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Magnetotransport and Tunneling Study of the Semimetals Bismuth and Graphite

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Title:
Magnetotransport and Tunneling Study of the Semimetals Bismuth and Graphite
Creator:
DU, XU ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Bismuth ( jstor )
Charge carriers ( jstor )
Electrical resistivity ( jstor )
Electrons ( jstor )
Graphite ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnets ( jstor )
Metalloids ( jstor )
Temperature dependence ( jstor )

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University of Florida
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University of Florida
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Copyright Xu Du. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text












MAGNETOTRANSPORT AND TUNNELING STUDY OF THE SEMIMETALS
BISMUTH AND GRAPHITE

















By

XU DU















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2004

































Copyright 2004

by

Xu Du



































To my parents















ACKNOWLEDGMENTS

I would like to express my sincere gratitude to the many individuals who

contributed to success of my work. First of all I would like to thank my research advisor,

Professor Arthur Hebard. Through his positive and open-minded attitude, and his

enthusiasm and optimism toward physics research, he created the legacy of the free,

vivid, intelligent, friendly and communicative research atmosphere in the lab. I feel really

lucky to be able to work in such environment. His experience, knowledge, and guidance

have been invaluable throughout my graduate career.

I would also like to thank Professor Dmitrii Maslov for his theoretic support.

Without the many useful discussions with him, and his constructive criticism, much of

my work would have gone nowhere. I would also like to express my deep appreciation to

Professor Andrew Rinzler. I truly benefited from his valuable opinions, his help with lab

facilities, and collaboration of some on his interesting and fruitful projects. I also want to

thank Professor Peter Hirschfeld, who gave me a better understanding of solid state

physics through his teaching; and Professor David Norton, for being on of my committee

members.

I am dearly thankful to current and former members of our group (Josh Kelly,

Jeremy Nesbitt, Partha Mitra, Ryan Rairigh, Sinan Selcuk, Guneeta Singh, Kevin

McCarthy, Quentin Hudspeth, Stephen Arnason, Nikoleta Theodoropoulou, and

Stephanie Getty), who provided a joyful working environment and great help. I would

especially like to thank Sinan Selcuk for his help on E-beam lithography. I also want to









thank Jamal Derakhshan (who worked as an REU student in the lab), for his help with the

bulk bismuth study.

My gratitude also goes to Professor Gray Ihas, and to Professor Amlan Biswas and

his students (Tara Dhakal, Jacob Tosado, and Sung-Hee Yun), who provided me great

help in using their facilities. I thank Zhuangchun Wu, Jennifer Sippel, and Amol Patil for

their kind help with my experiments. I also would like to thank Ronojoy Saha, for useful

discussions on high magnetic field transport. Many thanks go to the machine shop

personnel for their excellent work, which allowed my research work to move on

smoothly.

I would like to express my great appreciation and deep love to my parents for their

unconditional love and support. And finally, I would like to thank my dear wife, Zhihong

Chen, who was a graduate student in Professor Andrew Rinzler's group. Her knowledge

and intelligence have been of great help. She has shared my happiness and burden all

these years. Her love changes my life and makes me a better individual.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES .................................................... ............ .............. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

ABSTRACT ........ .............. ............. .. ...... .......... .......... xii

CHAPTER

1 GENERAL IN TRODU CTION ............................................................ ............... 1

2 MAGNETOTRANSPORT IN GRAPHITE ....................................... ............... 4

2.1 Overview of Classical Magnetotransport in Semimetals.......................................5
2.2 M ulti-B and M odel ................ .................................................. .....7
2.3 Sample Preparation and Characterizations............................................................9
2.3.1 Sample Preparation................... .................................. 9
2.3.2 Characterizations: Dingle Temperature and Landau Levels .....................11
2.4 Transport in the Classical R egion...................................... ........................ 17

3 MAGNETOTRANSPORT OF GRAPHITE IN THE ULTRA-QUANTUM FIELD..24

3.1 Transport D ata in the U ltra-Quantum Field................................... .....................24
3.2 In-Band Transport Behavior in the Ultra-Quantum Regime.............................27
3.3 Possible M odels in the Ultra-Quantum Regime ......................... .....................37

4 TUNNELING INTO BULK BISMUTH IN THE ULTRA-QUANTUM FIELD........41

4 .1 M o tiv atio n ........................................................................................................ 4 1
4 .2 E x p erim en ta l............................................................................................. .. 4 3
4.3 Results and D discussion .................................................... ...............47

5 ACHIEVING LARGE MAGNETORESISTANCE IN BISMUTH THIN FILMS .... 50

5 .1 In tro du ctio n ...................................... ............................ ................ 5 0
5.2 Experim mental ................................................................... .. ... ..... 54
5 .3 R esu lts an d D iscu ssion ........................................... ........................................ 55









6 METALLIC SURFACE STATES IN THE ULTRA-THIN BISMUTH FILMS........63

6.1 Introduction: Physics of the Ultra-Thin Bismuth Films ................................63
6.2 Transport Properties of the Ultra-Thin Bismuth Films............. ...............66
6 .2 .1 E x p erim ental ........... ... ....... ... .. .. ...... .. .................. .. ................ .. 66
6.2.2 M etallic Surface States .................................. ..... ............... 67
6.3 Control of the Surface States ........................................ .......................... 74

7 SURFACE SUPERCONDUCTIVITY IN ULTRA-THIN BISMUTH FILMS......... 80

7 .1 T ran sp ort E v id en ce ......... .. ............... ................. ............................................80
7.2 T unneling E evidence ........... ........................................................ ............... 84
7 .3 P o ssib le P ictu re ............................................................................................... 8 9

8 FUTURE W ORK ........... ............ ........................ .... .. ....... .............. 91

LIST OF REFEREN CES ......... .................. .............. .................................... 97

BIOGRAPHICAL SKETCH .............. ........... ................ 101
















LIST OF TABLES


Table page

1-1. Basic parameters of bismuth and graphite............... .......................1

5-1. Summary of results for different bismuth film growth conditions.........................61

6-1. Parameters for the simulating the effect of thickness and temperature on the
magnetic field dependence of the Hall resistivity in ultra-thin Bi films ................72

6-2. Parameters for the simulating the effect of Ge coating on the magnetic field
dependence of the Hall resistivity in ultra-thin Bi films .............................. 79
















LIST OF FIGURES


Figure page

2-1. Configuration of leads on graphite transport sample.................................... 10

2-2. Shubnikov-de Haas oscillations in graphite at indicated temperatures ...................11

2-3. The Landau level indices as a function of the inverse of magnetic field at different
low tem peratures ...................... .................... .. .. ........... ..... ....... 12

2-4. The amplitude of the ShdH oscillations as a function of the inverse of magnetic
fi eld at 2K ...................................................... ................... ... ....... ....... 14

2-5. Scaled ShdH oscillations amplitude as a function of the inverse of magnetic field in
different tem peratures. ........................................... ........... ..... ........15

2-6. Linear fit to the slopes of the scaled ShdH oscillation as function of temperature.. 15

2-7. Temperature dependence of the resistivity pxx for a graphite crystal in different
m magnetic fields. ....................................................................... 17

2-8. pxx and pxy versus applied magnetic field at the different temperatures .................20

2-9. Temperature dependence of mobility, relaxation time; and carrier density for the
bands indicated in the legends of each panel.. ................................................... 21

3-1. The magnetic field dependence of the longitudinal and Hall resistance of HOPG at
different tem peratures ...................... ................ .............................25

3-2. The temperature dependence of the longitudinal resistance of HOPG in different
m ag n etic fi eld s ..................................................... ................ 2 6

3-3. The ratio of the measured Hall resistance and longitudinal resistance as a function
of m magnetic field at 2K ............................................................................. .... .. 29

3-4. The Landau band dispersion relation of graphite in 12 Tesla field, calculated using
the SW M cC m odel. ............................................. .........................32

3-5. Estimation of carrier un-compensation in different magnetic fields at 2K ..............33

3-6. Shape of the in-band resistivity as a function of magnetic field at the indicated
tem peratures. .........................................................................34









3-8. Logarithmic plot of the shape of the in-band conductivity as function of
temperature in different strong magnetic fields ....................................... .......... 36

3-9. Model of the field induced Luttinger liquid with dressed impurity scattering........38

3-10. Scaled in-band conductivity as a function of temperature in magnetic fields above
th e U Q L .......................................................................... 3 9

4-1. Procedure for making tunnel junctions on bulk semimetal using photolithography
tech n iqu e. ......................................................... ................ 4 4

4-2. Mica mask method for making tunnel junctions on bulk semimetal.....................46

4-3. Microscopic picture of a Bi(bulk)-AlOx-Pb tunnel junction............................... 46

4-4. Differential conductance as a function of bias voltage in indicated strong magnetic
fields at 300m k. .......................................................................48

4-5. Differential conductance at low bias voltage in the magnetic fields indicated in the
leg en d .............................................................................. 4 8

5-1. Fermi surface and Brillouin zone of rombohedral bismuth ....................................50

5-2. Magnetotransport behavior of bulk single crystal bismuth. .....................................51

5-3. Magnetotransport behavior of a bismuth thin film .......................................... 52

5-4. X-ray diffraction pattern for a 4-um-thick Bi/Au film ..........................................56

5-5. Resistivity vs. temperature at 0 and 5T for two category-I Bi films ......................57

5-6. Resistivity vs. temperature at 0 and 5T for three category-II Bi films ...................58

5-7. Resistivity vs. temperature at 0 and 5T for two category-III Bi films....................60

6-1. Illustration of semimetal-to-semicinductor transition. ...........................................64

6-2. Resistivity vs. temperature for Bi film with indicated thicknesses. .......................67

6-3. Magnetoresistance at 5K for two different thicknesses Bi films..............................69

6-4. Hall Resistivity vs. magnetic field for (a) 180A and (b) 400A Bi films ..................70

6-5. Simulated Hall Resistivity vs. magnetic field for Bi(180A) and Bi(400A). ............73

6-6. Temperature dependence of resistivity for Bi(100A) and Bi(100A)/Ge..................75

6-7. Hall Resistivity vs. magnetic field for (a) Bi(100A) and (b) Bi(100A)/Ge ............77









6-8. Simulated Hall Resistivity vs. magnetic field for Bi(100A) and Bi(100A)/Ge ........78

7-1. Resistance vs. temperature in zero magnetic field for a 15nm bismuth film.. .........80

7-2. Example of resistance increases during the transition...........................................81

7-3. Sharp feature of resistance change observed in Hall resistivity measurements.......82

7-4. Film thickness dependence of the critical magnetic field at 4.5K .........................83

7-5. Differential conductance as a function of bias voltage in the superconducting gap
region for a Pb-AlOx-Bi(150A) tunnel junction. ............................................ 84

7-6. Differential conductance as a function of bias voltage in the superconducting gap
region at 300mk for a Pb-AlOx-Bi(1000A) tunnel junction...............................86

7-7. Differential conductance vs. bias voltage at 300mK in indicated low magnetic fields
perpendicular and parallel to the junction plane. ............. ....................... ......... 87

7-8. Differential conductance vs. bias voltage at 300mK in indicated strong magnetic
fields perpendicular and parallel to the junction plane.......................................... 88

7-9. A possible picture of surface superconductivity in ultra-thin bismuth films ..........90

8-1. Some examples of the sub-micron sized bismuth patterns ........... ...............93

8-2. Magnetic field dependence of the resistivity for a reservoir pattern ......................94

8-3. Measurements of a nano-cavity with a single grain-boundary in it .......................95















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

MAGNETOTRANSPORT AND TUNNELING STUDY OF THE SEMIMETALS
BISMUTH AND GRAPHITE

By

Xu Du

December 2004

Chair: Arthur F. Hebard
Major Department: Physics

Magnetotransport and tunneling studies on bulk crystals, thin films and patterned

nanostructures of semimetals reveal a surprising range of interesting behaviors. In our

study of ultrathin bismuth films, we found that the transport behavior is greatly affected

by the presence of metallic surface states, which become evident in the thinnest films and

are presumed to be responsible for a surface superconducting state seen in tunneling and

transport anomalies. We also studied bulk samples of both of these semimetals in

magnetic fields high enough to place all the carriers in the lowest Landau level. In this

ultraquantum regime, the apparent re-entrance in graphite from insulating to

metallic/superconducting behavior at low temperatures corresponds to the in-band

insulating behavior of carriers within a semiclassical 2-band model framework. This

analysis brings into question recently proposed explanations of a field-induced metal-

insulating transition and magnetic-field-induced superconducting fluctuations in graphite.














CHAPTER 1
GENERAL INTRODUCTION

A semimetal is a semiconductor with a small conduction band-valence band

overlap (instead of a gap). Semimetals have low Fermi energy. In contrast to

semiconductors, which are insulators at zero temperature (T = 0) where the carrier

concentration n = 0, semimetals have a finite conductivity at T = 0 where n is finite

because of the nonzero overlap of the conduction and valence bands. Semimetals are

metallic, with both electrons and holes contributing to electric conduction. Graphite and

bismuth are typical semimetals, with low Fermi energies and low carrier concentrations.

Table 1-1. Basic parameters of bismuth and graphite
Ef (meV) Carrier concentration (m 3)
(ne=np)
Bismuth -30 1023
Graphite -22 -1024


Semimetals have been of interest for many years, in many different aspects. A

major aspect of semimetal study is magnetotransport. Because of their small values of

carrier concentration, semimetals can be driven into the ultra-quantum regime, when only

the lowest Landau level remains occupied, with a magnetic field of-10 Tesla. In

addition, light cyclotron masses me in certain orientations of semimetals result in higher

cyclotron frequencies (eB/mn ) ensuring that quantum magneto-oscillations can be

observed in moderate magnetic fields and at moderate temperatures. High purity allows

the oscillations to survive the effects of disorder. Magnetotransport study of semimetals

in strong magnetic field allowed the Fermi surface to be mapped by quantum oscillations









in semimetals. In applications, the extremely large magnetoresistance of semimetals

makes them promising candidates for magnetic field sensors.

Another major aspect of semimetal study originated from the long Fermi

wavelength. By making bismuth thin films with thicknesses comparable to a Fermi

wavelength, one could study the energy band quantization because of quantum

confinement. Also, since the band mass is bigger for electrons than holes, as the size of

the bismuth structure decreases, the speed at which the conduction band shifts up will be

faster than that of the valence band. At a certain point, a gap opens up, and the

semimetal-semiconductor transition should happen. Existence of the transition has been

studied for many years, and is still not conclusive; mainly because of the existence of the

surface states, that smear out any sharp features of the transition.

Evidence of metallic surface states were found in films of superconducting bismuth

clusters, which indicates surface superconductivity because of the strongly increased

surface density of states. Further evidence of metallic surface states was found by angle

resolved photoemission spectroscopy (ARPES).

Our study of semimetals focused on two major aspects: 1) magnetotransport and

tunneling study of bulk single-crystal bismuth and graphite; and 2) the effect of bismuth

surface states on transport and bismuth surface superconductivity.

In the following chapters, we will show the motivation and our work on each aspect

of our study. Chapter 2 first of all describes the theoretic background of transport

behavior in semimetals. Then it explains the experimental details, sample characterization

and low field transport behavior analysis of graphite. Chapter 3 explains the high

magnetic field transport behavior of graphite, and proposes possible theoretic models.






3


Chapter 4 describes the magneto-tunneling measurements on bulk bismuth tunnel

junctions. Chapter 5 explains our work on achieving large magnetoresistance in Bi-Au

thin films. Chapter 6 describes the effect of the metallic surface states on the transport

properties of ultra-thin bismuth films. Chapter 7 describes the surface superconductivity

behavior we observed in the ultra-thin bismuth films. Chapter 8 discusses some possible

interesting future works on bismuth, including nano-meter sized bismuth structures and

spintronics applications.














CHAPTER 2
MAGNETOTRANSPORT IN GRAPHITE

Graphite is a typical semimetal, with a low Fermi energy (-22 meV) and low

carrier concentration (-3 x 1024 3 ). The zero-temperature conductivity in graphite

results from the small overlap between the conduction band and the valence band. The

Fermi level lies in the middle of the overlap, which makes graphite a typical compensated

2-band material.

Transport properties of graphite had been studied intensively since 1950s.

Recently, interest in magnetotransport in graphite was renewed because of the

observation of an effect that looks like a magnetic-field-induced metal-insulator

transition: the metallic temperature-dependence of the in-plane resistivity in zero field

turns into an insulating-like one when a magnetic field of a few tens of mTesla is applied

perpendicular to the basal (ab) plane. Increasing the field to about 1 Tesla produces a re-

entrance of the metallic behavior. It has been proposed that the low-field effect is caused

by a magnetic-field-induced excitonic insulator transition of Dirac fermions, 1,2 whereas

the high-field behavior is a manifestation of field-induced superconductivity.3' 4 It has

also been suggested that the apparent metal-insulator transition in graphite is similar to

that in 2D heterostructures (although the latter is driven by a field parallel to the

conducting plane). To elucidate these issues, we performed detailed measurements of

magnetoresistance in graphite and found data quite similar to data reported in 1-4 over

comparable temperature and field ranges. However, our interpretation is significantly

different from theirs.









2.1 Overview of Classical Magnetotransport in Semimetals

A combination of some unique features specific to semimetals [i.e., low carrier

density, high purity, small effective mass and equal number of electrons and holes

(compensation)] led to an unusual temperature dependence of the magnetoresistance even

in classically strong fields, defined by the condition

h/r < hco < kT (2-1)

where r is a scattering time of the carriers. Here we qualitatively compare a semimetal

with a conventional, high-density, uncompensated metal. To begin with, if the Fermi

surface is isotropic, a metal does not exhibit magnetoresistance because of the

cancellation between the longitudinal and Hall components of the electric field.5 In real,

anisotropic metals, this cancellation is broken, and as a result, magnetoresistance is finite

and proportional to (cor)2 in weak magnetic fields (oc <<1). In stronger fields

(oiz >> 1), classical magnetoresistance saturates.6 In contrast (Equation 2-14),

magnetoresistance of a compensated semimetal grows as B2 in both weak- and strong-

field regions.

In addition to the saturation effect described above, another factor that makes the

magnetoresistance much smaller in conventional metals than in semimetals is the higher

scattering rates and hence the smaller values of )ctz The impurity scattering rate in

semimetals is smaller than in conventional metals simply because semimetals are

typically much cleaner materials. The lower carrier density of semimetals also reduces

the rates of electron-phonon scattering compared to that of conventional metals. For

temperatures above the transport Debye temperature, which separates the regions of the









T- and T5 laws in the resistivity, 0 hkFS / k, where kF is the Fermi wave vector

and s is the speed of sound (both properly averaged over the Fermi surface), one can

estimate the electron-phonon scattering rate7 as

r 1 (ka,)(m* /m)k,T/h (2-2)

where a, is the atomic lattice constant, and m* and mo are respectively the effective mass

and the bare electron mass. In a conventional metal, kFaO 1 and m* i m In this case,

O is of the order of the thermodynamic Debye temperature hs /k Ba -few 100 K and

1/ r >> hAkT. Barring numerical factors, h / r < kT cannot be satisfied in a typical

metal. This means that as soon as it enters the classically strong field region,

magnetoresistance saturates and quantum magneto-oscillations start to show up. In a

low-carrier-density material (kFa <<1), 0' is much smaller (for Bi and graphite

O ~ few K) and also 1/ << h IkBT, which ensures that the inequality (2-1) can be

satisfied. Therefore, in a low-carrier-density compensated semimetal a wide interval of

temperatures and magnetic fields exists in which a) the scattering time is linear in T, in

accordance with Equation 2-2, b) we are in the regime of classically strong

magnetoresistance with essentially no signatures of quantum magneto-oscillations, as

specified by the inequality (2-1), and c) the magnetoresistance is large.

An additional feature that is crucial for interpreting the experimental data is that

the Fermi energies of graphite (EF = 22 meV)8 and bismuth (EF = 30 meV [holes])9 are

relatively low; and the temperature dependence of the resistivity is therefore a function of


SInequality (1) can be satisfied in a typical metal for T << R, when the (transport) time
1/ Ttr oc T << T. For an uncompensated metal, however, magnetoresistance saturates in this regime.









two temperature-dependent quantities, n(T) and -c(T). That materials are pure helps to

ensure that electron-phonon scattering is a dominant mechanism for resistance (in a

doped semiconductor, impurity scattering dominates).

2.2 Multi-Band Model

In the semiclassical theory of conduction in metals, DC electrical conductivity in a

multi-band system (in absence of a magnetic field) is described by

='-"(n) (2-3)
n
(n) = e2 dk r,n (k)) (k)i (k)(- f (k) (2-4)



where f is the Fermi function, and ( K) c(k)

h f

Since f 0, except when E is within kT of e, filled bands have no
OE

contribution to the conductivity. Only those partly filled bands that are close to the Fermi

level contribute to the conductivity.

In the presence of a classically strong magnetic field B = Bz (the Landau energy

level quantization is negligible), for an isotropic system in which all the occupied orbits

are closed, there will be no magnetoresistance because of the cancellation of Lorentz

force by the Hall components of the electric field. If an external electric field E = Exp is

applied, the induced current density will be j = coEx, where o0 is the in-plane zero

magnetic field conductivity. The Hall component of the electric field will be generated:

Ey = (o)r)E Then the definition: j = a E, hence










-oOE E
0 = ( (o or)Ex (2-5)
0 0

will yield

U0- 0-- C0D cT
1+(Ocr)2 1+(Or3)2
-C(o T (2-6)
or= 0
l+ (oc)2 1+((C0 )2
o o 1z (0)



This, in terms of resistivity, which is experimentally measured, is simply

p" -RB 0
p(B) = u(B) = RB pO 0 (2-7)
0 0 p"


where R = -is the Hall coefficient. Note that the longitudinal components of the
ne

resistivity have no field dependence.

Now consider such a system with more than one band. Each band contributes to the

conductivity of the system in parallel with the other bands. Then the total resistivity is in

terms of the resistivity of each band:



p p, Z RB p, 0 (2-8)
0 0 p:)


Even without calculating the resistivity above in detail, we can readily see that the

magnetic field dependence (which belongs to the off-diagonal components in the

resistivity tensor of each single band) may enter into the diagonal components of the total









resistivity. Thus the multiband system can have magnetoresistance even though none of

its band has magnetoresistance by itself.

For the simple (and most useful) case of a 2-band system, the resulting longitudinal

resistivity and Hall resistivity are:

p1, p (1 + pz)+(R p2 + R 1)B2 a)
p (2-9a)
(p1 + p22 +(R +R)2 B2

RR R(R, + R2)B3 + (2R2 +P2 R )B
p 2 (2-9b)
p(p + p2)2 +(R1 +R22 B2

For a system of more than 2 bands, it is convenient to describe the contribution of

each band in terms of conductivity. In a simple Drude model,

2 = u, (2-10a)
1 + (,B)2

enj ,2B
U e = (2-10b)
1 + (p, B)2

where, n, and /, are carrier density and mobility of the ith band. From the conductivity

tensor, we can then calculate the measured values of resistivy and Hall constant:

o, (B) a (B)/B
p(B) = xx and RH (B) = x(
U2 (B) + c2 (B) r2 (B) + U 2 (B)

2.3 Sample Preparation and Characterizations

2.3.1 Sample Preparation

The sample used in the study, a rectangular shaped highly oriented pyrolytic

graphite, with dimensions 2.4 mm wide by 8 mm long by 0.5 mm thick, was cut from a

bulk piece of highly oriented pyrolytic graphite (HOPG) using wire saw.









The HOPG sample has a mosaic spread, determined by X-rays, of 2 degrees. After

the sample was cut, it was glued onto a glass substrate. Figure 2-1 shows the

configuration of the measurement leads on the sample. The 4-terminal measurement leads

were connected to the sample applying silver paint. Because of the high in-plane/out-of-

plane conductivity ratio in layered graphite, we found it necessary to place the current

leads uniformly in contact with the sides of the sample. Thin-foil indium was coated

uniformly with silver paint, and was attached to the graphite end plates as current leads.

Gold wires with tiny loops were silver pasted with -2 mm separation to the edges of the

sample as voltage leads.

V" Hall











I+ V+ V- I-



Figure 2-1. Configuration of leads on graphite transport sample

Resistance measurements at 17 Hz were carried out using a Linear Research 700

resistance bridge. The sample was measured over the temperature range 2K- 350K in

fields as high as 17.5 Tesla. Low magnetic field measurements were carried out in a

Quantum Design Physical Property Measurement System (PPMS) with a 7 Tesla magnet.

High magnetic field measurements were carried out in a He3 refrigerator with a 17.5

Tesla magnet in the National High Magnetic Field Labs (NHFML). In all measurements,








the magnetic fields were applied perpendicular to the graphite basal plane (i.e., parallel to

the c-axis).

2.3.2 Characterizations: Dingle Temperature and Landau Levels

In the presence of strong magnetic fields, the energy bands of graphite split up into

Landau levels. With increasing magnetic fields, the interval between Landau levels

increases: As = ho)A = heB /m, where o), is the cyclotron frequency, and m is the

cyclotron mass When a Landau level moves across the Fermi surface, sharp features of

conductivity change appear. This caused the oscillatory behavior of resistivity in the

magnetic field sweep (Shubnikov-de Haas oscillations).


0.00 *
--.--2K f\. '""""'---

-5K


-0.05- -1


0. -

-0.10 4 1
0.2

0.0 -
-1 0 1 2 3 4 5 6 7 8
-0.15 B (Tesla)

0.0 0.5 1.0

1/B(Tesla-1)

Figure 2-2. Shubnikov-de Haas oscillations in graphite at indicated temperatures. The
oscillations are obtained by subtracting the background magnetoresistance
from the resistance vs. magnetic field curves. The inset shows the resistance
as a function of magnetic field at 2K.






12


Given the Fermi energy Ef, the number of Landau levels below the Fermi energy

Es
can be estimated as N = 1+ r Hence the period at which the one Landau level
heB / mI

moves across the Fermi surface goes like 1/B. As the magnetic field increases, every time

one Landau level shifts across the Fermi level, the resistivity will decrease because of the

high density of states at the bottom of the Landau band. And a valley will show up in the

resistivty vs. magnetic field plot. By plotting the valley positions of the ShdH oscillations

vs. 1/B, one gets evenly spaced points. By labeling these points one can count the number

of Landau levels lying below or across the Fermi surface. Since the band structure does

not change with temperature, the valley positions measured at different temperatures

should overlap well. Figure 2-3 below shows the Landau level indices as a function of the

inverse of magnetic fields at different low temperatures.




8 *
)* 2K
S7 M 5K
> A 10K
-J 6 -

5 -




2 -



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

1/B (Tesla1)

Figure 2-3. The Landau level indices as a function of the inverse of magnetic field at
different low temperatures









Figure 2-2 shows that the ShdH oscillations are most pronounced at very low

temperatures (kT << hio)). At higher temperatures, the oscillations are smeared by

phonon scattering. The relation between the amplitude of the ShdH oscillations and

temperature T at magnetic field B is shown in Equation 2-11.10


Ao B-52 exp a(T+ Td) (2-11)
B

Here Td, the Dingle temperature, describes the temperature-independent scattering

from impurities and dislocations. Dingle temperature is a useful parameter in

characterizing the quality of the graphite sample. The lower the Dingle temperature, the

less imperfection the sample has.

Dingle temperature can be measured from conductance-field sweep at different

temperatures. First, the background of the conductance vs. field curves is subtracted out,

and the oscillatory part is plotted vs. 1/B. Then the oscillation amplitudes are obtained by

subtracting the envelope curves of the oscillations valleys from the envelope curves for

the peaks (Figure 2-4).

Since only the shapes of those curves are important, we used arbitrary smooth

functions to fit the envelope curves. This method of interpolation give much more precise

oscillation amplitude values than simply subtracting the resistance values at the valleys

from those at the adjacent peaks (which do not correspond to the same field).

