatic source. The cosine term gives the modulation on the

detector signal as a function of x.

As mentioned earlier, with the interferometric technique all frequency components

are measured simultaneously. Eq. 3.4 can be generalized for a polychromatic source by

integrating it over all frequencies:

S(x) f ()[1 + cos(27vx)]dv (3.5)

At x = 0, the detector signal reaches its maximum value of

S(0) 2 f(v)dv (3.6)

This position corresponds to the zero optical path difference (or ZPD) where all

frequency components interfere constructively. As x -- o, on the other hand, the

coherence of the modulated light is completely lost, and the cosine term in Eq. 3.5 goes

to zero. Therefore, the detector signal oscillates around an average value:

S(oo) f (v)dv =S( (3.7)
n 2

The interferogram is the cosine modulation part of the detector signal:

F(x) = S(x) S(o) = f(v) cos(27vx)dv (3.8)

This is the cosine Fourier integral of the desired spectrum f(v) which can be recovered

by taking the inverse Fourier transform:

f(v) = 4 F(x) cos(27vx)dx (3.9)

3.2.2 Finite Retardation and Apodization

So far in our calculation, it was assumed that the spectrum is obtained after the

Fourier transform (FT) of the interferogram measured with an infinitely long optical

path difference (retardation). In practice the interferogram can not be measured

to infinite retardation, and it must be truncated. This type of truncation can be

manipulated mathematically by multiplying the complete interferogram by a truncation

function G(x) which vanishes outside the range of the data acquisition. Thus the actual

function which is transformed is the product of the interferogram and the truncation


Now according to the convolution theorem, the FT of the product of two functions,

i- F(x) and G(x), is the convolution of the FT of each function, f(v) and g(v), where

the convolution is defined by

f(v) *g() f(v')g( v')d' (3.10)

Hence, the calculated spectrum is the true spectrum convolved with the FT of the

truncation function.

In order to examine the effect of truncation, consider multiplying an interferogram

F(x) with the boxcar function G(x) which is defined by

(1 if x\ < L
G(x) = if < L (3.11)
0 if x > L,

Vv 1/

V cm

v,-1/2L v +1/2L

Figure 3-3: The sine function convolved with a single spectral line of wavenumber V1. L
is the maximum retardation.

where L is the maximum retardation. The FT of F(x) is the true spectrum f(v), while

the FT of G(x) is the sine function:

FT[G(x)] = 2Lsn(27 L) 2L sinc(27wL) (3.12)

When the sine function is convolved with a single spectral line of wavenumber

v1, the resultant spectrum is the sine function centered about v1, which is shown in

Figure 3-3. Thus the effect of convolution is to smooth out the narrow feature. The

single spectral line shape as a result truncation is sometimes called the instrument line

shape (ILS) function.

It can be shown that the first zeros on either side of v1 occur at v1 1/2L. Thus,

two spectral lines separated by 1/L are completely resolved. Thus the resolution is

limited by the maximum path difference on the interferogram. The value of 1/L (in

cm-1) is often used as a quick estimate of spectral resolution. The full width at half

maximum (FWHM) is sometimes used as an alternative estimate of resolution.

The sudden cutoff with the boxcar function introduces side lobes near sharp

features in the spectrum.4 Thus, it is desirable to use a weighted truncation function

that cuts off the interferogram in gentler fashion. This process, known as apodization,

reduces the ringing at the expense of a further reduction in resolution. For example, the

Happ-Genzel [24] is a simple apodization function given by

Gi(x) 0.54 + 0.46cos(7x/L) (3.13)

where L is still the maximum retardation. The FT of the Happ-Genzel is

FT[Gi(x)] 2L sinc(27vL) 0.54 (46(2v2 L)2 (3.14)

Again, convolution of this function with a single spectral line of wavenumber v1 is the

resultant spectrum which is shown in Figure 3-4 together with the FT of the boxcar

(i.e., since function). This figure clearly demonstrates that the use of a gentler truncation

function suppresses the side lobes while the resolution is reduced. The FWHM of the

spectrum using the Happ-Genzel and the boxcar are 0.91/L and 0.61/L, respectively.

Some of other popular apodization functions are the Norton-Beer (weak, medium,

strong) [25] and the Blackman-Harris (3-term, 4-term).5 Very nice discussion about

apodization functions can be found in Griffiths [22].

3.2.3 Sampling

In the use of a computer for data acquisition, the analog signal must be converted

to digitized data sets (i.e., A/D conversion) before any sort of manipulation can take

place. For this reason, the interferogram is sampled at small, equally spaced discrete

retardation, and the Fourier integral, Eq. 3.9, is approximated by a sum. This discrete

4 The first minimum drops off below zero by 22' of the height at central maximum.
The secondary maxima are also relatively large. These side lobes give rise to oscillation
which may appear as spurious features especially in the neighborhood of sharp spectral

5 Personally, I start with the Norton-Beer (medium) apodization function first, and
try others if ceratin improvement has to be made depending on the spectral features.

- --.. FT [Boxcar]
FT [Happ-Genzel]

"* : VI
- '# < 1 1
" ,-' I1/LT -*. "

V cm

v,-1/2L v +1/2L

Figure 3-4: The FT of the Happ-Genzel (HG) apodization function convolved with a
single spectral line of wavenumber v1. L is the maximum retardation. For comparison,
the FT of the boxcar sincec function) convolved with the same single line is shown. The
FWHM of the spectrum for the HG and the boxcar cases are 0.91/L and 0.61/L, respec-
tively. Note that the side lobes in the spectrum are suppressed by using the HG at the
cost of resolution.

nature can be handled mathematically by using the Dirac delta comb defined by

Im(x)= 6(x an) (3.15)

The Dirac delta comb is just a series of 6 functions at the integers. It has following


II(x + m) = U(x) (periodic), (3.16)

S(ax) (x= -) (scaling), (3.17)

i1(ax)T 0
FT[(axr)] (11 w Ci2'n/a (Fourier transform), (3.18)

iI(z) F(x) = F(n)6 (z- n) (sampling), (3.19)
) n=-

III(x)* F(x) > F(x- ) (replication). (3.20)

(A) f
!imax .. .. ......

0 A 2 v cm


I 0 I I o
1- v cm

Figure 3-5: The relation between spectrum replication and sampling rate. (A) Proper
choice of sampling rate such that the true spectrum is confined to one-half of the repli-
cation period (i.e., vmx < Av/2). In this situation, the periodic replicas do not overlap
and no error is introduced by sampling. (B) Improper choice of sampling rate. Overlap-
ping with .,I]i i ent replicated spectrum causes spurious result in the spectrum.

If the continuous (or analog) interferogram F(x) is sampled at intervals Ax, the

sampled (or digitized) interferogram F'(x) is given by

F'(x) II ( x) F(x) = x F(nAx)(x nAx) (3.21)

Then, the spectrum derived from the FT of F'(x) is

f'(7) -U f (V) f= ( nA), (3.22)

where Av = 1/Ax and f(v) = FT[F(x)].
Eq. 3.22 is the spectrum we actually obtain as data which is comprised of periodic
replicas of our desired spectrum f(v) with period Av. This replication, which obviously

arises from the sampling of the analog interferogram, raises an important issue of the
sampling frequency, which is peculiar to discrete sampling. Figure 3-5 illustrates the
relation between replication and sampling rate. As in the figure (B), if the highest

frequency, ,max, of the true spectrum exceeds the folding frequency, Av/2, then two
.,.li i:ent replicated spectra overlap, and too-high frequencies appear falsely at lower


Figure 3-6: The red curve of above-Nyquist frequency appears to possess the same set of
data points as the black curve of below-Nyquist frequency.

frequencies causing spurious result in the obtained spectrum. This is because higher

frequency waves can be drawn additionally through the same sampling points taken for

lower frequency waves as illustrated in Figure 3-6. This effect, known as spectral folding

or aliasing, can be prevented by insuring the condition:

Vmax <- (3.23)


Ax < An (3.24)

These conditions state that the highest frequency needs to be sampled at least twice per

wavelength, which is just the Nyquist sampling criterion. Therefore, it is experimentally

important either to ensure digitizing an interferogram at a high enough sampling rate

or to limit the range of frequency input to the detector using optical and/or electronic

filters. The responsivity of the detector often works as a kind of low-pass filter.

Following the above arguments, it is quite obvious that measurements of narrow

frequency range require smaller number of sampling points. If the number of points

is too small, the spectral shape may not be well defined. In such case, we can add

extra zero-valued data points at the end of the interferogram keeping the same sample

spacing. This technique, known as zero filling, effectively produces larger number

of spectrum points per resolution element. Since the points added are zero-valued,

spectrum resolution will not increase. It merely provides a smoother spectral line shape.

3.2.4 Phase Error and Correction

Until this point, we have assumed that the interferogram is perfectly symmetric

about the ZPD. In a real experiment, however, there often exists a phase error that

must be included to describe the actual measured (i.e., .,i-ii ". i ic) interferogram. It

mainly results from sampling errors, electronic filtering, and optical effects from various

parts of instrument optics as well as a sample. The effect of such error is to distort the

ILS function from the symmetric since function to an ..i-mmetric shape. This could lead

to negative spectrum or slight shift of sharp features. Therefore, it is important to have

schemes that could correct faulty effects from a calculated spectrum.

When the phase error is included, the interferogram given by Eq. 3.8 is modified


F(x) f (V)e-i(2-vx-dv = [fv ()ei]e- i2XFdv (3.25)

where 0 is the phase error (or phase spectrum) which can be frequency dependent. Note

that here we used the exponential notation for simplicity. Then, the the calculated

spectrum through the inverse complex FT is

f() =(V)ei F(x, -7dx. (3.26)

Hence, the ..ivmmetric interferogram yields a complex spectrum. The real part of the

spectrum, Re[f(v)], and the imaginary part of the spectrum, Im[f(v)], can be computed

by the cosine FT and the sine FT of the interferogram measured symmetrically on either

side of the zero retardation point (or centerburst), respectively. Our aim is to find the

phase error, 0, from which we apply some sort of phase correction scheme to determine

the true spectrum of interest, which is f(v) in Eq. 3.26.

Here we explain the simplest way to achieve the phase correction. First, we take

an interferogram between -L1 < x < L2 where x = 0 corresponds to the centerburst.

Since it is only required to calculate 0 at very low resolution, the distance L1, which

is determined by the phase resolution setting, can be smaller than the distance, L2,

6 The phase error is added to the phase angle of the interferogram as cos(2rvx 0) =
cos(27vx) cos 0 + sin(27vx) sin 0. The FT of a truncated sine wave is an odd function.
Thus, the added sine component is responsible for the ..i-, ii, I iir shape of an interfero-
gram, and the its FT causes distortion of the ILS function.


required to attain the desired resolution (I/L2).7 From the short double-sided region of

the interferogram (-Li < x < L1), the phase spectrum can be found from

(v) = arctan (m[f(f( (3.27)

Having calculated 0, the complex spectrum, f(v) may be corrected by multiplying it by

e-io cos 0 i sin 0 such that

f(v/)corrected f()ei e-io = f () (3.28)

In this way the recovered spectrum may be corrected for errors incurred as a result of

.,i-v,,ii. 1 i1. s in the measured interferogram.

There are several phase correction modes available among which the method

developed by Mertz is the most commonly used one. More detailed discussion of phase

correction methods can be found in various papers [26,27,28].

3.2.5 Step-Scan and Rapid-Scan Interferometers

There are in general two different kinds of interferometers depending on its scanner

(movable mirror) movements: step-scan interferometers and rapid-scan interferometers.
In step-scan interferometers, the scanner starts from its reference position, and steps

to equally spaced sampling positions. At each sampling position, the scanner is held

stationary and the detector output signal is integrated. The stepping continues until

the desired resolution has been achieved. Compared with rapid-scan interferometers,

step-and-integrate systems have unavoidable down time while moving to the next

sampling point and waiting for the scanner to be stable before actually starting data

acquisition. In addition, the fact that it takes longer time to complete single scan

makes step-and-integrate systems prone to be sensitive to slow variations in the source

intensity which could degrade spectrum especially at low frequency. Further, the

7 Typically the phase resolution is set to have 4 to 8 times lower than the spectral res-
olution. If the double-sided acquisition mode is used (i.e., L1 = L2), the phase resolution
setting is completely ignored for obvious reason.

systems generally require to use a chopper, and thus lose another half of the signal after

the interferometer. These characters makes step-scan technique rather inefficient, and in

general rapid-scan method is superior and adopted by the most of recent instruments.

In rapid-scan interferometers, the scanner moves at constant and sufficiently high

velocity. When a signal of a particular frequency v is sent as input to an interferometer,

it is modulated at a frequency v' which is related to the moving mirror velocity v as

V/ -v vv (3.29)
c A

where v' and v are in Hz, A in cm, v in cm/s, and v in cm-1. Hence, in order to be

able to analyze the signal at the detector it is necessary to know the scanner velocity

accurately. It can be calculated by measuring the modulated frequency of a laser

input with known wavelength. If a Helium-Neon (He-Ne) laser at 632.8 nm is used, for

example, its typical modulated frequency,8 -v- 10 kHz, corresponds to 0.6328 cm/s for

the scanner velocity.

Now, having a continuous source as input, all frequency components are modulated

(typically in the kHz range) according to Eq. 3.29. Since these modulation frequencies

are in the audio range, they can be easily amplified and filtered electronically. A low-

pass filter eliminates noise of much higher frequency than the modulation frequency

of the shortest wavelength in the spectrum and prevents aliasing. A high-pass filter,

on the other hand, may be used to force slow background modulation, such as source

fluctuation, to below its cutoff.

Another important factor in rapid-scan systems is the determination of the correct

time to start data acquisition. The position of the centerburst of continuous source

can be used as a reference point. This information is essential for data collection, since

the method of analysis relies on the cumulative addition (co-adding) of a number of

8 Some of modern instrument, such as Bruker IFS66v/S, can set the modulated fre-
quency of He-Ne laser to as fast as 200 kHz, which calculates to 12.66 cm/s of the scan-
ner velocity.


interferograms. Co-addition is a technique which improves the signal-to-noise (S/N)

ratio. Corresponding data points must be sampled at the same path difference on every

successive scan. The zero crossing points of the laser interferogram may be used as a

reference point to start and tri -. -.-r the sampling of analog data from the detector. The

signal increases linearly since it is ah--,v- coherent. Noise occurs randomly, thus the

signal increases faster than the noise. This process, called signal averaging, increases

the S/N ratio as square root of the number of scans co-added. Therefore, in order to

increase the S/N by a factor of 2, the number of scans must be increased by a factor of


3.3 Polarization Modulation

Up to this point of this section, we have been talking about interferometers of

amplitude separation type which use a partially transmitting and partially reflecting

beam-splitter with Michelson configuration. Despite clear advantages over dispersive

type of monochrometer, amplitude separation interferometers also exhibit some difficul-

ties especially for very far infrared. The main difficulty of this type is the low efficiency

and limited spectral range of thin-film dielectric beam-splitters such as Mylar which is

mostly used for far infrared. In order to overcome such disadvantages, different types of

interferometers were developed. One of them is a lamellar grating interferometer. Rather

than separating amplitude of incident light, a lamellar grating separates wavefront of

incident light. The efficiency of lamellar grating beam-splitter is nearly independent of

frequency and can be very high. Although the lamellar grating uses different method

to separate light from film beam-splitter, both are still intensity modulation type of

interferometers. We will not discuss the lamellar grating interferometer further, but

interested readers are encouraged to read several papers about this topic [22, 29].

There is another type of interferometer which is commercially available these d v-

It is the polarizing interferometer based on a concept by Martin and Puplett [30]. It

has similar configuration as Michelson, but uses rather unique approach to produce

modulation of incident light. A schematic diagram of polarizing (or Martin-Puplett)

interferometer is shown in Figure 3-7. Light from an unpolarized (or polarized) source is


Unpolarized >- /B
(or polarized) M2
source /

P1 v M -


"7 Detector

Figure 3-7: A schematic view of a Martin-Puplett interferometer. The collimated light
is linearly polarized at a polarizer P1 and travels to a polarizing beam-splitter B which
is aligned at an angle of 450 with respect to the plane of polarization after P1. The
beam-splitter separates two polarization components sending one component toward a
fixed rooftop mirror Ml and the other toward a movable rooftop mirror M2. On reflec-
tion, the polarization of each beam is rotated by 900, and two beams are recombined
at the beam-splitter. At the beam-splitter, the initially transmitted beam is completely
reflected, and initially reflected beam is completely transmitted. The recombined beam
heads to the second analyzing polarizer P2 and the beam is linearly polarized after P2
with an amplitude varying periodically with path difference.

linearly polarized at a polarizer P1 in the plane at ceratin orientation. It is then divided

into two polarization components by a polarizing beam-splitter B which is typically a

free-standing fine wire grid (or a grid on Mylar film). This type of beam-splitter has

been shown to have almost frequency independent and high efficiency of nearly 10(I' .

from effectively zero frequency up to roughly 1/2d (cm-1), where d (cm) is the spacing

of the wires [31]. The gird reflects the component of the incident light parallel to the

direction of the wires and transmit the component normal to the direction of the wires.

When the beam-splitter is oriented with the wire grids at an angle of 450 with respect to

P1, the incident polarized light is equally split sending one component to a fixed mirror

Ml and the other to a movable mirror M2. Both M1 and M2 are the 900 rooftop mirrors

which rotate the plane of polarization by 900 on reflection. Therefore, when two beams

come back to the beam-splitter, the one reflected initially is transmitted completely, and


(a) (b) (c)


2 -------------------^------ ^AAAAV VAAA/^_

Figure 3-8: Interferograms produced by a polarizing interferometer. (a) The interfero-
gram for parallel P1 and P2. (b) The inverted interferogram for crossed P1 and P2. (c)
The difference of (a) and (b), which is obtained by polarization modulation technique.
Note that the mean level of the interferogram is automatically eliminated.

the one transmitted initially is reflected completely. No beam is sent to the direction of

the source. The combined beam finally passes through the second polarizer P2 with its

polarization axis either parallel or perpendicular to that of Pl.

As M2 moves, a phase difference is introduced between two beams. For a monochro-

matic source, the initially linearly polarized beam is elliptically polarized after recombi-

nation at the beam-splitter with an ellipticity varying periodically with increasing path

difference. At the ZPD, the recombined beam has the same polarization as the incident

beam on the beam-splitter. After P2, the beam is plane polarized with an amplitude

that varies periodically with path difference in the same way as in a Michelson inter-

ferometer. Assuming an unpolarized source, the intensity at the detector is given by

I (x) = [1 + cos(27vx)] (3.30)


Ii(x) = [l cos(2vx)] (3.31)

where Jo is the intensity of the linearly polarized beam incident on the beam-splitter.

The case III is for parallel P1 and P2, and I_ is for crossed P1 and P2. For a source of

continuous spectrum, the output intensity yields an typical interferogram except that of

Ij is inverted (see Figure 3-8).


The complementary nature of the interferograms for the two orientations of one

polarizer with respect to the other can be utilized to introduce polarization modulation.

This is usually done by keeping P1 fixed and by dynamically switching the orientation

of P2 using a polarizing chopper.9 Then, by using standard Lock-in technique, the

detected signal is the difference between II and 1L, which is given for a monochromatic

source as

I(x) = I (x) I (x) = o cos(27vx) (3.32)

Thus, the phase modulation technique eliminates the mean level of the interferogram

which could introduce errors due to spurious fluctuations (see Figure 3-8). This and

wide range high efficiency of a polarizing beam-splitter makes the Martin-Puplett

interferometer advantageous for very far infrared measurements. One of spectrometers

we used is the polarization modulation type. Details of the specific instrument will be

discussed in C'! pter 6.

9 We can keep P2 and rotate P1 just as well. If we have a polarized input source,
however, P1 should be aligned such that the most light can go through, and use P2 for
the polarization modulation. There is also a technique called a double polarization mod-
ulation which uses two polarizing choppers for P1 and P2 [32,33].


4.1 Introduction

In 1911, soon after successfully liquifying helium in 1908, Kammerlingh Onnes

discovered that the electrical resistance of mercury suddenly drops to an unmeasur-

ably small value when it is cooled below 4.2 K [34]. This phenomenon was named as

superconductivity. In subsequent years, many more metals and metallic alloys were

found to be superconducting when cooled to below a certain critical temperature T,.

In 1933 Meissner and Ochsenfeld demonstrated another basic property of supercon-

ductor, perfect diamagnetism, which is known as the Meissner effect [35]. In 1935,

F. London and H. London developed a purely phenomenological description through

a modification of an essential equation of electrodynamics in such a way to explain

the Meissner effect [36]. They pointed out that superconductivity is a fundamentally

quantum mechanical phenomenon that is observed on a macroscopic scale, with an

energy gap between superconducting and normal state. Another phenomenological

theory was also developed by Ginzburg and Landau in 1950 [37]. Then, finally in 1957,

Bardeen, Cooper, and Schrieffer proposed a microscopic theory of superconductivity as

a phenomenon where electrons form pairs and an energy gap develops in the electronic

density of states around Fermi energy [38]. This so-called BCS theory remains as a valid

microscopic explanation for many simple superconductors (BCS superconductors).

A breakthrough in superconductivity research occurred in 1986 when Bednorz and

Miller found a copper oxide compounds of the Ba-La-Cu-O system which superconducts

at a substantially higher temperature (T, ~ 30 K) than previously known [39]. With

their work, a new era of superconductivity opened in this class of materials (high-T,

superconductors) which differs from conventional BCS superconductors. Now we know

various materials that have T, above the boiling point of liquid nitrogen (77 K).

The microscopic theory of superconductivity can not be described in the language

of the independent electron approximation, and relies on formal techniques.1 It is quite

extensive and highly specialized. Consequently, we will limit our survey of the theory

to qualitative descriptions of some of the in i.' concepts within the framework of BCS

theory. Details of the subject can be found in many places [40, 41, 42, 43, 44]. In the

following section, we will merely summarize a few basic properties of superconductors.

In C'!I pter 7, we will provide brief theoretical background for the infrared properties

of superconductors and the effects of localization on superconductivity, based on the

work by Mattis and Bardeen [45], and by Maekawa and Fukuyama [46], respectively. In

C'!I pter 8, we will discuss theory of nonequilibrium superconductivity.

4.2 Fundamentals of Superconductivity

4.2.1 Fundamental Phenomena

Vanishing DC Resistance

Of all the characteristics of superconductors, the absence of any measurable DC

electrical resistance is the most striking phenomenon. Above a critical temperature T, a

bulk superconducting specimen behaves completely as normal metal with DC resistivity

generally given by

p(T) po + BT (4.1)

where the first term arises from impurity and defect i ii. 1 ii.- and the second term

from phonon scattering. Below Tc, the metal becomes superconducting with no dis-

cernible DC resistivity (zero DC resistivity), and current flows in it without any

1 The second quantization description of many-body system is used to describe the
BCS theory, including the energy of the BCS ground state, giving the energy gap result-
ing from the electron pairing.


Table 4-1: Transition temperatures for several superconductors. Some of high-T, materi-
als are also listed.
Element Te(K) Compound Te(K) High-To T (K)
Mo 0.92 NiTi 10 Bao.75La4.25Cu505(3-y) 30
Al 1.2 NbN 15.2 La2_-SrCuO4 38
In 3.4 Nb3Sn 18.1 YB2C307 92
Hg 4.1 Nb3Ga 20.3 Bi2Sr2CaCu202 85
Pb 7.2 Nb3Ge 23.2 Bi2Sr2Ca2Cu3010 110
Nb 9.2 MgB2 39 HgBa2Ca2Cu30s 133

dissipation of energy. The transition of a bulk material is usually abrupt, and hap-

pens at very low temperature.2 Table 4-1 lists the transition temperatures of several

superconductors. The fact that there is no measurable resistivity allows us to pass large

current through a superconductor, and in turn, to create large magnetic field. However,

if the current density exceeds a critical current J,, a superconductor reverts to a normal

conductor (Silsbee effect). J, is related to whether the magnetic field created by the

current exceeds the critical field He above which superconductivity is destroyed. In an

AC electric field, superconductors at finite temperature no longer exhibit zero resistivity.

The resistivity increases with frequency. However, at temperatures well below T~, the

resistivity is still negligible provided that the frequency is not too large (< A/h, where

A is the energy gap.). 7.2.1 describes the response of superconductors in AC fields.

Meissner Effect

A superconductor expels magnetic flux, and hence acts like a perfect diamagnet.

This is another peculiar phenomenon of superconductivity known as the Meissner effect

(or Meissner-Ochsenfeld effect). Having p = 0 in the Maxwell equations for a perfect

conductor, we find OB/Ot = 0, and thus the magnetic flux is expected to remain

unchanged within the specimen. In superconducting state, however, we find not only

B = constant, but also B = 0, and the field penetrating the specimen (provided that

2 The thermal energy kBTc corresponding to the transition temperature is on the or-
der of a few meV or less. This is much smaller than the energy scale such as the Fermi
energy EF (~ 10 eV) and the Debye energy huoD (~ 0.1 eV) of the most of metals.

it is not too strong) prior to making the transition to superconducting state will be

expelled from the interior. A simple explanation for this effect is that the impinging

magnetic field induces shielding currents on the surface of a superconductor, which

are just enough to cancel the field in the interior. Since the superconductor has zero

resistivity, the currents (i.e., supercurrents) will persist even after the field stopped

changing. The distance which the supercurrents form a finite sheath into the specimen

is called the London penetration depth AL. Magnetic flux can also penetrate the same

distance into the material. For many simple, pure metals, this penetration depth is on

the order of 500 A. As mentioned above, if the field gets too large, however, the material

will eventually lose its superconducting state.

Magnetic Flux Quantization

It is another property of superconductors that the magnetic flux passing through

any area enclosed by a supercurrent in a closed loop can only take on values of integral

multiples of the so-called flux quantum (or fluxoid):

40 2.0679 10-7G cm2 (4.2)

where h is Planck's constant, c is the speed of light, and e is the elementary charge.

This flux quantization is a consequence of that the complex order parameter Q(r) =

i', '|. (introduced in the Ginzburg-Landau theory) is a single-valued function, and thus

its phase must change by 27 times an integer.

A similar effect occurs when a type II superconductor (refer to 4.2.3) is placed

in a magnetic field. At sufficiently high field strengths, some of the magnetic field may

penetrate the superconductor in the form of thin threads of material that have turned

normal. Each thread is in fact the central region ("core") of vortex of the supercurrent,

and carries a single flux quantum.

Josephson Effects

When two superconductors are in weak contact (e.g., separated by a thin insulating

oxide barrier (10 20 A), a normal conducting lv-r (100 1000 A), or a constriction),


Cooper pairs could tunnel from one to the other giving rise to a characteristic current

through the so-called Josephson junction.

When there is no voltage drop across the junction, a DC current will be generated,

given by

I = l sin (4.3)

where po is a constant phase difference (i.e., relative phase) of the BCS irn ivi-body

states in two superconductors, and Ic is the maximum current that can pass through the

junction before driving it to a resistive state. This is called the DC Josephson effect.

When a DC voltage U is applied between the junction, the relative phase pO evolves

with time as
(t) = (0) t (4.4)

This gives rise to an AC current given by

I = Isin[(0) t] (4.5)

2e= (4.6)

This is called the AC Josephson effect. Interesting interference phenomena arise when

two Josephson junctions are connected in parallel. These can be used for a very sensitive

magnetic field senor, known as the SQUID.

These are the effects where the characteristic high coherence of the Cooper pairs

becomes particularly evident.

Isotope Effect

When a constituent atom of a superconducting material is replaced by its isotope,

the critical temperature often changes with atomic mass M in accordance with the


M'T = constant ,



where a = 1/2 for the simplified BCS model. As will be shown later, the simple BCS

theory predicts that Tc is proportional to the Debye frequency LcD (hD is a measure

of the typical phonon energy), and thus this is expected since the Debye frequency is

proportional to the square root of the atomic mass for a simple metal.

4.2.2 Thermodynamic Properties

The transition of a metal from its normal state to its superconducting state is a

thermodynamic phase transition. Therefore, some sort of changes may be expected in

thermodynamic quantities as a specimen makes its transition. The electronic part of

specific heat, for example, increases discontinuously at T, from the linear temperature

dependence observed in normal state (T > Tc), and then at very low temperatures sinks

to below the value of the normal phase. At temperatures well below T,, the specific heat

decreases exponentially.

In thermodynamics, we can use a free energy F(T) to describe the stability of a

system at a given temperature. Naturally, the system tends to change its state toward

the lower free energy, and becomes stable at a minimum of the free energy. Below the

transition temperature, the free energy in the superconducting phase F, is reduced

below that in the normal phase F, (i.e., F, < F,) due to the condensation of Cooper

pairs, and thus the system naturally becomes superconducting. The critical temperature

is the temperature where F, crosses over F,.

As mentioned earlier, superconductivity is destroyed with a sufficiently large

magnetic field. When the amount of work has to be done to establish the magnetic field

of the screening currents that cancels the field in the interior becomes larger than the

reduction of free energy by turning into a superconducting phase, it is energetically

advantageous for a specimen to revert back to normal, allowing the field to penetrate.

The difference in free energy between normal and superconducting states is related to

the critical field He as

F (T)- F,(T) H(T) (4.8)

Thus, from He, we can calculate the free energy difference. Then using Eq. 4.8, we

can deduce a series of thermodynamical properties including the difference in entropy

between two phases, the latent heat of the transition, and the discontinuity in the

electronic specific heat. The latent heat of the transition L vanishes for the transition

in zero field, and thus the superconducting transition in zero field is of the second

order. When a magnetic field is present, however, there is a latent heat, and the nature

of the transition changes to the first order. The difference (F, Fn)T=o is called the

condensation energy.

4.2.3 Types of Superconductor

Superconductors are categorized as being one of two types: type I and type II. Type

I superconductors are mostly the simple (nontransition) metals and metalloids with low

T,. The BCS theory explains these superconductors quite well. Type II superconductors,

in contrast, are more complex (transition metals, intermetallic compounds, high-Tc,

and etc.), and often have a higher T,. One of the main difference between two types

of superconductors is the manner in which penetration occurs with increasing external

magnetic field strength. It generally depends also on the shape of the specimen, but

the clear distinction can be demonstrated with the simplest geometry of a long cylinder

(diameter > penetration depth) with its axis parallel to the applied field.

Type I

With the applied field below a critical field [H < Hc(T)], magnetic flux does not

penetrate the bulk of the type I cylindrical material, thus showing complete Meissner

effect (B = 0). When the applied field exceeds H,(T), the entire specimen becomes nor-

mal, and the field penetrates completely (B = H). This type of behavior is particularly

evident in the magnetization curve M(H). We know that the magnetic field inside any

material satisfies Eq. 2.18. Therefore, up to He, -47M increases in proportion to the

external field H. Then, at He it abruptly vanishes except for the very small diamagnetic

and paramagnetic effects of normal metals.


With more complex geometries, the fields at some macroscopic portions of specimen

necessarily exceed H,, and therefore, the sample exhibits an intermediate state with

some parts being normal while the rest li.ii;; superconducting.

Type II

For type II superconductors, there are three distinct phases depending on the

strength of the applied field. Below a lower critical field H~i(T), there is no flux pene-

tration just as for type I material. When the applied field exceeds an upper critical field

H,2(T), there is complete flux penetration, and the specimen becomes totally normal.

However, when the applied field is in between H~I(T) and H,2(T), there is a partial pen-

etration of flux into the specimen developing a rather complicated microscopic structure

of both superconducting and normally conducting regions. This phase is known as the

mixed state (or Shubnikov phase). The flux penetrates in the form of thin filaments

(referred as vortex lines). In the core of a filament, the field is high and the material is

normal. Each filament is surrounded by a superconducting screening current and en-

closes exactly one flux quantum o4. Current flows through the superconducting regions

and thus the material still has zero resistance. The vortex lines repel one another due

to the magnetic force between them, and thus they arrange themselves into an ordered

array of a triangular lattice. With increasing external field, the distance between the

vortex lines becomes smaller, and at H,2 they overlap completely. The magnetization

curve of type II superconductors is quite different from that of type I. Up to H,1, -47M

rises linearly with the applied field H just like type I. At H,1 partial penetration begins,

and the magnetization decreases monotonically with increasing field until it vanishes

completely at H,2. In contrast to the behavior of type I, the transition is not abrupt.

4.2.4 Length Scales

London Equation and Penetration Depth

In an effort to describe the observed behavior of the Meissner effect correctly, F.

and H. London -,ir.-. -1. '1 a condition (known as the London equation) that the local

magnetic field h(r) and the current density carried by superconducting electrons j,(r)


Vx j, h, (4.9)

where m is the effective mass of the superconducting electrons, and N, is the super-

conducting electron density (or superfluid density).4 This purely phenomenological

equation, together with the Maxwell equation V x h = 47j,/c yields


2j i, (4.11)

where the length scale AL, known as the London penetration depth, is defined by

AL(T) 4 2t)2 1 (4.12)

Eqs. 4.10 and 4.11 allow us to calculate the distribution of fields and currents within a

superconductor. For the simplest geometry of a semi-infinite superconductors occupying

the half space z > 0, the solutions of these equations decay exponentially showing

that both magnetic fields and currents in superconductors can exist only within a liv.

of thickness AL of the surface. Therefore, the London equation implies that when a

superconductor is in an external magnetic field, the surface current flows in a thin -1v- r

and keeps the interior field-free; i.e., the Meissner effect.

Coherence Length

The London equation (Eq. 4.9) assumes that the current density j,(r) at one

point r is related to the field h(r) (or the vector potential A(r)) at the same point.

3 Fields and currents are assumed to be weak and slowly varying on the length scale
of the coherence length of the superconductor.

4 The London brothers incorporated the two-fluid model of Gorter and Casimir [47].
The model separates the total density of conduction electrons N into a density of su-
perconducting electrons N, superfluidd density) and a density of normal electrons N,
(normal fluid density) such that N N= N + NV, Ns -- N as T -- 0, and N, = N when
T > T,. They assumed that only the superfluid participates in a supercurrent while the
normal fluid remain inert at T < Tc.

Thus, the London equation is a local equation. However, it is more general to assume

that j,(r) at one point r will depend on the vector potential A(r') at all neighboring

points r'. In order to describe this non-local effects, Pippard modified the London

equation, and introduced a length scale 1o, such that Ir r'i| < o [48]. This distance

o is one of fundamental lengths characterizing a superconductor, and is referred

to as the coherence length. In one context, it is used for the distance over which

the density of superconducting electrons N, varies significantly (from zero to full

thermodynamic value). In another context, it is used as the spatial extent of the pair

wave function (i.e., the size of a Cooper pair). In pure materials well below Te, however,

both coherence length definitions have the same value; using the uncertainty principle,

Pippard estimated the coherence length to be

o = (4.13)

where vU is the Fermi velocity, and Ao is the energy gap around the Fermi surface in

the superconducting state at absolute zero. Note that the Pippard's coherence length is

independent of temperature.

In the London model, it was assumed that the density of superconducting electrons

N, would have the full thermodynamic value right from the surface (i.e., 0o = 0). Since

j, and h vary on a scale AL, we might expect that the London's model is valid only for

AL > o0. In fact, this is the case, and Pippard's non-local model reduces to the London

model in such a limit. The materials that satisfy this condition (AL > 0) are the type

II superconductors, and Eq. 4.12 accurately calculate the penetration depth for the type


In type I materials, on the other hand, the penetration depth is much shorter

than the coherence length (AL < o). Thus, Ns does not reach its full value over the

penetration depth. This implies that not all of the electrons within the thickness o

from the surface contribute to the screening currents. For these materials the London

equation is inadequate. In order to calculate the penetration depth in the type I

materials more accurately, the Pippard's non-local model has to be used, and a rigorous

calculation gives

A 0.62A0o (4.14)

Consequently, the field penetrates type I materials deeper than the London value.

Ginzburg-Landau Theory

Another phenomenological approach proposed by Ginzburg and Landau (GL)

describes superconductivity in terms of a complex order parameter &(r) = &|(r)|e ,

whose magnitude |I(r) is a measure of the superconducting order at position r below

Tc [37, 49]. The order parameter ) is zero above T, and increases continuously as the

temperature falls below Tc. The physical significance of &(r) was not clear at the time

the GL theory was developed, but now we can interpret it as a wave function of a

particle of mass m*, charge q, and density N*, which are given by

m* = 2m (4.15)

q 2e, (4.16)

N* = |12 = ,/2 (4.17)

where m, e, and Ns are the effective electron mass, electron charge, and superfluid

density, respectively.

In the GL formalism, two temperature dependent characteristic lengths are intro-

duced: the coherence length (T) and the penetration depth A(T). The GL coherence

length defines the length scale over which Q(r) varies, and is given by

h h2 1/2
( 2T) 2 1) (4.18)

where a is a temperature dependent coefficient in the 12 term of the free energy

(See [40]). It is closely related to the Pippard coherence length o defined in Eq. 4.13. In

weak fields, the GL penetration depth is given by

( *C 2 1/2
A(T) 2q12 (4.19)
47q 21 b0

where ',, is the equilibrium order parameter well inside the material.

For a pure material near Tc, the microscopic calculation in the BCS approximation


(T) 0.74(o 1- 12 (4.20)

A(T) --/(0) (-- (4.21)
V2 Tj

Since both diverges in the same way as T To, it is practical to form their ratio

K = A (4.22)

The ratio K is known as the Ginzburg-Landau parameter of the material. For a pure

material, this is given by

S=0.96 A ) (4.23)

The difference in the behavior of two types of superconductors in a magnetic field

depends on whether the creation of interfaces between normal and superconducting

regions is energetically favorable, or not. Penetration of magnetic field reduces the field

energy penalty implicit in the Meissner effect. Thus, a material with large A favors

interfaces. Large coherence length means a greater extent of the superconducting state.

Therefore, with the associated energy gain from the superconducting condensation

energy, a large opposes interfaces. Interfacial energy changes sign at =t 1//2. When

V < 1/v2, the material is the type I, while r > 1//2, the material is the type II. In the

limit of r > 1, the GL theory reduces to the London theory.

Electron Mean Free Path

Another important length scale characterizing a superconductor is the electron

mean free path I in the normal state due to elastic scattering by disorder. In the

presence of disorder, we can define an effective coherence length (1), which is valid at

absolute zero,
1 1 1
S+ 1- (4.24)
(1) o I *
Depending of the size of 1 relative to o, we can think of two limiting cases concerning

the purity of a superconductor: clean limit and dirty limit [44]. Up to this point, we

have concerned only pure materials, and the mean free path has p1l i, d no role on

determining the characteristic length scales. In reality, however, the actual values of

both coherence length and penetration depth are somewhat modified from the values

defined above by mean free path effects.

In the clean limit (1 > co), Eq. 4.24 gives

(1) =o (4.25)

Thus, the coherence length and the penetration depth we discussed above can be used

without any modification.

In contrast, Eq. 4.24 gives for the dirty limit (1 < o),

(1) 1 (4.26)

Thus, the coherence length at absolute zero is completely determined by the mean free

path, the length that governs the transport properties of the material in the normal

state. In this limit, the relationship between current and magnetic field (i.e., vector

potential), and in turn, magnetic penetration depth are modified. Note that the London

penetration depth AL given in Eq. 4.12 is an expression for pure metals. The dirty limit

expression of the magnetic penetration depth is [40]

A AL( 1 (1 o). (4.27)

Thus, A increases as I becomes shorter. At temperatures near T,, the coherence length

and the penetration depth in the dirty limit are given by [40]
/ T\ 1/2
(T) 0.85(ol)/2 (4.28)

A(T) 0.62AL ( 1 (4.29)

Thus, the GL parameter for a dirty material is given by

S0.75AL(0) (4.30)

Consequently, as the mean free path becomes shorter, the coherence length becomes

smaller than that given in Eq. 4.13 and penetration depth becomes longer than the

London's defined in Eq. 4.12. In fact, it frequently happens that alloying a pure type I

superconductor transforms it into a type II superconductor. Many of superconductors in

the form of thin films, not to mention those in amorphous form, are in the dirty limit.

4.2.5 BCS Theory

In 1957, Bardeen, Cooper, and Schrieffer (BCS) proposed a microscopic theory

of superconductivity (now known as the BCS theory) [38]. A central result of the

BCS theory is the existence of an energy gap between the electron system in the

superconducting ground state and the excited states. Here, we will only describe the

underlying ideas, assumptions, and 1 i i"r predictions associated with the theory,

without any rigorous mathematical details.

Cooper Pairs

The ground state (T = 0 K) of a non-interacting Fermi gas of electrons corresponds

to the situation where all electron states with wave vector k within the Fermi sphere

(with EF h2k2/2m) are filled and all states outside are empty. If a pair of electrons

is added in states just above EF, the total energy of the system should increase by the

kinetic energy of the pair. However, Cooper recognized that if there is an attractive

interaction between the electrons, no matter how weak it is, they will form a bound

state, and adding a pair of electrons may reduce the total energy (kinetic plus potential

energy).5 Thus, the normal state becomes unstable to the formation of these paired

bound states [50].

5 Note that these two additional electrons are prevented from individually having
energy less than EF by the Pauli exclusion principle.


Since electrons have a repulsive Coulomb interaction,6 the attractive force must,

in theory, come from some interaction between the electrons that is mediated by some

other mechanism inherent in the material. Cooper argued that continuous exchange of a

virtual phonon between electrons occupying states kl and k2 provides a mechanism for

a weak attraction that results in the reduction of total energy.7 The probability of the

energy-reducing phonon exchange processes is maximum for the case ki = -k2 = k. It

is therefore sufficient to think that electrons with equal and opposite wave vectors form

a pair. This so-called Cooper pair can be represented by a two-particle wave function

given by

(ri, r2) akik-(r-r2), (4.31)
which is symmetric in spatial coordinates (ri, r2) upon exchange of electrons 1 and 2.

The range of summation is confined to

EF < < EF + Lj+ D (4.32)

ak is the probability amplitude for finding one electron in state k and the other in -k.

Since the states k < kF are already occupied, the Pauli principle requires ak = 0 for

k < kF. Because the Pauli principle applies to both electrons, the spin part of the whole

wave function must be antisymmetric. Therefore, a Cooper pair must be imagined as an

electron pair in which the two electrons ahv--, occupy states with opposite wave vectors

and opposite spins (k T, -k 1), (k' T, -k' 1), an so on.8

6 Screening reduces the natural Coulomb repulsion between two electrons, leading to
an effective interaction which is relatively short range compared with the unscreened
Coulomb potential.

7 Originally, the phonon-mediated attractive interaction was proposed by Frohlich in
1950 [51].

s This pair state is sometimes referred as an s-wave, spin-singlet pairing state
(i.e., L 0 and S 0).

BCS Ground State

The formation of a Cooper pair leads to an energy reduction. In a real material,

many more electrons participate in the Cooper pairing to achieve a new lower-energy

ground state. Because a Cooper pair is composed of two fermions with opposite spin,

it may be considered as a single entity that obeys Bose-Einstein statistics. Thus, at

T = 0 K, all Cooper pairs condense into an identical two-electron state even though the

individual electrons are being scattered continually between single electron states. The

Pauli principle limits the states into which the two interacting electrons, which make up

the pair, may be scattered.

In order to calculate the ground state (T = 0 K), BCS made several assumptions

for simplicity. First of all, just like Cooper did, the BCS theory is based on the free

electron approximation. Thus, the Fermi surface is spherical. They also simplified the

net attractive interaction between electrons by expressing the matrix element that

describes scattering of the electron pair from (k 1, -k 1) to (k' 1, -k' 1) and vice versa


Vk' -V/L3 for l D (4.33)
0 otherwise ,

where V is a positive constant, L3 is the volume of the system, h;D is the Debye energy,

and ( is the kinetic energy relative to the Fermi level, which is defined as

h= Ek. (4.34)

Further, they approximated the BCS ground state vector of the many-body system

of all Cooper pairs by the product of the identical pair state vectors. With all the

assumptions, they deduced the ground state energy of the superconductor, and showed

9 In some materials (e.g., Hg and Pb), the electron-phonon interaction is very strong.
In such case, the net interaction between electrons is quite complex and even retarded.
Since phonons are the origin of the coupling, the phonon structure of each material will
also influence the matrix element. The extension of the BCS theory to strong-coupling
superconductors is known as the strong-coupling theory [44].



E,+Ao EF+A(T)

E, N(E)/N(O) E, > N(E)/N(O)

(a) (b)

Figure 4-1: (a) Density of states in BCS ground state (at T = 0 K) relative to that in
normal state. A gap of 2Ao is developed around the Fermi level. The shaded region are
the states occupied by superconducting electrons. Note that no states are lost in the
phase transition. (b) Corresponding density of states at finite temperature (T < Tc).
Cooper pairs are thermally broken creating quasiparticles (or normal electrons) above
the gap, and the gap 2A(T) is smaller than the ground state value.

that the energy density of superconducting ground state is reduced from that of normal

state by N(0)A2/2, where N(0) is the single-spin electronic density of states at the

Fermi energy, and A is a half of the energy gap developed in the density of states

around the Fermi level at absolute zero. The energy gap is given by the solution to

1 D d -1 D
N(-O)V J WD d sinh-1 (. (4.35)

In the weak-coupling limit (i.e., N(O)V < 1),10 we find

A = 2h De-1/N()V (4.36)

Figure 4-1(a) shows the corresponding density of states in the vicinity of the Fermi

level. Note that it requires a minimum energy of 2A to break up a Cooper pair and

create two uncorrelated electrons. It is this gap that gives rise to the superconductor's

zero resistivity.

10 Empirically, the weak-coupling limit is justified when N(O)V < 0.3.






0.0 0.2 0.4 0.6 0.8 1.0
T/ T
Figure 4-2: Temperature dependence of the gap as a function of temperature in the
BCS approximation.

BCS Predictions

At temperatures above T = OK, thermally excited phonons become available to

scatter electrons in a Cooper pair. Thus, there is a finite probability of finding electrons

in the states above the gap 2A (see Figure 4-1). These excited electrons are called

quasiparticles. As the temperature increases, pair-breaking is progressively enhanced,

and finally Cooper pairs cease to exist at Tc. In a meantime, the gap 2A shrinks as T

increases, and completely closes at Tc. In the framework of the BCS theory, one can

deduce the temperature dependence of the gap A(T) as the solution to

1 [hwD T ]t [
t 1"( 24f ( 4.37)
N(O)V o 2 + A2

where f(E) is the Fermi function. Note that for T = 0, f(E) is zero (E being positive),

and one recovers Eq. 4.35. This equation defines an implicit relation between A and T,

and numerical analysis yields A(T) as shown in Figure 4-2. As the figure shows, the gap

develops quickly as the temperature is lowered below Tc, and opens up almost fully by a

half of T,. It is also convenient to remember that the result of numerical analysis can be

approximated by
A(T) cos (T ) (4.38)
A(0) 2 T,
At temperatures near Tc, the value of the gap can be approximated by

A(T) T \1/2
A() 174) 1 (4.39)
A (0) T,

By setting A = 0 in Eq. 4.37, an equation for T, is determined by

1 rfv^D t
V T tanh (4.40)
N(0)V o 2kBT,

For hwD > kBTc, numerical calculation yields

kBT 1.4De-1/N()V (4.41)

Then, by comparing Eq. 4.36 with Eq. 4.41, one can find the relationship between the

ground state gap Ao[= A(0)] and T, in the weak-coupling limit as

2 = 3.52 (4.42)

which is free from parameters such as V and wD.11 This relationship agrees quite well

(within about 10 percent with tunnelling experiments [52]) for the weakly coupled

superconductors. For those strong-coupling superconductors (e.g., Hg and Pb), the ratio

is larger than 3.52. For example, the ratio for Hg and Pb were experimentally found to

be 4.6 and 4.3, respectively. The strong-coupling theory [44] provides better agreement.

At finite temperature, the quasiparticle-occupation of the excited one-electron states

E = ((2 + A2)1/2 obeys Fermi statistics. Therefore, the density of quasiparticles at a

11 The Debye temperature OD (hence, Debye frequency UwD) may be deduced from
specific heat measurement, but the matrix element V is difficult to calculate precisely.
Therefore, parameter-free expression are desired.

function of temperature is given by

Nq = fka f(E)N( )d 2N(0) (4.43)
ka -O J0 + exp( +A2l2BT)

4.2.6 Eliashberg Formalism

In the BCS theory, the dynamic interaction induced among electrons by phonon

was crudely represented by using a dimensionless constant N(O)V as the strength of

electron-electron interaction and the Debye energy hwD as the maximum phonon energy;

the details of the electron-phonon coupling was not considered at all. Even though

the theory was quite successful for weak-coupling superconductors, it failed to treat

strong-coupling superconductors accurately. Eliashberg took a more general approach to

the electron-phonon coupling by taking into account the retarded nature of the phonon-

induced interaction and by properly treating the damping of the excitations [44, 53, 54,


The Eliashberg theory starts with two nonlinear coupled equations for the gap A

and the renormalization factor Z, which replace the BCS gap equation. In the Eliash-

berg equations, two parameters, p* and A, are introduced, where p* is known as the

Coulomb pseudopotential (or the renormalized Coulomb interaction parameter) which

describes a residual repulsive screened Coulomb interaction [56], and A is known as the

electron-phonon coupling constant12 which is related to the attractive interaction. The

constant A is defined by

A 2 QFQd (4.44)
where a2(Q) is the effective electron-phonon interaction, and F(Q) is the phonon density

of states with Q the frequency of the exchanged phonon. In the Eliashberg theory,

12 The parameter A is also known as mass-renormalization (or mass-enhancement)
parameter because the effective electron mass is modified by the electron-phonon in-
teraction as Z = 1 + A = m*/m. Superconductors are characterized according to the
magnitude of A, weak-coupling (A < 1), intermediate coupling (A < 1), strong coupling
(A > ).

a2(Q)F(Q) (the electron-phonon spectral density) is an important function that contains
all the relevant information about the electron-phonon interaction that gives rise to the
effective attractive interaction between electrons around the Fermi energy. In principle,
superconducting tunnelling measurements provide direct information on a2 ()F(Q).
For a simple model of a metal, a2(Q)F(Q) can be approximated at low frequencies by
the quadratic form bQ2, where b is a constant characteristic of a given material. For the
strong-coupling superconductors, a2 ()F(2)) shows significant low frequency structure.
From the Eliashberg equations, McMillan developed a much more quantitative
equation for T, than the BCS result (Eq. 4.41) [57]. The McMillan formula improved
later by Allen and Dynes is given by [58]

1kBTC -hexp rA + 02A) (4.45)
ksT, = exp (4.45)
1.2 A- p(1t+0.62A) '

where Qin is a characteristic phonon frequency defined by [58]

n,11 = exp +j2 In(Q) ,2() d (4.46)

Also from the Eliashberg equations, the ratio 2Ao/kBTc is expressed approximately
by [55]
2kB = 3.53 1 + 12.5 n n( ) (4.47)
kWT, 0n 2Tc
where 01, hQ= i/kB. This equation includes strong-coupling corrections in terms of the
parameter T1c/O1, and the universal BCS value is recovered for T1/01n 0. It shows
excellent agreement with experiment.


5.1 Synchrotron Radiation

5.1.1 Introduction

It is well known that an accelerated charged particle emits electromagnetic radia-

tion [9]. Synchrotron radiation is radiation emitted by a charge moving at relativistic

speed. It is a very stable, high flux, broadband light source. In addition, it has peculiar

properties such as polarization, pulsed time structure, angular collimation, and small

source size. It was first identified as a technical problem in accelerator physics, but its

properties make synchrotron vital for various fields of science.

Synchrotron facilities available around the world are based on the use of an

electron storage ring, a closed, high-vacuum chamber with a number of circular arc

and straight segments. In this section the theory and properties of synchrotron radiation

are summarized in particular for an electron making a circular trajectory at a dipole

bending magnet section of a storage ring. Synchrotron radiation from so-called insertion

devices, such as wigglers and undulators placed at straight segments, will not be

discussed here [59,60].

5.1.2 Radiated Power from a Bending Magnet

For a single nonrelativistic (v < c) accelerated particle with charge e, the total

instantaneous radiated power is given by Larmor formula:

2 e2
P = 3 112 ,(51)
3 C3

where v is the acceleration of electron.

Larmor formula can be generalized for arbitrary velocities by a series of Lorentz

transformations. For a particle of mass m in circular motion with velocity = v/c,

energy E, and radius of curvature (the bending radius) p, the relativistic (v C c)


generalization of the formula is [9,59,60,61]

2 e2c 4 4
P 44 (5.2)
3 p2

where 7 = E/mc2. The emitted power is proportional to the fourth power of the energy

and inversely proportional to the rest mass. This property explains why electrons are

used rather than other heavier charged particles such as protons. When electron travels

around a storage ring, it makes a circular trajectory and emits radiation only while it

experiences magnetic field at each bending magnet.1 Thus, the total time for it to have

radiative-energy loss per revolution is 27p//3c. Therefore, the radiative-loss per turn by

one electron is
4r e2
6E = 3 4 (5.3)
3 p
For a highly relativistic electrons ( -_ 1) Eq. 5.3 is expressed in practical units as2

6E(keV) 88.5 x Ge (5.4)

The synchrotron radiation spontaneously emitted from _,rn' i electrons in random

distribution (as from a storage ring) is generally incoherent. Figure 5-1 schematically

shows this situation as well as coherent radiation which can be found for example in the

coherent telahertz emission from micro-bunched electrons or in the free electron lasers

(FEL). In incoherent case, the total radiated power by N electrons in the ring is simply

N6E/T where T is the period of electron circulation around the ring. Therefore, the

total power radiated by ring with ring current i is given by3

Pi,,g(kW) 88.5 [E(GeV) A) 26.5[E(GeV)]3B(T)i(A) (5.5)

1 The bending radius p is related to the magnetic field B of the bending magnet,
which is given in practical units as p(m) 3.336E(GeV)/B(T). For the VUV ring with
E = 0.808 GeV and B = 1.41 T, the bending radius p is 1.91 m.
2 For the VUV ring, 6E ~ 20 keV per electron per revolution.

3 For the VUV ring, Pig, r 20 kW/amp of beam.

(a) Incoherent (b) Coherent

Eincoherent- N1/2Esingle Ecoherent NEsingle
P ~ NP P ~ N2P
incoherent NPsingle Pcoherent- single

Figure 5-1: (a) Incoherent radiation from N-electrons in random distribution,and (b)
coherent radiation from micro-bunched N-electrons.

(a) p<<1 (b) p~-1

radius -

4---- \
Electron orbit

Figure 5-2: Angular distribution of radiation emitted from an electron moving along a
circular orbit.

This equation tells us that the total intensity delivered to each beamline is proportional

to the beam current, thus the current signal can be used to normalize measured spectra

in order to compensate time varying intensity due to decay of beam current.

5.1.3 Angular Collimation and Polarization

For an electron circulating at nonrelativistic speed, the angular distribution of

emission is a dipole pattern which extends to a large range of angles (see Figure 5-2).

For a relativistic electron, however, the radiation is strongly concentrated to a narrow

angular range around a direction tangential to the orbit as shown in Figure 5-2. The

divergence of the vertical angle i is roughly estimated from 7-1


The instantaneous power (in cgs units of erg/[sec rad cm]) radiated per unit

wavelength and per unit vertical angle according to the Schwinger theory [61] is given by

d2P(A, ), t) 27 e2 4 A 8 ( + 2 2 X2 2
3 p 7 '(1+} LK2/3(0) + x K 3(0) (5.6)
d~d 3273 3 A 1 + X2 I/

where X = 7', = Ac(1 + X2)3/2/2A, and the subscripted K's are modified Bessel

functions of the second kind. The parameter Ac is called critical wavelength that

characterizes the spectral output of particular storage ring which is given by4

Ac 47p/373 (5.7)

Half the total power is emitted as photons of wavelength shorter and half longer than Ac.

Eq. 5.6 is the basic formula for the calculation of the characteristics of the synchrotron


The bending magnet radiation has a peculiar polarization property. The two terms

in the square brackets of Eq. 5.6 are associated with the parallel and perpendicular

components of the emitted power, respectively. At small vertical angles b the radiation

is predominantly polarized in the direction parallel to the electron's orbital plane, and at

S=- 0, it is completely linearly polarized. As the vertical angle increases, perpendicular

component starts showing up, but the parallel component is aliv-, the larger of two.

Both components are phase correlated, and as a consequence, the emission observed

above and below the electron's orbital plane (i.e., b / 0) is elliptically polarized. In

Figure 5-3 the normalized intensities of the parallel and perpendicular components are

plotted as a function of ) for the VUV ring of NSLS at three different photon energies.

This figure shows that the radiation is strongly concentrated at the critical wavelength,

but the vertical spread increases at longer wavelengths.

4 Alternative parameter used for the same purpose is the critical photon energy which
is given by hv = hc/A, 3hcy3/47p.


1.0 -,
\ dash://
08 solid: I
0. blue: 100 cm1
g ; green: 1000 cm-
a 0.6 -
S\ red: X
4 \c
N 0.4

Z 0.2 ,

0 10 20 30 40
w [mrad]

Figure 5-3: Angular distribution of parallel and perpendicular polarization compo-
nents at three different photon energies. The critical wavelength of the VUV ring is 19.9
A (~ 5,000,000 cm-1).

5.1.4 RF Cavity and Pulsed Nature

The accelerating fields inside the RF cavity system periodically acts on the circulat-

ing electrons to restore the energy lost due to emission. Because the RF field oscillates,

only electrons arriving at a particular time receive the proper acceleration. This leads

to form electron bunches which are contained in regularly spaced, imaginary contain-

ers so-called "RF b,. I [21]. Therefore, the light produced by the synchrotron is

pulsed. This pulsed time structure of the synchrotron radiation was exploited in our

time-resolved measurements. For the ordinary linear spectroscopic experiments, the time

constant of common detectors are much longer than the pulse repetition period of the

radiation. For example, the most commonly used far infrared detectors is a bolometer.

It is a thermal detector with a typical time constant on the order of milliseconds. Thus

such a detector sees pulses just as steady-state source of light.

The maximum number of buckets is determined by the RF frequency Vrf of the

cavity and the time To (or circumference D) for an electron to make one revolution

around the ring, which is given by

Nmbax rp rf D
N1V x Vrf To Vf (5.8)

where v is the velocity of the electron. Any integral number of buckets smaller than

Nbnax can be filled with electrons arbitrarily.5

Within a bunch, electrons are distributed randomly, and there is a slight spread in

energy from that of the average electron which is travelling around the ideal electron

path at the reference center of the bunch. All electrons in a bunch are moving at the

same speed (v ~ c) and subject to the same Lorentz force while passing through a

bending magnet. However, the electron with slightly higher (lower) energy has larger

(smaller) mass. As a consequence, it has slightly longer (shorter) orbital length than

the reference orbit, and thus arrives later (earlier) than the reference electron. The

accelerating field acts to electrons in such a way to bring the energy of all electrons

closer to that of reference every time a bunch enters the cavity. Figure 5-4 schematically

illustrates the field found by electrons arriving at cavity at different times.

The RF system is designed to regain only the energy lost by radiation for the

reference electron, but more (less) energy for electrons arriving earlier (later). This

causes longitudinal oscillations about the center of the bunch, which is referred as

synchrotron oscillations.6

5.1.5 Beam Lifetime

Even though the electron energy is maintained by the RF system, the electrons

have a finite lifetime due to two 1, i i" mechanisms [59]: the scattering of electrons

by residual gas particles in the vacuum chamber and Touschek effect (discussed be-

low). Therefore, refilling of electrons are regularly scheduled every a few hours. As

5 For the VUV ring, Nba = 9.
6 Besides the synchrotron oscillations, electrons in a bunch make transverse (both hor-
izontal and vertical) oscillatory motion which are called the betatron oscillations. Pairs
of quadrupole magnets are used to focus electrons toward the reference orbital.