After we get the amplitudes of the oscillations Aosc, we plot Ao B5/2 vs. 1/B,

and obtain straight lines in AocB5/2 vs. 1/B plots ((Figure 2-5)). The slopes of the

straight lines, according to Equation 2-11, simply correspond to a(T + T, ).









0.45 ., 1. ,


0.40


0.35 0


L 0.30 -
030.5






o 01
0.25 -




0.15 -
0.0 0.5 1.0 1.5 2.0
1/B (Tesia 1)
0.10 1 1 1 1 1 ,1
0.4 0.6 0.8 1.0 1.2 1.4 1.6

1/B

Figure 2-4. The amplitude of the ShdH oscillations as a function of the inverse of
magnetic field at 2K, obtained by subtracting the bottom envelope curves
from the top envelope curve of the oscillations, as indicated in the inset.

We then plot the values of the slopes vs. their corresponding temperatures, and

perform a linear equation fitting to the curve. The fitted straight line then intersects with

the negative side of the temperature axis, with the offset being the Dingle temperature Td.

For the graphite used in our study, the Dingle temperature we got from the analysis

described above is about 4.5K. This result suggests that the smearing of the ShdH

oscillations by the imperfections in the HOPG corresponds to the thermal smearing of

4.5K.












2K
0 5K
S10K
e 15K












0.4 0.6 0.8 1.0 1.2 1.4 1.6

1/B (Tesla1)


Figure 2-5. Scaled ShdH oscillations amplitude: Ln(AocB5/2),
of magnetic field in different temperatures.


as function of the inverse


-4x104


-6x104 -


0 2 4 6 8 10

T(K)


12 14 16


Figure 2-6. Linear fit to the slopes of the scaled ShdH oscillation as function of
temperature.


I-

S-8x104



-1x105


T, = 4.5 +1K


I I I I I I I I I I I I I I I


I I I I I I I I









ShdH oscillations also provide information about the Fermi surface. By measuring

ShdH oscillations in different magnetic field orientations, one can map out the extremal

of the Fermi surface. For graphite, by measuring ShdH oscillations and calculating the

volume of the Fermi pockets, we can then calculate the carrier concentration.

Here we simplify the problem by treating the Fermi surface of graphite as ellipsoids

with anisotropy ratio r. By measuring the ShdH oscillations with magnetic field parallel

to the c-axis of graphite, we get the period of the quantum oscillations to be

he
A(B') == (2-12)
E me

,where mc is the cyclotron mass for magnetic field along the c-axis. This provides the

information about the area of the extremal cross section of the Fermi surface with a plane

perpendicular to the magnetic field: Sx,, = 27i mc (in momentum space). Given the

anisotropy ratio r7, we can then calculate the volume of the ellipsoid and hence the

carrier concentration:


n /4 27N[A(B-1)]3/2 (2-13)


where N = 6 is the number of ellipsoids in the Brillouin zone. Taking 7 12-17, the

carrier concentration calculated from the measured oscillations period is

n 2 3 x 1024m3 This number corresponds to the zero temperature carrier

concentration in graphite. At low enough temperatures, one can separate the ShdH

oscillations periods from electrons and holes, and calculate the carrier concentration for

the two different carriers. Also, for each carrier group, there will be two sets of

oscillations because of the spins. At 2K however, we are not yet able to distinguish the









different periods result from the small effective mass difference between electrons and

holes, and from the spin splitting.

2.4 Transport in the Classical Region

In this section, we present a detailed study of low field magnetotransport in

graphite and show that the "unusual" behavior of the temperature and field-dependent

resistance, such as shown in Figure 2-7, can be described in a straightforward way by a

simple multi-band model that takes into account contributions to the conductivity from

the electron and hole carriers associated with the overlapping valence and conduction

bands.


1 E-6


0---
E


X
X
a-


1E-7


1E-8


100

T (K)


150


200


Figure 2-7. Temperature dependence of the resistivity px for a graphite crystal plotted on
a logarithmic axis at the magnetic fields indicated in the legend. The solid
lines are the fits to the data using the six parameters derived from the three
bands described in the text. The shadowed region on the inset and its mapping
onto the data in the main panel are described in the text.


1.5

1.0
I II
0.5

0.0
0 50 100 150 200
T (K)

* 0 mT
20 mT
40 mT
v 60 mT
80 mT
o 100 mT
A 200 mT









We use the qualifier "unusual" in describing the data of Figure 2-7, since on

lowering the temperature the resistance increases as it does in an insulator but then

saturates at lower temperatures. The non-trivial explanations of Kopelevich et.al 34 rely

heavily on such features as the Dirac spectrum of fermions and almost two-dimensional

transport, which are unique for graphite but not for Bi. That Bi and graphite behave

similarly suggests that these features are not responsible for the observed phenomena.

Our explanation for the insulating-like behavior in a magnetic field does not require more

exotic explanations of a magnetic-field-induced opening of an excitonic gap in the

spectrum of interacting quasiparticles.11 Instead, we propose that the uniqueness of the

low magnetic field transport behavior of semimetals lies in the existence of a wide

interval of temperatures and magnetic fields defined by the inequalities of Equation 2-1.

Our analysis of the experimental data confirms the inequalities .

Both pxx and py (see Figure 2-8) were measured in magnetic fields up to 0.2 Tesla

at different temperatures. A small field-symmetric component caused by slightly

misaligned electrodes was subtracted from the pxy(B) data. To fit the data, we adopt a

standard multi-band model.5 Each band has two parameters: resistivity p' and Hall

coefficient R, = Yq, where q, = +e is the charge of the carrier. In agreement with


earlier studies, we fix the number of bands to three.1 Two of the bands are the majority

electron and hole bands, and the third one is the minority hole band. Although the

presence of the third band is not essential for a qualitative understanding of the data, it is

necessary for explaining fine features in pxy. Our fitting routine incorporates both pxx(B)

and pxy(B) simultaneously by adjusting the six unknown parameters independently, until

the differences between the fitting curves and the experimental data are minimized.






19


Because the majority carriers in graphite derive from Fermi surfaces that have six-

fold rotational symmetry about the c-axis, we only need to deal with the 2x2

magnetoconductivity tensor with elements ,xx and cxy.

P, RB
Here, we define the conductivity ca R + ,RB and the
p2 + (RB)2 p2 +(RB)2 a

resistivity p, = m /nZe21, for the ithband. The total conductivity is simply a sum of the

contributions from all the bands: i = V ~r. The observable resistivity tensor is obtained
1=1..3

by inverting a p= :1

Qualitatively, the unusual temperature dependence of px displayed in Figure 2-7

can be understood for a simple case of a two-band semimetals, where p, reduces to


= PePh(Pe + Ph)+ (PRh + Ph Re) (2-14)
(Pe + Ph) +(R +Rh)2B2

Here pe (Ph) and R (Rh) are resistivity and Hall coefficient for the electron (hole) band,

respectively.

Assuming that pe,h oc Ta with a > 0, we find that for perfect compensation (i.e.,

ne = nh, where ne and nh are carrier density for the electron band and the hole band

respectively), Re = -Rh = R and the 2-band resistivity described by Equation 2-14 can

be decomposed into two contributions: a field-independent term o T" and a field-

dependent term oc R2 (T)B2 Ta At high T, the first term dominates and metallic

behavior ensues. At low T, R(T) oc 1/n(T) saturates and the second term dominates,

giving insulating behavior.














10-6





107
S5K 0 40K

15K v 100K
v 20K 100K
o 25K A 200K
10-8 1--
0.00 0.05 0.10 0.15 0.20
B (T)

3.0xl 07
.4x1- 5K o 40K
1.4x10" -
2.5x10.7 12x1- 10K 70K
15K v 100K
1.0x10io- 20K 150K
2.0x10.7 ~ 8.0x10 25K 4 200K
S 6.0x10

E 1.5x10-7 42x10
2.0x10i-
0.0
Q. 1.0x107 -2.0x10
0.00 0.01 0.02 0.03

5.0x10 -8


0.0


0.00 0.05 0.10 0.15 0.20
B (T)


Figure 2-8. xx and pxy versus applied magnetic field at the temperatures indicated in the
legend. The solid lines are determined by a fitting procedure described in the
text. The inset in the py plot magnifies the low-field region where the
contribution from the minority band is important.









350
S300
250
E 200
150
100
0 50
E 0

1.8E12
1.5
1.2
S 0.9
(D
c/ 0.6
T 0.3
0.0

10E24

E 8.0

S 6.0
(D
-0 4.0
(D
*E 2.0
0.0
0.0


50 100 150
Temperature


200


Figure 2-9. Temperature dependence of the fitting parameters: A) mobility; B) relaxation
time; and C) carrier density, for the bands indicated in the legends of each
panel.

The actual situation is somewhat more complicated because of the T-dependence of

the carrier concentration, the presence of the third band, and an imperfect compensation


'A I I I I
A --- band 1
-- -- band 2
-a-A- band 3


--- A b
A I



- -i- band 1 5
-*- band 2
-A- band 3








--- band 1
-*e- band 2
-A- band 3

C









between the majority bands. Results for the temperature-dependent fitting parameters are

shown in Figure 2-9, where band 1 corresponds to majority holes, band 2 to majority

electrons and band 3 to minority holes. The insulating-like behavior of the carrier density

with a tendency towards saturation at low temperatures is well reproduced. For the

majority bands, 1 and 2, the carrier concentrations are approximately equal and similar in

magnitude to literature values.12 The slope of the linear-in-T part of 1 = aexpkT/h

with axp = 0.065(3) (dashed line in Figure 2-9, panel A) is consistent with the electron-

phonon mechanism of scattering. To see this, we adopt a simple model in which carriers

occupying the ellipsoidal Fermi surface with parameters mab (equal to 0.055mo and 0.04

mo for electrons and majority holes, correspondingly), me (equal to 3mo and 6mo,

correspondingly) interact with longitudinal phonons via a deformation potential,

characterized by the coupling constant D (equal to 27.9 eV). In this model, the slope in

the linear-in-T dependence of r is given by7

theory= (2 /z) EFD2 /O S p h3 (2-15)

where m*= (mm)1/3 0.21mo both for electrons and holes, po = 2.27g/cm3 is the

mass density of graphite, and S = 2 x 106 cm / s is the speed of sound in the ab-plane.

(The numerical values of all parameters are taken from standard reference on graphite12)

With the above choice of parameters, atheo = 0.052 for both types of carriers. This value

is within 20% of the value found experimentally. Given the simplicity of the model and

uncertainty in many material parameters, especially the value of D, such an agreement

between the theory and experiment is quite satisfactory.









The solid lines through the data points in Figure 2-7 are calculated from the

temperature-dependent fitting parameters derived from our three-band analysis and

plotted in Figure 2-8. The shaded region (II) depicted in the inset of Figure 2-7 represents

those temperatures and fields that satisfy the inequalities of Equation 2-1. In region (I)

ShdH oscillations can be seen at sufficiently low T (our sample has a Dingle temperature

of 5K), and in region(II) the magnetoresistance is low. The boundary between (I) and (II)

reflects the rightmost of the inequality 2-1 and is determined by the relation

T > rheB/m *, where 7 = 5 has been chosen to represent the ratio kT/hIoi. A larger

value of 7 would decrease the slope of this boundary and diminish the area of (II). The

boundary between (II) and (III) reflects the leftmost inequality of 2-1 and is determined

by the relation B > m / e (T) where 1/r (T) is obtained from experimental fitting

parameters (Figure 2-9). In the main panel of Figure 2-7, we superimpose region (II),

again as shaded area, on the p,(T,B) data. Below the lower boundary co) < 1, and the

magnetoresistance is relatively small. The upper boundary is determined by the locus of

(B,T) points satisfying the rightmost inequality of 2-1 for 7 = 5. Clearly region (II),

constrained by the inequality 2-1, overlaps well with the metal-insulating like behavior of

graphite. We thus conclude that the semimetals graphite and, by implication, bismuth

share the common features of high purity, low carrier density, small effective mass and

near perfect compensation, and accordingly obey the unique energy scale constraints that

allow pronounced metal-insulating behavior accompanied by anomalously high

magnetoresi stance.














CHAPTER 3
MAGNETOTRANSPORT OF GRAPHITE IN THE ULTRA-QUANTUM FIELD

3.1 Transport Data in the Ultra-Quantum Field

The inequality discussed in chapter 2: h / z < Aho < kT defines two limits that are

satisfied within a wide temperature range in semimetals: h / r < hoi, (or o),z > 1) gives

rise to the large magnetoresistance in the semimetals; hoai < kT defines the "classically

strong" magnetic field (weak field), in which the number of Landau levels below the

Fermi level is so large that the quantum oscillations are well smeared by temperature,

hence the effect of quantization of the energy bands is negligible.

In stronger magnetic fields, there are only few Landau levels below the Fermi level

and A), < kBT is no longer satisfied. Eventually, when the magnetic field is so strong

that all the conduction electrons are all in the lowest Landau level, the so-called ultra-

quantum limit is reached. Above the ultra-quantum limit, the energy of the electrons is

fully quantized in the plane perpendicular to the field. The movement along the field lines

is free; hence the electrons in the system assume movement with spiral trajectories along

the field lines. Ideally, in the absence of scattering and interaction, a system in the ultra-

quantum regime should have zero conductance in the plane perpendicular to the magnetic

field, because the Lorentzian force confines the movement of the electron to the spiral

trajectories. However, with interactions and scattering, the electron can move along the

plane perpendicular to the field lines in a diffusive manner.









Figure 3-1 shows the strong magnetic field magneto-transport data taken from the

same HOPG used in the low magnetic field study.


0 5 10 15 20
B (T)


15K
B /20K
7.5K
10K

5K
40K



70K






0 5 10 15 20

B (T)


Figure 3-1. The magnetic field dependence of the
of HOPG at different temperatures


A) longitudinal; and B) Hall resistance


20K
15K
40K
10K
S7.5K
-5K
70K


1500


1200


900


S600


300


40




E
x20





0









2000



1500



S1000 17.5T
cn 16T
S/ 4T
12T
500 /10T
8T
6T
4T
0 I 1 T,
0 20 40 60 80 100

T (K)

Figure 3-2. The temperature dependence of the longitudinal resistance of HOPG in
different magnetic fields

From the magnetic field dependence of the longitudinal resistance data, we see that

the resistance increases roughly linearly with magnetic field, and tends to saturate in very

high magnetic field (>10 Tesla). For the curves taken at very low temperatures (T <15K),

we see ShdH oscillations on top of the magnetoresistance. The Hall resistance has much

smaller values than the longitudinal resistance in high magnetic fields, and has rather

complicated field dependence.

We are most interested in the temperature dependence of the resistance in different

strong magnetic fields. Here we see that, the resistance increases with decreasing

temperature for T >30K, similar to what is observed in the low (classical) magnetic

fields. For T <30K, however, the resistance plunges down with decreasing temperature in

strong magnetic fields.









The "metallic" behavior observed in strong magnetic fields and low temperatures

can not be explained by semi-classical transport theory. It was proposed that the high-

field transport behavior is a manifestation of field-induced superconductivity4. However,

we find a more conventional interpretation by considering graphite as a multi-band

system. Our strategy of analyzing the data relies on obtaining in-band transport behavior

from the experimentally measured data using the multi-band model. Then we will try to

understand the in-band transport, which represents the intrinsic physics of the graphite

system.

3.2 In-band Transport Behavior in the Ultra-Quantum Regime

In strong magnetic fields, the multi-band model still applies, except that we can no

longer take the resistivity and Hall coefficient of each single band to be field-independent

parameters, because of the strong quantum effect. Instead of curve fitting with field

independent parameters, we need to start by simplifying the multi-band model. Since the

contribution of the minority band vanishes in high fields, we can apply the simple 2-band

model, in which:

P1 2(1P + P2)+(pR2 + 2R )B2
(p1 + p2)2 +(R + R2)2B2

RR,(R, +R,)B3 +(Rp2 +R2p12)B
P (p + p2) +(RI +R,)2B2

Now we make the assumptions that, in high magnetic fields, the system is nearly

compensated, and the resistivities of each band are very close:

P2 P (3-la)

R, -R R (3-1b)









We will see that these assumptions are valid when applied to the experimental data

in high magnetic fields. With the simplification above, we have

p2 +RzB2
p, = 2p (3-2a)
4p2 + 2B2

-R2B2+p2
y = SB 2 (3-2b)
4p2+ 2B2

where 8 R1 + R2 is the difference of the electron and hole Hall coefficients.

The ratio of the two numerator terms p and RB in high magnetic fields gives rise

to two different pictures of transport properties in this region. In the first picture, we

assume that the magnetoresistance shown in px is mostly from the magnetoresistance of

each single band. An extreme case of this picture is when: p >> RB, and hence:

p, =p12 and p, = B/4.

In this case, the longitudinal resistivity of the 2-band system is essentially the same

as that of each single band. Accordingly, we see that in this picture, we do need the non-

trivial explanations for the non-saturating MR in a system with closed orbits, i.e., field

induced metal-insulator transition (MIT) and re-entrance to metallic behavior in

quantizing magetic fields.1-4, 13

In the second picture, we assume that the MR shown in px is mostly from the

diagonal (Hall) resistivity, (RB)2 >> p2 (or (o Zr)2 >> 1). Then the simplified forms of

the high field limit resistivities are:

2R2pB2
2R2B2 (3-3a)
4p2 +52B2










3R2B3
S4p2 +2B2


The ratio of the two measured parameters,





0.1





x

X 0.01







1E-3


- is plotted in the figure below:
p, 2p


6 12
BT)
Figure 3-3. The ratio of the measured Hall resistance and longitudinal resistance as a
function of magnetic field at 2K.

CB
It can be seen in the figure that, the ratio <<1 for field above the ultra-
P

quantum limit. This rules out the possibility ofun-compensation (i.e., 3 0 ) as a

mechanism for the "re-entrant" behavior in high magnetic fields and low temperatures,

since the second term in the denominator of both p, and p, can be neglected.

Now we have a even more simplified form of px and p :


R2B2
p, = (3-4a)
2p


(3-3b)









R2B3
P, = -- (3-4b)
4p2

It can be seen from these relations above that, the transport behavior in the second

picture is drastically different from that in the first picture. The 2-band longitudinal

resistivity is now proportional to the reciprocal of the resistivity of each single band!

Therefore the "re-entrance" to metallic behavior of the 2-band system as measured by

px would really correspond to a cross over from metallic to insulating behavior for each

band (as measured by p1,2) as the temperature drops.

The criterion for the second picture to be valid is that: RB >> p must be satisfied.

In relatively low magnetic fields (say, B 0.1 Tesla), taking the values of p 10 8 and

R ~ 10 6, we see that this criteria is well satisfied. Hence we are confident in ruling out

any non-trivial MIT mechanism in explaining the MIT-like behavior. Despite the

complicated field-dependence of p, the major mechanism for MR is the Hall resistivity

term RB.

R2B2
In very strong magnetic field, RB >> p is required for the result p, = to be
2p

valid. Hence we require that p >> RB. In the range of the magnetic field we are

studying, the carrier concentration increases slowly with increasing field12, hence the in-

band Hall coefficient R decreases with increasing field. Taking the experimentally

measured p, ~ 3 x 10-4 Om, and R < 10-6 QmT1, we can see that px >> RB is satisfied

for B ~ 10 Tesla. This indicates that our second picture is self-consistent according to the

experimental data.









From the analysis above, we see thatRB >> p is generally satisfied in the magnetic

CB
field range of our study. We also note that RB >> p and <<1 imply S << R. Hence
IP

our assumption that the system in strong magnetic field is nearly compensated is well

consistent with the experimental results.

Till now, we haven't made any assumption on the expressions of the resistivities

and the Hall coefficients of the two bands. Ideally, if we have the complete information

on the band structure of graphite in high magnetic field and the scattering mechanisms,

we can exactly calculate the conductivity tensor using Kubo formula. Such a procedure,

however, will be extremely complicated.

In the ultra-quantum limit (UQL), the problem may get simplified by the fact that

all the carriers are in the lowest Landau level at sufficiently low temperatures:

kBT << Ahco Ef This can be understood by examining the Landau band structure of

graphite in the ultra-quantum limit field.

The energy band quantization due to the magnetic field can be calculated using the

classical tight-binding SWMcC model.12 The dispersion relation from the calculation

shows that, in magnetic fields, each energy band separates into different Landau bands.

With increasing magnetic field, all other conduction bands and valence band move

further and further away from each other and from the Fermi level, while the lowest

(zeroth) Landau bands remains field in-dependent. For field above the ultra-quantum

limit (- 8 Telsa for graphite), the Fermi level runs across only the lowest (zeroth)

conduction band and valence band, which are the only Landau bands that contribute to

the conduction. Figure 3-4 shows the Landau band structure of graphite in 12 Tesla field,









obtained from the classical band structure calculation (SWMcC model). The field

independence of the zeroth Landau bands was confirmed by the recent high magnetic

field scan tunneling spectroscopy measurements on HOPG.14


0.10


0.05



0.00


-0.05


-0.10 1I / I iI
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Pzco/2


Figure 3-4. The Landau band dispersion relation of graphite in 12 Tesla field, calculated
using the SWMcC model.

From the Landau band structure shown in Figure 3-4, we can see that in strong

magnetic field, at sufficiently low temperature: kT << hao, E, the number of the

thermally excited electrons (and holes) in the higher Landau levels is negligible, hence

we can make the approximation that he carrier concentration does not change

significantly with temperature nor magnetic field. Since the high field limit of the Hall










coefficient is simply: R = 1, R will be roughly field and temperature independent in this
ne

region (compared with the factor of two resistivity change in this region).

With the approximation above, we have enough information to separate out the in-

band resistivity using the 2-band longitudinal resistivity. We can also estimate the un-

2 R2
compensation 5 in this high B low T region, by calculating p
p B 8

Figure 3-5 below shows the values calculated from experimental data:


106
106




10s








102



2 4 6 8 10 12 14 16 18

B (Tesla)
2 R2
Figure 3-5. as function of magnetic field at 2K, calculated from the
pB 8
experimental data.

p2 -R2
The saturation of is seen in the Figure at field above 8 Tesla (UQL for
pB 8

graphite). The weak field dependence can be attributed to the combination of weak









thermal excitation and field dependence ofun-compensation, and the saturation of 3.

Using the value ofR 10-6 / T (from the band structure calculation), we can estimate

that 38 10 17 Q/T. Hence our assumption of"near compensation" is well satisfied.

From the self-consistent approximations above, we reach a very simple high

magnetic field limit (B > BUQL) result: the resistivity of each single band is proportional

RB2 2B2
to the reciprocal of the 2-band resistivity in graphite: p = -- Figure 3-6 shows B-
P1 P1

calculated from measured data. The high field part of the curve will represent the

resistivity of a single band itself under our assumption that R is field-independent in high

magnetic field.











0 0.1



-570K
-7.5K



10
15K




20K
-25K
40K
-70K

1 10

B (Tesla)

Figure 3-6. as a function of magnetic field at the indicated temperatures, calculated
R
from the experimental data.









From Figure 3-6 above we can clearly see the tendency of scaling of curves in high

magnetic fields and low temperatures (curves become parallel to each other). This

indicates that in this region, we only need to care about the contribution from the lowest

Landau band, and neglect the thermally excited carriers in the higher index number

Landau levels. Figure 3-6 also gives us a range of such regime: B >10 Tesla and T<20K.

We note that this range satisfies the inequality for temperature and field independent

carrier density: kBT << ho E (at 10T, 20K: kBT ~ 0.9meV, hoA Ef 6meV)

With this simplification, we can directly analyze the temperature dependence of

resistivity in high fields and low temperatures assuming that the carrier densities of the

two bands are temperature independent, or R is T-independent. By treating R as a T-

independent and B-independent constant, we see that the in-band resistivity has a

temperature dependence p oc B2 / p,. The in-band conductivity scaler, which contains

the intrinsic physics about the interactions and scattering of the system, is

1 pa pa neZ2
simply o- = Here since o0 and because of the week
p (RB)2 B2 m*

temperature and magnetic field dependence of n and m* in the ultra-quantum region, this


implies that the relaxation time: r ~ Figure 3-8 shows the shape of the in-band
B2

conductivity o0 vs. T using logarithmic axis.

From Figure 3-8 we see that above the UQL field and for kBT << Ah E : 1) at

fixed temperature, the in-band conductivity o- decreases with increasing magnetic field;

2) at fixed magnetic field, the in-band conductivity co decreases with decreasing


temperature.









25

20

15


10
C ^ 4T
S C 6T
8T
5 10T
12T
14T
16T
17.5T
I I I
1 10 100

T (K)


Figure 3-8. Logarithmic plot of the shape of the in-band conductivity (- ) as function
B
of temperature in different strong magnetic fields, calculated from the
experimental data.

Using o0, we can also calculate the in-band conductivity tensor:

0 a0 1 1
= C 0 (3-5a)
l+ +(OZr)2 (c )2 B2Z pZ

S0) (3-5b)
1+ (+(0or)2 ot B

Since in fixed magnetic field, the relaxation time r decreases with decreasing

1
temperature, a, ~ will increase with decreasing temperature. The In-band Hall


conductivity is independent of the relaxation time.









3.3 Possible Models in the Ultra-Quantum Regime:

At this step, we have discovered the transport behavior directly resulting from the

physics of interactions in the graphite system in UQL field. Now we will try to

understand the physics that causes this transport behavior: 1) the relaxation time r

1.
decreases with decreasing temperature; 2) a, ~ increases with decreasing


temperature. In this section, we will consider two possible models that give this kind of

transport behavior: the magnetic field induced Luttinger liquid with impurity scattering,

and phonon delocalization (dephasing).

In the model of magnetic field induced Luttinger liquid with impurity scattering,

the scattering mechanism considered is the elastic scattering of the impurities dressed

with the Friedel potential,15 as illustrated in Figure 3-9. At zero temperature, the electrons

are localized by "dressed" impurities along the direction of the field lines. The potential

of the "dressed" impurities is considered as a tunnel barrier. With increasing temperature,

because of the increasing energy of the electrons, the effective scattering cross section of

the impurities will decrease, and the probability for the electrons to tunnel trough the

"dressed" impurity potential will increase. These will lead to an increasing relaxation

time (hence o, ) with increasing temperature. The temperature dependence of

conductivity along the field direction, ou coincides with that of o With increasing

conductivity along the field direction with increasing temperature, the conductivity along

LL 1
the plane perpendicular to the field will decrease with increasing temperature: o -


Since there is also classical magnetoresistance in the x-y plane, the field induced








Luttinger liquid behavior is a "secondary" correction to the semi-classical magneto-

transport behavior: a, = o0 + aj .






4'




V z


ICc


j impurity



Friedel oscillations



Figure 3-9. Model of the field induced Luttinger liquid with dressed impurity scattering.
The model of the field induced Luttinger liquid with impurity scattering predicts

that the correction of the conductance has power law temperature dependence, with

magnetic field dependent powers: 15

0L r T ~ (B) (along the field direction), (3-6a)

SLL 1/ r T- a(B) ( perpendicular to the field). (3-6b)

A most important prediction from the model is the field dependent power factor. To

check the field dependence of the experimental data, we plot scaled in-band conductivity









ao as function of temperature in different high magnetic field above the UQL. The

conductivities are normalized at the lowest temperature point:

20 I .
-10T
-12T
-14T
16 16T
16T
-17.5T











8

1

T(K)

Figure 3-10. Scaled in-band conductivity as a function of temperature in magnetic fields
above the UQL indicated in the legend.

It can be seen from the Figure 3-10 that, for field well above the UQL, the scaled

conductivities overlap perfectly. This indicates that the magnetic field dependence and

the temperature dependence of the conductivity can be separated: o- = F(B)G(T). This

is obviously contrary to the prediction from the magnetic field induced Luttinger liquid

theory, in which the power factor of the power law temperature dependence itself

depends on the magnetic field.