RF voltage RF voltage

-- RF system
Accelerating voltage 4th harmonic system
found by reference -- RF + 4th harmonic

0 Time 0 \ Time

Figure 5-4: The accelerating voltage as a function of time. The time = 0 corresponds to
the arrival of the reference electron. The RF system is designed to regain only the en-
ergy lost by radiation for the reference electron, but more (less) energy for electrons ar-
riving earlier (later). The right hand side shows the effect of using the higher harmonic
cavity system used in conjunction with the main RF system in the effort to increase the
lifetime. The 4th harmonic system is shown here. Each bunch sees a flat voltage which
stretches the bunch length.

explained in the previous subsection, the electrons oscillate around the reference or-

bit while orbiting around the ring: betatron oscillations (transverse oscillations) and

synchrotron oscillations (longitudinal oscillations). The Touschek effect is caused by

the scattering between transversely oscillating electrons inside each bunch. This type

of electron-electron scattering converts part of the transverse momentum into longi-

tudinal momentum that modifies the time at which the electron enters the RF cavity.

Then, those electrons which gained large enough longitudinal momentum are no longer

properly accelerated, and can be lost from the bunch. The effect is more severe when

electrons are packed tighter. Right after electron injection, electron density is the high-

est. Therefore, the Touschek effect is the dominant lifetime limiting mechanism at the

early stage of beam current decay with time-dependent decay time. As the electron

density decreases, the scattering by residual gas particles starts to take over, and at this

time, the decay becomes exponential that is represented by a single characteristic decay


A higher harmonic RF cavity can be used to flatten the potential in the main

RF bucket causing an increase in the bunch length with a consequent reduction of

intrabeam scattering and an improvement in the Touschek lifetime. Figure 5-4 shows

the effect of using the higher harmonic cavity system used in conjunction with the main

RF system in the effort to increase the lifetime.

5.1.6 Infrared Synchrotron Radiation

The Eq. 5.6 can be considerably simplified in the spectral range where wavelengths

A are much longer than the critical wavelength Ac. This condition (Ac/A < 1) is usually

satisfied in the entire infrared wavelengths for the most storage rings, and we can obtain

useful expressions valid for the infrared synchrotron radiation (IRSR) such as [62]

dP(A, t) 1W] (5.9)
= 8.6416 x 10-10iOpl/-7/3G (5.9)
dA cm
d2p(A, t) 5.2 x 10- 0ip2/3 -8/3H d c(5.10)
dA o rad cm

where p and A are both in cm, i (the ring current) in A, and 0 in mrad. The functions G

and H are defined as

G= 1 2.193 ) (5.11)

1 (p4/3
H [1 6.312 ) ]. (5.12)

For Ac/A < 1, G and H can be taken to be unity.

The vertical opening angle as a function of wavelength is given by

(A) 1.66188 (- G [rad] (5.13)

Note that T is twice of the angle defined for the angular divergence b (see Figure 5-2).

This is a useful expression when we determine the natural opening angle that is nec-

essary to collect the full-width at half-maximum of the power profile at a given wave-
length. Figure 5-5 shows T in the infrared spectral range as a function of wavenumber

using the VUV ring parameters.7

7 The U12IR's first mirror is capable of correcting light with 90 mrad of the vertical
opening angle.


CO 100

1 10 100 1000 10000

Frequency [cm1]

Figure 5-5: The natural opening angle of IRSR using the VUV ring parameters.

5.1.7 Source Comparison

The Eq. 5.5 gives the total radiated power from a storage ring in all directions

integrated over entire spectral range. Although the total power certainly indicates

certain aspect of the output capability, it does not describes superiorities of synchrotron

radiation over conventional thermal sources. For a practical point of view, the spectral

brightness b is a more useful source quality parameter since it takes into account the

source size as well as the angular distribution of synchrotron radiation. The brightness

of a light source is defined as

b() C x (v) (5.14)

where C is a constant, F(v) is the flux of photons, S is the source area, and f is the

solid angle of emission. It is intuitively obvious that a source with smaller size and

divergence has higher brightness just like a light beam from a laser is brighter than

that from a flame of candle. A small source size allows optics to focus photons to a

diffraction-limited size, and small divergence minimizes the loss of photons even with

reasonably small optical components. Therefore, a brighter source has a marked effect

S- Synchrotron
S10 -4 Blackbody

1 10 100 1000 10000

Frequency [cmn]

10a 8
1 10 100 1000 10000

Frequency [cm-1]

Figure 5-6: Spectral power calculated for a 2000 K blackbody source and synchrotron
radiation. For the synchrotron radiation, parameters for the VUV ring is used. This
shows the power advantage of the synchrotron radiation over the thermal source only in
the far infrared region.

on improving signal to noise ratio for various types of experiments such as microscopy

and surface science.

Figures 5-6 and 5-7 show calculated spectral power and brightness comparisons

between conventional thermal source and synchrotron radiation, respectively [21]. In the

plots, a blackbody source at temperature of 2000 K with its source size of 0.4 cm2 and

solid angle of 0.02 sr (f/3.5) is used. For the synchrotron radiation, the parameters of

the NSLS VUV ring are used.

Note that the synchrotron shows significantly lower output power than the thermal

source over most of the spectral range (between mid-IR and visible) where the globar is

commonly used (see Figure 5-6). The synchrotron has a power advantage only in the

very far infrared (< 100 cm-1). In terms of brightness, the synchrotron source has an

advantage over entire spectral range shown in Figure 5-7 over the thermal source, which

is obviously attributed to its small source size and angular collimation.

1 0 -1 ... . ... ... ...

10-2 ~ Synchrotron
E Blackbody

4 10-6


'O 10-9

1 10 100 1000 10000

Frequency [cm-1]

Figure 5-7: Spectral brightness calculated for a 2000 K blackbody source and syn-
chrotron radiation. For the synchrotron radiation, parameters for the VUV ring is used.
This shows the brightness advantage of the synchrotron radiation over the thermal
source in the entire spectral range.

5.2 Principle of Pump-Probe Studies

The pump-probe measurement is a valuable technique that determines the nonequi-

librium state of a system at various instants of time after some sort of stimulus has been

applied. The process is repeated for a wide range of time values to build up a complete

history of the sample's relaxation processes, namely the dynamics of the system. There

are av ii. i of excitation (pumping) methods commonly used that provide adequate en-

ergy density to create the desired density of excitations in the sample. Examples include

electrical current, electric field, magnetic field, or light pulses. Here we will discuss the

principle of the technique that uses near IR/visible laser pulses as excitation source and

synchrotron pulses as probe.

The purpose of this section is to provide a very simple idea of the technique

that would be helpful to know before going to the next chapter. The details of the

experimental technique are described in 6.6.


^ Sample
Probe (IRSR) 1 2 3

Figure 5-8: Principle of the pump-probe experiment. (1)Laser pulse creates photoexci-
tations in sample, which subsequently evolve with time. (2)After time At, broadband
IR pulse arrives and is partially absorbed (or reflected) by excitations. (3) IR pulse ana-
lyzed with or without a spectrometer, extracting details of excitations at a time At after
their creation.

5.2.1 Laser-Synchrotron Pump-Probe Measurement

Synchrotron radiation is a broadband bright source of light. Most people exploit

its brightness and overlook its temporal structure of the light pulses. The pump-probe

technique developed at the National Synchrotron Light Source (NSLS) of Brookhaven

National Laboratory utilizes the pulsed nature of synchrotron source especially at far

infrared where it offers both brightness and power advantage over conventional thermal

sources [1]. The short pulses of laser light are used to illuminate a sample, and create

photoexcitations. These excitations in the sample begin to relax immediately after the

arrival of laser pulse, and can appear as changes in the sample's optical properties. The

synchrotron pulse arrives at the sample at some point in time At after the pump, and

in 1v.. -the sample's response (e.g., transmission or reflection) at a time At into its

relaxation process. The experiments are performed by fixing the time difference between

the laser (pump) and the synchrotron (probe) pulses, and then measuring a spectrum

in the normal way. A fast detector is not required. The entire process repeats at a high

repetition rate (10's of MHz) in a manner similar to using a synchronized strobe light to

freeze a particular moment of a repetitive process, allowing the slowly responding human

eye to view it. This way a complete spectrum that represents a momentary snapshot of

the sample's state for a particular At can be measured. Various time differences between

pump and probe thus produce a set of data as a function of time and energy providing

greater insight into the relaxation process of the system. Figure 5-8 shows the principle

of the experiment.


The measured temporal response S(At) in a pump-probe experiment is determined

by the sample's impulse response function (the quantity of interest) as well as the

duration of the pump and probe pulses. When the sample has a linear response, S(At)

is given as
/+00 ft'
S(At) = dt' dt"Ipb(t + At)pump(t")G(t") (5.15)

where Iprobe(t) and Ipump(t) are temporal intensity profiles of the probe and pump pulses,

respectively, and G(t) is the impulse response function of the sample. Note that the

expression assumes that there is no self-excitation by probe pulses. This condition is

easy to achieve in practice. We can either use much less intense probe pulses than the

pump pulse or limit the spectral range of the probe below some photon energy threshold

using an optical filter. For G(t) = 6(t), the expression becomes a cross correlation

of the probe and pump pulses that defines the minimum temporal resolution. Note

that the temporal resolution is independent of the sensitivity and response time of the

spectrometer and detector.

In our pump-probe experiment, we used picosecond pump pulse that has signifi-

cantly shorter temporal profile than the synchrotron probe pulse of the pulse width on

the order of 1 ns. Hence we can take Ipump(t) o6(t), and Eq. 5.15 can be rewritten as

S(At) o= o dt'Iprobe(t' + At)G(t') (5.16)

This shows that the measured response is a convolution of the synchrotron pulse shape

with the sample's response. The nature of damping in a storage ring leads to a Gaussian

like electron distribution within a bunch and thus a Gaussian shaped probe intensity


5.2.2 Interferometry Using Pulsed Source

When pulsed light source is used for an interferometer, a beam-splitter divides every

pulse into two pulses. As a scanner mirror moves, one of the pulse is d. 1 ,i, .1 in time

relative to the other, so that the light travelling toward sample and detector consists

of two pulses for every pulse incident of the beam-splitter. When the delay is shorter


than the pulse duration, the two pulses overlap temporally, and cause interference. This

situation is clearly the same as the case of a continuous source. Now, if we run high-

resolution measurements to resolve a narrow spectral feature, one may think that we

may encounter a situation where the scanner moves far enough that the delay exceeds

the pulse duration and two pulses no longer overlap. However, this is not the case [1,21].

A sample with a narrow absorption feature will automatically lengthen a short

pulse. The lengthened pulses will overlap to cause interference. The amount of length-

ening is equivalent to the path difference necessary to resolve the feature's absorption

width. A Fabry-Perot interferometer is a simple way to picture how a short pulse can be

lengthened by a resonance. Therefore, regardless of the sharpness of a feature, there is

no limit to the spectral resolution using short pulsed source.

We can also think of it in the following way. The first pulse is incident on the

sample. If there is a narrow absorption feature, the sample absorbs that particular

fourier component from the pulse, and the absorption mode ilnl for awhile (ring

down time determined by the narrowness of the absorption mode). The second pulse is

then incident on the specimen, and the same fourier component is absorbed. But if the

mode is already imgini; (from the first pulse), then the particular fourier component

of the second pulse may be at the wrong phase (the second pulse tries to drive atoms

in one direction, but they are already moving in the opposite direction due to the first

pulse). Varying the time-delay of the interferometer will bring the proper mode in and

out of phase. So interference is observed. Observable interference will occur as long as

the mode keeps ringing. This is consistent with the longer path difference necessary

to achieve a higher spectral resolution. This path difference can be much longer than

the original pulse. In this picture, the sample has a "memory" determined by the

narrowness of the relevant absorption features. It remembers for a time long enough to

span the time between the two original pulses. There are other v- -- to picture this, but

the result is the same.8

5.2.3 Advantage of Laser-Synchrotron Technique

There are other sources of light that may be useful as a probe. For example,

tunable pulsed lasers, free-electron lasers (FELs), optical parametric oscillators (OPOs),

coherent THz pulses from a biased semiconductors illuminated by a femtosecond laser

are possible probe sources. Even though these can have higher temporal resolution

than the synchrotron, they have either restricted spectral range or stability issues.

A synchrotron, on the other hand, is a broadband, bright, and stable source. These

properties make the synchrotron suitable for ordinary spectroscopy over a broad spectral

range. For the time-resolved study, the synchrotron can follow the system that relaxes

through a wide range of energies. As will be described in the following chapter, pulse

width and repetition frequencies (PRF) are somewhat adjustable for various relaxation

time scales. All of these properties act as advantages of using synchrotron as a probing

source even at the expense of temporal resolution. The fact that our pump laser

(Ti:Sapphire) is tunable in wavelength, PRF, and power, adds flexibility to our pump-

probe system that can be very useful to investigate the dynamics of systems with time

scales from t100 ps to ~100 ns.

8 The explanation given here is based on a conversation with G.L. Carr.


6.1 Introduction

Pump-probe timing experiments were performed at the National Synchrotron Light

Source (NSLS) to study low-frequency dynamics in solids. Synchrotron radiation is a

broadband pulsed source. We took advantage of this pulsed nature to observe the state

of materials excited by a laser which is also pulsed and synchronized to the synchrotron

radiation. The Vacuum Ultraviolet (VUV) ring at the NSLS has two infrared beamlines,

U10A and U12IR, dedicated for solid state physics study. These are two beamlines

used for both time-resolved and linear spectroscopy, described in this dissertation. This

chapter will start with a description of the NSLS facility focusing on the properties

of the VUV ring and beamlines. Spectrometers at U10A and U12IR and pump laser

system are three principal pieces of instrumentation, and will be described separately

in detail followed by brief descriptions of the other apparatus such as the cryostat, our

home-made sample chamber, superconducting magnet, detectors, and fiber optics. The

experimental setup and techniques will be discussed toward the end of the chapter.

6.2 National Synchrotron Light Source

6.2.1 General

The NSLS is a user facility funded by the U.S. Department of Energy. Two

separate electron storage rings, an X-Ray ring (2.8 GeV, 300 mA) and a VUV ring

(800 MeV, 1.0 A), provide intense light spanning the electromagnetic spectrum from

the infrared through x-rays. The properties of this light, and the specially designed

experimental stations, called beamlines, allow scientists in many fields of research

to perform experiments not otherwise possible at their own laboratories. The NSLS

currently has 56 X-Ray and 23 VUV operational beamlines for performing a wide range

Full Text




Copyright2004 by HidenoriTashiro




ACKNOWLEDGMENTS Overthepastfewyears,Ihavereceivedfullofsupport,encoura gement,andvaluableadvicefrommanypeople.Here,Iwouldliketoacknowledg esomeindividualswho helpedmeinvariousways.Iwouldliketoexpressmyforemostand sinceregratitude toProfessorDavidB.Tanner,myresearchadvisor.Myworkcouldn otpossiblyhave beencompletedwithouthisguidanceandsupport.Henotonlyco ntinuouslyencouragedmetokeepgoingbutalsosuppliedmejusttherightamountof pressuretoget thingsdone.ThemanythingsIhavelearnedfromhimwillbemy treasure.Iamtruly gratefultoProfessorChristopherJ.Stantonforintroducingm etoDr.Tanner.Iwould likeequallytothankmysupervisorycommitteemembers,ArthurF .Hebard,Hai-Ping Cheng,StephenJ.Hagen,andStephenJ.Pearton,fortheirgui danceandreadingthis dissertation. IwouldalsoliketoexpressmyappreciationtoProfessorDavidH.R eitzeforhis occasionaladviceandhisstep-by-stepinstructionforusingand maintainingthelaser systemusedinmywork,JohonM.Graybealforsupplyingmeasetofsam plesand informationaboutthem,andCharlesPorterforprovidingme withalgorithmswhich helpedthedataanalysis. IwouldliketoexpressmyspecialthankstoG.LawrenceCarr,wh owasmydaily advisorattheNationalSynchrotronLightSource(NSLS).Hisgui dance,input,and patiencemademyprojecttogosmoothly.IequallythankRicar doLobo,whoisthe authorofvaluableprogramsusedfordataacquisitionandanal ysis,forhelpingmeto startmyprojectduringhisvisitattheNSLS.Additionalthanksg otoJiufengJ.Tu, ChristopherHomes,LaszloMihaly,DiyarTalbayev,GregoryD.Sm ith,RandyJ.Smith, andallofwhohelpedmeattheBrookhavenNationalLaboratory (BNL). IamdearlygratefultoallofmypastandpresentcolleaguesinD r.Tanner'sgroup fortheircooperation,conversations,andmostlyfriendship.I amalsothankfultoKevin iv


T.McCarthy,StephenArnason,ZhihongChen,SuzetteA.Pabit,Am olPatel,Susumu Takahashi,andNaokiMatsunaga. Iwouldalsoliketoacknowledgethehelpofthemembersofthem achineshop, electronicshop,andcryogenicsteamoftheUniversityofFlori daPhysicsDepartment, aswellasthemembersoftechnicalstaattheNSLS.Acknowledge mentalsogoesto thePhysicsDepartmentstafortheirassistance,especiallyJill KirkpatrickandDarlene LatimerfortakingcareofthebureaucraticdetailswhileIw asattheBNL. Finally,Iwouldliketoaddressmyexceptionalthankstomypa rents,Hiroyukiand KimikoTashiro,fortheirsupportovermyentirelife.Finally ,mydeepestappreciationis duetomywifeandtwosons,Yasuko,Mitsuru,andHikaru,fortheirc onstantsupport andpatience.Theymademylifesomuchfun. v


TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................iv LISTOFTABLES ....................................x LISTOFFIGURES ...................................xii ABSTRACT .......................................xv CHAPTER1INTRODUCTION .................................1 1.1IntroductoryRemarks ...........................1 1.2Motivation ..................................2 1.3Organization .................................3 2OPTICALPROPERTIES .............................4 2.1Introduction .................................4 2.2OpticalPhenomena .............................4 2.3InteractionofLightwithMatter ......................7 2.4ExperimentalDeterminationofOpticalConstants ............15 2.4.1RerectionandTransmissionataPlaneInterface .........15 2.4.2Kramers-KronigDispersionRelations ...............16 2.4.3RerectionandTransmissionatTwoParallelInterfaces ......18 2.4.4OpticsinThinFilmonaSubstrate ................20 2.4.5PhotoinducedAbsorption ......................23 2.5MicroscopicModels .............................23 2.5.1LorentzModel ............................24 2.5.2FreeCarrierResponseandDrudeModel .............27 2.5.3Drude-LorentzModel ........................31 2.5.4SumRules ..............................31 3FOURIERSPECTROSCOPY ...........................32 3.1Introduction .................................32 3.2FourierTransformInterferometry .....................33 3.2.1GeneralPrinciples ..........................33 3.2.2FiniteRetardationandApodization ................36 3.2.3Sampling ...............................38 3.2.4PhaseErrorandCorrection .....................41 3.2.5Step-ScanandRapid-ScanInterferometers ............43 3.3PolarizationModulation ..........................45 vi


4SUPERCONDUCTIVITY .............................49 4.1Introduction .................................49 4.2FundamentalsofSuperconductivity ....................50 4.2.1FundamentalPhenomena ......................50 4.2.2ThermodynamicProperties .....................54 4.2.3TypesofSuperconductor ......................55 4.2.4LengthScales ............................56 4.2.5BCSTheory .............................62 4.2.6EliashbergFormalism ........................68 5SYNCHROTRONRADIATIONANDPUMP-PROBETECHNIQUE .....70 5.1SynchrotronRadiation ...........................70 5.1.1Introduction .............................70 5.1.2RadiatedPowerfromaBendingMagnet ..............70 5.1.3AngularCollimationandPolarization ...............72 5.1.4RFCavityandPulsedNature ...................74 5.1.5BeamLifetime ............................75 5.1.6InfraredSynchrotronRadiation ..................77 5.1.7SourceComparison .........................78 5.2PrincipleofPump-ProbeStudies .....................80 5.2.1Laser-SynchrotronPump-ProbeMeasurement ...........81 5.2.2InterferometryUsingPulsedSource ................82 5.2.3AdvantageofLaser-SynchrotronTechnique ............84 6EXPERIMENT ...................................85 6.1Introduction .................................85 6.2NationalSynchrotronLightSource ....................85 6.2.1General ................................85 6.2.2VacuumUltravioletRing ......................86 6.2.3BeamlinesU12IRandU10A ....................90 6.3Spectrometers ................................93 6.3.1BrukerIFS66v/S ..........................94 6.3.2BrukerIFS125HR .........................97 6.3.3SciencetechSPS-200 .........................98 6.4PumpLaserSystem .............................100 6.4.1SystemOverview ..........................100 6.4.2Mode-locked,Solid-StateTi:SapphireLaser ............101 6.4.3OpticsandLightDistribution ...................104 6.4.4Laser-SynchrotronSynchronization ................107 6.5OtherExperimentalComponents .....................111 6.5.1OxfordOptistatBathCryostat ...................111 6.5.2Ox-BoxCustom-madeSampleChamber ..............116 6.5.3OxfordInstrumentsVertical-boreSuperconductingMa gnet ...118 6.5.4Detectors ...............................120 6.5.5RatioBox ..............................126 6.5.6FiberOpticCableandPulseDelivery ...............127 vii


6.6ExperimentalTechniquesandSetups ...................129 6.6.1PhotoinducedMeasurements ....................130 6.6.2LaserInsertion ............................136 7OPTICALCONDUCTIVITYOF -MoGeTHINFILMS ............140 7.1Introduction .................................140 7.2Background .................................141 7.2.1InfraredPropertiesofSuperconductors ..............141 7.2.2EectsofDisorderupon2DSuperconductivity ..........145 7.2.32DModelSystems ..........................147 7.3ExperimentalDetails ............................148 7.3.1Samples ...............................148 7.3.2Measurements ............................151 7.4Analysis ...................................151 7.5Discussion ..................................154 7.6Conclusion ..................................156 8TIME-RESOLVEDSTUDYOF -MoGeTHINFILMS .............159 8.1Introduction .................................159 8.2Background .................................160 8.2.1NonequilibriumSuperconductivity .................160 8.2.2TemperatureDependenceofLifetimes ...............165 8.3ExperimentalDetails ............................169 8.3.1Time-resolvedMeasurements:QuasiparticleDecay ........171 8.3.2PhotoinducedGapShiftMeasurements ..............174 8.3.3FluenceDependence .........................176 8.3.4SpectrallyAveragedFarInfraredTransmission ..........178 8.4AnalysisandDiscussion ..........................179 8.4.1RelaxationTimes ..........................179 8.4.2PhotoinducedGapShift .......................185 8.5Conclusion ..................................187 9MAGNETO-OPTICALSTUDYOF -MoGeTHINFILMS ..........189 9.1Introduction .................................189 9.2TransmittanceRatioinMagneticFields ..................190 9.3RelaxationTimesinMagneticFields ...................191 10SUMMARYANDCONCLUSION .........................192 APPENDIXAVUVSTORAGERINGPARAMETERS .....................197 BINFRAREDBEAMLINES .............................199 B.1InfraredProgramsatNSLSVUVring ...................199 B.2HightResolutionFar-infraredSpectraatU12IR .............200 viii


CLASERSAFETYANDOPERATINGPROCEDURES .............201 C.1HazardousBeamControl ..........................201 C.2PersonalProtectiveEquipment ......................201 C.2.1EyeProtection ............................201 C.2.2SkinProtection ...........................202 C.3LaserSafetyTraining ............................202 C.4Alignment ..................................202 C.4.1GrossAlignment ...........................202 C.4.2FineAlignment ...........................202 C.4.3AtBeamlineEndstation .......................203 C.5DailyOperationProcedure .........................203 C.6OptimizationoftheDownstreamOptics .................205 C.7ANSILaserClassications .........................206 DUSEFULINFORMATION .............................209 D.1FrequencyRanges ..............................209 D.2EnergyandPressureUnitsConversion ..................211 D.3Gas-phaseContamination .........................211 REFERENCES ......................................213 BIOGRAPHICALSKETCH ...............................219 ix


LISTOFTABLES Table page 4{1Transitiontemperaturesforseveralsuperconductors .............51 6{1OperationmodesoftheVUVring .......................88 6{2Frequencyrangesofvarioussources ......................95 6{3Frequencyrangesofvariousbeam-splitters ..................96 6{4Frequencyrangesofvariousdetectors .....................96 6{5SpecicationsoftheSPS-200 ..........................101 6{6SpecicationoftheMira ............................103 6{7PropertiesofOxfordcryostatwindows .....................114 6{8Characteristicsofberopticcable .......................127 7{1 -MoGelmparameters ............................149 7{2 T s = T n ttingparameters ...........................152 7{3Valuesof N s and ................................155 8{1 -MoGelmusedfortimingexperiment ....................170 8{2Parametersforthetimingexperiment .....................171 8{3Fluencedependencedata ............................177 8{4 e and A atvarioustemperatures .......................180 8{5Materialparametersin R 0 and B 0 .......................182 8{6 r R 0 B 0 ,and R 0 = B 0 ............................182 8{72 0 B 0 = R 0 ....................................183 8{8RelaxationtimesandmaterialparametersfromKaplan etal. ........184 8{9Photoinducedgapshifts .............................186 A{1VUVstorageringparameters ..........................197 A{2VUVstoragering'sarcsourceparameters ...................198 A{3VUVstoragering'sinsertiondeviceparameters ................198 x


A{4NSLSlinacparameters .............................198 A{5NSLSboosterparameters ............................198 A{6NSLSboostermagneticelements ........................198 B{1InfraredbeamlinesoftheVUVring ......................199 D{1Infraredspectralregions .............................209 D{2Frequencyrangesofconventionallightsources ................209 D{3Frequencyrangesofdetectors ..........................209 D{4Spectralrangesofbeam-splitters ........................210 D{5Transmissionrangeofopticalwindowandltermaterials ..........210 D{6Relationsbetweenenergyunits .........................211 D{7Relationsbetweenpressureunits ........................211 D{8Absorptionpeaksduetoair ...........................212 xi


LISTOFFIGURES Figure page 2{1Rerectionandtransmissionattwoparallelinterfaces ............18 2{2Rerectionandtransmissionwithathinlmonasubstrate .........21 3{1Spectrometerclassication ...........................32 3{2SchematicviewofaMichelsoninterferometer .................34 3{3Sincfunctionconvolvedwithasinglespectralline ..............37 3{4ComparisonoftheHapp-Genzelandboxcarapodization ..........39 3{5Relationbetweenspectrumreplicationandsamplingrate ..........40 3{6Twosinewavesdrawnthroughthesamesamplingpoints ..........41 3{7SchematicviewofaMartin-Puplettinterferometer ..............46 3{8Interferogramsproducedbyapolarizinginterferomete r ...........47 4{1DensityofstatesforaBCSsuperconductor ..................65 4{2VariationofthegapwithtemperatureintheBCSapproxim ation .....66 5{1Incoherentandcoherentradiations ......................72 5{2Angulardistributionoftheradiation .....................72 5{3Angulardistributionofpolarizationcomponents ...............74 5{4RFandhigherharmoniccavityvoltages ...................76 5{5NaturalopeningangleofIRSRwiththeVUVringparameters .......78 5{6Powercomparisonbetweenblackbodyandsynchrotron ...........79 5{7Brightnesscomparisonbetweenblackbodyandsynchrotron .........80 5{8Principleofthepump-probeexperiment ...................81 6{1Changeofthepulsewidthduringthedetunedmodeoperatio n .......89 6{2ElevationviewofU12IRbeamline .......................91 6{3Transmittedpowerwithandwithoutalightcone ..............92 6{4Emissionspectraofconventionalinternalsources ...............95 xii


6{5BrukerIFS125HR ...............................98 6{6SciencetechSPS-200 ..............................99 6{7OpticalschematicoftheMiralaserhead ...................102 6{8OpticalLayoutoftheU6lasersystem .....................105 6{9Eectsofthepulsepickers ...........................107 6{10Timingscheme .................................108 6{11Synchronizedlaserandsynchrotronpulses ..................110 6{12OxfordOptistatbathcryostat .........................111 6{13Custommadesamplecompartment ......................117 6{14O-axisparaboloidalrerector .........................118 6{15Oxfordmagnetsetup ..............................119 6{16Transferfunction ................................121 6{17Classicationofdetectors ............................122 6{18Compositebolometer ..............................124 6{19Structureandproleoftypicalopticalbercable ..............128 6{20Experimentalsetupfortimingexperiment ..................134 6{21Newditheringscheme ..............................135 6{22LaserinsertionsetupswithOptistat ......................137 6{23LaserinsertionsetupwithHeli-tran ......................138 6{24Couplingofdiodelaserwithopticalbercable ................138 7{1Mattis-Bardeenrelativeconductivityandtransmittance ...........143 7{2 R 2 vs.1 =d ....................................149 7{3 T c =T c 0 vs. R 2 ..................................150 7{4Measuredtransmittanceratio .........................152 7{5Mattis-Bardeentto T s = T n .........................153 7{6Measuredrerectanceratio ...........................154 7{7Mattis-Bardeentto R s = R n .........................155 7{8Opticalconductivitiesof -MoGe .......................156 7{9 N s vs. R 2 ...................................157 xiii


7{10 N s vs. T c .....................................158 8{1Simpliedrelaxationprocesses .........................163 8{2Universaltemperaturedependenceoflifetimes ................168 8{3Dierential,integratedsignalvs.time .....................172 8{4Quasiparticledecaysignalvs.timeandmodelfunction ...........173 8{5Pointsonadecaycurveforgapshiftmeasurements .............175 8{6Fluencedependenceof e ...........................176 8{7Spectrallyaveragedfar-IRtransmittionvs. T=T c ...............178 8{8Quasiparticledecaysignalfor16.5nmlm ..................180 8{9 e vs. T=T c ...................................181 8{102 0 B 0 = R 0 vs. T c ................................183 8{11 R 0 = B 0 vs. b ...................................185 8{12Photoinducedspectralchanges .........................186 9{1Measuredtransmittanceratioinmagneticelds ...............190 9{2Quasiparticledecaysignalinmagneticeldforthe33nml m .......191 B{1Highresolutionfar-IRsynchrotronspectraatU12IR .............200 xiv


AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy TIME-RESOLVEDINFRAREDSTUDIESOFSUPERCONDUCTING MOLYBDENUM-GERMANIUMTHINFILMS By HidenoriTashiro December2004 Chair:DavidB.TannerMajorDepartment:Physics Superconductingamorphousmolybdenum-germanium( -MoGe)thinlmsshow progressivelyreducedtransitiontemperatures T c asthethicknessisreduced.This suppressionhasbeenexplainedintermsofelectronlocalizati oneectsandreduced screening.Thisdissertationpresentstheresultsofbothlinear spectroscopyandtimeresolvedstudiesofasetof -MoGelmstounderstandmorefullythisweakened superconductingstate.Theobservedopticalconductivityshow sthepresenceofan energygap.Theeectsofreducedthicknessintheselmsareto depress T c andthe superruiddensity,whilemaintainingthenormal-stateresistiv ity.Alloftheresultsfrom ourlinearopticalmeasurementsappeartobeconsistenttrivia llywiththoseexpected forweaktointermediatecouplingdirtylimitsuperconducto rs.Ourtime-resolved studyrevealstheoverallrelaxationofthesamplesatatimesca leontheorderof100 ps.Thetemperaturedependenceoftherelaxationtimeseemsto beconsistentwith thepredictionbasedonweak-couplingBCStheoryforalllms wemeasuredwithout changinganymaterialparametersfordierentthickness.The applicationofmagnetic elddidnotchangetherelaxationtimes,whichwasunexpecte d. xv


CHAPTER1 INTRODUCTION 1.1IntroductoryRemarks Spectroscopyisaveryusefultechniqueforinvestigatingthep ropertiesofvarious typesofmaterials.Informationonthematerialisencodedin theradiationspectrum modiedthroughinteractionwiththematerial.Theextensiv eenergyrangecoveredby electromagneticradiationallowsustostudymanypropertie s( e.g. ,electronic,magnetic, lattice,andsoon)dependingonthefrequencyranges.Forexam ple,theconductivity peakforfreecarriersiscenteredatzerofrequency.Lattic evibrations( i.e. ,phonons) interactwithelectromagneticradiationatfar-infraredf requencies.Electronictransitions acrosstheenergygapofasemiconductorlikeSihappeninthene arinfrared.Transitions fromcorelevelsrequireevenhigherenergies.Insuperconduc tors,anenergygap developsintheelectronicdensityofstatesaroundFermiener gy.Thetypicalenergy scaleofthisgapisinmeV,whichcorrespondstothefrequencies betweenmicrowaveand farinfrared.Thus,opticalstudiesinthisfrequencyrangepr ovideanimportanttoolfor investigatingsuperconductors. Asynchrotronisasourceofhighbrightnesselectromagneticra diationemittedfrom electronsorbitingaroundaclosedpathofastoragering.Itis averybroad-bandsource, extendingfromtheveryfarinfraredtothehardx-ray.Becau seelectronsarebunched astheytravel,theradiationemittedfromthesynchrotronsou rceispulsed.Beamlines U10AandU12IRaretwobeamlinesontheVUVringattheNationalSync hrotron LightSource(NSLS)ofBrookhavenNationalLaboratory(BNL)de dicatedtosolid-state physicsexperiments.TheNSLSprovidesapowerful,tunable,ne ar-infrared/visible mode-lockedTi:Sapphirelaser,whichproducespulsesofafew picosecondsduration synchronizedtothesynchrotronpulses.Aspecimencanbeexcited (pumped)bylaser pulses,andprobedbyinfraredsubnanosecond-durationsynchrot ronpulsesfromthe 1


2 VUVring[ 1 2 ].Dependingonthemodeofsynchrotronoperation,thisfacil ityprovides auniqueopportunityformeasuringtransientphenomenaontim escalesinafew100 psupto170nsrangeoverabroadspectralregion.Withpropert uningofexcitation energy,thedynamicsofvarioussystemscanbeinvestigated:qu asiparticlerelaxation inconventionalBCSsuperconductors,recombinationoftheph otogeneratedelectronholeplasmainsemiconductors,dynamicsofthephotodopedpola ronsandsolitons inconductingpolymers,andrelaxationofphotoinducedcond uctivityeectsinthe insulatingphaseofhighT c superconductors,tonameafew.Useofasuperconducting magnetalsoallowsustostudymaterialsinmagneticelds. 1.2Motivation Thisdissertationdescribesopticalstudiesonasetofsupercondu ctingamorphous molybdenum-germanium( -MoGe)thinlmsdepositedonthickinsulatingsubstrates. AllmeasurementswereperformedatthetwobeamlinesontheNSLS VUVring.Itis wellknownthatincreasingdisorderleadstolocalizationand therelatedenhancements oftherepulsiveelectron-electronCoulombinteraction[ 3 4 5 ].Itisalsowellknown thatinsuperconductors,twoelectronsformapairduetophono nmediatedattractive interaction.Thus,theenhancedCoulombinteractioninhere ntlycompeteswithsuperconductivity. -MoGeisawell-studieddisorderedsystemusedforstudyingtheint erplay betweensuperconductivityanddisorder.Itstransportproper tieswereinvestigatedby Graybeal[ 6 7 8 ],andshowedprogressivelyreducedtransitiontemperatures( i.e. ,a weakenedsuperconductingstate)asthethicknessisdecreased. Inthiswork,wehaveset outtounderstandthissystemevenfurther,andpossiblyndsomesor tofconnections betweenreducedtransitiontemperatureandthedegreeofdiso rder.Forthispurpose, werstundertookathoroughstudyoflinearspectroscopyinthef arinfrared.Then, thedynamicsofthesystemwasstudiedbyapump-probetechnique .Drivingsuperconductorstononequilibriumstatescorrespondstobreaking Cooperpairs,producing excitationscalledquasiparticles.Inreturningtoequilibr ium,thesequasiparticlesrecombineintopairs,releasingenergyusuallyasphonons.Therateat whichthisrelaxation


3 progressesinvolvestheinteractionbetweenquasiparticles,w hichisoffundamental interestforanytheoryofsuperconductivity. 1.3Organization InChapter 2 wereviewabasictheoryofopticalproperties.Someofthecomm on techniquesandmodelsusedforextractingopticalparameter sfromexperimentsare discussed.Chapter 3 providestheconceptsofFourierspectroscopy.Twotypesof interferometersaredescribed:amplitudemodulationandpo larizationmodulation. Chapter 4 summarizesafewfundamentalpropertiesofsuperconductorsw ithinthe frameworkofBCStheory.TherstpartofChapter 5 describesthepropertiesof synchrotronradiation.Comparisontoconventionalthermalso urcesrevealsadvantages ofusingsynchrotronradiationforspectroscopicstudyparticul arlyinthefarinfrared frequencies.Thesecondpartofthechapterisintendedtointr oducethebasicideaof thepump-probetechniqueusingalaserasthepumpsourceandsync hrotronradiation astheprobesource.Readingthispartofthechapterpriortor eadingthefollowing chaptersisrecommended.Chapter 6 describesindetailallexperimentalcomponents, techniques,andsetupsusedinthisproject.Thespecicsofthea pparatussuchasthe NSLSVUVstoragering,beamlines,spectrometers,lasersystem,andoth ersareall includedinthischapter.Chapters 7 and 8 discussourexperimentalresultsoflinear andtime-resolvedstudyon -MoGethinlms,respectively.Eachchaptercontainsa theoreticalbackgroundspecictotheanalysisusedinthechap ter.Finally,Chapter 9 showstheresultsofourmostrecentexperimentofmagneto-opti calmeasurements. Inopticalstudies,dierentenergyunitsarequitecommonfor thedierenttechniques,frequencyranges,anddisciplines.Intheinfraredspect ralregion,themost commonlyusedunitisthewavenumbergivenbycm 1 .Itisdenedasthefrequencyin Hzdividedbythespeedoflightincm/sorthereciprocalofwave lengthincm.Inthis dissertation,thewavenumberisusedinterchangeablywithfre quencyorenergysincethe valuesfortheseunitsaresimplyrelated.


CHAPTER2 OPTICALPROPERTIES 2.1Introduction Inthischapter,wewillprovideageneralbackgroundofthet heoryofoptical properties.Thechapterbeginswithveryintroductorydiscussi onofanumberof phenomenathatcanoccuraslightpropagatesthroughamediu mandthecoecients thatareusedtoquantifythem.ThebasicresultsofMaxwell'seq uationsalsowillbe summarized.Inthefollowingsection,wewillshowseveraltechn iquesforextracting opticalparametersfromexperimentaldatameasuredonsample sincommonforms.At theendseveralmicroscopicmodelswillbedevelopedtoexplai ntheopticalphenomena thatareobservedinthesolidstate.Themainpurposeofthechapt eristogiveabasic theoryandtechniquesingeneralterms,andalsotobeusedasqui ckreference. 2.2OpticalPhenomena Inthesimplestway,rerection,transmission,andpropagationar ethethreesimplestgroupsofopticalphenomenathatareobservedinsolidstat ematerials.Whena lightwavepropagatinginonemediumencountersanothermed ium,someofthelightis rerectedfromtheinterface,whiletheresttransmitsintothe mediumandpropagates throughit.Thelightalsoexperiencesavarietyofopticalph enomenaduringthepropagationwithineachmedium.Thelightraysarebentattheinte rfaceduetothechange inthevelocityofthelightwaveindierentmedia.Thisiskn ownasrefraction,andis describedbySnell'slaw.Thelightmaybeattenuatedasitpro pagatescausedbyprocessessuchasabsorptionorscattering.Absorptionoccursifthefr equency( i.e. ,energy ofphoton)ofthelightisresonantwiththetransitionenergyo ftheatomsandelectrons inthemedium.Hence,absorptioncausesreductionofthenumber ofphotonsinthe forwarddirection.Intheeventofscattering,thelightbeam isre-directedinotherdirectionscausedbythepresenceofimpurities,defects,orinhomoge neities.Thisobviously 4


5 causesattenuationintheoriginaldirectioninananalogous waytoabsorptioneven thoughthenumberofphotonsisunchanged.Scatteringcanal soaccompanychangesin frequencyofthelight.Ifthefrequencyofthescatteredligh tischanged,itissaidtobe inelastic;ifitisunchanged,itissaidtobeelastic.Luminesce nceisthephenomenonof spontaneousemissionoflightbytheexcitedatomsinamedium.T helightisemittedin alldirections,andhasadierentfrequencytotheincomingl ight. Withconventionalsourcesoflight,opticalpropertiesared escribedbylinearoptics, whereitisassumedthatquantitiessuchastherefractiveindex ,absorptioncoecient, andrerectivityareindependentoflightintensity.Thisisb asedonanapproximation thatisonlyvalidinthelowintensitylimit,andpractically everythingwewillbe discussinginthisdissertationfallsintotherealmoflinearopt ics.Whenhighintensity lightpropagatesthroughamedium,anumberofnonlinearphe nomenacanoccur. Frequencydoublingandtriplingareexamplesofthesenonlin eareects,andarerealized throughtheuseoflasers.Thisisthesubjectofnonlinearoptics, whereitallowsthe electricsusceptibilityaswellasallthepropertiesthatfol lowfromittovarywiththe strengthoftheelectriceldofthelightbeam.Eventhoughno nlinearopticsisan interestingsubjectinitsownright,itwillnotbediscussedfurt her. Opticalphenomenacanbequantiedbyanumberofopticalcon stants(optical parameters)thatdescribethemacroscopicbehaviorofthemedi um.Thererectionfrom aninterfaceofdierentmediaisdescribedbythererectance R ,denedastheratio ofthepowerofrerectedlighttothatofincidentlightonthe surface.Transmission throughtheinterface,ontheotherhand,isdescribedbythet ransmittance T ,dened astheratioofthetransmittedpowertotheincidentpower.At theeveryinterfacethe lightwaveencounters,conservationofenergyrequiresthat R + T =1 : (2.1) Therefractionisdescribedbytherefractiveindex n ,denedas n = c v ; (2.2)


6 where c and v arethespeedoflightinfreespaceandinthemedium,respective ly. Therefractiveindexdependsonthefrequencyofthelightwa ve,whichisknown asdispersion,andcharacterizesthepropagationofthelight throughatransparent ( i.e. ,non-absorbing)medium. Theabsorptionoflightbyamediumisquantiedbytheabsorpti oncoecient denedasthefractionofthepowerabsorbedinaunitlengthof themedium.Interms ofadierentialequation,theeectofabsorptionisgivenby dI = dx I ( x ) ; (2.3) where I ( x )istheintensityoflightatposition x .Thesolutiontothisequationisthe exponentialdecayoflightintensityasitpropagatesthroug hthemedium: I ( x )= I 0 e x : (2.4) Thefrequencydependenceoftheabsorptioncoecientisrespo nsibleforthecolorof materials. Rayleighscatteringiscausedbyvariationsoftherefractive indexofthemediumon alengthscalesmallerthanthewavelengthofthelight.Asmenti onedearlier,scattering hassimilarattenuationeectasabsorption,andtheintensity oflightasitpropagates canbeexpressedby I ( x )= I 0 e N s z ; (2.5) where N isthenumberofscatteringcenterperunitvolume,and s isthescattering cross-sectionofthescatteringcenter. Thefrequencydependentrefractiveindex n ( )andabsorptioncoecient ( ) arethetwoimportantquantitiesthatcharacterizetheprop agationoflightwaveina mediumsincetheydescribethedispersiveandabsorptivenatureo famaterialinthe mostdirectway.Therefractionandabsorptionofamediumcanb edescribedbya


7 singlequantitycalledthecomplexrefractiveindex: 1 ~ N = n + i; (2.6) where n (realpartof ~ N )istherefractiveindexdenedinEq. 2.2 ,and (imaginary partof ~ N )istheextinctioncoecientthatisdirectlyrelatedtothe absorptioncoecient aswillbediscussedshortly. Afewexamplesofopticalquantitiesthathavebeendiscussedsof arprovide descriptionsoftheopticalphenomenaonlyfromthepointofv iewofthererection, transmission,andpropagation,andoerthemostusefulinformat iontomanufacturers ofopticalelements.Thefrequencydependentrerectance R ( )andtransmittance T ( ) aretheexperimentallymeasurablequantitiesfromwhichoth erparameterssuchasthe complexrefractiveindex ~ N canbederived.Themicroscopicmodels,however,usually enableustocalculateotherparameterssuchasthecomplexdi electricfunction~ and complexconductivity~ ratherthan ~ N .Therelationshipbetween~ and ~ N provides directconnectionbetweenmicroscopicmodelsofmaterialsa ndpropagationpropertiesof electromagneticwaves. 2.3InteractionofLightwithMatter Thissectionsummarizestheprincipalresultsofelectromagne tismthataresucient forthestudyofopticalpropertiesofsolids.Detailsofthesubj ectcanbefoundinmost booksonopticsandelectromagnetism[ 9 10 11 12 13 14 15 16 ].CGSunitsareused throughoutunlessotherwisespecied. Theresponseofamaterialtoexternalelectricelds E ischaracterizedbyafew macroscopicvectors:polarization P ,electricdisplacement D ,andcurrentdensity J Withinthelinearapproximation,thesethreevectorsarepro portionaltotheelds E 1 Theuseofcomplexquantitiessuchasthecomplexrefractivein dex,dielectricconstant,andconductivitytodescribepropertiesofmediumnatu rallyarrivesfromtheuse ofcomplexsolutionstotheMaxwell'sequations.Thispointwi llbeclearafterreading x 2.3


8 ( e.g. P / E ),andtheirproportionalityconstantsarethelinearresponse functions whichdescribepropertiesofthesolid-statesystemitselfandare independentofthe drivingforce.Thelinearresponseisformulatedintimeandspa ce.Sincetheresponseis, ingeneral,frequencyandwavevectordependentanditiscon venienttohandleharmonic functions,adiscussioninFourierspace,bothwithrespecttotime andcoordinates,is moreappropriate.Thus,ratherthanstudyingtheresponsefunct iondirectly,thelinear relationbetweentheFouriertransformofthedrivingforcea ndtheFouriertransform ofthesystemresponseisconsidered.Also,wewillusethetime-varyi ng E inthe formofexp( i!t ),andexpresstheproportionalityconstantasacomplexquant ityto accountforthephaseshiftbetweentwoelds:areal(animagina ry)partrepresentsthe responseofamediumin(outof)phasewiththeappliedelectric eld.Inaddition,the assumptionofisotropic,homogeneousmediumwasmadetosimplif ythediscussions. 2 Thepolarizationisdenedasthenetdipolemomentperunitv olume.Withinthe assumptionmadeabove,themicroscopicdipoles(bothpermanen tandinduceddipole moments)tendtoaligninthedirectionoftheexternaleld.T hisallowsustodenea polarizationas P =~ e E ; (2.7) where~ e isthecomplexelectricsusceptibilityofthemedium,whichis oneofthemost fundamentalresponsefunction. Theelectricdisplacementofthemediumisdenedby D E +4 P : (2.8) Theabovetwoequationscanbecombinedtogiveanalternativ eexpression: D =~ E ; (2.9) 2 Anisotropiccrystalshavenonequivalentopticalpropertiesa longdierentcrystalline axes.Thephenomenonofbirefringenceisanexampleofoptica lanisotropy.Insuchmaterials,theproportionalityconstantsmustberepresentedbya tensor.


9 where ~ 1 + i 2 =1+4 ~ e : (2.10) Theparameter~ isacomplexdielectricconstant(ordielectricfunction). Whentimevarying E isapplied,thereisanassociatedmotionofeachelementof charge.Thisleadstoarelationshipbetweenthecurrentdensi ty J andpolarizationas J = @ P @t = i! P : (2.11) Inasimilarwayto P and D ,thecurrentdensitycanalsobewrittenas J =~ E ; (2.12) where ~ 1 + i 2 : (2.13) Theparameter~ isthecomplexconductivityofthemedium.Generally,thecu rrent density J isthesumoftwocontributions:onearisingfromthemotionofch argesthat arefreetomovethroughthemedium J free ,andtheotherarisingfromchargesthatare restrictedtolocalizedmotion J bound Opticalconstantssuchas~ e ,~ ,and~ representtheresponseofamedium( i.e. ,responsefunctions)toaperturbingeldoffrequency 3 Alloftheseparameters, however,arenotindependent.Theyareallinterrelatedtoo neanother.Ascanbe seenfromEq. 2.10 ,~ e and~ providethesameinformation.UsingEqs 2.7 2.10 2.11 and 2.12 ,wecanndausefulrelationshipbetween~ and~ : ~ =1+ i 4 ~ ; (2.14) 3 Theresponsefunctionsshouldbeconsideredasafunctionofboth frequency andwavevector k .However,theexplicitdependenceoftheresponsefunctionson k ( i.e. ,wavelength),theso-calledspatialdispersion,canbeneglect edincasetheelds couldbeaveragedoveraunitcell.Spatialdispersionariseswh enevertherelationbetween D and E isnotexactlylocalwith D ataparticularpointdeterminedsolelyby E atthatpoint.


10 orexplicitly 1 = 2 4 ; (2.15) 2 = (1 1 ) 4 : (2.16) Thusopticalmeasurementsof~ ( )areequivalenttoconductivitymeasurementsof~ ( ). Incaseourinterestisintheopticalresponsesduetothefreecar riergasinmaterials suchasmetalsanddopedsemiconductors,opticaldataarefrequ entlydiscussedinterms oftheconductivityratherthanthedielectricconstant. Later,wewillshowtheconnectionbetweentheopticalparame tersdescribedhere andthepropagationconstantsofelectromagneticwavesinam edium,namelythe complexrefractiveindex ~ N Theresponseofamaterialtoexternalmagneticeldsischarac terizedinasimilar way.Themagnetization M isdenedasthenetmagneticmomentperunitvolume,and isproportionaltomagneticeldstrength H : M =~ m H : (2.17) Theparameter~ m isthemagneticsusceptibility. Themagneticruxdensity B isdenedby B = H +4 M : (2.18) Theabovetwoequationscanbecombinedtogive B =~ H ; (2.19) where ~ =1+4 ~ m : (2.20) Theparameter~ isthepermeabilityofthemedium.Atopticalfrequencies,an y paramagneticorferromagneticmomentscannotfollowthera pidoscillationsofmagnetic eldbecauseoftheirlongrelaxationtimes.Theremainingdia magneticmomentsare


11 sosmallastohavenoappreciableeectonopticalbehavior.Th us,unlesswestudy magneto-opticalphenomena,wecanset B = H Thestartingpointforthetreatmentofinteractionbetweene lectromagneticelds andmatteriscontainedwithinthefourMaxwell'sequations fortheaverageelds. 4 In theabsenceofexternalcharges,theseequationsaregivenby r ~ E =0 ; (2.21) r H =0 ; (2.22) r E = 1 c @ H @t ; (2.23) r H = 1 c @ ~ E @t ; (2.24) where~ isthecomplexdielectricconstantdenedinEq. 2.14 ,whichallowsthecurrent densityarisingfromfreecarriers( i.e. ,Ohm'slaw)tobeincludedinEq. 2.24 Weconsiderthesolutioncorrespondingtoaplanewaveoftheang ularfrequency : E H = E 0 H 0 exp[ i ( ~ k x !t )] ; (2.25) whereaconstantamplitudevector E 0 isingeneralcomplex.Thecomplexwavevector ~ k wasusedtodescribeenergydissipationofthewave.Substitution ofEq. 2.25 intothe Maxwell'sequationsyields ~ ~ k E =0 ; (2.26) ~ k H =0 ; (2.27) ~ k E = c H ; (2.28) ~ k H = c ~ E : (2.29) 4 Theuseoftheso-calledmacroscopicMaxwell'sequationscanbe justiedasfollows. Inopticalmeasurements,featuresthatcanbeprobedarethesiz eoftheorderofawavelengthoflightorlarger.Sinceasolidcontainsnumerousato mswithinthelengthscale ofthewavelengthoflight,itcanbetreatedasacontinuousm edium


12 Herewehaveassumedanisotropic,homogeneous,andnon-magnetic mediumsothat ~ hasnospatialvariation.Equationsareseparatelycorrectfo rboththerealand imaginaryparts.Theseequationsarecombinedtoyieldarelat ionbetweenthewave vectorandfrequencyknownasthedispersionrelationship: ~ k ~ k = 2 c 2 ~ : (2.30) Inanon-absorbingmediumofrefractiveindex n ,thewavelengthofthelightis reducedbyafactor n comparedtothefreespacewavelength 0 (=2 c=! ).Therefore, wavevector k isgivenby k = 2 0 =n = !n c : (2.31) Thisleadstothephasevelocity !=k = c=n inEq. 2.2 .Thewavevectorcanbe generalizedtothecaseofanabsorbingmediumbyallowing n (andasaresult k too)tobecomplex: ~ k = c ~ N = c ( n + i ) : (2.32) Eq. 2.30 andEq. 2.32 allowustorelatethepropagationpropertiesoflightthrou gha mediumtotheresponseofthemediumintheelectromagneticel dsas ~ N = p ~ ; (2.33) orexplicitly 1 = n 2 2 ; (2.34) 2 =2 n; (2.35) and n = 1 p 2 h 1 + 21 + 22 1 2 i 1 2 ; (2.36) = 1 p 2 h 1 + 21 + 22 1 2 i 1 2 : (2.37)


13 Eqs. 2.26 2.29 alsoshowthat E H ,and ~ k aremutuallyperpendicular( i.e. ,transversewaves),andthescalarrelationbetween E and H isgivenby H = p ~ E = ~ NE: (2.38) Theratio(4 =c ) E=H iscalledthewaveimpedance ~ Z : ~ Z Z 0 E H = 1 p ~ = 1 ~ N ; (2.39) where Z 0 isthewaveimpedanceoffreespace,whichhasavalueof Z 0 =4 =c (or377n inSI). OnsubstitutingEq. 2.32 intoEq. 2.25 ,wendthattheeldsattenuatease !x=c Theopticalintensityoflightisproportionaltotheabsolute squareoftheelectric eld 5 ( I / E E ).Thus,fromEq. 2.4 ,wendthat = 2 c = 4 0 ; (2.40) where and aretheabsorptionandextinctioncoecients,respectively.T hepenetration(orskin)depth isthecharacteristiclengthoftheelds'penetrationintoa mediumdenedby 2 = c : (2.41) Theaveragerateofdissipationofelectromagneticenergyden sityis W = h Re( E ) Re( J ) i = 1 2 Re( E J )= 1 2 1 j E j 2 : (2.42) Thusonlythecurrentthatisinphasewith E contributestoanenergyloss,and 1 representstheresistiveresponse( i.e. ,absorptionthataccompaniestheenergyloss) ofthemediumintheelds.Theoutofphasecurrent,ontheother hand,doesnot accompanytheenergyloss,and 2 describesthereactiveresponse.The~ iscalledthe 5 Thetime-averagedenergyrowintheelectromagneticwaveis calculatedfromthe realpartofthecomplexPoyntingvector S = 1 2 c 4 ( E H ).Themagnitudeofthisvector givestheintensityofthelightwaveproportionaltothesquar eoftheeld.


14 opticalconductivitysincetheresponseconcernedherearisesf romtransitionsasaresult ofphotonabsorption. Wehavesofarconsideredonlytransversewaves( i.e. k ? E ).However,Eq. 2.26 canalsobesatisedforlongitudinalwaves( i.e. k k E )foranyfrequency l provided that~ ( l )=0.Atthisfrequency,longitudinalwavescanpropagateth roughamedium andcontributetotheenergylossthatisproportionaltotheso -calledlossfunction denedas Im 1 ~ = 2 21 + 22 = 2 n ( n 2 + 2 ) 2 : (2.43) ThelongitudinalwavescanexciteLOphononmodesattheLOfr equencies. Inanonlyweaklyabsorbingmedium( i.e. n ),Eqs. 2.34 and 2.35 simplifyto n = p 1 ; (2.44) = 2 2 n = 0 1 cn : (2.45) Theseequationstellusthattherefractiveindex n isapproximatelydeterminedby 1 whiletheabsorptionismainlydeterminedby 2 (or 1 ). Thepurposeofthissectionhasbeentoprovidesomeoftheoptica lconstants thatdescribeopticalpropertiesofamedium,aswellasthere lationshipbetweenthese constants.TherelationssuchasEq. 2.30 and 2.33 aretheconnectionsbetweenthe macroscopicopticalparameterssuchas ~ N andquantitiesthatcanbecalculatedby microscopictheorysuchas~ .Notethatalltheopticalconstantsdescribedherearein generalfrequencydependentprovidinginformationabouth owphotonsofparticular energy, ~ ,interactwithelectrons,phonons,andotherexcitationsint hesystem. Detailedanalysisofopticalconstantsallowsustounderstand variouspropertiesof solids.Forexample,knowledgeoftheelectronicpropertieso fsolidsisthekeyto understandingmostoftheirphysicalandchemicalproperties.


15 2.4ExperimentalDeterminationofOpticalConstants Informationaboutsolidmaterialsisoftenobtainedbystudyi ngtheelectromagnetic wavesrerectedfromand/ortransmittedacrossinterfacesbet weenmaterialswithdifferentopticalproperties.Thiscanbedonebyconsideringthe boundaryconditionsof the E and H eldsandtheenergyconservation.Inexperiment,weusuallym easurethe fractionofenergyrerected[ i.e. ,rerectance, R ( )]fromand/ortransmitted[ i.e. ,transmittance, T ( )]throughaspecimen.Theformofaspecimenusuallydetermines which measurementtechniquehastobeemployed.Ourmaingoalistod educethedielectric functionaswellasotherfunctionsdirectlyrelatedtoit.I nthissection,wewilldiscussa fewexamplesofsimpleproceduresusedfordeterminingoptica lconstants. 2.4.1RerectionandTransmissionataPlaneInterface Werstconsiderthetransmissionandrerectionoflightataplane interface betweentwomediawithdierentrefractiveindices, ~ N 1 and ~ N 2 .Forsimplicity,wewill assumethelightisincidentnormaltotheinterface.Then,the boundaryconditions requirethatthetangentialcomponentsoftheelectricandm agneticeldsareconserved suchthat E i + E r = E t and H i H r = H t ; (2.46) where i r ,and t refertothecomponentsoftheincident,rerected,andtransm itted elds,respectively.Usingtherelationbetween E and H fromEq. 2.38 theboundaryconditionsyieldtheamplitudererectioncoecientand amplitudetransmission coecientas 6 ~ r = E r E i = ~ N 1 ~ N 2 ~ N 1 + ~ N 2 ; (2.47) and ~ t = E t E i =1+~ r = 2 ~ N 1 ~ N 1 + ~ N 2 : (2.48) 6 Foranarbitraryangleofincidenceamoregeneraltreatment isrequired.Thererectionandtransmissioncoecientsarethengivenbyformulaekno wnasFresnel'sequations[ 9 ].


16 Thererectance(orrerectivity)istheintensityrerectionc oecient.Ifthelightis incidentonamediumfromavacuumside,thererectanceothem ediumissimply givenby R =~ r ~ r = (1 n ) 2 + 2 (1+ n ) 2 + 2 ; (2.49) wherewehaveused ~ N 1 =1and ~ N 2 = n + i .Thisisthevalidequationforthe single-bouncererectancemeasuredfromathickcrystal( i.e. ,abulkmaterial)withits thicknessmuchgreaterthanthepenetrationdepth( d ). FromEq. 2.49 ,itisobviousthatrerectancedataalonecannotdetermineb oth n and .Itisingeneralnotpossibletodeterminebothcomponentsfro mthemeasurementofjustoneopticalparameter,suchasrerectance.Theref ore,weneedseparate measurementofeither n or bysomeothermeans,ortodosomethingelseinconjunctionwithrerectancemeasurement.TheKramers-Kronigr elationsoerpractical solutiontothisproblemasdiscussedbelow.2.4.2Kramers-KronigDispersionRelations TheKramers-Kronigrelations(KK)areintegralrelationshi psbetweenrealand imaginarypartsofacomplexfunction,suchasthelinearrespo nsefunctions~ ( ), ~ ( ),and ~ N ( ),asaresultofinvokingthelawofcausalityandapplyingthec omplex analysis. 7 Oneoftherequirementsfortherelationshipstobevalidisth attheresponse functionvanishesfor !1 .TheKKrelationsforthecomplexrefractiveindexandthe complexdielectricfunctionmaybestatedasfollows: n ( ) 1= 2 P Z 1 0 0 ( 0 ) 0 2 2 d! 0 ; (2.50) ( )= 2 P Z 1 0 n ( 0 ) 1 0 2 2 d! 0 ; (2.51) 7 Inaphysicalsystem,responsefunctionsmustsatisfy G ( )= G ( ).Forthedielectricfunction,thisconditionleadsto 1 ( )= 1 ( )and 2 ( )= 2 ( ).Inother words, 1 ( )isanevenand 2 ( )isanoddfunctionofthefrequency


17 and 1 ( ) 1= 2 P Z 1 0 0 2 ( 0 ) 0 2 2 d! 0 ; (2.52) 2 ( )= 2 P Z 1 0 1 ( 0 ) 1 0 2 2 d! 0 ; (2.53) where P standsfortheCauchyprincipalvalueoftheintegral.Simila rrelationsare availableforotherlinearresponsefunctions.Fromtheserelat ionsweseethatifthereal partofaresponsefunctionisknownoveranentirefrequencyra nge(0

18 d Medium 1, N 1 Medium 2, N 2 Medium 3, N 3 RT Figure2{1:Rerectionandtransmissionattwoparallelinterfa ces.Thethicknessofthe secondmediumis d .Weassumethecaseofnormalincidence,butthebeamsaredrawn atanangleforaclarity.FromEq. 2.49 andEq. 2.55 ,wecandetermine n ( )and ( ),thedielectricfunction, andallotherrelatedfunctions. 8 2.4.3RerectionandTransmissionatTwoParallelInterface s Ifthelightisincidentonaplaneinterface(betweenmedium 1andmedium2), andtransmittedthroughthesecondparallelplaneinterface( betweenmedium2and medium3),theexpressionsoftransmittanceandrerectancebec omemorecomplicated sincenowwehavetoconsiderthemultiplererectionaswellasa bsorptionabsorption withinthesecondmedium.ThissituationisdepictedinFigure 2{1 .Therstandthird mediaareassumedtobenon-absorbing,andspanthesemi-innitesp acewiththeir complexrefractiveindex ~ N 1 and ~ N 3 ,respectively.Thesecondmediumhasitsthickness d withtherefractiveindex ~ N 2 .Weagainassumenormalincidenceforsimplicity.Then, thegeneralformulaefortheresultantamplitudetransmissiona ndrerectioncoecients 8 Thistechniqueisquitepractical,yettherequirementsofw iderangemeasurement canbeinconvenientinsomesituation.Oneofthetechniquecal ledellipsometrycandeterminesimultaneouslybothrealandimaginarypartsofthedi electricfunctionovera limitedfrequencyrange,andmayserveasanalternativemeth odtoconsider[ 17 ].


19 includingmultiplererectionsare ~ t = ~ t 12 ~ t 23 e i ~ [1+(~ r 23 ~ r 21 e i 2 ~ )+(~ r 23 ~ r 21 e i 2 ~ ) 2 + ] = ~ t 12 ~ t 23 e i ~ 1 ~ r 23 ~ r 21 e i 2 ~ (2.58) and ~ r =~ r 12 + ~ t 12 ~ r 23 ~ t 21 e i ~ [1+(~ r 21 ~ r 23 e i 2 ~ )+(~ r 21 ~ r 23 e i 2 ~ ) 2 + ] = ~ r 12 +~ r 23 e i 2 ~ 1 ~ r 21 ~ r 23 e i 2 ~ ; (2.59) where~ r ij and ~ t ij aretheamplitudererectionandtransmissioncoecientsbetwe en mediums i and j asgivenbyEqs. 2.47 and 2.48 ,and ~ isthecomplexphasedepthof thesecondmediumwhichisdenedby ~ = c ~ N 2 d = c n 2 d + i 2 d; (2.60) where istheabsorptioncoecientdenedbyEq. 2.40 .FromEqs. 2.58 and 2.59 ,the resultanttransmittanceandrerectanceareobtained: T = n 3 n 1 j ~ t j 2 = n 3 n 1 j ~ t 12 j 2 j ~ t 23 j 2 e d 1+ j ~ r 23 j 2 j ~ r 21 j 2 e 2 d 2 j ~ r 23 jj ~ r 21 j e d cos (2.61) and R = j ~ r j 2 = j ~ r 12 j 2 + j ~ r 23 j 2 e 2 d +2 j ~ r 23 jj ~ r 12 j e d cos 1+ j ~ r 23 j 2 j ~ r 21 j 2 e 2 d 2 j ~ r 23 jj ~ r 21 j e d cos (2.62) with =2 c n 2 d + 23 + 21 ; (2.63) where ij isthephaseshiftuponrerectionateitherinterface.Thecosin etermleads tointerferencefringesinthespectrumduetomultipleinter nalrerectioninthesecond medium.Whenthesecondmediumisthick( d )orwedged,thereisnocoherence amongmultiplererections.Inalowresolutionmeasurement,th osefringesarenot resolved,andthetransmittanceorrerectanceisaveragedove rthephaseangleforthe


20 partialbeamsas T ave = n 3 n 1 j ~ t 12 j 2 j ~ t 23 j 2 e d 1 j ~ r 23 j 2 j ~ r 21 j 2 e 2 d (2.64) and R ave = j ~ r 12 j 2 + j ~ r 23 j 2 e 2 d 2 j ~ r 23 j 2 j ~ r 12 j 2 e 2 d 1 j ~ r 23 j 2 j ~ r 21 j 2 e 2 d : (2.65) Whenathicksampleofthickness d withcomplexrefractiveindex~ n ismeasuredin avacuum,itisstraightforwardtondtheaveragedtransmitta nceandrerectance: T ave = (1 R s ) 2 (1+ 2 =n 2 ) e d 1 R 2 s e 2 d (2.66) and R ave = R s (1+ T ave e d ) ; (2.67) where R s isthesingle-bouncererectancegivenbyEq. 2.49 .Experimentsofthistype areveryimportantandareoftenappliedtomeasureabsorption coecientsofsolids. Whenwavelengthsofincidentlightarecomparabletothethi ckness d ,theinterferencefringesareresolvedwithsucientlyhighresolutionmeasu rements.FromEq. 2.63 itisapparentthatthespectrumexhibitsperiodicfringesof thefrequencyspacing betweentwosuccessivefringesgivenby = 1 2 nd ; (2.68) where isincm 1 and d isincm.Thisequationissometimesusefultodeterminethe thicknessofthesamplefromthefringespacing,andviceversa.F orexample,mylarlms usedasabeam-splitterinfarinfraredhavetherefractiveind exbetween1 : 64and1 : 67. Then,wecanexpectthattherstminimumfor6 mmylarbeam-splitterappearsnear 500cm 1 2.4.4OpticsinThinFilmonaSubstrate Astructureofathinlmofthickness d ( wavelengthorpenetrationdepth)laid onathickbutnon-absorbing(orweaklyabsorbing)substratewit hrefractiveindex n andthickness x isquitecommonintheopticalexperiment.Figure 2{2 showsthe schematicdiagramofthesituation.Againweonlyconsidertheno rmalincidencetoa


21 n x d RT vacuumvacuum thin filmsubstrate Figure2{2:Rerectionandtransmissionwithathinlmonaweakl yabsorbingsubstrate.Thethicknessofthethinlmandthesubstrateare d and x respectively.We assumethecaseofnormalincidence,butthebeamsaredrawnatan angleforaclarity. sampleinfreespace.Itisobviouslymorecomplicatedsincenoww earedealingwitha fourlayeredstructure.Insuchcase,theKKtechniqueisinappl icable.However,itis possibletoextractlinearresponsefunctionsfrommeasurements ofbothrerectanceand transmittanceoveranitefrequencyrange. Atrstweconsiderthecasewhenmultiplererectionsinsidethesu bstratemaybe neglected.Thissimpliesthesituationsignicantlysincethe thicknessofthesubstrate x becomesunimportant,andthesystemcanbeconsideredasathree layeredstructure (vacuum-lm-substrate)justliketheonediscussedabove.Then,f romEqs. 2.58 and 2.59 withthefollowingapproximations: j ~ N 2 j ~ N 1 =1 ; j ~ N 2 j ~ N 3 n 3 n ( 3 n 3 ) ; d wavelengthorskindepth ; (2.69) itcanbeshownthatthetransmittanceacrossthelmintosubstrat eandthererectance fromthelmaregivenbytheGlover-Tinkhamequations[ 18 19 ]: T f = 1 1+ ~ y n +1 2 4 n ( y 1 + n +1) 2 + y 2 2 (2.70) and R f = ( y 1 + n 1) 2 + y 2 2 ( y 1 + n +1) 2 + y 2 2 ; (2.71)


22 wherenistherefractiveindexofthesubstrate, y 1 and y 2 aretherealandcomplex partofthedimensionlesscomplexadmittanceofthelm,~ y ,respectively.~ y isrelated tothecomplexconductivity~ = 1 + i 2 ofthelmby~ y = Z 0 ~ d where Z 0 isthe impedanceoffreespace(4 =c incgs;377ninmks).Thus,theopticalbehaviorofa lmisdeterminedbyitselectricalpropertiesofthelmtha tismodiedbythesurface eects. Theactualmeasuredtransmittanceandrerectanceareinruenc edbymultiple internalrerectionswithinthesubstrateofthethickness x andtheabsorptioncoecient .Then,thesystemisafourlayeredstructurewithvacuumasthef orthmedium.Ifa substrateisthick( x )orwedged,coherenceamongmultiplererectionsarelost,th e measuredtransmittanceandrerectancearesimpliedto T = T f (1 R s ) e x 1 R s R 0 f e 2 x (2.72) and R R f + T 2 f R s e 2 x 1 R s R 0 f e 2 x ; (2.73) where R 0 f isthesubstrate-incident(backside)rerectionofthelm: R 0 f = ( y 1 n +1) 2 + y 2 2 ( y 1 + n +1) 2 + y 2 2 (2.74) and R s isthesingle-bouncererectanceofthesubstrategivenbyEq. 2.49 .Foraweakly absorbingsubstratesuchthat c= 2 n R s maybeapproximatedas R s 1 n 1+ n 2 : (2.75) Thisisusuallysatisedformeasurementsatlowtemperatureand lowfrequencies. Frommeasurementsofthetransmittanceandrerectanceoftheb aresubstrate,we canndtheindexofrefraction n andtheabsorptioncoecient ofsubstrateusing Eqs. 2.66 and 2.67 .Notethattheterm 2 =n 2 inEq. 2.66 canbeneglectedforaweakly absorbingsubstrate.


23 Withtheknowledgeofsubstrate'sopticalproperties, 1 and 2 andinturnall otherresponsefunctionscanbeextractedbyinvertingEqs. 2.72 2.74 aftermeasuring bothtransmittanceandrerectanceofthelm-on-substratesyste m.Forastructure withmorelayers( e.g. ,vacuum-lm-buer-substrate-vacuum),theanalysisbecomes progressivelymorecomplicated.Amoregeneraldiscussionofthe opticalresponsefrom multi-layersisgivenin[ 15 20 ]. 2.4.5PhotoinducedAbsorption Inthephotoinducedmeasurements,weareinterestedinchanges intheoptical behaviorofasampleinphotoexcitedstatewithrespecttonon-e xcitedstate(ground state).Forasampleintheformoflmwiththickness d depositedonasubstrate,it wouldbeidealtohaveasubstratematerialthatisinsensitiveto thephotoexcitation. Insuchcase,thephotoinducedchangeinthetransmittance, T ,isduetothephotoinducedabsorptionbythelmitself.Ifthemeasurementisdonein lowresolution,the transmittancethroughthelmintosubstrateisgivenbyEq. 2.66 ,anditcanbeshown thatthenormalizedphotoinducedtransmittanceiswrittena s T T = T T 0 T 0 ( ) d; (2.76) where T and T 0 arethetransmittanceofthelminexcitedstateandgroundstat e, respectively.Notethatthenegativeofthequantity T = T iscustomaryusedasthe photoinducedsignal. 2.5MicroscopicModels Uptothispoint,wehavenotdescribedtheopticalphenomenafr omamicroscopic pointofview.Therearevariousmicroscopicmodelsthattryt oexplaintheoptical behaviorobservedexperimentally.Thesemodelsmaybeclassie daseitherclassical, semiclassical,orquantummechanical,dependingonhowwetrea tinteractionbetween lightandmatter. Intheclassicalmodel,bothlightandmatteraretreatedclassic ally.Thedipole oscillatormodel(Drude-Lorentzmodel),whichwillbediscusse dshortly,isaexample


24 ofaclassicaltreatment.Thismodelhasbeenproventobeverysu ccessful,andisoften usedforunderstandingthegeneralopticalpropertiesofmedi um. Inthesemiclassicalapproach,theatomsinthemediumaretreat edquantum mechanically,whilethelightisstilltreatedasaclassicalel ectromagneticwave.The absorbtioncoecientoroscillatorstrengthduetotransitionb etweentwostatesortwo bandscanbecalculatedusingFermi'sgoldenrule,whichrequ iresknowledgeofthewave functionsofthestates. Inthecompletelyquantumapproach,thelightisalsotreated quantummechanically,namelyasphotons.Feynmandiagramscanbedrawntorep resenttheinteraction processesbetweenphotonsandatoms. Inthissection,wewilldiscussonlyafewofmicroscopicmodelst hatarecommonly usedduringanalysis.2.5.1LorentzModel Inasolid,therearevariousprocesses(orexcitations)thatcon tributetothe dielectricfunctionwhich,inturn,describesitsopticalbe haviors.Forexample,free carrierabsorptionandphonon(includingmulti-phonon)abso rptionarethetypical processesatfar-andmid-infraredfrequencies. 9 Inthespectralrangeofnear-infrared andultraviolet,processessuchasexcitonsandfundamentalab sorptionacrossthe energygap,interbandtransitions,andplasmaabsorptionmaybe seen.InthevacuumultravioletandX-rayspectralregion,thetransitionsofthec oreelectronscanbe expectedtodominatethedielectricfunction.Attheveryhi ghenergiesbeyondnuclear excitations,nothingcanrespondtothedrivingeld,andthed ielectricfunctionbecomes unitysincethemediumdoesnotpossessanypolarization.Notetha talltransitions requiretheconservationofenergyandmomentum. 9 Inprinciple,magneticexcitationscouldexistatevenlower energies.


25 TheLorentzmodelisasimple,yetveryusefulclassicalmodeldie lectricfunction thatcanbederivedforasetofdampedharmonicoscillators.Whe naharmonicoscillatorwithmass m ,charge q ,dampingconstant r ,andresonantfrequency 0 isexcitedby aharmonicelectriceldoftheform E ( t )= E 0 e i!t ,theequationofmotionisgivenby m r + mr r + m! 2 0 r = q E ( t ) : (2.77) Thesecondtermmodelstheenergylossmechanismoftheoscillati ngdipole.Notethat theresonantfrequency 0 isatransverseoscillatorfrequencythatiscoupledtothe transverseelectriceld.Insertingasolutionoftheform r = r 0 e i!t intoEq. 2.77 yields r = q m 1 2 0 2 ir! E : (2.78) Ifthereare N oscillatorsperunitvolume,theresonantcontributiontothe macroscopicpolarizationis P = Nq r = Nq 2 m 1 2 0 2 ir! E : (2.79) Notethattheisotropicmediumisassumedhere.Then,thesusceptib ilityarisingfrom theoscillatoris ~ = Nq 2 m 1 2 0 2 ir! : (2.80) Thetotalpolarizationisgivenby P total =~ e E =(~ 1 ) E ; (2.81) where 1 isthebackgroundsusceptibilitythatarisesfromthepolariza tionduetoall theotheroscillatorsathigherfrequencies. ThedielectricfunctionisdeterminedfromEq. 2.10 : ~ ( )= 1 + 2 p 2 0 2 ir! ; (2.82) wherewehavedenedthehighfrequencylimitof~ ( )as 1 =1+4 1 ; (2.83)


26 andtheplasmafrequency p as 2 p = 4 Nq 2 m : (2.84) Notethatthesubscript 1 shouldbeunderstoodascontributionsaboveacertain resonance.Separatingtherealandimaginaryparts,weobtain 1 ( )= 1 + 2 p ( 2 0 2 ) ( 2 0 2 ) 2 +( r! ) 2 ; (2.85) 2 ( )= 2 p r! ( 2 0 2 ) 2 +( r! ) 2 : (2.86) Fromtheseequations,itisstraightforwardtoseethat 1 graduallyincreasesfromthe value 1 + 2 p =! 2 0 asfrequenciesincreasetoward 0 ,andpeaksat 0 r= 2.Ittakes sharpnegativeslope,passingthrough 1 at 0 ,andbottomsat 0 + r= 2.Asfrequencies increasefurther,itnallyapproachesthehighfrequencyli mitof 1 .Asmentioned brieryintheearliersectionofthischapter,thefrequencyf orwhich 1 ( )=0islabelled as l atwhichelectromagneticwavesarecoupledtothelongitudi nalcomponentof theoscillator.Comparedwith 1 2 hasasimplebellshapewithastrongpeakat 0 andthefullwithathalfmaximumgivenby r .Notethatboth 1 and 2 varyonthe frequencyscaleof r ,andthedampingoftheoscillatorhastheeectofbroadening .It isingeneralthatmaterialishighlyabsorbingneartheresona nce,forobviousreason, stronglyrerectingbetween 0 and l ,andtransparentatfrequenciesfurtherawayfrom theresonancewhere 1 doesnotvarystrongly. Eq. 2.82 canbegeneralizedtoanarbitrarynumberofdierentoscilla torsas ~ ( )= 1 + X j 2 pj 2 j 2 ir j ; (2.87) where j r j ,and pj aretheresonantfrequency,dampingconstant,andplasma frequencyoftheoscillatoroftype j ,respectively.Theplasmafrequencyisdenedby 2 pj = 4 N j q 2 j m j ; (2.88) where N j q j ,and m j arethenumberdensity,eectivecharge,andeectivemassoft he oscillatoroftype j ,respectively.Thesevalesmustbeappropriatelychosentoacco unt


27 forthedierentoscillators.Forexample,inthecaseofaphono n, p istheionplasma frequencywith q and m astheeectivechargeandthereducedmassoftheparticular latticevibrationmode. AcorrespondingquantummechanicalversionofEq. 2.87 canbewrittenas ~ ( )= 1 + 2 p X j f j 2 j 2 ir j ; (2.89) wherewehaveintroducedaoscillatorstrength f j inordertoaccountforthestrength oftheresponseofdierenttransitionstotheperturbingelect riceld.Inthequantum picture, j isthetransitionfrequencybetweentwostateswhicharesepara tedinenergy by ~ j ,and r j istheuncertainty(orwidth)inenergyoftheinitialandna lstates oftransition.Theoscillatorstrength f j isrelatedtotheprobabilityofaquantum mechanicaltransition,whichcanbecalculatedusingFermi's goldenrule. 10 Itsatisesa sumrule X j f j =1 : (2.90) Theoscillatorstrengthprovidesusanexplanationforthedi erentabsorptionstrength indierenttransitions.2.5.2FreeCarrierResponseandDrudeModel TheequationofmotiongiveninEq. 2.77 canalsobeusedtoderivethedielectric responseoffreecarriersofcharge q andeectivemass m bytakingtherestoringforce termzero( i.e. 0 =0): m v + m v = q E ( t ) ; (2.91) 10 Fermi'sgoldenruleshowsthatthetransitionratebetweentwo statesisproportional tothesquareofamatrixelementandalsotoadensityofstatesfor boththeinitialand nalstates.Theoscillatorstrengthandabsorptioncoecientar erelatedtothequantummechanicaltransitionrate.


28 wherewehaveexpressedthedampingconstant r asareciprocalofthecollisiontime thatcharacterizeslossofmomentumofcarriersduetoscatter ing. 11 Thisisasimple equationbasedontheDrudemodel.Insertingasolutionofthefo rm v = v 0 e i!t into Eq. 2.91 yields v = q m 1 1 i! E : (2.92) For N freecarriersperunitvolume,thecurrentdensityisthen j = Nq v = Nq 2 m 1 1 i! E =~ E : (2.93) Thus,theacconductivitybasedontheDrudemodelis ~ D ( )= 0 1 i! ; (2.94) where 0 isthedcconductivitydenedas 0 = Nq 2 m : (2.95) Therealandimaginarypartsare D 1 = 0 1+ 2 2 ; (2.96) D 2 = 0 1+ 2 2 : (2.97) FromEq. 2.14 ,thedielectricfunctionforthefreecarriersisgivenby ~ D ( )=1 2 p 2 + i!= ; (2.98) 11 Atypicalvalueof forametalordopedsemiconductorisintherange 10 14 10 13 seconds,whichcorrespondto 3000cm 1 and 300cm 1


29 where pD istheDrudeplasmafrequencydenedby 12 2 pD = 4 Nq 2 m : (2.99) Eq. 2.98 isobviouslythesameexpressionasEq. 2.82 with 0 =0, r = 1 ,and 1 =1. NotethatEq. 2.98 assumesthatonlyfreecarrierscontributetothedielectricf unction. Whenotherprocessesathigherfrequenciesgivecontribution s,theunityshouldbe replacedby 1 .Therealandimaginarypartsare D 1 =1 2 pD 2 1+ 2 2 ; (2.100) D 2 = 2 pD (1+ 2 2 ) : (2.101) Inthelimitoflowfrequencywhere 1 issatised,wecanobtainfollowing relations: D 1 2 pD 2 ; (2.102) D 2 2 pD =! =4 0 =! D 1 ; (2.103) n ( 2 = 2) 1 = 2 ; (2.104) R 1 2 =n 1 (2 != 0 ) 1 = 2 : (2.105) Eq. 2.105 isknownastheHagen-Rubensrelation.Fromthesecondexpressio nwecan ndtheabsorptioncoecient: = 2 c p 8 0 c (2.106) 12 TheDrudeplasmafrequency pD isrelatedtothedcconductivityandcarriermobility as 0 = 2 pD = 4 = Nq .Themobility(aratioofthecarrierdriftvelocityto theeld)isgivenby = q=m .Formetals, 0 is(nearly)independentoftemperature assumingthat doesnot(oronlyweakly)varywithtemperature,andisusedto characterizemetals.Forsemiconductors,ontheotherhand,theirc arrierdensitiescanbe variedbychangingthetemperatureorthedopantconcentrat ion.Therefore,themobility ismoreconvenientquantitytocharacterizesemiconductorsi ncethecarrierdensityis takenout.


30 ortheskindepth: = 2 c p 2 0 : (2.107) Therefore,theskindepthisinverselyproportionaltothesqua rerootofdcconductivity andfrequency.Thisimpliesthatamaterialwithhigherdcco nductivityallowsshorter penetrationofaceldsforagivenfrequency. Asanotherlimitingcase,consideranundampedfreecarriersyste mlikeaperfect conductor.Inthisspecialcase,theDrudewidth 1 =0andthedielectricfunctionis realandgivenby D 1 = 1 2 pD 2 ; (2.108) D 2 = D 1 =0( 6 =0) : (2.109) Herewehaveused 1 justforthepurposeofgenerality.Thisequationtellsusthat D 0forfrequenciesbelowtheplasmaedge( ! pD = p 1 ).Then,thecomplex refractiveindex ~ N ispurelyimaginaryandthusthererectance R is1inthisfrequency rangeandthesystemsuddenlybecomestransparentabovetheplasm aedge. 13 This, so-calledaplasmarerection,happenswithoutlossofenergysin cethereisnoresistive current( i.e. D 1 =0)associatedwiththisfreecarrierresponse. Inrealmaterials,thedamping 1 hasanon-zerovaluewhichmaybededuced fromdcconductivityorothermeasurements.Theeectoftheda mpingmaybesmall butresultsinslightlylessrerectanceaswellasbroadeningof theplasmaedge.The rerectancemaybeevenlowerandhavestructuresduetoothera bsorptionprocesses suchasinterbandtransitions.Ifotherprocessesoccursnearthe plasmaedge,sharp onsetoftransmissionmaynotbeobserved. 13 Formetals, 1 1andthererectanceisveryhighforfrequenciesupto pD .For semiconductors,ontheotherhand, 1 canbelargeandasaresulttheplasmaedgeat pD = p 1 islowerthan pD


31 2.5.3Drude-LorentzModel WhenboththeDrudeandtheLorentztypesofdielectricrespon seisobservedina spectrum,thetotaldielectricfunctioncanbeexpressedasthe sumofvariousdierent processesthatcauseapolarization: ~ ( )= 1 + X j 2 pj 2 j 2 ir j ! 2 p 2 + i!= : (2.110) ThisrelationiscalledtheDrude-Lorentzmodel,andcanbeu sedinttingtheexperimentalrerectancedataforextractingopticalparameters.Un liketheKK-methods,the ttingdatawiththemodelfunctioncanbeemployedinanite frequencyrangeaslong aswehaveawelldenedbackgroundcontribution 1 beyondthemeasuredfrequency range.2.5.4SumRules In x 2.5.1 ,weintroducedthenotionofoscillatorstrength f .Usingquantum mechanics,itcanbeshownthatthetotalabsorptionbyalltransi tionsforthewhole frequencyrangeisconstant,andcanbeexpressedbythe f -sumrule: Z 1 0 1 ( ) d! = X j 2 p f j 8 : (2.111) Thistellsthatthetotalareundertherealpartoftheconduc tivityisindependentof temperature,phasetransition,photo-excitations,andsoon.T hereexistseveralother sumrules,butwewillnotdiscussthemhere. Thesumruleisoftenappliedtoacertainprocess.Ifonlyfreeca rriersareconcerned,Eq. 2.111 isrewrittenas Z 1 0 1 ( ) d! = 2 pD 8 : (2.112) Thisisanexceptionallyusefulequationtoseehowthespectral weightshiftsintothe deltafunctionatzerofrequencyasasuperconductorexperie ncesphasetransition.


CHAPTER3 FOURIERSPECTROSCOPY 3.1Introduction Aspectrometerisaninstrumentthatisdesignedtoyieldspectra linformation containedintheelectromagneticwavesunderstudy.Thereex istseveraltypesof spectrometersusedforanumberofresearchelds.Figure 3{1 showstheclassication ofspectrometers.Ofall,thescanningtwo-beaminterferometr ictypesareprobablythe mainstreaminstrumentnowowingtotheirvariousadvantagesw hichwillbeexplained shortly. Themonochrometerspatiallyseparatestheindividualfreque ncycomponents bymeansofadispersiveelementsuchasprismordiractiongrati ng.Anindividual frequencycomponentisselectedbyaslit,anditsintensityisse quentiallysampled.A powerspectrumisproducedaftermeasuringoverallfrequenci esofinterest.Although thistypeofinstrumentisstillusedcommonly,especiallyforne arinfraredandvisible spectroscopy,theyhavemetwithseverallimitations.Themaind icultycomesfrom theirslowscanningprocess.Becausethemonochrometermeasurese achfrequency individually,ittakesalongtime(typically10minutesorm oredependingonthe signaltonoiseaswellasresolution)tocompleteasinglescan.Th einterferometric Spectrometer Dispersingspectrometer(monochrometer)Interferencespectrometer(interferometer) Prism spectrometerDiffraction spectrometerTwin-beam interferometerMulti-beam interferometer Michelson interferometer(amplitude separation)Lamellar grating interferometer(wavefront separation)Martin-puplett interferometer(polarization separation)Fabry-Perot interferometerEtalon Figure3{1:Classicationofspectrometers. 32


33 techniquesweredevelopedtoovercomesomeofthelimitation sencounteredwith dispersiveinstruments. Theinterferometerisaninstrumentthatcandividetheincom ingbeamoflight intotwopathsandthenrecombinethetwobeamsaftera(optic al)pathdierence(or retardation)hasbeenintroduced.Theserecombinedbeamspr oduceinterferenceand theresultingsignalisdetected.Themeasuredsignalasafuncti onofpathdierence, calledaninterferogram,istheFouriertransformofthepowe rspectrumoftheincident light.Thusitcanbeinverse-Fouriertransformedtoyieldthe powerspectrum.However, becauseofthefactthatthedetectedsignalmustbetreatedmath ematicallybefore obtainingmeaningfulspectrum,certaincaremustbetakentoa voidintroducingerrors intothespectrum. Interferometrictechniquehastwobasicadvantagesoverdisp ersivemethods.The factthatnearlyalwaysthetotalintensityhitsthedetector duringthewholeperiodof measurementimprovesthesignal-to-noise(S/N)ratio,particu larlyforweakradiation sources.Thisisknownasthethroughput(orJacquinot)advant age.Theinterferometer measuresallfrequencycomponentssimultaneously.Thisleads toconsiderablemultiplex (orFellgett)advantageallowingquickdataacquisitionand higherS/N. 3.2FourierTransformInterferometry 3.2.1GeneralPrinciples Thegeneralprincipleofinterferometrycanbeunderstoodby consideringasimpliedMichelsoninterferometer[ 21 22 23 ],whichisshownschematicallyinFigure 3{2 Considerthatamonochromaticplanewaveoftheform E S = E 0 e i (2 x !t ) ; (3.1)


34 R Source Detector M1 M2 Beam splitter (Movable) x /2 ((w), f(w)) r 11 ((w), f(w)) r 22 ((w),(w)) rt Figure3{2:AschematicviewofasimpliedMichelsoninterfero meter.Thelighttravels tothebeam-splitterwithitsamplitudererectivity r andtransmissivity t .Thepartially rerectedbeamtravelstowardthexedmirror(M1)thathasth ererectivity r 1 and introducesphaseshift 1 .Thepartiallytransmittedbeamtravelsavariabledistance towardthemovablemirror(M2)with r 2 and 2 .Thebeamsarerecombinedatthe beam-splitterafteraopticalpathdierence x hasbeenintroduced,andhalfofthetotal beamreturnstothesource,andtheotherhalfproceedstoadet ector.Asamplecanbe placebetweeninterferometerandthedetector.isincidentonthebeam-splitterfromthesource.Here, isthewavenumber. 1 This beam-splitterpartiallyrerectsthebeamtowardthexedmir rorM1,andtransmits theresttowardthemovablemirrorM2. 2 Aftertravellingtheirrespectivepaths,the twobeamsarerecombinedatthebeam-splitter,andtheresulta ntbeamproceeds toadetector.Ifthebeam-splitterhastheamplitudererecta nce r andamplitude transmittance t 3 theresultingeldemergingfromtheinterferometertowardt he 1 Thewavenumberisdenedas =1 = =2 k = != 2 c 2 Hereforsimplicity,wewillassumethatbothM1andM2areequival entperfect rerectors.Theamplitudererectivityisacomplexnumberwhi chcanbeexpressedas r = E r =E i = j r ( ) j e i ( ) ,where E r isthererectedeld, E i istheincidenteld,and ( ) isthephaseshift.Foraperfectmirror, j r j =1and = 3 Foranidealbeam-splitter, r = t =2 1 = 2 .Theyarebothfrequencydependent.


35 detectorisasuperpositionofeldsfromtwobeamswhichisgiv enby E D = rtE 0 [ e i (2 x 1 !t ) + e i (2 x 2 !t ) ] ; (3.2) where x 1 and x 2 arethetotaldistancesofrespectivebeam'sopticalpath(seeF igure 3{2 ).Sincetheenergyreachingthedetectorisproportionalto E D E D ,thetime averageddetectorsignalcanbewrittenas S ( x )= 1 2 I 0 ( )[1+cos(2 x )] ; (3.3) wherewehavedenedtheopticalpathdierence, x = x 2 x 1 ,thebeam-splitter eciency, =4 j rt j 2 ,andthesourceintensity, I 0 ( ).Thisexpressionmaybesimplied to S ( x )= f ( )[1+cos(2 x )] ; (3.4) where f ( )isanarbitraryspectralinputthatdependsonlyon S ( x )isthedetector signalforamonochromaticsource.Thecosinetermgivesthemod ulationonthe detectorsignalasafunctionof x Asmentionedearlier,withtheinterferometrictechniqueal lfrequencycomponents aremeasuredsimultaneously.Eq. 3.4 canbegeneralizedforapolychromaticsourceby integratingitoverallfrequencies: S ( x )= Z 1 0 f ( )[1+cos(2 x )] d : (3.5) At x =0,thedetectorsignalreachesitsmaximumvalueof S (0)=2 Z 1 0 f ( ) d : (3.6) Thispositioncorrespondstothezeroopticalpathdierence( orZPD)whereall frequencycomponentsinterfereconstructively.As x !1 ,ontheotherhand,the coherenceofthemodulatedlightiscompletelylost,andthec osineterminEq. 3.5 goes tozero.Therefore,thedetectorsignaloscillatesaroundana veragevalue: S ( 1 )= Z 1 0 f ( ) d = S (0) 2 : (3.7)


36 Theinterferogramisthecosinemodulationpartofthedetect orsignal: F ( x )= S ( x ) S ( 1 )= Z 1 0 f ( )cos(2 x ) d : (3.8) ThisisthecosineFourierintegralofthedesiredspectrum f ( )whichcanberecovered bytakingtheinverseFouriertransform: f ( )=4 Z 1 0 F ( x )cos(2 x ) dx: (3.9) 3.2.2FiniteRetardationandApodization Sofarinourcalculation,itwasassumedthatthespectrumisobt ainedafterthe Fouriertransform(FT)oftheinterferogrammeasuredwithani nnitelylongoptical pathdierence(retardation).Inpracticetheinterferogr amcannotbemeasured toinniteretardation,anditmustbetruncated.Thistypeof truncationcanbe manipulatedmathematicallybymultiplyingthecompletein terferogrambyatruncation function G ( x )whichvanishesoutsidetherangeofthedataacquisition.Thus theactual functionwhichistransformedistheproductoftheinterfero gramandthetruncation function. Nowaccordingtotheconvolutiontheorem,theFToftheproduc toftwofunctions, say F ( x )and G ( x ),istheconvolutionoftheFTofeachfunction, f ( )and g ( ),where theconvolutionisdenedby f ( ) ?g ( )= Z 1 1 f ( 0 ) g ( 0 ) d 0 : (3.10) Hence,thecalculatedspectrumisthetruespectrumconvolvedw iththeFTofthe truncationfunction. Inordertoexaminetheeectoftruncation,considermultipl yinganinterferogram F ( x )withtheboxcarfunction G ( x )whichisdenedby G ( x )= 8><>: 1if j x j

37 n cm -1 n 1 n 1 -1/2 L n 1+1/2 L 1/ L Figure3{3:Thesincfunctionconvolvedwithasinglespectrall ineofwavenumber 1 L isthemaximumretardation.where L isthemaximumretardation.TheFTof F ( x )isthetruespectrum f ( ),while theFTof G ( x )isthesincfunction: FT[ G ( x )]=2 L sin(2 L ) 2 L =2 L sinc(2 L ) : (3.12) Whenthesincfunctionisconvolvedwithasinglespectrallineo fwavenumber 1 ,theresultantspectrumisthesincfunctioncenteredabout 1 ,whichisshownin Figure 3{3 .Thustheeectofconvolutionistosmoothoutthenarrowfeat ure.The singlespectrallineshapeasaresulttruncationissometimescal ledtheinstrumentline shape(ILS)function. Itcanbeshownthattherstzerosoneithersideof 1 occurat 1 1 = 2 L .Thus, twospectrallinesseparatedby1 =L arecompletelyresolved.Thustheresolutionis limitedbythemaximumpathdierenceontheinterferogram. Thevalueof1 =L (in cm 1 )isoftenusedasaquickestimateofspectralresolution.Theful lwidthathalf maximum(FWHM)issometimesusedasanalternativeestimateofre solution.


38 Thesuddencutowiththeboxcarfunctionintroducessidelobe snearsharp featuresinthespectrum. 4 Thus,itisdesirabletouseaweightedtruncationfunction thatcutsotheinterferogramingentlerfashion.Thisproce ss,knownasapodization, reducestheringingattheexpenseofafurtherreductioninre solution.Forexample,the Happ-Genzel[ 24 ]isasimpleapodizationfunctiongivenby G 1 ( x )=0 : 54+0 : 46cos( x=L ) ; (3.13) where L isstillthemaximumretardation.TheFToftheHapp-Genzelis FT[ G 1 ( x )]=2 L sinc(2 L ) 0 : 54 0 : 46(2 L ) 2 (2 L ) 2 2 : (3.14) Again,convolutionofthisfunctionwithasinglespectralline ofwavenumber 1 isthe resultantspectrumwhichisshowninFigure 3{4 togetherwiththeFToftheboxcar ( i.e. ,sincfunction).Thisgureclearlydemonstratesthattheuseo fagentlertruncation functionsuppressesthesidelobeswhiletheresolutionisreduce d.TheFWHMofthe spectrumusingtheHapp-Genzelandtheboxcarare0 : 91 =L and0 : 61 =L ,respectively. SomeofotherpopularapodizationfunctionsaretheNorton-B eer(weak,medium, strong)[ 25 ]andtheBlackman-Harris(3-term,4-term). 5 Verynicediscussionabout apodizationfunctionscanbefoundinGriths[ 22 ]. 3.2.3Sampling Intheuseofacomputerfordataacquisition,theanalogsignalm ustbeconverted todigitizeddatasets( i.e. ,A/Dconversion)beforeanysortofmanipulationcantake place.Forthisreason,theinterferogramissampledatsmall,e quallyspaceddiscrete retardation,andtheFourierintegral,Eq. 3.9 ,isapproximatedbyasum.Thisdiscrete 4 Therstminimumdropsobelowzeroby22%oftheheightatcent ralmaximum. Thesecondarymaximaarealsorelativelylarge.Thesesidelobes giverisetooscillation whichmayappearasspuriousfeaturesespeciallyintheneighb orhoodofsharpspectral features. 5 Personally,IstartwiththeNorton-Beer(medium)apodization functionrst,and tryothersifceratinimprovementhastobemadedependingon thespectralfeatures.


39 n cm -1 n 1 n 1 -1/2 L n 1+1/2 L 1/ L FT[Boxcar] FT[Happ-Genzel] Figure3{4:TheFToftheHapp-Genzel(HG)apodizationfunctio nconvolvedwitha singlespectrallineofwavenumber 1 L isthemaximumretardation.Forcomparison, theFToftheboxcar(sincfunction)convolvedwiththesamesing lelineisshown.The FWHMofthespectrumfortheHGandtheboxcarcasesare0 : 91 =L and0 : 61 =L ,respectively.Notethatthesidelobesinthespectrumaresuppressedbyusi ngtheHGatthe costofresolution.naturecanbehandledmathematicallybyusingtheDiracdelta combdenedby( x )= 1 X n = 1 ( x n ) : (3.15) TheDiracdeltacombisjustaseriesof functionsattheintegers.Ithasfollowing properties:( x + m )=( x ) (periodic) ; (3.16)( ax )= 1 j a j 1 X n = 1 x n a (scaling) ; (3.17) FT[( ax )]= 1 j a j a = 1 j a j 1 X n = 1 e i 2 n=a (Fouriertransform) ; (3.18)( x ) F ( x )= 1 X n = 1 F ( n ) ( x n )(sampling) ; (3.19)( x ) ?F ( x )= 1 X n = 1 F ( x n )(replication) : (3.20)


40 f '() n n cm -1 DnD =1/ x 0 n max Dn / 2 n cm -1 0 n max (A)(B) Figure3{5:Therelationbetweenspectrumreplicationandsam plingrate.(A)Proper choiceofsamplingratesuchthatthetruespectrumisconnedto one-halfofthereplicationperiod( i.e. max = 2).Inthissituation,theperiodicreplicasdonotoverlap andnoerrorisintroducedbysampling.(B)Improperchoiceof samplingrate.Overlappingwithadjacentreplicatedspectrumcausesspuriousresulti nthespectrum. Ifthecontinuous(oranalog)interferogram F ( x )issampledatintervals x ,the sampled(ordigitized)interferogram F 0 ( x )isgivenby F 0 ( x )= x x F ( x )= x 1 X n = 1 F ( n x ) ( x n x ) : (3.21) Then,thespectrumderivedfromtheFTof F 0 ( x )is f 0 ( )= 1 ?f ( )= 1 X n = 1 f ( n ) ; (3.22) where =1 = x and f ( )=FT[ F ( x )]. Eq. 3.22 isthespectrumweactuallyobtainasdatawhichiscomprisedof periodic replicasofourdesiredspectrum f ( )withperiod .Thisreplication,whichobviously arisesfromthesamplingoftheanaloginterferogram,raisesan importantissueofthe samplingfrequency,whichispeculiartodiscretesampling.Fi gure 3{5 illustratesthe relationbetweenreplicationandsamplingrate.Asinthegur e(B),ifthehighest frequency, max ,ofthetruespectrumexceedsthefoldingfrequency, = 2,thentwo adjacentreplicatedspectraoverlap,andtoo-highfrequenc iesappearfalselyatlower


41 Figure3{6:Theredcurveofabove-Nyquistfrequencyappearst opossessthesamesetof datapointsastheblackcurveofbelow-Nyquistfrequency.frequenciescausingspuriousresultintheobtainedspectrum.T hisisbecausehigher frequencywavescanbedrawnadditionallythroughthesamesam plingpointstakenfor lowerfrequencywavesasillustratedinFigure 3{6 .Thiseect,knownasspectralfolding oraliasing,canbepreventedbyinsuringthecondition: max 2 (3.23) or x min 2 : (3.24) Theseconditionsstatethatthehighestfrequencyneedstobesam pledatleasttwiceper wavelength,whichisjusttheNyquistsamplingcriterion.There fore,itisexperimentally importanteithertoensuredigitizinganinterferogramatah ighenoughsamplingrate ortolimittherangeoffrequencyinputtothedetectorusingo pticaland/orelectronic lters.Theresponsivityofthedetectoroftenworksasakindof low-passlter. Followingtheabovearguments,itisquiteobviousthatmeasur ementsofnarrow frequencyrangerequiresmallernumberofsamplingpoints.Ift henumberofpoints istoosmall,thespectralshapemaynotbewelldened.Insuchcase ,wecanadd extrazero-valueddatapointsattheendoftheinterferogra mkeepingthesamesample spacing.Thistechnique,knownaszerolling,eectivelypr oduceslargernumber ofspectrumpointsperresolutionelement.Sincethepointsad dedarezero-valued, spectrumresolutionwillnotincrease.Itmerelyprovidesasmoo therspectrallineshape. 3.2.4PhaseErrorandCorrection Untilthispoint,wehaveassumedthattheinterferogramisperf ectlysymmetric abouttheZPD.Inarealexperiment,however,thereoftenexi stsaphaseerrorthat mustbeincludedtodescribetheactualmeasured( i.e. ,asymmetric)interferogram.It


42 mainlyresultsfromsamplingerrors,electronicltering,and opticaleectsfromvarious partsofinstrumentopticsaswellasasample.Theeectofsuche rroristodistortthe ILSfunctionfromthesymmetricsincfunctiontoanasymmetricsh ape.Thiscouldlead tonegativespectrumorslightshiftofsharpfeatures.Therefore ,itisimportanttohave schemesthatcouldcorrectfaultyeectsfromacalculatedspe ctrum. Whenthephaseerrorisincluded,theinterferogramgivenbyE q. 3.8 ismodied to 6 F ( x )= Z 1 0 f ( ) e i (2 x ) d = Z 1 0 [ f ( ) e i ] e i 2 x d ; (3.25) where isthephaseerror(orphasespectrum)whichcanbefrequencydep endent.Note thathereweusedtheexponentialnotationforsimplicity.The n,thethecalculated spectrumthroughtheinversecomplexFTis ~ f ( ) f ( ) e i = Z 1 1 F ( x ) e i 2 x dx: (3.26) Hence,theasymmetricinterferogramyieldsacomplexspectrum .Therealpartofthe spectrum,Re[ ~ f ( )],andtheimaginarypartofthespectrum,Im[ ~ f ( )],canbecomputed bythecosineFTandthesineFToftheinterferogrammeasuredsymm etricallyoneither sideofthezeroretardationpoint(orcenterburst),respectiv ely.Ouraimistondthe phaseerror, ,fromwhichweapplysomesortofphasecorrectionschemetodeter mine thetruespectrumofinterest,whichis f ( )inEq. 3.26 Hereweexplainthesimplestwaytoachievethephasecorrection. First,wetake aninterferogrambetween L 1 x L 2 where x =0correspondstothecenterburst. Sinceitisonlyrequiredtocalculate atverylowresolution,thedistance L 1 ,which isdeterminedbythephaseresolutionsetting,canbesmallertha nthedistance, L 2 6 Thephaseerrorisaddedtothephaseangleoftheinterferogram ascos(2 x )= cos(2 x )cos +sin(2 x )sin .TheFTofatruncatedsinewaveisanoddfunction. Thus,theaddedsinecomponentisresponsiblefortheasymmetricsh apeofaninterferogram,andtheitsFTcausesdistortionoftheILSfunction.


43 requiredtoattainthedesiredresolution(1 =L 2 ). 7 Fromtheshortdouble-sidedregionof theinterferogram( L 1 x L 1 ),thephasespectrumcanbefoundfrom ( )=arctan Im[ ~ f ( )] Re[ ~ f ( )] : (3.27) Havingcalculated ,thecomplexspectrum, ~ f ( )maybecorrectedbymultiplyingitby e i =cos i sin suchthat ~ f ( ) corrected = f ( ) e i e i = f ( ) : (3.28) Inthiswaytherecoveredspectrummaybecorrectedforerrors incurredasaresultof asymmetriesinthemeasuredinterferogram. Thereareseveralphasecorrectionmodesavailableamongwhic hthemethod developedbyMertzisthemostcommonlyusedone.Moredetailed discussionofphase correctionmethodscanbefoundinvariouspapers[ 26 27 28 ]. 3.2.5Step-ScanandRapid-ScanInterferometers Thereareingeneraltwodierentkindsofinterferometersd ependingonitsscanner (movablemirror)movements:step-scaninterferometersandra pid-scaninterferometers. Instep-scaninterferometers,thescannerstartsfromitsrefere nceposition,andsteps toequallyspacedsamplingpositions.Ateachsamplingposition,t hescannerisheld stationaryandthedetectoroutputsignalisintegrated.Thest eppingcontinuesuntil thedesiredresolutionhasbeenachieved.Comparedwithrapid -scaninterferometers, step-and-integratesystemshaveunavoidabledowntimewhilem ovingtothenext samplingpointandwaitingforthescannertobestablebeforeac tuallystartingdata acquisition.Inaddition,thefactthatittakeslongertimet ocompletesinglescan makesstep-and-integratesystemspronetobesensitivetoslowvar iationsinthesource intensitywhichcoulddegradespectrumespeciallyatlowfrequ ency.Further,the 7 Typicallythephaseresolutionissettohave4to8timeslowerth anthespectralresolution.Ifthedouble-sidedacquisitionmodeisused( i.e. L 1 = L 2 ),thephaseresolution settingiscompletelyignoredforobviousreason.


44 systemsgenerallyrequiretouseachopper,andthusloseanother halfofthesignalafter theinterferometer.Thesecharactersmakesstep-scantechniq ueratherinecient,andin generalrapid-scanmethodissuperiorandadoptedbythemostof recentinstruments. Inrapid-scaninterferometers,thescannermovesatconstantan dsucientlyhigh velocity.Whenasignalofaparticularfrequency issentasinputtoaninterferometer, itismodulatedatafrequency 0 whichisrelatedtothemovingmirrorvelocity v as 0 = v c = v = v ; (3.29) where 0 and areinHz, incm, v incm/s,and incm 1 .Hence,inordertobe abletoanalyzethesignalatthedetectoritisnecessarytoknow thescannervelocity accurately.Itcanbecalculatedbymeasuringthemodulatedf requencyofalaser inputwithknownwavelength.IfaHelium-Neon(He-Ne)laserat632 .8nmisused,for example,itstypicalmodulatedfrequency, 8 say10kHz,correspondsto0.6328cm/sfor thescannervelocity. Now,havingacontinuoussourceasinput,allfrequencycompon entsaremodulated (typicallyinthekHzrange)accordingtoEq. 3.29 .Sincethesemodulationfrequencies areintheaudiorange,theycanbeeasilyampliedandltered electronically.Alowpassltereliminatesnoiseofmuchhigherfrequencythanthem odulationfrequency oftheshortestwavelengthinthespectrumandpreventsaliasing .Ahigh-passlter, ontheotherhand,maybeusedtoforceslowbackgroundmodulati on,suchassource ructuation,tobelowitscuto. Anotherimportantfactorinrapid-scansystemsisthedetermina tionofthecorrect timetostartdataacquisition.Thepositionofthecenterbursto fcontinuoussource canbeusedasareferencepoint.Thisinformationisessentialf ordatacollection,since themethodofanalysisreliesonthecumulativeaddition(coadding)ofanumberof 8 Someofmoderninstrument,suchasBrukerIFS66v/S,cansetthem odulatedfrequencyofHe-Nelasertoasfastas200kHz,whichcalculatesto12.6 6cm/softhescannervelocity.


45 interferograms.Co-additionisatechniquewhichimprovest hesignal-to-noise(S/N) ratio.Correspondingdatapointsmustbesampledatthesamepath dierenceonevery successivescan.Thezerocrossingpointsofthelaserinterferogra mmaybeusedasa referencepointtostartandtriggerthesamplingofanalogdat afromthedetector.The signalincreaseslinearlysinceitisalwayscoherent.Noiseoccu rsrandomly,thusthe signalincreasesfasterthanthenoise.Thisprocess,calledsignal averaging,increases theS/Nratioassquarerootofthenumberofscansco-added.The refore,inorderto increasetheS/Nbyafactorof2,thenumberofscansmustbeincre asedbyafactorof 4. 3.3PolarizationModulation Uptothispointofthissection,wehavebeentalkingaboutinte rferometersof amplitudeseparationtypewhichuseapartiallytransmittinga ndpartiallyrerecting beam-splitterwithMichelsonconguration.Despiteclearadv antagesoverdispersive typeofmonochrometer,amplitudeseparationinterferomete rsalsoexhibitsomedicultiesespeciallyforveryfarinfrared.Themaindicultyofth istypeistheloweciency andlimitedspectralrangeofthin-lmdielectricbeam-split terssuchasMylarwhichis mostlyusedforfarinfrared.Inordertoovercomesuchdisadvant ages,dierenttypesof interferometersweredeveloped.Oneofthemisalamellargr atinginterferometer.Rather thanseparatingamplitudeofincidentlight,alamellargrat ingseparateswavefrontof incidentlight.Theeciencyoflamellargratingbeam-split terisnearlyindependentof frequencyandcanbeveryhigh.Althoughthelamellargrating usesdierentmethod toseparatelightfromlmbeam-splitter,botharestillintensi tymodulationtypeof interferometers.Wewillnotdiscussthelamellargratinginte rferometerfurther,but interestedreadersareencouragedtoreadseveralpapersabou tthistopic[ 22 29 ]. Thereisanothertypeofinterferometerwhichiscommercial lyavailablethesedays. ItisthepolarizinginterferometerbasedonaconceptbyMart inandPuplett[ 30 ].It hassimilarcongurationasMichelson,butusesratheruniquea pproachtoproduce modulationofincidentlight.Aschematicdiagramofpolariz ing(orMartin-Puplett) interferometerisshowninFigure 3{7 .Lightfromanunpolarized(orpolarized)sourceis


46 M1 M2 P1 P2 B x Unpolarized (or polarized) source Detector Figure3{7:AschematicviewofaMartin-Puplettinterferome ter.Thecollimatedlight islinearlypolarizedatapolarizerP1andtravelstoapolar izingbeam-splitterBwhich isalignedatanangleof45 withrespecttotheplaneofpolarizationafterP1.The beam-splitterseparatestwopolarizationcomponentssending onecomponenttowarda xedrooftopmirrorM1andtheothertowardamovablerooftop mirrorM2.Onrerection,thepolarizationofeachbeamisrotatedby90 ,andtwobeamsarerecombined atthebeam-splitter.Atthebeam-splitter,theinitiallytra nsmittedbeamiscompletely rerected,andinitiallyrerectedbeamiscompletelytransmi tted.Therecombinedbeam headstothesecondanalyzingpolarizerP2andthebeamisline arlypolarizedafterP2 withanamplitudevaryingperiodicallywithpathdierence linearlypolarizedatapolarizerP1intheplaneatceratino rientation.Itisthendivided intotwopolarizationcomponentsbyapolarizingbeam-split terBwhichistypicallya free-standingnewiregrid(oragridonMylarlm).Thistype ofbeam-splitterhas beenshowntohavealmostfrequencyindependentandhighecie ncyofnearly100% fromeectivelyzerofrequencyuptoroughly1 = 2 d (cm 1 ),where d (cm)isthespacing ofthewires[ 31 ].Thegirdrerectsthecomponentoftheincidentlightparal leltothe directionofthewiresandtransmitthecomponentnormaltoth edirectionofthewires. Whenthebeam-splitterisorientedwiththewiregridsatanan gleof45 withrespectto P1,theincidentpolarizedlightisequallysplitsendingonec omponenttoaxedmirror M1andtheothertoamovablemirrorM2.BothM1andM2arethe90 rooftopmirrors whichrotatetheplaneofpolarizationby90 onrerection.Therefore,whentwobeams comebacktothebeam-splitter,theonererectedinitiallyis transmittedcompletely,and


47 0 I 0 I 0 2 (a)(b)(c) Figure3{8:Interferogramsproducedbyapolarizinginterf erometer.(a)TheinterferogramforparallelP1andP2.(b)Theinvertedinterferogramf orcrossedP1andP2.(c) Thedierenceof(a)and(b),whichisobtainedbypolarizati onmodulationtechnique. Notethatthemeanleveloftheinterferogramisautomaticall yeliminated. theonetransmittedinitiallyisrerectedcompletely.Nobeam issenttothedirectionof thesource.Thecombinedbeamnallypassesthroughthesecondpo larizerP2withits polarizationaxiseitherparallelorperpendiculartothat ofP1. AsM2moves,aphasedierenceisintroducedbetweentwobeams.Fo ramonochromaticsource,theinitiallylinearlypolarizedbeamisellip ticallypolarizedafterrecombinationatthebeam-splitterwithanellipticityvaryingperi odicallywithincreasingpath dierence.AttheZPD,therecombinedbeamhasthesamepolari zationastheincident beamonthebeam-splitter.AfterP2,thebeamisplanepolarize dwithanamplitude thatvariesperiodicallywithpathdierenceinthesamewaya sinaMichelsoninterferometer.Assuminganunpolarizedsource,theintensityatthed etectorisgivenby I k ( x )= I 0 2 [1+cos(2 x )](3.30) or I ? ( x )= I 0 2 [1 cos(2 x )] ; (3.31) where I 0 istheintensityofthelinearlypolarizedbeamincidentonth ebeam-splitter. Thecase I k isforparallelP1andP2,and I ? isforcrossedP1andP2.Forasourceof continuousspectrum,theoutputintensityyieldsantypicali nterferogramexceptthatof I ? isinverted(seeFigure 3{8 ).


48 Thecomplementarynatureoftheinterferogramsforthetwoo rientationsofone polarizerwithrespecttotheothercanbeutilizedtointrodu cepolarizationmodulation. ThisisusuallydonebykeepingP1xedandbydynamicallyswitc hingtheorientation ofP2usingapolarizingchopper. 9 Then,byusingstandardLock-intechnique,the detectedsignalisthedierencebetween I k and I ? ,whichisgivenforamonochromatic sourceas I ( x )= I k ( x ) I ? ( x )= I 0 cos(2 x ) : (3.32) Thus,thephasemodulationtechniqueeliminatesthemeanleve loftheinterferogram whichcouldintroduceerrorsduetospuriousructuations(see Figure 3{8 ).Thisand widerangehigheciencyofapolarizingbeam-splittermakes theMartin-Puplett interferometeradvantageousforveryfarinfraredmeasurem ents.Oneofspectrometers weusedisthepolarizationmodulationtype.Detailsofthespe cicinstrumentwillbe discussedinChapter 6 9 WecankeepP2androtateP1justaswell.Ifwehaveapolarizedi nputsource, however,P1shouldbealignedsuchthatthemostlightcangothro ugh,anduseP2for thepolarizationmodulation.Thereisalsoatechniquecalle dadoublepolarizationmodulationwhichusestwopolarizingchoppersforP1andP2[ 32 33 ].


CHAPTER4 SUPERCONDUCTIVITY 4.1Introduction In1911,soonaftersuccessfullyliquifyingheliumin1908,Kamm erlinghOnnes discoveredthattheelectricalresistanceofmercurysuddenlyd ropstoanunmeasurablysmallvaluewhenitiscooledbelow4.2K[ 34 ].Thisphenomenonwasnamedas superconductivity.Insubsequentyears,manymoremetalsandme tallicalloyswere foundtobesuperconductingwhencooledtobelowacertaincri ticaltemperature T c In1933MeissnerandOchsenfelddemonstratedanotherbasicprope rtyofsuperconductor,perfectdiamagnetism,whichisknownastheMeissnere ect[ 35 ].In1935, F.LondonandH.Londondevelopedapurelyphenomenologicald escriptionthrough amodicationofanessentialequationofelectrodynamicsinsu chawaytoexplain theMeissnereect[ 36 ].Theypointedoutthatsuperconductivityisafundamentall y quantummechanicalphenomenonthatisobservedonamacroscop icscale,withan energygapbetweensuperconductingandnormalstate.Anotherp henomenological theorywasalsodevelopedbyGinzburgandLandauin1950[ 37 ].Then,nallyin1957, Bardeen,Cooper,andSchrieerproposedamicroscopictheory ofsuperconductivityas aphenomenonwhereelectronsformpairsandanenergygapdev elopsintheelectronic densityofstatesaroundFermienergy[ 38 ].Thisso-calledBCStheoryremainsasavalid microscopicexplanationformanysimplesuperconductors(BCS superconductors). Abreakthroughinsuperconductivityresearchoccurredin198 6whenBednorzand MullerfoundacopperoxidecompoundsoftheBa-La-Cu-Osyste mwhichsuperconducts atasubstantiallyhighertemperature( T c 30K)thanpreviouslyknown[ 39 ].With theirwork,aneweraofsuperconductivityopenedinthisclass ofmaterials(highT c superconductors)whichdiersfromconventionalBCSsupercon ductors.Nowweknow variousmaterialsthathave T c abovetheboilingpointofliquidnitrogen(77K). 49


50 Themicroscopictheoryofsuperconductivitycannotbedescrib edinthelanguage oftheindependentelectronapproximation,andreliesonfo rmaltechniques. 1 Itisquite extensiveandhighlyspecialized.Consequently,wewilllimit oursurveyofthetheory toqualitativedescriptionsofsomeofthemajorconceptswith intheframeworkofBCS theory.Detailsofthesubjectcanbefoundinmanyplaces[ 40 41 42 43 44 ].Inthe followingsection,wewillmerelysummarizeafewbasicpropert iesofsuperconductors. InChapter 7 ,wewillprovidebrieftheoreticalbackgroundfortheinfra redproperties ofsuperconductorsandtheeectsoflocalizationonsupercon ductivity,basedonthe workbyMattisandBardeen[ 45 ],andbyMaekawaandFukuyama[ 46 ],respectively.In Chapter 8 ,wewilldiscusstheoryofnonequilibriumsuperconductivity. 4.2FundamentalsofSuperconductivity 4.2.1FundamentalPhenomenaVanishingDCResistance Ofallthecharacteristicsofsuperconductors,theabsenceofan ymeasurableDC electricalresistanceisthemoststrikingphenomenon.Aboveacr iticaltemperature T c a bulksuperconductingspecimenbehavescompletelyasnormalm etalwithDCresistivity generallygivenby ( T )= 0 + BT 5 ; (4.1) wherethersttermarisesfromimpurityanddefectscattering, andthesecondterm fromphononscattering.Below T c ,themetalbecomessuperconductingwithnodiscernibleDCresistivity(zeroDCresistivity),andcurrentrows initwithoutany 1 Thesecondquantizationdescriptionofmany-bodysystemisusedt odescribethe BCStheory,includingtheenergyoftheBCSgroundstate,givi ngtheenergygapresultingfromtheelectronpairing.