Another possible model is the phonon delocalization (dephasing) model. In this

model, the electrons are delocalized by phonon scattering. Hence with increasing









temperature, the conductivity along the magnetic field increases, while the conductivity

perpendicular to the magnetic field decreases. The phonon delocalization mechanism

predicts the same trend of the temperature dependence of the conductance:16

o7 T ~ T (along the field direction), (3-7a)

o ~ 1 / Tr T- (perpendicular to the field). (3-7b)

The phonon delocalization mechanism differs from the field induced Luttinger

liquid by an exponent which is now independent of the magnetic field. So the phonon

delocalization mechanism agrees better with the experimental data. However, as far as we

know, there is no complete theory for phonon delocalization, and there is no theoretical

prediction of the values of the power factors. Further work needs to be done to

quantitatively understand the transport behavior of graphite in the UQL region.














CHAPTER 4
TUNNELING INTO BULK BISMUTH IN THE ULTRA-QUANTUM FIELD

4.1 Motivation

In the previous chapters, we have discussed the magneto-transport properties of

semimetal graphite. Graphite is a relatively simple system for transport study in a sense

that, in its hexagonal Brillouin zone, the Fermi surface comprises six cigar shaped

pockets with their long axis parallel to the c axis. Transport in graphite is roughly 2-D

due to the weak coupling between the graphene layers and large ratio of out-of-plane and

in-plane effect mass. All these factors simplify the transport study of graphite so that it

can be treated as an isotropic 2-D system, in which only two major groups of carriers,

electrons and holes (each has one single mobility), need to be considered.

Bismuth, on the other hand, is much more complicated. In the Brillouin zone of

bismuth, there are 3 electron pockets and 1 hole pocket (see Figure 5-1 in chapter 5).

None of the pockets is parallel to any of the others, hence Bismuth is 3-D in all

orientations. And in most of the orientations, every pocket contributes carriers with

different mobility. Hence in bismuth, one needs to consider up to 4 majority bands, each

with different effective mass and different mobility. This makes the magneto-transport

study of bismuth very complicated.

Rather than studying transport, we studied the magneto-tunneling properties of

bismuth in strong magnetic fields. The original intent of our work was to study the

possible high magnetic field induced 1-D (Luttinger Liquid) behavior. The tunnel

junctions used in the study comprise metal-insulator-semimetal trilayer structure. In zero









field, we are simply tunneling from a 3-D metal into a 3-D semimetal. Tunneling theory

predicts that the measured differential conductance has a very weak dependence on the

density of states of any of the electrodes and depends mainly on the properties of the

tunnel barrier. When a strong magnetic field is applied, the energy levels in the semimetal

separate into different Landau bands. When the magnetic field is strong enough so that all

the electrons in the semimetal are in the lowest Landau level, the semimetal enters the

ultra-quantum regime, and the tunneling is between a 3-D Fermi liquid in the normal

metal and "Landau tubes" in the semimetal. In this regime, the semimetal has an

essentially ID character with 1D Landau tubes aligned along the magnetic field and

perpendicular to the tunnel junction area.

For 1-D systems, one can no-longer describe the physics using the Fermi liquid

theory, because of the strong perturbation of Coulomb interaction due to the lack of

screening and phase space for scattering. Instead the 1-D system will be described by the

Luttinger liquid theory. The tunneling experiment provides an opportunity to discover the

enhanced density of states predicted by the Luttinger liquid theory.

When tunneling into a 1-D system, the tunneling theory predicts that the measured

differential conductance across the junction has strong dependence on the density of

states in the 1-D electrode. The proposed magnetic field induced Luttinger Liquid

theory15 states that: for magnetic field induced LL connected to 3D reservoirs by tunnel

barriers:

dI/dV ocT(B) (when eV << kBT), (4-la)

dI/dV ocVa(B) (when eV >> kBT). (4-1b)









The reason that a semimetal is used as the electrode of interest is that, the magnetic

field for the semimetals to reach the ultra-quantum limit is relatively obtainable. For

example, -10 Tesla is needed to drive bismuth into the ultra-quantum regime. For most

metals however, the magnetic field needed will be >104 Tesla, because of their high

Fermi energies and large cyclotron masses.

4.2 Experimental

Tunnel junctions are made on freshly cleaved bismuth crystals. A big piece of pre-

cut bismuth crystal is dropped into liquid nitrogen bath. After the bismuth crystal

equilibrates with the liquid nitrogen, a Razor blade was used to cleave the crystal and

expose a fresh and smooth surface of the trigonal plane. The piece of cleaved bismuth

with smooth surface was then taken out of the liquid nitrogen bath and warmed up in pure

nitrogen gas flow. Through this way, the cleaved surface will maintain its freshness and

there will be no condensation on the surface when it is warming up.

On the surface of bismuth, we have 2 methods for making tunnel junctions. Figure

4-1 shows the procedure using photolithography to make tunnel junction on top of the

semimetal surface. First of all, we define an undercut photoresist pattern on top of the

junction area we choose. Then a thick layer of AlOx was RF sputter deposited onto the

sample as separation layer. This layer prevents shorting outside the tunneling area

resulting from the surface roughness, and from the force of the contact leads. Then we

performed lift-off and opened up the tunneling area. Then we deposit the tunnel barrier,

followed by the counter-electrode, through shadow masks. Finally we pasted gold wire

on top of the counter-electrode above the separation layer.













UV expose


photoresist


semimetal


UV expose









semimetal


semnimetal


- opitcle mask






semimetal


photoresist


200A
Aluminum








develop


Aluminum Oxide 400A
RF sputtering


40um


semimeta


Ilift-off
l in Aceton


Stunneling barrier
semimetal Aluminum Oxide -10A
semimetal


Pb
counter
-electrode


sI- e"


semimetal


Figure 4-1. Procedure for making tunnel junctions on bulk semimetal using
photolithography technique.









This method using photolithography is convenient for defining the junction area.

Using the precise alignment function of the photomask aligner, it is easy to find the ideal

smooth junction area and place the photoresist pattern on top of it. The shortcoming of

this method is that after liftoff, there is always some residue of photoresist left on the

surface. The residue can be mostly cleaned by prolonged treatment with UV ozone

cleaner (-40 minutes). But the cleaning process can cause unwanted oxidation.

Figure 4-2 shows another way we used to make junctions on the surface of

semimetals. This is what we called mica mask method. Essentially it is a shadow mask

method used in RF sputtering. Through conventional shadow masks, sputtering fails to

yield sharp edges due to the high Ar pressure during the deposition. Here in the mica

mask method, an extremely thin mica foil is placed on top of the semimetal. The sheet of

mica will attach itself by Van der waals force to the semimetals surface. We can also

attach another layer of mica on top of it to get an undercut. Then we RF sputter thick

AlOx film as a separation layer. After removing the mica foils, we get a very sharp edge

of the AlOx film, due to the intimate contact of the mica mask to the semimetal surface.

Then we deposit the tunneling barrier and the counter-electrode, and finally put on the

gold wires as measurement leads.

The mica mask method, without using any lithography technique, is fast and

convenient. Also, there is no contamination from the resist polymers to the junction

surface. The shortcoming of the mica mask method is that, without lithography and

precise alignment, it is relatively hard to obtain well defined tunneling area.

Figure 4-3 shows the microscope picture of a tunnel junction made on the surface

of bulk bismuth.








AlOt
N lica foil S eri
Sputtering


III.


Thermnal A1C
(barrier)


iFI Pb


7+


V-


Figure 4-2. Procedure for making tunnel junctions on bulk semimetal using mica mask
method.


Figure 4-3. Microscopic picture of a Bi(bulk)-AlOx-Pb tunnel junction.









Measurements of the tunnel junctions are carried out in a He3 refrigerator with an

18 Tesla superconducting magnet in the National High Magnetic Field Labs (NHMFL).

Differential conductance, dl / dV, is measured using a double lock-in amplifier technique,

in which one lock-in amplifier is used together with a feedback circuit to keep the small

AC (500Hz) excitation voltage, dV, across the tunnel junction constant, while the other

lock-in amplifier measures the AC voltage response of a standard resistor in series with

the junction, from which dl can be calculated. A slow DC ramping signal is summed with

the AC excitation voltage to apply the bias voltage across the tunnel junction.

4.3 Results and discussion

Figure 4-4 shows the magneto-tunneling result for a Bi(bulk)-AlOx-Pb tunnel

junction made through mica mask method. The figure shows the differential tunneling

conductance vs. bias voltage in different magnetic fields. The inset shows the Pb

superconducting gap, from which we can see that the junction has low leakage and

reasonable good quality.

The differential conductance vs. bias voltage sweep shows an asymmetric "V"

shaped background, resulting from the asymmetry in the energy dependence of the

density of states in bismuth. On top of the background there are small features of

oscillations. The oscillations, which are most pronounced in the -20 to 20 mV range,

show no obvious magnetic field dependence and are characterized by 2nd derivative

d21
conductance V- to be mostly symmetric with the bias voltage. Hence they do not
dV

likely correspond to the density of states features in bismuth, but ratherto phonon

excitations.












600







400

> 400
'1


-80 -60 -40 -20 0
v (mV)


20 40 60 80


Figure 4-4. Differential conductance as a function of bias voltage in indicated strong
magnetic fields at 300mk. Inset: Pb superconducting gap feature in zero
magnetic field at 300mK.


-3 -2 -1 0
v (mV)


1 2 3


Figure 4-5. Differential conductance at low bias voltage in the magnetic fields indicated
in the legend.









Figure 4-5 shows the differential conductance at near zero bias voltage. In zero

magnetic field, the differential conductance shows a peak at zero bias. As the magnetic

field increases, the peak at zero bias separates into two peaks, which move towards

higher bias voltages and leaves a valley at zero bias. The change of dI/dV with magnetic

field appears to be as if the field opens up a gap at the Fermi level.

The feature observed at low bias voltages can be explained by considering the

Zeeman splitting of the spins. Since /u 5.8 x 105 e V / Tesla, the g factor corresponds to

our experimentally observed splitting (e.g., -1.6mV at 14 Tesla) is -2. Another possible

mechanism for the low bias voltage feature is the field induced Luttinger liquid behavior.

In this picture, the strong magnetic field opens up a Coulomb gap at the Fermi energy.

The Luttinger liquid behavior enhances the density of states in the gap. Detailed analysis

requires knowledge of the classical "background" of the differential conductance. Tilting

of magnetic field is required to tell if the zero bias feature is a spin effect or an orbital

effect.

Except for the features at bias voltage V < 3mV, a major feature of the differential

conductance show in Figure 4-4 is the lack of field dependence. The curves we took at 2,

4, 8 and 14 tesla overlap almost perfectly. This result contradicts the prediction from the

field induced Luttinger liquid theory, in which the magnetic field dependence enters the

power factors of the power law dependence. There are also some concerns about our

measurements. For example, the surface states in bismuth might prevent the tunneling

measurements from probing the intrinsic properties of bismuth single crystal.














CHAPTER 5
ACHIEVING LARGE MAGNETORESISTANCE IN BISMUTH THIN FILMS

5.1 Introduction

Semimetal bismuth has been interest of study for many years, because of it many

special properties. Figure 5-1 shows the Fermi surface and the brillouin zone of

rombohedral bismuth. The highly anisotropic Fermi surface consists of tiny hole pockets

and electron pockets, which occupy only a few thousandth of the volume of the Brillouin

zone. Hence bismuth has very low carrier density (-1023 m-3) and low Fermi energy

(-25meV). Also because of the small Fermi momentum, the chance of phonon scattering

is very low. Hence bismuth has an extremely long phonon mean free path at low

temperatures (-mm at 4.2K).


trigonal (7)

hole pocket



electron pocket (A)




bisectrix (y)
binary (x) L


Blillouin zone of Bi


Figure 5-1. Fermi surface and Brillouin zone of rombohedral bismuth









These unusual properties of bulk single crystal bismuth give rise to a huge

magnetoresistance.17-19 Figure 5-2(a) shows the magnetic field dependence of the

resistivity at 5K for single crystal (99.9995% pure) bulk bismuth. At the 7 Tesla, the

resistivity is 6 orders of magnitude higher than the zero field resistivity. Figure 5-2 (b)

shows the temperature dependence of resistivity in zero magnetic field. Bulk single

crystal bismuth is metallic, with resistivity decreasing with decreasing temperature.


-7
a&bdO
Q015 -





6r1\. I -
(0 K \-7



u o 2 o0

QOCO] QO

I lt tI t t lt lt lt I 1 I I I I1 0I I I I I
-8-6 4-2 0 2 4 6 8 0 50 101502 02503

B(1) T(0)

Figure 5-2. (a): Magnetic field dependence of the reisistivity at 5K for bulk single crystal
bismuth; (b) resistivity as function of temperature in zero magnetic field.

The extremely large magneto-resistance makes bismuth a promising candidate for

applications, such as magnetic field sensors. Many efforts have been carried out in order

to make bismuth thin films that have quality comparable to the bulk material.2022

However, it was found that bismuth thin films made by normal technique, such as

thermal evaporation, yield bismuth films with very small magnetoresistance.23 24 These









films may even behave in a non-metallic manner with the resistance increasing with

decreasing temperature.








S4 6 8 0





BdO
3& 1 1 I I -




-8-64-202468 0 100 200 300
B(1) T(H

Figure 5-3. (a): Magnetic field dependence of the reisistivity at 5K for a 1.5um bismuth
film thermally evaporated onto glass substrate; (b) resistivity as function of
temperature in zero magnetic field.

Figure 5-3 shows the typical transport behavior of thermally evaporated bismuth

thin films on glass substrates (thickness between 800-10000A). The resistance generally

increases with decreasing temperature. And the magnetoresistance at low temperatures is

much lower (MR(7T)<10) than that of the bulk bismuth (MR(7T)-105).

The major reason for the differences between bulk single crystal bismuth and the

bismuth films is the small grain size in the films. The gains, generally with size of

-1000A, are actually not small compare to that of normal metals. However, bismuth has

a very long phonon mean free path, due to the small Fermi momentum. Application of

the Matthiessen's rule shows that the scattering in the films is dominated by temperature

independent gain boundary scattering (except when the temperature is very high, e.g.

higher than room temperature, and the phonon mean free path is shorter than the grain









size). The temperature dependence of the resistivity is mainly from that of the carrier

concentration, which, due to the small Fermi energy of bismuth, decreases significantly

as the temperature drops.

Magnetoresistance in bismuth films is also limited by grain boundary scattering.

This can be illustrated from a simplified 2-band model, in which the electrons and holes

are compensated, and the resistivity in the magnetic field simply goes like:

p(H)- p(O) RH 2H2 (r)2 (5.1)
p(O) p(0)2

Hence the small relaxation time due to grain boundaries scattering leads to small

magnetoresi stance.

To make high quality bismuth thin films, it is necessary to make the grain sizes

large. A judicious combination of lattice-matched substrates and carefully regulated post-

deposition thermal annealing provides a strategy for growing bismuth films with large

grains. In early work on bismuth films thermally deposited onto mica substrates,2 it was

found that post deposition annealing close to the bismuth melting temperature caused the

helium temperature resistance to decrease by a factor of 15 when compared with

unannealed films. In addition, MR for fields perpendicular to the film surface is

significantly improved with annealing. Epitaxial films of bismuth having a trigonal

orientation have been grown on BaF2(111) (3.6% lattice mismatch)25 and CdTe( 11)

(0.7% lattice mismatch). 22 In the latter case, post-deposition annealing at 3 oC below the

melting temperature of Bi lead to significant increases in the MR.

An alternative approach, which has been found to give large MR in Bi films 1-20

,um thick, is the technique of electro-chemical deposition from aqueous solutions of

Bi(N03)3 5H20 .20,21 An underlying Au layer, patterned onto a silicon substrate, serves as









the working electrode for the electrodeposition. As is the case for vacuum-deposited Bi

films, 22,23 post-deposition annealing of the electrodeposited films close to the melting

temperature of Bi leads to a small resistivity and a large increase in MR (2.5 at room

temperature and 3800 at 5K for the thickest 20um film in a perpendicular 5T magnetic

field 20). For technological applications, electrodeposition is economical and well suited

for large-scale production. Similar advantage would likewise hold for thermal deposition,

provided ultrahigh vacuum and specialized groeth techniques, such as MBE, are not

required.

We studied the thermally deposited bismuth films on pre-deposited gold thin films

followed by post-annealing processes. We find that, upon annealing, the Au from the Au

underlayer rapidly diffuses into the bismuth, giving rise to a film with large-crystal grains

oriented with trigonal axis perpendicular to the plane of the film and having

magnetotransport properties comparable to those grown by electro-depositions.20' 21 We

show that improvements of MR are only for annealing temperature higher than the 241

C eutectic temperature of the BiAu solid solution and below the 271 C melting

temperature of Bi. This 30 C annealing window provides considerable latitude when

compared to the narrow annealing window of a few C confirmed here and reported

previously for pure Bi films.23

5.2 Experimental

All of our samples are prepared by thermally evaporating 99.999% pure Bi onto

pre-cleaned glass substrates at 5E-7 torr base pressure. In the cases where heated

substrates are needed, the substrates are glued onto a variac controlled heater with silver

paste. Then the shadow mask is glued onto the substrates using the same silver paste.









Substrate temperature is read from a thermometer, and is manually controlled by

adjusting the output voltage of the variac.

Three categories of samples are prepared: (I) two pure bismuth films (1 um thick)

grown at 150 C, followed by annealing at 2650C and 270 C for 6 hours; (II) three

bismuth films (1 um thick) grown simultaneously on pre-deposited gold films (350 A) at

150 C, followed by annealing at 238 C, 243C and 251 C, respectively for 6 hours; (III)

bismuth films (1 um) grown on pre-deposited gold films (350A) at room temperature,

followed by annealing at 251 C for 6 hours. Annealing is performed in a quartz vacuum

tube furnace with temperature calibrated with respect to the observed melting of a small

bismuth crystal placed in close proximity to the samples.

Measurements of resistance vs. temperature at different magnetic fields are carried

out in a Quantum Design Physical Property Measurement System (PPMS). In all of the

measurements, the magnetic field is applied perpendicular to the film.

5.3 Results and Discussion

We characterized the crystal structure of the Bi/Au films by X-ray diffraction.

Figure 5-4 shows the X-ray diffraction pattern of bismuth (001) planes. The sharp lines

indicate that the film is well c-axis oriented. The inset of Figure 5-4 shows a schematic of

the relevant portion of the Bi(Au) phase diagram.26 A small amount of gold in Bismuth

reduces the melting point, and the lowest melting temperature, the eutectic temperature,

occurs at 241 C for the Bi.868Au.132 composition. In our experiment, the mass ratio of the

Bi and Au is controlled by the thickness of the 2 films. Thus a pre-deposited 360-A-thick

Au layer mixed by annealing into a 1-um-thick Bi film represents a solid solution

(vertical dashed line) with stoichiometry Bi0.93Auo.07. All of the Bi/Au films here are at

this composition.













5000 i

271 442 'C
L
4000 -



3000 r'c

.- 9O 934 1DO
BU)
Mass Percent Bismuth
w 2000



1000



0


0 20 40 60 80 100 120
2*theta


Figure 5-4. X-ray diffraction pattern for a 4-um-thick Bi/Au film grown at 150 oC and
annealed at 251 C. Inset: the relevant portion of the Bi/Au phase diagram and
corresponding annealing temperatures (indicated by the crosses) for Bi and
Bi/Au films discussed in this chapter.

Figure 5-5 shows the resistance vs. temperature at 0 and 5 Tesla for the two

category-I pure bismuth films (no Au underlayer) annealed at 265 C and 270 C,

respectively. We note that a small difference of annealing temperature at close to the 271

C melting point of bismuth produces a drastic change of the properties of the films. The

film annealed at 270 C just starts to melt and is recrystalized during the slow cool-down.

As observed through the quartz tube, the film develops a shiny surface just below the

melting temperature, but at higher temperature begins to fully melt and ball up. The

positive slope in the resistance-temperature curve in zero magnetic field indicates that the

film is metallic. Also, at 5T, the MR=286 at 5K indicates the good quality of the film.










I I I I I I I
B A A) TG = 20C, TA = 265C
B) Te = 20C, TA = 270C
S 100 A
a, : B = 5T







0 B= OT
A

B

0 50 100 150 200 250 300
Temperature (K)

Figure 5-5. Temperature dependence of the resistivity at 0 and 5T for two category-I Bi
films

In contrast, the film annealed at 265 C does not change its appearance during the

annealing process. The resistance of this film shows a characteristic minimum at near

200K 23 and then increases as the temperature is further lowered. In addition, the MR of

this film is much smaller than the one annealed at 270 C through out the whole

temperature range. These results are in accord with previous studies,22 23 which have

shown that post-annealing at melting point followed by re-crystallization is an effective

way to get high quality bismuth films. However, the temperature control must be accurate

to a few oC and must not be allowed to go above the melting point where there will be a

loss of film adhesion leading to agglomeration and discontinuity between grains.











I I I I
SB A) T. = 150C, T =238G
100 C "-.. B) T_ =150C, T =243
A' A. C)T_ = 150C, T =251
B( B=5T


S10


(-
0 A B=OT

1 -B
:C ---. ,

0 100 200 300
Temperature (K)

Figure 5-6. Temperature dependence of the resistivity at 0 and 5T for three category-II
films grown simultaneously on 150 C substrates and them annealed
separately at respective temperatures of 238, 243 and 251 C.

For samples in category-II, the presence of a gold underlayer leads to completely

different behavior. Figure 5-6 shows the effect of annealing temperature on the quality of

these films. The three Bi(lum)-Au(350A) films are grown at 150 oC, and then annealed at

238 C, 243C and 251C respectively, as indicated by the crosses in the Fig.5-4 inset.

Prior to each post-deposition anneal, a gold color can be observed from the back

side of each glass substrate. Afer 243 C and 251 C anneal, the gold color is gone and the

underside of each Bi/Au film is silver color and indistinguishable from the underside of a

pure Bi film. These color changes indicate that during the annealing, the gold atoms no

longer remain segregated beneath the bismuth film but diffuse into the bismuth. For an

annealing temperature of 238 C, which is below the eutectic temperature of 241 C, all

of the film remains in the solid form, and the surface texture of the film does not change









during the anneal. In addition, the temperature dependent resistance is nonmetallic and

the 5K MR is low (MR=37). In contrast, for the two anneals above the eutectic

temperature, the films undergo a definite change in appearance in which they become

shiny and remain metallic after cooldown, the temperature-dependant resistance becomes

progressively more metallic, and both films exhibit significantly larger MR.

MR(5K)=130 for the 243 C anneal and MR(5K)=327 for the 251 C anneal. We note

that the MR of our 251 C annealed Bi/Au film is higher than the MR(5K)=250 of a

comparable 1-um-thick "single-crystal" film grown by electrodeposition.20' 21 Further

increase of the annealing temperature to slightly above 160 C but well below the 271 C

Bi melting temperature leads to sever melting and loss of electrical connectivity, as

would be expected from the intersection of the vertical dashed line with the solid/liquid

phase boundary shown in the Figure 5-4 inset.

The plots in Figure 5-7 for the category-III films show the effect of growth

temperature on transport properties for the same anneal conditions. The film grown at

150 C shows metallic temperature dependence at zero field and has MR(5K)=327. The

film grown at room temperature however, shows rather complicated temperature-

resistance dependence. The resistance drops a little bit as the temperature sweep from

300K to about 200K, then increase a lot from 200K to 5K. The resistance-temperature

curve suggests that the film has grain sizes comparable to phonon mean free path at

200-300K. At low temperatures, the grain boundary scattering dominates and the

resistance increases due to decrease of carrier concentration. The magnetoresistance of

the room temperature grown film, MR(5K) = 34, is also significantly lower than the film

grown at 150 C. For pure Bi films grown on CdTe substrates, growth temperature in the






60


range of 80 C to 220 C are required to obtain epitaxial behavior.22 For our Bi/Au films,

the higher growth temperature promotes the growth of larger grains, thus facilitating the

effectiveness of the annealing procedure by starting with larger grains. We also note, as

show by x-ray diffraction pattern in Figure 5-4 for a 4-um-thick Bi/Au film grown at 150

C and annealed at 251 C, that the annealed films exhibit a pronounced single-crystal

orientation with trigonal axis oriented perpendicular to the film plane. Similar behavior

has been noted for annealed electrodposited films.



B A) TG = 150C, TA = 252C
100 A B) T= 20C, T = 252C
SB = 5T












0 100 200 300

Temperature (K)



Figure 5-7. Temperature dependence of the resistivity at 0 and 5T for two category-III
films

Optical microscopy verifies a smoother topography and larger grain size (1-lOum)

for the films annealed at high temperature and exhibiting a large MR. This result is

consistent with the aforementioned conclusions that large grain size achieved either by

epitaxy and/or annealing is a prerequisite for large MR. The primary factors that affect
.L_ 10
U)


r--


B

0 100 200 300
Temperature (K)



Figure 5-7. Temperature dependence of the resistivity at 0 and 5T for two category-Ill
films

Optical microscopy verifies a smoother topography and larger grain size (1-10um)

for the films annealed at high temperature and exhibiting a large MR. This result is

consistent with the aforementioned conclusions that large grain size achieved either by

epitaxy and/or annealing is a prerequisite for large MR. The primary factors that affect









the quality ofBi/Au films are the growth temperature and the annealing temperature. A

moderate growth temperature (-150 C) encourages the formation of large grains, but

should not be so high as to cause the film to agglomerate and to become discontinuous.

Table 5-1. Summary of results for different bismuth film growth conditions
Sample Growth Annealing Metallic MR (5K: 300K)
temperature temperature
Pure Bi 150 C 265 C N 39 : 3
270 C Y 283:4
Bi (lum)- Room temp. 251 C N 34: 3
Au(350A) 238 oC N 37 : 3
150 C 243 C N 130 :3
251 C Y 327:3


We summarize our results in Table 1.The effect of the diffusion of the Au into Bi

during the post-deposition annealing process can be qualitatively understood by referring

to the phase diagram depicted in the Figure 5-4 inset. If equilibrium is assumed, then for

isothermal (tie line) drawn at a given annealing temperature, application of the "lever

rule" for binary phase diagram will determine a gold-rich melted phase and a bismuth-

rich solid (unmelted) phase. It is the presence of this melted phase that facilitates grain

boundary migration and grain growth resulting in the high MR that we have observed.

We suspect that this melted phase is most likely associated with grain boundaries

although detail microcompositional analysis would be necessary to verify such a

scenario.

In oversimplified terms, the Au can be thought as a lubricant that facilitates the

growth of large grains during the post-deposition anneal. However, one should not forget

that Au is a impurity that gives rise to increased carrier scattering and associated lower

MR, thereby preventing the MR from approaching the hig values reported in single

crystals.17, 19 Accordingly, the use of annealed Bi/Au bilayers to obtain large MR requires






62


judicious balance between using enough gold to assure large grain growth, but not using

too much gold that additional scattering compromises that MR. We believe that these

considerations also apply to the Bi/Au films deposited by electrodeposition technique

reported previously by Yang et al. 20,21














CHAPTER 6
METALLIC SURFACE STATES IN ULTRA-THIN BISMUTH FILMS

6.1 Introduction: Physics of the Ultra-Thin Bismuth Films

Bismuth, like many other semimetals, has a very long electron wavelength, due to

its low Fermi energy (-25 meV) and small effective mass (-0.005me). One can estimate

h
the wavelength to be: A ~ 102 A. When the size of sample is comparable or


smaller than this length scale, one will need to consider the effect of the sample boundary

on the band structure. This is where the so-called quantum size effect become important.