51 Table4{1:Transitiontemperaturesforseveralsuperconducto rs.SomeofhighT c materialsarealsolisted. Element T c (K) Compound T c (K) High-T c T c (K) Mo0.92 NiTi10 Ba 0 : 75 La 4 : 25 Cu 5 O 5(3 y ) 30 Al1.2 NbN15.2 La 2 x Sr x CuO 4 38 In3.4 Nb 3 Sn18.1 YB 2 C 3 O 7 92 Hg4.1 Nb 3 Ga20.3 Bi 2 Sr 2 CaCu 2 O 2 85 Pb7.2 Nb 3 Ge23.2 Bi 2 Sr 2 Ca 2 Cu 3 O 10 110 Nb9.2 MgB 2 39 HgBa 2 Ca 2 Cu 3 O 8 133 dissipationofenergy.Thetransitionofabulkmaterialisusual lyabrupt,andhappensatverylowtemperature. 2 Table 4{1 liststhetransitiontemperaturesofseveral superconductors.Thefactthatthereisnomeasurableresistivit yallowsustopasslarge currentthroughasuperconductor,andinturn,tocreatelarg emagneticeld.However, ifthecurrentdensityexceedsacriticalcurrent J c ,asuperconductorrevertstoanormal conductor(Silsbeeeect). J c isrelatedtowhetherthemagneticeldcreatedbythe currentexceedsthecriticaleld H c abovewhichsuperconductivityisdestroyed.Inan ACelectriceld,superconductorsatnitetemperaturenolo ngerexhibitzeroresistivity. Theresistivityincreaseswithfrequency.However,attemperat ureswellbelow T c ,the resistivityisstillnegligibleprovidedthatthefrequencyis nottoolarge( = ~ ,where istheenergygap.). x 7.2.1 describestheresponseofsuperconductorsinACelds. MeissnerEect Asuperconductorexpelsmagneticrux,andhenceactslikeape rfectdiamagnet. Thisisanotherpeculiarphenomenonofsuperconductivitykn ownastheMeissnereect (orMeissner-Ochsenfeldeect).Having =0intheMaxwellequationsforaperfect conductor,wend @ B =@t =0,andthusthemagneticruxisexpectedtoremain unchangedwithinthespecimen.Insuperconductingstate,howe ver,wendnotonly B =constant,butalso B =0,andtheeldpenetratingthespecimen(providedthat 2 Thethermalenergy k B T c correspondingtothetransitiontemperatureisontheorderofafewmeVorless.Thisismuchsmallerthantheenergyscalesu chastheFermi energy E F ( 10eV)andtheDebyeenergy ~ D ( 0 : 1eV)ofthemostofmetals.


52 itisnottoostrong)priortomakingthetransitiontosupercond uctingstatewillbe expelledfromtheinterior.Asimpleexplanationforthisee ctisthattheimpinging magneticeldinducesshieldingcurrentsonthesurfaceofasup erconductor,which arejustenoughtocanceltheeldintheinterior.Sincethesup erconductorhaszero resistivity,thecurrents( i.e. ,supercurrents)willpersistevenaftertheeldstopped changing.Thedistancewhichthesupercurrentsformaniteshe athintothespecimen iscalledtheLondonpenetrationdepth L .Magneticruxcanalsopenetratethesame distanceintothematerial.Formanysimple,puremetals,thisp enetrationdepthison theorderof500 A.Asmentionedabove,iftheeldgetstoolarge,however,them aterial willeventuallyloseitssuperconductingstate.MagneticFluxQuantization Itisanotherpropertyofsuperconductorsthatthemagneticr uxpassingthrough anyareaenclosedbyasupercurrentinaclosedloopcanonlytake onvaluesofintegral multiplesoftheso-calledruxquantum(orruxoid): 0 = hc 2 e =2 : 0679 10 7 Gcm 2 ; (4.2) where h isPlanck'sconstant, c isthespeedoflight,and e istheelementarycharge. Thisruxquantizationisaconsequenceofthatthecomplexord erparameter ( r )= j j e i (introducedintheGinzburg-Landautheory)isasingle-valu edfunction,andthus itsphasemustchangeby2 timesaninteger. AsimilareectoccurswhenatypeIIsuperconductor(referto x 4.2.3 )isplaced inamagneticeld.Atsucientlyhigheldstrengths,someofthe magneticeldmay penetratethesuperconductorintheformofthinthreadsofma terialthathaveturned normal.Eachthreadisinfactthecentralregion(\core")of vortexofthesupercurrent, andcarriesasingleruxquantum.JosephsonEects Whentwosuperconductorsareinweakcontact( e.g. ,separatedbyathininsulating oxidebarrier(10 20 A),anormalconductinglayer(100 1000 A),oraconstriction),


53 Cooperpairscouldtunnelfromonetotheother,givingriseto acharacteristiccurrent throughtheso-calledJosephsonjunction. Whenthereisnovoltagedropacrossthejunction,aDCcurrent willbegenerated, givenby I = I c sin '; (4.3) where isaconstantphasedierence( i.e. ,relativephase)oftheBCSmany-body statesintwosuperconductors,and I c isthemaximumcurrentthatcanpassthroughthe junctionbeforedrivingittoaresistivestate.Thisiscalledt heDCJosephsoneect. WhenaDCvoltage U isappliedbetweenthejunction,therelativephase evolves withtimeas ( t )= (0) 2 eU ~ t: (4.4) ThisgivesrisetoanACcurrentgivenby I = I c sin[ (0) !t ] ; (4.5) with = 2 eU ~ : (4.6) ThisiscalledtheACJosephsoneect.Interestinginterferenc ephenomenaarisewhen twoJosephsonjunctionsareconnectedinparallel.Thesecanbe usedforaverysensitive magneticeldsenor,knownastheSQUID. Thesearetheeectswherethecharacteristichighcoherenceo ftheCooperpairs becomesparticularlyevident.IsotopeEect Whenaconstituentatomofasuperconductingmaterialisrepla cedbyitsisotope, thecriticaltemperatureoftenchangeswithatomicmass M inaccordancewiththe relation M T c =constant ; (4.7)


54 where =1 = 2forthesimpliedBCSmodel.Aswillbeshownlater,thesimpleBC S theorypredictsthat T c isproportionaltotheDebyefrequency D ( ~ D isameasure ofthetypicalphononenergy),andthusthisisexpectedsince theDebyefrequencyis proportionaltothesquarerootoftheatomicmassforasimpleme tal. 4.2.2ThermodynamicProperties Thetransitionofametalfromitsnormalstatetoitssupercondu ctingstateisa thermodynamicphasetransition.Therefore,somesortofchange smaybeexpectedin thermodynamicquantitiesasaspecimenmakesitstransition. Theelectronicpartof specicheat,forexample,increasesdiscontinuouslyat T c fromthelineartemperature dependenceobservedinnormalstate( T>T c ),andthenatverylowtemperaturessinks tobelowthevalueofthenormalphase.Attemperatureswellbe low T c ,thespecicheat decreasesexponentially. Inthermodynamics,wecanuseafreeenergy F ( T )todescribethestabilityofa systematagiventemperature.Naturally,thesystemtendstochan geitsstatetoward thelowerfreeenergy,andbecomesstableataminimumofthefr eeenergy.Belowthe transitiontemperature,thefreeenergyinthesuperconducti ngphase F s isreduced belowthatinthenormalphase F n ( i.e. F s

55 Thus,from H c ,wecancalculatethefreeenergydierence.ThenusingEq. 4.8 ,we candeduceaseriesofthermodynamicalpropertiesincluding thedierenceinentropy betweentwophases,thelatentheatofthetransition,andthedi scontinuityinthe electronicspecicheat.Thelatentheatofthetransition L vanishesforthetransition inzeroeld,andthusthesuperconductingtransitioninzero eldisofthesecond order.Whenamagneticeldispresent,however,thereisalat entheat,andthenature ofthetransitionchangestotherstorder.Thedierence( F s F n ) T =0 iscalledthe condensationenergy.4.2.3TypesofSuperconductor Superconductorsarecategorizedasbeingoneoftwotypes:ty peIandtypeII.Type Isuperconductorsaremostlythesimple(nontransition)metals andmetalloidswithlow T c .TheBCStheoryexplainsthesesuperconductorsquitewell.Ty peIIsuperconductors, incontrast,aremorecomplex(transitionmetals,intermetall iccompounds,highT c andetc.),andoftenhaveahigher T c .Oneofthemaindierencebetweentwotypes ofsuperconductorsisthemannerinwhichpenetrationoccurs withincreasingexternal magneticeldstrength.Itgenerallydependsalsoontheshapeo fthespecimen,but thecleardistinctioncanbedemonstratedwiththesimplestgeom etryofalongcylinder (diameter > penetrationdepth)withitsaxisparalleltotheappliedel d. TypeI Withtheappliedeldbelowacriticaleld[ H

56 Withmorecomplexgeometries,theeldsatsomemacroscopicpor tionsofspecimen necessarilyexceed H c ,andtherefore,thesampleexhibitsanintermediatestatewit h somepartsbeingnormalwhilethereststayingsuperconducting.TypeII FortypeIIsuperconductors,therearethreedistinctphasesdep endingonthe strengthoftheappliedeld.Belowalowercriticaleld H c 1 ( T ),thereisnoruxpenetrationjustasfortypeImaterial.Whentheappliedeldexce edsanuppercriticaleld H c 2 ( T ),thereiscompleteruxpenetration,andthespecimenbecome stotallynormal. However,whentheappliedeldisinbetween H c 1 ( T )and H c 2 ( T ),thereisapartialpenetrationofruxintothespecimendevelopingarathercomplic atedmicroscopicstructure ofbothsuperconductingandnormallyconductingregions.Thi sphaseisknownasthe mixedstate(orShubnikovphase).Theruxpenetratesinthefor mofthinlaments (referredasvortexlines).Inthecoreofalament,theeldi shighandthematerialis normal.Eachlamentissurroundedbyasuperconductingscreen ingcurrentandenclosesexactlyoneruxquantum 0 .Currentrowsthroughthesuperconductingregions andthusthematerialstillhaszeroresistance.Thevortexline srepeloneanotherdue tothemagneticforcebetweenthem,andthustheyarrangethe mselvesintoanordered arrayofatriangularlattice.Withincreasingexternaleld ,thedistancebetweenthe vortexlinesbecomessmaller,andat H c 2 theyoverlapcompletely.Themagnetization curveoftypeIIsuperconductorsisquitedierentfromthato ftypeI.Upto H c 1 4 M riseslinearlywiththeappliedeld H justliketypeI.At H c 1 partialpenetrationbegins, andthemagnetizationdecreasesmonotonicallywithincreasi ngelduntilitvanishes completelyat H c 2 .IncontrasttothebehavioroftypeI,thetransitionisnotabr upt. 4.2.4LengthScalesLondonEquationandPenetrationDepth InaneorttodescribetheobservedbehavioroftheMeissnereec tcorrectly,F. andH.Londonsuggestedacondition(knownastheLondonequatio n)thatthelocal magneticeld h ( r )andthecurrentdensitycarriedbysuperconductingelectron s j s ( r )


57 satisfy 3 r j s = N s e 2 mc h ; (4.9) where m istheeectivemassofthesuperconductingelectrons,and N s isthesuperconductingelectrondensity(orsuperruiddensity). 4 Thispurelyphenomenological equation,togetherwiththeMaxwellequation r h =4 j s =c yields r 2 h = h 2L ; (4.10) r 2 j s = j s 2L ; (4.11) wherethelengthscale L ,knownastheLondonpenetrationdepth,isdenedby L ( T )= mc 2 4 N s ( T ) e 2 1 = 2 : (4.12) Eqs. 4.10 and 4.11 allowustocalculatethedistributionofeldsandcurrentsw ithina superconductor.Forthesimplestgeometryofasemi-innitesupe rconductorsoccupying thehalfspace z> 0,thesolutionsoftheseequationsdecayexponentiallyshowin g thatbothmagneticeldsandcurrentsinsuperconductorscan existonlywithinalayer ofthickness L ofthesurface.Therefore,theLondonequationimpliesthatw hena superconductorisinanexternalmagneticeld,thesurfacecu rrentrowsinathinlayer andkeepstheinterioreld-free; i.e. ,theMeissnereect. CoherenceLength TheLondonequation(Eq. 4.9 )assumesthatthecurrentdensity j s ( r )atone point r isrelatedtotheeld h ( r )(orthevectorpotential A ( r ))atthesamepoint. 3 Fieldsandcurrentsareassumedtobeweakandslowlyvaryingont helengthscale ofthecoherencelengthofthesuperconductor. 4 TheLondonbrothersincorporatedthetwo-ruidmodelofGort erandCasimir[ 47 ]. Themodelseparatesthetotaldensityofconductionelectrons N intoadensityofsuperconductingelectrons N s (superruiddensity)andadensityofnormalelectrons N n (normalruiddensity)suchthat N = N s + N n N s N as T 0,and N n = N when T>T c .Theyassumedthatonlythesuperruidparticipatesinasupercur rentwhilethe normalruidremaininertat T

58 Thus,theLondonequationisalocalequation.However,itismo regeneraltoassume that j s ( r )atonepoint r willdependonthevectorpotential A ( r 0 )atallneighboring points r 0 .Inordertodescribethisnon-localeects,Pippardmodiedt heLondon equation,andintroducedalengthscale 0 ,suchthat j r r 0 j 0 [ 48 ].Thisdistance 0 isoneoffundamentallengthscharacterizingasuperconduct or,andisreferred toasthecoherencelength.Inonecontext,itisusedforthedi stanceoverwhich thedensityofsuperconductingelectrons N s variessignicantly(fromzerotofull thermodynamicvalue).Inanothercontext,itisusedasthespa tialextentofthepair wavefunction( i.e. ,thesizeofaCooperpair).Inpurematerialswellbelow T c ,however, bothcoherencelengthdenitionshavethesamevalue;usingth euncertaintyprinciple, Pippardestimatedthecoherencelengthtobe 0 = ~ v F 0 ; (4.13) where v F istheFermivelocity,and 0 istheenergygaparoundtheFermisurfacein thesuperconductingstateatabsolutezero.NotethatthePippar d'scoherencelengthis independentoftemperature. IntheLondonmodel,itwasassumedthatthedensityofsupercondu ctingelectrons N s wouldhavethefullthermodynamicvaluerightfromthesurfac e( i.e. 0 =0).Since j s and h varyonascale L ,wemightexpectthattheLondon'smodelisvalidonlyfor L 0 .Infact,thisisthecase,andPippard'snon-localmodelredu cestotheLondon modelinsuchalimit.Thematerialsthatsatisfythiscondition ( L 0 )arethetype IIsuperconductors,andEq. 4.12 accuratelycalculatethepenetrationdepthforthetype II. IntypeImaterials,ontheotherhand,thepenetrationdepthi smuchshorter thanthecoherencelength( L 0 ).Thus, N s doesnotreachitsfullvalueoverthe penetrationdepth.Thisimpliesthatnotalloftheelectron swithinthethickness 0 fromthesurfacecontributetothescreeningcurrents.Forthese materialstheLondon equationisinadequate.Inordertocalculatethepenetrati ondepthinthetypeI materialsmoreaccurately,thePippard'snon-localmodelh astobeused,andarigorous


59 calculationgives 3 =0 : 62 2L 0 : (4.14) Consequently,theeldpenetratestypeImaterialsdeeperth antheLondonvalue. Ginzburg-LandauTheory AnotherphenomenologicalapproachproposedbyGinzburgandL andau(GL) describessuperconductivityintermsofacomplexorderparam eter ( r )= j ( r ) j e i whosemagnitude j ( r ) j isameasureofthesuperconductingorderatposition r below T c [ 37 49 ].Theorderparameter iszeroabove T c andincreasescontinuouslyasthe temperaturefallsbelow T c .Thephysicalsignicanceof ( r )wasnotclearatthetime theGLtheorywasdeveloped,butnowwecaninterpretitasawa vefunctionofa particleofmass m ,charge q ,anddensity N ,whicharegivenby m =2 m; (4.15) q =2 e; (4.16) N = j j 2 = N s = 2 ; (4.17) where m e ,and N s aretheeectiveelectronmass,electroncharge,andsuperruid density,respectively. IntheGLformalism,twotemperaturedependentcharacteristi clengthsareintroduced:thecoherencelength ( T )andthepenetrationdepth ( T ).TheGLcoherence lengthdenesthelengthscaleoverwhich ( r )varies,andisgivenby ( T )= ~ 2 2 m j j 1 = 2 ; (4.18) where isatemperaturedependentcoecientinthe j j 2 termofthefreeenergy (See[ 40 ]).ItiscloselyrelatedtothePippardcoherencelength 0 denedinEq. 4.13 .In weakelds,theGLpenetrationdepthisgivenby ( T )= m c 2 4 q 2 j 2 0 j 1 = 2 ; (4.19) where 0 istheequilibriumorderparameterwellinsidethematerial.


60 Forapurematerialnear T c ,themicroscopiccalculationintheBCSapproximation gives ( T )=0 : 74 0 1 T T c 1 = 2 ; (4.20) ( T )= 1 p 2 L (0) 1 T T c 1 = 2 : (4.21) Sincebothdivergesinthesamewayas T T 0 ,itispracticaltoformtheirratio = ( T ) ( T ) : (4.22) Theratio isknownastheGinzburg-Landauparameterofthematerial.F orapure material,thisisgivenby =0 : 96 L (0) 0 : (4.23) Thedierenceinthebehavioroftwotypesofsuperconductors inamagneticeld dependsonwhetherthecreationofinterfacesbetweennorma landsuperconducting regionsisenergeticallyfavorable,ornot.Penetrationof magneticeldreducestheeld energypenaltyimplicitintheMeissnereect.Thus,amaterial withlarge favors interfaces.Largecoherencelengthmeansagreaterextentof thesuperconductingstate. Therefore,withtheassociatedenergygainfromthesupercondu ctingcondensation energy,alarge opposesinterfaces.Interfacialenergychangessignat =1 = p 2.When < 1 = p 2,thematerialisthetypeI,while > 1 = p 2,thematerialisthetypeII.Inthe limitof 1,theGLtheoryreducestotheLondontheory. ElectronMeanFreePath Anotherimportantlengthscalecharacterizingasuperconduct oristheelectron meanfreepath l inthenormalstateduetoelasticscatteringbydisorder.Inthe presenceofdisorder,wecandeneaneectivecoherencelengt h ( l ),whichisvalidat absolutezero, 1 ( l ) = 1 0 + 1 l : (4.24) Dependingofthesizeof l relativeto 0 ,wecanthinkoftwolimitingcasesconcerning thepurityofasuperconductor:cleanlimitanddirtylimit[ 44 ].Uptothispoint,we


61 haveconcernedonlypurematerials,andthemeanfreepathhas playednoroleon determiningthecharacteristiclengthscales.Inreality,how ever,theactualvaluesof bothcoherencelengthandpenetrationdeptharesomewhatmod iedfromthevalues denedabovebymeanfreepatheects. Inthecleanlimit( l 0 ),Eq. 4.24 gives ( l )= 0 : (4.25) Thus,thecoherencelengthandthepenetrationdepthwediscusse dabovecanbeused withoutanymodication. Incontrast,Eq. 4.24 givesforthedirtylimit( l 0 ), ( l )= l: (4.26) Thus,thecoherencelengthatabsolutezeroiscompletelydete rminedbythemeanfree path,thelengththatgovernsthetransportpropertiesofthe materialinthenormal state.Inthislimit,therelationshipbetweencurrentandmag neticeld( i.e. ,vector potential),andinturn,magneticpenetrationdeptharemod ied.NotethattheLondon penetrationdepth L giveninEq. 4.12 isanexpressionforpuremetals.Thedirtylimit expressionofthemagneticpenetrationdepthis[ 40 ] = L 0 l 1 = 2 ( l 0 ) : (4.27) Thus, increasesas l becomesshorter.Attemperaturesnear T c ,thecoherencelength andthepenetrationdepthinthedirtylimitaregivenby[ 40 ] ( T )=0 : 85( 0 l ) 1 = 2 1 T T c 1 = 2 ; (4.28) ( T )=0 : 62 L 0 l 1 = 2 1 T T c 1 = 2 : (4.29) Thus,theGLparameterforadirtymaterialisgivenby =0 : 75 L (0) l : (4.30)


62 Consequently,asthemeanfreepathbecomesshorter,thecoher encelengthbecomes smallerthanthatgiveninEq. 4.13 andpenetrationdepthbecomeslongerthanthe London'sdenedinEq. 4.12 .Infact,itfrequentlyhappensthatalloyingapuretypeI superconductortransformsitintoatypeIIsuperconductor.Ma nyofsuperconductorsin theformofthinlms,nottomentionthoseinamorphousform,ar einthedirtylimit. 4.2.5BCSTheory In1957,Bardeen,Cooper,andSchrieer(BCS)proposedamicr oscopictheory ofsuperconductivity(nowknownastheBCStheory)[ 38 ].Acentralresultofthe BCStheoryistheexistenceofanenergygapbetweentheelectr onsysteminthe superconductinggroundstateandtheexcitedstates.Here,wewil lonlydescribethe underlyingideas,assumptions,andmajorpredictionsassociated withthetheory, withoutanyrigorousmathematicaldetails.CooperPairs Thegroundstate( T =0K)ofanon-interactingFermigasofelectronscorresponds tothesituationwhereallelectronstateswithwavevector k withintheFermisphere (with E F = ~ 2 k 2 F = 2 m )arelledandallstatesoutsideareempty.Ifapairofelectro ns isaddedinstatesjustabove E F ,thetotalenergyofthesystemshouldincreasebythe kineticenergyofthepair.However,Cooperrecognizedthati fthereisanattractive interactionbetweentheelectrons,nomatterhowweakitis,th eywillformabound state,andaddingapairofelectronsmayreducethetotalener gy(kineticpluspotential energy). 5 Thus,thenormalstatebecomesunstabletotheformationofthese paired boundstates[ 50 ]. 5 Notethatthesetwoadditionalelectronsarepreventedfromin dividuallyhaving energylessthan E F bythePauliexclusionprinciple.


63 SinceelectronshavearepulsiveCoulombinteraction, 6 theattractiveforcemust, intheory,comefromsomeinteractionbetweentheelectronst hatismediatedbysome othermechanisminherentinthematerial.Cooperarguedthat continuousexchangeofa virtualphononbetweenelectronsoccupyingstates k 1 and k 2 providesamechanismfor aweakattractionthatresultsinthereductionoftotalenerg y. 7 Theprobabilityofthe energy-reducingphononexchangeprocessesismaximumforthe case k 1 = k 2 = k .It isthereforesucienttothinkthatelectronswithequalando ppositewavevectorsform apair.Thisso-calledCooperpaircanberepresentedbyatwo-p articlewavefunction givenby ( r 1 ; r 2 )= X k a k e i k ( r 1 r 2 ) ; (4.31) whichissymmetricinspatialcoordinates( r 1 ; r 2 )uponexchangeofelectrons1and2. Therangeofsummationisconnedto E F < ~ 2 k 2 2 m

64 BCSGroundState TheformationofaCooperpairleadstoanenergyreduction.I narealmaterial, manymoreelectronsparticipateintheCooperpairingtoach ieveanewlower-energy groundstate.BecauseaCooperpairiscomposedoftwofermionsw ithoppositespin, itmaybeconsideredasasingleentitythatobeysBose-Einsteinst atistics.Thus,at T =0K,allCooperpairscondenseintoanidenticaltwo-electro nstateeventhoughthe individualelectronsarebeingscatteredcontinuallybetwe ensingleelectronstates.The Pauliprinciplelimitsthestatesintowhichthetwointeract ingelectrons,whichmakeup thepair,maybescattered. Inordertocalculatethegroundstate( T =0K),BCSmadeseveralassumptions forsimplicity.Firstofall,justlikeCooperdid,theBCStheor yisbasedonthefree electronapproximation.Thus,theFermisurfaceisspherical. Theyalsosimpliedthe netattractiveinteractionbetweenelectronsbyexpressingt hematrixelementthat describesscatteringoftheelectronpairfrom( k ; k # )to( k 0 ; k 0 # )andviceversa as 9 V kk 0 = 8><>: V=L 3 for j k j ; j k 0 j < ~ D 0otherwise ; (4.33) where V isapositiveconstant, L 3 isthevolumeofthesystem, ~ D istheDebyeenergy, and isthekineticenergyrelativetotheFermilevel,whichisde nedas = ~ 2 k 2 2 m E F : (4.34) Further,theyapproximatedtheBCSgroundstatevectorofthe many-bodysystem ofallCooperpairsbytheproductoftheidenticalpairstatev ectors.Withallthe assumptions,theydeducedthegroundstateenergyofthesupercon ductor,andshowed 9 Insomematerials( e.g. ,HgandPb),theelectron-phononinteractionisverystrong. Insuchcase,thenetinteractionbetweenelectronsisquiteco mplexandevenretarded. Sincephononsaretheoriginofthecoupling,thephononstruc tureofeachmaterialwill alsoinruencethematrixelement.TheextensionoftheBCStheo rytostrong-coupling superconductorsisknownasthestrong-couplingtheory[ 44 ].


65 NE ()/ N (0) EE F E F + D 0 E F + D 0 1 NE ()/ N (0) EE F 1 E+(T) F D E-(T) F D(a) (b)electron-likethermalquasiparticles hole-likethermallquasiparticles NTkT qB ()~exp(-2/) D 0 Figure4{1:(a)DensityofstatesinBCSgroundstate(at T =0K)relativetothatin normalstate.Agapof2 0 isdevelopedaroundtheFermilevel.Theshadedregionare thestatesoccupiedbysuperconductingelectrons.Notethatnost atesarelostinthe phasetransition.(b)Correspondingdensityofstatesatnitete mperature( T

66 0.0 0.2 0.4 0.6 0.8 1.0 D ( T )/ D (0)T / T c Figure4{2:Temperaturedependenceofthegapasafunctiono ftemperatureinthe BCSapproximation.BCSPredictions Attemperaturesabove T =0K,thermallyexcitedphononsbecomeavailableto scatterelectronsinaCooperpair.Thus,thereisaniteproba bilityofndingelectrons inthestatesabovethegap2(seeFigure 4{1 ).Theseexcitedelectronsarecalled quasiparticles.Asthetemperatureincreases,pair-breakingis progressivelyenhanced, andnallyCooperpairsceasetoexistat T c .Inameantime,thegap2shrinksas T increases,andcompletelyclosesat T c .IntheframeworkoftheBCStheory,onecan deducethetemperaturedependenceofthegap( T )asthesolutionto 1 N (0) V = Z ~ D 0 d p 2 + 2 [1 2 f ( p 2 + 2 )] ; (4.37) where f ( E )istheFermifunction.Notethatfor T =0, f ( E )iszero( E beingpositive), andonerecoversEq. 4.35 .Thisequationdenesanimplicitrelationbetweenand T andnumericalanalysisyields( T )asshowninFigure 4{2 .Asthegureshows,thegap developsquicklyasthetemperatureisloweredbelow T c ,andopensupalmostfullybya halfof T c .Itisalsoconvenienttorememberthattheresultofnumerical analysiscanbe


67 approximatedby ( T ) (0) = vuut cos 2 T T c 2 # : (4.38) Attemperaturesnear T c ,thevalueofthegapcanbeapproximatedby ( T ) (0) 1 : 74 1 T T c 1 = 2 : (4.39) Bysetting=0inEq. 4.37 ,anequationfor T c isdeterminedby 1 N (0) V = Z ~ D 0 d tanh 2 k B T c : (4.40) For ~ D k B T c ,numericalcalculationyields k B T c =1 : 14 ~ D e 1 =N (0) V : (4.41) Then,bycomparingEq. 4.36 withEq. 4.41 ,onecanndtherelationshipbetweenthe groundstategap 0 [ (0)]and T c intheweak-couplinglimitas 2 0 k B T c =3 : 52 ; (4.42) whichisfreefromparameterssuchas V and D 11 Thisrelationshipagreesquitewell (withinabout10percentwithtunnellingexperiments[ 52 ])fortheweaklycoupled superconductors.Forthosestrong-couplingsuperconductors( e.g. ,HgandPb),theratio islargerthan3.52.Forexample,theratioforHgandPbwereex perimentallyfoundto be4.6and4.3,respectively.Thestrong-couplingtheory[ 44 ]providesbetteragreement. Atnitetemperature,thequasiparticle-occupationofthee xcitedone-electronstates E =( 2 + 2 ) 1 = 2 obeysFermistatistics.Therefore,thedensityofquasiparticle sata 11 TheDebyetemperature D (hence,Debyefrequency D )maybededucedfrom specicheatmeasurement,butthematrixelement V isdiculttocalculateprecisely. Therefore,parameter-freeexpressionaredesired.


68 functionoftemperatureisgivenby N q = X k f k = Z 1 1 f ( E ) N ( ) d 2 N (0) Z 1 0 d 1+exp( p 2 + 2 =k B T ) : (4.43) 4.2.6EliashbergFormalism IntheBCStheory,thedynamicinteractioninducedamongele ctronsbyphonon wascrudelyrepresentedbyusingadimensionlessconstant N (0) V asthestrengthof electron-electroninteractionandtheDebyeenergy ~ D asthemaximumphononenergy; thedetailsoftheelectron-phononcouplingwasnotconsider edatall.Eventhough thetheorywasquitesuccessfulforweak-couplingsuperconduct ors,itfailedtotreat strong-couplingsuperconductorsaccurately.Eliashbergtoo kamoregeneralapproachto theelectron-phononcouplingbytakingintoaccounttheret ardednatureofthephononinducedinteractionandbyproperlytreatingthedampingof theexcitations[ 44 53 54 55 ]. TheEliashbergtheorystartswithtwononlinearcoupledequat ionsforthegap andtherenormalizationfactor Z ,whichreplacetheBCSgapequation.IntheEliashbergequations,twoparameters, and ,areintroduced,where isknownasthe Coulombpseudopotential(ortherenormalizedCoulombinter actionparameter)which describesaresidualrepulsivescreenedCoulombinteraction[ 56 ],and isknownasthe electron-phononcouplingconstant 12 whichisrelatedtotheattractiveinteraction.The constant isdenedby 2 Z 1 0 2 (n) F (n) n d n ; (4.44) where 2 (n)istheeectiveelectron-phononinteraction,and F (n)isthephonondensity ofstateswithnthefrequencyoftheexchangedphonon.IntheE liashbergtheory, 12 Theparameter isalsoknownasmass-renormalization(ormass-enhancement) parameterbecausetheeectiveelectronmassismodiedbythe electron-phononinteractionas Z =1+ = m =m .Superconductorsarecharacterizedaccordingtothe magnitudeof ,weak-coupling( 1),intermediatecoupling( 1),strongcoupling ( > 1).


69 2 (n) F (n)(theelectron-phononspectraldensity)isanimportantfu nctionthatcontains alltherelevantinformationabouttheelectron-phononint eractionthatgivesrisetothe eectiveattractiveinteractionbetweenelectronsaround theFermienergy.Inprinciple, superconductingtunnellingmeasurementsprovidedirectinf ormationon 2 (n) F (n). Forasimplemodelofametal, 2 (n) F (n)canbeapproximatedatlowfrequenciesby thequadraticform b n 2 ,where b isaconstantcharacteristicofagivenmaterial.Forthe strong-couplingsuperconductors, 2 (n) F (n)showssignicantlowfrequencystructure. FromtheEliashbergequations,McMillandevelopedamuchmore quantitative equationfor T c thantheBCSresult(Eq. 4.41 )[ 57 ].TheMcMillanformulaimproved laterbyAllenandDynesisgivenby[ 58 ] k B T c = ~ n ln 1 : 2 exp 1 : 04(1+ ) (1+0 : 62 ) ; (4.45) wheren ln isacharacteristicphononfrequencydenedby[ 58 ] n ln exp + 2 Z 1 0 ln(n) 2 (n) F (n) n d n : (4.46) AlsofromtheEliashbergequations,theratio2 0 =k B T c isexpressedapproximately by[ 55 ] 2 0 k B T c =3 : 53 1+12 : 5 T c ln 2 ln ln 2 T c # ; (4.47) where ln = ~ n ln =k B .Thisequationincludesstrong-couplingcorrectionsinter msofthe parameter T c = ln ,andtheuniversalBCSvalueisrecoveredfor T c = ln 0.Itshows excellentagreementwithexperiment.


CHAPTER5 SYNCHROTRONRADIATIONANDPUMP-PROBETECHNIQUE 5.1SynchrotronRadiation 5.1.1Introduction Itiswellknownthatanacceleratedchargedparticleemitse lectromagneticradiation[ 9 ].Synchrotronradiationisradiationemittedbyachargemo vingatrelativistic speed.Itisaverystable,highrux,broadbandlightsource.Ina ddition,ithaspeculiar propertiessuchaspolarization,pulsedtimestructure,angul arcollimation,andsmall sourcesize.Itwasrstidentiedasatechnicalprobleminacce leratorphysics,butits propertiesmakesynchrotronvitalforvariouseldsofscienc e. Synchrotronfacilitiesavailablearoundtheworldarebased ontheuseofan electronstoragering,aclosed,high-vacuumchamberwithanu mberofcirculararc andstraightsegments.Inthissectionthetheoryandproperties ofsynchrotronradiation aresummarizedinparticularforanelectronmakingacircula rtrajectoryatadipole bendingmagnetsectionofastoragering.Synchrotronradiati onfromso-calledinsertion devices,suchaswigglersandundulatorsplacedatstraightsegm ents,willnotbe discussedhere[ 59 60 ]. 5.1.2RadiatedPowerfromaBendingMagnet Forasinglenonrelativistic( v c )acceleratedparticlewithcharge e ,thetotal instantaneousradiatedpowerisgivenbyLarmorformula: P = 2 3 e 2 c 3 j v j 2 ; (5.1) where v istheaccelerationofelectron. Larmorformulacanbegeneralizedforarbitraryvelocities byaseriesofLorentz transformations.Foraparticleofmass m incircularmotionwithvelocity = v=c energy E ,andradiusofcurvature(thebendingradius) ,therelativistic( v c ) 70


71 generalizationoftheformulais[ 9 59 60 61 ] P = 2 3 e 2 c 2 4 r 4 (5.2) where r = E=mc 2 .Theemittedpowerisproportionaltothefourthpowerofthe energy andinverselyproportionaltotherestmass.Thispropertyexpla inswhyelectronsare usedratherthanotherheavierchargedparticlessuchasproto ns.Whenelectrontravels aroundastoragering,itmakesacirculartrajectoryandemit sradiationonlywhileit experiencesmagneticeldateachbendingmagnet. 1 Thus,thetotaltimeforittohave radiative-energylossperrevolutionis2 =c .Therefore,theradiative-lossperturnby oneelectronis E = 4 3 e 2 3 r 4 : (5.3) Forahighlyrelativisticelectrons( 1)Eq. 5.3 isexpressedinpracticalunitsas 2 E (keV) 88 : 5 [ E (GeV)] 4 (m) : (5.4) Thesynchrotronradiationspontaneouslyemittedfrommanyele ctronsinrandom distribution(asfromastoragering)isgenerallyincoherent .Figure 5{1 schematically showsthissituationaswellascoherentradiationwhichcanbe foundforexampleinthe coherenttelahertzemissionfrommicro-bunchedelectronsor inthefreeelectronlasers (FEL).Inincoherentcase,thetotalradiatedpowerby N electronsintheringissimply NE=T where T istheperiodofelectroncirculationaroundthering.There fore,the totalpowerradiatedbyringwithringcurrent i isgivenby 3 P ring (kW) 88 : 5 [ E (GeV)] 4 (m) i (A) 26 : 5[ E (GeV)] 3 B (T) i (A) : (5.5) 1 Thebendingradius isrelatedtothemagneticeld B ofthebendingmagnet, whichisgiveninpracticalunitsas (m) 3 : 336 E (GeV) =B (T).FortheVUVringwith E =0 : 808GeVand B =1 : 41T,thebendingradius is1.91m. 2 FortheVUVring, E 20keVperelectronperrevolution. 3 FortheVUVring, P ring 20kW/ampofbeam.


72 (a) Incoherent ENEPNP incoherentsingleincoherentsingle ~~ 1/2 (b) Coherent ENEPNP coherentsinglecoherentsingle ~~ 2 Figure5{1:(a)Incoherentradiationfrom N -electronsinrandomdistribution,and(b) coherentradiationfrommicro-bunched N -electrons. Acceleration Electron orbit (a)<<1 b (b)~1 b q y Bending radius q Figure5{2:Angulardistributionofradiationemittedfroman electronmovingalonga circularorbit.Thisequationtellsusthatthetotalintensitydeliveredtoe achbeamlineisproportional tothebeamcurrent,thusthecurrentsignalcanbeusedtonorma lizemeasuredspectra inordertocompensatetimevaryingintensityduetodecayofbe amcurrent. 5.1.3AngularCollimationandPolarization Foranelectroncirculatingatnonrelativisticspeed,theang ulardistributionof emissionisadipolepatternwhichextendstoalargerangeofan gles(seeFigure 5{2 ). Forarelativisticelectron,however,theradiationisstrong lyconcentratedtoanarrow angularrangearoundadirectiontangentialtotheorbitassh owninFigure 5{2 .The divergenceoftheverticalangle isroughlyestimatedfrom r 1


73 Theinstantaneouspower(incgsunitsoferg/[secradcm])radi atedperunit wavelengthandperunitverticalangleaccordingtotheSchw ingertheory[ 61 ]isgivenby d 2 P ( ; ;t ) dd = 27 32 3 e 2 c 3 c 4 r 8 (1+ X 2 ) 2 K 2 2 = 3 ( )+ X 2 1+ X 2 K 2 1 = 3 ( ) ; (5.6) where X = r ; = c (1+ X 2 ) 3 = 2 = 2 ; andthesubscripted K 'saremodiedBessel functionsofthesecondkind.Theparameter c iscalledcriticalwavelengththat characterizesthespectraloutputofparticularstoragering whichisgivenby 4 c =4 = 3 r 3 : (5.7) Halfthetotalpowerisemittedasphotonsofwavelengthshorte randhalflongerthan c Eq. 5.6 isthebasicformulaforthecalculationofthecharacteristic softhesynchrotron radiation. Thebendingmagnetradiationhasapeculiarpolarizationpr operty.Thetwoterms inthesquarebracketsofEq. 5.6 areassociatedwiththeparallelandperpendicular componentsoftheemittedpower,respectively.Atsmallverti calangles theradiation ispredominantlypolarizedinthedirectionparalleltothe electron'sorbitalplane,andat =0,itiscompletelylinearlypolarized.Astheverticalangl eincreases,perpendicular componentstartsshowingup,buttheparallelcomponentisalw aysthelargeroftwo. Bothcomponentsarephasecorrelated,andasaconsequence,th eemissionobserved aboveandbelowtheelectron'sorbitalplane( i.e. 6 =0)isellipticallypolarized.In Figure 5{3 thenormalizedintensitiesoftheparallelandperpendicula rcomponentsare plottedasafunctionof fortheVUVringofNSLSatthreedierentphotonenergies. Thisgureshowsthattheradiationisstronglyconcentrateda tthecriticalwavelength, buttheverticalspreadincreasesatlongerwavelengths. 4 Alternativeparameterusedforthesamepurposeisthecriticalp hotonenergywhich isgivenby h c = hc= c =3 hcr 3 = 4


74 010203040 0.0 0.2 0.4 0.6 0.8 1.0 dash://solid: ^ blue:100cm-1green:1000cm-1red: lcNormarizedIntensityy [mrad] Figure5{3:Angulardistributionofparallelandperpendicul arpolarizationcomponentsatthreedierentphotonenergies.Thecriticalwavele ngthoftheVUVringis19 : 9 A( 5 ; 000 ; 000cm 1 ). 5.1.4RFCavityandPulsedNature TheacceleratingeldsinsidetheRFcavitysystemperiodicall yactsonthecirculatingelectronstorestoretheenergylostduetoemission.Becauset heRFeldoscillates, onlyelectronsarrivingataparticulartimereceivethepro peracceleration.Thisleads toformelectronbuncheswhicharecontainedinregularlyspa ced,imaginarycontainersso-called\RFbuckets"[ 21 ].Therefore,thelightproducedbythesynchrotronis pulsed.Thispulsedtimestructureofthesynchrotronradiation wasexploitedinour time-resolvedmeasurements.Fortheordinarylinearspectrosco picexperiments,thetime constantofcommondetectorsaremuchlongerthanthepulserep etitionperiodofthe radiation.Forexample,themostcommonlyusedfarinfraredde tectorsisabolometer. Itisathermaldetectorwithatypicaltimeconstantontheord erofmilliseconds.Thus suchadetectorseespulsesjustassteady-statesourceoflight. ThemaximumnumberofbucketsisdeterminedbytheRFfrequen cy rf ofthe cavityandthetime T 0 (orcircumference D )foranelectrontomakeonerevolution


75 aroundthering,whichisgivenby N max b = rf T 0 = rf D v ; (5.8) where v isthevelocityoftheelectron.Anyintegralnumberofbucket ssmallerthan N max b canbelledwithelectronsarbitrarily. 5 Withinabunch,electronsaredistributedrandomly,andther eisaslightspreadin energyfromthatoftheaverageelectronwhichistravelling aroundtheidealelectron pathatthereferencecenterofthebunch.Allelectronsinabu ncharemovingatthe samespeed( v c )andsubjecttothesameLorentzforcewhilepassingthrougha bendingmagnet.However,theelectronwithslightlyhigher(l ower)energyhaslarger (smaller)mass.Asaconsequence,ithasslightlylonger(shorter)o rbitallengththan thereferenceorbit,andthusarriveslater(earlier)thant hereferenceelectron.The acceleratingeldactstoelectronsinsuchawaytobringthee nergyofallelectrons closertothatofreferenceeverytimeabunchentersthecavit y.Figure 5{4 schematically illustratestheeldfoundbyelectronsarrivingatcavityat dierenttimes. TheRFsystemisdesignedtoregainonlytheenergylostbyradiati onforthe referenceelectron,butmore(less)energyforelectronsarri vingearlier(later).This causeslongitudinaloscillationsaboutthecenterofthebunc h,whichisreferredas synchrotronoscillations. 6 5.1.5BeamLifetime EventhoughtheelectronenergyismaintainedbytheRFsystem, theelectrons haveanitelifetimeduetotwomajormechanisms[ 59 ]:thescatteringofelectrons byresidualgasparticlesinthevacuumchamberandTouscheke ect(discussedbelow).Therefore,rellingofelectronsareregularlyschedu ledeveryafewhours.As 5 FortheVUVring, N max b =9. 6 Besidesthesynchrotronoscillations,electronsinabunchmake transverse(bothhorizontalandvertical)oscillatorymotion,whicharecalledt hebetatronoscillations.Pairs ofquadrupolemagnetsareusedtofocuselectronstowardther eferenceorbital.


76 RFsystem 4thharmonicsystem RF+4thharmonic RF voltage Time RF voltage Time Accelerating voltagefound by referenceelectron 0 0 Figure5{4:Theacceleratingvoltageasafunctionoftime.T hetime=0correspondsto thearrivalofthereferenceelectron.TheRFsystemisdesigned toregainonlytheenergylostbyradiationforthereferenceelectron,butmore(l ess)energyforelectronsarrivingearlier(later).Therighthandsideshowstheeectofu singthehigherharmonic cavitysystemusedinconjunctionwiththemainRFsystemintheeo rttoincreasethe lifetime.The4thharmonicsystemisshownhere.Eachbunchseesa ratvoltagewhich stretchesthebunchlength.explainedintheprevioussubsection,theelectronsoscillate aroundthereferenceorbitwhileorbitingaroundthering:betatronoscillations(t ransverseoscillations)and synchrotronoscillations(longitudinaloscillations).TheTo uschekeectiscausedby thescatteringbetweentransverselyoscillatingelectronsinsi deeachbunch.Thistype ofelectron-electronscatteringconvertspartofthetransve rsemomentumintolongitudinalmomentumthatmodiesthetimeatwhichtheelectron enterstheRFcavity. Then,thoseelectronswhichgainedlargeenoughlongitudina lmomentumarenolonger properlyaccelerated,andcanbelostfromthebunch.Theeec tismoreseverewhen electronsarepackedtighter.Rightafterelectroninjecti on,electrondensityisthehighest.Therefore,theTouschekeectisthedominantlifetimeli mitingmechanismatthe earlystageofbeamcurrentdecaywithtime-dependentdecayt ime.Astheelectron densitydecreases,thescatteringbyresidualgasparticlesstart stotakeover,andatthis time,thedecaybecomesexponentialthatisrepresentedbyasi nglecharacteristicdecay time. AhigherharmonicRFcavitycanbeusedtorattenthepotential inthemain RFbucketcausinganincreaseinthebunchlengthwithaconseque ntreductionof


77 intrabeamscatteringandanimprovementintheTouscheklifet ime.Figure 5{4 shows theeectofusingthehigherharmoniccavitysystemusedinconju nctionwiththemain RFsystemintheeorttoincreasethelifetime.5.1.6InfraredSynchrotronRadiation TheEq. 5.6 canbeconsiderablysimpliedinthespectralrangewherewavel engths aremuchlongerthanthecriticalwavelength c .Thiscondition( c = 1)isusually satisedintheentireinfraredwavelengthsforthemoststorage rings,andwecanobtain usefulexpressionsvalidfortheinfraredsynchrotronradiatio n(IRSR)suchas[ 62 ] dP ( ;t ) d =8 : 6416 10 10 i 1 = 3 7 = 3 G W cm ; (5.9) d 2 P ( ; ;t ) d =0 =5 : 2 10 10 i 2 = 3 8 = 3 H W radcm ; (5.10) where and arebothincm, i (theringcurrent)inA,and inmrad.Thefunctions G and H aredenedas G = 1 2 : 193 1 r 2 2 = 3 ; (5.11) H = 1 6 : 312 1 r 4 4 = 3 : (5.12) For c = 1, G and H canbetakentobeunity. Theverticalopeningangleasafunctionofwavelengthisgiv enby ( )=1 : 66188 1 = 3 G [rad] : (5.13) Notethatistwiceoftheangledenedfortheangulardiverge nce (seeFigure 5{2 ). Thisisausefulexpressionwhenwedeterminethenaturalopenin ganglethatisnecessarytocollectthefull-widthathalf-maximumofthepowerp roleatagivenwavelength.Figure 5{5 showsintheinfraredspectralrangeasafunctionofwavenumb er usingtheVUVringparameters. 7 7 TheU12IR'srstmirroriscapableofcorrectinglightwith90m radofthevertical openingangle.


78 110100100010000 10 100 1000 Y [mrad]Frequency[cm -1 ] Figure5{5:ThenaturalopeningangleofIRSRusingtheVUVringp arameters. 5.1.7SourceComparison TheEq. 5.5 givesthetotalradiatedpowerfromastorageringinalldirec tions integratedoverentirespectralrange.Althoughthetotalpow ercertainlyindicates certainaspectoftheoutputcapability,itdoesnotdescribes superioritiesofsynchrotron radiationoverconventionalthermalsources.Forapractical pointofview,thespectral brightness b isamoreusefulsourcequalityparametersinceittakesintoacc ountthe sourcesizeaswellastheangulardistributionofsynchrotronra diation.Thebrightness ofalightsourceisdenedas b ( )= C F ( ) S n ; (5.14) where C isaconstant, F ( )istheruxofphotons, S isthesourcearea,andnisthe solidangleofemission.Itisintuitivelyobviousthatasourcew ithsmallersizeand divergencehashigherbrightnessjustlikealightbeamfromal aserisbrighterthan thatfromarameofcandle.Asmallsourcesizeallowsopticstofo cusphotonstoa diraction-limitedsize,andsmalldivergenceminimizesthe lossofphotonsevenwith reasonablysmallopticalcomponents.Therefore,abrightersou rcehasamarkedeect


79 110100100010000 10 -8 10 -7 10 -6 1x10 -5 1x10 -4 10 -3 SpectralPower[W/cm -1 ]Frequency[cm -1 ] Synchrotron Blackbody Figure5{6:Spectralpowercalculatedfora2000Kblackbody sourceandsynchrotron radiation.Forthesynchrotronradiation,parametersforth eVUVringisused.This showsthepoweradvantageofthesynchrotronradiationoverth ethermalsourceonlyin thefarinfraredregion.onimprovingsignaltonoiseratioforvarioustypesofexperim entssuchasmicroscopy andsurfacescience. Figures 5{6 and 5{7 showcalculatedspectralpowerandbrightnesscomparisons betweenconventionalthermalsourceandsynchrotronradiati on,respectively[ 21 ].Inthe plots,ablackbodysourceattemperatureof2000Kwithitssourc esizeof0 : 4cm 2 and solidangleof0.02sr(f/3.5)isused.Forthesynchrotronradiat ion,theparametersof theNSLSVUVringareused. Notethatthesynchrotronshowssignicantlyloweroutputpower thanthethermal sourceovermostofthespectralrange(betweenmid-IRandvisibl e)wheretheglobaris commonlyused(seeFigure 5{6 ).Thesynchrotronhasapoweradvantageonlyinthe veryfarinfrared( 100cm 1 ).Intermsofbrightness,thesynchrotronsourcehasan advantageoverentirespectralrangeshowninFigure 5{7 overthethermalsource,which isobviouslyattributedtoitssmallsourcesizeandangularcoll imation.


80 110100100010000 10 -9 10 -8 10 -7 10 -6 1x10 -5 1x10 -4 10 -3 10 -2 10 -1 SpectralBrightness[W/(srcm -1 )]Frequency[cm-1] Synchrotron Blackbody Figure5{7:Spectralbrightnesscalculatedfora2000Kblack bodysourceandsynchrotronradiation.Forthesynchrotronradiation,paramet ersfortheVUVringisused. Thisshowsthebrightnessadvantageofthesynchrotronradiati onoverthethermal sourceintheentirespectralrange. 5.2PrincipleofPump-ProbeStudies Thepump-probemeasurementisavaluabletechniquethatdete rminesthenonequilibriumstateofasystematvariousinstantsoftimeaftersomesort ofstimulushasbeen applied.Theprocessisrepeatedforawiderangeoftimevalue stobuildupacomplete historyofthesample'srelaxationprocesses,namelythedynamic softhesystem.There areavarietyofexcitation(pumping)methodscommonlyusedt hatprovideadequateenergydensitytocreatethedesireddensityofexcitationsinthe sample.Examplesinclude electricalcurrent,electriceld,magneticeld,orlight pulses.Herewewilldiscussthe principleofthetechniquethatusesnearIR/visiblelaserpulse sasexcitationsourceand synchrotronpulsesasprobe. Thepurposeofthissectionistoprovideaverysimpleideaofthe technique thatwouldbehelpfultoknowbeforegoingtothenextchapter .Thedetailsofthe experimentaltechniquearedescribedin x 6.6


81 Probe (IRSR) Pump (Laser) Sample Spectrometer 1 2 3 Figure5{8:Principleofthepump-probeexperiment.(1)Lase rpulsecreatesphotoexcitationsinsample,whichsubsequentlyevolvewithtime.(2)Afte rtime 4 t ,broadband IRpulsearrivesandispartiallyabsorbed(orrerected)byexc itations.(3)IRpulseanalyzedwithorwithoutaspectrometer,extractingdetailsofe xcitationsatatime 4 t after theircreation.5.2.1Laser-SynchrotronPump-ProbeMeasurement Synchrotronradiationisabroadbandbrightsourceoflight. Mostpeopleexploit itsbrightnessandoverlookitstemporalstructureoftheligh tpulses.Thepump-probe techniquedevelopedattheNationalSynchrotronLightSourc e(NSLS)ofBrookhaven NationalLaboratoryutilizesthepulsednatureofsynchrotron sourceespeciallyatfar infraredwhereitoersbothbrightnessandpoweradvantageo verconventionalthermal sources[ 1 ].Theshortpulsesoflaserlightareusedtoilluminateasample,a ndcreate photoexcitations.Theseexcitationsinthesamplebegintorel aximmediatelyafterthe arrivaloflaserpulse,andcanappearaschangesinthesample's opticalproperties.The synchrotronpulsearrivesatthesampleatsomepointintime t afterthepump,and analyzesthesample'sresponse( e.g. ,transmissionorrerection)atatime t intoits relaxationprocess.Theexperimentsareperformedbyxingth etimedierencebetween thelaser(pump)andthesynchrotron(probe)pulses,andthenmea suringaspectrum inthenormalway.Afastdetectorisnotrequired.Theentirep rocessrepeatsatahigh repetitionrate(10'sofMHz)inamannersimilartousingasynchr onizedstrobelightto freezeaparticularmomentofarepetitiveprocess,allowingt heslowlyrespondinghuman eyetoviewit.Thiswayacompletespectrumthatrepresentsamo mentarysnapshotof thesample'sstateforaparticular t canbemeasured.Varioustimedierencesbetween pumpandprobethusproduceasetofdataasafunctionoftimean denergyproviding greaterinsightintotherelaxationprocessofthesystem.Figur e 5{8 showstheprinciple oftheexperiment.


82 Themeasuredtemporalresponse S ( t )inapump-probeexperimentisdetermined bythesample'simpulseresponsefunction(thequantityofinter est)aswellasthe durationofthepumpandprobepulses.Whenthesamplehasalinea rresponse, S ( t ) isgivenas S ( t )= Z + 1 1 dt 0 Z t 0 1 dt 00 I probe ( t 0 + t ) I pump ( t 00 ) G ( t 00 ) ; (5.15) where I probe ( t )and I pump ( t )aretemporalintensityprolesoftheprobeandpumppulses, respectively,and G ( t )istheimpulseresponsefunctionofthesample.Notethatthe expressionassumesthatthereisnoself-excitationbyprobepulse s.Thisconditionis easytoachieveinpractice.Wecaneitherusemuchlessintensepr obepulsesthanthe pumppulseorlimitthespectralrangeoftheprobebelowsomepho tonenergythreshold usinganopticallter.For G ( t )= ( t ),theexpressionbecomesacrosscorrelation oftheprobeandpumppulsesthatdenestheminimumtemporalr esolution.Note thatthetemporalresolutionisindependentofthesensitivity andresponsetimeofthe spectrometeranddetector. Inourpump-probeexperiment,weusedpicosecondpumppulsetha thassignicantlyshortertemporalprolethanthesynchrotronprobepul seofthepulsewidthon theorderof1ns.Hencewecantake I pump ( t )= I 0 ( t ),andEq. 5.15 canberewrittenas S ( t )= I 0 Z + 1 1 dt 0 I probe ( t 0 + t ) G ( t 0 ) : (5.16) Thisshowsthatthemeasuredresponseisaconvolutionofthesynch rotronpulseshape withthesample'sresponse.Thenatureofdampinginastoragerin gleadstoaGaussian likeelectrondistributionwithinabunchandthusaGaussiansha pedprobeintensity prole.5.2.2InterferometryUsingPulsedSource Whenpulsedlightsourceisusedforaninterferometer,abeam-sp litterdividesevery pulseintotwopulses.Asascannermirrormoves,oneofthepulseisde layedintime relativetotheother,sothatthelighttravellingtowardsamp leanddetectorconsists oftwopulsesforeverypulseincidentofthebeam-splitter.Whe nthedelayisshorter


83 thanthepulseduration,thetwopulsesoverlaptemporally,an dcauseinterference.This situationisclearlythesameasthecaseofacontinuoussource.No w,ifwerunhighresolutionmeasurementstoresolveanarrowspectralfeature,o nemaythinkthatwe mayencounterasituationwherethescannermovesfarenoughth atthedelayexceeds thepulsedurationandtwopulsesnolongeroverlap.However,th isisnotthecase[ 1 21 ]. Asamplewithanarrowabsorptionfeaturewillautomaticallyl engthenashort pulse.Thelengthenedpulseswilloverlaptocauseinterferenc e.Theamountoflengtheningisequivalenttothepathdierencenecessarytoresolvet hefeature'sabsorption width.AFabry-Perotinterferometerisasimplewaytopictur ehowashortpulsecanbe lengthenedbyaresonance.Therefore,regardlessofthesharpn essofafeature,thereis nolimittothespectralresolutionusingshortpulsedsource. Wecanalsothinkofitinthefollowingway.Therstpulseisinci dentonthe sample.Ifthereisanarrowabsorptionfeature,thesampleabsor bsthatparticular fouriercomponentfromthepulse,andtheabsorptionmode\rin gs"forawhile(ring downtimedeterminedbythenarrownessoftheabsorptionmode) .Thesecondpulseis thenincidentonthespecimen,andthesamefouriercomponenti sabsorbed.Butifthe modeisalready\ringing"(fromtherstpulse),thentheparti cularfouriercomponent ofthesecondpulsemaybeatthewrongphase(thesecondpulsetries todriveatoms inonedirection,buttheyarealreadymovingintheopposited irectionduetotherst pulse).Varyingthetime-delayoftheinterferometerwillbr ingthepropermodeinand outofphase.Sointerferenceisobserved.Observableinterfer encewilloccuraslongas themodekeepsringing.Thisisconsistentwiththelongerpath dierencenecessary toachieveahigherspectralresolution.Thispathdierencec anbemuchlongerthan theoriginalpulse.Inthispicture,thesamplehasa\memory"d eterminedbythe narrownessoftherelevantabsorptionfeatures.Itremembersf oratimelongenoughto


84 spanthetimebetweenthetwooriginalpulses.Thereareotherwa ystopicturethis,but theresultisthesame. 8 5.2.3AdvantageofLaser-SynchrotronTechnique Thereareothersourcesoflightthatmaybeusefulasaprobe.Fo rexample, tunablepulsedlasers,free-electronlasers(FELs),opticalpar ametricoscillators(OPOs), coherentTHzpulsesfromabiasedsemiconductorsilluminatedby afemtosecondlaser arepossibleprobesources.Eventhoughthesecanhavehighertemp oralresolution thanthesynchrotron,theyhaveeitherrestrictedspectralran georstabilityissues. Asynchrotron,ontheotherhand,isabroadband,bright,andst ablesource.These propertiesmakethesynchrotronsuitableforordinaryspectro scopyoverabroadspectral range.Forthetime-resolvedstudy,thesynchrotroncanfollow thesystemthatrelaxes throughawiderangeofenergies.Aswillbedescribedinthefoll owingchapter,pulse widthandrepetitionfrequencies(PRF)aresomewhatadjustab leforvariousrelaxation timescales.Allofthesepropertiesactasadvantagesofusingsync hrotronasaprobing sourceevenattheexpenseoftemporalresolution.Thefactthat ourpumplaser (Ti:Sapphire)istunableinwavelength,PRF,andpower,add srexibilitytoourpumpprobesystemthatcanbeveryusefultoinvestigatethedynamicso fsystemswithtime scalesfrom 100psto 100ns. 8 TheexplanationgivenhereisbasedonaconversationwithG.L. Carr.

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CHAPTER6 EXPERIMENT 6.1Introduction Pump-probetimingexperimentswereperformedattheNationa lSynchrotronLight Source(NSLS)tostudylow-frequencydynamicsinsolids.Synchr otronradiationisa broadbandpulsedsource.Wetookadvantageofthispulsednatur etoobservethestate ofmaterialsexcitedbyalaserwhichisalsopulsedandsynchroni zedtothesynchrotron radiation.TheVacuumUltraviolet(VUV)ringattheNSLShastwoin fraredbeamlines, U10AandU12IR,dedicatedforsolidstatephysicsstudy.Thesearetw obeamlines usedforbothtime-resolvedandlinearspectroscopy,describedi nthisdissertation.This chapterwillstartwithadescriptionoftheNSLSfacilityfocusi ngontheproperties oftheVUVringandbeamlines.SpectrometersatU10AandU12IRandp umplaser systemarethreeprincipalpiecesofinstrumentation,andwill bedescribedseparately indetailfollowedbybriefdescriptionsoftheotherapparat ussuchasthecryostat,our home-madesamplechamber,superconductingmagnet,detector s,andberoptics.The experimentalsetupandtechniqueswillbediscussedtowardthee ndofthechapter. 6.2NationalSynchrotronLightSource 6.2.1General TheNSLSisauserfacilityfundedbytheU.S.DepartmentofEnerg y.Two separateelectronstoragerings,anX-Rayring(2.8GeV,300mA)and aVUVring (800MeV,1.0A),provideintenselightspanningtheelectromagn eticspectrumfrom theinfraredthroughx-rays.Thepropertiesofthislight,an dthespeciallydesigned experimentalstations,calledbeamlines,allowscientistsinma nyeldsofresearch toperformexperimentsnototherwisepossibleattheirownlabo ratories.TheNSLS currentlyhas56X-Rayand23VUVoperationalbeamlinesforperf ormingawiderange 85

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86 ofexperiments[ 63 ].Thepropertiesofsynchrotronradiationanditsadvantage sover othersourcesoflightaredescribedinthepreviouschapter. ElectronsareinjectedintotheNSLSstorageringsfroma750Me Vboostersynchrotronfedbya120MeVlinac.Theelectronsarerstproduce dina100keVtriode electrongun.Thegunispulsedattheboosterrevolutionperio d,94.6ns,seventimes perboostercycle.Eachpulseis5nslongandsuppliesabout17mi crobunchesinthe linac.Afteraccelerationinthelinac,thebeamisinjectedi ntotheboosteronseven successiveturns.Multi-turninjectionintheboosterisaccompl ishedinthefollowing way:Thebeamisderectedintotheboosterbyaseptummagnet.Th erstlinacpulse goesaroundtheboosterandreturnstotheinjectionpointjust asthesecondpulseis comingoutoftheseptum.Thetwopulsesmergeintoaboosterbunc handcontinueto circulate.Thisprocessisrepeateduntilallsevenlinacpulse sareinjected.Duringthe injectionprocess,theeldofapulsedmagnetpreventsthecirc ulatingbeamfromstrikingtheseptum.Theeldofthemainboostermagnetsisalsoincre asingslightlyduring injectiontoplaceeachlinacpulseonaslightlydierentorbi tfromitspredecessors. Afterinjection,themagneticeldoftheboosterincreasestom aintainaconstantorbit radiusastheradiofrequency(RF)acceleratingcavityboost stheelectronenergyto750 MeV.Atmaximumenergy,akickermagnetispulsedtosendthebeam pastaseptum andintotheX-RayorVUVstoragering.Afterextraction,thebooste rmagnetsramp downtotheirinjectionsettings.Theboostercycletakes1.2sec ondsfromoneinjection tothenext.Inthefuture,theinjectioncyclewillbedecrea sedto0.5secondsandthe maximumenergywillberaisedto800MeVforimprovedstorageri nginjection. 6.2.2VacuumUltravioletRing TheVUVringisthesmallerofthetwostorageringswithacircumfe renceof51.0 meters.Itcontains8dipolebendingmagnetsforforcingelec tronstotravelinaclosed orbitalongwith24quadrupoleand12sextupolefocusingmagne tsforthepractical operation.SeeAppendix A forparametersoftheVUVringaswellastheinjection system.Eachbendingmagnetallowstwobeamextractionportso neachofwhich singlebeamlineisattachedinordertodeliverthelighttoit sexperimentalendstation.

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87 Someofbeamlines,suchasU10,aresplitintomultiplebeamlines. Therearealsotwo insertiondevicesinastraightportionofthestoragering.The ringisnormallyoperated atanenergyof0.808GeVwithamaximumcurrent(averagedove rbunches)of1.0 A(1 : 06 10 12 electrons)atinjection.Electronscirculatetheringevery 170.2nsat relativisticspeed,andemitsynchrotronradiationateachben dingmagnet;thesource sizeinroutineoperationis536to568 mhorizontallyand170to200 mvertically. TheRFcavityoftheVUVringrunsatthefrequencyof52.887MHz,a ndit supports9RFbuckets,whicharedeterminedbytheringcircumf erenceandtheRF periodasdiscussedin x 5.1.4 ;anarbitrarynumberofbucketscanbelledwithelectron bunches.Thisleadstoavarietyofoperationmodestheringca noperate.Table 6{1 liststhemodescurrentlyavailablewiththeirparameters.In so-called9-bunchmode, allofthe9bucketsarelledproducing52.9MHzofthepulserep etitionfrequency (PRF),whichisthesameastheRFcavityfrequency.Thismode, however,isnotthe onecommonlyusedbecauseoftheinstabilitiesintheelectrono rbitcausedbyllingall buckets.Forday-to-dayoperation,7-bunchmodeisused.Itru nsatthesamemaximum averagecurrent(1000mA)asthe9-bunchmodeprovidinghighe stintensityoflight pulseat52.9MHztouserswithmaximumstabilityandreliability .The7-bunchmode hasseveraldierenttypes(stretched,detuned,compressed,an dwiggled)depending ontheshapeandmotionofelectronbunches,andthedetailsoft hesetypesaregiven below.Whenwesayroutineoperation(ornormalmode),itisth e7-bunchstretched mode.In3-bunchsymmetricmode,everythirdbucketislledp roducingthePRF of17.6MHz.Insingle-bunchmode,onlyasinglebucketislled, andthePRFisthe sameastherateatwhichsingleelectronbunchgoesaroundonec ycleofitsclosed orbit.Boththe3-bunchandsingle-bunchmodesrunatlowerav eragecurrentthanthe onefornormaloperationbecauseofbunchinstabilitycausedby llingmoreelectrons ( i.e. ,increasingCoulombinteractionbetweenelectrons)inagive nbucket.Runningthe ringatloweraveragecurrentcausestheloweroutputintensit yofemittedlight,and manyoftheotheruserswhodependdeeplyontheoutputintensit yofthebeamsuer fromthelowersignaltonoiseratiointheirdata.However,these modesareusefulto

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88 Table6{1:OperationmodesoftheVUVring.Inthellpattern,1 and0indicatelled andemptybucket,respectively. I max isamaximumcurrentatinjectionforeachmode, l b arangeofelectronbunchlength,PRFthepulsedradiationfre quency, T PRF thetime betweenpulses,and pw arangeofpulsewidth.Themodessuitedforthetimingexperimentsareindicatedbytheasterisk. Mode FillPattern I max (mA) l b (cm) PRF(MHz) T PRF (ns) pw (ns) 9b 111111111 1000 72-36 52.9 18.9 2.4-1.2 7b-stretched 111111100 1000 72-36 52.9 18.9 2.4-1.2 7b-detuned* 111111100 800 30-18 52.9 18.9 1.0-0.6 7b-compressed* 111111100 200 15-9 52.9 18.9 0.5-0.3 7b-wiggled 111111100 1000 72-36 52.9 18.9 2.4-1.2 3b-symmetric* 100100100 600 40-20 17.6 56.7 1.4-0.7 1b* 100000000 400 60-30 5.9 170.2 2.0-1.0 studythetransientphenomenaofthetimescalelongerthantheP RFperiodofnormal operation.Inordertominimizetheimpactontheotherusers,o perationmodesdierent fromthenormalonemustberequestedattheweeklyuser'smeetin g.Thetiming operation,whichusestypicallyeither7-bunchdetunedorco mpressedbyrequest,are scheduledaheadfortwodaysineverymonthfortheuserswhoper formtime-resolved experiments.Weutilizedthemostofthebeamtimesscheduledfo rthetimingandthe nightshiftsduringstudy,maintenance,andweekendsforthew orkdescribedonthis dissertation. Duringroutineoperation,anotherRFsystemcalledthe\4thha rmoniccavity"(a RFcavitywithitsresonantfrequency4timesthatoftheRF,52 .9MHz)isturnedon tostretchanelectronbunchinthelongitudinaldirectionof itsorbit.This,inturn,diminishesTouschekeect,andincreasesthelifetimeofelectro nsallowinghigheraverage beamcurrentandfewerlls.Abroaderelectronbunch,howeve r,implieslongerpulse widththatresultsinlowertimeresolutionsincethedurationo fthesynchrotronpulses determinesthetemporalresolutionofthetime-resolvedmeasu rementsasdescribedin thepreviouschapter.Forthe7-bunchstretchedmode,arange ofthepulsewidthis between2.0and1.0nsdependingonthebeamcurrentoftherin g.Thepulsewidth hereisdenedasthefullwidthathalfmaximum(FWHM)ofthepu lseintensity.See Table 6{1 forthepulsewidthoftheothermodes,too.

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89 192021222324 600 700 800 900 1000 1100 PulseWidth[ps]Time[hours] Figure6{1:Changeofthepulsewidthemittedfromall7bunche sduringdetunedmode. Asthecurrentdecays,electronsarere-injectedinthestorage ringat10:30PM. Inordertoprovidebettertemporalresolutionforthetime-r esolvedexperiments, theRFgroupattheNSLShasdevelopedtwospecial7-bunchmodes otherthanthe stretchedmode;detunedandcompressed.Inthe7-bunchdetuned mode,the4th harmoniccavityisturnedoleavingthebunchesunstretched providingthepulse widthofbetween1.0and0.6ns.Figure 6{1 showsthetimedependentpulsewidth, emittedfromall7bunchesduringthedetunedmode.Thismode canberunatafull current,buthasapproximatelyhalfthelifetimeofthatach ievedinnormal7-bunch stretchedmode.Inthe7-bunchcompressedmode,the4thharmoni ccavityisturned on,butruns180degreesoutofphasewithrespecttothephaseused forthestretched mode.Thiscompressesthebunches,andmakesthepulsewidthofbe tween0.5and 0.3ns;thebesttemporalresolutionachievableattheNSLS.Adra wbackofusingthe compressedmodeisthatitmustberunatconsiderablylowercurre ntcausingworse signaltonoiseratio(S/N).Hence,anappropriatemodemustbesele ctedforthetimeresolvedmeasurements,negotiatingtherequiredtemporalreso lutionandenoughS/N forthesystemunderinvestigation.The7-bunchwiggledmodeis understudy(atthe timeofwritingthisdissertation)forfutureoperations.This modeshakestheelectron

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90 buncheshorizontallyatacertainfrequency,andeectivel yremovestheringintrinsic interferencefringesthatshowupintheveryfarendofinfrar edspectrum(2to20cm 1 ) takenathighresolution( < 1cm 1 ).SeeFigure B{1 inAppendix B.2 .Thedetailofthe techniqueremainstobestudiedfurther.Inalltypesofthe7bunchmode,therearetwo emptybuckets,butthiswillnotcauseanyproblemtobothspectr oscopicandtiming measurements.Duringmeasurements,thestateofasystemisprobedb ycollecting thesynchrotronpulsesthathadinteractionwiththesampleatt hedetector.Since thedetectordoesnotseethepulseduringtwomissingpulses,there isnoeectonthe spectra. TheVUVringistheworld'sbrightestsourceofinfraredbecauseof itshighstored beamcurrent.Inaddition,thereal-timeglobalorbitfeedb acksystemimprovesthe stabilityoftheclosedorbit,byimplementingafeedbacksystem baseduponharmonic analysisoftheorbitmovementsandthecorrectionmagnetic elds.Thestablebeam iscrucialfortheacceleratorbasedFourier-Transformspectr oscopy.Thesetwofactors, brightnessandstability,alongwithitsrexibilityofoperat ionmodesmaketheNSLS VUVringauniquefacilityprovidingsolutionstomanyscientists forcertaintypesof experimentsotherwiseimpossibleinthelowendofthespectralr ange. 6.2.3BeamlinesU12IRandU10A Forthestudiesdescribedinthisdissertation(bothlinearandt ime-resolved measurements),twoofsixinfraredbeamlines(U12IRandU10A)that arededicated tosolidstatephysicsinvestigationswereused.RefertoAppendix B foradditional informationaboutotherNSLSinfraredprograms. U12IRwasdesignedspecicallyforoptimalperformanceinthef arinfrared[ 64 ]. Asexplainedinthepreviouschapter,synchrotronradiationi semittedintoincreasinglylargerangleasthewavelengthbecomeslonger.This,as aresult,requireslarger extractionopticsforecientlycollectingthefarinfrare d.U12IRhasagold-coated, 1 1 RerectorswithgoldcoatingareusedforexcellentIRrerecta ncewithUVrejection.

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91 Diamond window 1.1 cm aperture Collimating Box Light coneSpectrometerFar-infrared detector Mirror M2 & M315 cm aperture Rough vacuum UHV Mirror M1 6 cm aperture Orbit plane Ring chamber Figure6{2:TheelevationviewofU12IRbeamlinewiththeSPS200spectrometerattached.ThediamondwindowseparatesanUHVsectionandaroughva cuumsection. water-cooled,SiCextractionmirrorthathasanangularacc eptanceof90H 90Vmrad. Itcollects100%ofverticallyspreadinglightdowntoapprox imately30cm 1 ,butfalling to40%at2cm 1 .InFigure 6{2 ,M1correspondstothismirror.Thecombinationof M1andtwoaluminum-coatedPyrexmirror,M2andM3,directst hebeamtowardan 11mmaperture,wedgedCVDdiamondwindow 2 (17.7mmdiameter,0.5 wedgeangle, 0.35mmthick)thatseparatestheultrahighvacuum(UHV)oftheri ngside( 10 9 Torr)andaroughvacuumofthecollimatingopticsandendstat ion( 20mTorrbase). M1andM2areplanemirrors,whileM3isanellipsoidalmirrorth atfocusesthesource toapointjustbeyondthediamondwindow. 2 Polycrystallinediamondsynthesizedbychemicalvapordeposit ion(CVD)isa uniquematerialforopticalexperimentsbecauseofitsextre melygoodtransparency overawidespectralrange.Thisiscausedbythefactthatdiamo ndisapurelycovalent crystal,whichmeansthatitsopticalphononscannotinterac tdirectlywithlightwaves. Dependingonthelevelofpurity,diamondistransparentfrom itsfundamentalcut-oat 220nmtothefarinfrared.Thereisalsoanintrinsicmulti-pho nonabsorptionbandin thewavelengthrangeof2.5 mto6.7 m[ 65 ].

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92 110100100010000 Power/BW[ m W/cm -1 ]Frequency[cm-1] 0.0010.0100.1001.000 Energy[eV] 0.001 0.01 0.1 1 10Far-IR 1 mW Mid-IR 17 mW Near-IR 52 mW Vis-UV 87 mW Without light cone With light cone Figure6{3:Powertransmittedthroughthediamondwindowatt heU12IRbeamline withandwithoutthelightcone.Theabsorptionaround2600cm 1 isduetotwophononabsorptioninthediamondwindow.Thebandat12500cm 1 isduetoelectronic transitionsontheAlmirrors.Theedgeat 15000cm 1 isduetoabsorptioninthegold coatingoftherstmirror.Thebandgapofthediamondwindowi sresponsiblefora cutoat 40000cm 1 Atthelongestwavelengths,diractioncausesappreciableloss. Alargelightcone justupstreamofthediamondwindowhelpscollectingthelonge stwavelengthsand guidethemintothecollimatingmirrorbox.Theconeistaper edfroma62mmdiameter entranceaperturedowntothe11mmapertureofthediamondwi ndow,anditsinterior wallisgold-coatedforimprovedIReciency.Powerdeliver edthroughthediamond windowwithandwithoutthelightconeisshowninFigure 6{3 Thebeamthroughthediamondwindowiscollimatedbyaspheric almirror(8-inch focallength)inthecollimatingbox,andre-focusedbyanoaxisparaboloidrerector (6-inchfocallength)attheveryendoftheU12IRbeamlinejust infrontofendstation's entrance.Dependingonexperimentalrequirementsofeache xperiment,aparticulartype

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93 ofspectrometerordirectconnectionofourcustommadesamplec hamberisselected asanendstation.Detailsofspectrometersandthesamplechamb erwillbedescribed shortly. U10AisanotherinfraredportlocatedadjacenttotheU12IR,an dmayalsobeused forpump-probestudies.IthasaBrukerIFS66v/Sspectrometer. Bycombiningsynchrotronandspectrometer'sinternalsource,widerange,hig hbrightnessIRspectroscopy ispossibleatthisbeamline. TheOpticalcongurationoftheU10beamlineisfollowing.At wo-mirrorsystem (M1andM2)collectsandre-imagesthesynchrotroninfraredso urceatapointjust beyondasimilardiamondwindowastheU12IRwhichseparatesthe UHVofthering sideandaroughvacuumofamirrorbox.M1isagold-coated,wat er-cooled,plane extractionmirrormadefromsiliconwithanangularacceptan ceof40H 40Vmrad (100%verticalcollectiondownto240cm 1 ).M2isaglassellipsoidwithanaluminum rerectivecoating.Thedeliveredspectralrangeextendsfro mapproximately10cm 1 tobeyond40,000cm 1 .Atthemirrorboxtheinfraredbeamissplitintotwo(onefor U10AandanotherforU10B),thenthebeamsenttowardU10Aiscolli matedwithan aluminizedo-axisparaboloidtoadiameterofeither14mmo r8mmandtransported underroughvacuumthroughaKBr(orpolyethylene)window.T hecollimatedbeamis thenrefocusedintothespectrometerusingano-axisparaboli cmirror;thismirroris identicaltothemirrorusedtocollimatethelightfromthein ternalsourcesresultingin asymmetricarrangement,allowingtheusertochangebackandf orthbetweensources whiletheinstrumentisundervacuumwithoutanylossinalignm ent. 6.3Spectrometers In x 3.1 ,theclassicationofcommonlyusedspectrometerswaslisted(see Figure 3{1 ).TheNSLShasmanyspectrometers,includingallthreetypesof two-beam interferometer(Michelson,Lamellar,andMartin-Puplett) .Inthissectiononlythe specicinstrumentusedforourexperimentswillbedescribedin detail.

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94 6.3.1BrukerIFS66v/S TheBrukerIFS66v/S(Bruker66)isaFouriertransforminterf erometerwithrapid andstep-scanoptions.Withproperchoiceofthesource,beam-spl itter,anddetector, itcancoverthefullspectralrangefromtheveryfarinfrared ( > 5cm 1 )uptothe vacuumUV( < 55,000cm 1 )withspectralresolutionof 0.1cm 1 .Thefriction-free airbearingscannermakesitpossibletoachieveverystablerapi dscanattherate greaterthan100spectra/sec.DigitalSignalProcessing(DSP)e lectronicsprovide precisescannercontrolandinstrumentautomationforsource,a perture,anddetector selections.Beam-splittersarechangedmanually.Combiningt heveryfastscanrate capabilitywithsuperiorprecision,spectroscopywithhighsign altonoiseratio(S/N)is possibleeveninthefarinfrared.Theinstrumentoperatesunde rvacuum( < 3mbar) torecordspectrafreefromabsorptionfromH 2 OandCO 2 vaporinthefarandmid infrared.SeeAppendix D formoreinformationaboutthisgas-phasecontamination. Thespectrometerhasanexternalporttoextractcollimatedl ight,whichallowsusers toattachspecializedexperimentalsetup;suchasacustommadesa mplechamberora magnet. AHe-Nelaser(633nm,nominal1mW)isusedtocontrolthepositiono fthe movingmirror(thescanner)andtocontrolthedataacquisitio nprocess.ThemonochromaticbeamproducedbythisHe-Nelaserismodulatedbytheinter ferometertoproduce asinusoidalsignal.Aphotodiodedetectorisplacedatbothout putsoftheinterferometer:laserdetectorAandB.Signalsfromthesetwodetector saremonitoredwith anoscilloscope,andthetheamplitudeofsignalsareusedtoopti mizethealignmentof thebeam-splitter.Whenthebeam-splitterisnotalignedprop erly,theamplitudecan becometoosmalltocontrolthescanner,andthendataacquisiti onwillbeinterrupted. Therearethreeinternalsources(Hgarclamp,SiCglobar,andT ungstenlampfor thefarinfrared,midinfrared,andvisibleregions),aswella sthebroad-bandexternal synchrotronsource.Figure 6{4 andTable 6{2 showthefrequencyrangesofthevarious sources.

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95 104 40 100 1000 10000 40000 13000 250004000400 UV VISNIRMIRFIR WAVENUMBER (cm) -1EMISSION 1m m 250nm 10m m 100m m 1mm WAVELENGTH (HeNe)LASER =0.6328m lm Tungsten Globar Hg-Arc Figure6{4:EmissionspectraofthreeinternalsourcesoftheBru ker66. Table6{2:FrequencyrangesoflightsourcesusedfortheBruke r66. Source Range(cm 1 ) PrimaryApplication Mercury-arclamp 10-700 FarIR SiCglobar 100-6000 MidIR Tungstenlamp 4000-40000 VIS Synchrotron(U10A) 2-40000 IRtoVIS Avarietyofbeam-splittersanddetectorscancoverwholeran geofspectrum. Tables 6{3 and 6{4 showsthefrequencyrangesofsetsofbeam-splittersanddetect ors, respectively,availableforthebenchweused.Detailsofvari ousdetectorswillbe describedin x 6.5.4 TheBruker66isaveryreliablespectrometerforwiderangeof spectrum,and thusitisthemainspectrometerusedinourworkatthebeamline sU10AandU12IR. However,thebeam-splittereciencyaswellasdiractionloss duetospectrometer's smallopticssetthelimittothelongestwavelengthofspectrumt hatwecanmeasure condentlywiththisspectrometer.Withthethickest(50 m)Mylarbeam-splitter availableathandandverysensitive1.5Kbolometer,theBruke r66attachedonthe U12IRbeamline,whichisthebetteroftwobeamlinesforthefa rinfrared,iscapableof

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96 Table6{3:Frequencyrangesofbeam-splittersusedfortheBru ker66. Beam-splitter Type Range(cm 1 ) PrimaryApplication Mylar3.5 m T201 125-750 FarIR Mylar6 m T202 80-450 FarIR Mylar12 m T203 40-220 FarIR Mylar23 m T204 20-110 FarIR Mylar50 m T205 10-50 FarIR Ge/Mylar T222 30-680 FarIR Ge/KBr T303 370-7800 MidIR Ge/KBr(WideRange) T304 400-10000 MidIR QuartzVIS T501 9000-25000 VIS Table6{4:FrequencyrangesofdetectorsusedfortheBruker6 6.PEinthewindow collumstandsforpolyethylene. Detector Window Temperature(K) Range(cm 1 ) PrimaryApplication SiBolometer Quartz+scatterlayer 1.8(LHe-pumped) 2-100 FarIR Si:BBolometer PE 4.2(LHe) 10-600 FarIR DTGS PE 300(ambient) 10-600 FarIR DTGS KBr 300(ambient) 400-7000 MidIR Si:B KRS-5 4.2(LHe) 350-4000 MidIR Ge:Cu KRS-5 4.2(LHe) 350-4000 MidIR MCT 77(LN 2 ) MidIR InSb Sapphire 77(LN 2 ) 1850-15000 NearIR SiPhotodiode 300(ambient) 9000-28000 VISandUV GaPPhotodiode 300(ambient) 18000-33000 VISandUV producingacceptablespectrumdownto 20cm 1 .Theuseoffar-IRlowpasslter suchasFluorogold(thecut-ofrequencydependsofthethick nessofthelter)pushes thelimitdownbyimprovingtheS/Neventhoughtheoverallen ergygoingintothe detectordecreases.Witha50cm 1 cut-olter,areasonablespectrumdownto 12cm 1 wascondentlyobtained.ThereisathickerMylarbeam-split ter(125 m) availableonthemarket.Thismayimprovetheperformanceat evenlongerwavelengths. TheVUVringisrigidlyattachedontheexperimentalroorwhere therearemany sourcesoflowfrequencynoises(mostnotably,mechanicalnoise duetothepumps). Thesecouldcausemodulationoflightatthosefrequencies,anda retranslatedinto noiseinthespectrum.Thismayposesomeproblemswhiletakingda taespecially inthefarinfrared.Asmentionedabove,theBruker66isabenc hthatiscapableof scanningatexceptionallyfastspeed.Bychoosingthescanspeedfa stenough,wecan pushtheseundesirablelowfrequencynoisesbelowthespectralra ngeofinterest.For example,ifweuse200kHzofthescanvelocity,60Hznoiseshowsupat 4.74cm 1 in

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97 thespectrum,andinfactthistechniquewasusedwhenwewerefo cusingonthespectral rangebetween10and20cm 1 usingtheBruker66. Foreachmeasurement,oneneedtospecifyascanvelocity.Hereis atechniqueI personallyusetodetermineanoptimalscanvelocity.Iwoulduse thefastestvelocity thatdoesnotsignicantlyreducethesignalattheupperfrequ encyrange.Onewayto determinethisistostartatamodestscanvelocity,say20kHzfort hefarinfrared,and increasethisinmultiplesoftwo.Whenthesignalintheupperh alfofthespectralrange fallsbyafactorof1.4( i.e. ,dropsto70%ofthepreviousscanathalfthevelocity),then wehavejustfoundtheoptimalvelocity.Wecanestimateitbased onthemodulation frequencyoftheupperspectralendandcompareittothedetec tor'stimeconstant.At fasterspeeds,thelossinsignalexceedstheincreasednumberofsca nssothattheS/N pertimeisnowfalling.Atslowerspeeds,thesignaldoesnotincr easeverymuch,butwe getfewerscanspertime,sothenoisestartstoincrease(assumingit israndomnoise). 6.3.2BrukerIFS125HR TheBrukerIFS125HR(Bruker125)isanotherFouriertransform interferometer thatwasattachedontheU12IRbeamlineduringthemaintenanc eperiodinDecember 2003.Thisinstrumentoerstheultimatelyhighresolution(r esolvedlinewidthsof < 0.001cm 1 )acrossentirespectrumfrom5cm 1 inthefar-IRto50,000cm 1 inthe UV.Thehighresolutionisachievedbyitsextraordinarylongsca nner( 5m)armthat createslongpathdierence(retardation).Itsglide-bear inginterferometerwithhybrid scannerconstructionenhancesvelocitystability.Thesourcec hambercanaccommodate upto4internalsourcesaswellastheexternalsynchrotronsour cethroughasource inputport.Thereare4internaland2externaldetectorsacc essiblethrougheachoftwo dierentsamplecompartmentswithoutbreakingvacuum.Itha smodularconstruction justlikeolderBrukerIFS113vsothatuserscanremovewholesamp leanddetector compartmentstosetuptheirownexperiment.Thereareafewex tractionportsthrough whichthebeamcanbebroughtintoothercompartmentssuchasa custommadesample chamber,microscope,andmagnet.Withalltheadvantagestha tU12IRoers,this rexibilityaswellasitscapabilitytoresolvehighlycomple xspectraintodiscretelines

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98 5.7m 2.22m Sources Spectometer Sample Detectors Scanner arm SynchrotronFigure6{5:DiagramoftheBruker125. makesU12IRtheuniqueplacethatcouldattractmanyuser'scon siderationfortheyears tocome.Figure 6{5 showsthediagramoftheBruker125. Thisspectrometeruses80mmoptics,approximatelytwiceaslar geasthosein theBruker66(40mm)collectingmorelightoflongwavelengt hs.Thescannerand xedmirrorarethefullcubiccornermirror.Wealsohave125 mMylarbeamsplitter(usefulrangefrom5to22cm 1 )fortheBruker125.Useofthisbeam-splitter, synchrotronsource,andlargeopticscertainlyimproveperfo rmanceatlowfrequency achievingreasonablespectrumdownto 5cm 1 6.3.3SciencetechSPS-200 TheSciencetechSPS-200isaMartin-Puplett(roofmirrorpo larizing)interferometerthatcanoperateinthreemodes:Michelson(amplitudemod ulation),Polarizing (polarizationmodulation),orMixed.Inmostcase,weoptedto usethePolarizing modeforitsbetterperformanceatthelongestwavelength.Re ferto x 3.3 fortheprincipleofpolarizationmodulationtechnique.Figure 6{6 showstheopticallayoutofthe spectrometer.IntheMichelsonmode,thepolarizingchopperC 2andpolarizerP1are removed,andintensitybeam-splitterisused.ThereareMylarb eam-splittersofseveral

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99 S1 M1 C1 I1 M2 M3 P1 BS1 M4 TRAVEL400mm M5 M7 C2 M6 Tilt Rotary Shear Lateral Shear ExternalAdjustments S2 Sample and detector Beamline Figure6{6:SchematicofopticallayoutoftheSciencetechS PS-200.BS1polarizingor intensitybeam-splitter;C1intensitychopper;C2polarizing chopper;I1iris;M1concave mirror;M2andM6ratmirrors;M3andM7paraboloidalmirrors;M 4andM5rooftop mirrors;P1xedpolarizer;S1Hg-Xelamp;S2synchrotronlightso urce. dierentthicknessavailable.Theoperationislikethatofa standardMichelsonasin Bruker66.Thelowestfrequencyobtainablewiththismodeisl imitedmainelybythe beam-splittereciency,butitcouldmeasureupto1250cm 1 thatisdeterminedby thesamplingfrequencyaccordingtotheNyquisttheorem.Inthe polarizingmode,the intensitybeam-splitterisreplacedbyapolarizingbeam-spli ttermadeofanaluminum gridof2500lines/cm(4 mpitch)onathinMylarsubstrate.Atlowfrequency,the beam-splittereciencyisalmost100%:theonlylimitingfact orsarethesizeofthe mirrorsandthesource'spowerspectrum.Usingasensitivedetecto rlike1.5KSi:B bolometerandlteringouthighfrequencies,thelowestfrequ encyof2cm 1 areachievableforthisinstrument.Synchrotronbrightnessispreserved bythelargeinputoptics(9 cmdiameter f= 2).Athighfrequenciestheeciencyofthebeam-splitterand polarizers fallstozeroatthemaximumfrequencyproportionaltothespa cing.TheSPS-200can takespectraupto600cm 1 ,butatthatfrequency,theBruker66outperformsthe SPS-200,andthusweusedthisinstrumentmainlybelow100cm 1 .Aninputpolarizer

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100 P1andapolarizingchopperC2areadded.Theinputintensityc hopperC1isremoved. Mixedmodeisbarelyused.Itusesamplitudemodulationwithpo larizers(including beam-splitter)inplace.Itpasseslowersignalattheexitthane itherpuremode,butthe spectralrangeisthecombinationofbothmodes. Inthegure,M4andM5aretheroofmirrors.M4cantravelupto4 00mm ( i.e. ,80cmofretardation)achievingamaximumresolutionof0.00 6cm 1 .Thishigh resolutioncapabilityrevealedverynespectralcontentsin trinsictothesynchrotron radiationatU12IR.Thescannercanoperateineitherthestep-sc anorquasi-rapid-scan mode.TheSciencetechMD500electronicsboxcontrolsthech opperandscanner.The instrumenthasasingleinternalsource(Highpressure100WHg-Xelam p),andthe externalinputportisconnectedtotheU12IRbeamlineforthe synchrotronradiation. Modulatedlightgoesthroughtheexternaloutputportontow hichvariousexperimental setups( e.g. ,samplechamberormagnet)canbeattached.Thespectrometer' svacuum boxhasthreeexternaladjustmentknobsinordertoalignthe xedmirrorM5whileit isinvacuum.Althoughtheplanemirrorneedsonlytwotiltadj ustments,therooftop mirrorrequiresthree:tiltintheaxisperpendiculartothe mirroropticalaxisandthe roofedge,lateralshear,androtaryshear.Oncealignedoptim ally,theSPS-200works betterthanBrukerbenchesespeciallybelow20cm 1 owingtoitsusebiggeroptics(90 mm)combinedwithhighereciencyofthepolarizingbeam-spl itter.SeeTable 6{5 for thespecicationoftheSPS-200. 6.4PumpLaserSystem 6.4.1SystemOverview ThepumplasersystemislocatedinsidetheU6beamlinehutchonthe VUVroor alongwithothercomponentsthatarenecessaryfortime-resolv edexperiments.The titanium-dopedsapphire(Ti:Sapphire)solid-statelaserprod uceslightpulsesofshort durationthatcanbesynchronizedwiththatofthesynchrotron .Theprimarypurposeof wholelasersystemistophotoexcitematerials,andtointroduce anarbitrarytime-delay betweenlaserandthesynchrotronprobepulsesusedtoinvestigat ethetransientstate ofsamples.Ourgoalistoobtaincompleteinfraredspectraduri ngthevariousphases

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101 Table6{5:SpecicationsoftheSPS-200. OPERATINGMODES Michelson amplitudemodulation Polarizing polarizationmodulation Mixed amplitudemodulationwithpolarizers SOURCES Internal Highpressure100WHg-Xelamp(arcsize:1.3mm) External Synchrotronradiation Chopperfrequency 20-175Hz SCANNER Stepscanandquasi-rapidscan 40cm(80cmretardation) Min.scanstep 2.54 m Max.unapodizedresolution 0.006cm 1 Spectralrange 2-600cm 1 inpolarizingmode OPTICS f/ 2.5 Beamsize 60cm 2 Roofmirrorsangle 90 2arcsec. POLARIZERS Beam-splitter,polarizerandpolarizingchopper Gridmaterial Aluminum Conductorscrosssection 2 mwide,0.4 mthick Linepitch 4 m Substrate Mylar,12 m MichelsonmodeBeam-splitter 3 mMylar oftherelaxationprocessthatfollowsimmediatelyafterthe laserpulse.Theprinciple oflaser-synchrotron(pump-probe)experimentsaredescribed inthepreviouschapter (see x 5.2 ).ThelaserpulsesarecoupledintoopticalberattheU6hutch, anddelivered totheendstationsatbeamlines(mainlyU10AandU12IR)whereac tualexperiments areperformed.Detailsofeachcomponentoflasersystemaredesc ribedinthefollowing subsections.Inaddition,becauseofthenatureofdangerinvolv edinusingtheclass 4laserssuchasTi:SapphireandNd:YVO 4 lasers,theissuesoflasersafetyandthe operatingproceduresarealsodescribedinAppendix C 6.4.2Mode-locked,Solid-StateTi:SapphireLaser TheCoherentMiraModel900-Pisamode-lockedultrafastlaser thatusestitaniumdopedsapphireasagainmedium.AnopticalschematicoftheMira isshownin Figure 6{7 .Itproducesapproximately2to3psdurationpulesofnearin fraredlight. Apulserepetitionrate(PRF)ofacommercialMiraunitis76MHz ,buttheoneinthe U6hutchismodiedtooperatewithaPRFof105.8MHz,twicetheVUV RFsystem fundamentalfrequency(52.9MHz).Attemptwasmadetoproduc ealongercavityfor

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102 Pump Beam M8 M9 GTI M5 L1 M4 M3 M2 Slit M1 output coupler Ti:S BRF StarterFigure6{7:TheopticalschematicoftheMiralaserhead. producingstablepulsesat52.9MHzwithoutsuccess.Thewavelengt histunablefrom approximately700to1000nm.Aparticularsetofoptics(S-,M -,L-,andX-Wave optics)determinesitstuningrange.Themaximumaveragepow eroutputisnearly1 Watt(or10nJperpulse).TheMiraisopticallypumpedwithaCo herentVerdipump laser.TheVerdiisacompactsolid-statediode-pumped,freque ncy-doubledNd:Vanadate (Nd:YVO 4 )laserthatprovidessingle-frequencyCWgreen(532nm)outpu tatpower levelupto5.5Watts. 3 TheVerdiisequippedwithPowerTrack TM softwaretoinsure stableoperation.SeeTable 6{6 fortheMiralaserspecication. Pulsesinthepicosecondregimecanbegeneratedbymode-locki ng[ 66 ].The simplestwaytovisualizemode-lockedpulsesisagroupofphoton sclumpedtogether andalignedinphaseastheyoscillatethroughthelasercavity. Thebasicideaisthe following.Emissionspectraofthemode-lockedlaser( e.g. ,Miralaser)aregenerally composedofmanydierentlongitudinalmodescausedbymodula tionoftheloss orphaseofanopticalelementinthelasercavity.Whenthesereso nantmodesare coupledinphase,thelaseryieldatrainofveryshortpulses,othe rwisecontinuouswith 3 TheVerdiusesaLBOnon-linearcrystalforthesecondharmonicg eneration.Since therefractiveindexistemperaturedependent,thetempera tureofthecrystaliselevated fortheoptimalphasematching( 150 C).

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103 Table6{6:ThespecicationoftheMiralasersystemontheVUVexper imentalroor. Pumplaser CWNd:YVO 4 laser(532nm) Pumppower < 6Watts Pulserepetitionfrequency 105.8MHz Pulseduration < 3ps Averagemaximumpower < 1Watt Peakenergyperpulse 10nJ Spectralrange: SWopticsset 720to810nm MWopticsset 800to910nm LWopticsset 900to980nm XWopticsset 710to900nm ructuatingamplitude.Thecouplingofthemodesisobtained bymodulationofthe gainintheresonator,andcanbeactive(electro-opticmodul ationofthelossesorofthe pumpintensity),orpassive(withasaturableabsorber).Thetech niqueusedtomodelocktheMiralaserisreferredtoasKerrLensMode-locking(K LM).Theopticalcavity isspecicallydesignedtoutilizechangesinthespatialprol eofthebeamproducedby self-focusingfromtheopticalKerreect 4 inthetitanium:sapphirecrystal.Thisselffocusingresultsinhigherroundtripgaininthemode-locked( highpeakpower)versus CW(lowpeakpower)operationduetoanincreasedoverlapbetw eenthepumpedgain proleandthecirculatingcavitymode.Inaddition,anaper tureisplacedataposition withinthecavitytoproducelowerroundtriplossinmode-loc kedversusCWoperation ( i.e. ,alocationwherethemode-lockedbeamdiameterissmallerth anthatoftheCW beam). Titanium-dopedsapphire(Al 2 O 3 :Ti)hasthebroadesttuningrangeofanyconventionallaser[ 67 ].Themediumcontainsontheorderof0.1%titanium,addedin theform ofTi 2 O 3 toproducethedesiredTi 3+ ion,whichreplacesaluminuminthecrystallattice. OpticalpumpingexcitesTi 3+ fromgroundstatetoavibrationallyexcitedsublevelof 4 TheKerreectisoneoftheelectro-opticeectsthatinduce sbirefringenceinan isotropictransparentsubstancewhenplacedinanelectriceld .Itisoftenreferredtoas thequadraticelectro-opticeectsincetheinducedbirefri ngenceisproportionaltothe squareoftheeld[ 10 ].

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104 theupperlaserlevel.Theionthendropstoalowersubleveloft heupperlaserlevel beforemakingthelasertransitiontoavibrationallyexcited sublevelofthegroundstate. Vibrationalrelaxationreturnstheiontothegroundstate.Th estronginteractionbetweenthetitaniumatomandhostcrystal,combinedwiththelar gedierenceinelectron distributionbetweenthetwoenergylevels,leadstoabroadtr ansitionlinewidth. TheMiralasercanbeoperatedinoneofthreemodes;continuous wave(CW), mode-locked(ML),or -locked( L)modes.CWmodeisusedatstartupofdailyoperation,andtheinternalpowermeterreadingallowsaquickal ignmentofthelasercavity. Thismodecanbeusefulforsteadystatephotoinducedexperimen ts.Whenmirrors aresucientlyaligned,theMiraisputinMLmode,andsynchron izedtotheringRF forpump-probeexperiments.Detailsofthesynchronizationsc hemewillbedescribed shortly.Inordertomode-lockthesystem,theeectivecavityl engthshouldbeadjusted toanintegermultipleofthelaserwavelength( i.e. ,theresonantwavelength).This canbeachievedbyeitherchangingmicrometersettingofabir efringentlter(BRF)or movingtheratcavitymirror(M3)mountedonthePZTactuator .TheBRFprovides smoothlasercavitytuning,andallowstheusertoselectasingler esonantmode.For picosecondpulses,theintracavitygroupvelocitydispersion(G VD)compensationisrequired.ThismodeisgeneratedviaaGries-TournoisInterfer ometer(GTI)endmirrorin thecavity. Lmodeusesaproprietaryservolooptomonitorandadjusttothec orrect valueoftheGVD.ItcanbeimaginedasanautomatedversionofML mode. 6.4.3OpticsandLightDistribution Figure 6{8 showstheopticallayoutoftheU6lasersystem.Aninterlocksystem onthehutchdoorwayentranceisinterfacedtoashutterdirec tlyattheexitaperture ofthelaser.Ifanunauthorizedpersonentersthehutchwithou tproperlytriggering theinterlockbypasselectronics,theshutterwillcloseforthe beamtobecontained insidethelaser'sownenclosure,andreducethepossibilityofin jurytotheunprotected eyes.Whenthesafetyshutterisopened,thebeamgoesthroughaF aradayisolator.A Faradayisolatorisaunidirectionalopticaldevicebasedont heFaradayeect.Atthe heartofaFaradayisolatorisaFaradayrotator.Faradayrota torsutilizehighstrength,

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105 1 2 3 4 5 6 7 8 9 10 11 Optical Delay 12 Figure6{8:TheopticallayoutoftheU6lasersystem.(1)Nd:YVO 4 laser(Verdi), (2)Ti:Sapphirelaser(Mira),(3)Photodiode,(4)SafetyShu tter,(5)Faradayisolator, (6)Cylindricallenses,(7)EOM/2pulsepicker,(8)EOM/Npulsep icker,(9)Photodiode,(10)Fibercoupler,(11)1/2waveplate,(12)Opticalb ercable. rareearthpermanentmagnetsinconjunctionwithasingle,hi ghdamagethreshold opticalelementtoproduceauniform45 polarizationrotation.WhenaFaradayrotator isplacedbetweenalignedpolarizers,itbecomesaFaradayiso lator.Faradayisolatorsare usedtopreventdestabilizingback-rerectionsofdownstreamo pticsfromreenteringthe lasercavity.Followingtheisolatorisapairofcylindricall ensesthatareusedtocorrect foranastigmatisminthehorizontaldirection.Thecauseofthi sastigmatismisnot known,butitisprobablyduetoaslightdivergencecreatedin thecustommadecavity. Allmirrorsontheopticalbencharebroadbandrerectorswith amulti-layerdielectric coatingforanenhancedrerectanceinthenearinfrared. Thepulserepetitionfrequency(PRF)ofthelaser,105.8MHz,is reducedtothat ofthesynchrotronbyselectivelypassingeverytwopulsesthroug hadevice,whichis whatwecallthe\divide-by-twopulsepicker"(orEOM/2,ConO pticsmodel360-40),

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106 madebyConopticsInc.ItcontainsaPockelscell 5 andapolarizingbeam-splitter. ThePockelscellcausespolarizationofeveryotherpulsetobe rotatedby90 ,andthe subsequentpolarizingbeam-splittertransmitspulsesincertai npolarizationdirection, andrerectsonesinorthogonaldirection.Whenthemodulato risproperlytriggered, everyotherpulseisselected(orrejected)producing52.9MHzP RFthatmatcheswith thesynchrotronPRF.Ratherthanwastingrejectedpulses,theya rerecycledtorecover someofthelostpower.Thisisachievedbyintroducinganoptic aldelay(of9.45ns)so thattherecycledpulseisrejoinedtotheonesrightbehindit selfafterthepolarization hasbeenproperlyrotatedbackbyahalf-waveplate.Thewell stackedpulseshaveas highas80%additionalpowercomparedtonon-recycledsingle pulse. Asecondpulseselectionsystem,the\divide-by-Npulsepicker"(o rEOM/N, ConOpticsmodel350-60),isusedtomatchthevariousbunchpa tternssuchassymmetric3-bunchoperation(17.6MHz)andsinglebunchoperation(5 .9MHz).Figure 6{9 showstheeectsofpulseselection.ThesecondEOMhasthesameope ratingprinciple asthe\divide-by-twopulsepicker",buttheelectronicshas asixdecadethumbwheel switchtoprovidealltherequiredtiming.Rejectedpulsesfro mthismodulatoraresent toabeamdump,andarenotrecycled. Thelaserpulsesarecoupledintoanopticalbercableusingasta ndardber coupler,andaretransportedtoaparticularbeamlineendstat ionoveradistanceof approximately30m.Becauseopticalbercableisanimportan tcomponentofour experiment,thesomedetailsaredescribedin x 6.5.6 AlthoughnotshownontheFigure 6{8 ,anINRADM/N5-505UltrafastHarmonic GenerationSystemcanbeinsertedinfrontofthebercouplert oextendsthefundamentaltuningrangeofTi:Sapphirelasertoshorterwavelengt hsthroughtheuseof 5 APockelscellisanelectro-opticalmodulator(EOM)whiche xploitsanothervery importantelectro-opticaleectknownasthePockelseect .ThePockelseectisalinearelectro-opticaleect,inasmuchastheinducedbirefrin genceisproportionaltothe rstpoweroftheappliedelectriceldandthereforetheappl iedvoltage.Theeectexistsonlyincertaincrystalsthatlackacenterofsymmetry[ 10 ].

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107 050100150200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (D)(C) (B) (A)PhotodiodeSignal[a.u.]Time[ns] Figure6{9:Eectsofthepulsepickingonthe105.8MHzlaserpul setrain.Thecurve (A)showsthecasewithoutanypulsepicker,thatis105.8MHzpulset rain.(B)shows thepulseswiththe\divide-by-twopulsepicker"inuse.(C)and (D)arethecaseofusingthe\divided-by-Npulsepicker"for3bunchsymmetricandsi nglebunchmode.Note thatvariationsinpeakheightareduetoundersamplingbythe digitaloscilloscopeand donotrerectpowerructuationsinthelaser.nonlinearmixingcrystals.Fromafundamentalrangeof700nmt o900nm,forexample,secondharmonicwavelengthscanbegeneratedfrom350nm to450nm,andthird harmonicwavelengthsfrom233nmto300nmcanbeproduced.6.4.4Laser-SynchrotronSynchronization TheSynchro-lock900isanaccessorytotheMira900unit.Itisd esignedtoallow thesynchronizationofthelaserpulsesfromaMirawithastablee xternalfrequency sourceorwithastableinternalcrystaloscillator.Thesystemuses threecavitylength actuatorsintheMiraheadtocontrolthelaserfrequency:ahi gh-frequencypiezo-electric transducer(PZT),alowfrequencygalvonometerdrivendelay lineandadiscretestepmotordrive.Thesystemmonitorsthelaseroutputwithapickoa ndphotodiode mountedontheopticaltabledirectlybeyondtheMiraoutput bezel.TheSynchro-lock systemmixesthepulsedlasersignalwiththereferencesignal,and anerrorsignalis

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108 HPPulse Generator (Delay) Phase Shifter (Delay) Function Generator (Phase Modulation) Synchro-Lock 900 Control Box Mira 900 /2 Pulse Picker /N Pulse Picker Conoptics Model 305 Electronics (Delay) Conoptics Model 10 Electronics (Delay) 53 MHz RF DC Power Supply Beamline U6 hutch Figure6{10:Blockdiagramofthetimingscheme.Althoughnotsh own,coaxialcables arealsodelayelements.usedtocorrectthecavitylength.Whentheerrorsignalisdriv entozero,thelaseris synchronizedtothereference.Theelectronicsarecontaine dinacontrollerboxwhich ismonitoredandcompletelycontrolledbyacomputer.Theco mputercontinuously measuresanddisplaysthelaserfrequencyandhasafullyautoma tic\one-touch"lock acquisitionoperationmode. Inthecaseof7-bunchoperation,the52.9MHzsignalfromtheRFc avityofthe VUVringservesastheexternalfrequencysourcefortheSynchrolock900.Forthe time-resolvedmeasurements,aspecicphaserelation(delay)be tweenthelaserand synchrotronpulsesatthesamplelocationmustbeestablishedinac ontrolledway. Figure 6{10 showsthetiming( i.e. ,bothsynchronizationanddelay)schemewehave optedtouse.SeealsoFigure 6{20 formorevisualizedversionofthisdiagram.There areseveralwaystointroducetimedelay.Dierentlengthsof coaxialcable 6 canbe 6 Signalsthroughthetransmissionlinetravelsatarate(charac teristicdelaytime) givenby t d = p LC ,where L isaseriesinductanceperunitlengthand C isaparallel

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109 usedasdelayelements.Finecontrolofdelaytimeisachieveda sfollowing.TheRF ringsignalisfedintoapulsegenerator(HP81101A)atthebeamli ne(U10A/U12IR), andtriggersitsoastogeneratevoltagepulsesofthesame52.9M Hzfrequencywitha variabledelay-settinginincrementsaslowas0.01ns.Thedel ayedRFsignalfromthe generatoroutputistransmittedtotheSynchro-Lockcontrol boxthroughcoaxialcables fordesiredtiming.Wecanalsouseavoltagecontrolledphaseshif terbetweenthepulse generatorandtheSynchro-Lockcontrolboxforanadditiona ldelaythatcanbeutilized indierencespectrameasurementsaswillbediscussedin x 6.6 Inordertosetaccuratelythepump-to-probedelaytime,thec oincidencebetween thelaserandsynchrotronpulsesmustbemeasuredatthesampleloca tion.Thedelaysettingofthepulsegeneratorisadjusteduntilthepeaksofbot hpulsesdetectedby aGeavalanchephotodiode(APD;nearinfraredfastdetectorof 150psrisetime) overlapcompletelyonthedisplayofa1GHzdigital(electroni csrise/falltimeof400ps) fastoscilloscope.Becausethesynchrotronradiationisabroadb andsourcecontaining spectrumbeyondnearinfrared,asingledetectorcanbeusedtod etectbothlaserand synchrotronpulses.Wedenethiscoincidenceasthe\zero"del aypoint( t d =0),and canintroduceanarbitrarytimedelaywithrespecttothecoin cidence.Figure 6{11 showseachsynchrotronpulsearrivingatthesamplelocation4ns afterlaserpulse ( i.e. t d =+4.0ns). Thereisasignicantmodicationthatwasmadefromtheiniti aldesignofsynchronizationmethod.Sincethelaserrunsat105.8MHzwhichisexac tlytwiceasfastasthe PRFofsynchrotron,everyotherpulseshadtobeselectedbytheE OM/2pulsepicker asdiscussedearlier.Acorrectsetoftwopossiblepulsetrains(one spassedandothers rejected)shouldbechosenforapropertiming.TheConOpticsM odel10electronics forEOM/2pulsepickerprovidesabiasadjustknobforaneasysele ctionofapulse capacitanceperlength.Forexample,thepopular50ncoaxia lcablehasadelaytimeof 4.2ns/m.

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110 140160180200220 -2 0 2 4 6 8 10 DetectorsignalTime[ns] Synchrotron Laser 4 nsFigure6{11:Synchronizedlaserandstorageringpulses.Itshows thesynchrotronpulse arriving4nsafterthelaserpulseatthesamplelocation.train.Ifthewrongsetisselected,laserpulsesarrive9.45nso thecoincidence.When thesynchronizationschemewasdesignedinitially,thephotod iodesignalfromthelaser outputdividedby2wasusedtodrivethepulsepicker,anddelay wascontrolledfrom theSynchro-lock900program.Inthisconguration,althou ghthepulsepickerwas operatinginphasewiththelasernomatterwhattimedelaywasse t,thecorrectsetof pulsetrainhadtobeconrmedeverytimethelaserloosemode-lo ckduetoinstabilities. Withthenewconguration,thedelayedRFsignalfromtheHP811 01Adrivesthe Synchro-lock900andthepulsepickertogether,eliminating thepossibilityofhoppingto thewrongsetofpulses. Whenthesynchrotronisoperatingineitherthesingle-buncho r3-bunchsymmetric operation,the5.9MHzRF/9signalavailableatbeamlinecanbe usedinplaceofthe RFsignalforthe7-bunchoperation.DelayedRF/9signaloutof theHP81101Acanbe decomposedintoitsharmonics,thenwithahelpofbandpasslte r,the9thharmonics canserveasanexternalfrequencysourcefortheSynchro-lock 900,andthefundamental orthe3rdharmonicscanbeusedtodrivetheEOM/Npulsepicker.

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111 Nitroogen fill/exhaust Needle valve OVC valve Sample space Wiring connections Helium transfer tube entry Helium reservoir exhaust Level probeentry Sample spacepumping line Sample space 77K radiationshield Heat exchanger Figure6{12:SchematicoftheOxfordOptistatbathcryostat.T opofthecryostatis ttedwithservicesnecessarytorunthecryostat. ThefunctiongeneratorshowninFigure 6{10 allowslaserpulsestobeditheredwith respecttothesynchrotronpulses,makingquasi-dierentialmea surementspossible.See x 6.6 forthedetailofthistechnique. 6.5OtherExperimentalComponents 6.5.1OxfordOptistatBathCryostat TheOxfordOptistatcryostatisaveryimportantcomponentofo urpump-probe measurement.Itisabathcryostatwithabuiltinvariabletemp eraturefacility.The sample,mountedonasamplerod(insert),isloadedthroughthet opofthecryostat, andcanbecooleddirectlybythecryogen(dynamicsystem).The sampletemperature iscontinuouslyvariablebetween1.5and320K.Figure 6{12 showsaschematicofthe Optistat.

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112 Thecryostatcontainsa2.5-litterliquidheliumreservoiras wellasaliquidnitrogen reservoir.Thereisnoneedtorellthecryostatcontinuouslyf romastoragedewar foraconvenientperiodofoperation.Liquidheliumissuppli edfromthereservoirto thesamplespace(20mmdiameter)throughaneedlevalve,regul atingtherowtobe optimizedtosuittheoperatingrequirements.Theheliumreser voirhastwonecks,used fordierentservices.Oneisusedfortheneedlevalvedriverod .Theotheracceptsa9.6 mmdiameterliquidheliumtransfertubeusedtolltheheliumr eservoiranda4.8mm diameterheliumlevelprobe(dipstick).Anon-return(one-w ay)valveisconnectedto theexhaustportoftheheliumreservoir.Aliquidnitrogenrese rvoirisusedtocoolthe radiationshieldaroundtheliquidheliumreservoirandsample space.Thisshieldsthe lowtemperaturepartsofthesystemfromroom-temperaturethe rmalradiation.Italso shieldsthesamplespacefromtheliquidheliumreservoirsothatt heheliumevaporation rate(boilo)isnotaectedbyhighsampletemperatures.Then itrogenreservoir hasthreevents.Oneofthenecksisttedwithanon-returnval ve.Thisensuresthat evaporatingnitrogengascanbereleasedsafelyfromthereserv oireveniftheothernecks areaccidentallyblockedbyicecondensedfromtheatmosphere Thesamplespaceisbuiltintothecryostat.Itusesacontinuousr owofliquid heliumfromthereservoirtoprovidecoolingpower.Thus,thesa mpleisindirectcontact withthecryogeneitheringasorliquidphase,providingmuch fastercoolingrateto thesampleandsampleholderblockthanthetypeofcryostatthat reliescoolingon heatconductionthroughacoldnger( e.g. ,theAPDHelitran).Inourpump-probe experiment,thebathcryostatsuitedthebetteroftwotypessin ceitcouldminimizethe thermalexcitationduetothelocalheatingbythepumplaser. Aheatexchangeratthe bottomofthesamplespaceisttedwithaheaterandRhFetemper aturesensor,sothat thetemperatureoftheheliumrowingthroughthesamplespacec anbecontrolledby atemperaturecontroller.TheITC502temperaturecontroll erisusedtooptimizethe heliumrowrateandheaterpowerautomaticallyovermostofth eoperatingtemperature range.TheneckofthesamplespaceisttedwithanNW25rangethr oughwhich theheliumispumped.Byreducingthepressureinthesamplespace temperatures

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113 downto1.5Kcanbeobtained.Apumpingmanifoldcontrolsthe pumprate,and thus,thepressureinsidethesamplespace.Theenvironmentaround thesampleaects theparametersuchasphononescaperate(themeaningofwhichw illbeclearlater) obtainedfromtheexperiment.Insteadofpumpinghard,weused thehighestheliumgas pressurepossible(justbelowambientpressureifpossible)forthede siredtemperature. Thisensuresthatasmanyheliumatomsrowbythesampleaspossibl ehelpingecient heattransfer. Thetopofthecryostatisttedwithotherservicesincludinga nelectricalaccess totheheatexchangerandavacuumvalvewithapressurerelieff ortheoutervacuum chamber(OVC). Thesampleholderisattachedatthetipoftheinsert.Wemadesev eralsample holderswithvarioussizedapertures.Oneholderweusedmostacc ommodatesthree samplesverticallyinseries;eachparticularsamplecanbeselec tedbymovingtheinsert upordown.Theinsertcanberotatedabouttheverticalaxis(t heshaft)providing anotherdegreeoffreedomforthealignmentofthesample.The rearetwomulti-pin electricalfeedthroughesattheotherendoftheinsert.Onei sconnectedtoanother temperaturecontroller(ScienticInstrumentsModel9650) thatmeasuresthetemperaturenearthesampleviaadiodethermometerembeddedinthesa mpleholderand appliesvoltagetoaheater(Teroninsulatedconstantanwire, 0.005"diameter)whichis wrappedaroundthesampleholder.Oncethetemperaturesetpoi ntontheITC502are changed,thetemperaturereadingontheSIModel9650quickl yandcloselyfollowsthat oftheITC502.Thisimplieshowecientlydynamicsystemcanad justthetemperature ofthesample.Theheaterontheinsertservesanotherusefulpurp ose.Sincewecan warmthesampleandholderto300Kindependentlyfromthesampl espace,samplescan bechangedtoanothersetofthreesampleswithoutwarmingenti resamplespace. Therearethreepairsofwindows:theoutervacuumwindows,mid dle77Kradiationshieldwindows,andinnersamplespacewindows.SeeFigur e 6{12 fortheir locations.Thecryostathastwomutuallyperpendiculardirec tionsinwhichexperiment canbeperformed.Wecan,thus,putonesetofwindows(totalof6 windows)inone

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114 Table6{7:PropertiesofwindowsfortheOxfordOptistatcryo stat. location material dimensions range(cm 1 ) Outervacuum Quartz 42mmdiameter,1mmthick 0-250 and2700-65000 KBr 42mmdiameter,3mmthick,wedged30mins. 400-40000 Polyethylene 42mmdiameter,variousthickness 0-700 Middleshield Sapphire 26mmdiameter,1mmthick 0-350(below50K) and2000-65000 KBr 26mmdiameter,1mmthick,wedged30mins. 400-40000 Innersample Sapphire 0-350(below50K) and2000-65000 ZnSe 720-17000 directionanddierentsetinanotherdirectionsothatwidespe ctralrangeofoptical measurementscanbedonejustbyrotatingthecryostatby90degr ees.Theouter vacuumwindowssealtheOVCbyO-rings,andareretainedbyfour screws.Wehave Quartzwindows,KBrwindows,andpolymer(polyethyleneandpo lypropylene)windows. Theradiationshieldwindowsareheldinplacebywireclips.Th epressurefromthe wireclipshouldbesucienttomakesurethatthewindowiscoole dproperly. 7 These middlewindows,however,donotserveasvacuumwindow,andare optional.Wehave sapphireandKBrwindows,butchoosenottousethemiddlewindows inordertodeliverasmuchlightaspossibletothesample.This,ofcourse,cause smoreheatloadthat limitsthelowesttemperatureaccessiblefortheexperiment. 8 Theinnersamplespace windowsareindiumsealed.WehaveapairofsapphireandZnSewi ndowsattachedon theblockofsamplespace.Table 6{7 summarizespropertiesofwindowsfortheOptistat. Forthefarinfraredmeasurements(below250cm 1 ),apairofquartzvacuumwindows, sapphireradiationshieldwindows(optional),andsapphiresam plespacewindowscanbe used.Atroomtemperature,themulti-phononprocessesinsapphi relimitthespectral range.Asthetemperatureislowered,theacousticphononsstar tfreezingoutand,asa result,thespectralrangestartsopeningup.Below100K,sapphi reopensitsrangeup 7 Lightlygreasingthewindowmountwithvacuumgrease,suchasApi ezon`N'will improvesthethermalcontactbetweentheshieldandthewindo w.Thiscandecreasethe liquidheliumconsumptionsignicantly. 8 Theamountofheatradiatedfromonebodytoanotheratadier enttemperatureis proportionaltothedierencebetweenthefourthpowerofth eirtemperatures.