Bismuth provides great convenience in studying the quantum size effect in many

aspects. Ultra thin bismuth films, with their thicknesses comparable to the electron wave

length (- 300 A), have been of great interest in the study of the quantum size effect and

semimetal-semiconductor transition. Ogrin, Lutski, and Elinson, in their study of the

magnetotransport of the bismuth thin films in 1965,27 produced the first clear

experimental evidence for quantum size effect in any solids. Oscillatory behaviors in both

resistivity and Hall coefficient were observed with decreasing film thickness, due to the

quasi-2D sub-bands passing across the Fermi level. Since then, quantum size effect in

bismuth thin films has been intensively studied both theoretically and experimentally.28-34

The existence of the thickness dependent quasi-2D sub-bands resulting from quantum

confinement has been generally accepted.

One important prediction as a result of the quantum size effect is the so-called

semimetal-to-semiconductor (SMSC) transition. The SMSC transition happens when the









energy shift due to the quantum confinement becomes large enough so that the lowest

electron sub-band rises above the top of the highest hole sub-band, due to their difference

band masses. The critical thickness of the thin film for the transition to happen, given by

most of the theoretical calculation, is between 230 -340 A.28, 35, 36










/ 38meV

13.6meV









Figure 6-1. Illustration of semimetal-to-semicinductor transition.

Despite the numerous experimental investigations carried out to look for the SMSC

transition,27 28 31 37-40 the existence of the SMSC transition remains ambiguous. Chu and

co-workers argued against the SMSC transition, and proposed theory that the boundary

condition for the electron wave function is that the gradient of the wave function (rather

than the wave function itself) vanish at the sample boundary. Hence the ground state

electron and hole energies depend only weakly on the thickness of the films, and the

conduction band and the valence band remains overlaped.

The arguments against the SMSC transition mainly focused on the lack of sharp

transitions in transport properties (resistivity, Hall coefficient, magnetoresistance...).









Hoffman et al., in their work that stand for the SMSC transition, pointed out that the main

reason for the absence of the sharp transition in the previous works is that people failed to

take into account the effect of the surface carrier and surface conductivity, which may be

important and even dominating when the bismuth films are very thin.39 The simple model

that takes into account the surface carrier is that the surface acts like a high carrier density

conductor in parallel with the bulk part of the film, and the averaged carrier concentration

is simply: n = n, + n, / d, where n, and n, are bulk intrinsic carrier density and surface

sheet carrier density.38 When the film is thin, n, will dominate, and the effect of the

surface conductivity has to be seriously considered.

Further evidences of metallic surface states were found in a very different bismuth

system. In 1991, B. Weitzel and H. Micklitz discovered superconductivity in granular

systems built from rhombohedral Bi clusters.41 They explained their result as surface

superconductivity due to the strongly increased surface density of states and suggested

photoelectron spectroscopic study on bismuth surfaces to further confirm their proposal.

Angle resolved photoemission spectroscopy (ARPES) since then had been a major

tool people used to probe the surfaces of bismuth. The experiments were carried out by

several groups.42-47 Consistent results were obtained, indicating the existence of metallic

surface states in bismuth (111) and (110) surfaces. Christian R. Ast and Hartmut Horchst

reported in 2001 a surface carrier density associated with the surface state of Bi (111) to

have sheet densities of p, = 1.1 x 1013 cm2 for holes and n = 5.5 x 1012 cm2 for

electrons.47 In 2003, Gayone et. al reported their study on the temperature dependence of

the surface states linewidth and the strong energy dependence of the electron-phonon

coupling strength on Bi (100) surface.48









6.2 Transport Properties of the Ultra-Thin Bismuth Films

6.2.1 Experimental

Bismuth films are thermally evaporated from 99.9999% pure bulk bismuth in a

high vacuum chamber of-6E-7 torr at a rate of 1-2 A/sec, through shadow masks. In the

cases where in situ measurements are required, bismuth or gold contact pads are

thermally evaporated through shadow masks. Then the substrate pre-deposited with

contact pads is fixed onto a special designed sample holder, where the Hall-bar shadow

mask is installed aligned with the contact pads, and the gold wires are attached to the

contacts and connected to leads which enable electrical measurements from outside the

vacuum chamber.

Tunnel junctions on thin bismuth films were made using standard cross stripe

geometry. Mica is used as substrate and bismuth is used for base electrodes, so that lattice

match between bismuth and mica can be achieved. AlOx is used for tunnel barriers. Lead

is used for top electrodes, so that when the samples are cooled down to below the lead

superconducting temperature, the superconducting gap can be used to characterize the

quality of the junction. In making of a Bi-AlOx-Pb junction, a thin bismuth film stripe as

base electrode is deposited onto mica substrate through a shadow mask. The film is then

taken out of the vacuum chamber, and the shadow mask is removed. The film is then

immediately put back into vacuum, and the aluminum oxide tunnel barrier (- 10 A) is

coated through thermal evaporation of aluminum in the oxygen pressure of 2E-5 torr, at a

rate of about 1 A /sec. A cross stripe of lead as counter electrode is then deposited

through a shadow mask. Typical working junction resistances are in the range of

10-10000Q.









6.2.2 Metallic Surface States

In the study of the Bi/Au films, the thicknesses of these films are in the order

micrometers. Films with these thicknesses can be treated as bulk polycrystal bismuth, in a

sense that there's no band structure change due to the quantum confinement, and the

surface effect is not significant. When the (pure) bismuth films get really thin (e.g.,

thickness Fermi wavelength), quantum size effect will take place, and the effect of the

surface states needs to be seriously considered.

Figure 6-2 shows the temperature dependence of resistivity, for films with

thicknesses indicated in the legend.

1.0x105
400A

9.0x106-


8.0x10-6 310A


S7.0x106 -

150A
6.0x106


5.0x1 06
0 50 100 150 200 250 300
T (K)

Figure 6-2. Resistivity vs. temperature for Bi film with indicated thicknesses.

It can be seen from the figure that, for films with thickness -400A, the resistivity

increases as the temperature drops, and reaches a maximum at ~40K. As the film

becomes thinner, the temperature for the resistivity maximum shifts higher. Also the









resistivity in the low temperature region (say, T <150K) decreases with decreasing film

thickness. For the films with thickness < 150 A, the resistivity drops monotonically from

room temperature with decreasing temperature, showing metallic behavior.

The metallic behavior (positive R-T slope) can be explained by considering the

existence of the metallic surface states. A simplified model is to treat the whole film as

two separate films in parallel: a very thin metallic-like film on the surface with sheet

resistance R, and thickness ts, and the "intrinsic" film underneath it with resistivity

1 (t- t_)
p, and thickness t t. The measurered resistivity for is then p = -+ t.
R, A )

The resistivity of the bulk (intrinsic) part of the film, p, has negative R-T slope.

And when the film is thick, it has low resistance and hence will dominate the total

resistance of the film. When the film gets thin, the contribution of the metallic surface

becomes increasingly important, and the R-T curve starts to show a maximum, which

moves to higher and higher temperature with decreasing film thickness. Eventually, when

the film thickness reaches 150A or thinner, the surface states will dominate the

temperature dependence and the R-T curve shows positive slope throughout the

temperature range of measurements (4.5K-300K).

The magneto-transport of ultra-thin bismuth films is studied under the framework

of classical muti-band model, as described in the previous chapters. The classical

magnetoresistance of the ultra thin bismuth film can be roughly estimated to be (mr)2.

Here, the small thickness of the films leads to very short mean free path of grain

boundary scattering, and hence a very small MR.











0.020



0.015 31 OA



s 0.010 180A



0.005



0.000

-6 -4 -2 0 2 4 6
B (T)

Figure 6-3. Magnetoresistance vs. magnetic field at 5K for two Bi films with indicated
thicknesses.

Figure 6-3 shows the MR of two bismuth films with thicknesses 180 and 310 A. It

can be seen from the figure that the classical MR increases with the film thickness, due to

the increasing grain size with the film thickness. The sharp dip at low fields can not be

explained by classical theory, and is due to anti-localization, originating from the strong

spin-orbit interaction in bismuth.

The in-balance or non-compensation of the positive and negative carriers in the thin

bismuth films is revealed by the field dependence of the Hall resistivity. Figure 6-3

shows the field dependence of Hall resistivity at indicated temperatures for 2 bismuth

thin films, 180 A and 400A thick. For both films, the measured Hall resistivities are not

linear with the magnetic field. Also the zero field slope of the Hall resistivity has strong

temperature dependence. For the 180A film, the low field p vs. field curve even


changes sign from 5K to 150K.









8x10 (a) 1s8A

6x10

4x10

S2x10 -

0- 75K

-2x10 -

-4x108 '* *
0 2 4 6 8
B(T)



6x07 (b) 400A
6x10 --
150K

4x10-7


.2x10- -
75K

0 -5K

0 2 4 6 8
B(T)


Figure 6-4. Hall Resistivity vs. magnetic field at indicated temperatures for (a) 180A and
(b) 400A Bi films
The Hall resistivity p, observed can be qualitatively understood through the

simple 2-band model expression,









RR2(R1 + R2)B3 + (R1,p2 + R2p12)B
(P1 + P2 )2 + (R, + R 2 (6-B2

We can see that, for a 2-band system with electron band and hole band, the Hall

resistivity is in general not linear with the magnetic field. Also the Hall resistivity itself

does not give enough information about the in-band carrier concentration of each band.

From Equation 6-1, we get the low field and high field limit of the Hall resistivity:

(R p 2 + R 2 )B
p, (H 0) = p2 (6-2a)
(pI + 2)2

RRzB
p (H ) = 2 (6-2b)
(R, +R2)

Hence the zero field Hall resistance slope by itself does not give any information

on the carrier density of the films. In fact, it does not even give the information about

whether the film is n-type or p-type, due to the complication from the in-band resistivity

(or mobility). However, the high-field limit of the Hall resistance does indicate the carrier

type of the film (or, the sign of R + R2). The change of slope (even the sign of the slope)

from low field to high field gives rise to the curvature observed in the Hall resistivity

measurement. One can see from the p vs. field curves that, even though the low field

data shows strong temperature dependence (even change of sign), the sign of the

extrapolated high field limit of the Hall resistivity slope is temperature and thickness

independent, indicating the type of the films, n-type in this case, does not change with

temperature, nor the film thickness.

A more detailed analysis of the Hall resistivity data yields information about the


un-balance of the carriers, defined by the compensation factor: A = -h And the
ne +nh









thickness and temperature dependence of the Hall resistivity can be qualitatively

understood by considering the change of compensation factor (due to the n-type surface

states) with temperature and thickness. To simulate the field dependence of the Hall

resistivity, we adopt a 4-band model, with a bulk electron band, a bulk hole band, a

surface electron band, and a surface hole band. Figure 6-5 shows the simulation results

for the magnetic field dependence of the Hall resistivity for 180A thick and 400 A thick

bismuth films. In the simulation, we assume that the mobility of the carriers does not

change with temperature, due to the fact that the grain boundary scattering dominates

over phonon scattering. Because of the complication of the energy band quantization due

to the quantum size effect, we can not calculate in detail the temperature dependence of

the carrier density. In the simulation, we assume different values of carrier density for the

bulk part of the films, and assume the sheet surface carrier density to be temperature and

thickness independent. The parameters used in the simulations are listed below:

Table 6-1. Parameters for the simulating the effect of thickness and temperature on the
magnetic field dependence of the Hall resistivity in ultra-thin Bi films
Surface (10A) In the film
Carrier density Mobility Carrier density Mobility
ns (m-2) / 2 -) n i (m-3) /S (m2V 1)
400A e 2x1017 0.031 ao(5x1023) 0.119
h 1.4 x 1017 0.036 a (5x1023) 0.138
180A e 2x1017 0.031 a*(5x 1023) 0.056
h 1.4 x 1017 0.036 a (5x1023) 0.063


Note from the listed parameters that: 1) the surface sheet carrier density and mobility are

the same for both films; 2) in bulk (intrinsic) part of the films, the carrier density of

electrons is equal to that of the holes; 3) since we cannot calculate the carrier density in

the films, we modulate its number by adjusting the parameter a.






73


2x10-8


a=1.5
E 0-



-2x108 a=0.8
c/)

^ -4x108 -O
c (a) 180 A a=0.3


-6x108 a=0.01
0 1 2 3 4 5 6 7 8
B (T)


3x107
(b) 400 A a=1.5

C: 2x107

> a=0.8
1, 0-7




-1x107


0 1 2 3 4 5 6 7 8
B (T)



Figure 6-5. Simulated Hall Resistivity vs. magnetic field at indicated temperatures for (a)
180A and (b) 400A Bi films, with fitting parameters described in the text.









Comparing the calculated magnetic field dependence of the Hall resistivity with

the data, we see that the simulations yield the major features in the experimental results.

From the simulations, we get the physical picture about the carriers in the ultra-thin films.

The bulk part of the film is compensated, like in bulk bismuth. The surface of the film,

however, has a high sheet carrier density and is uncompensated. As the film gets thinner,

or the temperature gets lower, the number of compensated carriers in the bulk part of the

film decreases. Hence the degree of un-compensation, due to the existence of the un-

compensated surface carrier, will increase.

6.3 Control of the Surface States

All the bismuth films discussed in the previous section are measured after removal

from the vacuum chamber. Even though the oxidation of bismuth at room temperature is

insignificant, we will still need to consider the effect of oxygen on the surface of the film.

For comparison, we have carried out in-situ measurements on bismuth thin films. A

thin bismuth film is deposited onto a mica substrate pre-deposited with contact pads, and

measured without breaking vacuum. The substrate is mounted on a cold stage and cooled

down from room temperature to -100K, and the resistance vs. temperature is recorded.

The sample is then warmed up to room temperature, and a small amount of oxygen is

introduced into the chamber for 10 minutes. The chamber is then evacuated, and the

sample is cooled down again to 100 K, with resistance vs. temperature recorded.

The in-situ measurement of freshly deposited bismuth films shows that for ultra-

thin bismuth films measured in vacuum, the resistance increases with decreasing

temperature. What is different for the ultra-thin bismuth films from the thicker bismuth

films (-um) is that, the ratio of the resistance increase, say, R(100K)/R(300K), is much

smaller in the ultra-thin bismuth films (<10%) than in the thicker films (-200%). Hence









the existence of the metallic surface states is intrinsic, while the sheet carrier density

originated from the surface states is very sensitive to the surface condition.

We have seen that oxygen has a significant effect on the surface states. To study the

surface "terminated" ultra-thin bismuth film, we coated the bismuth films with Ge. Ge is

known to be a material that, when deposited as thin films, creates dangling bounds and

nucleation sites. When thin metals films are deposited onto predeposited atomically

smooth Ge thin films, the metal film growth nucleates at the Ge dangling bounds. Thus

the metal films tend to be very smooth.

Here we deposit a few monolayers to Ge right after the deposition of bismuth thin

film, without breaking vacuum. The idea is that the dangling bounds of the Ge film may

bind with the surface states in the bismuth film, and terminate the surface of the bismuth

film from being affected by the air.


-- Bi (100A)
Bi (1 0A)/Ge (8A)
1.4x103 -



C 1.3x103



1.2x103



0 50 100 150 200 250 300
T(K)

Figure 6-6. Temperature dependence of resistivity for Bi(100A) and Bi(100A)/Ge.

Figure 6-6 shows the resistance vs. temperature curves of two 100A thick bismuth

films simultaneously grown and measured. The only difference between the 2 films is









that, one sample is coated with a few angstrom of Ge in situ, right after the deposition of

bismuth film. The two samples show completely different temperature dependence of

resistance. The bare bismuth film shows positive resistance-temperature slope, while Ge

coated film shows negative resistance-temperature slope. We also coated the bare

bismuth film with Ge after it's taken out of vacuum. No significant change of transport

behavior was observed. We conclude that the change happens at the Bi-Ge interface,

rather than in the Ge film itself.

The Hall resistivity measurements provide more information on the in-balance of

the carriers. From Figure 6-7, we can readily see the big differences in the carrier

distribution between the bare and the Ge coated bismuth films. Comparison with Figure

6-4 reveals that the magnetic field dependence of the Hall resistivity for the bare 100 A

Bi film at 75K, 150K and 250K resembles that of the bare 180 A Bi film at 5K, 75K and

150K. And the magnetic field dependence of the Hall resistivity for the Ge coated 100 A

Bi film at 75K, 150K and 250K resembles that of the bare 400 A Bi film at 5K, 75K and

150K (the smaller curvature here is due to the smaller mobility in the thinner films). This

comparison suggests that the Ge coated Bi film has better compensation than the bare Bi

film with the same thickness.

Simulations results with a 4-band model described earlier are shown in Figure 6-8,

with fitting parameters listed in table 6-2. Note from the listed parameters that: 1) the

effect of the surface carrier is adjusted by setting the thickness of the surface layer; 2) in

bulk (intrinsic) part of the films, the carrier density of electrons is equal to that of the

holes; 3) The carrier density of the bulk (intrinsic) part of the film is modulated by

adjusting parameter a.






77


3x108 -

(a) 2 50 K
-8
S2x10 -


1x108 -
UB
) 0-
^ I-150K

I -1x108 -
75K
-2x10 -8
0 1 2 3 4 5 6 7 8
B (T)


1.5x10-7
250


1.0x10-7 (b)

150K


w 5.0x10 -


75K

0.0

0 1 2 3 4 5 6 7 8
B (T)



Figure 6-7. Hall Resistivity vs. magnetic field at indicated temperatures for (a) Bi(100A)
and (b) Bi(100A)/Ge films.






78



1x10 -

a=4.0

0 -



S-1x108 -
W a=1.5
I (a)(

-2x108 a=0.8


0 1 2 3 4 5 6 7 8
B (T)



a=4.0
2x107 a=4.
(b)


> 1x10-7



5 0 a=1.5
n 5x108 -



0 a=0.8

0 1 2 3 4 5 6 7 8
B (T)




Figure 6-8. Simulated Hall Resistivity vs. magnetic field at indicated temperatures for (a)
Bi(100A) and (b) Bi(100A)/Ge films, with parameters described in the text.









Table 6-2. Parameters for the simulating the effect of Ge coating on the magnetic field
dependence of the Hall resistivity in ultra-thin Bi films
Surface (3A for Bi/Ge, 10A for In the film (100A)
Bi)
Carrier density Mobility Carrier density ni Mobility
ns (m-2) / (2V is -1) (m-3) (m V -s 1)
Bi/G e 2x1017 0.031 a (5x1023) 0.047
e h 1.4x1017 0.036 a (5x1023) 0.054
Bi e 2x1017 0.031 a (5 x 1023) 0.047
h 1.4x1017 0.036 a*(5x1023) 0.054


The simulations suggest that, the effect of the Ge layer on the Hall resistivity is

equivalent to reducing the density of the surface carriers. It is also due to the reduction or

neutralization of the metallic surface states so that the carriers in the bulk part of the films

again dominate the transport and give rise to the negative resistance-temperature slope

shown in Figure 6-6.

The mechanism through which exposure of the bismuth film to the air increases the

surface sheet carrier density is still not known. However we believe that results obtained

from the Ge coated bismuth ultra-thin films opens the possibility of passivating the

surface and even neutralizing surface states. These results should be important for

studying the nanoscopic bismuth systems, such as bismuth nanowires, in which the effect

of the surface state becomes very significant.














CHAPTER 7
SURFACE SUPERCONDUCTIVITY IN ULTRA-THIN BISMUTH FILMS

7.1 Transport Evidence

In the previous chapter, we have studied the effect of the metallic surface states on

the transport of ultra-thin bismuth films. For the films of certain thickness, a closer look

at the transport data at low temperatures reveals some very un-expected features. Figure

7-1 shows the zoom-in of the temperature dependence of the resistance for a 15nm thick

bismuth film. We can clear see a sharp drop of the resistance at about 5.6K. The inset

shows the magnetic field dependence of resistance for the same film. We also see a sharp

decrease of R below some critical field of 200mT.



768





764 770

765

I 760
760 -1.0 -0.5 0.0 0.5 1.0
II B (T)
6 (K)8 10


Figure 7-1. Resistance vs. temperature in zero magnetic field for a 15nm bismuth film.
Inset: resistance vs. magnetic field at 4.5K for the same sample.









The sharp feature of resistance change has been observed reproducibly in samples

with thickness within certain ranges. The resistance will either jump down or jump up

below a critical temperature and critical field by very small amount. Figure 7-2 here

shows an example in which the resistance jumps up at below a critical temperature and

critical field.


799.5
804



799.0 -
801



798.5
-1.0 -0.5 0.0 0.5 1.0
B (T)


798.0 '-
I i I
6 T (K) 8


Figure 7-2. Example of resistance increases during the transition. The main figure shows
resistance vs. temperature in zero magnetic field for a 15nm bismuth film.
Inset: resistance vs. magnetic field at 4.5K for the same sample.

We also observed sharp feature of resistance increase or decrease in the Hall

resistivity measurements (see Figure 7-3). Since the features are even with the magnetic

field, they are really from the longitudinal resistance pickup due to the misalignment of

the Hall leads. But the change of resistance at the transition is much bigger percentage

wise. We also find that the critical field for such feature decreases with increasing








temperature, and the relation satisfies the T dependence of the critical field in

superconductors:


B,(T)= B(O) 1 T-


1.5







0.5
0.5


(7.1)


-0.5 0.0 0.5 1.0
B (T)


Figure 7-3. Sharp feature of resistance change observed in Hall resistivity measurements
at indicated temperatures for a 15nm thick bismuth film. Inset: critical
magnetic field as a function of T2.


By measuring bismuth films with different thickness, we map out the thickness

dependence of the critical magnetic field for the resistance transition. From Figure 7-4 we

can see that at 4.5K, the transition happens for films with thickness smaller than -16nm,

and for films with thickness -40nm. In fact the critical field vs. thickness plot suggests

oscillating thickness dependence of the critical magnetic field.










0.8

A
0.6

4\A



S0.2 A


0.0- A-A- A

0 10 20 30 40 50
Thickness (nm)
Figure 7-4. Film thickness dependence of the critical magnetic field at 4.5K.


The features we have observed for these bismuth films together with the absence of

a full transition to a zero-resistance state are suggestive to that superconductivity occurs

only in certain portions of the films. The bulk rhombohedral bismuth is not

superconducting (T,<50 mK). But there are several reported superconducting phases of

bismuth: high-pressue phases of Bi called Bi II, III and V with Tc =3.9, 7.2, and 8.5K

respectively,4951 fcc Bi with T, with Tc<4K,52 amorphous Bi with T,=6K, and granular

system of Bi clusters, with T, -2-6K depending on the size of the clusters.41

X-ray diffraction (XRD) analysis shows that our films are rhombohedral. To make

the amorphous bismuth films that show superconductivity, one needs to deposit bismuth

onto liquid Helium cooled substrate. These amorphous bismuth films lose their

superconductivity when annealed up to room temperature. Hence we believe that






84


amorphous phase is not the reason for the superconductivity observed in our room

temperature deposited and 200 C annealed films.

7.2 Tunneling Evidence

To further probe the properties of our films, we performed tunneling

measurements. The samples we studied are standard cross-bar Pb-I-Bi junctions

described in chapter 6. Figure 7-5 shows the curves of tunneling conductance vs. bias

voltage at indicated different temperatures, for a Pb-AlOx-Bi(150A) junction.


1.5





1.0
0-




0.5


-6 -4 -2 0 2 4
V (mV)


Figure 7-5. Differential conductance as a function of bias voltage in the superconducting
gap region at indicated temperatures, for a Pb-AlOx-Bi(150A) tunnel junction

A major feature of differential conductance at T<5.5K is the existence of two

superconducting gaps. With increasing temperature, the two gaps move to some bias

voltage in between, and the intensity of the inner gap drops rapidly. At T>5.5K, the









conductance spectrum recovers the shape of normal metal-insulator-superconductor

tunneling, with a single superconducting gap from Pb counter-electrode. With further

increase of temperature, the superconducting gap vanishes as Pb electrode loses its

superconductivity.

The double gap feature in the bias voltage dependence of differential conductance

is very typical for superconductor-insulator-superconductor (S-I-S') tunnel junction, with

the DOS peaks correspond to Ai+ A2 and A1- A2, where A1 and A2 are the BCS gaps of Pb

and Bi, respectively. It should be noted that the values of the superconducting gaps

determined from the data above turn out to be bigger than what they should be (e.g.,the

standard value for Pb is Ai=1.4meV). Two possible reasons may cause the "enhanced"

gap size. First of all, the sheet resistance of the bismuth film (-500Q) is comparable to

the junction resistance itself, and hence will contribute to the measured result as a series

resistor. Second, electrons may first of all tunnel into a surface state, and then lose energy

when they travel into the bulk part of the film. Hence the existence of surface states may

cause voltage drop at the bismuth-AlOx interface.

Another surprising feature in the dI/dV vs. V characteristic is the conductance

maximum for temperature lower than -7K. This feature doesn't not reproduce for all the

samples. The reason for its existence is not well understood.

As a comparison, Figure 7-6 shows the superconducting gap feature of a Pb-AlOx-

Bi(1000l) tunnel junction. We see no evidence of superconductivity in the transport

measurements of the 1000A thick bismuth films. The tunneling measurement, we also see

the standard Pb superconducting gap in the differential conductance vs. bias voltage






86


sweep, but no evidence of the smaller gap seen in the sample with smaller thickness of

the Bi electrode.


2


















-4 -2 0 2 4
V (mV)

Figure 7-6. Differential conductance as a function of bias voltage in the superconducting
gap region at 300mk, for a Pb-AlOx-Bi(1000A) tunnel junction.

We also measured the tunneling conductance of our samples in various magnetic

fields. Shown in Figure 7-7 is the tunneling conductance vs. bias in different low

magnetic fields perpendicular to and parallel to the junction area. For the perpendicular

field, as the field increases, the gaps decrease in size and move towards each other. In a

field higher 200mT, only one gap is left. Since at 200mT, Pb already loses its

superconductivity, the gap is the Bi superconducting gap. A similar characteristic is

observed in the measurements with magnetic field applied parallel to the junction plane,

except that the changes occur within a wider field range. The differences between the







87


results with magnetic field parallel and perpendicular to the junction suggest that the

double gap feature is not a spin effect.


1.5



C
3
2 1.0





0.5


1.5







e-0
.-1




0.5


-4 -2 0 2
v (mV)


-4 -2 0 2 4
v (mV)


4 6


Figure 7-7. Differential conductance vs. bias voltage at 300mK in indicated low magnetic
fields perpendicular and parallel to the junction plane.







88



1.5
S300mT

500mT

1T


4 1.0
> 4T
-10T
18T

Perp. field

0.5-

-6 -4 -2 0 2 4 6
V (mV)



300mT

500mT

1T


4 1.0
> 4T
10T



Para. field
0.5

-6 -4 -2 0 2 4 6
v (mV)




Figure 7-8. Differential conductance vs. bias voltage at 300mK in indicated strong
magnetic fields perpendicular and parallel to the junction plane.