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115 to350cm 1 ,buttheabsorptionduetophononsinquartzlimitstheupperl imitofthe rangewiththissetofwindows. PreparationofthecryostatstartsbypumpingtheOVC.Ithasto bepumped tohighvacuum(typically10 4 or10 5 Torr).Weusedaturbo-molecularpump backedbyarotarypump.Incasethatthesystemisbadlycontamin atedwithwater vapor,therotarypump(ideallywithagasballastfacility)sh ouldbeusedrstto roughout.Beforewecooldownthesystem,itisveryimportantt oremoveairfrom theheliumreservoirandsamplespaceinordertoreducethechan ceoffreezingand blockingthenarrowcapillarytubeandneedlevalve.Thisis donebyopeningthe needlevalvewidefollowedbypumpingthereservoirandsample spaceandrushing heliumgasfromsamplespacepumpingline(orthereservoirexha ust).Oncethisis done,thecryostatcanbepre-cooledbyllingtheliquidnitr ogenreservoirwithliquid nitrogen.Wenormallywaitatleast12hoursbeforeliquidhel iumistransferredinto thesystem.Duringthepre-coolingprocess,thepumpcanbeeithe rleftonorturned o.Ipersonallyoptedtoturnitobecausethenliquidnitroge nevaporatesquicker resultinginfastercoolingrateforthesystem. 9 Justbeforestartllingliquidhelium,I lltheheliumreservoirandsamplespacewithheliumgasonemor etimetoatmospheric pressure,closethevalveforthepumpingline,andsettheneedle valvetothe30%of maximumopenedstate.Nowwecanlltheliquidfromastoragedew arthrougha transfertube.Thenon-returnvalveonthereservoirexhaustsho uldberemovedbefore insertingtransfertubeintothetubeentry.Atthebeginning, theboiledheliumcreates aplumeofexhaustgas,butwhenliquidstartstocollect,thegas rowratewilldrop noticeably.Whenthereservoirislledcompletely,thereis suddenexcessivecoldgas 9 Liquidnitrogenhasahighlatentheatofevaporation:thati s,averylargeamount ofheatisrequiredtoevaporateit.Thus,fastercoolingisach ievedbyevaporatingthe liquidnitrogen.Liquidhelium,ontheotherhand,hasavery lowlatentheat.However,heliumgashasaveryhighenthalpy.Itmeansitisverye asytogenerategas,but itismuchmorediculttowarmthatgasup,andthecoldgaspro videsahighcooling power.

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116 (noticeablydenserexhaustgas),andthentheheliumtransfersho uldbestopped.The dipstick 10 shouldindicatealevelof 10cm.Thesoundfromtheexhaustalsochanges slightly.Therearetwowaysofcoolingthesample:pumpfastwit hnarrowneedlevalve openingorpumpslowwithwideneedlevalveopening.Therowra teshouldbeadjusted manuallytogetasuitablecoolingrate(typically2to3Kperm inute).Ittypically takes60to90minutestocoolthesamplespaceto10Korbelow.On cethetemperature approachesthedesiredsetpoint,wecanswitchtotheautomatic temperaturecontrol. TheITC502isathree-termcontroller.Automatictemperatur econtrolisoptimizedby settingthebestvaluesforproportional(P),integral(I),an dderivative(D)constants. Fortheexperimentatthelowestpossibletemperature,thesampl espacecanbelled withliquidheliumbyopeningtheneedlevalvewideopen.The n,byclosingtheneedle valveandbypumpingasfastaspossible,wemanagedtocoolthesam pledownaslow as1.8K. 11 Thisisasingleshotmethod.Oncetheliquidispumpedoutcompl etely fromthesamplespace,thetemperaturestartsgoesupsuddenly,a ndthentheprocess hastoberepeated.6.5.2Ox-BoxCustom-madeSampleChamber Inordertoaddrexibilitytoourexperimentalsetup,wedesign edacustommade samplecompartment(wehavebeencallingitasthe\Ox-Box".) thatbecameanother keycomponentofourexperiment.TheOx-Boxhasansourceinpu tportandan externaldetectorport.Wemadeseveralrangesthatconnectt hesourceinputport totheBruker66'sexternalport,theSPS-200'sexternalpor t,ordirectlytheU12IR beamline.Theexternaldetectorportisdesignedtosupportva rioustypesofbolometers. 10 Narrowtubewiththinmembraneoverahousingatthetopend,used asasimple levelprobeforliquidhelium.Thethermalgradientsetupint hetubeleadstothermal oscillationswhicharefeltbythevibrationofthemembrane. Thefrequencyoftheoscillationwiththelowerendinliquidheliumisnoticeablyl owerthanthatwhenitisin coldgas,allowingtheliquidleveltobedeterminedeasily. 11 Thesuperruidtransitionofheliumoccursatthe point(2.2K)belowwhichthe liquidbecomescalmandhardlydistinguishablefromvacuum.

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117 M1 M2 M3 M4 M5 M6 M7 M8 Bolometer Input port M3M4 (A)Top (B) Side Figure6{13:Diagramofourcustommadesamplecompartment,th eOx-Box.(A)Top viewoftheOx-Boxshowingitsopticallayout.(B)Sideviewwi ththeOptistatmounted onthebox.OnlyM3andM4areshown.Therearetwowidesideopeningsthroughwhichwecanaccessthe insideforminor adjustmentofopticswithoutremovingthetopcoverofthebox .Theseopeningsare coveredbyacrylicwindowsthatsometimeshelpalignmentwhi letheboxisevacuated. Duringthepump-probeexperiments,thesewindowsmustbecover edbymetallicplates inordertoavoidanypotentialleakofhazardousclass4laserl ightfromthebox.An opticalbercableisinsertedthroughaholeonthesidettedw ithanNW25range. ThereisanotherutilityholewithanNW25rangetowhichwecan connectavacuum pump,drynitrogenline,pressuregauge,ventvalve,and/orel ectricalfeedthroughfora chopper.Figure 6{13 showsthediagramoftheOx-Box. MirrorsM2,M7,andM8aregoldcoatedplanemirrors.MirrorM1 ,M3,M4,M5, andM6areallo-axisparaboloidwhicharecutfromalargepa raboloidalrerector (RhodiumcoatedonanelectroformedNickelfromOpti-forms,I NC.)withfocallength f =2 : 35inches.SeeFigure 6{14 .MirrorsM1andM5arethetrue90 paraboloid thatrerectsthecenterofincomingrayat90 towarditsfocus.Ascanbeseenfrom Figure 6{14 (B),thesemirrorshaveaneectivefocallengthof2 f (4.7inches).Mirrors M3,M4,andM6,arecutsuchthattheireectivefocallengthar elongerthan2 f

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118 f y x y = x/4f 2 f 2f Focus OpticalAxis 90 f 2f Focus OpticalAxis 90 (A)(B) (C) Figure6{14:(A)Crosssectionofaparaboloid.(B)O-axispara boloidalrerectorwith thecenterofrayrerectedat90 .(C)O-axisparaboloidalrerectorwiththecenterof rayrerectedatgreaterthan90 asshowninFigure 6{14 (C).Theyaresittingonthestagethatcanbemovedby micrometerinonedirectionforanalignmentpurpose. TheOx-Boxisdesignedtodobothtransmittanceandrerectance measurements insequencewithoutbreakingvacuumifitisdesired.MirrorM8 canbesledinto thepositionsothatitdeliversrerectedlightoasampletodet ectorblockingthe transmittedlightoM4.Forthetransmittancemeasurement,M8 isjustremovedso thatonlytransmittedlightgoesintodetector.Theboxisalso designedtoacceptboth theOxfordOptistatandAPDHelitrancryostats.6.5.3OxfordInstrumentsVertical-boreSuperconductingM agnet Opticalmeasurementscanbeperformedinmagneticeldofupt o16Teslawithan Oxfordinstrumentsvertical-boresuperconductingmagnet.T hecollimatedlightfromthe extractionportoftheBruker125spectrometer 12 isdeliveredtoaparaboloidrerector atthebottomofthemagnet.Then,afterpassingaseriesofwedge dquartzwindows,the lightisfocusedatasamplemountedatthetipofavariabletemp eratureinsert(VTI), 12 TheBruker125usesHe-Nelaser.SinceHe-Nelaseroperateswithanel ectricaldischargeinaplasmatubecontainingamixtureofheliumandneon gases,strongmagnetic eldcouldcauseinstabilityofthelaser,andspectrometerstops scanning.Thus,the magnetshouldbeplacedatcertaindistancefromthespectromet er.

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119 SpectrometerBruker 125 Bolometer Brass elbow Paraboloid VTI Superconducting magnet Polyethylene window Quartz window Stainless steellight-pipe (A) VTIHeaterTemperaturesensorLight-conesSample Wedged quartzwindow Teflon centeringring (B) H Figure6{15:(A)AschematicoftheOxfordinstrumentsvertical -boresuperconductingmagnetconnectedtotheBruker125atU12IR.(B)Moredetai ledviewofthe sample-space.Asampleissandwichedbetweentwolight-cornsin Faradaygeometry formagneto-opticalmeasurement,andplacedatthecenterof themagnet. andnallyguidedtoadetectorthroughlight-pipes(14mmin nerdiameter)witha polyethylenewindowattheveryend.Figure 6{15 (A)showsaschematicofthesetup. Copperisoneofcommonlyusedmaterialforlight-pipesbecau seofitshigh conductivitytominimizethererectionlossinthepipe.Howev er,thehighconductivity ofcopperalsoresultsinhighthermalconductivity.Inordert oreduceheatload,a stainlesssteellight-pipeisusedforthesectionthatgoesintot hecoldenvironmentof themagneteventhoughrerectionlossinastainless-steelpipeis higherthanthatina copperpipe. ThesampleismountedattheendoftheVTIsuchthatstaticeldise itherparallel (Faradaygeometry)orperpendicular(Voigtgeometry)toth epropagationofthe electromagneticwaves.IntheFaradaygeometry,thesampleis sandwichedbetweentwo light-cones,andplacedatthecenterofthemagnet.Thisisth eonlygeometryweused forourmeasurements.Figure 6{15 (B)showsadetailedviewofthesample-spaceofthe VTI.

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120 6.5.4Detectors Opticaldetectorsaretheradiationtransducersthatconver tradiationpowerinto anelectricalsignalortoanotherphysicalquantity( e.g. ,heatorresistance)thatcanbe convertedtoanelectricalsignal.Inthissubsection,someofth egeneralcharacteristics ofdetectors(sensitivity,linearity, etc .)thatplayanimportantroleindeterminingthe accuracyandprecisionattainableinspectroscopicmethodsar epresentedrst[ 68 ]. Then,theoperatingprinciplesofseveraltypesofthemostcom monlyuseddetectorsare discussed.Wewill,nally,describethedetailofthe1.8KSibol ometer,whichisour principalfar-infrareddetectorforourtime-resolvedexpe riments. DetectorCharacteristics Opticaldetectorsvarywidelyincharacteristicssuchassensit ivity,linearity,spectral response,responsespeed,noisegure,andsoon.Beforewegoontode scribevarious typesofdetectorsandtheiroperatingprinciples,itmaybew orthmentioningsome termsthatcharacterizeopticaldetectors. Thesensitivityofadetectorcanbedescribedinseveralways.The responsivity R ( )istheratioofthermssignaloutput X (voltage,current,etc)tothermsincident radiantpowerevaluatedataparticularwavelengthandinc identpower: R ( )= X= : (6.1) Thesensitivity Q ( )istheslopeofaplotof X vs.: Q ( )= dX=d : (6.2) Aplotof R vs. or Q vs. iscalledthespectralresponseofthedetector.The functionalrelationshipbetween X andisknownasthetransferfunction.If Q is constantandindependentof,thedetectorissaidtobelinear ( i.e. ,theoutput electricalsignalislinearlyproportionaltotheinputopti calpower).Figure 6{16 illustratesthesimpliedtransferfunction.Nodetectorhasac onstantsensitivityunder allconditionsofuse.Forinstance,thesensitivityofphotodio dedetectorsfallsoabove acertainincidentpowerlevelduetosaturationeects.Thus,d etectorsexhibitlinearity

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121 X () l Fl () Slope = sensitivity (/) dXd F Responsivity (/) at given incident power X F Linearity :() constant, independent of Q lF Linear dynamic Range Figure6{16:Simpliedtransferfunction. overalimitedrangeofincidentradiantpower.Lineardyna micrange(orlinearity range)referstothetotalrangeofincidentopticalpowerle velsoverwhichthedetector outputvarieslinearlywithincidentpower. Detectorsalsovarywidelyintheirabilitytodetectrapidch angesinincident radiantpower.Quantitatively,theresponsetimeisdenedas: =1 = 2 c ; (6.3) where c istherollofrequencyatwhich R ( )hasfallento0.707ofitsmaximumvalue (3-dBpoint)whenasinusoidalinputoffrequency c isincidentonthedetector. Thenoiseequivalentpower(NEP)andthedetectivityareother importantquantitiesthatcharacterizethedetectorperformance.NEP(inw atts)isdenedasthe incidentradiantpower,ataspeciedwavelengthandbandpass thatwillproducea anoutputsignalfromadetectorthatisequivalenttotheinhe rent(background)noise inthatdetector.TheNEPdependsonthetypeofdetector,surfa ceareaofdetector, andwavelength.Thedetectivity(incmHz 1 = 2 W 1 )denotedas D (termedDstar)isa measureofminimumdetectabilityusedforcomparisonbetweend ierentdetectors. D

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122 isinverselyproportionaltoNEP,andisgivenas: D =( A ) 1 = 2 = NEP : (6.4) whereAisthesurfaceareaofdetector(incm 2 )and isthenoiseequivalentbandwidth(inHz).DetectorTypes Generally,detectorsfallintotwomajorcategories:therma ldetectorsandphoton detectors.Figure 6{17 showstheclassicationofcommonlyuseddetectors. Thermaldetectorssensethechangeintemperaturethatisprod ucedbythe absorptionofincidentradiation( i.e. ,photonenergy).Thetemperaturechangeis convertedintoanelectricalsignalbymethodsthatdependon thespecictransducer. Thermaldetectorshaveanearlyuniformspectralresponsethat isdeterminedbythe absorptioncharacteristicsoftheirresponsiveelementsandwi ndowmaterials.Photon detectors,ontheotherhand,respondtoincidentphotonarriv alratesratherthanto photonenergies.Thespectralresponseofthesedetectorswithwa velength,buttheir majoradvantageoverthermaldetectorsistheirfasterrespon setime.Photondetectors canalsodetectlowerradiantpowersthanthermaldetectorsi nmanycases. Abolometerisoneofthemostwidelyusedthermaldetectorfort hefarinfrared region.Itisarelativelyslowdetector(thermaltimeconstan tontheorderof1ms),and exhibitsawidelinearityrange.Itisatypeofresistancether mometerconstructedfroma Optical Detector Thermal DetectorPhoton Detector Optical ExcitationPhotoemission Thermal Effect BolometerPyroelectricPneumaticThermocouplePhotodiodePhototransisterPhotoconductorLinearArray SensorPhotoTubePhoto MultiplierTube Figure6{17:Classicationofopticaldetectors.

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123 dopedsemiconducting(germaniumorsilicon)sensingelementsup portedinavacuumby leadwiresattachedtothecooledsubstrate.Adopedsiliconist hemostcommonlyused materialsthesedays,andexhibitarelativelylargechangein resistanceasafunction oftemperature.Theresponsiveelementiskeptsmall( i.e. ,smallerheatcapacity)for highersensitivity.Inmanyapplicationsitisnecessarytoincr easetheeectiveareaof abolometerwithoutincreasingthethermaltimeconstant.Thi sisaccomplishedby bondingthesiliconsensingelementtoalarge,thinplateofeit hersilicon,sapphire, ordiamond.Thesematerialshaveexceptionallylowheatcapa cityandhighthermal conductivityatlowtemperatures.Athinconductinglayerof NichromeorBismuthis vacuumdepositedontoonefaceofthethinsubstrateinordertoi ncreaseabsorption ofinfraredradiation.Thistypeofbolometerisknownascom positebolometer(See Figure 6{18 ).Astheoperatingtemperatureislowered( 4.2K),thesensingelement approachesclosertothesensitivitylimitsofthermaldetecto rssetbyfundamental thermalructuations,anddetectorperformancebecomesbett er. 13 Theoperating temperatureisdeterminedbydewarbathtemperature.Arstst ageJ-FETpreamplier isplacednearthesensingelementmountedonthecoldplateofd ewar.Itrequiresan operatingtemperatureof60Kormore.Thisisaccomplishedby maintainingasmall currentrowthroughaheaterresistor.Thereisusuallyaserieso fwedgedopticallters infrontofthesensingelement. Apyroelectricdetectorisanotherexampleofathermaldete ctor.Deuterated triglycinesulfate[DTGS:(NH 2 CH 2 COOH) 3 H 2 SO 4 ]crystalhasgoodpyroelectric 13 Wecancheckifthebolometerisstillcold(inotherword,ifth ereisstillLHeleft inthedewar)bymeasuringtheresistanceattheBIASTESTBNCconne ctorwiththe BIASswitchturnedo.Thismeasurestheresistanceofthebolomet erandloadresistorconnectedinseries.Obviously,theseresistancesdependonap articularsystem,but bolometerresistanceisusuallyontheorderof k natroomtemperatureandloadresistanceisnear10Mn.Whenbolometerisatitsoperatingtemper ature,themeasured resistanceshouldreadhigherthanloadresistance.Asbolometerst artswarmingup, thebolometer'sresistancebecomessmaller,andthemeasuredre sistanceapproachesthe valueofloadresistance.

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124 MetalAbsorber Si:B thermometer bonded to substrate Diamond or sapphire substrate Lead Wire Support if needed Figure6{18:Aschematicofcompositebolometer. properties.Whenplacedinanelectriceld,asurfacecharger esultsfromalignment ofelectricdipoles.WhenincidentradiationheatstheDTGS, achangeinsurface chargeresults(pyroelectriceect),whichisrelatedtothe incidentradiantpower.The outputcurrentisproportionaltotherateoftemperaturech angeofthematerial;the detectordoesnotrespondtoconstantradiantenergylevels.Th epyroelectricdetector isfasterthanbolometerbecauseonlycharge-reorientationl imitstheresponsespeedfor modulatedinputs.Thecrystalexhibitsstrongabilityinthemo stpartofinfraredregion. WehavetwoDTGSdetectors:farandmidinfraredDTGSs.Thedie rencebetween thesetwodetectorsarethewindowmaterials(Polyethylenea ndKBr,respectively).See Table 6{4 fortheirspectralranges.Thecrystalhasthehighestpyroelect riccoecient valueatroomtemperature,andthusthedetectorisusedatamb ienttemperature. Thereareotherexamplesofthermaldetectorssuchasthermoc oupleandpneumatic (Golay)detectors,butsincewedidnotencountertousethesedet ectors,wewillnot discussaboutthemfurther. Photondetectorscanbebroadlyclassiedasphotoemissivetype andopticalexcitationtype.Photomultipliertubeisanexampleofthephoto emissiontypewhichhas sensitivityintheregionfromultraviolettovisiblelight.Ph otodiode,phototransistor, photoconductivedetectors,andlineararraysensorsareexamp lesofopticalexcitation types.Ofthesedetectors,photodiodesandphotoconductorsar eprobablythemost commonlyadopteddetectorsinopticalmeasurements.Theoper atingprinciplesofthese twodetectorsarediscussedbelow.

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125 Inaphotodiode,absorptionofelectromagneticradiationby ap(i)n-junctiondiode exciteselectronsfromthevalencebandtotheconductionba nd.Thusasingleelectronholepairperphotonarecreatedinthedepletionregion.Ift herateofphotoinduced chargecarriercreationgreatlyexceedsthatduetothermal excitation,thelimiting currentunderreversebiasisdirectlyproportionaltothein cidentradiantpower. Therefore,thephotodiodeactsasacurrentsource,andthevo ltagedropacrossaload resistorismeasured.Thistypeofdetectorshowsexcellentline arityoverwiderange ofincidentradiantpower.Itisaroomtemperaturedetector thatisusedforthenear infraredandUV-Visspectralrange.Itisalsoaveryfastdetector, andthusoftenused todetectveryshortpulses.Theavalanchephotodiode(APD)isuse dwherebothfast responseandhighsensitivityarerequired.APDisoperatedinthe reversebreakdown regionofthepnjunction.WehaveSiphotodiodedetectorand GeAPD. Aphotoconductivedetectorismadeofanintrinsicsemiconduc tormaterial ( e.g. ,CdS,PbS,PbSe,InAs,InSb,Hg-Cd-Te(MCT),Pb-Sn-Te,etc.)or anextrinsicsemiconductormaterial( e.g. ,Si:B,Ge:Cu,Ge:Au,Ge:Hg,Ge:Cd,Ge:Zn,etc).When incidentphotonsareabsorbed,electron-holepairsarecrea tedandincreaseconductivity.Thusitactslikeavariableresistorasafunctionofradia ntpower.Typically,the detectorisputinserieswithavoltagesourceandloadresistor, andthevoltagedrop acrosstheloadresistorismeasured.Wehavea77KInSb,77KMCT,4 .2KSi:B, and4.2KGe:Cu.SeeTable 6{4 fortheirspectralrange.Coolingisnecessarytoavoid thermalexcitationofelectronsintotheconductionband.T histypeofdetectortendsto benonlinear.MCT'snonlinearityiswellknown.1.8KSiBolometer The1.8KSibolometer(pumpedLHe 4 )isaverysensitivedetectorusedprincipally forthetime-resolvedexperimentsintheveryfarinfrared.T herefore,wewilldescribe thisparticulardetectoralittlemoredetail.Thesensingele mentismountedonanL shapedblockwhichisxedonagoldplatedcoppercoldworksurf aceofthedewar (InfraredLaboratoriesInc.,ModelHDL-8Dewar).ThemodelHD L-8dewarisascaledupversionofthecommonlyusedInfraredLaboratories'HD-3andHD L-5units,and

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126 feature8.12inchdiametercoldworksurfacewithincreasedli quidheliumandnitrogen capacity(2.8and2.5liters,respectively).Theheliumdewar canbepumpedtoreduce theoperatingtemperaturedownto1.8Kforhighersensitivity .Whenitispumped, theholdtimeforLHeisabout50hours.Thedewarhasawedgedwhi tepolyethylene withadiamondscatterlayerasthevacuumlteraswellasafar infraredlowpass lterthatismadeofaquartzcrystallaminatedwithcombinat ionsofantirerection (AR)coatingandGarnetpowderscatteringlayerthatlimitsth espectralrangeforthis detector( < 100cm 1 ). 14 Thefarinfraredlterisacooledtocryogenictemperature. Thetransmissioncharacteristicsofalayerofdielectricpowde risdeterminedbythe size,distribution,refractiveindex,andthickness.Thedewar alsohasaWinstoncone condenserformaximumconcentrationandecientreceptiono ffarinfraredandsubmillimeterradiation.Thedetectorisplacedinacavityatt herearofthecondenser. Theconeisconstructedfromnickelandisthengoldplatedtoe nhancererectivityand thermalcooling. Otherthanthe1.8KSibolometer,wehaveSibolometerandSi: B/Sisubstrate compositebolometersoperatingat4.2(LHe 4 ),andSibolometerat0.3K(LHe 3 ).The LHe 3 systemisthemostsensitiveofall,butitrequiresarepairatthis moment. 6.5.5RatioBox Thenumberofelectronsinthestorageringcontinuouslydecay swithtime.Since theintensityofthesynchrotronlightisproportionaltother ingcurrent,theintensity goesdownwithtimeaswell.Unlesswerecordtheringcurrentev erysooften,this couldbetroublesomeforameasurementthattakeslongtimeorr equiresmanyscans foraveraging.Inordertoavoidthiscomplication,weusedaso -calledtheratioboxthat normalizesthedetectorsignalwiththeringcurrent.Therin gcurrentof1000mAis convertedto10V,anddeliveredtothebeamlineareathrougha BNCcable.Theratio 14 Weoftenusedaruorogold(aglass-lledTeron)lowpassltertof urtherincrease sensitivityatlowfrequencyend.Thecut-ofrequencydepend sonthickness,butitit typicallyaround60cm 1

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127 Table6{8:Characteristicsoftheberopticcable(SpecTran SpecialtyOpticsCo.Model F-MFD)usedtotransferlaserpulsestobeamlines. Attenuation 3.2dB/kmat850nm 0.9dB/kmat1300nm Bandwidth 160MHz-kmat850nm 500MHz-kmat1300nm CoreDiameter 62.5 3 m CladdingDiameter 125 2 m CoatingDiameter 250 15 m Numericalaperture 0.275 IndexProle Graded boxtakesthedetectorsignalasanumeratorandthecurrentsig nalasadenominator. Thedenominatorsignalisampliedbyagainof10withanopera tionalamplier.The divisionisdonebyananalogmultiplierchipindividermode, anditsoutputsignal becomesindependentoftime.Theratioboxwasanecessarycomp onentforournotonly thetime-resolvedmeasurements,butalsotheordinarylinearme asurements. 6.5.6FiberOpticCableandPulseDelivery Theparticularopticalberweused(SpecTranSpecialtyOpti csCo.ModelFMFD)isastandardcommunicationgrade,multi-modetypewith agradedrefractive indexprole.Itisoptimizedfortransmissionat850and1300nm .Table 6{8 showsthe specicationsofthisberopticcable.Thebareopticalber consistsofacentralcore (madeofsilica, n 1 : 45),acladdingwithslightlylowerrefractiveindex,andala yerof acrylicbuer.Thelargecore(62.5 mdiameter)hasagraded-indexprole(asopposed toastep-index)inordertosupportspropagationofanumberof modes( e.g. ,linear, sinusoidal,helical,andcombinationsofthese).Ingraded-in dexbers,therefractive indexofthecorevariesgraduallyasafunctionofradialdist ancefromthebercenter. Eachmodecarriesonlyafractionofthetotalopticalpower. Thegraded-indexproleis achievedwithindex-modifyingdopantssuchasGeO 2 .Forgreaterprotection,theberis incorporatedintocablethathasapolyethyleneinnersheath (900 mdiameter),Kevlar reinforcingstrands,andanopaquePVCouterjacket(3mmdiame ter).Thelightexiting theberexpandsintoaconewitha30 vertex.SMAtypeopticalberconnectorsare axedtotheendofthebercable.Cablescanthenbeconnecte dwithascrew-type

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128 CoreCladding Acrylate Coating (250m dia.) m Bare Cabled Bare Fiber PE inner sheath (900m dia.) m Kevlar Strands PVC Outer Jacket (3 mm) n1n2a b Graded-IndexFigure6{19:Structureoftypicalopticalbercableandgra ded-indexprole. union.Figure 6{19 showsthestructureofthetypicalopticalbercableaswellas the graded-indexprole. NotallthelaserpowerinjectedintotheopticalberattheU6la serhutchis deliveredtothesample.Dispersion,absorption,andscattering arethethreeproperties ofopticalbersthatcauseattenuation,oramarkeddecreasei ntransmittedpower[ 69 ]. Absorptionandscatteringareobviouscausesofattenuationtha tneednoexplanation. Therearethreemaintypesofdispersion:chromaticdispersion( alsoknownasmaterial dispersion),wave-guidedispersion,andmodaldispersion.Thech romaticdispersion resultsfromthefactthattherefractiveindexofthebermed iumvariesasafunctionof wavelength.Inordertosatisfytheuncertaintyprinciple,ap ulseoflightofduration t p mustnecessarilycontainaspreadoffrequencygivenapproximat elyby: 1 t p : (6.5) Hencethepulsesarebroadenedintimeastheypropagatethroug hthemedium.This canbecomeaseriousproblemwhenattemptingtotransmitverysh ortpulsesover

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129 alongdistancewithoutsomesortofdispersioncompensation.Inou rsystemwith 2pspulsesat 800nm,theintrinsicdispersionis0.12ps/mresultinginapulse broadeningofnearly4psasthepulsetravelsapproximately3 0mfromtheU6hutch totheU12IRorU10Abeamline[ 1 ].However,sincethisbroadeningismuchsmaller thanthesubnanosecondsynchrotronpulses,wearesafeinneglecti ngthiseectinour time-resolvedexperiment. Wave-guidedispersionalsocausessignalsofdierentwavelengt hstospread,similar tochromaticdispersion.Itresultsfromthefactthatnotallof thelightareconned tothecore.Somefractionofthelightactuallypropagatest hroughtheinnerlayerof thecladding.Becausetherefractiveindexofthecladdingis lowerthanthatofthe core,thesmallfractionoflighttravelsfastercausingspreadi ngofsignalswithdierent frequencies. Modaldispersionisrelatedtothefactthatapulseoflighttran smittedthrougha multi-modeberopticcableiscomposedofseveralmodesoflig ht.Sincetheraysofthe lightpulsearenotperfectlyfocusedtogetherintoonebeam,e achmodeoflighttravelsa dierentpath,someshortandsomelong.Asaresult,themodeswill notbereceivedat thesametime,andthesignalwillbedistortedorevenlostoverlo ngdistances. Thereareotherplacesthatcouldcauselossofphotons.Forinsta nce,inecient couplingofbeamandber,andkinksandsharpbendoftheberc ablearesome examplesoflossyparts.Fromourexperience,morethan70%ofth einputlaserpower atthehutch(justbeforethebercoupler)canbedeliveredwi thecientlycoupled andwell-preparedundamagedbercable.Wewereabletodeli vertheaveragepoweras highas500mW,whichcorrespondsto 10nJperpulse(power/PRF)for52.9MHz excitationfrequency,or3nJ/cm 2 laserruence(apulseenergy/beamsize)for1/4inches diameterlaserspot. 6.6ExperimentalTechniquesandSetups Uptothispoint,eachcomponentofcomplexexperimentalsetup wasdiscussed separately.Forthetime-resolvedexperiment,allofthesecom ponentshavetoworkasa

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130 unit.Inthissection,wewilldiscussourvarioustechniquesof steady-stateandpumpprobetime-resolvedmeasurements,andseeanoverallpictureof theexperimentalsetup. Attheend,themethodsofthelaserinsertionatthesamplelocat ionaredescribed aswell.Somesubtletiesinvolvedinusinginterferometersusi ngthepulsedsourcewas discussedin x 5.2.2 6.6.1PhotoinducedMeasurementsSteady-statePhotoinducedMeasurements Insteady-statephotoinducedmeasurements,wesimplylookforch angesinpropertieswhenmaterialisopticallypromotedtoitsexcitedeq uilibriumstatefromthe groundstate.Bytakingadierencebetweensignalswithlaser( excitationsource)on ando,wecanmeasureaspectrumshowingeectssuchasphotoindu cedabsorption, bleaching,orheatingduetophotoexcitation.Thereareseve ralwaystocollectdata forthistypeofmeasurement.Firstofall,thelasercanbeplace dineithertheCW modeormode-locked.Sourcecanbeeithersynchrotronorinte rnalsource,butin caseofusingsynchrotron,itismoreecienttosynchronizethel aserandsynchrotron pulsesneartheircoincidence.Foraspectrometerthatiscapa bleofdoingstep-scan ( e.g. ,SPS-200),thelasercanbechoppedandadierencesignalisdi rectlygenerated byalock-inamplierateachscannerlocation.Aftercompleti ngthewholescan,the photoinducedspectrum( e.g. T or R )isproduced.Thismethodallowsthestudy ofweakersignals,butithasthedrawbackofbeingsusceptibleto falsesignalscaused bysampleheatingifthesamplehasanytemperaturedependence initstransmittance (orrerectance).Alternatively,wecansimplytakespectraint henormalfashionwith thelaseronando,andthentakethedierence.Inthecaseofaf astscanbenchlike theBruker66,datapointsaretakenonthekHzscalewithscanrep etitionrateon theorderofseconds.Therefore,thesecondmethodistheonlyop tionavailable.Normallyphotoinducedspectroscopyrequiresonetodetectverysm allchanges,andthus high-sensitivitydierentialmethodsshouldbeemployedwhen everpossible. Tobemorespecicforthesteady-statemeasurementwiththeBruk er66or125, spectrawiththelaseronandoaretakenbyopeningandclosingt hesafetyshutterin

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131 theU6laserhutch(seeFigure 6{8 ),respectively.Theshutterisdesignedtobeopened (orclosed)remotelybysupplying5(or0)voltsDCtotheelectr onics.Thiscanbe donebyaDCpowersupplyconnectedtothecontrolpanelattheb eamline.Sincethe photoinducedsignalsaretypicallyverysmall,manysetsofscan smustbetakento suppressnoisesasmuchaspossible.Wetookaseriesofinterleaved setsofscanswith laseronandobyusingamacrofortheOPUSNT(thesoftwarefordata acquisition andcontrollingBrukerbench)thatsendscommandstotheDCpo wersupplythrough aserialcable.ForNsetsofMscans,themacroroutinewaswritten totakedatasa follows: Mscanslasero(1)Mscanslaseron(1)Mscanslaseron(2)Mscanslasero(2)Mscanslasero(3) .. Mscanslaseron(N-1)Mscanslaseron(N)Mscanslasero(N) TheM Nspectraforeachonandocasesarethenaveraged.Thisschemew as usedminimizethelongtermdriftsindetectors,source,andoth ercomponents.In manycases,normalizedquantitiessuchas T = T ( T on T off ) = T off and R = R ( R on R off ) = R off aredenedasthephotoinducedsignal,andwewilluse thesequantitiestorepresentphotoinducedsignalsaswell. Asanexampleofthistypeofmeasurement,photoinducedtransmi ttancemeasurementsofasemiconductinglmmayrevealDruderesponsefrom thephotoinduced absorptionatfarinfraredthatiscalculatedfrom T = T (see x 2.4.5 ).Photoinduced

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132 bleachingmayalsobeobservednearthefundamentalabsorption edgeifthelmsisthin enough. 15 Pump-probeTime-resolvedMeasurements Pump-probetime-resolvedmeasurementsarealsophotoinduced measurements,with anaddedcomplexityforobviousreasons.Onetechniqueistosim plysynchronizethe laserwiththeringandthentakeaseriesofphotoinducedspectr aasdescribedabove. Eachspectrumistakenwithaparticulardelaybetweenthelase randsynchrotron pulses,andspectraforasetofdierentdelaysrevealsthetimee volutionofthespectra forthephotoexcitedsystem.Thetransmittanceandrerectance canofcoursebeused tocomputeanintrinsicresponsefunction,suchastheopticalco nductivity,whichis nowalsoafunctionoftime.Thisisaveryusefultechniquesince ittakestimevarying spectroscopic( i.e. ,frequencydependent)data.However,itrequiresimpractic allylong time(longertimethaneverycomponentofexperimenttorema instable)tocollect closelyspaceddataset,andthereforespectraforonlyafewdela ysettingaremeasured. Forexample,thedelayissetsuchthatsynchrotronpulsearrives earlierthanlaserpule ( i.e. ,negativedelayinourdenition),andthenthesinglebeamspe ctrumofnon-excited stateistakenrst.Thisspectrumisusedasareferencespectrumm uchlikespectrum withlaserointhesteady-statemeasurement.Then,thedelayis shiftedbyusinga DCpowersupplytoapplyacertainvoltagetothevoltagecontr olledphaseshifterthat comesaftertheHPpulsegenerator(seeFigure 6{10 ),andthesinglebeamspectrum atdierentdelayismeasured.Thesemeasurementsarerepeated asinthesteady-state measurement,anddierencespectrumfromthereferencespectr umiscalculated. Inanothermethodoftime-resolvedmeasurement,weintroduce aphasemodulation forthelaserpulses.Thiscreatesaditherofthepump-probedel aybyasmallamount t (= t 2 t 1 )aroundagivendelaysetting t (=[ t 2 + t 1 ] = 2).Thechanges 15 Forabulksample,itisdiculttoobservebreachingduetoitsa bruptabsorption edgenearthebandgap.

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133 intheopticalresponse S ( t )( e.g. ,transmissionorrerection)ofthesampledueto modulation t isgivenbythetimederivativeofEq. 5.16 as: S ( t )= I 0 @ @t Z + 1 1 dt 0 I probe ( t 0 + t ) G ( t 0 ) t: (6.6) Thissignalisdetectedasalock-inamplieroutput.Whenasma llmodulationamplitudeisused,thistechniquegivesessentiallythetimederivat iveofthephotoinduced signal.Itisadierential(ormoreaccuratelyquasi-dieren tial)techniquethatismore sensitivetothoseopticalpropertieschangingonthenanosecon dtimescalebutless sensitivetoslowthermaleects.Becausethelaserisirradiating thesampleatveryhigh frequency(53MHz),thesampledoesnotundergotherelatively largeswingintemperaturethatmayhappenincaseofusingalowfrequencychopperfo rthesteady-state measurement.Aseriesofderivativesignalsaretakenataclosel yspaceddelaysettings, andafterafullsetisobtained,theycanbeintegratedtogive thetimedependence (dynamics)ofthephotoinducedresponseofthesample.Wesometim esreferredthis techniqueastheZPDdecaymeasurement.Figure 6{20 showstheoverallexperimental setupforthisparticulartypeofexperimentwiththerowofRF signalandpulsesof bothpumplaserandsynchrotron.Notethataspectrometerisnotsh owninfrontof thesampleforsimplicityofthediagram.Withthistechnique, weeitherconnectedthe Ox-BoxdirectlytotheU12IRbeamlineorstoppedthescannerofsp ectrometeratsome xedposition(ideallyatthezeropathdierence,ZPD).With thissetup,weobviously cannottakespectroscopicdata, 16 butinsteadwemeasurethefrequency-integrated, singlebeamspectrumshapeweightedsignal( i.e. ,thephotoexcitedresponseaveraged overtheentirespectralrangeoftheexperiment)fromwhichw ecandeterminethe overalltimedecayofthesample. 16 Thedithertechniquecanbeemployedtotakespectroscopicdat eifweuseastepscaninterferometer.Theditherfrequencythatcanbeusedisl imitedtoafewhundred HzduetoPZTresponseasexplainedinthecontext.Thisismanyor dersslowerthan thesamplingrateoftherapidscaninterferometermakingunsui tabletousethistechniqueforacquiringspectroscopicdata.

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134 Pulse Generator Coherent Synchro-Lock Coherent Mira Ti:Sapphire laser ~100 Hz Func. Gen. Lock-in Driver Driver 52.9 MHz RF 52.9 MHz RF + Delay Control signals Sample Reference signal 52.9, 17.6, 5.9 MHz synchrotron IR pulses /2 divider/N divider 105.8 MHz52.9 MHz l /2 52.9, 17.6 or 5.9 MHz laser pulses Phase mod.sig. IR detector signal Time-dep. Sig. (Derivative) VUV Ring Fiber optic cable Figure6{20:Schematicofexperimentalsetupwiththemostofm ajorcomponents.It showstherowofelectricalsignalsandpump-probepulses.Noteth ataspectrometeris notshowninfrontofthesampleforsimplicity. Insidethelaserhead,laserditheringisachievedbyoscillatin gthemirror(M3in Figure 6{7 )mountedonthepiezoelectrictransducer(PZT)withafuncti ongenerator. Thecavitylengthischangedinthiswaymodulatingthephaseo flaserpulseswith respecttothatofsynchrotron.Theamountofphasemodulation( ditheramplitude) isdeterminedbythevoltageapplied.Wetypicallychosetouse ditheramplitudeless thanthesynchrotronpulsewidth( e.g. < 700psfordetunedmode).Forthedierential measurementstheidealdithersignalisasquarewave.However,a sweincreasethe ditheramplitude,westartdrivingthePZTbeyonditsmaximum safecurrent,and eventuallylosemode-lock.ThislossoccursbecausethePZTcan notmovethedistance askedforataglowrateofthesquarewave.The20MHzfunctiongen eratorweused certainlyexceedsthebandwidthofthePZT.Fromthestandpoi ntofintrinsicnoise, ahighfrequencyditherwouldbethemostdesirable.However,be causeofthelimited

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135 DC BiasTee RF(53MHz) Function Generator(Square Wave)Signal Out HPPulse Generator (delay) DC Power Supply Phase Shifter (delay) Lock-inAmplifier Reference OutDetector Signal Synchro-Lock 900 Control Box BeamlineU6 HutchAC DC AC+DCFigure6{21:Newschemeofcontrollingditheramplitudeandfr equencyatbeamline. bandwidthofthePZT,highfrequencyditherisnotpossible,an dwetypicallydithered atnear100Hz. Thefunctiongeneratorusedforthedelaymodulationwasconn ecteddirectlyto theSynchro-Lock900controlboxandplacedintheU6laserhutc htodrivethePZT directly(SeeFigure 6{10 ).Inordertocontroltheditheramplitudeandfrequency convenientlyatbeamline,dierentschemeisproposed.Figur e 6{21 showsthisnew method.Inthismethod,aDCbiasteeisintroducedbeforethe pulsegenerator.Itis abroadbandbiasinsertionteewithaDCblockingcapacitorth atisdesignedtohave averylowcutofrequencyofontheorderofonly1kHzwhenusedw ithaninductor. ItpassesfastRFsignalwithDCosetsuppliedfromthefunctiongen erator'ssquare wavewithaminimumofwaveformdistortion.Thus,ifwesettotri ggertheHPpulse generatoratthezerocrossingonthepositiveslopeside,theinpu tsignalfromthebias teeshiftstheRFsignalasthesquarewavechangesitsstateresult ingintheoutputwith delayanddithersignalalltogetheratthebeamline. Thereareafewissuesthatcouldyieldproblemsintheditherme thod.Firstofall, thefactthattheopticalpathinthelaserheadisfolded(seeFi gure 6{7 )couldmove thebeamacrosstheoutputslitasM3movesbackandforthfordit hering.Ifthebeam isnotwellcenteredattheslitthiscausesapowerructuationt hatisinphasewiththe dithersignal,andcanbemisinterpretedasspurioussignal.Thi ssignaltypicallyshows upasaDCoset,mainlyduetoheatingeect.Itismoretrouble someifthesample showsalargesteady-statephotoinducedresponsecomparedtothe time-dependent

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136 response.Byopeningtheslitsucientlywide,thistypeofructu ationcanbereduced. Thereisanothersourceofpowerructuationinthesystem.Asexpl ainedearlier,the EOM/2pulsepickerpasseseveryotherpulsesof105.8MHzpulsespro ducedbythe laser.IfthephaseoftherotatingpolarizerwithinthePockel scellisproperlyadjusted, suchthatthetimedelayisditheredsymmetricallyaroundthep eaktransmission throughthepolarizer,anotherpowerructuationisintrodu ced.Whenthepulsepicker's phaseisnotproperlyadjusted,thewrongsetofpulsesstartsshowi ngupinthesignal fromthediodeafterthepulsepicker.Weadjustedthephasesotha tthewrongpulses disappearandtherightpulseshavethehighestpeakintensityju stbylookingata oscilloscopescreen,butitwouldbemoreaccuratetouselock-in technique.Thedriftof thepulsepickerphaseduringmeasurementcausedtroublefromti metotime. Asmentionedbrieryin x 5.2.1 ,thepump-probemeasurementrequiresthattherebe negligibleexcitationbytheprobepulsesascomparedtothose createdbypumppulses. Becausethesynchrotronisabroadbandsource,itmaycontainsub stantialnumberof photonsthathavehigherenergythanthethresholdforexcita tion.Thisproblemcanbe eliminatedeitherbyusingopticallowpasslterstoconstrain thespectralrangebelow thethresholdenergyorbyusingabandpasslterstorestrictthe spectralrangetothe regionofinterest.Forexample,weusedablackpolyethylene( polyethylenewithcarbon ller)lmjustafterthediamondwindowforthefarinfraredm easurements. 6.6.2LaserInsertion Inordertophotoexcitethesample,pulsesoflaserlightmustbeb roughttothe samelocationasthoseofthesynchrotron.Thisisachievedbyx ingtheendofthe opticalberatafewcentimetersinfrontofthesample.Thela serlightexitingthe bernaturallydivergesata 30degreeangle,andilluminatesanareaupto1cm indiameteronthespecimen.Smallerregionscanbeilluminat edbyeithermoving

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137 Opti-stat cryostat Sample NIR mirror Lens Optical Fiber Cable Synchrotron pulses Windows Opti-stat cryostat Sample Optical Fiber Cable Synchrotron pulses Windows (B) (A) Figure6{22:Schematicrepresentationofthelaserinsertionse tupswithOptistatcryostat.(A)SetupfortheOx-Box.(B)SetupfortheBruker66'ssamp lechamber. theberclosertothesample,orusingsmalllensesorGRIN(gradie ntindex) 17 rods tofocusthelaserlight.Asmallerlaserbeamspotwasusedwheneve rhigherenergy density(ruence)wasnecessaryforagivensample.Weattachedasm allcollimating asphericlens[ThorlabsInc,Fibercollimationpackage,ARCo ating600-1050nm,NA =0.25, f (mm)=11.00]attheveryendoftheberwithSMAconnector,an dsimply maintaineditrelativelyclosetothespecimen.Thismakesthe beamspotapproximately 1/4inchesindiameter.Figure 6{22 (A)showsaschematicrepresentationofthislaser insertionsetup.Theberendiscontainedcompletelywithint hevacuumbox,andthus itiscompletelysafetooperatethelaser.Thisschemewasadopt edforthemostofour pump-probeexperimentswithOptistatintheOx-Box.Noeortw asmadetofocusthe 17 Gradientindexmicrolenseshavearadiallyvaryingindexofr efractionthatcauses anopticalraytofollowasinusoidalpropagationpaththrough thelens.Theycombine refractionattheendsurfacesalongwithcontinuousrefract ionwithinthelens.

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138 Cryostat shroud Sample holder Cold finger Stainless steel fiber guide Optical fiber cable Figure6{23:SchematicofthelaserinsertionsetupwithHeli-tr ancryostat.Fiberoptic cableisfednearthetopofthecryostat. ~1mWDiode laser Last portion of optical fiber cable Sample Sample holder Fiber coupler Figure6{24:Lowpowerreddiodelasercoupledtoopticalber cablewithSMAconnector.Thissourceoflightcanbefedintothesamplevacuumcompa rtmentorheli-tranto checkalignmentofthelightbeamontothesample.light.IncaseofusingthesamplechamberoftheBruker66,there isnotenoughspace toplacetheberendunionwithoutbendingthecablesharply. Therefore,amirrorhas tobeusedtoplacetheberendatmoreconvenientplace.Thisc aseisillustratedin Figure 6{22 (B).Wehavelenseswithseveraldierentfocallengthtoadjust thespot sizeatthespecimen. WhentheAPDHeli-tranisused,thebercableisbroughtnearthe samplethrough asmallstainlesssteeltubewhichisbenttopointthebeamexitin gthebareberonto thesample.SeeFigure 6{23 .Thespotsizeinthissetupisdeterminedbythedistance fromthebertiptothesample. Alignmentoftheopticalber,especiallytoplacetheendofth eberattheproper distancefromthesample,isdonewiththeaidofalowpowervisib lelightsource(0.95 mWdiodelaserat670nm),coupledintothelastportionoftheb ercablewithan

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139 SMAconnector(seeFigure 6{24 ).Thepositionofthebertipisadjusteduntilthe beamspotisvisuallycenteredonthesample.Thisavoidsperfor mingalignmentwith theTi:Sapphirelaser.

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CHAPTER7 OPTICALCONDUCTIVITYOF -MoGeTHINFILMS 7.1Introduction BCStheoryhasbeenproventobeverysuccessfulinexplainingqu antitativelythe phenomenaofsuperconductivity[ 38 ].Thebasicideaisthattheinteractionbetween electronsandlatticevibrations( i.e. ,phonons)causesanattractiveforcebetweenthe electronsbelowacertaintransitiontemperature T c .Thisinturncausestheelectrons toformpairsandalsocausesagaptodevelopinthespectrumofel ectronicexcitations aroundtheFermienergy.Thus,itrequiresanenergyofatleast theenergygap2to breakpairs.InweakcouplingBCSsuperconductors,theenergyg apatabsolutezero temperature,2 0 ,isknowntobeapproximately3.5 k B T c .Astemperatureisincreased toward T c ,electronpairsarebroken,andthegapcloses.Farinfraredspe ctroscopywas thersttechniquetoprovetheexistenceofthegap[ 70 71 ],andcandetectthisenergy gapevolutionwithtemperature.MattisandBardeenapplied theBCStheorytoderive expressionsforthecomplexopticalconductivityfordirtyli mitBCSsuperconductors, fromwhichthetransmissionofelectromagneticradiationthro ughthinsuperconducting lmscanbecalculated[ 45 ].Theircalculationshowsthatatabsolutezerotemperature therealpartofopticalconductivity 1 ( )inthesuperconductingstateiszeroforphotonenergiessmallerthanthegap2 0 .Thisareathusremovedfromtheconductivity, bythesumrule,showsupasadeltafunctionatzerofrequency[ 72 73 ].Theimaginary partoftheopticalconductivity 2 ( )canbecalculatedfrom 1 ( )byKramers-Kronig equations,andthedeltafunctionof 1 ( =0)givesadominant1/ formto 2 ( )at lowfrequencies.Thissortofbehaviorisobservedinmostmetall icsuperconductors. Inthepresenceofdisorder,itiswellknownthatmetallicsystem sexperiencesignificantalterationintheirphysicalproperties.Anomalousdiu sionleadstolocalizationof electronsandarelatedenhancementoftheCoulombinteract ionviareducedscreening 140

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141 ( i.e. ,anincreasein ,therenormalizedCoulombinteractionparameter)[ 4 74 ].With asucientdegreeofdisorder,abulkmetallicsystemcouldevene xperiencesomeform ofmetal-insulatortransition.Inasystemoflowerdimensions,th ecouplingtodisorder increases,andpronouncedeectsareexpected. Inthecaseofsuperconductors,theeectsoflocalizationandt herelatedenhancementintheCoulombinteractioninherentlycompetewithatt ractiveinteraction,thus withsuperconductivity[ 46 75 76 77 ].Therewereanumberofstudiesreportedon thissubjectin80's,beforetheemergenceofhightemperature superconductors,but manyquestionshavenotbeenanswered,yet.Theoriespredictt hatsuchcompetition reducesthetransitiontemperature.Ofparticularinteresta retwo-dimensional(2D) superconductorsinwhichthedegreeofdisordercanbeadjusted byvaryingtheappropriatematerialparameters.Inanideal2Dsystem,therelev antparameteristhe sheetresistance R 2 whichcanbecontrolledwiththelmthickness d .Severaltransport experimentsshowedasharpreductionin T c withincreasing2Ddisordereveninthe weaklylocalizedregime[ 6 7 8 78 79 80 81 82 83 ].AmorphousMoGelmisthoughtto beoneofthebettersystemsthatservesasamodelforstudyingthi sinterplaybetween superconductivityanddisorder.Here,afterbriefdiscussionofa ppropriatetheoretical background,wepresentourresultsofthefarinfraredtransmit tanceandrerectance measurementsonasetofthin -MoGelmswithdierentthicknessinbothnormaland superconductingstates.Astrongsuppressionof T c withincreasing R 2 isobservedas expectedfromtheoryandearliertransportexperiments.Adet ailedanalysisusingthe Mattis-Bardeenexpressionsfortheopticalconductivityisdi scussed.Thedependenceof superconductingparameters,suchassuperruiddensityandmagne ticpenetrationdepth, on R 2 isalsoreported. 7.2Background 7.2.1InfraredPropertiesofSuperconductors Spectroscopictechniqueshavebeenwidelyusedforthestudyof superconductors. Sincetheenergygap2intheelectronicexcitationspectrum correspondstothe energyofphotonsinfarinfrared,itisofparticularintere sttoseehowsuperconductors

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142 behavesinthisfrequencyrange.Here,wewillconsiderasample intheformofthin lmonasubstrate,andlookattheratiooftransmittancethroug hthelminthe superconductingstatetothatinthenormalstate.Thisformofsa mpleisadvantageous (overabulksample)becauseamoderatechangeintheconductiv itycanproducea relativelylargechangeintransmission. Asdiscussedin x 2.4.4 ,thetransmittancethroughathinlm(providedthat multiplererectionswithinasubstrateisneglected)isgiven bytheGlover-Tinkham equation: T = 1 1+ Z 0 ~ d n +1 2 4 n 4 c 1 d + n +1 2 + 4 c 2 d 2 ; (7.1) where d isthethicknessofthelm, n istherefractiveindexofthesubstrate,and Z 0 is theimpedanceoffreespace(4 =c incgs;377ninmks).Thus,thetransmittanceisdirectlyrelatedtotheelectricalpropertiesofthematerial ( i.e. ,thecomplexconductivity ~ ). Forathinlminitsnormalstate,onecangenerallyexpectthe electronscattering frequency(1 = )tobemuchhigherthanthefrequenciesofinteresthere( i.e. 1 inEq. 2.94 ),leadingtoarealandfrequency-independentnormal-state conductivity n Then,thetransmittanceinthenormalstate T n shouldalsobeaconstantdepending onlyon n d T n = 1 1+ Z 0 n d n +1 2 : (7.2) TheDCresistancepersquareofthelm, R 2 =1 = n d = n =d ,canbemeasured experimentally.Thus, T n canbedeterminedfrom R 2 ,andviceversa. Byformingtheratioofthetransmittanceinthesuperconducti ngstatetothatin thenormalstate,onecanndtherelativetransmittance T s = T n as T s T n = 1 h T 1 = 2 n +(1 T 1 = 2 n ) 1 n i 2 + h (1 T 1 = 2 n ) 2 n i 2 : (7.3)

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143 0123 0 1 2 3 s 1 s 2ConductivityRatio(s s/s n)Frequency( w / w g ) 0123 0 1 2 3 T =0.99 T c T =0KTransmittanceRatioFrequency( w / w g ) (/) TT snFigure7{1:TheMattis-Bardeencalculationsforthefrequen cyandtemperaturedependenceofconductivityratioandthetransmittanceratioi nthedirtylimit.Thetemperaturesfortheconductivityratioareat0K,0 : 6 T c ,0 : 8 T c ,0 : 9 T c ,and0 : 99 T c .The temperaturesforthetransmittanceratioareat0K,0 : 5 T c ,0 : 6 T c ,0 : 7 T c ,0 : 8 T c ,0 : 9 T c ,and 0 : 99 T c .Calculationsat T =0Kareshowninthickerlines.Thesampleisassumedto have R 2 =50n, T c =10K,2 0 =24 : 46cm 1 ,1 = =1000cm 1 ,and n =3 : 05. Thus,theformsof 1 ( ) = n and 2 ( ) = n determinethefrequencydependenceofthe relativetransmittance T s = T n .BasedonBCStheory, 1 MattisandBardeendeducedthe frequencydependenceof 1 ( ) = n and 2 ( ) = n [ 45 ].TheMattis-Bardeencalculation wasdoneintheextremeanomalouslimitandareapplicablefo rdirtysamples( l= 0 1),whichisusuallythecaseforthinlms,andsoonappliedtothe transmissionofthin lms[ 18 70 71 88 89 ].Figure 7{1 showsthetheoreticalfrequencyandtemperature dependenceof 1 ( ) = n 2 ( ) = n ,and T s = T n forasamplewith R 2 =50n, T c =10K, 2 0 =24 : 46cm 1 ,1 = =1000cm 1 ,and n =3 : 05.Thetemperaturedependenceinthe Mattis-BardeencalculationarisesfromtheBCStemperatured ependenceofthegap. 1 Transmissionmeasurementsinvolvingstrong-couplingsupercond uctorswereshown todeviatefromtheMattis-Bardeenpredictions[ 84 ].Namintroducedacorrectionfor strongcouplingtotheMattis-Bardeentheory[ 85 86 ].Thiscorrectionwasshowntohave betteragreementwiththeexperimentaldata.Later,Ginsber g,Harris,andDynespublishedarstordercorrectionforstrongcouplingthatonlytak esthechangein 2 ( ) intoaccount[ 87 ].

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144 At T =0K, 1 =0forallfrequenciesbelowthegapfrequency( g 2 = ~ )except at =0whereithasadeltafunctiongivenby 1 ( )= 2 ps 4 ( ) ; (7.4) where ps istheplasmafrequencywhichisrelatedtothedirty-limitma gneticpenetrationdepth (giveninEq. 4.27 )by ps = c= .Notethat ps issmallerthanthe plasmafrequency p whichisrelatedtotheLondonpenetrationdepth L by p = c= L Photonswithenergylessthan2donothaveenoughenergytobr eakCooperpairs, andthustheyarenotabsorbed.Fromthedeltafunctionof 1 ,theKramers-Kronig relationsdeterminethedominantpartof 2 atlowfrequencies( ! g )tobe 2 ( )= 2 ps 4 = c 2 4 2 = N s e 2 m! : (7.5) Since 1 =0below g ,thelosslessinductiveresponse 2 ,whichgoesas1 =! ,governs thebehaviorofthelminthelow-frequencyrange,andinfac t,themonotonicdecrease of 2 resultsintheinitialriseof T s = T n withfrequency.Notethatatlowfrequencythe ratiogoestozeroas 2 duetothediverging 2 .Thetemperaturevariationofthe low-frequencylimit( ! g )of 2 = n isgivenbyasimpleanalyticform 2 n = ~ tanh 2 k B T : (7.6) AssumingthattheBCStemperaturedependenceof( T ),thisequationtogetherwith Eq. 7.5 providesthetemperaturedependenceofthesuperruiddensity N s ( T ).For ! g ,photonshave,atleast,theminimumenergyrequiredtobreak Cooperpairs, andalossyconductivitycomponent 1 appearsandincreaseswithfrequency.Near thegapfrequency g ,both 1 and 2 aresmallresultinginahighertransmissionin superconductingstatethaninnormalstatearound g .Then,as 1 becomesmoreand moreeective, T s = T n startsfallinggradually.Thepeakpositionin T s = T n roughly representsthesuperconductinggapfrequency g ;thisisespeciallytrueforlmswith large R 2 .Infact,apeakintheexperimental T s = T n curvewasanearlyevidencefor thepresenceofanenergygapintheexcitationspectrumofsuper conductors.Ifwe

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145 observe 2 closelynearthegapfrequency,itstartsfallingfasterthan1 =! .Thisisdue toanother 2 contributionfromtheedgeof 1 near g .For ! g ,thepresenceofthe gapbecomesunimportant,andsuperconductingandnormalstat esactessentiallythe same.Atthesefrequencies, 2 becomesnegligiblysmall(solongas 1 = ),andthe lossyconductivity 1 determinesthebehaviorofthelm. Atnitetemperatures(0
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146 thelocalizationandelectron-electroninteractiontheor yproposedbyMaekawaand Fukuyama(MF)[ 46 ].Theirtheorydescribesthequantumcorrectionstothetheo ryof dirtysuperconductorsbyAnderson[ 90 ]andGorkov[ 91 ], 3 andisvalidfor2Dsystem intheweaklylocalizedregime.AccordingtotheMFtheory,th eCoulombinteraction isenhancedandthedensityofstatesofelectronsaredepressedb ylocalizationeects resultinginaconsiderablereductionof T c .Theapproximateresultoftheirperturbation calculationisgivenby ln T c T c 0 = s 2 ln 5 : 4 0 l T c 0 T c 2 s 3 ln 5 : 4 0 l T c 0 T c 3 ; (7.8) where s = g 1 N (0) ~ 2 E F 0 ; (7.9) and T c 0 isthetransitiontemperatureofthebulkmaterial, 0 and l arethecoherence lengthcorrespondingto T c 0 andthemeanfreepathassociatedwithelasticscattering, respectively, g 1 isaparameterthatrepresentstherepulsiveCoulombinteract ion ( g 1 > 0),and N (0)isthedensityofstatesatFermilevel.Theproduct g 1 N (0)isan eectivecouplingconstantwithvalueof 1associatedwiththeCoulombinteraction. E F istheFermienergyand 0 istheelasticscatteringtime.Thesheetresistanceis givenby R 2 = ~ 2 e 2 E F 0 : (7.10) Thersttermontheright-hand-sideofEq. 7.8 isduetothecorrectiontothedensityof statesduetotheCoulombinteraction.Thesecondtermisdueto theenhancementof thescreenedCoulombinteraction( i.e. ,therenormalizedCoulombinteractionparameter mentionedin x 4.2.6 )particularyatlongwavelength,whichisaconsequenceof Proximityeectsmightberelevanttosomeextent,butexperi mentalresultsargue againsttheirbeingtheprimarycauseoftheobservedphenomena .Anothermodelisa simplepair-breakingduetoretardationeects. 3 Intheirtheory, T c isnotaectedbystaticandnonmagneticdisorder.

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147 thedynamicalnatureofthescreeningbroughtaboutbythediso rder.Foramorphous lmswheretheratio 0 =l isexpectedtobemuchlargerthan1,thesecondtermisthe predominantcontributiontothe T c reduction.TheMFtheorypredictsinitiallinear decreasein T c withincreasing R 2 providedthat R 2 isnottoolarge.Thisis,infact, whatobservedalmostuniversallyinvarious2Ddisorderedsystems. By2D,itismeant thatthelmthickness d issmallerthanthethermaldiusionlength( ~ D=k B T ) 1 = 2 whichisthelengthscaleoflocalization,ontheorderof 100 A.Here, D isthe diusionconstantofelectronsgivenby[ e 2 N (0) R 2 d ] 1 .Notethatlmsmightremain3D withrespecttotheelectroniclengthparametersuchasmeanfr eepath l 7.2.32DModelSystems Inordertoexamineinterplaybetweenlocalizationandsuper conductivitywithless complications,itwouldbeidealtousemodelsystemswhichhavef ollowingproperties: 1.Theeectscanbeobservedclearly.2.Themeasureofdisordercanbechangedinacontrolledway.3.Themeasureofdisordercanbechangedoverawiderange.4.Allotherpropertiesarexed.5.Bulkpropertiesarethoroughlycharacterized. Therelevantmeasureofdisorderis1 =E F 0 ,whichisproportionalto in3D and R 2 in2D(seeEq. 7.10 ).Thus,byusinglmswithvarious R 2 ,yetkeeping constant,onecanunambiguouslystudythe2Deectspredictedb ytheMFtheory. 4 Homogeneous,amorphousthinlmscanbegrowntopossessproperti esdesiredfor modelsystems.Sincethistypeofmaterialhasrelativelyhighr esistivity,lmswith sucientlylarge R 2 areaccessibleforahomogeneousmaterial.Also R 2 canbevaried bysimplychanginglmthickness d whilemaintainingthebulkproperty relatively 4 Intypicalthinlmsystems,evenifthenormalstateelectronicp ropertiesdonot change,thelmresistivityoftenchangeswiththelmthickne ss,particularlyfor d 100 A.

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148 constant(providedthat d l ).Theuseofamorphouslmsmakesinterpretationof experimentconsiderablysimplerbecausealldisorder-sensitive band-structure(smearing) eectshavemostlikelyalreadyoccurredandarenotfurthera lteredbyreducingthe lmthickness.Sampleuniformityisalsoimportanttoseparateo uteectson T c dueto localizationandinteractioneectsfromthoseresultingfro mthestructuresbuiltinto thelm( e.g. ,oxidation,clusteringand/orpercolation,orgrossinhomog eneities),which couldalsoinruencetheelectronscreeningatlongwavelength sand T c suppression.As longas ismaintainedconstant,theeectsfromthosestructuresshould benegligible. MoGeisaveryinterestingsystemofatransitionmetal(TM)-meta lloidmixture, inwhichthecompositioncanbevariedwidelyfromalowcoordi nation-numberrandom tetrahedralnetworkinthe -Gelimittoahighcoordination-numbermetallicglasslimit forMo-richalloys.Theamorphousphaseexistsoveraverybroad compositionrange, from20-100%Ge.ThechoiceofaTM-metalloidalloy,insteado faTM-TMalloy,leads tostabilityoftheamorphousphaseatroomtemperature.Inthe formofthinlms, materialswithproperGecontentssatisfythepropertiesliste dabove,andcanserveas modelsystems. 7.3ExperimentalDetails 7.3.1Samples Forouropticalstudy,fourlmsof -MoGewithdierentthicknessweregrown byco-magnetronsputteringfromelementaltargetsinaUHVsystem ontorapidly rotating(3rev/secor1 Adeposited/rev)single-crystalr-cutsapphiresubstrates(1 mmthick, n 3 : 05).A75 A -Geunderlayerwasrstlaiddownonthesubstrates toensuresmoothnessofthesubsequentlydepositedMoGelms.Rapid rotationofthe substrateduringdepositionseemstoimproveuniformcoverageo fultrathinlms.For lmspreparedinsimilarfashion,nosignofcrystallineinclusio nswereobservedbyx-ray andtransmissionelectronmicroscopy. ParametersforthelmsweusedarelistedinTable 7{1 .Thethickness d was determinedfromaquartzthicknessmonitor.Wedene T c asthetemperatureatwhich transmissionthroughalminsuperconductingstate T s becomesindistinguishable

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149 Table7{1:Filmparametersfor -Mo 79 Ge 21 thinlms. R 2 wasopticallydetermined from T n ,andgapparameter 0 isdeterminedbytting T s = T n datatothecurvecalculatedfromtheMattis-Bardeentheory. Film d (nm) T c (K) R 2 (n)2 0 (cm 1 )2 0 =k B T c A4.3 < 1 : 8505-B8.34.526011.53.68 C16.56.113116.53.89 D336.96918.53.85 0 100 200 300 400 500 R o [ W ]1/ d [nm-1] Ourfilms J.M.Graybeal'stransport Figure7{2:Sheetresistanceasafunctionofinverselmthickn essforfourlmsusedin ouropticalstudyaswellaslmsusedfortransportmeasurementb yJ.M.Graybealfor comparison.fromthatinnormalstate T n .Thethinnestlm(43 A)didnotgosuperconducting atthelowesttemperatureaccessibletooursystem(1.8K).Theshee tresistance R 2 ofalllmswasdeterminedfromEq. 7.2 withanextrapolationof T n to =0 (See x 7.2.1 ).Figure 7{2 shows R 2 asafunctionofinverselmthickness d forlms usedinouropticalstudy.Forcomparison,thesameplotforlmsu sedinGraybeal's transportmeasurementisalsoshown. 5 Thedatapointsforourslmsfallnicelyalong 5 NotethatourlmswerepreparedbyJ.M.Graybealinthesameway ashepreparedforhisthesiswork.Dierencesarethatthelmswith21 %Geusedforhisthesis weredepositedon -Si 3 N 4 substratewith10 A -Geunderlayerand28 A -Sioverlayer.

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150 0100200300400500 0.5 0.6 0.7 0.8 0.9 1.0 M-FTheory T c / T c0R o [ W ] Ourfilms J.M.Graybeal'stransport Figure7{3:Suppressionoftransitiontemperatureswithincre asingsheetresistancefor fourlmsusedinouropticalstudy.Forcomparison,Graybeal's transportresultisalso plottedwithttotheMaekawa-Fukuyamatheory. T c 0 isthebulkvalue(7.25Kfor Mo 79 Ge 21 ).Errorbaronthedatapointsarisesfromerrorincalibratio noftemperature controlleraswellaserrorindetermining T c value. astraightlinethatincludestheorigin,andthusthebulkresi stivity ofourlms hasaconstantvalue( 220 4 n cm)demonstratingsuccessfulrealizationof2D modelsystem.Thefactthat isessentiallyconstantimpliesthatthestructuresbuilt intothelmareinsignicant.Theresistivitywedeterminedhe rewasallmeasuredat 10K,butthetransportmeasurementsonlmspreparedinthesame wayshoweda logarithmicincreaseinresistivityasthetemperatureisdecr eased,whichistypically seenindisorderedsystemsintheweaklylocalizedregime.Aswill bedescribedlater,the gapparameter 0 isdeterminedbytting T s = T n datatothecurvecalculatedfromthe Mattis-Bardeentheory. InFigure 7{3 T c =T c 0 isplottedasafunctionof R 2 where T c 0 isthebulkvalueof thetransitiontemperature(7.25KforMo 79 Ge 21 ).Forcomparison,Graybeal'stransport resultisalsoplottedwithttotheMaekawa-Fukuyamatheory. Thegureshowsthat T c isstronglyandlinearlyreducedwithincreasing R 2 .Forthethinnestlmthatwent superconducting( d =8.3nm), T c issuppressedby 40%withoutshowingnoticeable

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151 changesinthebulkproperty .InGraybeal'swork[ 6 8 ],itwasshownthattherate of T c depressionincreaseswithincreasingGecontent.Ourresultappe arstofallmore rapidlythanthatofGraybeal'swith21%Gecontent.Itmight havebeenintendedto make21%Gecomposition,butourresultisclosertotheGraybeal 'sresultwith23% Gecomposition.Theoverallbehavior,however,seemstobecon sistentwithGraybeal's transportresultswhichwasinterpretedaslocalizationeec tsin2Dsuperconductors. 7.3.2Measurements Thefarinfraredmeasurementswereperformedattwobeamline s,U10AandU12IR oftheNSLSVUVring.U12IR,equippedwithaSciencetechSPS200Ma rtin-Puplett interferometer,wasusedforfrequencybetween5and50cm 1 .Forfrequenciesabove 20cm 1 ,aBrukerIFS-66v/Srapid-scanFourier-transforminterfero meteratU10Awas used.Abolometeroperatingat1.4Kprovidedanexcellentsensi tivityatfarinfrared, butawindowonthebolometerlimitedourmeasurementstofreq uenciesbelow100 cm 1 .TheOxfordInstrumentsOptistatbathcryostatmadeuspossiblet oreducethe temperatureofsamplesdownto1.8K.Detailsofeachexperime ntalapparatusare describedinChapter 6 .Thetransmittance T ( )andrerectance R ( )offourlms weretakenatvarioustemperaturesbelow T c ,andtheresultswerenormalizedtothe respectivenormalstatetransmittanceandrerectancetakenat 10K. 7.4Analysis Figure 7{4 showsourmeasurementsof T s = T n asafunctionoffrequencyatseveral temperaturesforthreelmsthatbecamesuperconducting.Th emeasuredratioappears tohavetheexpectedshape.Asmentionedearlier,thepeakin T s = T n isameasure ofthesuperconductinggap.Fromthegure,wecanclearlyobse rveshrinkinggapas temperatureincreasestoward T c foreachlm.Althoughnotshown,atagivenreduced temperature T=T c thegapshiftstolowerenergywithdecreasingthicknessoflms ( i.e. ,increasing R 2 )asexpectedsince T c goesdownwithdecreasingthickness,too. Themeasured T s = T n werettedtoEq. 7.3 usingEq. 7.2 for T n ontheright-handsideandtheMattis-Bardeenconductivityratioexpressions, 1 ( ) = n and 2 ( ) = n RecallthattheMattis-Bardeencalculationisvalidforadir tylimitBCSsuperconductor.

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152 020406080100 0 1 2 D Frequency[cm-1] 2.2K, 4.0K, 5.0K 6.0K, 6.5K 0 1 2 CTransmittanceRatio 2.2K, 4.0K, 5.0K 5.5K, 6.0K 0 1 2 B 2.2K, 3.0K 3.5K, 4.0K (/) TT snFigure7{4:MeasuredtransmittanceratioofthreeMoGelmsat severaltemperatures. Table7{2:Parametersusedforthettingof T s = T n curves. ParameterDescriptionvaluesnote n refractiveindexofr-cutsapphire3.05 v f Fermivelocity10 8 cm/sestimated l meanfreepath3 Aestimated 1 = scatteringrate17,600cm 1 1 = v F = 2 cl p plasmafrequency70,000cm 1 2 p = n 60 1 resistivity220 4 n cm Figure 7{5 showsthetfor T s = T n measuredat2.2Kforthreelms.Inthet,we allowedonlythegap2 0 tobeafreeparameter.Weused T c and R 2 valuesdetermined fromourmeasurementsforeachlmasdescribedabove.Otherpa rametersnecessary tocalculate T n aregiveninTable 7{2 .Themeanfreepathisestimatedtobe 3 Afor -MoGe,whichisroughlyoneinteratomicspacing.Thettothe dirtylimitBCS predictionlooksquitegood.Fromthet,thevalueof2 0 inTable 7{1 wasdetermined. Then,thisgivestheratio2 0 =k B T c ,whichwasfoundtobeslightlyhigherthanthe BCSweakcouplinglimitof3.5foralllms.Thevariationin2 0 =k B T c withthicknessis

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153 020406080100 0 1 2 DFrequency[cm-1]0 1 2 CTransmittanceRatio0 1 2 B Experimentat2.2K Mattis-Bardeen (/) TT snFigure7{5: T s = T n comparedwiththecalculationaccordingtoMattis-Bardeene xpressionsofconductivity.Only T s = T n takenat2.2Kareshownforeachlm. notnearlyasmuchasthe T c reduction(seeTable 7{1 ),andwemaysafelyconcludethat theyareidenticalwithinexperimentalerror. Inordertodeducetheopticalconstants,transmittancemeasure mentinanarrow spectralrangealoneisnotsucient.Therefore,wemeasuredre rectanceinthesame frequencyrangeasthetransmittancemeasurements.Figure 7{6 showstherelative rerectance R s = R n forthreelms.Justlike T s = T n R s = R n alsottedquitegoodtothe Mattis-BardeencalculationasshowninFigure 7{7 Fromthemeasuredtransmittanceandrerectanceofthelms,wew ereableto extractthecomplexconductivity 1 ( )and 2 ( )foreachlmatvarioustemperatures usinganalgorithmbasedontheapproachesofPalmerandTinkha m[ 89 ]andGlover andTinkham[ 71 ].Figure 7{8 (a)shows 1 ( )and 2 ( )ofeachlmat2.2K.For comparison,theBCSMBconductivityarealsoshown,and 1 tswellforallthree MoGelms.Thegapof2isnowquiteevident.Also, 1 approachesthenormalstatevalueforeachlmathigherfrequencies.Allthreelmsha veapproximatelythe

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154 01020304050 1.0 1.5 D Frequency[cm -1 ] 2.2K 4.0K 5.0K 6.0K1.0 1.5 CReflectanceRatio 2.2K 4.0K 5.0K 5.5K1.0 1.5 B 2.2K (/) RR snFigure7{6:MeasuredrerectanceratioofthreeMoGelmsatsev eraltemperatures. samenormal-stateconductivity, n ,of 4300n 1 cm 1 obtainedfromtransmittance measurementoflminnormalstate.Thisisveryclosetothevalu efoundbytransport measurement. 2 ( )hascorrectlineshapeof 1 =! atlowfrequenciesbuttendstobe abovetheBCScurveespeciallyathigherfrequencies. Thetemperaturedependenceoftherealandimaginarypartof theopticalconductivityforthe16.5nmlmisshowninFigure 7{8 (b).Therealpart 1 showstheclosing ofthegap2asthetemperatureapproaches T c .Theimaginarypart 2 shows 1 =! behaviorbelowthegap,andthereductionoftheinductivere sponseasthesuperruid densitybecomessmaller. 7.5Discussion Fromthe1 =! behaviorof 2 ( )belowthegap,wecanndthemagneticpenetrationdepth ( T )andsuperruiddensity N s ( T )usingEqs. 7.5 and 4.12 inwhich L is replacedby .Theyareproportionalto( 2 ) 1 = 2 and 2 ,respectively.Figure 7{9 showsthedependenceof and N s onthesheetresistance R 2 deducedfromthelow

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155 01020304050 1.0 1.5 D Frequency[cm-1] 1.0 1.5 CReflectanceRatio1.0 1.5 Experimentat2.2K Mattis-Bardeen B (/) RR snFigure7{7: R s = R n comparedwiththecalculationaccordingtoMattis-Bardeene xpressionsofconductivity.Only R s = R n takenat2.2Kareshownforeachlm. Table7{3:Valuesof N s and forthreelmsmeasuredat2.2K. Film d (nm) T c (K) R 2 (n) N s ( 10 21 cm 3 ) (nm) B8.34.52601.12768 C16.56.11311.49687 D336.9691.66652 frequencypartof 2 forthreelmsat2.2K.Table 7{3 liststhevalues.Strong T c suppressionwithincreasing R 2 wasdiscussedearlier(seeFigure 7{3 ).Asmentionedearlier, thiseecthasbeattributedtoelectronlocalizationand/o rCoulombinteractioneects. Dependenceofthesuperruiddensity N s issimilartothatof T c ,butrelationof N s to thelocalizationeectshasnotbeenunderstood,yet. Inaplotoftheconductivityratio 1 ( ) = n ( )asafunctionofphotonenergy ( i.e. ,frequency),missingarea, A ,of 1 ( )belownormalstateconductivity n ( )can beestimatedas n 2 C ,andthus n T c C 0 where C and C 0 aresomeconstants. Thisisareasonableapproximationespeciallyforweaklycoup ledBCSsuperconductors inthelowtemperaturelimit.Bythesumrule,thismissingareag oestotheareaunder

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156 01020304050 0 5000 10000 0 5000 10000 0 5000 10000 DFrequency[cm-1] CConductivity[ W -1 cm -1 ] B s1 s2 sn BCS 0 2000 4000 6000 50000 100000 s1[ W-1cm-1] 2.2K 4.0K 5.0K 5.5K 10K 01020304050 0 5000 10000 C Frequency[cm-1]s2[ W-1cm-1] s2wg/ w (a)(b)Figure7{8:(a)Opticalconductivitiesofthreelmsextrac tedfromthetransmittance andrerectancedataat2.2K.TheBCS,Mattis-Bardeentisalso shownforcomparison.(b) 1 ( )and 2 ( )atvarioustemperaturesfor16.5nmMoGelm.Insetshows 1 =! behaviorof 2 belowthegapfrequency. deltafunctionof 1 at =0,andhencetothesuperruiddensity N s .Thereforewe expecttohaverelationsas A N s T c .AsshowninFigure 7{10 ,ouropticalresults appeartogivelinearrelationshipbetween N s and T c .Alloftheresultsfromouroptical measurementsaretriviallyconsistentwiththoseexpectedforw eaktointermediate couplingBCSsuperconductors.Wedidnotndanyindication,a tleastoptically,that maybepeculiartotheeectsoflocalizationindisorderedsup erconductorsexceptthe strongsuppressionof T c with R 2 7.6Conclusion Inconclusion,wehavestudiedthefarinfraredconductivity~ ( )extractedfrom transmittanceandrerectancemeasurementsofthinamorphous superconducting MoGelmsofvariousthicknessesatvarioustemperatures.Allth reelmsmeasured appeartobeingeneralagreementwithdirtylimitBCSsuperco nductorswiththeratio 2 0 =k B T c slightlyhigherthanthatofweakcouplingvalue3.5.Strongr eductionof T c withincreasingsheetresistance( i.e. ,withdecreasingthickness)thatwasexplainedas

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157 50100150200250300 600 650 700 750 800 l [nm]Ro[ W ] 50100150200250300 1.0 1.5 2.0 N s [x10 21 cm -3 ]Ro[ W ] Figure7{9: R 2 dependenceofmagneticpenetrationdepth andsuperruiddensity N s forthreelmsmeasuredat2.2K.theeectsoflocalizationandrelatedenhancementofCoulo mbinteractionwasalso observed.Thesuperruiddensityalsodecreasesas R 2 increases,buthowtheseare relatedtotheeectoflocalizationremaintobeanswered.Th innerlmswithmore severelocalizationeectsmayrevealstrongdeviationfromt heresultsexpectedfor

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158 1.2 1.4 1.6 1.8 N s [x10 21 cm -3 ]Tc[K] Figure7{10:Relationshipbetween N s and T c of -MoGelms. asuperconductinglmwithoutelectronlocalization.Exper imentforthinnerlms, however,willbeharderbecause T c willbeloweredfurtherrequiringHe3system.

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CHAPTER8 TIME-RESOLVEDSTUDYOF -MoGeTHINFILMS 8.1Introduction Superconductivityisaphenomenonwhereelectronscondense intoboundpairs andanenergygapdevelopsintheelectronicdensityofstates bothaboveandbelow theFermienergy.ThetypicalenergyscaleisseveralmeV.Asweha veshowninthe lastchapter,thisenergygap,akeyparameterindicatingthe overallstrengthofthe superconductingstate,canbesensedbyfarinfraredspectroscopy .Electronictransitions acrossthefullgap(energy2)correspondtobreakingpairs,pr oducingexcitations calledquasiparticles.Breakingasignicantnumberofpairsl eadstoanonequilibrium conditionwherethesuperconductingstateisweakened(indic atedbyasmallerenergy gap)orevendestroyed.Thereturntoequilibriuminvolvesth erecombinationofthe excessquasiparticlesintopairs,releasingenergyofatleast2 (perrecombination event),usuallyasaphonon.Therateforthisprocessinvolves theinteractionbetween quasiparticles,whichisoffundamentalinterestforanytheor yofsuperconductivity. Kaplan etal. havecalculatedtherecombinationtimeforanumberofeleme ntal BCSsuperconductors,obtainingcharacteristicvaluesinthe1 0to100picosecondrange, dependingonmaterialandtemperature[ 92 ].Butthesystemdoesnottrulyrelaxon thistimescale,sincetheresultingexcess2phononsusuallybre akotherpairsat asimilarrate.Sotheexcessenergyistemporarilytrappedina coupledsystemof quasiparticlesandphonons.Thecoupledsystemeventuallyrela xesasthephonons escapetootherpartsofthespecimen. Weperformedtime-resolvedpump-probespectroscopytofollow thiscomplicated relaxationprocess.Thetechniqueexploitsthepulsednatureo fsynchrotronradiation. Amode-lockednear-IR/visiblelaser,synchronizedtothesynch rotronpulses,serves asanexcitation(pump)source.Thephotonsfromthelaserbrea kasmallfractionof 159

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160 Cooperpairs,leadingtoexcessquasiparticlesandanonequili briumstatethatevolves withtime.Thisnonequilibriumstatecanbesensedasasmallweak ening(downward shift)ofthesuperconductingenergygap,andincreasedfarinf raredabsorption.Our uniquepump-probetechniqueallowsustofollowthisabsorpt ion(andthereforethe excessquasiparticles)asafunctionoftime.Themagnitudeoft hissignalatagiventime providesameasureoftheexcessquasiparticleatthattime,and completeresultyields adecaycurveoftheexcessquasiparticledensityfromwhichwec andeducetheoverall relaxationtimeforthesample. Inthischapter,webeginwithasynopsisoftherelevantbackgr oundfornonequilibriumsuperconductivityanditsrelaxation,followedbyanex perimentaldetails.Then, wepresentourresultsoftime-resolvedstudyon -MoGelms.Resultsofmeasured quasiparticlesignalsatvarioustemperaturebelow T c areanalyzedusingthetheoriesby Kaplan etal. [ 92 ]andRothwarfandTaylor[ 93 ].WealsopresentourresultofphotoinducedgapshiftwithananalysisbasedonthetheorybyOwenandSc alapino[ 94 ]. 8.2Background 8.2.1NonequilibriumSuperconductivity Inasuperconductoratanynitetemperaturebelow T c ,quasiparticleexcitations arecontinuouslygeneratedbythermalprocess.Atagiventempe rature(withoutother externalagitation),thereisacompensatingrecombination processsuchthatCooper pairsandquasiparticlesareinthermodynamicequilibrium. Apairstate( k ; k # )is brokenwhenatleastoneoftheelectronsisscatteredintoanew state k 0 where k 0 # isunoccupied. Anyexternalsourceofenergy,suchasphotons,thatcansupplyatl east2of energycanintroduceexcessquasiparticlesintothesystem.Thi sleadstoanexcited statewherethesuperconductingstateisweakened.Experiment sbyTestardishowed thatlightofsucientintensitycanevenconvertthesupercond uctingstatetothe normalstate[ 95 ].Evidencewaspresentedthattheseeectswerenotduetosimpl e lattice-heatingtoatemperatureabove T c ,butwereapparentlyduetoanexcessnumber ofquasiparticlesinducedbyphotonabsorption.OwenandScal apinodevelopeda

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161 modelofsuperconductorsinwhichthenumberofquasiparticle sislargerthanthat ofthermalequilibriumshowingthatanexcessquasiparticlepo pulationreducesthe superconductor'sorderparameter(theenergygap)[ 94 ].Theirresultforthereducedgap attemperatures T T c is 0 3 = 8<: 0 2 + n 2 # 1 = 2 n 9=; 2 ; (8.1) where n isthedimensionlessexcessquasiparticlenumberdensityinunit sof4 N (0) 0 ( = N pairs ). 1 Forsmall n ,thisreducesto 0 1 2 n: (8.2) TheOwen-Scalapinomodelwaslaterconrmedqualitatively byParkerandWilliams frommeasurementsonsuperconductingtunneljunctionsillum inatedwithoptical radiation[ 96 ]andbySai-Halasz etal. frommicrowavererectivitymeasurements[ 97 ]. Whenthenumberofexcessquasiparticlesbecomessucientlyla rge,thegapshrinksto thepointwhereitcollapsestozero,eventhoughtthelattice temperatureisstillbelow T c Whenthesupplyofenergyisstopped,thesystemrelaxestowardth ermodynamic equilibrium.Thedetailsofthisrelaxationprocessesarequi tecomplex.Inanexperiment wherelightpulseofenergyconsiderablylargerthan2( e.g. ,near-IRorvisiblepulse laser)isusedasexcitation,high-energyquasiparticlesarec reated,whichquickly( femtoseconds)relaxtolowerenergybydominantelectron-ele ctron( e e )scattering. Then,theselow-energyquasiparticlesrelaxfurthernowmain lyviaelectron-phonon ( e ph )scattering( picoseconds),eventuallypopulatingquasiparticlestatesnea r thegapedge[ =( T )].Onthewaydowntothegapedge,additionalquasiparticles arecreated(thisprocessiscalledmultiplication).Anoneq uilibriumstateexistsuntil 1 4 N (0) 0 ( = N pairs )isthedensityofelectronsstronglyaectedbyenteringthe pairedstate.Notethathere N (0)isgiveninunitsofstates/eV cm 3

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162 allexcessquasiparticlesrecombineintopairsandrejointhe condensate.The e e 2 and e ph 3 scatteringtimesareexpectedtobeconsiderablyshorterthant heaverage timeforaquasiparticletondamateandrecombine.Therecom binationisprimarily accompaniedbyphononemission,thustransferringenergytoth elattice. 4 EarlyBCS calculationsofthetheoreticalquasiparticlelifetime(or recombinationtime) R for temperatureswellbelow T c werecarriedoutbySchrieerandGinsberg[ 100 ],Rothwarf andCohen[ 101 ],andLucasandStephen[ 102 ].Theytookintoaccountthe e ph interactionandtheavailabilityofquasiparticlesindeter miningatemperaturedependent R ,butignoredphononsarisingfromtherecombinationevents.I naddition,theexcess quasiparticledensity N q wasassumedtobesmallcomparedwiththermalquasiparticle density N q ( T ).RothwarfandTaylor,then,recognizedthattheeectofph ononsemitted viarecombinationevents[ 93 ].Theseextraphonons,possessingenergiesofatleast2, canbreakpairsandrecreatequasiparticleswhichsubsequentl yrecombineintopairs, releasingphononsagain.Inotherwords,theexcessenergyiste mporarilytrappedin acoupledsystemofquasiparticlesandphonons.Thisprocess,theso -calledphonon trappingeect,continuesuntiltheseexcessphononsarelostt otheirsurroundings, leadingtoanenhancedapparent(oreective)lifetime e .Phononscanbelostby variousmechanisms.Forexample,theycanescapeintothesubstra te,orthehelium bath,orthesampleholder,ortheymaydecaybyinelasticscatte ringotherthanpairbreaking[ 103 104 ].Notethatiftheenergyofthephononsisdegradedbelow2,i t 2 Electron-electronscatteringcanplayasignicantroleinme talswithlargeDebye temperaturesandlowsuperconductingtransitiontemperatur essuchasAl[ 98 ]. 3 Theelectron-phononscatteringtimeincreasesasthetempera tureisloweredowing toadecreaseinthethermalphononpopulation. 4 Itisalsopossibleforrecombinationtotakeplaceviaphotonem issionsimultaneouslysatisfyingenergyandmomentumconservation.However,Bur stein etal. calculated thelifetimeforradiativerecombinationismuchtoolongto bethedominantmodeof decay[ 99 ].Forexample,thelifetimeforrecombinationbyphotonemi ssionforleadat2 Kwascalculatedtobe0.4seconds,whilethatbyphononemissionc alculatedbySchriefferandGinsbergforleadat1.44Kwas43nanoseconds[ 100 ].

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163 paired electrons & thermal quasiparticles excessquasiparticles D excess 2phonons D ~picoseconds ~femtoseconds high-energy (~ ) quasiparticles EFlow-energy quasiparticles &phonons EDebye 1/ tR1/ tB h n bottleneckphonon escape 1/ tg Figure8{1:Simpliedmodelofmulti-steprelaxationprocesse sofquasiparticlescreated byabsorptionofphotonsofenergymuchlargerthan2.iseectivelylost,sinceitcannolongerbreakpairs.Theeect ivetimeforphonons tobelostoutoftheenergyrange ~ n 2(orphonon-escapetime) r dependsin complicatedwaysonthelmgeometry,theacousticmatchingo fthelmtosubstrate, andotherenvironmentsaroundthelm,butitisroughlyinde pendentoftemperature. Further,if r isconsiderablylongerthanothertimeconstants,theexcessqua siparticles andphononcannearlyequilibrateonatimescale 1 eq 1 R + 1 B ; (8.3) where R istheintrinsicquasiparticlelifetime(orrecombinationti me)and B isthe phononlifetimeagainstpari-breaking(orpair-breakingti me).Thebottleneckinthe completerelaxationprocessesofquasiparticlesisthephonon escapetime.Figure 8{1 schematicallyshowsamodelofrelaxationprocesseswithalight pulseasanexcitation. Inanexperimentwhereaxedenergyisdepositedinthesuperco nductor,this internalequilibriumconditionallowsonetodeterminehow theexcessenergyisdistributedbetweentheexcessquasiparticlesandexcessphonons. Forexample,consider thecasewherethequasiparticlelifetimeisextremelylongco mparedtothephonon lifetime( R B ).Pairsofquasiparticlesarerarelyremovedbyrecombinati onevents,

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164 andthefewthatdooccurarequicklyfollowedbypair-breaki ngeventsthatrestorethe quasiparticlepairs.Soonewouldndtheexcessenergyinthefo rmofalargenumberof quasiparticlesandveryfewphonons.Thiscanbequantiedas N q ( T ) N q (0) = 0 ( T ) 1+2 B ( T ) R ( T ) 1 ; (8.4) where N q (0)isthequasiparticledensityifalltheexcessenergyisinth eformof quasiparticles.Thefactorof2isaconsequenceofthefactthat pair-breakingcreates twoquasiparticleswhileremovingonlyonephonon.Aswillbed iscussedin x 8.2.2 ,the expectedtemperaturedependenceof R and B impliesthattheexcessenergyresides mostlyinexcessphononsfortemperaturesnear T c ,shiftingpredominantlytoexcess quasiparticlesastemperaturesbecomelowerandlower.Atab solutezero,theexcess energyisentirelyintheformofexcessquasiparticles. RothwarfandTaylarderivedasetofdierentialequationsth atdescribethe behaviorofquasiparticleandphonondensitiesinsuperconduc tingthinlmsinteracting withanexternalquasiparticleinjectionmechanism.Inthewe akperturbationlimit ( i.e. N q N q ( T )),theseequationscanbelinearizedtoobtaincoupledequat ions forexcessquasiparticledensity N q [ N total q N q ( T )]andexcessphonondensity N n [ N total n N n ( T )]as @N q @t = I 0 + 2 N n B 2 N q R ; (8.5) @N n @t = N q R N n B N n r ; (8.6) where I 0 isthequasiparticlesinjectionratepercm 3 .Inanexperimentusingalight pulseasapair-brakingmechanism, I 0 =0afterphotonabsorption.Thetimedependent solution[ 98 ]oftheRothwarf-Taylorequationsappropriateforsuchanex perimentisan

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165 exponentialdecayofexcessquasiparticledensitywithaneec tivelifetimegivenby 5 e r + R 2 1+ r B : (8.7) Atlowtemperature, R 6 istypicallylargecomparedto r and e isapproximated by R r = 2 B provided r B .Attemperaturenear T c R becomessmalland e approaches r .Atheoreticalcalculationoftemperaturedependentlifet imesbyKaplan etal. [ 92 ]isdiscussedinthefollowingsubsection. 8.2.2TemperatureDependenceofLifetimes Inmostsuperconductors,inelasticphononprocesses,particularl yrecombination withphononemission,isthemostdominantlow-energy( i.e. ,neargapedge)quasiparticlerelaxationprocesses.Kaplan etal. derivedtheenergyandtemperaturedependence oftheintrinsicrecombinationtime R ( !;T )intermsoftheelectron-phononspectral density 2 (n) F (n) 7 [ i.e. ,thephonondensityofstates F (n)weightedbythesquare ofthematrixelementoftheelectron-phononinteraction 2 (n)]aswellasthephonon pair-breakingtime B (n ;T )intermsof 2 (n) 8 foradirtylimitsuperconductorinor verynearthermalequilibrium[ 92 ].Theirresultsare 1 R ( !;T )= 2 ~ (1+ )[1 f ( )] Z 1 + d n 2 (n) F (n)Re n [(n ) 2 2 ] 1 = 2 1+ 2 (n ) [ n (n)+1] f (n ) ; (8.8) 5 ForlargerperturbationstheRothwarf-Taylorequationsar enonlinearandcannotbe solvedanalytically. 6 For T T c R istypicallymuchgreaterthan1ns. 7 Inprinciple, 2 F canbedeterminedfromsuperconductingtunnellingmeasureme nts. 8 Thephonondensityofstates F (n)caninprinciplebeobtainedfromneutronscatteringexperiments.Then, 2 (n)canbededucedbytakingratioof 2 (n) F (n)from tunnellingmeasurementsto F (n).

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166 and 1 B (n ;T )= 4 N (0) 2 (n) ~ N i Z n d! ( 2 2 ) 1 = 2 (n )+ 2 [(n ) 2 2 ] 1 = 2 [1 f ( ) f (n )] ; (8.9) where isthequasiparticleenergyrelativetotheFermienergy,nis thephononenergy, f ( )istheFermi-Diracdistribution, n (n)istheBose-Einsteindistribution, isthe electron-phononcouplingconstant(ormass-enhancementpara meter;see x 4.2.6 ), N (0) isthesingle-spinelectronicdensityofstatesattheFermiener gy,and N i istheion numberdensity. Forasimplemodelofametalatsucientlylowfrequencies, 2 (n)isknow toapproachaconstant,and F (n)isproportionalton 2 asfoundfromtheDebye model[ 105 ].Thus,thelow-frequencybehaviorof 2 F canbeapproximatedby 9 2 (n) F (n)= b n 2 ; (8.10) where b isaconstantthatcharacterizesagivenmaterial.Usingthisfo rmof 2 (n) F (n) inEq. 8.8 ,Kaplan etal. demonstratedthattherecombinationrateforquasiparticles at thegapedge[ =( T )]foraBCSweak-couplingsuperconductorfollowsauniversal temperaturedependence: 1 R ( ;T )= 1 R 0 F R ( ;T ) ; (8.11) where F R ( ;T )canbeexpressedintermsofthemodiedBesselfunctions K 0 and K 1 as F R ( ;T )= k B T c 3 1 X n =1 ( 1) n +1 +exp n k B T (" 4+ 3 k B T n +2 k B T n 2 # K 1 n k B T + 4+ k B T n K 0 n k B T ) : (8.12) 9 Somematerials,suchasstrong-couplingsuperconductorsPband Hg,showsignicantstructurein 2 (n) F (n)evenatlowfrequencies.

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167 InEq. 8.11 ,asimplescalefactor R 0 isdenedas R 0 (1+ ) ~ 2 b ( k B T c ) 3 : (8.13) Thisparametercontainsallthenecessarymaterialconstantsa ndhandlesvariationsbetweendierentmaterials.Atlowtemperatures,theseriesinEq 8.12 rapidlyconverges, leadingtolow-temperaturebehavior: 1 R ( ;T ) = 1 R 0 1 = 2 2 0 k B T c 5 = 2 T T c 1 = 2 e 0 =k B T : (8.14) Thus, R increasesexponentiallyas e 0 =k B T atlowtemperatures,rerectingtheexponentialdecreaseinthermalequilibriumquasiparticledensity,w hichgoesas N q ( T ) T T c 1 = 2 e 0 =k B T : (8.15) Therefore,weseethat R ( T ) 1 =N q ( T ),whichisnotdiculttoimaginesincetwo quasiparticlesmustmeeteachotherbeforecondensingintoapa ir. Inasimilarfashion,fromEq. 8.9 ,Kaplan etal. showedthatthepair-breakingrate forphononsatthethresholdenergyn=2( T )alsofollowsauniversaltemperature dependence: 1 B (2 ;T )= 1 B 0 F B (2 ;T ) ; (8.16) where F B (2 ;T )= 0 [1 2 f ()] : (8.17) InEq. 8.16 ,anothercharacteristicscalefactor R 0 containsalltherelevantmaterial parametersspecictoagivematerial,andisdenedas B 0 ~ N i 4 2 N (0) h 2 i 0 ; (8.18) where h 2 i isanaverage 2 denedby 3 h 2 i Z 1 0 2 (n) F (n) d n : (8.19)

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168 0.250.500.751.00 0.0 0.5 1.0 1.5 Lifetimes[ns]Temperature( T / Tc) tR tB teff Figure8{2:Universaltemperaturedependenceofintrinsicrec ombinationtime R phononpair-breakingtime B ,andeectiverelaxationtime e ,whicharevalidfor weak-couplingBCSsuperconductorswitharbitrarilyspecie dparameters R 0 and B 0 Thephononescapetimehereis0.2ns.Notethatthesecurvesareob tainedusingthe approximateformof 2 (n) F (n)= b n 2 Since2phononsmustndavailablepairsbeforebreakingthe m,wecanimagine thatthepair-breakingtimeroughlybehavesas B ( T ) 1 =N s ( T ),where N s ( T )isthe superruiddensity.Thefactor[1 2 f ()]inEq. 8.17 rerectsthissortofbehavior. Figure 8{2 showstheuniversaltemperaturedependenceofintrinsicrecom bination time R andphononpair-breakingtime B usingEqs. 8.11 and 8.16 witharbitrarily speciedparameters R 0 and B 0 .Theeectiverelaxationtime e isalsoplottedusing Eq. 8.7 with0.2nsforthephononescapetime r .Attemperaturesnear T c ,wenotice that e approaches r asmentionedearlier.Notethatthesecurvesareobtainedusing theapproximateformof 2 (n) F (n)= b n 2 ,andarevalidforweak-couplingBCS superconductors. SubstitutingEqs. 8.11 and 8.16 intoEq. 8.4 yieldsthequasiparticlefraction N q ( T ) =N q (0)asafunctionoftemperature.Theshapeofacurveforagive nmaterial dependsontheratio R 0 = B 0 .Thus,measurementsof N q ( T ) =N q (0)canbeusedto

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169 determinethisratio R 0 = B 0 ,whichischaracteristictoaspecicmaterial,andcompare itwiththeory. Therateatwhichtherecombinationeventsoccurisameasureo fthecoupling betweenelectronsandphononsinBCSsuperconductors.Sincet hisistheinteraction responsibleforthepairingprocess,anunderstandingofthereco mbinationiscentralto anunderstandingofsuperconductivity.Unfortunately,itisq uitediculttomeasure themostintrinsicquantities R and B individuallywithreasonableaccuracy.However, wecandeterminetheratio R = B inareliablefashion.Time-resolvedspectroscopycan beausefultoolforstudyingthefundamentalpropertiesofsupe rconductorsinboth equilibriumandnonequilibriumstates. 8.3ExperimentalDetails Therehavebeenseveralexperimentaleortstomeasuretheee ctiverelaxation time e insuperconductors.Mostofthemeasurementsperformedearlier ,however, arerestrictedinsomeways.Forexample,Jaworski etal. indirectlydetermined e insuperconductingPblmsfortemperatureswellbelow T c frommeasurementson opticallyilluminatedtunneljunctions[ 106 ].Hu etal. carriedoutdirecttime-resolved studiesonSn-oxide-Sntunneljunctionsexcitedbylaserpulse s,buttemporalresolution wastensofnanosecondsatbest[ 107 ].Transientelectricalphotoresponsemeasurement onNblmsbyJohnsonwasadirectmeasurementofthelifetimewit hsubnanosecond temporalresolutionattemperaturesnear T c [ 108 ].Thetime-resolvedtechniqueatthe NSLSallowsdirectmeasurementswithatimeresolutionontheor derof100psovera widetemperaturerange( e.g. ,between0 : 28 T c
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170 Table8{1:Filmparametersfor -MoGethinlmsusedforthetime-resolvedexperiment. Film d (nm) T c (K) R 2 (n)2 0 (cm 1 )2 0 =k B T c B8.34.526011.53.68 C16.56.113116.53.89 D336.96918.53.85 absorption(typicallybelow60cm 1 foroursamples),andthepump-probetechnique providesaschemetofollowthepair-breakingandrecombinat ionprocessasafunctionof time.Amode-lockedTi:Sapphirelaser,synchronizedtothesyn chrotronpulses,serveas anexcitation(pump)source.Thelaserwastunedtoemitpulsedl ightatawavelength of 810nm(12,300cm 1 or1.53eV),wherethelasersystemseemstorunstably forlongperiodsoftime.Aneutraldensitylterwasusedtoadj ustthelaserpower deliveredtothesample.Weused14mW(0.25nJ/pulse)powerleve l(measuredatthe veryendofbercable)forreasonswhichwillbediscussedlateri nthissection.Because ourfocusisinfarinfrared,alltime-resolvedexperimentsw ereperformedatU12IRwith eitheraBruker66,Bruker125,orOx-Box(ourcustommadesampl ecompartment) connectedtothebeamline.Forprobepulses, 10 7-bunchdetunedmodewiththepulse widthsbetween0.6and1.0nswasusedforthemostofmeasurement s.Whenhigher temporalresolutionwasnecessary,weswitchedtothe7-bunchco mpressedmodewith pulsewidthsbetween0.3and0.5ns.Allsamplesweremountedvert icallyinserieson acopperclampwhichisinsertedintheOxfordOptistatbathcry ostat.Thesample spaceofthecryostatwascontinuouslypumpedtoachievetemper aturebelow4.2K. A1/4-inch( 6mm)diameteraperturedenedthesampleareailluminatedby laser andsynchrotronpulses.Thefar-infraredtransmissionthroughth elmswasdetected byabolometeroperatedat1.4K.Table 8{2 summarizestheparametersusedforthe time-resolvedexperiment.Detailsofeachexperimentalcom ponent,technique,andsetup arediscussedinChapter 6 10 Inordertoavoidpossibleexcitationbyprobepuleitself,abla ckpolyethylenelter wasplacedbeforethesample.

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171 Table8{2:Parametersusedforthetime-resolvedexperiment. Pumplaserwavelength 810nm(12,300cm 1 or1.53eV) Pumplaserpower 14mW(0.26nJ/pulse) Synchrotronoperationmode 7-bunchdetune(800mA),pulsewidth(0.6-1ns) 7-bunchcompressed(200mA),pulsewidth(0.3-0.5ns) Aperture(illuminationarea) 1/4-inch( 6mm)diameter Ditheramplitude 0.4ns Ditherfrequency 100Hz 8.3.1Time-resolvedMeasurements:QuasiparticleDecay Todeterminetheoveralldecaytime( i.e. e )ofthesamplewedoaspectrally integratedmeasurement.Forthismeasurement,theOx-Boxisd irectlyconnectedtothe beamlinebypassingtheinterferometer,oralternativelythe interferometermodulator (scanner)isleftataxedposition.Withthissetup,wemeasuret hetransmittance averagedovertheentirespectralrangeoftheexperiment,we ightedbythesource spectrumandinstrumenttransmission( i.e. ,thesinglebeamspectrum).Aswillbe justiedin x 8.3.4 ,themagnitudeofthissignalprovidesameasureoftheexcess quasiparticledensity.Inthemeasurements,thearrivalofthep umppulserelative tothatofthesynchrotronpulsewasmodulated(dithered)abou tadelaysetpoint andthedierential(derivative)transmittancesignalwasme asuredwithalock-in amplier.Weusedaditheramplitudeof 0.4ns.Ateachdelaysetpoint,thesignal wasaverageduntiltheS/Nbecamereasonable,thenthedelayw assteppedtothenext setpoint.Aftermeasuringforarelevantrangeofthepump-to-p robedelaytimearound coincidence(zerorelativedelaytime),thederivativesign alwasintegratedtondthe photoinducedchangeinthespectrallyaveragedtransmissionas afunctionoftime. Theresultingcurverepresentsadecayoftheexcessquasipartic lesfromwhichwecan determine e .Asanexample,Figure 8{3 showsboththedierentialandintegrated signalsforthe16.5nmlmat2.2K.Wehavebeencallingthisty peofmeasurementthe quasiparticledecaymeasurement.Moredetailsofthistechni quearediscussedin x 6.6.1 ModellingQuasiparticleDecaySignal Assumingthatonlyonekindoftimeconstantdeterminestheovera llrelaxationof thesample,thedecayofthecoupledsystemcanbeexpressedasasimp leexponential

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172 -10123456 -10 0 10 20 MoGe16.5nmat2.2KPhoto-inducedIRsignal[arb.]Time[ns] differential integrated Figure8{3:PhotoinducedierentialIRsignalvs.pump-to-pr obedelaytimeandthe resultofintegrationfor16.5nmlmmeasuredat2.2K.decay: G ( t )= 8><>: 0 t
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173 -10123456 0 10 20 Photo-inducedIRSignal[arb.]Time[ns] experiment fit exponentialdecay Figure8{4:Quasiparticledecaysignal(integratedsignalshow ninFigure 8{3 withat tothemodelfunction.Curveinbluedotsisasimpleexponenti aldecay. andtheGaussiansynchrotronpulse: 12 S ( t )= I probe ( t ) G ( t )= A p 2 Z 1 1 exp t 0 t 0 e exp ( t t 0 ) 2 2 2 dt 0 ; (8.22) whichhasananalyticalsolutionof S ( t )= A 2 exp 2 2 2 e t t 0 e 1 erf p 2 e t t 0 p 2 ; (8.23) whereerf( z )istheerrorfunction. Theintegratedsignalfrommeasurementsisttedtothemodelf unction 8.23 ,from whichwecandeducetheparameters e A ,and t 0 .Figure 8{4 demonstratesthe resultoftstothe16.5nmlmmeasuredat2.2K.Theexcellenta greementreassures usinouruseofasimpledecaymodelwithasinglecharacteristict ime.Thegurealso showssimpleexponentialdecay,drawnusingtheparameters A t 0 ,and e foundby tting.Wecannoticethatthepeakheightoftheintegratedsi gnalisconsiderably 12 Here,thepumppulseisassumedtoberepresentedbyadeltafunctio n.Thisisreasonablesincethepumppulseismuchshorterthantheprobepulse.

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174 smallerthanthemagnitudeoftheexponentialdecay A ;moreover,itisshiftedfrom t 0 atwhichtheexponentialdecaybeginsfalling.Thesearethec onsequenceofthe synchrotronpulsewidthbeingclosetotheeectivelifetimeof theexponentialdecay ( i.e. e 2 ).Inthelimitof e 2 ,theshapeoftheintegratedsignalcomesto coincidewiththesimpleexponentialdecay.Intheotherlimi t, e 2 ,theshape oftheintegratedsignalapproachesthatofthesynchrotronpu lse,andthedecayof thesamplecannolongerberesolved.Thus,thetime-resolutiono fourexperimentis determinedbythesynchrotronpulsewidth,whichisabout0.3n satbest. Asanalternativewaytoextractingthettingparameters,wec anusethederivative ofEq. 8.23 tottherawdierentialsignal: S 0 ( t )= S 0 0 A 2 e exp 2 2 2 e t t 0 e ( 1 erf p 2 e t t 0 p 2 r 2 e exp p 2 e t t 0 p 2 2 #) ; (8.24) wherewehaveaddedanotherconstantparameter S 0 0 .Thisprocedureissometimes easierordesirablebecauseitcantakecareofanosetthatmayex istinthederivative signal.8.3.2PhotoinducedGapShiftMeasurements Inprinciple,wecanobtainspectraateachpointonthequasipa rticledecay curve,whichwouldrevealhowphotoinducedspectrumchanges asafunctionoftime ( i.e. ,time-resolvedspectroscopy).Unfortunately,thesephotoindu cedspectralchanges areminusculeinmanycases,requiringagreatdealofaveraging toreachanacceptableS/N.Insteadofdoingthis,wemeasuredthephotoinducedsign al T = T where T ( T b T a )isthechangeintheinfraredtransmissionofalmbetweentwop oints onthedecaycurveshowninFigure 8{5 .Thewaywedothisexperimentissimilarto thatofsteady-statephotoinducedmeasurementsdescribedin x 6.6.1 .Insteadofopening andclosingashutter,wesupplyacertainDCvoltage(typically between1to2volts) tothevoltagecontrolledphaseshifterinordertomovethepum p-to-probedelaytime fromthepoint a tothepoint b .See x 6.4.4 formoredetailsonthesetup.Aseriesof

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175 0 0 Photo-inducedIRSignalTime a b Figure8{5:Twopointsbetweenwhichphotoinducedtransmitt ancechangeweretaken onadecaycurveforaphotoinducedgapshiftmeasurements.transmittancespectraweretakeat a and b inaninterleavedorder,and T = T was calculatedafterward.Notethatwedonotneedtoditherthepu mp-to-probedelaytime forthisexperiment,andobviouslytheinterferometermodul atorneedstobescanning unlikethequasiparticledecaymeasurement. Usingthesameparametersasfor T s = T n (seeChapter 7 ),wecant T = T allowingachangeintheenergygap asanonlyttingparameter.AsinEqs. 8.1 and 8.2 isrelatedtotheexcessquasiparticledensity N q [ 94 96 ].Foraweakperturbationandtemperaturesmuchlowerthan T c ,itcanbeshownafollowingapproximate relationship: = 2 N q 4 N (0) 0 = 2 N q N pairs : (8.25) Forthisexperiment,theBruker125atU12IRwasused.A125 mMylarbeamsplitterwaschosenforabettereciencybelow25cm 1 .Wealsousedathickruorogold lterinfrontofthedetectortocutoeverythingabove30cm 1 toimprovesensitivity aslowfrequencyaspossible.Forthesynchrotron,wetriedboth 7-bunchdetunedand

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176 050100150200 0.5 1.0 1.5 2.0 MoGe8.3nmat T =2.0K(0.44 T c )t eff [ns]Power[mW] Figure8{6:Fluencedependenceof e for8.3nmlmmeasuredataxedtemperature of2.0K(=0 : 44 T c ). compressed,butdetunedmodewaselectedsimplybecauseofbette rS/Nowingto higherbeamcurrent.8.3.3FluenceDependence AspointedoutbyRothwarfandTaylor[ 93 ],theassumptionofsmallexcess quasiparticledensitycomparedtothethermalquasiparticled ensitycaneasilybecome invaliddependingontheinjectionlevelsandtemperatureu sed.Althoughthepropertie ofhighlynonequilibriumstateofsuperconductorsisaveryin terestingsubjectinitsown right,wemadesurethatweareintheweakperturbationlimit[ i.e. N q N q ( T )]in ourquasiparticledecaymeasurementssincethatiswhatthethe oriesdiscussedabove arebasedon.Thisconditioncanbesatisedbyusingsucientlylo wlaserruence ( i.e. ,excitationpowerperpulse)andtemperaturesnottoolow.In ordertoassurethat wearetrulyinthedesiredlimit,wemeasuredtheruencedepend enceof e onthe8.3 nmlminasuperruidenvironmentat2.0K.AsFigure 8{6 shows, e isindependent ofthelaserpowerbelow 25mW,andbecomesfasterathigherpowers( i.e. ,higher quasiparticleinjection).Table 8{3 lists e and A foundbytting.Whenquasiparticles areoverlypopulatednearthegapedge,itiseasierforthemto ndmateswithwhichto

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177 Table8{3:Fluencedependenceof e and A .Thepowershownherewasmeasuredat theveryendofberopticcableatthebeamline. Power(mW) e (ns) A (arb.) 8 1.77 14.3 12 1.74 21.9 16 1.80 29.1 23 1.73 42.8 35 1.54 65.1 70 1.04 124 140 0.73 246 210 0.63 291 recombine,resultinginfasterrelaxation.Thetemperature2 .0Kcorrespondsto0 : 44 T c forthe8.3nmlm.Fromthisresult,wedecidedtouse14mW(0.26 nJperpulse)for thelaserpowerandtemperaturesabove0 : 44 T c foralllms. Wecancrudelyestimatetheexcessquasiparticledensityasfoll ows.Within 0.26nJperpulse,becauseofrerectivelossduetoseveralwindow sandsampleitself andotherlosses,let'ssay1/8oftheenergyisabsorbedinthelm. If1/2ofthe energyappearsasexcesslow-energyquasiparticles,approxim ately5 10 10 pairsare broken.For16.5nmlm(2 0 =16 : 5cm 1 )witha6mmaperture,thisresultsin N q 2 10 17 cm 3 excessquasiparticledensity.Using N (0)=0 : 33states = eV atom forMo 3 Ge[ 109 ]and6 : 42 10 22 atoms = cm 3 forMo,thisnonequilibriumdensity correspondsto0.2%ofthedensityofelectronsstronglyaecte dbyenteringthepaired state, N pairs = 4 N (0) 0 =8 : 5 10 19 cm 3 13 Thethermalquasiparticledensityforlow reducedtemperaturesisgivenby[ 96 ] N q ( T ) = 4 N (0)[ ( T ) k B T= 2] 1 = 2 e ( T ) =k B T : (8.26) At T =3K,wend N q ( T ) 10 18 cm 3 .Thus,theexcessquasiparticledensityis estimatedtobeabout20%ofthethermalquasiparticledensity, whichisnottoobad 13 Notethat N (0)isthesinglespindensityofstatesaswehavebeenusingthroug hout,andnottobeconfusedwiththenumberdensityat T =0K.