Full Text

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MAGNETOTRANSPORT AND TUNNELING STUDY OF THE SEMIMETALS BISMUTH AND GRAPHITE By XU DU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Xu Du

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To my parents

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iv ACKNOWLEDGMENTS I would like to express my sincere gratitude to the many individuals who contributed to success of my wo rk. First of all I would like to thank my research advisor, Professor Arthur Hebard. Through his positive and open-minded attitude, and his enthusiasm and optimism toward physics resear ch, he created the legacy of the free, vivid, intelligent, friendly and communicative re search atmosphere in the lab. I feel really lucky to be able to work in such envir onment. His experience, knowledge, and guidance have been invaluable throughout my graduate career. I would also like to than k Professor Dmitrii Maslov for his theoretic support. Without the many useful discussions with him, and his constructive criticism, much of my work would have gone nowhere. I would al so like to express my deep appreciation to Professor Andrew Rinzler. I truly benefited fr om his valuable opinions his help with lab facilities, and collaborati on of some on his interesting and fr uitful projects. I also want to thank Professor Peter Hirschfeld, who gave me a better understanding of solid state physics through his teaching; a nd Professor David Norton, for being on of my committee members. I am dearly thankful to current and former members of our group (Josh Kelly, Jeremy Nesbitt, Partha Mitra, Ryan Ra irigh, Sinan Selcuk, Guneeta Singh, Kevin McCarthy, Quentin Hudspeth, Stephen Arnason, Nikoleta Theodoropoulou, and Stephanie Getty), who provide d a joyful working environm ent and great help. I would especially like to thank Sinan Selcuk for hi s help on E-beam lithography. I also want to

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v thank Jamal Derakhshan (who worked as an REU student in the lab), for his help with the bulk bismuth study. My gratitude also goes to Professor Gray Ihas, and to Professor Amlan Biswas and his students (Tara Dhakal, Jacob Tosado, a nd Sung-Hee Yun), who provided me great help in using their facilitie s. I thank Zhuangchun Wu, Jennife r Sippel, and Amol Patil for their kind help with my experiments. I also would like to thank Ronojoy Saha, for useful discussions on high magnetic field transpor t. Many thanks go to the machine shop personnel for their excellent work, which allowed my research work to move on smoothly. I would like to express my gr eat appreciation and deep love to my parents for their unconditional love and support. And finally, I would like to thank my dear wife, Zhihong Chen, who was a graduate st udent in Professor Andrew Rinzlers group. Her knowledge and intelligence have been of great help. She has shared my happiness and burden all these years. Her love changes my lif e and makes me a better individual.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT......................................................................................................................x ii CHAPTER 1 GENERAL INTRODUCTION.......................................................................................1 2 MAGNETOTRANSPORT IN GRAPHITE...................................................................4 2.1 Overview of Classical Magnetotransport in Semimetals.........................................5 2.2 Multi-Band Model...................................................................................................7 2.3 Sample Preparation and Characterizations..............................................................9 2.3.1 Sample Preparation........................................................................................9 2.3.2 Characterizations: Dingle Te mperature and Landau Levels.......................11 2.4 Transport in the Classical Region..........................................................................17 3 MAGNETOTRANSPORT OF GRAPHITE IN THE ULTRA-QUANTUM FIELD..24 3.1 Transport Data in th e Ultra-Quantum Field...........................................................24 3.2 In-Band Transport Behavior in the Ultra-Quantum Regime.................................27 3.3 Possible Models in the Ultra-Quantum Regime....................................................37 4 TUNNELING INTO BULK BISMUTH IN THE ULTRA-QUANTUM FIELD........41 4.1 Motivation..............................................................................................................41 4.2 Experimental..........................................................................................................43 4.3 Results and Discussion..........................................................................................47 5 ACHIEVING LARGE MAGNETORESISTANCE IN BISMUTH THIN FILMS....50 5.1 Introduction...........................................................................................................50 5.2 Experimental .........................................................................................................54 5.3 Results and Discussion.........................................................................................55

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vii 6 METALLIC SURFACE STATES IN THE ULTRA-THIN BISMUTH FILMS........63 6.1 Introduction: Physics of the Ultra-Thin Bismuth Films.......................................63 6.2 Transport Properties of the Ultra-Thin Bismuth Films.........................................66 6.2.1 Experimental...............................................................................................66 6.2.2 Metallic Surface States...............................................................................67 6.3 Control of the Surface States................................................................................74 7 SURFACE SUPERCONDUCTI VITY IN ULTRA-THIN BISMUTH FILMS.........80 7.1 Transport Evidence...............................................................................................80 7.2 Tunneling Evidence..............................................................................................84 7.3 Possible Picture.....................................................................................................89 8 FUTURE WORK.........................................................................................................91 LIST OF REFERENCES...................................................................................................97 BIOGRAPHICAL SKETCH...........................................................................................101

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viii LIST OF TABLES Table page 1-1. Basic parameters of bismuth and graphite..................................................................1 5-1. Summary of results for differe nt bismuth film growth conditions...........................61 6-1. Parameters for the simulating the effect of thickness and temperature on the magnetic field dependence of the Hall re sistivity in ultrathin Bi films..................72 6-2. Parameters for the simulating the e ffect of Ge coating on the magnetic field dependence of the Hall resistivit y in ultra-thin Bi films..........................................79

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ix LIST OF FIGURES Figure page 2-1. Configuration of leads on graphite transport sample................................................10 2-2. Shubnikov-de Haas oscillations in graphite at indicated temperatures....................11 2-3. The Landau level indices as a function of the inverse of magnetic field at different low temperatures......................................................................................................12 2-4. The amplitude of the ShdH oscillati ons as a function of th e inverse of magnetic field at 2K.................................................................................................................14 2-5. Scaled ShdH oscillations amplitude as a function of the inverse of magnetic field in different temperatures..............................................................................................15 2-6. Linear fit to the slopes of the scaled ShdH oscillation as f unction of temperature..15 2-7. Temperature dependence of the resistivity xx for a graphite crystal in different magnetic fields.........................................................................................................17 2-8. xx and xy versus applied magnetic field at the different temperatures...................20 2-9. Temperature dependence of mobility, relaxation time; and carrier density for the bands indicated in the le gends of each panel...........................................................21 3-1. The magnetic field dependence of the longitudinal and Hall re sistance of HOPG at different temperatures..............................................................................................25 3-2. The temperature dependence of the l ongitudinal resistance of HOPG in different magnetic fields.........................................................................................................26 3-3. The ratio of the measured Hall resist ance and longitudinal resistance as a function of magnetic field at 2K.............................................................................................29 3-4. The Landau band dispersion relation of gr aphite in 12 Tesla fi eld, calculated using the SWMcC model...................................................................................................32 3-5. Estimation of carrier un-compensati on in different magnetic fields at 2K..............33 3-6. Shape of the in-band re sistivity as a function of ma gnetic field at the indicated temperatures.............................................................................................................34

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x 3-8. Logarithmic plot of the shape of the in-band conductivity as function of temperature in different strong magnetic fields.......................................................36 3-9. Model of the field induced Lutti nger liquid with dressed impurity scattering........38 3-10. Scaled in-band conductivity as a functi on of temperature in magnetic fields above the UQL....................................................................................................................39 4-1. Procedure for making tunnel junctions on bulk semimetal using photolithography technique..................................................................................................................44 4-2. Mica mask method for making tunnel junctions on bulk semimetal........................46 4-3. Microscopic picture of a Bi(bulk)-AlOx-Pb tunnel junction....................................46 4-4. Differential conductance as a function of bias voltage in indicated strong magnetic fields at 300mk.........................................................................................................48 4-5. Differential conductance at low bias voltage in the magne tic fields indicated in the legend.......................................................................................................................48 5-1. Fermi surface and Brillouin zone of rombohedral bismuth......................................50 5-2. Magnetotransport behavior of bulk single crystal bismuth......................................51 5-3. Magnetotransport behavior of a bismuth thin film...................................................52 5-4. X-ray diffraction pattern for a 4-um-thick Bi/Au film.............................................56 5-5. Resistivity vs. temperature at 0 and 5T for two category-I Bi films........................57 5-6. Resistivity vs. temperature at 0 and 5T for three category-II Bi films.....................58 5-7. Resistivity vs. temperature at 0 and 5T for two category-III Bi films......................60 6-1. Illustration of semimeta l-to-semicinductor transition..............................................64 6-2. Resistivity vs. temperature for Bi film with indi cated thicknesses..........................67 6-3. Magnetoresistance at 5K for two different thicknesses Bi films..............................69 6-4. Hall Resistivity vs. magnetic fiel d for (a) 180 and (b) 400 Bi films..................70 6-5. Simulated Hall Resistivity vs. ma gnetic field for Bi(180) and Bi(400).............73 6-6. Temperature dependence of resi stivity for Bi(100) and Bi(100)/Ge..................75 6-7. Hall Resistivity vs. magnetic fiel d for (a) Bi(100) and (b) Bi(100)/Ge .............77

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xi 6-8. Simulated Hall Resistivity vs. magne tic field for Bi(100) and Bi(100)/Ge........78 7-1. Resistance vs. temperature in zero magnetic field for a 15nm bismuth film...........80 7-2. Example of resistance in creases during the transition..............................................81 7-3. Sharp feature of resi stance change observed in Hall resistivity measurements.......82 7-4. Film thickness dependence of th e critical magnetic field at 4.5K............................83 7-5. Differential conductance as a function of bias voltage in the superconducting gap region for a Pb-AlOx-Bi( 150) tunnel junction......................................................84 7-6. Differential conductance as a function of bias voltage in the superconducting gap region at 300mk for a Pb-AlO x-Bi(1000) tunnel junction....................................86 7-7. Differential conductance vs. bias voltage at 300mK in indicated low magnetic fields perpendicular and parallel to the junction plane......................................................87 7-8. Differential conductance vs. bias volta ge at 300mK in indi cated strong magnetic fields perpendicular and para llel to the junction plane.............................................88 7-9. A possible picture of surface superc onductivity in ultra-thin bismuth films...........90 8-1. Some examples of the s ub-micron sized bismuth patterns.......................................93 8-2. Magnetic field dependence of the resistivity for a re servoir pattern........................94 8-3. Measurements of a nano-cavity with a single grain-boundary in it.........................95

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MAGNETOTRANSPORT AND TUNNELING STUDY OF THE SEMIMETALS BISMUTH AND GRAPHITE By Xu Du December 2004 Chair: Arthur F. Hebard Major Department: Physics Magnetotransport and tunneli ng studies on bulk crystals, thin films and patterned nanostructures of semimetals reveal a surpri sing range of interesti ng behaviors. In our study of ultrathin bismuth films, we found that the transport behavior is greatly affected by the presence of metallic surface states, whic h become evident in th e thinnest films and are presumed to be responsible for a surf ace superconducting state s een in tunneling and transport anomalies. We also studied bulk samples of both of these semimetals in magnetic fields high enough to place all the ca rriers in the lowest Landau level. In this ultraquantum regime, the apparent re-ent rance in graphite from insulating to metallic/superconducting behavior at low temperatures corresponds to the in-band insulating behavior of carriers within a se miclassical 2-band model framework. This analysis brings into question recently propos ed explanations of a field-induced metalinsulating transition and magnetic-field-induced superconducting fluctuations in graphite.

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1 CHAPTER 1 GENERAL INTRODUCTION A semimetal is a semiconductor with a small conduction band-valence band overlap (instead of a gap). Semimetals have low Fermi energy. In contrast to semiconductors, which are insula tors at zero temperature ( T = 0 ) where the carrier concentration n = 0 semimetals have a finite conductivity at T = 0 where n is finite because of the nonzero overlap of the conduc tion and valence bands. Semimetals are metallic, with both electrons and holes contri buting to electric conduction. Graphite and bismuth are typical semimetals, with low Ferm i energies and low carrier concentrations. Table 1-1. Basic parameters of bismuth and graphite Ef (meV) Carrier concentration (m-3) (ne=np) Bismuth ~30 ~2310 Graphite ~22 ~2410 Semimetals have been of interest for many years, in many different aspects. A major aspect of semimetal study is magnetotran sport. Because of their small values of carrier concentration, semimetals can be driv en into the ultra-quantum regime, when only the lowest Landau level remains occupied, with a magnetic field of ~10 Tesla. In addition, light cyclotron masses cm in certain orientations of semimetals result in higher cyclotron frequencies (cm eB/) ensuring that quantum ma gneto-oscillations can be observed in moderate magnetic fields and at moderate temperatures. High purity allows the oscillations to survive th e effects of disorder. Magnetotr ansport study of semimetals in strong magnetic field allowed the Fermi surface to be mapped by quantum oscillations

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2 in semimetals. In applications, the extremely large magnetoresistance of semimetals makes them promising candidates for magnetic field sensors. Another major aspect of semimetal study originated from the long Fermi wavelength. By making bismuth thin films with thicknesses comparable to a Fermi wavelength, one could study the energy band quantization because of quantum confinement. Also, since the band mass is bigge r for electrons than holes, as the size of the bismuth structure decreases, the speed at which the conduction band shifts up will be faster than that of the valence band. At a certain point, a gap opens up, and the semimetal-semiconductor transition should happ en. Existence of the transition has been studied for many years, and is still not conclu sive; mainly because of the existence of the surface states, that smear out any sharp features of the transition. Evidence of metallic surface states were found in films of su perconducting bismuth clusters, which indicates surface superconducti vity because of the strongly increased surface density of states. Further evidence of metallic surface states was found by angle resolved photoemission sp ectroscopy (ARPES). Our study of semimetals focused on two major aspects: 1) magnetotransport and tunneling study of bulk single-crystal bismuth a nd graphite; and 2) the effect of bismuth surface states on transport and bismuth surface superconductivity. In the following chapters, we will show the motivation and our work on each aspect of our study. Chapter 2 first of all desc ribes the theoretic b ackground of transport behavior in semimetals. Then it explains the experimental details, sample characterization and low field transport behavior analysis of graphite. Chapter 3 explains the high magnetic field transport behavi or of graphite, and proposes possible theoretic models.

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3 Chapter 4 describes the magneto-tunneling measurements on bulk bismuth tunnel junctions. Chapter 5 explains our work on achieving large magnetoresistance in Bi-Au thin films. Chapter 6 describes the effect of the metallic surface states on the transport properties of ultra-thin bismuth films. Ch apter 7 describes the surface superconductivity behavior we observed in the ultra-thin bism uth films. Chapter 8 discusses some possible interesting future works on bismuth, includ ing nano-meter sized bi smuth structures and spintronics applications.

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4 CHAPTER 2 MAGNETOTRANSPORT IN GRAPHITE Graphite is a typical semimetal, with a low Fermi energy (~22 meV) and low carrier concentration (~3 2410 3 m ). The zero-temperature conductivity in graphite results from the small overlap between th e conduction band and the valence band. The Fermi level lies in the middle of the overlap, which makes graphite a typical compensated 2-band material. Transport properties of graphite had been studied intensively since 1950s. Recently, interest in magnetotransport in graphite was renewed because of the observation of an effect that looks lik e a magnetic-field-induced metal-insulator transition: the metallic temperature-dependence of the in-plane resistivity in zero field turns into an insulating-like one when a magnetic field of a few tens of mTesla is applied perpendicular to the basal (ab) plane. Increa sing the field to about 1 Tesla produces a reentrance of the metallic behavior. It has been proposed that the low-field effect is caused by a magnetic-field-induced ex citonic insulator transition of Dirac fermions,1, 2 whereas the high-field behavior is a manifest ation of field-induced superconductivity.3, 4 It has also been suggested that the a pparent metal-insulator transition in graphite is similar to that in 2D heterostructures (although the latter is driven by a field parallel to the conducting plane). To elucidate these issues, we performed detailed measurements of magnetoresistance in graphite and found da ta quite similar to data reported in 1-4 over comparable temperature and field ranges. Ho wever, our interpretation is significantly different from theirs.

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5 2.1 Overview of Classical Magnetotransport in Semimetals A combination of some unique features specific to semimetals [i.e., low carrier density, high purity, small effective mass and equal number of electrons and holes (compensation)] led to an unusual temperatur e dependence of the magnetoresistance even in classically strong fields, defined by the condition T kB c / (2-1) where is a scattering time of the carriers. He re we qualitatively compare a semimetal with a conventional, high-den sity, uncompensated metal. To begin with, if the Fermi surface is isotropic, a metal does not e xhibit magnetoresistance because of the cancellation between the longitudinal and Hall components of the electric field.5 In real, anisotropic metals, this cancellation is broken, and as a result, magnetoresistance is finite and proportional to 2 c in weak magnetic fields (1c ). In stronger fields (1c ), classical magnetoresistance saturates .6 In contrast (Equation 2-14), magnetoresistance of a compensated semimetal grows as 2 B in both weakand strongfield regions. In addition to the saturation effect descri bed above, another f actor that makes the magnetoresistance much smaller in conventional metals than in semimetals is the higher scattering rates and hence the smaller values of c. The impurity scattering rate in semimetals is smaller than in conventional metals simply because semimetals are typically much cleaner materials. The lower carrier density of semimetals also reduces the rates of electronphonon scattering compared to that of conventional metals. For temperatures above the transport Debye temperature, which sepa rates the regions of the

PAGE 18

6 T and 5Tlaws in the resistivity, / D FBksk where kF is the Fermi wave vector and s is the speed of sound (both properly averaged over the Fermi surface), one can estimate the electronphonon scattering rate7 as / ) / )( (0 0 1T k m m a kB F (2-2) where 0a is the atomic lattice constant, and m* and m0 are respectively the effective mass and the bare electron mass. In a conventional metal,1 ~0a kF and 0* m m In this case, D is of the order of the thermodynamic Debye temperature 0/Bska ~few 100 K and 11/BkT Barring numerical factors, T kB / cannot be satisfied in a typical metal. This means that as soon as it en ters the classically strong field region, magnetoresistance saturates and quantum magneto-oscillations start to show up. In a low-carrier-density material (01Fka ), D is much smaller (for Bi and graphite ~D few K) and also 11/BkT which ensures that the inequality (2-1) can be satisfied. Therefore, in a low-carrier-density compensated semimetal a wide interval of temperatures and magnetic fields exists in which a) the scattering time is linear in T, in accordance with Equation 2-2, b) we ar e in the regime of classically strong magnetoresistance with essentially no signa tures of quantum magneto-oscillations, as specified by the inequality (2-1), and c) the magnetoresistance is large. An additional feature that is crucial for in terpreting the experimental data is that the Fermi energies of graphite ( EF = 22 meV)8 and bismuth ( EF = 30 meV [holes])9 are relatively low; and the temperat ure dependence of the resistivity is therefore a function of Inequality (1) can be satisfied in a typical metal for D T when the (transport) time 51trTT /. For an uncompensated metal, however, magnetoresistance saturates in this regime.

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7 two temperature-dependent quantities, n(T) and (T) That materials are pure helps to ensure that electron-phonon scattering is a dominant mechanism for resistance (in a doped semiconductor, impurity scattering dominates). 2.2 Multi-Band Model In the semiclassical theory of conduction in metals, DC electrical conductivity in a multi-band system (in absence of a magnetic field) is described by n n) ( (2-3) ) ( 3 2 ) () )( ( ) ( )) ( ( 4 k n n n n nnf k v k v k dk e (2-4) wherefis the Fermi function, and k k k vn n ) ( 1 ) (. Since 0 f except when is within T kB of F filled bands have no contribution to the conductivity. Only those partly filled bands that are close to the Fermi level contribute to the conductivity. In the presence of a classi cally strong magnetic field z B B (the Landau energy level quantization is negligible), for an isotr opic system in which all the occupied orbits are closed, there will be no magnetoresist ance because of the can cellation of Lorentz force by the Hall components of the electric field. If an external electric field x E Ex is applied, the induced current density will bex E jx0 where 0 is the in-plane zero magnetic field conductivity. The Hall component of the electric field will be generated: x c yE E ) ( Then the definition: E j hence

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8 0 ) ( 0 00 x c x xE E E (2-5) will yield ) 0 ( 0 0 0 ) ( 1 ) ( 1 0 ) ( 1 ) ( 12 0 2 0 2 0 2 0 zz c c c c c c (2-6) This, in terms of resistivity, which is experimentally measured, is simply 0 0 0 10 0 0 0 ) ( ) (ZRB RB B B (2-7) where ne R 1 is the Hall coefficient. Note that the longitudinal composnents of the resistivity have no field dependence. Now consider such a system with more th an one band. Each band contributes to the conductivity of the system in para llel with the other bands. Then the total resistivity is in terms of the resistivity of each band: ii zi i i i i iB R B R1 0 0 0 1 10 0 0 0 (2-8) Even without calculating the resistivity above in detail, we can readily see that the magnetic field dependence (which belongs to the off-diagonal components in the resistivity tensor of each single band) may ente r into the diagonal components of the total

PAGE 21

9 resistivity. Thus the multiband system can have magnetoresistance even though none of its band has magnetoresistance by itself. For the simple (and most useful) case of a 2-band system, the resulting longitudinal resistivity and Hall resistivity are: 2 2 2 1 2 2 1 2 1 2 2 2 2 1 2 1 2 1) ( ) ( ) ( ) ( B R R B R Rxx (2-9a) 2 2 2 1 2 2 1 1 2 2 2 2 1 3 2 1 2 1) ( ) ( ) ( ) ( B R R B R R B R R R Rxy (2-9b) For a system of more than 2 bands, it is convenient to describe the contribution of each band in terms of conductivity. In a simple Drude model, i i i i xxB en2) ( 1 (2-10a) i i i i xyB B en2 2) ( 1 (2-10b) where, in and i are carrier density and mobility of the ith band. From the conductivity tensor, we can then calculate the measured values of resistivy and Hall constant: ) ( ) ( ) ( ) (2 2B B B Bxy xx xx and ) ( ) ( / ) ( ) (2 2B B B B B Rxy xx xy H 2.3 Sample Preparation and Characterizations 2.3.1 Sample Preparation The sample used in the study, a recta ngular shaped highly oriented pyrolytic graphite, with dimensions 2.4 mm wide by 8 mm long by 0.5 mm thick, was cut from a bulk piece of highly oriented pyrolytic graphite (HOPG) using wire saw.

PAGE 22

10 The HOPG sample has a mosaic spread, de termined by X-rays, of 2 degrees. After the sample was cut, it was glued onto a glass substrate. Figure 2-1 shows the configuration of the measurement leads on th e sample. The 4-terminal measurement leads were connected to the sample applying silver paint. Because of the high in-plane/out-ofplane conductivity ratio in la yered graphite, we found it nece ssary to place the current leads uniformly in contact with the sides of the sample. Thin-foil indium was coated uniformly with silver paint, and was attached to the graphite end plates as current leads. Gold wires with tiny loops were silver pasted with ~2 mm separation to the edges of the sample as voltage leads. I+I-V+V-VHall Figure 2-1. Configuration of l eads on graphite transport sample Resistance measurements at 17 Hz were carried out us ing a Linear Research 700 resistance bridge. The sample was measured over the temperature range 2K~ 350K in fields as high as 17.5 Tesla. Low magnetic field measurements we re carried out in a Quantum Design Physical Property Measuremen t System (PPMS) with a 7 Tesla magnet. High magnetic field measurements were carrie d out in a He3 refrigerator with a 17.5 Tesla magnet in the National High Magnetic Fi eld Labs (NHFML). In all measurements,

PAGE 23

11 the magnetic fields were applied perpendicular to the graphite basal plane (i.e., parallel to the c-axis). 2.3.2 Characterizations: Dingle Temperature and Landau Levels In the presence of strong magnetic fields, the energy bands of graphite split up into Landau levels. With increasing magnetic fields, the interval between Landau levels increases: c cm eB / where c is the cyclotron frequency, and cm is the cyclotron mass When a Landau level moves acr oss the Fermi surface, sharp features of conductivity change appear. This caused the oscillatory beha vior of resistivity in the magnetic field sweep (Shubnikov-de Haas oscillations). Figure 2-2. Shubnikov-de Haas os cillations in graphite at indicated temperatures. The oscillations are obtaine d by subtracting the background magnetoresistance from the resistance vs. magnetic field curves. The inset shows the resistance as a function of magnetic field at 2K. 0.00.51.0 -0.15 -0.10 -0.05 0.00 2K 5K 10K 15KRosc()1/B(Tesla-1) -1012345678 0.0 0.2 0.4 0.6 0.8 R ()B (Tesla)

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12 Given the Fermi energy Ef, the number of Landau levels below the Fermi energy can be estimated as c fm eB E N / 1 Hence the period at which the one Landau level moves across the Fermi surface goes like 1/ B As the magnetic field increases, every time one Landau level shifts across the Fermi level, the resistivity will decrease because of the high density of states at the bottom of the Landau band. And a valley will show up in the resistivty vs. magnetic field pl ot. By plotting the valley positi ons of the ShdH oscillations vs. 1/ B one gets evenly spaced points. By labeling these points one can count the number of Landau levels lying below or across the Fermi surface. Since the band structure does not change with temperature, the valley positions measured at different temperatures should overlap well. Figure 2-3 below shows th e Landau level indices as a function of the inverse of magnetic fields at different low temperatures. Figure 2-3. The Landau level i ndices as a function of the i nverse of magnetic field at different low temperatures 0.20.40.60.81.01.21.41.6 2 3 4 5 6 7 8 Number of Landau Levels 1/B (Tesla-1) 2K 5K 10K

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13 Figure 2-2 shows that the ShdH oscillat ions are most pronounced at very low temperatures (c BT k ). At higher temperatures, the oscillations are smeared by phonon scattering. The relation between the am plitude of the ShdH oscillations and temperature T at magnetic fiel d B is shown in Equation 2-11.10 B T T B Ad osc) ( exp ~2 / 5 (2-11) Here Td, the Dingle temperature, describes th e temperature-inde pendent scattering from impurities and dislocations. Dingle temperature is a useful parameter in characterizing the quality of the graphite sa mple. The lower the Dingle temperature, the less imperfection the sample has. Dingle temperature can be measured from conductance-field sweep at different temperatures. First, the background of the conduc tance vs. field curves is subtracted out, and the oscillatory part is pl otted vs. 1/B. Then the osci llation amplitudes are obtained by subtracting the envelope curves of the oscillations valleys fr om the envelope curves for the peaks (Figure 2-4). Since only the shapes of those curves are important, we used arbitrary smooth functions to fit the envelope curves. This method of interpolation give much more precise oscillation amplitude values than simply subt racting the resistance values at the valleys from those at the adjacent peaks (which do not correspond to the same field). After we get the amplitudes of the oscillations Aosc, we plot 2 / 5B Aoscvs. B / 1, and obtain straight lines in 2 / 5B Aoscvs. B / 1 plots ((Figure 2-5)). The slopes of the straight lines, according to Equa tion 2-11, simply correspond to ) (dT T

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14 Figure 2-4. The amplitude of the ShdH osc illations as a functi on of the inverse of magnetic field at 2K, obtained by subt racting the bottom envelope curves from the top envelope curve of the osci llations, as indicated in the inset. We then plot the values of the slope s vs. their corresponding temperatures, and perform a linear equation fitting to the curve. The fitted straight line then intersects with the negative side of the temperature axis, w ith the offset being the Dingle temperature Td. For the graphite used in our study, the Dingle temperature we got from the analysis described above is about 4.5K. This result suggests that the sm earing of the ShdH oscillations by the imperfections in the HOPG corresponds to the thermal smearing of 4.5K. 0.40.60.81.01.21.41.6 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Amp. osc1/B 0.00.51.01.52.0 -0.5 0.0 0.5 Conductance_osc (-1)1/B (Tesla-1)

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15 Figure 2-5. Scaled ShdH os cillations amplitude: Ln(2 / 5B Aosc), as function of the inverse of magnetic field in di fferent temperatures. 0246810121416 -1x105-8x104-6x104-4x104 Td4.51111 0.99368a*(T+Td)T ( K ) Figure 2-6. Linear fit to th e slopes of the scaled ShdH oscillation as function of temperature. 0.40.60.81.01.21.41.6 -4 -3 -2 -1 0 1 2K 5K 10K 15Kln(amp.*B^(5/2))1/B (Tesla-1) K Td1 5 4

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16 ShdH oscillations also provide informa tion about the Fermi surface. By measuring ShdH oscillations in different magnetic fiel d orientations, one can map out the extremal of the Fermi surface. For graphite, by measur ing ShdH oscillations and calculating the volume of the Fermi pockets, we can then calculate the carrier concentration. Here we simplify the problem by treating the Fermi surface of graphite as ellipsoids with anisotropy ratio By measuring the ShdH oscillations with magnetic field parallel to the c-axis of graphite, we get the pe riod of the quantum oscillations to be c fm E e B ) (1 (2-12) where cm is the cyclotron mass for magnetic fiel d along the c-axis. This provides the information about the area of the extremal cr oss section of the Fermi surface with a plane perpendicular to the magnetic field: c f extrm E S 2 (in momentum space). Given the anisotropy ratio we can then calculate the volume of the ellipsoid and hence the carrier concentration: 2 / 3 1 2 / 3) ( 4 3 1 B N h e n (2-13) where 6 N is the number of ellipsoids in the Brillouin zone. Taking ~ 12-17, the carrier concentration calc ulated from the measured oscillations period is 3 2410 3 2 ~ m n This number corresponds to the zero temperature carrier concentration in graphite. At low enough te mperatures, one can separate the ShdH oscillations periods from elect rons and holes, and calculate the carrier concentration for the two different carriers. Also, for each carrier group, there will be two sets of oscillations because of the spins. At 2K how ever, we are not yet able to distinguish the

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17 different periods result from the small eff ective mass difference between electrons and holes, and from the spin splitting. 2.4 Transport in the Classical Region In this section, we pres ent a detailed study of low field magnetotransport in graphite and show that the unusual behavi or of the temperatur e and field-dependent resistance, such as shown in Figure 2-7, can be described in a st raightforward way by a simple multi-band model that takes into accoun t contributions to the conductivity from the electron and hole carriers associated with the overlapping valence and conduction bands. 050100150200 1E-8 1E-7 1E-6 0 mT 20 mT 40 mT 60 mT 80 mT 100 mT 200 mT 050100150200 0.0 0.5 1.0 1.5 IB (Tesla)T (K)II III xx ( m)T (K) Figure 2-7. Temperature depe ndence of the resistivity xx for a graphite crystal plotted on a logarithmic axis at the magnetic fiel ds indicated in the legend. The solid lines are the fits to the data using th e six parameters derived from the three bands described in the text. The shadow ed region on the inset and its mapping onto the data in the main panel are described in the text.

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18 We use the qualifier unusual in describing the data of Figure 2-7, since on lowering the temperature the resistance increas es as it does in an insulator but then saturates at lower temperatures. The nontrivial explanations of Kopelevich et.al 3,4 rely heavily on such features as the Dirac spectrum of fermions and almost two-dimensional transport, which are unique for graphite but not for Bi. That Bi and graphite behave similarly suggests that these features are not responsible for the observed phenomena. Our explanation for the insulating-like behavior in a magnetic field does not require more exotic explanations of a magnetic-field-indu ced opening of an excitonic gap in the spectrum of interact ing quasiparticles.11 Instead, we propose that the uniqueness of the low magnetic field transport be havior of semimetals lies in the existence of a wide interval of temperatures and magnetic fields defined by the inequalities of Equation 2-1. Our analysis of the experimental data confirms the inequalities Both xx and xy (see Figure 2-8) were measured in magnetic fields up to 0.2 Tesla at different temperatures A small field-symmetric component caused by slightly misaligned electrodes was subtracted from the xy(B) data. To fit the data, we adopt a standard multi-band model.5 Each band has two parameters: resistivity i xx and Hall coefficient i i in q R 1 where e qi is the charge of the carrier. In agreement with earlier studies, we fix the number of bands to three.1 Two of the bands are the majority electron and hole bands, and the third one is the minority hole band. Although the presence of the third band is not essential for a qualitative understandi ng of the data, it is necessary for explaining fine features in xy. Our fitting routine incorporates both xx(B) and xy(B) simultaneously by adjusting the six unknown parameters independently, until the differences between the fitting curves and the experimental data are minimized.

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19 Because the majority carriers in graphite derive from Fermi surfaces that have sixfold rotational symmetry about the c -axis, we only need to deal with the 2x2 magnetoconductivity tensor with elements xx and xy. Here, we define the conductivity 2 2 2 2) ( ; ) ( B R B R B Ri i i i xy i i i i xx and the resistivity *2/iiiimne for the ith band. The total conductivity is simply a sum of the contributions from all the bands: 13..i i The observable resistivity tensor is obtained by inverting : 1 Qualitatively, the unusual temperature dependence of xx displayed in Figure 2-7 can be understood for a simple case of a two-band semimetals, where xx reduces to 2 2 2 2 2 2B R R B R Rh e h e e h h e h e h e xx (2-14) Here ) (h e and ) (h eR Rare resistivity and Hall coefficien t for the electron (hole) band, respectively. Assuming thata h eT with 0 a, we find that for perfect compensation (i.e., h en n where en and hnare carrier density for the el ectron band and the hole band respectively), R R Rh e and the 2-band resistivity described by Equation 2-14 can be decomposed into two contri butions: a field-independent terma T and a fielddependent term aT B T R/ ) (2 2. At high T, the first term dominates and metallic behavior ensues. At low T, ) ( / 1 ) (T n T R saturates and the seco nd term dominates, giving insulating behavior.

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20 Figure 2-8. xx and xy versus applied magnetic field at the temperatures indicated in the legend. The solid lines are determined by a fitting procedure described in the text. The inset in the xy plot magnifies the lowfield region where the contribution from the minority band is important. 0.000.050.100.150.20 10-810-710-6 40K 70K 100K 100K 200K 5K 10K 15K 20K 25Kxx ( m)B (T)0.000.050.100.150.20 0.0 5.0x10-81.0x10-71.5x10-72.0x10-72.5x10-73.0x10-7 40K 70K 100K 150K 200K 5K 10K 15K 20K 25Kxy (m)B (T)0.000.010.020.03 -2.0x10-90.0 2.0x10-94.0x10-96.0x10-98.0x10-91.0x10-81.2x10-81.4x10-8 xy (m)B (T)

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21 0501001502000.0 2.0 4.0 6.0 8.0 10E24 band 1 band 2 band 3Temperaturecarrier density (m-3)0.0 0.3 0.6 0.9 1.2 1.5 1.8E12 band 1 band 2 band 3 (sec-1)0 50 100 150 200 250 300 350 band 1 band 2 band 3mobility (m2V-1s-1) Figure 2-9. Temperature dependence of the fitting parameters: A) mobility; B) relaxation time; and C) carrier density, for the bands indicated in the legends of each panel. The actual situation is somewhat more complicated because of the T-dependence of the carrier concentration, the presence of the third band, and an imperfect compensation A B C

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22 between the majority bands. Results for the temperature-dependent fitting parameters are shown in Figure 2-9, where band 1 corresponds to majority holes, band 2 to majority electrons and band 3 to minority holes. The insu lating-like behavior of the carrier density with a tendency towards saturation at low temperatures is well reproduced. For the majority bands, 1 and 2, the carrier concentrations are approximately equal and similar in magnitude to literature values.12 The slope of the linear-in-T part of /exp 1T kB with ) 3 ( 065 0exp (dashed line in Figure 2-9, panel A) is consistent with the electronphonon mechanism of scattering. To see this, we adopt a simple model in which carriers occupying the ellipsoidal Fermi surface with parameters mab (equal to 0.055m0 and 0.04 m0 for electrons and majority holes, correspondingly), mc (equal to 3m0 and 6m0, correspondingly) interact with longitudinal phonons via a deformation potential, characterized by the coupling constant D (equal to 27.9 eV). In this model, the slope in the linear-in-T dependence of is given by7 3 2 0 2 3/ *) ( ) / 2 ( ab F theors D E m (2-15) where 0 3 / 1 221 0 ) ( m m m mc ab both for electrons and holes, 3 0/ 27 2 cm g is the mass density of graphite, and s cm Sab/ 10 26 is the speed of s ound in the ab-plane. (The numerical values of all parameters ar e taken from standard reference on graphite12) With the above choice of parameters, 052 0 theor for both types of carriers. This value is within 20% of the value found experimentally. Given the simplicity of the model and uncertainty in many material parameters, especially the value of D such an agreement between the theory and experi ment is quite satisfactory.

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23 The solid lines through the data points in Figure 27 are calculated from the temperature-dependent fitting parameters de rived from our three-band analysis and plotted in Figure 2-8. The shaded region (II) de picted in the inset of Figure 2-7 represents those temperatures and fields that satisfy the inequalities of Equation 2-1. In region (I) ShdH oscillations can be seen at sufficientl y low T (our sample has a Dingle temperature of 5K), and in region(II) th e magnetoresistance is low. The boundary between (I) and (II) reflects the rightmost of the inequality 2-1 and is determined by the relation / m eB T where 5 has been chosen to represent the ratio c BT k /. A larger value of would decrease the slope of this boundary and diminish the area of (II). The boundary between (II) and (III) refl ects the leftmost inequality of 2-1 and is determined by the relation ) ( / T e m B where ) ( / 1 T is obtained from e xperimental fitting parameters (Figure 2-9). In the main panel of Figure 2-7, we s uperimpose region (II), again as shaded area, on the ) ( B Txx data. Below the lower boundary 1 c, and the magnetoresistance is relatively small. The upper boundary is determin ed by the locus of (B,T) points satisfying the rightm ost inequality of 2-1 for 5 Clearly region (II), constrained by the inequality 2-1, overlaps we ll with the metal-insulating like behavior of graphite. We thus conclude that the semime tals graphite and, by implication, bismuth share the common features of high purity, lo w carrier density, small effective mass and near perfect compensation, and accordingly obe y the unique energy scal e constraints that allow pronounced metal-insulating beha vior accompanied by anomalously high magnetoresistance.

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24 CHAPTER 3 MAGNETOTRANSPORT OF GRAPHITE IN THE ULTRA-QUANTUM FIELD 3.1 Transport Data in the Ultra-Quantum Field The inequality discussed in chapter 2: T kB c /defines two limits that are satisfied within a wide temperature range in semimetals: c / (or ) 1 c gives rise to the large magnetoresistance in the semimetals; T kB c defines the classically strong magnetic field (weak fi eld), in which the number of Landau levels below the Fermi level is so large that the quantum oscillations are well smeared by temperature, hence the effect of quantization of the energy bands is negligible. In stronger magnetic fields, there are onl y few Landau levels below the Fermi level and T kB c is no longer satisfied. Eventually, when the magnetic field is so strong that all the conduction electrons are all in the lowest Landa u level, the so-called ultraquantum limit is reached. Above the ultra-quantum limit, th e energy of the electrons is fully quantized in the plane perpendicular to the field. The movement along the field lines is free; hence the electrons in the system a ssume movement with sp iral trajectories along the field lines. Ideally, in the absence of scatte ring and interaction, a system in the ultraquantum regime should have zero conductance in the plane perpendicu lar to the magnetic field, because the Lorentzian force confines the movement of the el ectron to the spiral trajectories. However, with interactions and s cattering, the electr on can move along the plane perpendicular to the fiel d lines in a diffusive manner.

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25 Figure 3-1 shows the strong magnetic field magneto-transport data taken from the same HOPG used in the low magnetic field study. 05101520 0 300 600 900 1200 1500 70K 40K 20K 15K 10K 7.5KRxx (m)B ( T ) 5K 05101520 0 20 40 70K 40K 20K 15K 10K 7.5K 5KRxy (m)B (T) Figure 3-1. The magnetic field dependence of the A) longitudinal; and B) Hall resistance of HOPG at different temperatures B A

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26 020406080100 0 500 1000 1500 2000 4TRab (m) T (K)17.5T 16T 14T 12T 10T 8T 6T 1T Figure 3-2. The temperature dependence of the longitudinal resi stance of HOPG in different magnetic fields From the magnetic field dependence of the longitudinal resistance data, we see that the resistance increases roughly linearly with magnetic field, a nd tends to saturate in very high magnetic field (>10 Tesla). For the curves taken at very low temperatures (T <15K), we see ShdH oscillations on top of the ma gnetoresistance. The Hall resistance has much smaller values than the longitudinal resistan ce in high magnetic fields, and has rather complicated field dependence. We are most interested in the temperature dependence of the resistance in different strong magnetic fields. Here we see that, the resistance increases with decreasing temperature for T >30K, similar to what is observed in the low (classical) magnetic fields. For T <30K, however, the resistance pl unges down with decreas ing temperature in strong magnetic fields.

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27 The metallic behavior observed in str ong magnetic fields and low temperatures can not be explained by semi-classical trans port theory. It was proposed that the highfield transport behavior is a manifestation of field-induced superconductivity4. However, we find a more conventional interpretation by co nsidering graphite as a multi-band system. Our strategy of analyzing the data re lies on obtaining in-band transport behavior from the experimentally measured data usi ng the multi-band model. Then we will try to understand the in-band transport, which repres ents the intrinsic phys ics of the graphite system. 3.2 In-band Transport Behavior in the Ultra-Quantum Regime In strong magnetic fields, the multi-band model still applies, except that we can no longer take the resistivity and Hall coefficient of each single band to be field-independent parameters, because of the strong quantum ef fect. Instead of curv e fitting with field independent parameters, we need to start by simplifying the multi-band model. Since the contribution of the minority band vanishes in high fields, we can a pply the simple 2-band model, in which: 2 2 2 1 2 2 1 2 2 1 2 2 2 1 2 1 2 1) ( ) ( ) ( ) ( B R R B R Rxx 2 2 2 1 2 2 1 2 1 2 2 2 1 3 2 1 2 1) ( ) ( ) ( ) ( B R R B R R B R R R Rxy Now we make the assumptions that, in hi gh magnetic fields, the system is nearly compensated, and the resistivitie s of each band are very close: 2 1 (3-1a) R R R 2 1 (3-1b)

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28 We will see that these assumptions are vali d when applied to the experimental data in high magnetic fields. With th e simplification above, we have 2 2 2 2 2 24 2 B B Rxx (3-2a) 2 2 2 2 2 24 B B R Bxy (3-2b) where 2 1R R is the difference of the electron and hole Hall coefficients. The ratio of the two numerator terms and R Bin high magnetic fields gives rise to two different pictures of tr ansport properties in this regi on. In the first picture, we assume that the magnetoresistance shown in xx is mostly from the magnetoresistance of each single band. An extreme case of this picture is when: RB and hence: 2 / xx and 4 / Bxy In this case, the longitudinal resistivity of the 2-band system is essentially the same as that of each single band. Accordingly, we see that in this pict ure, we do need the nontrivial explanations for the nonsaturating MR in a system with closed orbits, i.e., field induced metal-insulator transition (MIT) a nd re-entrance to metallic behavior in quantizing magetic fields.1-4, 13 In the second picture, we assume that the MR shown in xx is mostly from the diagonal (Hall) resistivity, 2 2) ( RB (or 1 ) (2 c). Then the simplified forms of the high field limit resistivities are: 2 2 2 2 24 2 B B Rxx (3-3a)

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29 2 2 2 3 24 B B Rxy (3-3b) The ratio of the two measured parameters, 2 Bxx xy is plotted in the figure below: Figure 3-3. The ratio of the measured Hall resistance and longitudinal resistance as a function of magnetic field at 2K. It can be seen in the figure that, the ratio 1 B for field above the ultraquantum limit. This rules out the possibility of un-compensation (i.e., 0 ) as a mechanism for the re-entrant behavior in high magnetic fields and low temperatures, since the second term in the denominator of both xx and xy can be neglected. Now we have a even more simplified form of xx and xy : 22 2B Rxx (3-4a) 612 1E-3 0.01 0.1 xy/xxB (T)

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30 2 3 24 B Rxy (3-4b) It can be seen from these relations above that, the transport be havior in the second picture is drastically different from that in the first picture. The 2-band longitudinal resistivity is now proportional to the reciprocal of the resi stivity of each single band! Therefore the re-entrance to metallic behavi or of the 2-band system as measured by xx would really correspond to a cross over from metallic to insulating behavior for each band (as measured by 2 1 ) as the temperature drops. The criterion for the second pi cture to be valid is that: RB must be satisfied. In relatively low magnetic fields (say B ~ 0.1 Tesla), ta king the values of 810 ~and 610 ~R we see that this criteria is well satisf ied. Hence we are confident in ruling out any non-trivial MIT mechanism in explaini ng the MIT-like beha vior. Despite the complicated field-dependence of the major mechanism for MR is the Hall resistivity term RB In very strong magnetic field, RB is required for the result 22 2B Rxx to be valid. Hence we require that RBxx In the range of the magnetic field we are studying, the carrier concen tration increases slowly with increasing field12, hence the inband Hall coefficient R decreases with increasing fiel d. Taking the experimentally measured mxx 410 3 ~, and 1 610 mT R we can see that RBxx is satisfied for B ~ 10 Tesla. This indicates that our sec ond picture is self-consis tent according to the experimental data.

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31 From the analysis above, we see that RB is generally satisfied in the magnetic field range of our study. We also note that RB and 1 B implyR Hence our assumption that the system in strong ma gnetic field is nearly compensated is well consistent with the experimental results. Till now, we havent made any assumption on the expressions of the resistivities and the Hall coefficients of the two bands. Ideally, if we have the complete information on the band structure of graphi te in high magnetic field and the scattering mechanisms, we can exactly calculate the conductivity tensor using Kubo formula. Such a procedure, however, will be extremely complicated. In the ultra-quantum limit (UQL), the probl em may get simplified by the fact that all the carriers are in the lowest Landau level at sufficiently low temperatures: f c BE T k This can be understood by examining the Landau band structure of graphite in the ultra-quantum limit field. The energy band quantization due to the ma gnetic field can be calculated using the classical tight-binding SWMcC model.12 The dispersion relati on from the calculation shows that, in magnetic fields, each energy band separates into different Landau bands. With increasing magnetic field, all othe r conduction bands and valence band move further and further away from each other a nd from the Fermi level, while the lowest (zeroth) Landau bands remains field in-dep endent. For field above the ultra-quantum limit (~ 8 Telsa for graphite), the Fermi level runs across only the lowest (zeroth) conduction band and valence band, which are th e only Landau bands that contribute to the conduction. Figure 3-4 shows the Landau band structure of graphite in 12 Tesla field,

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32 obtained from the classical band structure calculation (SWMcC model). The field independence of the zeroth Landau bands was confirmed by the recent high magnetic field scan tunneling spectr oscopy measurements on HOPG.14 Figure 3-4. The Landau band disp ersion relation of graphite in 12 Tesla field, calculated using the SWMcC model. From the Landau band structure shown in Figure 3-4, we can see that in strong magnetic field, at suffici ently low temperature: f c BE T k the number of the thermally excited electrons (and holes) in the higher Landau levels is negligible hence we can make the approximation that he carrier concentra tion does not change significantly with temperature nor magnetic field. Since the high field limit of the Hall c0 v0 v1 v2 c1 c2 c3 e h h

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33 coefficient is simply: ne R 1 R will be roughly field and temperature independent in this region (compared with the f actor of two resistivity ch ange in this region). With the approximation above, we have enough information to separate out the inband resistivity using the 2-ba nd longitudinal resistivity. We can also estimate the uncompensation in this high B low T region, by calculating 2 2R Bxy xx Figure 3-5 below shows the values ca lculated from experimental data: Figure 3-5. 2 2R Bxy xx as function of magnetic fiel d at 2K, calculated from the experimental data. The saturation of 2 2R Bxy xx is seen in the Figure at field above 8 Tesla (UQL for graphite). The weak field dependence can be attributed to the combination of weak 24681012141618 101102103104105106 xx 2/(B*xy)B (Tesla)

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34 thermal excitation and field dependence of un-compensation, and the saturation of Using the value of R ~ 10-6 T / (from the band structure calc ulation), we can estimate that 1710 ~T / Hence our assumption of near co mpensation is well satisfied. From the self-consistent approximations above, we reach a very simple high magnetic field limit ( B > BUQL) result: the resistivity of each single band is proportional to the reciprocal of the 2-band resistivity in graphite: xxB R 2 2 Figure 3-6 shows xxB2 calculated from measured data. The high field part of the curve will represent the resistivity of a single band itself under our assumption that R is field-independent in high magnetic field. Figure 3-6. xxR B2 as a function of magnetic field at th e indicated temperatures, calculated from the experimental data. 110 0.1 5K 7.5K 10K 15K 20K 25K 40K 70KB2 / RxxB (Tesla)

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35 From Figure 3-6 above we can clearly see the tendency of scaling of curves in high magnetic fields and low temperatures (curve s become parallel to each other). This indicates that in this region, we only need to care about the cont ribution from the lowest Landau band, and neglect the thermally ex cited carriers in the higher index number Landau levels. Figure 3-6 also give s us a range of such regime: B >10 Tesla and T <20K. We note that this range satisfies the inequa lity for temperature and field independent carrier density: f c BE T k (at 10T, 20K: meV T kB9 0 ~, meV Ef c6 ~ ) With this simplification, we can direc tly analyze the temperature dependence of resistivity in high fields and low temperatures assuming that the carrier densities of the two bands are temperature independent, or R is T-independent. By treating R as a Tindependent and B-independent constant, we see that the in-band resistivity has a temperature dependence xxB /2 The in-band conductivity scaler, which contains the intrinsic physics about the interactions and scat tering of the system, is simply 2 2 0~ ) ( 1B RBxx xx Here since *2 0m ne and because of the week temperature and magnetic field dependence of n and m* in the ultra-quantum region, this implies that the relaxation time: 2~ B xx Figure 3-8 shows the shape of the in-band conductivity 0 vs. T using logarithmic axis. From Figure 3-8 we see that above the UQL field and for f c BE T k : 1) at fixed temperature, the in-band conductivity 0 decreases with increasing magnetic field; 2) at fixed magnetic field, the in-band conductivity 0 decreases with decreasing temperature.

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36 Figure 3-8. Logarithmic plot of the shape of the in-band conductivity (~ 2 B xx) as function of temperature in different strong magnetic fields, calculated from the experimental data. Using 0 we can also calculate th e in-band conductivity tensor: xx c c xxB 1 ~ 1 ~ ) ( ) ( 12 2 0 2 0 (3-5a) Bc c c xy1 ~ ) ( 10 2 0 (3-5b) Since in fixed magnetic fi eld, the relaxation time decreases with decreasing temperature, 1 ~xx will increase with decreasing temperature. The In-band Hall conductivity is independent of the relaxation time. 110100 5 10 15 20 25 17.5T 16T 14T 12T 10T 8T 6Txx/B2T (K)4T

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37 3.3 Possible Models in the Ultra-Quantum Regime: At this step, we have disc overed the transport behavior directly resulting from the physics of interactions in the graphite system in UQL field. Now we will try to understand the physics that causes this tran sport behavior: 1) the relaxation time decreases with decreasing temperature; 2) 1 ~xx increases with decreasing temperature. In this section, we will consider two possible models that give this kind of transport behavior: the magnetic field induced Luttinger liquid with impurity scattering, and phonon delocalization (dephasing). In the model of magnetic field induced Luttinger liquid with impurity scattering, the scattering mechanism considered is the elastic scattering of the impurities dressed with the Friedel potential,15 as illustrated in Figure 3-9. At zero temperature, the electrons are localized by dressed impurities along the di rection of the field lines. The potential of the dressed impurities is considered as a tunnel barrier. With increasing temperature, because of the increasing energy of the electr ons, the effective scattering cross section of the impurities will decrease, and the probabi lity for the electrons to tunnel trough the dressed impurity potential will increase. Th ese will lead to an increasing relaxation time (hence 0 ) with increasing temperature. The temperature dependence of conductivity along the field direction, LL zz, coincides with that of 0 With increasing conductivity along the field direct ion with increasing temperat ure, the con ductivity along the plane perpendicular to the field will decrease with increasing temperature: 1 ~LL xx. Since there is also classical magnetoresist ance in the x-y plane, the field induced

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38 Luttinger liquid behavior is a secondary correction to the semi-classical magnetotransport behavior: LL xx xx xx ) 0 (. Figure 3-9. Model of the field induced Luttinge r liquid with dressed impurity scattering. The model of the field induced Luttinger liquid with impurity scattering predicts that the correction of the conductance has power law temperature dependence, with magnetic field dependent powers: 15 ) (~ ~B LL zzzzT (along the field direction), (3-6a) ) (~ / 1 ~B LL xxxxT ( perpendicular to the field). (3-6b) A most important prediction from the model is the field dependent power factor. To check the field dependence of the experimental data, we plot scaled in-band conductivity c pz Friedel oscillations impurity

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39 o as function of temperature in different high magnetic field above the UQL. The conductivities are normalized at the lowest temperature point: Figure 3-10. Scaled in-band c onductivity as a function of te mperature in magnetic fields above the UQL indicated in the legend. It can be seen from the Figure 3-10 that for field well above the UQL, the scaled conductivities overlap perfectly. This indica tes that the magnetic field dependence and the temperature dependence of the conductivity can be separated: ) ( ) (0T G B F This is obviously contrary to the prediction from the magnetic field induced Luttinger liquid theory, in which the power factor of th e power law temperature dependence itself depends on the magnetic field. Another possible model is the phonon deloca lization (dephasing) model. In this model, the electrons are delocalized by phonon scattering. Hence with increasing

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40 temperature, the conductivity along the magne tic field increases, while the conductivity perpendicular to the magnetic field decr eases. The phonon delocalization mechanism predicts the same trend of the temp erature dependence of the conductance:16 zzTDP zz ~ ~ (along the field direction), (3-7a) xxTDP xx ~ / 1 ~ (perpendicular to the field). (3-7b) The phonon delocalization mechanism differs from the field induced Luttinger liquid by an exponent which is now independent of the ma gnetic field. So the phonon delocalization mechanism agrees better with the experimental data. However, as far as we know, there is no complete theory for phonon de localization, and ther e is no theoretical prediction of the values of the power factors. Further wo rk needs to be done to quantitatively understand the transport beha vior of graphite in the UQL region.

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41 CHAPTER 4 TUNNELING INTO BULK BISMUTH IN THE ULTRA-QUANTUM FIELD 4.1 Motivation In the previous chapters, we have disc ussed the magneto-trans port properties of semimetal graphite. Graphite is a relatively simple system for transport study in a sense that, in its hexagonal Brillouin zone, the Fermi surface comprises six cigar shaped pockets with their long axis parallel to the c axis. Transport in gr aphite is roughly 2-D due to the weak coupling between the graphene layers and large rati o of out-of-plane and in-plane effect mass. All these factors simplif y the transport study of graphite so that it can be treated as an isotropic 2-D system, in which only two major groups of carriers electrons and holes (each has one single mobility), need to be considered. Bismuth, on the other hand, is much more complicated. In the Brillouin zone of bismuth, there are 3 electron pockets and 1 hole pocket (see Figure 5-1 in chapter 5). None of the pockets is parallel to any of the others, hence Bismuth is 3-D in all orientations. And in most of the orientations, every pocke t contributes carriers with different mobility. Hence in bismuth, one need s to consider up to 4 majority bands, each with different effective mass and different mobility. This makes the magneto-transport study of bismuth very complicated. Rather than studying transport, we studi ed the magneto-tunneling properties of bismuth in strong magnetic fields. The origin al intent of our work was to study the possible high magnetic field induced 1-D (Luttinger Li quid) behavior. The tunnel junctions used in the study comp rise metal-insulator-semimetal trilayer structure. In zero

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42 field, we are simply tunneli ng from a 3-D metal into a 3D semimetal. Tunneling theory predicts that the measured differential conductance has a very weak dependence on the density of states of any of the electrodes and depends mainly on the properties of the tunnel barrier. When a strong magnetic field is applied, the energy levels in the semimetal separate into different Landa u bands. When the magnetic field is strong enough so that all the electrons in the semimetal are in the lo west Landau level, the semimetal enters the ultra-quantum regime, and the tunneling is between a 3-D Fermi liquid in the normal metal and Landau tubes in the semimetal. In this regime, the semimetal has an essentially 1D character with 1D Landau tubes aligned along the magnetic field and perpendicular to the tunnel junction area. For 1-D systems, one can no-longer desc ribe the physics using the Fermi liquid theory, because of the strong perturbation of Coulomb interaction due to the lack of screening and phase space for scattering. Inst ead the 1-D system will be described by the Luttinger liquid theory. The tunneling experime nt provides an opportunity to discover the enhanced density of states predic ted by the Luttinger liquid theory. When tunneling into a 1-D system, the tunneli ng theory predicts that the measured differential conductance across the junction has strong dependence on the density of states in the 1-D electrode. The propo sed magnetic field induced Luttinger Liquid theory15 states that: for magnetic field induced LL connected to 3D reservoirs by tunnel barriers: dI/dV Ta(B) (when e V << kBT ), (4-1a) dI/dV Va(B) (when e V >> kBT ). (4-1b)

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43 The reason that a semimetal is used as the electrode of interest is that, the magnetic field for the semimetals to reach the ultraquantum limit is relatively obtainable. For example, ~10 Tesla is needed to drive bism uth into the ultra-quant um regime. For most metals however, the magnetic field needed will be >104 Tesla, because of their high Fermi energies and large cyclotron masses. 4.2 Experimental Tunnel junctions are made on freshly cleaved bismuth crystals. A big piece of precut bismuth crystal is dropped into liquid nitrogen bath. After the bismuth crystal equilibrates with the liquid nitrogen, a Razor blade was us ed to cleave the crystal and expose a fresh and smooth surface of the tr igonal plane. The piece of cleaved bismuth with smooth surface was then taken out of th e liquid nitrogen bath and warmed up in pure nitrogen gas flow. Through this way, the cl eaved surface will maintain its freshness and there will be no condensation on th e surface when it is warming up. On the surface of bismuth, we have 2 me thods for making tunnel junctions. Figure 4-1 shows the procedure usi ng photolithography to make tunne l junction on top of the semimetal surface. First of all, we define an undercut photore sist pattern on top of the junction area we choose. Then a thick layer of AlOx was RF sputte r deposited onto the sample as separation layer. This layer pr events shorting outside the tunneling area resulting from the surface roughness, and from the force of the contact leads. Then we performed lift-off and opened up the tunneling area. Then we deposit the tunnel barrier, followed by the counter-electrode, through shad ow masks. Finally we pasted gold wire on top of the counter-electrode above the separation layer.