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178 0.0 0.2 0.4 0.6 0.8 1.0 NormalizedMagnitudeTemperature( T / Tc) Experiment(33nm) Ns(weak-coupling) Ns(strong-coupling) Figure8{7:Spectrallyaveragedfar-IRtransmittionvs. T=T c .Temperaturedependence ofsuperruiddensityforbothweak-andstrong-couplingBCSsupe rconductorareshown insolidlines.tobeconsideredastheweakperturbationlimit.Thisestimate willbecomparedtothe resultsfromthegapshiftmeasurements.8.3.4SpectrallyAveragedFarInfraredTransmission InChapter 7 ,wehaveseenthat T s = T n asafunctionoftemperaturegivesanidea ofhowthegapandthesuperruiddensityvarywithtemperature. Wementionedearlier thatthespectrally-averagedfar-infraredtransmissionprovi desameasureoftheexcess quasiparticledensity.Inthequasiparticledecaymeasurement s( x 8.3.1 ),wemeasurethe photoinducedchangeinthespectrallyaveragedtransmission T ave ,butwanttoknow N s (andinversely,thefreezingoutofthequasiparticles, i.e. N s is N q ).Inorderto determinetherelationshipbetween T ave and N s ,welookedatthespectrally-averaged far-infraredtransmissionasafunctionoftemperature.Figur e 8{7 showsthenormalized resultaftersubtractingthetransmissioninthenormalstate,tha tis, T ave ( T ) = T ave (0), where T ave ( T )= h T s ( T ) T n i .ThemeasurementsweredoneatU12IRwiththeOxBoxconnecteddirectly.Thesynchrotronlightwaschoppedat 100Hz,andthedetector signalwasmeasuredbyalock-inamplier.Then, T ave ( T ) = T ave (0)canbecompared

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179 with N s ( T )sothatadenitiverelationshipcanbedetermined.Ascanbesee ninthe gure, T ave ( T ) = T ave (0)closelycoincidewith N s ( T )givingarelationship: T ave ( T ) = T ave (0) N s ( T ) : (8.27) Thus,thespectrallyaveragedfar-infraredtransmission T ave sensesthenumberofpairs. Thistellsthatthespectrallyaveragedphotoinducedtransmi ssion T ave issimplyrelated to N s and N q as T ave N s = N q : (8.28) Therefore,ourmeasurementsofthephotoinducedtransmissionc hangeisameasureof theexcessnumberofquasiparticles,includingasafunctionof time. NotethattherelationshipgivenbyEq. 8.27 isnothingbutanempiricalobservation.ItalsohasbeenobservedforothersimpleBCSsupercondu ctors,suchasPb andNbN.Thisisprobablybecausethespectrallyaveragedtransmi ssionismuchmore sensitivetothechangesinareaunder T s = T n curvenearthepeakpositionthanthatat frequencieswhere T s = T n goesunder1becausethesinglebeamintensityapproacheszero inthesamespectralregion. 8.4AnalysisandDiscussion 8.4.1RelaxationTimes Wetookdataonthequasiparticledecayforallthreelmsfort emperatures between0.44 T c and T c .Alldatacouldbettedreasonablywellbythemodeldecay function,especiallyfordatatakenatlowreducedtemperatu rewheretheamplitude ofthesignalandtheS/Nwerehigher.Figure 8{8 showsthequasiparticledecaycurve ( i.e. ,photoinducedIRsignalafterintegrationofdierentialsig nal)asafunctionof pump-to-probedelaytimeforthe16.5nmforvarioustempera tures.Itisplottedona semi-logscale,withstraightlinesdrawnasaguidetotheeye,i llustratingchangesin slope.Notethatthecurvesinthegureshowsonlytheearlypart ofthedecaycurves. Theentiredecayprocessends(i.g.,becomesindistinguishabl efromthenon-excited state)after4to6nsfromexcitation.Theothertwolmsshowed similarcurves.Unlike

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180 012 0.1 1 10 PhotoinducedIRSignal[arb.]Time[ns] 0.44 T c 0.57 T c 0.67 T c 0.77 T c 0.89 T c Figure8{8:Quasiparticledecaysignalvs.pump-to-probedela ytimeforthe16.5nm lmplottedonasemi-logscale.Thelinesaredrawnasaguideto theeye,illustrating changesinslope.Thegraphshowsonlytheearlypartofthedeca ycurves. theresultsofJohnson[ 108 ]andCarr etal. [ 2 ],wedidnotndatwo-componentdecay: excessquasiparticledecayfollowedbyheatrow.Thisisproba blybecauseoursamples wereindirectcontactwithheliumgas,whichismuchmoreeci enttoextractheatfrom thesystemthanthecoolingthatreliesonthermalconductiont hroughcopperblock. Fromthet,wededucedparameters e A ,and t 0 inEq. 8.23 .Table 8{4 shows e and A forallthreelmsforvarioustemperatures.Thevaluesofpul sewidth2 : 35 Table8{4:Fitparameters e and A forallthreelmsatvarioustemperatures.Fitsare basedonthemodelfunctiongivenbyEq. 8.23 .Theuncertaintyindetermining e by ttingwasasmuchas 0 : 1ns. 33nm 16.5nm 8.3nm T=T c e (ns) A (arb.) T=T c e (ns) A (arb.) T=T c e (ns) A (arb.) 0.45 1.18 3.30 0.44 1.29 10.8 0.44 1.76 15.0 0.58 0.75 3.02 0.57 0.68 10.3 0.51 1.10 14.7 0.67 0.63 2.56 0.67 0.55 9.85 0.58 0.84 14.5 0.72 0.60 2.50 0.77 0.50 8.60 0.67 0.64 14.2 0.78 0.59 2.23 0.88 0.47 7.35 0.78 0.51 13.3 0.88 0.57 1.75 0.88 0.46 12.2 0.94 0.61 1.63

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181 0.0 0.5 1.0 1.5 2.0 t eff [ns]Temperature( T / T c ) 8.3nm 16.5nm 33.0nm Figure8{9: e vs. T=T c forthreelms.SolidlinesarethetstoEq. 8.7 usingtheuniversaltemperaturedependenceof R and B (Eqs. 8.12 and 8.17 )derivedbyKaplan et al. Allthreelmsweretusingthesamematerialparametersinthesc alefactors R 0 and B 0 (Eqs. 8.13 and 8.18 ).Thesame T c and2 0 valuesasfor T s = T n tswereused foreachlm.foundfromthetwereconsistentwiththeFWHMofthesynchrotron pulsesatthetime ofmeasurement. InFigure 8{9 e vs. T=T c (fromTable 8{4 )forthreelmsareshown,along withtstoEq. 8.7 usingtheuniversaltemperaturedependenceof R ( T )and B ( T ) (Eqs. 8.12 and 8.17 )derivedbyKaplan etal. Allthreelmsshowsimilarbehavior.As thetemperatureincreasestoward T c ,therearemanythermalquasiparticlesavailable forexcessquasiparticlestorecombinewith,resultinginashor tintrinsicrecombination time R .Wesee e decreasesgraduallyandnallyapproachesconstantvalue.Ph onon escapetime r canbeestimatedfromthisconstantvalueof e near T c .Wefound 500 psasthevalueof r .Therearesomevariationamongthelms,whichisprobablydue tovariationinheliumenvironmentatdierentsamplelocati on.Asthetemperature decreases,ontheotherhand,excessquasiparticleshavemorean dmorediculttimeto ndtheirmatesfromthermalquasiparticlesleadingtorapid lyincreasing R .Then, e isapproximatedby R r = 2 B .Below0 : 5 T c R wasfoundtoexceed1nsforalllms.

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182 Table8{5:Materialparametersthatcontributetothechara cteristicscalefactors R 0 and B 0 (Eqs. 8.13 and 8.18 ).Thesameparametervalueswerefoundtotreasonably wellforallthreelmsasshowninFigure 8{9 Z 1 (0)[=1+ ] 1.3 0.1 10 3 b (meV 2 ) 9 1 h 2 i (meV) 0.9 0.1 N (0)(10 22 states/eV) 2 0.2 N i (10 22 cm 3 ) 6 0.5 Table8{6:Relaxationtimes r R 0 ,and B 0 ,andtheratio R 0 = B 0 forallthreelms. ValueswerefoundfromttingofFigure 8{9 .Estimateduncertaintiesof R 0 B 0 ,and R 0 = B 0 inttingwere 10ps, 10ps,and0.05,respectively. Film d (nm) r (ps) R 0 (ps) B 0 (ps) R 0 = B 0 B 8.3 420 259 78 3.3 C 16.5 450 104 54 1.9 D 33 540 72 48 1.5 Forthetting,weneedthematerialparametersthatgointot hecharacteristic quasiparticleandphonontimes( R 0 and B 0 )inEqs. 8.13 and 8.18 .Unfortunately,the valuesoftheseparameters[ N (0), N i , b ,and h 2 i ]forMo 79 Ge 21 (thecomposition ofoursamples)arenotavailableanywhere.Thus,werstlookeda tdataforthe16.5 nmlm,andvariedthevaluesof b ,and h 2 i whilekeepingthevaluesof N (0) and N i xedtothoseforMo 3 Ge( 2 10 2 states/eV)andforMo( 6 10 22 cm 3 ),respectively.Then,wefoundthatthesameparametervalues werefoundtot reasonablywellfortheothertwolmsjustbychanging T c and2 0 totheappropriate valuesandadjustingonly r .Thematerialparametersusedinthettingaregivenin Table 8{5 .Theseparametersdeterminethecharacteristictimes R 0 B 0 ,andtheirratio R 0 = B 0 .Table 8{6 liststhemincluding r .Thettingwasverysensitivetothevalues of b ,and h 2 i changingthevaluesof R 0 and B 0 byafactorof2.Thus,thevalues of R 0 and B 0 shouldnotbetakenasaccuratenumbers.However,theratio R 0 = B 0 is muchlesssensitivetotheparameters.Thisgivesuscondenceto believethatthevalue of R 0 = B 0 isclosetotruevalueintrinsictothelms.

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183 45678 2 3 4 5 6 7 8 9 10 20 2 D 0 t B0 / t R0 [cm -1 ]T c [K] Figure8{10:2 0 B 0 = R 0 vs. T c .Forcomparison, T 3 c dependencepredictedbythetheory isshowninblue( i.e. ,aslopeof3).Theerrorbararetheresultsofthepropagationo f errorsindetermining2 0 and R 0 = R 0 Table8{7:2 0 B 0 = R 0 forallthreelms. Film d (nm) 2 0 (cm 1 ) T c (K) 2 0 B 0 = R 0 (cm 1 ) B 8.3 11.5 4.5 3.5 C 16.5 16.5 6.1 8.6 D 33 18.5 6.9 12 Assumingthatmaterialparametersarethesameforallthreelms, thetheory predicts: 2 0 B 0 R 0 / T 3 c : (8.29) Figure 8{10 showsourexperimentallydetermined2 0 B 0 = R 0 vs. T c alongwithadotted linewithslopeof3forcomparisonthetheoreticalprediction .Table 8{7 liststhevalues. Wefoundourresultisconsistentwiththetheory,supportingthe analysisofKaplan et al [ 92 ]. Inordertocompareourexperimentallydeterminedrelaxati ontimesforMoGelms withthoseforothermaterials,thecalculatedvaluesbyKapla n etal. [ 92 ]aregivenin Table 8{8 .ThevaluesforMoGeareonthesimilarorderofthoseforPb,In, andNb. Thetablealsoshowssomeassociatedmaterialparameters.Inaneo rttondsome

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184 Table8{8:Characteristicquasiparticleandphonontimes( R 0 and B 0 )andtheirratio aswellassomeassociatedparametersgiveninthepaperbyKapla n etal. Notethat Z 1 (0)=1+ Material R 0 (ns) B 0 (ns) R 0 = B 0 Z 1 (0) 10 3 b (meV 2 ) h 2 i (meV) Pb 0.196 0.034 5.76 2.55 5.72 1.34 In 0.799 0.169 4.73 1.81 9.43 0.913 Sn 2.30 0.110 20.9 1.72 2.32 1.14 Hg 0.0747 0.135 0.55 2.63 78.4 0.833 Tl 1.76 0.205 8.59 1.80 13.2 0.666 Ta 1.78 0.0227 78.4 1.69 1.73 1.38 Nb 0.149 0.00417 35.7 2.84 4.0 4.6 Al 438. 0.242 1810 1.43 0.317 1.93 Zn 780. 2.31 338 1.34 0.420 0.596 sortofregularity,wehaveplotted R 0 = B 0 against b asshowninFigure 8{11 14 Itis interestingtoseethatdatapointsforallthematerialslisted inthetableaswellasour MoGefallinthevicinityofthelineof b 3 = 2 ,buttherelationshipbetween R 0 = B 0 and b arenotclear. Indevelopingthetheoryofquasiparticleandphononlifetim es,Kaplan etal. assumedweak-couplingBCSsuperconductors,constant 2 ,andsimpleDebyelike n 2 dependenceofthephonondensityofstates.Ourresultsforthere laxationtime measurementsareconsistentwiththistheory,suggestingthatth ematerialparameters appeartobeconstantforallthreelmsofMoGe.Fromthemeasur ementsofnormal statetransmittance(seeChapter 7 ),wefoundtheresistivitystaysalsoconstantfor allthreelmsdespitetheirthicknesschange.Theresultsofop ticalconductivity measurementswerealsoconsistentwiththetheoryofweaktointe rmediatecoupling BCSsuperconductors.Onceagain,wedidnotndanyindication oftheeectsof localizationandrelatedenhancementofCoulombinteracti onwhichoughttochange somematerialparametersastheseeectsincrease.Aswemention edinChapter 7 ,our samplesmaynotbethinenoughtorevealsizabledeviationsfro mthesimpletheory andvariationsinmaterialconstants,butwestillcannotexpla inthesuppressionof 14 ThiswasdoneoriginallybyG.L.CarrattheBrookhavenNation alLaboratory.

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185 10 -4 10 -3 10 -2 10 -1 10 -1 10 0 10 1 10 2 10 3 10 4 8.3 16.5 33 Hg Tl In Pb Nb Sn Ta Zn Al a 2 ( W ) F ( W )= b W 2 t R0 / t B0 ~ b -3/2 tR0 /tB0b [meV -2 ] Figure8{11: R 0 = B 0 vs. b .ValuesforMoGelmsarededucedfromexperiment,while othermaterialsarethecalculatedvaluesbyKaplan etal [ 92 ]. T c asthethicknessofthelmsdecreases.However,thefactthatwec anhandlethe time-resolveddataforallthreelmswiththesamematerialpa rameterswhilechanging T c withdierentthicknessprovidesustotesttheanalysismadeby Kaplan etal. aswe sawinFigure 8{10 .ThisaddsuniquenesstotheMoGelms.Itwouldbeinterestingt o performsameopticalexperimentsonMoGelmswiththicknesst hinnerthanwehave now,butthen,theexperimentbecomesmoredicultas T c goesdownfurther.Weneed LHe 3 opticalcryostattothis. 8.4.2PhotoinducedGapShift Wealsomeasuredthephotoinducedchangesinfarinfraredtran smissionspectrum (photoinducedsignal) T = T betweentwopointsonthequasiparticledecaycurve showninFigure 8{5 .Detailsofthisexperimentwerediscussedin x 8.3.2 .Figure 8{12 showstheresultsfortwolaserpowers,23mW(0.43nJ/pulse)and53 mW(1nJ/pulse), onthe16.5nmlmat3K.ThesolidlinesaretheBCStsusingthesa meparameters determinedfromthe T s = T n measurements,butallowingthegaptobereducedby fromitsfullvalueatagiventemperature.Inmakingthet s,weassumedthat temperatureremainedconstantthroughout.Althoughthedata aresomewhatnoisy,

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186 101520 -0.02 0.00 0.02 23mW 53mW Frequency[cm-1] -/dTTFigure8{12:Photoinducedspectralchangesforthe16.5nml mat3K.Theresultsof twodierentlaserenergies23mW(blacksquares)and53mW(redc ircles)areshown, alongwithBCStsassuming0.6%and1.33%reductioninthegap, respectively. Table8{9:Photoinducedgapshiftsandfractionsofpairsbro kenbyphotoexcitationat twolaserpowersforthe16.5nmlmat3K. Power(mW) Energy(nJ/pulse) (cm 1 ) N q =N pairs (%) 23 0.43 0.05 0.30 53 1.0 0.11 0.67 gapshiftsof =0.05cm 1 and0.11cm 1 gaveacceptabletsfor23and53mW powerlevels,respectively.Notethatthetheoreticalcurvesa resmoothedtohavethe sameresolutionastheexperimentaldata.UsingEq. 8.25 ,wefoundthatapproximately 0.30and0.67%oftheCooperpairswerebrokenbyphotoexcita tion,respectively.These resultsareinreasonableagreementwiththeestimate(0.2%)we madein x 8.3.3 .The resultsaresummarizedinTable 8{9 .In x 8.3.3 ,wecalculated N pairs 8 : 5 10 19 cm 3 and N q ( T =3K) 10 18 cm 3 .Then,thedensityofexcessquasiparticlescreated by53mWofpowerleveliscomparabletothethermalpopulatio n( i.e. ,weareinthe intermediateperturbationlimit).Thisleadstoahigherpr obabilityforquasiparticles tondeachothercausingfasterrelaxationasweobservedinthe ruencedependence measurements(seeFigure 8{6 ).

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187 8.5Conclusion Inthischapter,wereportedtheresultsoftime-resolvedfari nfraredstudieson superconducting -MoGethinlmsofthreedierentthicknesses.Weusedamodelockednear-IR/visiblelasertoexcite(pump)thesystemandsync hrotronradiationto probethedynamicsofexcessquasiparticlesandphononswitha timeresolutionuptoa fewhundredpicoseconds. Thespectrally-averagedphotoinducedfar-infraredtransmi ssionwasfoundtobe closelyrelatedtothechangeinsuperruiddensitycausedbythep hotoexcitation.This empiricalobservationgivesusadeniterelationshipbetwee nthephotoinducedtransmission( i.e. ,far-infraredabsorptionchanges)andtheexcessnumberofqua siparticles. Bymeasuringthephotoinducedtransmissionatvarioustimingbe tweenpumpand probe,wewereabletoobservethedecay( i.e. ,relaxation)oftheexcessquasiparticlesas afunctionoftime,andtodeduceaneectiverelaxationtime atvarioustemperatures below T c usingaconvolutionofasimpleexponentialdecayandaGaussiansy nchrotron pulsestructureasattingfunction.Then,thetemperaturede pendenceofthiseective relaxationwasanalyzedbyusingtheeectiverelaxationtim eexpressionfoundasa solutionoftheRothwarf-Taylorequationsfornonequilibri umsuperconductorsandthe universaltemperaturedependenceoftheintrinsicrecombina tiontime(quasiparticle lifetime)andthepair-breakingtime(phononlifetime)der ivedbyKaplan etal. Thereasonabletstotheorywereachievedwithoutvaryinganymater ialparametersinallthree lms,andsomeinformationaboutthecharacteristicquasiparti cleandphononlifetimes wereobtained.Ourresultsarequiteconsistentwiththetheory byKaplan etal. ,which isbasedonweak-couplingBCStheory.Justasintheresultsofli nearspectroscopy,it seemsnoneofmaterialparametersarechanging,but T c changeswiththickness.We thinkthatthisisnotunderstoodcompletelyyet.Thinnerlm smayrevealsomeinterestingdeviationsrelatedtoelectronlocalizationandrela tedCoulombinteractions,but theexperimentbecomesprogressivelymoredicultas T c goesdownfurther. Wealsoreportedthephotoinducedchangesinthefar-infrare dtransmissionspectrumasalmisexcitedfromequilibrium.Theresultsoftwoex citationpowerlevels

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188 werettedwiththeBCScalculationallowingonlythesuperco nductingenergygapto changebyasmallamount.BasedonthetheorybyOwenandScalapi no,wededuced thedensityofexcessquasiparticles,whichisinreasonableagre ementwiththeestimate basedonthephotonruence.

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CHAPTER9 MAGNETO-OPTICALSTUDYOF -MoGeTHINFILMS 9.1Introduction Whentheappliedmagneticeldisbetweenalowercriticale ld H c 1 ( T )and anuppercriticaleld H c 2 ( T ),typeIIsuperconductorsallowapartialpenetration ofmagneticruxintothespecimen,developingnormalstatereg ionsintheformof thinlaments,whicharereferredasvortexlines.Thetypesof superconductorswere describedin x 4.2.3 .Asthestrengthoftheappliedeldincreasestoward H c 2 ( T ),a largerfractionoftheareaturnsnormal,andat H c 2 ( T ),thespecimenrevertscompletely tonormal. Inthedirtylimit( l 0 ,where l isthemeanfreepathand 0 isthePippard coherencelengthgiveninEq. 4.13 ),theeectivecoherencelength(Eq. 4.24 )isdeterminedbythemeanfreepath(seeEq. 4.26 ).Themeanfreepathofahighlydisordered materialismuchshorterthanitsLondonpenetrationdepth,a ndthustheyareusuallya typeIIsuperconductorhavingacondition 1,where istheGLparameterdened inEq. 4.22 -MoGethinlmswiththemeanfreepathofroughlyatomicspaci ng( 3 A)areextremelydirtymaterial,andaretypeIIsuperconducto rs. Wehaverecentlyattachedavertical-boresuperconductingm agnettothespectrometer,Bruker125HR,atthebeamlineU12IR.Withthismagne t,wetriedmagnetoopticalmeasurementsonthe33nm -MoGelm.Becausethesetupwasnewtoall ofus,ourexperimentservedthreepurposes:testinghowtheentir esystem,includingthespectrometer,themagnet,andothercomponents,funct ionswiththemagnet running,ndingthecorrecttiming( i.e. ,ndingalaser-synchrotroncoincidencefortimeresolvedexperiment),andconductingmeasurementsatthesame time.Throughthe measurements,wefoundseveralmattersthatshouldbeconsidered foroptimizinglater 189

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190 20406080100 1.0 1.5 2.0 TransmittanceRatioFrequency[cm -1 ] 0T 1T 2T 4T 6T 8T 10T (/) TT snFigure9{1:Measuredtransmittanceratioforthe33nmlmat3K inmagneticelds. experiments.Inthischapter,wemerelyshowourdataforthetr ansmittanceratioand quasiparticledecayinappliedexternalelds,andnospecicc onclusionsarediscussed. 9.2TransmittanceRatioinMagneticFields Figure 9{1 showstheelddependenttransmittanceratio T s ( H ) = T n ofthe33 nm -MoGelmat T =3K.Thenormalstatetransmittance, T n ,wasmeasuredat T =8Kwithoutapplyingtheeld.Theshapesoftheratioaregene rallythesameas weobservedintemperaturedependenceofthetransmittancera tiodiscussedin x 7.4 Asthestrengthoftheeldisincreased,thepeakheightdiminish edasiftemperature wereincreasedtoward T c ,andatapproximately10tesla,thetransmittancebecomes indistinguishablefromthatinthenormalstate.Thus, H c 2 ( T =3K)forthislmis approximately10tesla.Unlikethetemperaturedependentdat a,however,thepeak positionof T s ( H ) = T n staysnearlyatthesamefrequency.Althoughnotshownhere, thecurveforzeroeldtswelltotheMattis-Bardeentheory[ 45 ]onlyifwesetthe temperatureto4K,butotherwiseusingthesameparametersprev iouslydetermined (seeChapter 7 ).Atthetimeofmeasurements,wehadsomedoubtsincalibration ofthe

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191 0123 0.1 1 MoGe33nmat3.0KPhoto-inducedIRSignal[arb.]Time[ns] 0T 4T 8T Figure9{2:Quasiparticledecaysignalvs.pump-to-probedela ytimeinmagneticeld forthe33nmlmplottedinsemi-logscale.Thelinesaredrawnf oraguide.Thegraph showsonlyearlypartofthedecaycurves.temperaturesensor.Thus,itmighthavewellbeen4Kratherthan 3K.Obviously,the sensorhastobecalibratedbeforenextexperimentisconducte d. 9.3RelaxationTimesinMagneticFields Figure 9{2 showsthequasiparticledecaycurvesofthe33nm -MoGelm.Again thetemperaturereadingmeasuredusingthesamesensorwas T =3K.Withthe magneticeldapplied,quasiparticlesmaydiusetowardvort exeswherethereisno gapandthusnobottleneck.Astheeldstrengthincreases,norma lstateregionsgrow, andthereforeweexpectfasterrelaxationtime.However,what weobservedisaeld independenteectiverelaxationtime, 1 : 1ns,foralleldsupto8T.Notethat thevalueoftheeectiverelaxationtimeherecannotbecomp areddirectlytothat foundinthepreviouschapterbecausethesampleisincomplete lydierenthelium gasatmosphere,andalsobecausewecannotdenethetemperatur ewell.Aswecan seeinFigure 9{1 ,thetransmittanceat8Tisbarelydistinguishablefromthenor mal stateimplyingthatthemostofregionsareconvertedtothenor malstate.Thiseld independencecannotbeexplainedatthismoment.

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CHAPTER10 SUMMARYANDCONCLUSION Thisdissertationhaspresentedresultsofanextensivefarinfra redstudieson superconducting -MoGethinlmsofincreasingdisorder.Bothlinearandtime-r esolved measurementshavebeenmade.Inthischapter,wesummarizethe resultsofour experiments,andendthisdissertationwithconcludingremark s. OpticalConductivity Theopticalconductivity~ ( )of -MoGethinlmsofvariousthicknesseson athicksubstrateatvarioustemperatureswereextractedfrom transmittanceand rerectancemeasurementsinthefarinfrared.Threeoutoffou rlmsturnedtobe superconductingbythelowesttemperatureaccessibletoouropt icalcryostat.Transition temperature T c forthesethreelmswerefoundtobereducedprogressivelywith decreasingthickness(orincreasingdisorder).Thesuppressionof T c hasbeenattributed totheeectsoflocalizationandrelatedenhancementofCou lombinteraction,which inherentlycompetewiththeattractiveinteractionthatbo undselectronsintopairs.The sheetresistance,whichisarelevantmeasureofdisorderinourl ms,wasdetermined fromthenormalstatetransmittance.Itwasfoundtobeinversel yproportionaltothe thicknessofthelms,provingthenormal-statebulkresistivity remainsconstantfor allthreelms.Thissuggeststhattheobserved T c reductionisa2Deectsandnot duetothechangesinbulkproperties.Theobserved T c reductionasafunctionofsheet resistancewasconsistentwiththeMaekawa-Fukuyamatheory[ 46 ]onlyifweassume thattheGecontentsofourlmsaresomewherebetween21%Ge,w hichwasprobably intendedduringsynthesis,and25%Ge,whichrepresentsthestoic hiometriccompound Mo 3 Ge. Therelativetransmittanceaswellasrerectanceofathinlm [ i.e. ,aratioof transmittancethough(orrerectancefrom)alminitssuperco nductingstatetoits 192

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193 normalstate]aredeterminedbythefrequencydependenceoft hecomplexconductivity, ascalculatedbyMattisandBardeen[ 45 ]basedontheweak-coupling,dirtylimit BCStheory.Ourexperimentally-observedrelativetransmitt anceandrerectancewere generallyconsistentwiththeMattis-Bardeentheory,andfrom thetwedetermined theenergygap2 0 .Theratio2 0 =k B T c wasfoundtoberoughlythesameforallthree lmswiththevalueslightlyhigherthantheBCSresultof3.5. Therealpartoftheopticalconductivity 1 ( )extractedfromtransmittance andrerectancemeasurementsclearlydisplayedthepresenceof anenergygap,which shrinksastemperatureapproaches T c .Theimaginarypart 2 ( )representsthelossless inductiveresponseofthelms.Itshowedroughlya1 =! behaviorbelowthegap frequency2 = ~ ,andbecomessmallerasfewerelectronsparticipateinpairi ng.Fromthe 1 =! behaviorof 2 ( ),superruiddensityandmagneticpenetrationdepthwerededu ced. Wefoundthatthesuperruiddensitydecreasesasthesheetresistan ceincreases,but howthesearerelatedtotheeectoflocalizationremaintobe answered.Thesuperruid densityalsoseemstodependlinearlyon T c ,whichistriviallyconsistentwiththe expectationfortheweaktointermediatecouplingdirty-li mitBCSsuperconductors. Theeectofreducedthicknessinthesesamplesistodepress T c andthesuperruid density.Atthesametime,thenormal-stateconductivityappea rsunchanged.Thinner lmswithlower T c andmoreseverlocalizationeectsmayrevealstrongdeviatio nfrom theresultsexpectedforasuperconductinglmwithoutelectr onlocalization. Time-resolvedStudy Quasiparticledynamicswasexploredbyusingauniquepump-pr obetechnique, usingapulsefromaTi:Sapphirelaserasanexcitationsourceand apulseofsynchrotron radiationasaprobesource.Photonsofthepumppulsebreakasig nicantamountof Cooperpairsleadingtoanonequilibriumcondition,whichc anbesensedasasmall decreaseintheenergygapandincreaseinthefarinfraredabsor ption.Weempirically foundthatthespectrallyaveragedtransmittancecloselyfoll owsthetemperature dependenceofthesuperruiddensitypredictedbyBCStheory.T hisgaveadenite

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194 relationshipbetweenthephotoinducedchangeinthespectral lyaveragedtransmittance andtheexcessnumberofquasiparticles. Byadjustingtherelativedelaytimebetweenthepumpandthep robepulses,we followedtherelaxationofexcessquasiparticlesasafunctio noftime.Thequasiparticle decaysignalwasmodelledasaconvolutionofasimpleexponent ialdecayandasynchrotronGaussiantemporalstructure,fromwhichwededucedth eeectiverelaxation timeatvarioustemperaturebelow T c forthesamethree -MoGethinlmsusedinthe linearspectroscopicmeasurements.Thetemperaturedependenc eoftheeectiverelaxationtimewasanalyzedusingasolutionoftheRothwarf-Taylo requations[ 93 ],which describethebehaviorofexcessquasiparticlesandphononsinn onequilibriumsuperconductors,andtheuniversaltemperaturedependenceofintrinsi cquasiparticlelifetime (recombinationtime) R andphononlifetime(pair-breakingtime) B ,whichwerederivedbyKaplan etal. [ 92 ]basedonweak-couplingBCStheory.Notethatboththeories assumesweakperturbationlimit, i.e. ,thecaseoftheexcessquasiparticledensitymuch smallerthanthethermalquasiparticledensity,andwemadesure thatwewereinthis limitduringourexperiments.In R and B ,simplescalefactors R 0 and B 0 containall thematerialparametersrelevanttothetheory,andwewerea bletotourdatawithout varyingthematerialparametersfordierentthickness.Justa stheresultsoflinear spectroscopy,itseemsnoneofmaterialparametersarechangin g,but T c changeswith thickness,andwethinkthisisnotunderstoodcompletelyyet.T hevaluesof R 0 and B 0 wereontheorderof100psand10ps,respectively,whichwerein thesamerangeasfor materialslikePb,In,andNb.Inmakingthets,weobservedthat theratio R 0 = B 0 is lesssensitivetothechangeinthematerialparametersthanind ividual R 0 and B 0 ,and webelievethattheratiorepresentsthepropertiesof -MoGelmsmoreaccurately.The theorybyKaplan etal. predictsthattheratio2 0 B 0 = R 0 isproportionalto T 3 c .This isexactlywhatweobservedinourexperiments,supportingthet heory.Wealsolooked athowthecalculatedvaluesof R 0 = B 0 forvariouselementalsuperconductors[ 92 ]are relatedto b ,where b isaconstantintheassumption 2 F (n)= b n 2 .Wesawtheratio ofallmaterialsincludingourMoGefallssomewhereinthevic inityoftheline b 3 = 2 .We

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195 didnotdrawstrongconclusionsforthisobservationbecausewed onotreallyknowthe correctvalueof b forMoGe. Wealsomeasuredthephotoinducedsuperconductinggapshiftfor twolaser ruences.Assumingthattemperatureremainedconstantbeforeand afterexcitation, BCStsusingthesameparametersfoundinlinearspectroscopyme asurementsallowed ustondasmallreductionintheenergygapbyexcitationwith laserpulses.Basedon thetheorybyOwenandScalapino[ 94 ],wedeterminedthatapproximately0.30%and 0.67%ofpairswerebrokenbyphotoexcitationwith0.43and1 .0nJ/pulseofruence, respectively.Theresultswereroughlyinagreementwiththee stimatebasedonthe photonruenceitself.Magneto-opticalMeasurements Ourmostrecentexperimentsoftransmittanceaswellasquasipa rticledecayin magneticeldsweredescribed.TheexperimentsweredoneatU1 2IRwithrecently installednewspectrometerBruker125HR.Avertical-boresuper conductingmagnet wasconnectedtothespectrometerforthersttime.Thus,ourex perimentserved totesthowtheentiresystem,includingthespectrometer,thema gnet,andallother setup,functionswiththemagnetrunning,tondacorrecttim ing( i.e. ,ndingalasersynchrotroncoincidence)forthenewopticalpath,andtocon ductmeasurementsatthe sametime.Throughthemeasurements,wefoundseveralmattersth atshouldbeconsideredforoptimizinglaterexperiments.Nonetheless,weobserved thattransmittanceratio of -MoGe(33nmlm)at T =3Kbehavesasiftemperaturewereincreasedtoward T c astheeldstrengthwasincreasedexceptthepeakpositionrema inedstationary,and approximately10teslaofmagneticelddrivesthesupercondu ctingstatetoitsnormal state. Withthemagneticeldapplied,quasiparticlesmaydiusetow ardvortexeswhere thereisnogapandthusnobottleneck.Astheeldstrengthincr eases,normalstate regionsglow,andthereforefasterrelaxationtimeisexpect ed.However,whatwe observediseldindependenteectiverelaxationtime( 1 : 1nsforalleldsupto8 tesla),whichcannotbeexplainedatthismoment.

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196 ConcludingRemarks Overall,thisdissertationdescribesthefacilityattheNSLSan dresultsofour experimentsconductedthereon.Useoflargeexperimentalfac ilitiessuchassynchrotrons couldprovideopportunitiestoconductexperimentsthatmi ghtbeuniqueorthatcan notbedonewithinanordinarylaboratoryscale.TheNSLS,inpa rticular,oersarange ofcapabilitiesofperformingopticalstudiesowingtoitshi ghbrightness,broadband characteristics,andpulsednature.Withthelaser-synchrotron pump-probetechnique developedattheNSLS,wecanfollowthedynamicsofvarioussyst emsthatevolveas fastasatimescaleofafewhundredpicoseconds.Althoughthisisn otparticularlyhigh speedcomparedwiththecoherentterahertzlaserspectroscopyt echnique[ 110 ],there aremanyphenomenathatcanbestudiedatthistimescale.Throu ghthisdissertation webelievethatwewereabletoshowhowthesynchrotronsourceca nbeusefulfor investigatingfundamentalpropertiesofsolidstatephysics.

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APPENDIXA VUVSTORAGERINGPARAMETERS Thisappendixcontainstablesofvariousparametersforthe NSLSVUVstorage ring,thelinac,andtheboosterring[ 63 ].TheparametersfortheNSLSX-raystorage ringarenotlistedhere. TableA{1:TheVUVstorageringparameters(2002). NormalOperatingEnergy0.808GeVPeakOperatingCurrent(multibunchOps.)1.0amp(1 : 06 10 12 e ) Circumference51.0metersNumberofBeamPortsonDipoles18NumberofInsertionDevices2MaximumLengthofInsertionDevices 2.25meters c (E c ) 19.9 A(622eV) B( ) 1.41Tesla(1.91meters) ElectronOrbitalPeriod170.2nanosecondsDampingTimes x = y =13msec; z =7msec Lifetime@200mAwith52MHz360min(with211MHzBunchLengthening)(590min)LatticeStructure(Chasman-Green)SeparatedFunction,Quad,DoubletsNumberofSuperperiods4MagnetComplement 8<: 8Bending(1.5meterseach)24Quadrupole(0.3meterseach)12Sextupole(0.2meterseach) 9=; NominalTunes( x ; y )3.14,1.26 MomentumCompaction0.0235RFFrequency52.887MHzRadiatedPower20.4kW/ampofBeamRFPeakVoltagewith52MHz(with211MHz)80kV(20kV)DesignRFPowerwith52MHz(with211MHz)50kW(10kW)SynchrotronTune( s )0.0018 NaturalEnergySpread( e /E)5.0 10 4 ,I b < 20mA BunchLength(2 )9.7cm(I b < 20mA) (2L rms with211MHzBunchLengthening)(36cm) NumberofRFBuckets9TypicalBunchMode7HorizontalDampedEmittance( x )160nm-rad VerticalDampedEmittance( y ) 0.35nm-rad(4nm-radinnormalops.) PowerperHorizontalMilliradian(@1A)3.2Watts 197

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198 TableA{2:TheVUVstoragering'sarcsourceparameters(2002). BetatronFunction( x ; y )1.18to2.25m,10.26to14.21m DispersionFunction( x ; 0 x )0.500to0.062m,0.743to0.093m x;y = 0 x;y = 2-0.046to1.087,3.18to-0.96 r x;y =(1+ 2 x;y )/ x;y 0.738to0.970m 1 ,1.083to0.135m 1 SourceSize( x ; y )536to568 m, > 60to > 70 m (170-200 minnormalops.) SourceDivergence( 0 x ; 0 y )686to373 rad,19.5to6.9 rad (55-20 radinnormalops.) TableA{3:TheVUVstoragering'sinsertiondeviceparameters(20 02). BetatronFunction m( x ; y )11.1m,5.84m SourceSize( x ; y )1240 m, > 45 m(220 minnormalops.) SourceDivergence( 0 x ; 0 y )112 rad, > 7.7 rad(22 radinnormalops.) y isadjustable TableA{4:TheNSLSlinacparameters(2002). InjectionEnergy100keVFinalEnergy120MeVNumberofSections3NumberofKlystrons3Frequency2856MHz TableA{5:TheNSLSboosterparameters. InjectionEnergy120MeV MinimumVerticalBeta1.73m ExtractionEnergy750MeV MaximumDispersion1.21m Circumference28.35m MinimumDispersion0.41m NumberofSuperperiods4 MomentumCompaction0.106 DipoleBendingRadius1.91m PeakRFVoltage25kV NominalHorizontalTune2.42 RFFrequency52.88MHz NominalVerticalTune1.37 HorizontalAcceptance1.66E-04m-rad MaximumHorizontalBeta8.63m VerticalAcceptance6.11E-05m-rad MinimumHorizontalBeta1.01m MomentumAcceptance 0.0025 MaximumVerticalBeta5.26m TableA{6:TheNSLSboostermagneticelements(eldsat750MeV). NameTypeQuantityB(kG)B'(kG/m)B"(kG/m)EectiveLength(m) BBDipole813.099-7.97-1251.5Q1Quadrupole468.820.3Q2Quadrupole493.600.3SFSextupole41223.70.2

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APPENDIXB INFRAREDBEAMLINES Mostofthesynchrotronfacilitiesallovertheworldaredesign edandoptimized forproducingphotonsmainlyintheVUVthroughX-raysspectrum. Meanwhile, theuseofthelowenergyendsofsynchrotronradiationarealsor ealized,andseveral facilities(notablytheVUVringatNSLSandUV-SORinOkazaki,Jap an)nowfacilitate beamlinesoptimizedforfarinfrared.Asthesubjectssuchasth ecoherentsynchrotron radiation(CSR)andtheTHzradiationstartgettingattention ,needsofinfrared beamlineshavebeenincreased,andevenawholestorageringop timizedforlong wavelengthiscurrentlyunderconstructionattheAdvancedLi ghtSource(ALS)ofthe Lawrence-BerkeleyNationallaboratory.Thisappendixserve sadditionalinformation aboutinfraredbeamlinesattheNSLSVUVring. B.1InfraredProgramsatNSLSVUVring TableB{1:Examplesofresearchsubjectsandaliationsofresp ectiveinfraredbeamlinesoftheVUVring. Beamline ResearchSubjectsandAliation Ultra-highpressurespectroscopy/microscopy U2A CarnegieInstitutionofWashington Proteindynamicsandbiologicalmicroscopy U2B AlbertEinsteinCollegeofMedicine Surfacevibrationalspectroscopy/advancedmicroscopy U4IR NSLS/UniversityofWisconsin-Milwaukee Dynamicsofcomplexmetals U10A BrookhavenNationalLaboratory,NSLS Biological/ruorescence-assistedmicroscopy U10B NSLS,NorthropGrummanandCanadianLightSource Time-resolved,high-eld,far-IRspectroscopy U12IR UniversityofFlorida,StonyBrook,NSLS TherearesixbeamlinescurrentlyoperationalasapartofVUVri ngforlong wavelengthstudies.Table B{1 listsexamplesofresearchsubjectsandthealiationof respectivebeamlines. 199

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200 481216202428 0 5 10 15 20 Intensity[a.u.]Wavenumber[cm -1 ] 1cm -1 0.1cm -1 Frequency[THz] 50001000015000 23456 0 5 10 15 20 0.1cm-1 0.01cm-10.100.15 ebunch at t 1 ebunch at t 2 Ring bend IR emission Chamber wall reflection (A) (B)FigureB{1:(A)Highresolutionfarinfraredspectraofsynchrotr onradiationtakenat U12IR.Theinsetintheuppergraphdisplaystheinterferogramf or0.1cm 1 resolution. Lowergraphshowsspectraonanexpandedscalewithevenhigherr esolutionof0.01 cm 1 .(B)Modelofexplainingtheobservedfringes. B.2HightResolutionFar-infraredSpectraatU12IR ThespectraofsynchrotronradiationtakenwiththeSPS-200re vealapeculiar characteristicinhighresolutionfarinfraredspectraatU12IR .Figure B{1 (A)displays fringesinthespectratakenat0.01cm 1 ,0.1cm 1 and1cm 1 resolutions.Insetin thetopplotistheinterferogramwith0.1cm 1 resolution,andshowspeaksotherthan thecenterburstduetothefringeperiodof 1cm 1 .Asthevacuumchamberofthe VUVringchangesslightlyitsdimensionduetotemperaturevaria tion,thefringepattern moveswithtime.Thismaybetroublesomeforthosewhowanttota kehighresolution spectrum.Figure B{1 (B)showsapossibleexplanationfortheobservedfringes.

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APPENDIXC LASERSAFETYANDOPERATINGPROCEDURES C.1HazardousBeamControl Controlofthelaserradiationisaccomplishedbythreesystems:( 1)theU6hutch whichhousesthelaserandaccessoryoptics,(2)theopticalberc ableand(3)the endstationenclosure.TheU6hutchallowsproperlytrainedand protectedpersonnel toworkwiththelasersystemoptics.Allhazardsarecontainedwit hinthehutch. ProtectivegogglesforbothTi:SapphireandNd:YVO 4 wavelengthsarelocatedinside theU6hutch.Aninterlocksystemonthehutchdoorwayentranceis interfacedtoa shutterdirectlyattheexitapertureofthelaser.Entrybyuna uthorizedpersonneltrips theinterlockcausingthebeamtobecontainedinsidethelaser' sownenclosure.The doorwayentrancetotheU6hutchwillcarrystandardpostingsfo rboththepulsed Ti:Sapphirelaser,andtheparticularpumplaser(presentlya6 WNd:YVO 4 solidstate laser).Asignwiththemessage"INTERLOCKED"willilluminatetoi ndicatethat thedoorsysteminterlockisfunctioning.Thebercablerunsa longthecabletrays abovetheVUVringitself,andisthereforenotnormallyaccessibl etounauthorized personnel.TheKevlarreinforcingbersprovideprotection againstinadvertentbreakage. ThecablesendatthebeamlinewithanSMAtypeconnector.Ari gidscrew-type unionconnectorattachesthecabletoanothercablewhichen dsinsideanenclosed sample/experimentchamber,orafastopticaldetectorenclosu re. C.2PersonalProtectiveEquipment C.2.1EyeProtection TheU6beamlinehutchhas3pairsofprotectiveeyeglassesforth eVerdiNd:YVO 4 laser,theTi:Sapphirelaser,andGaAswavelengths(GlendaleM odel2175;OD > 9for 190nm-520nm;OD > 5for750nm-850nm)and2pairsofprotectiveeyewearfor theTi:SapphirelaserandGaAslaserdiodes(GlendaleModel222 6OD > 5for750nm 201

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202 -850nm).Duringnormaloperations,personnelenteringthehu tcharerequiredtowear protectiveglassesforTi:Sapphirelaserlight. TheNd:YVO 4 beamiscompletelycontainedduringbothalignmentandoper ational modesoftheU6laser,thuseyeprotectionisnotnecessary.Forro utinemaintenanceand servicing,protectiveeyewearwillbeworn.C.2.2SkinProtection NoUVlightispresentlyavailablefromthelaser.Exposingskintot helaserbeam willbeavoided,howevernospecialprotectivewearismandat ed. C.3LaserSafetyTraining OperatorsofANSIClass3band4laser( e.g. ,MiraandVerdi)mustcomplete sucienttrainingtoassurethattheycanidentifyandcontrolt heriskspresentedby thelasersystemstheyuse.QualiedLaserOperatorsmustcomplete theBNLlaser safetytrainingcourse,system-specicorientationwiththesyste mowner/operator,anda baselinemedicaleyeexaminationpriortolasersystemoperatio n. C.4Alignment C.4.1GrossAlignment DuringinitialdailyalignmentoftheTi:Sapphirelaser,pro tectiveeyewearwill bewornforanygrossalignments.PhotosensitiveIRsensorcardsor IRimageconverter/viewerswillbeusedtolocatethebeamandsteeritthro ughthevariousoptical elements.C.4.2FineAlignment Forcriticalnealignments( e.g. ,throughthePockelscells),itisnecessarytobe abletodetectthebeamdirectlyusingwhitebusinesscardwitho utprotectiveeyewear. Duringtheseoperations,thelaserwillbeoperatedatreducedp ower( < 100mW)to reduceriskofeyeinjurytoanacceptablylowlevel.Neutralde nsityltersonrigid opticalmountswillbeplacedupstreamoftheopticstobealig ned,andthepower levelmeasuredtoensurelowoutputbeforesuchalignmentisbeg un.Followingisthe proceduretobeexercisedduringnealignment.

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203 1.WithglassesON,setTi:Sapphirelaserto700nmendoftuningran ge(increases visibilityatlowpowers). 2.ImmediatelydownstreamoftheOpticalIsolator,xsupportf orNDlters, followedbythelaserpowermeterimmediately. 3.InstallNDlterstoreducepowertobelow100mW.4.Closelasershutterandplacepowermeter(ormetalbeamblock )downstreamof opticalsectiontobealigned. 5.UsingIRviewer,conrmthatthebeamisfallingontoeachopt icalelement. Surveytheroomforextraneousrerectionsandeliminateany 6.Performminoralignment(withoutglassesasneeded),usingd iusewhitecard (tocheckaccuratepositioning).Mirrorsorotheropticsshal lNOTbeusedtoobserve anypartsofthebeampath. 7.Returntousinglaserglasses/gogglesbeforeremovingtheneut raldensity lter(s).C.4.3AtBeamlineEndstation AlignmentofberopticsatbeamlinewillONLYbedoneusinglowp ower( 1 mW)reddiodealignmentlaser.Thesolid,opaquemetalvacuume nclosurewillbe securelyfastened(boltedand/orundervacuum)anytimethepo ssibilityexistsfor > 5 mWlaserpowerthroughthebercableintotheenclosure.Alllase rlightcomingfrom theU6hutchthroughthebercablewillbecontainedatalltim es. C.5DailyOperationProcedure Thissectionservesasachecklistforturningon/othelasersyste m,mode-locking, andsynchronizing.Adetaileddescriptionofthelaseroperati on,alignmentofinternal optics,maintenance,andtroubleshootingcanbefoundintheo perator'smanualsfor MiraandVerdi.TheSynchro-Lock900operator'smanualprov idesthroughinstruction forsynchronizingthelasertotheVUVring.Followinginstructio nisforthesystem atthetimeofwritingthisdissertation,andtheprocedureissu bjecttochangedue tomodicationofsynchronizationmethodandopticssetup.Iw illassumethatthe

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204 alignmentthroughthedownstreamopticsandecientcouplin gintotheopticalberare wellestablishedbeforethedailyoperation. Toturnonthesystem;1.Checkiftheairconditioningsysteminsidethehutchisturne don,andthe temperaturessetatbetween73and75 F.Thisisfortheoptimalstabilityofthelaser system. 2.Verifythatthechillerwatertemperatureissettoapproxi mately20 C,andthe waterlevelisbetweenahighandlowlevelmarker.Thenturni ton.Thebaseplate temperatureshouldbemaintainedbelow55 C. 3.MakesurethatU6hutchdoorLaserHazardpostingsisinplace,an dthe INTERLOCKandwarninglightoperational. 4.Insurethatthesafetyshutterisclosed.5.Beamblock(blackmetalmonumentorpowermeterhead)shoul dbeinstalledin beampathupstreamofbercoupler.Wheneverthebeamistransf erredtoabeamline, makesurethatthebeamiscompletelycontainedwithinanendst ationenclosure. 6.TurnthekeyswitchonthepowersupplyfrontpaneltoON,andop entheVerdi shutter(Thisshutterisnormallykeptopen).Ifnecessary,adju stthePOWERADJUST knobforthedesiredoutputpowerlevel.Wenormallyoperatea t5.0watts,butitcanbe ashighas5.5watts. 7.Warmthesystemforapproximately1houruntiltheunitachie vesstable emission. 8.WiththemodeselectswitchontheMiracontrollerintheCWpo sition,optimize theoutputpowerbyadjustingpumpmirrorcontrolsandGTIali gnmentcontrols.Make surethebeamiswellcenteredattheslit. 9.Atthispoint,alltheotherelectronics,suchasdigitalosci lloscope,pulse-picker powersupply,functiongenerator,andbiasofphotodiodeaft erthepulse-picker,canbe powered.Aslongastheamplitudeoffunctiongeneratorissett oitsminimum, 1.53 mV,itcanbeleftpoweredallthetime.

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205 10.Mode-lockedoperationcanbeestablishedbyswitchingthem odeselectionto eitherMLor L.AdjusttheBRFmicrometeruntiltheCWcomponentintheoutpu t beambecomesnegligible. 11.Turnonthepulsegeneratoratthebeamline.TheSynchro-l ock900software allowsthesynchronizationoftheopticalpulsesfromMirawit hastableexternal frequencysource,whichisinthiscasethepulsegenerator. Toturnofthesystem;1.Unlockthesynchronizationfromthesoftware.2.SwitchthemodeselectiontotheCW.3.TurnothekeyswitchoftheVerdipowersupply.4.Turnothechiller.5.Eithersettheamplitudeoffunctiongeneratortominimumo rturnito. 6.Allotherelectronicscanbeturnedo.Especiallydonotfor gettoturnothe biasofbothphotodiodeandpowermeter. C.6OptimizationoftheDownstreamOptics Thissectiondescribestheproceduretofollowforoptimizing thethroughputofthe downstreamoptics[ 21 ]. 1.Maximizethepowerpassingthroughtheopticalisolatorbyad justingthemirror rerectingthebeamintotheisolator. 2.Withtherejectedbeamblocked,maximizethepowerpassingt hroughtherst pulse-pickerbyadjustingtheinputmirrorandthepulse-picke rposition.Thispartof alignmentissensitiveandthebeamshouldpassthroughthecente rofentranceaperture withnodistortions.Thismustbedonewiththelasermode-locked andthepulse-picker operating. 3.Maximizethepowerthroughthesecondpulsepickerbyadjusti ngitsposition andtheinputmirror(alsousedfortherstpulse-picker).Afewi terationsofstep2and 3maybenecessarytoachieveagoodalignment. 4.Onceagoodalignmentoftheprimarypulsehasbeenachieved ,unblockthe rejectedpulsebeamandalignitvisuallyuntilitisontheedge ofthemirrorusedto

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206 re-insertitintothemainbeampathatthesameheightasthepri marybeam.Then adjustthatmirroruntilthesecondbeamiscoincidentwiththe primarybeamatapoint afterthesecondpulsepicker. 5.Placeapowermeteratthepositionwherethetwobeamsoverl ap.Adjustthe half-waveplatetomaximizethepower. 6.Blockthesecondbeamandoptimizethecouplingoftherstin totheoptical berusingtheadjustmentsofthebercoupler.Thenunblockth esecondbeamand optimizeitscouplingbyadjustingthere-insertionmirror.B ylookingatthebeamgoing intoaberthroughanIRviewer,optimalcouplingcanbeeasil yachieved. Asacriterionofgoodalignment,amaximumpowerjustinfronto fbercoupler shouldbeabove400mW.Underidealconditionsamaximumofnear ly500mWcanbe achieved.Usingwellprepared,undamagedbercableandSMAty peconnectors,more than70%ofpowergoingintothebershouldbedeliveredtothe beamlineprovidedthe ecientcouplinghasachieved. C.7ANSILaserClassications Lasersafetystandardsarederivedfromgovernmentmandatedre gulationsand voluntarystandards.Thestandardrequiresthatlasersbeprope rlyclassiedand labelledbythemanufacturer.Thus,formostlasers,measurement sorcalculations todeterminethehazardclassicationarenotnecessary.Inaddi tion,thestandard establishescertainengineeringrequirementsforeachclassa ndrequireswarninglabels thatstatemaximumoutputpower.Extensiverecommendationsf orthesafeuseoflasers havebeendevelopedbytheAmericanNationalStandardsInstitu te(ANSIZ136.1-2000). Theappropriateclassisdeterminedfromthewavelength,pow eroutput,andduration ofpulse(ifpulsed).Classicationisbasedonthemaximumaccessib leoutputpower. Therearefourlaserclasses,withClass1representingtheleasthaz ardous.Alllasers, exceptClass1,mustbelabelledwiththeappropriatehazardcl assication. Class1 Class1laserdevicescannotproducedamagingradiationlevel stotheeyeevenif viewedaccidentally.Prolongedstaringatthelaserbeamhowe ver,shouldbeavoidedas

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207 amatterofgoodIndustrialHygienepractice.Thisclasshasapo weroutputlessthan 0.4 Wforcontinuouswave(CW)lasersoperatinginthevisiblerang e.Acompletely enclosedlaserisclassiedasaClass1laserifemissionsfromtheenc losurecannot exceedlimitsforaClass1laser.Iftheenclosureisremoved, e.g. ,duringrepair,control measuresfortheclassoflasercontainedwithinarerequired.Class2 Class2lasersareincapableofcausingeyeinjurythedurationo ftheblink,or aversionresponse(0.25sec).Althoughtheselaserscannotcauseeye injuryunder normalcircumstances,theycanproduceinjuryifvieweddirec tlyforextendedperiods oftime.Class2lasersonlyoperateinthevisiblerange(400-70 0nm)andhavepower outputsbetween0.4 Wand1mWforCWlasers.ThemajorityofClass2lasersare helium-neondevices.Class3a Class3alaserscannotdamagetheeyewithinthedurationofthe blinkoraversion response.However,injuryispossibleifthebeamisviewedthroug hbinocularsorsimilar opticaldevices,orbystaringatthedirectbeam.Poweroutput sforCWlasersoperating inthevisiblerangearebetween1-5mW.Class3b Class3blaserscanproduceaccidentalinjuriestotheeyefrom viewingthedirect beamoraspecularlyrerectedbeam.Class3blaserpoweroutputs arebetween5-500 mWforCWlasers.ExceptforhigherpowerClass3blasers,thisclass willnotproduce ahazardousdiusererectionunlessviewedthroughanoptical instrument. Class4 Class4lasersarethemosthazardouslasers.Exposuretotheprimar ybeam,specularrerections,anddiusererectionsarepotentiallytothe skinandeyes.Inaddition, class4laserscanigniterammabletargets,createhazardousai rbornecontaminantsand usuallycontainapotentiallylethalhighvoltagesupply.The poweroutputforCWlasers

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208 operatinginallwavelengthrangesisgreaterthan500mW.All pulsedlasersoperating intheocularfocusregion(400nmto1,400nm)shouldbeconside redClass4. UnknownClass Laserclassicationcanbedeterminedbymeasuringtheoutputir radianceor radiantexposureusinginstrumentstraceabletotheNationalBu reauofStandards. Thesemeasurementsshouldonlybeperformedbyqualiedpersonn el.Thelaserclass canalsobedeterminedfromcalculations.ForCWlasers,thewave lengthandaverage poweroutputmustbeknown.Classicationofpulsedlasersrequir esthefollowing information:wavelength,totalenergyperpulse(orpeakpow er),pulseduration,pulse repetitionfrequency(PRF),andemergentbeamradiantexpo sure.Inadditiontothe aboveinformationlasersourceradianceandmaximumviewinga nglesubtendedbythe lasermustbeknownforextended-sourcelasers,suchasinjectionl aserdiodes.Detailed informationonclassifyinglasersmaybefoundintheANSIZ136.12000.

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APPENDIXD USEFULINFORMATION Thisappendixcontainstablesofvarioususefulinformation D.1FrequencyRanges Thissectioncontainsfrequencyrangesofinfraredspectralr egions,conventional lightsources,beam-splitters,detectors,andopticalwindowsa ndlters. TableD{1:Infraredspectralregions. Region Wavelength( m) Wavenumber(cm 1 ) Frequency(Hz) Near-IR 0.78-2.5 4000-12800 1.2 10 14 -3.8 10 14 Mid-IR 2.5-50 200-4000 6.0 10 12 -1.2 10 14 Far-IR 50-1000 10-200 3.0 10 11 -6.0 10 12 TableD{2:Frequencyrangesofconventionallightsources. Source Range(cm 1 ) PrimaryApplication Mercury-arclamp 10-700 FarIR SiCglobar 100-6000 MidIR Tungstenlamp 4000-40000 VIS TableD{3:Frequencyrangesofdetectors.PEinthewindowcol lumstandsfor polyethylene. Detector Window Temperature(K) Range(cm 1 ) PrimaryApplication Si:BBolometer PE 4.2(LHe) 10-600 FarIR DTGS PE 300(ambient) 10-600 FarIR DTGS KBr 300(ambient) 400-7000 MidIR Si:B KRS-5 4.2(LHe) 350-4000 MidIR Ge:Cu KRS-5 4.2(LHe) 350-4000 MidIR MCT 77(LN 2 ) MidIR InSb Sapphire 77(LN 2 ) 1850-15000 NearIR SiPhotodiode 300(ambient) 9000-28000 VISandUV GaPPhotodiode 300(ambient) 18000-33000 VISandUV 209

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210 TableD{4:Spectralrangesofbeam-splittermaterials. Material Type Range(cm 1 ) PrimaryApplication MetalMesh 20(lines/mm) 40 $ 750 FarIR 8 40 $ 750 FarIR Mylar 3.5( m) 125 $ 750 FarIR 6.0 80 $ 450 FarIR 12.0 40 $ 220 FarIR 23.0 20 $ 110 FarIR 50.0 15 $ 55 FarIR 125.0 5 $ 22 FarIR Gecoated 30 $ 680 FarIR KBr Gecoated 370 $ 7800 MidIR Gecoated(widerange) 400 $ 10000 MidIR CsI Gecoated 200 $ 5000 MidIR ZnSe 500 $ 5000 MidIR Quartz 9000 $ 25000 VIS TableD{5:Transmissionrangeofopticalwindowandltermater ials. Material Transmissionrange(cm 1 ) KBr 400|40,000 NaCl 625|40,000 SiO 2 (Quartz) 0|250 2700|65,000 Sapphire 0|350(below50K) 2000|65,000 Diamond 10|45,000 ZnSe 720|17,000 ZnS 830|17,000 CsI 200|40,000 GaAs 600|5,500 AgCl 360|10,000 AgBr 290|22,000 Polyethylene < 600 CaF 2 1,200|66,000 BaF 2 740|67,000 MgF 2 1,250|87,000 LiF 1,700|95,000 CdTe 400|20,000 KRS-5(Thallium-Bromide-Iodide) 250|20,000 Fluorogold < 60(dependsonthickness) Blackpolyethylene < 600

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211 D.2EnergyandPressureUnitsConversion Thissectionprovidesrelationsbetweenvariousenergyandp ressureunits. TableD{6:Relationsbetweenenergyunits. 1meV=8 : 0658cm 1 =242GHz=11 : 6K 1cm 1 =0 : 124meV =30GHz=1 : 439K 1THz=33 : 33cm 1 =4 : 133meV =48K 1K=0 : 695cm 1 =0 : 086meV TableD{7:Relationsbetweenpressureunits. 1atmosphere(atm) 101 ; 325Pa 1 : 01325bar =760torr=14 : 7psi 1013 : 25millibar 1bar 100 ; 000Pa =0 : 987atm 1millibar 100Pa 1hPa =0 : 75torr 1torr(mmHg,0 C)=133 : 32Pa =1 : 32 10 3 atm 1millitorr(micron)=1 : 32 10 6 atm 1psi(lbf/in. 2 )=6894 : 8Pa D.3Gas-phaseContamination Waterandcarbondioxidearethetwomostcommonlydetectedco ntaminantpeaks inaspectrumtakeninair.Evacuationofthesamplecompartmen tshouldsignicantly reducetheabsorptionduetothesemolecules.Purgingthesample compartmentwith drynitrogencanalsoreducethelevelsofthesecontaminantsb utnotaseectivelyas

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212 evacuation.Table D{8 showsthepeaklocationsofwatervaporandcarbondioxideabsorptions. TableD{8:AbsorptionpeaksforH 2 OandCO 2 vapor. Contaminant PeakLocation(cm 1 ) Notes WaterVapor 3480-3960 Seriesofsharppeaks (H 2 O) 1300-1950 Seriesofsharppeaks < 500 Seriesofsharppeaks CarbonDioxide 2280-2390 Usuallyunresolveddoubletwithcentralmaximum (CO 2 ) (transmission)at2348cm 1 665-672 Sharpminimumat667cm 1 (transmission)

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BIOGRAPHICALSKETCH HidenoriTashirowasborninKudamatsuCityofYamaguchi-prefe cture,Japan. Hidenoriwasanexceedinglyathleticchild,anddedicatedmo stofhistimeplaying soccer,baseball,andgolf.HeattendedtheKudamatsuHighSchool inthesametown hewasraised.In1996,hegraduatedwithaB.S.inphysicsfromt heStateUniversity ofNewYorkatBualo.In1997hestartedattendingtheUniversity ofFlorida,and receivedhisM.S.inphysicsin2001.Inthesameyearhebeganwo rkingforProfessor DavidTanner,andin2002,hemovedtoUpton,NewYork,toworkfu ll-timeforhis dissertationprojectattheNationalSynchrotronLightSource ofBrookhavenNational Laboratory.Aftertwoyears,hecamebacktoFloridaandcomple tedhisPh.D.in2004. Hemethisextremelykindwife,Yasuko,whenhecametotheUnited Statesfortherst time,andtwoboys,MitsuruandHikaru,werebornsincethen. 219