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44 Figure 4-1. Procedure for making tunne l junctions on bulk semimetal using photolithography technique.

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45 This method using photolithography is conve nient for defining the junction area. Using the precise alignment function of the phot omask aligner, it is easy to find the ideal smooth junction area and place the photoresist pattern on top of it. The shortcoming of this method is that after liftoff, there is always some residue of photoresist left on the surface. The residue can be mostly clean ed by prolonged treatment with UV ozone cleaner (~40 minutes). But the cleaning process can cause unwanted oxidation. Figure 4-2 shows another way we used to make junctions on the surface of semimetals. This is what we called mica mask method. Essentially it is a shadow mask method used in RF sputtering. Through conve ntional shadow masks, sputtering fails to yield sharp edges due to the high Ar pressure during the deposition. Here in the mica mask method, an extremely thin mica foil is pl aced on top of the semimetal. The sheet of mica will attach itself by Van der waals for ce to the semimetals surface. We can also attach another layer of mica on top of it to get an undercut. Then we RF sputter thick AlOx film as a separation layer. After rem oving the mica foils, we get a very sharp edge of the AlOx film, due to the intimate contac t of the mica mask to the semimetal surface. Then we deposit the tunneling barrier and th e counter-electrode, a nd finally put on the gold wires as measurement leads. The mica mask method, without using a ny lithography technique, is fast and convenient. Also, there is no contamination from the resi st polymers to the junction surface. The shortcoming of the mica mask method is that, w ithout lithography and precise alignment, it is relatively ha rd to obtain well de fined tunneling area. Figure 4-3 shows the microscope picture of a tunnel junction made on the surface of bulk bismuth.

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46 Figure 4-2. Procedure for making tunnel j unctions on bulk semimetal using mica mask method. Figure 4-3. Microscopic picture of a Bi(bulk)-A lOx-Pb tunnel junction.

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47 Measurements of the tunnel junctions are ca rried out in a He3 refrigerator with an 18 Tesla superconducting magnet in the Natio nal High Magnetic Field Labs (NHMFL). Differential conductance,dV dI /, is measured using a double lock-in amplifier technique, in which one lock-in amplifier is used together with a feedback circuit to keep the small AC (500Hz) excitation voltage, dV across the tunnel junction constant, while the other lock-in amplifier measures the AC voltage respon se of a standard resistor in series with the junction, from which dI can be calculated. A slow DC ramping signal is summed with the AC excitation voltage to apply the bi as voltage across the tunnel junction. 4.3 Results and discussion Figure 4-4 shows the magneto-tunneling re sult for a Bi(bulk)-AlOx-Pb tunnel junction made through mica mask method. Th e figure shows the differential tunneling conductance vs. bias voltage in different magnetic fields. The inset shows the Pb superconducting gap, from which we can s ee that the junction has low leakage and reasonable good quality. The differential conductance vs. bias vo ltage sweep shows an asymmetric V shaped background, resulting from the asymmetry in the energy dependence of the density of states in bismuth. On top of the background there are small features of oscillations. The oscillati ons, which are most pronounced in the -20 to 20 mV range, show no obvious magnetic field depe ndence and are characterized by 2nd derivative conductance 2 2d V I d to be mostly symmetric with the bias voltage. Hence they do not likely correspond to the density of states features in bismuth, but ratherto phonon excitations.

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48 Figure 4-4. Differential conducta nce as a function of bias voltage in indicated strong magnetic fields at 300mk. Inset: Pb superconducting gap feature in zero magnetic field at 300mK. Figure 4-5. Differential conducta nce at low bias voltage in the magnetic fields indicated in the legend. -80-60-40-20020406080 400 600 2T 4T 8T 14TdI/dV (Arb. unit)V (mV) -4-2024 0 1 2 dI/dV (Arb. unit)V (mV)-3-2-1012 3 2T 4T 8T 14TdI/dV (Arb. unit)V (mV)

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49 Figure 4-5 shows the differential conductan ce at near zero bias voltage. In zero magnetic field, the differential conductance s hows a peak at zero bias. As the magnetic field increases, the peak at zero bias sepa rates into two peaks, which move towards higher bias voltages and leaves a valley at zero bias. The change of dI/dV with magnetic field appears to be as if the field opens up a gap at the Fermi level. The feature observed at low bias voltages can be expl ained by considering the Zeeman splitting of the spins. Since Tesla eVB/ 10 8 5 ~5, the g factor corresponds to our experimentally observed splitting (e.g., ~1 .6mV at 14 Tesla) is ~2. Another possible mechanism for the low bias voltage feature is the field induced Luttinger liquid behavior. In this picture, the strong magnetic field opens up a Coulomb gap at the Fermi energy. The Luttinger liquid behavior enhances the density of states in the gap. Detailed analysis requires knowledge of the classical background of the differential conductance. Tilting of magnetic field is required to tell if the zero bias feature is a spin effect or an orbital effect. Except for the features at bias voltage mV V 3 a major feature of the differential conductance show in Figure 4-4 is the lack of field dependence. The curves we took at 2, 4, 8 and 14 tesla overlap almost perfectly. Th is result contradicts the prediction from the field induced Luttinger liquid theory, in which the magnetic field dependence enters the power factors of the power law dependence. There are also some concerns about our measurements. For example, the surface stat es in bismuth might prevent the tunneling measurements from probing the intrinsi c properties of bismuth single crystal.

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50 CHAPTER 5 ACHIEVING LARGE MAGNETORESISTANCE IN BISMUTH THIN FILMS 5.1 Introduction Semimetal bismuth has been interest of study for many years, because of it many special properties. Figure 5-1 shows the Fermi surface a nd the brillouin zone of rombohedral bismuth. The highly anisotropic Fermi surface consists of tiny hole pockets and electron pockets, which occupy only a fe w thousandth of the volume of the Brillouin zone. Hence bismuth has very low carrier density (~1023 m-3) and low Fermi energy (~25meV). Also because of the small Fe rmi momentum, the chance of phonon scattering is very low. Hence bismuth has an ex tremely long phonon mean free path at low temperatures (~mm at 4.2K). Figure 5-1. Fermi surface and Brillo uin zone of rom bohedral bismuth

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51 These unusual properties of bulk single cr ystal bismuth give rise to a huge magnetoresistance.17-19 Figure 5-2(a) shows the magne tic field dependence of the resistivity at 5K for single crystal (99.9995% pure) bulk bism uth. At the 7 Tesla, the resistivity is 6 orders of ma gnitude higher than the zero fiel d resistivity. Figure 5-2 (b) shows the temperature dependence of resist ivity in zero magnetic field. Bulk single crystal bismuth is metallic, with resistivity decreasing with decreasing temperature. Figure 5-2. (a): Magnetic fiel d dependence of the reisistivity at 5K for bulk single crystal bismuth; (b) resistivity as function of temperature in zero magnetic field. The extremely large magneto-resistance ma kes bismuth a promising candidate for applications, such as magnetic field sensors. Many efforts have been carried out in order to make bismuth thin films that have quality comparable to the bulk material.20-22 However, it was found that bismuth thin films made by normal technique, such as thermal evaporation, yield bismuth film s with very small magnetoresistance.23, 24 These 050100150200250300 0.0 2.0x10-74.0x10-76.0x10-78.0x10-7 Resistivity ( m)T (K) -8-6-4-202468 0.000 0.005 0.010 0.015 R es i s ti v it y ( m ) B (T)

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52 films may even behave in a non-metallic manner with the resist ance increasing with decreasing temperature. Figure 5-3. (a): Magnetic fiel d dependence of the reisistivity at 5K for a 1.5um bismuth film thermally evaporated onto glass subs trate; (b) resistiv ity as function of temperature in zero magnetic field. Figure 5-3 shows the typical transport behavior of thermally evaporated bismuth thin films on glass substrates (thickness between 800~10000A). The resistance generally increases with decreasing temperature. And th e magnetoresistance at low temperatures is much lower (MR(7T)<10) than that of the bulk bismuth (MR(7T)~105). The major reason for the differences betw een bulk single crystal bismuth and the bismuth films is the small grain size in the films. The gains, generally with size of ~1000A, are actually not small co mpare to that of normal metals. However, bismuth has a very long phonon mean free path, due to th e small Fermi momentum. Application of the Matthiessens rule shows that the scattering in the films is dominated by temperature independent gain boundary scatte ring (except when the temperature is very high, e.g. higher than room temperature, and the phonon m ean free path is shorter than the grain -8-6-4-202468 1x10-52x10-53x10-54x10-5 R es i s ti v it y ( m ) B (T)0100200300 3x10-66x10-69x10-61x10-5 Resistivity ( m)T (K )

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53 size). The temperature dependence of the resis tivity is mainly from that of the carrier concentration, which, due to the small Ferm i energy of bismuth, decreases significantly as the temperature drops. Magnetoresistance in bismuth films is al so limited by grain boundary scattering. This can be illustrated from a simplified 2-band model, in which the electrons and holes are compensated, and the resistivity in the magnetic field simply goes like: 2 2 2 2) ( ~ ) 0 ( ~ ) 0 ( ) 0 ( ) ( c HH R H (5.1) Hence the small relaxation time due to gr ain boundaries scatte ring leads to small magnetoresistance. To make high quality bismuth thin films, it is necessary to make the grain sizes large. A judicious combination of lattice-matc hed substrates and caref ully regulated postdeposition thermal annealing provides a strate gy for growing bismut h films with large grains. In early work on bismuth films thermally deposited onto mica substrates,23 it was found that post deposition annealing close to the bismuth melting temperature caused the helium temperature resistance to decrease by a factor of 15 when compared with unannealed films. In addition, MR for fiel ds perpendicular to the film surface is significantly improved with a nnealing. Epitaxial films of bismuth having a trigonal orientation have been grown on BaF2(111) (3.6% lattice mismatch)25 and CdTe(111) (0.7% lattice mismatch). 22 In the latter case, post-deposi tion annealing at 3 C below the melting temperature of Bi lead to significant increases in the MR. An alternative approach, which has been found to give large MR in Bi films 1-20 m thick, is the technique of electro-chemi cal deposition from aqueous solutions of Bi(NO3)3 5H2O .20, 21 An underlying Au layer, patterned onto a silicon substrate, serves as

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54 the working electrode for the electrodeposition. As is the case for vacuum-deposited Bi films, 22, 23 post-deposition annealing of the electrodeposited films close to the melting temperature of Bi leads to a small resistivity and a large increase in MR (2.5 at room temperature and 3800 at 5K for the thickest 20um film in a perpendicular 5T magnetic field 20). For technological applications, elec trodeposition is economical and well suited for large-scale production. Similar advantag e would likewise hold fo r thermal deposition, provided ultrahigh vacuum and specialized gr oeth techniques, such as MBE, are not required. We studied the thermally deposited bismut h films on pre-deposite d gold thin films followed by post-annealing processes. We find that, upon annealing, the Au from the Au underlayer rapidly diffuses into th e bismuth, giving rise to a f ilm with large-crystal grains oriented with trigonal axis perpendicu lar to the plane of the film and having magnetotransport properties comparable to those grown by el ectro-depositions.20, 21 We show that improvements of MR are only for annealing temperature higher than the 241 C eutectic temperature of the BiAu so lid solution and below the 271 C melting temperature of Bi. This 30 C annealing window provides considerable latitude when compared to the narrow annealing window of a few C confirmed here and reported previously for pure Bi films.23 5.2 Experimental All of our samples are prepared by th ermally evaporating 99.999% pure Bi onto pre-cleaned glass substrates at 5E-7 torr base pressure. In the cases where heated substrates are needed, the substrates are glued onto a variac controlled heater with silver paste. Then the shadow mask is glued onto th e substrates using the same silver paste.

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55 Substrate temperature is read from a thermometer, and is manually controlled by adjusting the output vo ltage of the variac. Three categories of samples are prepared: (I) two pure bismuth films (1 um thick) grown at 150 C, followed by annealing at 265C and 270 C for 6 hours; (II) three bismuth films (1 um thick) grown simultaneous ly on pre-deposited gold films (350 ) at 150 C, followed by annealing at 238 C, 243C and 251 C, respectively for 6 hours; (III) bismuth films (1 um) grown on pre-deposited gold films (350) at room temperature, followed by annealing at 251 C for 6 hours. A nnealing is performed in a quartz vacuum tube furnace with temperature calibrated with respect to the observed melting of a small bismuth crystal placed in close proximity to the samples. Measurements of resistance vs. temperature at different magnetic fields are carried out in a Quantum Design Physical Property Me asurement System (PPMS). In all of the measurements, the magnetic field is applied perpendicular to the film. 5.3 Results and Discussion We characterized the crystal structure of the Bi/Au films by X-ray diffraction. Figure 5-4 shows the X-ray diffraction pattern of bismuth (00l) planes. The sharp lines indicate that the film is well c-axis oriented. The inset of Figure 5-4 shows a schematic of the relevant portion of th e Bi(Au) phase diagram.26 A small amount of gold in Bismuth reduces the melting point, and the lowest melti ng temperature, the eutectic temperature, occurs at 241 C for the Bi.868Au.132 compostion. In our experiment, the mass ratio of the Bi and Au is controlled by the thickness of the 2 films. Thus a pre-deposited 360--thick Au layer mixed by annealing into a 1-um-thi ck Bi film represents a solid solution (vertical dashed line) with stoichiometry Bi0.93Au0.07. All of the Bi/Au films here are at this composition.

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56 Figure 5-4. X-ray diffraction pa ttern for a 4-um-thick Bi/A u film grown at 150 C and annealed at 251C. Inset: the relevant portion of the Bi/Au phase diagram and corresponding annealing temp eratures (indicated by the crosses) for Bi and Bi/Au films discussed in this chapter. Figure 5-5 shows the resistance vs. temp erature at 0 and 5 Tesla for the two category-I pure bismuth films (no Au unde rlayer) annealed at 265 C and 270 C, respectively. We note that a small difference of annealing temperature at close to the 271 C melting point of bismuth produces a drastic change of the properties of the films. The film annealed at 270 C just starts to melt a nd is recrystalized during the slow cool-down. As observed through the quartz tube, the film develops a shiny surface just below the melting temperature, but at higher temperat ure begins to fully melt and ball up. The positive slope in the resistance-temperature cu rve in zero magnetic field indicates that the film is metallic. Also, at 5T, the MR=286 at 5K indicates the good quality of the film. 020406080100120 0 1000 2000 3000 4000 5000 Intensity2*theta Bi_Au (00l)

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57 Figure 5-5. Temperature dependence of the resi stivity at 0 and 5T for two category-I Bi films In contrast, the film annealed at 265 C does not change its appearance during the annealing process. The resistance of this f ilm shows a characteristic minimum at near 200K 23 and then increases as the temperature is further lowered. In addition, the MR of this film is much smaller than the on e annealed at 270 C through out the whole temperature range. These results are in accord with previous studies,22, 23 which have shown that post-annealing at melting point fo llowed by re-crystallization is an effective way to get high quality bismuth films. Howeve r, the temperature control must be accurate to a few C and must not be allowed to go above the melting point where there will be a loss of film adhesion leading to agglomer ation and discontinu ity between grains. 050100150200250300 1 10 100 B B A A B = 0T B = 5TSheet Resistance ()Temperature (K) A) TG = 20C, TA = 265C B) TG = 20C, TA = 270C

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58 Figure 5-6. Temperature dependence of the resi stivity at 0 and 5T for three category-II films grown simultaneously on 150 C substrates and them annealed separately at respective temp eratures of 238, 243 and 251 C. For samples in category-II, the presence of a gold underlayer l eads to completely different behavior. Figure 5-6 shows the effect of annealing temperature on the quality of these films. The three Bi(1um)-Au(350) films are grown at 150 C, and then annealed at 238 C, 243C and 251C respectively, as indicat ed by the crosses in the Fig.5-4 inset. Prior to each post-deposition anneal, a gol d color can be observed from the back side of each glass substrate. Afer 243 C a nd 251 C anneal, the gold color is gone and the underside of each Bi/Au film is silver color and indistinguishable from the underside of a pure Bi film. These color changes indicate th at during the annealing, the gold atoms no longer remain segregated beneath the bismuth film but diffuse into the bismuth. For an annealing temperature of 238 C, which is belo w the eutectic temperature of 241 C, all of the film remains in the solid form, and th e surface texture of the film does not change 0100200300 1 10 100 B = 0T B = 5T C C B B A A A) TG = 150C, TA = 238C B) TG = 150C, TA = 243C C) TG = 150C, TA = 251CSheet Resistance ()Temperature (K)

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59 during the anneal. In addition, the temperat ure dependent resistance is nonmetallic and the 5K MR is low (MR=37). In contrast for the two anneals above the eutectic temperature, the films undergo a definite cha nge in appearance in which they become shiny and remain metallic after cooldown, th e temperature-dependant resistance becomes progressively more metallic, and both f ilms exhibit significantly larger MR. MR(5K)=130 for the 243 C anneal and MR (5K)=327 for the 251 C anneal. We note that the MR of our 251 C a nnealed Bi/Au film is highe r than the MR(5K)=250 of a comparable 1-um-thick single-crys tal film grown by electrodeposition.20, 21 Further increase of the annealing temperature to s lightly above 160 C but well below the 271 C Bi melting temperature leads to sever melti ng and loss of electrical connectivity, as would be expected from the in tersection of the vert ical dashed line with the solid/liquid phase boundary shown in the Figure 5-4 inset. The plots in Figure 5-7 for the category -III films show the effect of growth temperature on transport properties for the sa me anneal conditions. The film grown at 150 C shows metallic temperature dependen ce at zero field and has MR(5K)=327. The film grown at room temperature however, shows rather complicated temperatureresistance dependence. The resistance drops a little bit as the te mperature sweep from 300K to about 200K, then increase a lot fr om 200K to 5K. The resistance-temperature curve suggests that the film has grain sizes comparable to phonon mean free path at 200~300K. At low temperatures, the grain boundary scattering dominates and the resistance increases due to decrease of ca rrier concentration. The magnetoresistance of the room temperature grown film, MR(5K) = 34, is also significantly lower than the film grown at 150 C. For pure Bi films grown on CdTe substrates, growth temperature in the

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60 range of 80 C to 220 C are requi red to obtain epitaxial behavior.22 For our Bi/Au films, the higher growth temperature promotes the grow th of larger grains, thus facilitating the effectiveness of the annealing procedure by star ting with larger grains We also note, as show by x-ray diffraction pattern in Figure 54 for a 4-um-thick Bi/Au film grown at 150 C and annealed at 251 C, that the anneal ed films exhibit a pronounced single-crystal orientation with trigonal axis oriented perpendicular to the film plane. Similar behavior has been noted for annealed electrodposited films. Figure 5-7. Temperature dependence of the re sistivity at 0 and 5T for two category-III films Optical microscopy verifies a smoother t opography and larger grain size (1-10um) for the films annealed at high temperature and exhibiting a large MR. This result is consistent with the aforementioned conclusions that large grain size achieved either by epitaxy and/or annealing is a pr erequisite for large MR. The primary factors that affect 0100200300 1 10 100 B = 5T B = 0T B B A ASheet Resistance ()Temperature (K) A) TG = 150C, TA = 252C B) TG = 20C, TA = 252C

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61 the quality of Bi/Au films are the growth te mperature and the annealing temperature. A moderate growth temperature (~150 C) encour ages the formation of large grains, but should not be so high as to cause the film to agglomerate and to become discontinuous. Table 5-1. Summary of re sults for different bismuth film growth conditions Sample Growth temperature Annealing temperature Metallic MR (5K : 300K) 265 C N 39 : 3 Pure Bi 150 C 270 C Y 283 : 4 Room temp. 251 C N 34 : 3 238 C N 37 : 3 243 C N 130 : 3 Bi (1um)Au(350) 150 C 251 C Y 327 : 3 We summarize our results in Table 1.The e ffect of the diffusion of the Au into Bi during the post-deposition annealing proce ss can be qualitatively understood by referring to the phase diagram depicted in the Figure 5-4 inset. If equ ilibrium is assumed, then for isothermal (tie line) drawn at a given anneal ing temperature, applic ation of the lever rule for binary phase diagram will determine a gold-rich melted phase and a bismuthrich solid (unmelted) phase. It is the presen ce of this melted phase that facilitates grain boundary migration and grain growth resulting in the high MR that we have observed. We suspect that this melted phase is most likely associated with grain boundaries although detail microcompositional analysis w ould be necessary to verify such a scenario. In oversimplified terms, the Au can be t hought as a lubricant that facilitates the growth of large grains during the post-depos ition anneal. However, one should not forget that Au is a impurity that gives rise to in creased carrier scattering and associated lower MR, thereby preventing the MR from appro aching the hig values reported in single crystals.17, 19 Accordingly, the use of annealed Bi/A u bilayers to obtain large MR requires

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62 judicious balance between using enough gold to assure large grain growth, but not using too much gold that additional scattering co mpromises that MR. We believe that these considerations also apply to the Bi/Au films deposited by electrodeposition technique reported previously by Yang et al. 20, 21

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63 CHAPTER 6 METALLIC SURFACE STATES IN ULTRA-THIN BISMUTH FILMS 6.1 Introduction: Physics of th e Ultra-Thin Bismuth Films Bismuth, like many other semimetals, has a very long electron wavelength, due to its low Fermi energy (~25 meV) and small effective mass (~0.005me). One can estimate the wavelength to be: 2 *10 ~ 2 ~f eE m h When the size of sample is comparable or smaller than this length scale, one will need to consider the effect of the sample boundary on the band structure. This is where the so-c alled quantum size effect become important. Bismuth provides great convenience in studying the quantum size effect in many aspects. Ultra thin bismuth films, with their thicknesses comparable to the electron wave length (~ 300 ), have been of great interest in the study of the qua ntum size effect and semimetal-semiconductor transition. Ogrin, Lutski, and Elinson, in their study of the magnetotransport of the bismuth thin films in 1965,27 produced the first clear experimental evidence for quantum size effect in any solids. Oscillat ory behaviors in both resistivity and Hall coefficient were observed with decreasing film thickness, due to the quasi-2D sub-bands passing across the Fermi le vel. Since then, quantum size effect in bismuth thin films has been intensively studied both theoretically and experimentally.28-34 The existence of the thickness dependent quasi-2D sub-bands resulting from quantum confinement has been generally accepted. One important prediction as a result of the quantum size effect is the so-called semimetal-to-semiconductor (SMSC) transition The SMSC transition happens when the

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64 energy shift due to the quantum confinemen t becomes large enough so that the lowest electron sub-band rises above the top of the hi ghest hole sub-band, due to their difference band masses. The critical thickness of the thin film for the transition to happen, given by most of the theoretical calcul ation, is between 230 ~340 .28, 35, 36 Figure 6-1. Illustration of semimetal-to-semicinductor transition. Despite the numerous experimental investig ations carried out to look for the SMSC transition,27, 28, 31, 37-40 the existence of the SMSC tr ansition remains ambiguous. Chu and co-workers argued against the SMSC transi tion, and proposed theory that the boundary condition for the electron wave function is that the gradient of the wave function (rather than the wave function itself) vanish at the sample boundary. Hence the ground state electron and hole energies depend only weakly on the thickness of the films, and the conduction band and the valence band remains overlaped. The arguments against the SMSC transiti on mainly focused on the lack of sharp transitions in transport properties (resistivit y, Hall coefficient, magnetoresistance).

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65 Hoffman et al., in their work that stand for the SMSC transition, pointed out that the main reason for the absence of the sharp transition in the previous works is that people failed to take into account the effect of the surface carrier and surface conduc tivity, which may be important and even dominating when the bismuth films are very thin.39 The simple model that takes into account the surf ace carrier is that the surface ac ts like a high carrier density conductor in parallel with the bulk part of th e film, and the averaged carrier concentration is simply:d n n ns i/ where inand sn are bulk intrinsic carr ier density and surface sheet carrier density.38 When the film is thin, ns will dominate, and the effect of the surface conductivity has to be seriously considered. Further evidences of metallic surface states were found in a very different bismuth system. In 1991, B. Weitzel and H. Mickl itz discovered superconductivity in granular systems built from rhom bohedral Bi clusters.41 They explained their result as surface superconductivity due to the strongly increase d surface density of states and suggested photoelectron spectroscopic study on bismuth surfaces to further confirm their proposal. Angle resolved photoemission spectroscopy (A RPES) since then had been a major tool people used to probe th e surfaces of bismuth. The experiments were carried out by several groups.42-47 Consistent results were obtained, indicating the existence of metallic surface states in bismuth (111) and (110) surf aces. Christian R. Ast and Hartmut Horchst reported in 2001 a surface carrier density associ ated with the surface st ate of Bi (111) to have sheet densities of 2 1310 1 1 cm psfor holes and 2 1210 5 5 cm ns for electrons.47 In 2003, Gayone et. al reported their study on the temperature dependence of the surface states linewidth and the str ong energy dependence of the electron-phonon coupling strength on Bi (100) surface.48

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66 6.2 Transport Properties of th e Ultra-Thin Bismuth Films 6.2.1 Experimental Bismuth films are thermally evaporated from 99.9999% pure bulk bismuth in a high vacuum chamber of ~6E-7 torr at a rate of 1~2 /sec, through shadow masks. In the cases where in situ measurements are required, bismuth or gold contact pads are thermally evaporated through shadow masks. Then the substrate pre-deposited with contact pads is fixed onto a special designed sample holder, where the Hall-bar shadow mask is installed aligned with the contact pads, and the gold wires are attached to the contacts and connected to leads which enable electrical measurements from outside the vacuum chamber. Tunnel junctions on thin bismuth films we re made using standard cross stripe geometry. Mica is used as substrate and bismuth is used for base electrodes, so that lattice match between bismuth and mica can be achieved. AlOx is used for tunnel barriers. Lead is used for top electrodes, so that when th e samples are cooled down to below the lead superconducting temperature, the superconductin g gap can be used to characterize the quality of the junction. In making of a Bi-AlOxPb junction, a thin bismuth film stripe as base electrode is deposited onto mica substrate through a sha dow mask. The film is then taken out of the vacuum chamber, and the sh adow mask is removed. The film is then immediately put back into vacuum, and the aluminum oxide tunnel barrier (~ 10 ) is coated through thermal evaporation of aluminum in the oxygen pressure of 2E-5 torr, at a rate of about 1 /sec. A cross stripe of lead as counter electrode is then deposited through a shadow mask. Typical working j unction resistances are in the range of 10~10000

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67 6.2.2 Metallic Surface States In the study of the Bi/Au f ilms, the thicknesses of thes e films are in the order micrometers. Films with these thicknesses can be treated as bulk polycrystal bismuth, in a sense that theres no band structure change due to the quantum confinement, and the surface effect is not significant. When the (pure) bismuth films get really thin (e.g., thickness ~ Fermi wavelength), quantum size effect will take place, and the effect of the surface states needs to be seriously considered. Figure 6-2 shows the temperature depende nce of resistivity, for films with thicknesses indicated in the legend. 050100150200250300 5.0x10-66.0x10-67.0x10-68.0x10-69.0x10-61.0x10-5 (m)T (K)150310400 Figure 6-2. Resistivity vs. temperature for Bi film with indicated thicknesses. It can be seen from the figure that, for films with thickness ~400, the resistivity increases as the temperature drops, and r eaches a maximum at ~40K. As the film becomes thinner, the temperature for the resistivity maximum shifts higher. Also the

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68 resistivity in the low temper ature region (say, T <150K) decr eases with decreasing film thickness. For the films with thickness < 150 the resistivity drops monotonically from room temperature with decreasing temper ature, showing metallic behavior. The metallic behavior (positive R-T slope) can be explained by considering the existence of the metallic surface states. A simp lified model is to treat the whole film as two separate films in parallel: a very thin me tallic-like film on the surface with sheet resistance Rs and thickness ts and the intrins ic film underneath it with resistivity i and thickness st t. The measured resistivity for is then t t t Ri s s 1) ( 1 The resistivity of the bulk (i ntrinsic) part of the film,i has negative R-T slope. And when the film is thick, it has low re sistance and hence will dominate the total resistance of the film. When the film gets thin, the contribution of the metallic surface becomes increasingly important, and the R-T curve starts to show a maximum, which moves to higher and higher temperature with decreasing film thickness. Eventually, when the film thickness reaches 150A or thi nner, the surface states will dominate the temperature dependence and the R-T cu rve shows positive slope throughout the temperature range of measurements (4.5K~300K). The magneto-transport of ultra-thin bism uth films is studied under the framework of classical muti-band model, as described in the previous chapters. The classical magnetoresistance of the ultra thin bismut h film can be roughly estimated to be ~2) ( c. Here, the small thickness of the films leads to very short mean free path of grain boundary scattering, and hence a very small MR.

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69 -6-4-20246 0.000 0.005 0.010 0.015 0.020 MRB (T) 310180 Figure 6-3. Magnetoresistance vs. magnetic fiel d at 5K for two Bi films with indicated thicknesses. Figure 6-3 shows the MR of two bismuth films with thicknesses 180 and 310 It can be seen from the figure that the classical MR increases with the film thickness, due to the increasing grain size with the film thickne ss. The sharp dip at low fields can not be explained by classical theory, and is due to anti-localization, orig inating from the strong spin-orbit interaction in bismuth. The in-balance or non-compensa tion of the positive and nega tive carriers in the thin bismuth films is revealed by the field de pendence of the Hall resistivity. Figure 6-3 shows the field dependence of Hall resistivit y at indicated temperatures for 2 bismuth thin films, 180 and 400 thick. For both film s, the measured Hall resistivities are not linear with the magnetic field. Also the zero field slope of the Hall resistivity has strong temperature dependence. For the 180 film, the low field xy vs. field curve even changes sign from 5K to 150K.

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70 02468 -4x10-8-2x10-80 2x10-84x10-86x10-88x10-8 xy (m)B (T)150K 75K 5K 02468 0 2x10-74x10-76x10-7 150K 75Kxy (m)B (T)5K Figure 6-4. Hall Resistivity vs magnetic field at indicated temperatures for (a) 180 and (b) 400 Bi films The Hall resistivity xy observed can be qualitati vely understood through the simple 2-band model expression, (b) 400 (a) 180

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71 2 2 2 1 2 2 1 2 1 2 2 2 1 3 2 1 2 1) ( ) ( B R R B R R B R R R Rxy (6-1) We can see that, for a 2-band system with electron band and hole band, the Hall resistivity is in general not linear with the ma gnetic field. Also the Hall resistivity itself does not give enough information about the in-b and carrier concentration of each band. From Equation 6-1, we get the low field a nd high field limit of the Hall resistivity: 2 2 1 2 1 2 2 2 1) ( ) ( ) 0 ( B R R Hxy (6-2a) 2 1 2 1) ( R R B R R Hxy (6-2b) Hence the zero field Hall resistance slope by itself does not give any information on the carrier density of the films. In fact it does not even give the information about whether the film is n-type or p-type, due to the complication from the in-band resistivity (or mobility). However, the high-field limit of the Hall resistance does indicate the carrier type of the film (or, the sign of 2 1R R ). The change of slope (even the sign of the slope) from low field to high field gives rise to the curvature observed in the Hall resistivity measurement. One can see from the xy vs. field curves that, even though the low field data shows strong temperature dependence (e ven change of sign), the sign of the extrapolated high field limit of the Hall resi stivity slope is temperature and thickness independent, indicating the type of the films, n-type in this case, does not change with temperature, nor the film thickness. A more detailed analys is of the Hall resistivity data yields information about the un-balance of the carriers, defi ned by the compensation factor: h e h en n n n And the

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72 thickness and temperature depe ndence of the Hall resistivity can be qualitatively understood by considering the change of compen sation factor (due to the n-type surface states) with temperature and thickness. To simulate the field dependence of the Hall resistivity, we adopt a 4-band model, with a bulk electron band, a bulk hole band, a surface electron band, and a surface hole band. Figure 6-5 shows the simulation results for the magnetic field dependence of the Ha ll resistivity for 180 thick and 400 thick bismuth films. In the simulation, we assume that the mobility of the carriers does not change with temperature, due to the fact that the grain boundary scattering dominates over phonon scattering. Because of the compli cation of the energy band quantization due to the quantum size effect, we can not calcula te in detail the temperature dependence of the carrier density. In the simulation, we assume different values of carrier density for the bulk part of the films, and assume the sheet surface carrier density to be temperature and thickness independent. The parameters used in the simulations are listed below: Table 6-1. Parameters for the simulating th e effect of thickness and temperature on the magnetic field dependence of the Hall resistivity in ultra-thin Bi films Surface (10) In the film Carrier density ns (m-2) Mobility s (1 1 2 s V m) Carrier density ni (m-3) Mobility s (1 1 2 s V m) e1710 2 03 01 ) 10 5 (23 a 0.119 400 h1710 4 1 0.036 ) 10 5 (23 a 0.138 e1710 2 03 01 ) 10 5 (23 a 0.056 180 h1710 4 1 0.036 ) 10 5 (23 a 0.063 Note from the listed parameters that: 1) the surface sheet carrier density and mobility are the same for both films; 2) in bulk (intrinsic ) part of the films, the carrier density of electrons is equal to that of the holes; 3) since we cannot ca lculate the carrier density in the films, we modulate its number by adjusting the parameter a

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73 012345678 -6x10-8-4x10-8-2x10-80 2x10-8 a=1.5 a=0.8 a=0.3Hall Resistivity ( m)B ( T ) a=0.01 012345678 -1x10-70 1x10-72x10-73x10-7 Hall Resistivity ( m)B (T)a=1.5 a=0.8 a=0.3 a=0.01 Figure 6-5. Simulated Hall Resistivity vs. magne tic field at indicated temperatures for (a) 180 and (b) 400 Bi films, with fitti ng parameters described in the text. (a) 180 (b) 400

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74 Comparing the calculated magnetic field de pendence of the Hall resistivity with the data, we see that the simulations yield the major features in the experimental results. From the simulations, we get the physical picture about the carriers in the ultra-thin films. The bulk part of the film is compensated, like in bulk bismuth. The surface of the film, however, has a high sheet carrier density and is uncompensated. As the film gets thinner, or the temperature gets lower, the number of compensated carriers in the bulk part of the film decreases. Hence the degree of un-compen sation, due to the existence of the uncompensated surface carrier, will increase. 6.3 Control of the Surface States All the bismuth films discussed in the prev ious section are measured after removal from the vacuum chamber. Even though the oxi dation of bismuth at room temperature is insignificant, we will still need to consider the effect of oxygen on the surface of the film. For comparison, we have carried out in-situ measurements on bismuth thin films. A thin bismuth film is deposited onto a mica s ubstrate pre-deposited with contact pads, and measured without breaking vacuum. The substr ate is mounted on a cold stage and cooled down from room temperature to ~100K, and th e resistance vs. temperature is recorded. The sample is then warmed up to room temperature, and a small amount of oxygen is introduced into the chamber for 10 minutes. The chamber is then evacuated, and the sample is cooled down again to 100 K, w ith resistance vs. temperature recorded. The in-situ measurement of freshly deposited bi smuth films shows that for ultrathin bismuth films measured in vacuum, the resistance increases with decreasing temperature. What is different for the ultrathin bismuth films from the thicker bismuth films (~um) is that, the ratio of the resistance in crease, say, R(100K)/R(300K), is much smaller in the ultra-thin bismuth films (<10% ) than in the thicker films (~200%). Hence

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75 the existence of the metallic surface states is intrinsic, wh ile the sheet carrier density originated from the surface states is ve ry sensitive to the surface condition. We have seen that oxygen has a significant effect on the surface states. To study the surface terminated ultra-thin bismuth film, we coated the bismuth f ilms with Ge. Ge is known to be a material that, when deposited as thin films, creates dangling bounds and nucleation sites. When thin metals films are deposited onto predeposited atomically smooth Ge thin films, the metal film grow th nucleates at the Ge dangling bounds. Thus the metal films tend to be very smooth. Here we deposit a few monolayers to Ge ri ght after the deposition of bismuth thin film, without breaking vacuum. The idea is that the dangling bounds of the Ge film may bind with the surface states in the bismuth f ilm, and terminate the surface of the bismuth film from being affected by the air. Figure 6-6. Temperature dependence of re sistivity for Bi(100) and Bi(100)/Ge. Figure 6-6 shows the resistance vs. temper ature curves of two 100 thick bismuth films simultaneously grown and measured. Th e only difference betw een the 2 films is 050100150200250300 1.2x1031.3x1031.4x103 Bi (100) Bi (100)/Ge (8)Resistance ( )T(K)

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76 that, one sample is coated with a few angstrom of Ge in situ right after the deposition of bismuth film. The two samples show comple tely different temperature dependence of resistance. The bare bismuth film shows pos itive resistance-temperature slope, while Ge coated film shows negative resistance-temperature slope. We also coated the bare bismuth film with Ge after its taken out of vacuum. No significant change of transport behavior was observed. We conclude that the change happens at the Bi-Ge interface, rather than in the Ge film itself. The Hall resistivity measurements provide more information on the in-balance of the carriers. From Figure 6-7, we can read ily see the big differe nces in the carrier distribution between the bare and the Ge co ated bismuth films. Comparison with Figure 6-4 reveals that the magnetic field dependen ce of the Hall resistivity for the bare 100 Bi film at 75K, 150K and 250K resembles that of the bare 180 Bi film at 5K, 75K and 150K. And the magnetic field dependence of the Hall resistivity for the Ge coated 100 Bi film at 75K, 150K and 250K resembles that of the bare 400 Bi film at 5K, 75K and 150K (the smaller curvature here is due to th e smaller mobility in the thinner films). This comparison suggests that the Ge coated Bi film has better compensation than the bare Bi film with the same thickness. Simulations results with a 4-band model de scribed earlier are s hown in Figure 6-8, with fitting parameters listed in table 6-2. Note from the listed parameters that: 1) the effect of the surface carrier is adjusted by sett ing the thickness of the surface layer; 2) in bulk (intrinsic) part of the film s, the carrier density of electrons is equal to that of the holes; 3) The carrier density of the bulk (i ntrinsic) part of the film is modulated by adjusting parameter a.

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77 012345678 -2x10-8-1x10-80 1x10-82x10-83x10-8 250K 150KHall Resistivity ( m)B (T)75K 012345678 0.0 5.0x10-81.0x10-71.5x10-7 250K 150K 75KHall Resistivity ( m)B (T) Figure 6-7. Hall Resistivity vs. magnetic field at indicated te mperatures for (a) Bi(100) and (b) Bi(100)/Ge films. (a) (b)

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78 012345678 -2x10-8-1x10-80 1x10-8 Hall Resistivity ( m)B (T)a=4.0 a=1.5 a=0.8 012345678 0 5x10-81x10-72x10-7 a=4.0 a=1.5Hall Resistivity ( m)B (T)a=0.8 Figure 6-8. Simulated Hall Resistivity vs. magne tic field at indicated temperatures for (a) Bi(100) and (b) Bi(100)/G e films, with parameters described in the text. (b) (a)

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79 Table 6-2. Parameters for the simulating th e effect of Ge coating on the magnetic field dependence of the Hall resistivity in ultra-thin Bi films Surface (3 for Bi/Ge, 10 for Bi) In the film (100) Carrier density ns (m-2) Mobility s (1 1 2 s V m ) Carrier density ni (m-3) Mobility s (1 1 2 s V m ) e 1710 2 03 01 ) 10 5 (23 a 0.047 Bi/G e h 1710 4 1 0.036 ) 10 5 (23 a 0.054 e 1710 2 03 01 ) 10 5 (23 a 0.047 Bi h 1710 4 1 0.036 ) 10 5 (23 a 0.054 The simulations suggest that, the effect of the Ge layer on the Hall resistivity is equivalent to reducing the density of the surface carriers. It is also due to the reduction or neutralization of the metallic su rface states so that th e carriers in the bulk part of the films again dominate the transport and give rise to the negative resistance-temperature slope shown in Figure 6-6. The mechanism through which exposure of the bismuth film to th e air increases the surface sheet carrier density is still not known. However we believe that results obtained from the Ge coated bismuth ultra-thin films opens the possibility of passivating the surface and even neutralizing surface states These results should be important for studying the nanoscopic bismuth systems, such as bismuth nanowires, in which the effect of the surface state becomes very significant.

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80 CHAPTER 7 SURFACE SUPERCONDUCTIVITY IN ULTRA-THIN BISMUTH FILMS 7.1 Transport Evidence In the previous chapter, we have studied the effect of the meta llic surface states on the transport of ultra-thin bismuth films. For the films of certain thickness, a closer look at the transport data at low temperatures re veals some very un-expected features. Figure 7-1 shows the zoom-in of the temperature de pendence of the resistance for a 15nm thick bismuth film. We can clear see a sharp drop of the resistance at about 5.6K. The inset shows the magnetic field dependence of resistan ce for the same film. We also see a sharp decrease of R below some critical field of ~200mT. Figure 7-1. Resistance vs. temperature in ze ro magnetic field for a 15nm bismuth film. Inset: resistance vs. magnetic field at 4.5K for the same sample. 6810 760 764 768 R ()T ( K ) -1.0-0.50.00.51.0 760 765 770 R( )B (T)

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81 The sharp feature of resist ance change has been observe d reproducibly in samples with thickness within certain ranges. The re sistance will either jump down or jump up below a critical temperature and critical field by very small amount. Figure 7-2 here shows an example in which the resistance ju mps up at below a cri tical temperature and critical field. Figure 7-2. Example of resistance increases during the transition. The main figure shows resistance vs. temperature in zero magnetic field for a 15nm bismuth film. Inset: resistance vs. magnetic field at 4.5K for the same sample. We also observed sharp feature of re sistance increase or decrease in the Hall resistivity measurements (see Figure 7-3). Si nce the features are even with the magnetic field, they are really from the longitudina l resistance pickup due to the misalignment of the Hall leads. But the change of resistance at the transition is mu ch bigger percentage wise. We also find that the critical field for such feature decreases with increasing 68 798.0 798.5 799.0 799.5 R ( )T ( K ) -1.0-0.50.00.51.0 801 804 R ()B (T)

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82 temperature, and the relation satisfies th e T dependence of the critical field in superconductors: 21 ) 0 ( ) (c c cT T B T B (7.1) Figure 7-3. Sharp feature of resistance change observed in Hall resistivity measurements at indicated temperatures for a 15nm thick bismuth film. Inset: critical magnetic field as a function of T2. By measuring bismuth films with differen t thickness, we map out the thickness dependence of the critical magnetic field for the resistance transition. From Figure 7-4 we can see that at 4.5K, the transition happens for films with thickness smaller than ~16nm, and for films with thickness ~40nm. In fact th e critical field vs. th ickness plot suggests oscillating thickness dependence of the critical magnetic field. -0.50.00.51.0 0.5 1.0 1.5 4.5K 5K 6KRxy ( )B ( T ) 16182022242628 5.0x1011.0x1021.5x1022.0x102 Bc ( mT)T2 (K2)

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83 Figure 7-4. Film thickness dependence of the critical magnetic field at 4.5K. The features we have observed for these bi smuth films together with the absence of a full transition to a zero-resis tance state are suggestive to that superconduc tivity occurs only in certain portions of the films. The bulk rhombohedral bismuth is not superconducting ( Tc<50 mK). But there are several reported superconducting phases of bismuth: high-pressue phases of Bi called Bi II, III and V with Tc =3.9, 7.2, and 8.5K respectively,49-51 fcc Bi with Tc with Tc<4K,52 amorphous Bi with Tc=6K, and granular system of Bi clusters, with Tc ~2-6K depending on the size of the clusters.41 X-ray diffraction (XRD) analysis shows th at our films are rhombohedral. To make the amorphous bismuth films that show supe rconductivity, one need s to deposit bismuth onto liquid Helium cooled substrate. Th ese amorphous bismuth films lose their superconductivity when annealed up to room temperature. Hence we believe that 01020304050 0.0 0.2 0.4 0.6 0.8 Critical field (T)Thickness ( nm )

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84 amorphous phase is not the reason for th e superconductivity observed in our room temperature deposited and 200 C annealed films. 7.2 Tunneling Evidence To further probe the properties of our films, we performed tunneling measurements. The samples we studied are standard cross-bar Pb-I-Bi junctions described in chapter 6. Figure 7-5 shows th e curves of tunneling conductance vs. bias voltage at indicated different temperat ures, for a Pb-AlOx-Bi(150A) junction. -6-4-20246 0.5 1.0 1.5 10K 8K 7.1K 6.5K 5.5K 4K 0.3KdI/dV (Arb. unit)V (mV) Figure 7-5. Differential conducta nce as a function of bias voltage in the superconducting gap region at indicated te mperatures, for a Pb-AlOxBi(150) tunnel junction. A major feature of differen tial conductance at T<5.5K is the existence of two superconducting gaps. With increasing temperat ure, the two gaps move to some bias voltage in between, and the intensity of th e inner gap drops rapidly. At T>5.5K, the

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85 conductance spectrum recovers the shape of normal metal-in sulator-superconductor tunneling, with a single superconducting gap fr om Pb counter-electrode. With further increase of temperature, the superconducti ng gap vanishes as Pb electrode loses its superconductivity. The double gap feature in the bias voltage dependence of differential conductance is very typical for superconductor-insulator-s uperconductor (S-I-S) tunnel junction, with the DOS peaks correspond to 1+ 2 and 12, where 1 and 2 are the BCS gaps of Pb and Bi, respectively. It should be noted that the va lues of the superconducting gaps determined from the data above turn out to be bigger than what they should be (e.g.,the standard value for Pb is 1=1.4meV). Two possible reasons may cause the enhanced gap size. First of all, the sheet resistance of the bismuth film (~500 ) is comparable to the junction resistance itself, and hence will c ontribute to the measured result as a series resistor. Second, electrons may first of all tunne l into a surface state, and then lose energy when they travel into the bulk part of the film. Hence the existence of surface states may cause voltage drop at the bismuth-AlOx interface. Another surprising feature in the dI/dV vs. V characteristic is the conductance maximum for temperature lower than ~7K. This feature doesnt not reproduce for all the samples. The reason for its exis tence is not we ll understood. As a comparison, Figure 7-6 shows the s uperconducting gap feature of a Pb-AlOxBi(1000 ) tunnel junction. We see no evidence of superconductivity in the transport measurements of the 1000 thick bismuth films. The tunneli ng measurement, we also see the standard Pb superconducting gap in th e differential conductance vs. bias voltage

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86 sweep, but no evidence of the smaller gap seen in the sample with smaller thickness of the Bi electrode. Figure 7-6. Differential conducta nce as a function of bias vol tage in the superconducting gap region at 300mk, for a Pb-A lOx-Bi(1000) tunnel junction. We also measured the tunneling conducta nce of our samples in various magnetic fields. Shown in Figure 7-7 is the tunneli ng conductance vs. bias in different low magnetic fields perpendicular to and parallel to the junction area. For the perpendicular field, as the field increases, the gaps decrea se in size and move towards each other. In a field higher 200mT, only one gap is left. Since at 200mT, Pb already loses its superconductivity, the gap is the Bi superc onducting gap. A similar characteristic is observed in the measurements with magnetic fi eld applied parallel to the junction plane, except that the changes occur within a wide r field range. The differences between the -4-2024 0 1 2 dI/dV (Arb. unit)V (mV)

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87 results with magnetic field pa rallel and perpendicular to th e junction suggest that the double gap feature is not a spin effect. -6-4-20246 0.5 1.0 1.5 dI/dV (Arb. unit)V (mV)200mT 120mT 90mT 60mT 20mT 0T 40mT -6-4-20246 0.5 1.0 1.5 200mT 120mT 90mT 60mT 20mT 0TdI/dV (Arb. unit)V (mV)40mT Figure 7-7. Differential conducta nce vs. bias voltage at 300mK in indicated low magnetic fields perpendicular and para llel to the junction plane. Perp. field Para. field

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88 -6-4-20246 0.5 1.0 1.5 18TdI/dV (Arb. unit) V ( mV ) 10T 4T 1T 500mT 300mT -6-4-20246 0.5 1.0 1.5 10T 4T 1T 500mT 300mTdI/dV (Arb. unit)V (mV) Figure 7-8. Differential conduc tance vs. bias voltage at 300mK in indicated strong magnetic fields perpendicular and parallel to the junction plane. Para. field Perp. field

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89 In stronger magnetic field, the zero bias conductivity maximum disappears at ~500mT. With further increase of field, the gap feature slowly weakens out, but still can be seen even at the highest field (18T). Wh ether or not the gap-like feature in the very strong magnetic field is due to superc onductivity is still not well understood. 7.3 Possible Picture Putting together all the evidence for s uperconductivity both in transport and in tunneling measurement, we find that even t hough the resistance cha nge at the transition temperature is very small as observed in the transport measurement, the tunneling characteristic of S-I-S junction is well pronounc ed. This suggests that the overall area of the superconducting domains of the film is la rge at the surface of the film. However, these domains are only weakly coupled by mean s of tunneling. Hence for some films, the resistances exhibit sharp increase rather than decrease at the superconducting transition temperature, similar to what happens to th e tunneling resistance of a S-I-N junction at Tc. A possible model for the superconductivity is that the superconducting domains are separated from each other and normal (non-superconducting) domains by grain boundaries that serve as tunnel barriers. Depending on the th ickness and height of the barrier, the domains coupled with each ot her and the normal domains either through normal tunneling (which give s low conductance below Tc) or point contact(which gives high conductance below Tc). The average of these two kinds of effects, combined with the fact that the grain boundaries contribute to most of the film resistance, explains the small effect is observed in the transport meas urements. However, since the total area of the superconducting domains is large, pronoun ced evidence is obser ved in the tunneling measurements, which are more sensitive to the surface area than the transport

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90 measurements. This model can also explain th e much larger percentage resistance change observed in the Hall resistivity measurements. In those measurements, because the voltage leads are very close (the misalignmen t of the Hall leads are small), the number of grains that get averaged is small, and hence th e fluctuation of the average resistance at the superconducting transition is large. So the overa ll percentage effect of the resistance change during the tr ansition is large. Figure 7-9. A possible physical picture of surface superconduc tivity in ultra-thin bismuth films There are still many pending questions th at need to be answered. The major question that needs theoretical explanati on is, why does the s uperconducting transition happen only in films with certain thickness. Si nce the thickness depende nce of the critical magnetic field shows oscillating behavior, th e surface superconductivity might be related to quantum confinement in the ultra-thin bismuth films. Then, theoretically how does the QSE in the Bi thin films affect the su rface DOS and surface electron-phonon coupling. s n G V GG VV substrate

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91 CHAPTER 8 FUTURE WORK There are many interesting future directio ns that can be followed in the study of bismuth. In this chapter we will briefly show two possible directions: the low dimensional bismuth nano-structures, and the ph ysics and applications of the strong spinorbit coupling in bismuth. In the previous chapters, we have di scussed 3-D and 2-D bismuth structures. Further decreasing the dimensionality, we can envision 1-D or 0-D bismuth systems. Bismuth is a promising candidate for studying such structures mainly because bismuth has a very long Fermi wavelength. This makes it relatively easy to get reduced dimensionality without pushing the limit of the lithography technique too much. Also bismuth has very long phonon mean free path which makes it a good system for studying ballistic transport. Figure 8-1 shows some examples of the s ub-micron sized bismuth patterns we have made. All the patterns are made using sta ndard e-beam lithography technique described below: 1) A 6% copolymer is spinned onto the Si substrates at 2500 RPM for 60 sec. The coated substrate is then baked in a 140C convection furnace for 30 min. 2) A3 PMMA is spinned on top of the copolymer coated substrates at 4000 RPM for 60 sec, followed by 170C baking in a convection furnace for 30 min. 3) E-beam exposure: 30 kV beam with 30 um aperture is used. For line features, an exposure dose of 200~240 2/ cm Cis used. While for gap

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92 features, exposure dose of 3402/ cm C was found to give the best resolution. 4) Developing: for line features (with 200~2402/ cm Cdose), 1:1 (MIBK:IPA) developer was used, w ith developing time of 30 seconds; while for gap features, 1:3 (MIBK:IPA) developer was used, with developing time of 30 seconds. After the lithography procedure above, th e substrates with PMMA patterns were treated with oxygen glow discharge (150 mT orr oxygen, 500V) for 2 min to remove polymer and solvent residue. Then the bi smuth films are thermally deposited from 99.999% pure bismuth source, onto the Si substr ates, with typical th ickness of ~100 nm. The substrates are kept at 100 C during the deposition to achieve large grain sizes. Liftoff was carried out in 9:1 Methol yen chloride: acetone solution. For very thin bismuth films, PMMA itself, without the copolymer layer gives better resolution on bismuth patterns. In this case, A3 PMMA is spinned at ~3500 RPM onto the Si substrate, followed by 170 C baking in a convection furnace for 60 min. For exposure, 10kV beam with 30 um aperture is used, with exposure dose of ~120 2/ cm C. The patterns are developed in 1:3 (MIBK:IPA) solution for 30 sec. For thick bismuth films (>100 nm), however, copolymer layer is necessary for creating a proper undercut, thereby facilitating smooth lift-off after the metallization. To make bismuth patterns on th e insulating substrates (e.g., mica), it is necessary to deposit a thin layer of bismuth (~15 nm) be fore the e-beam lithography procedure to prevent charging. Then, after going through the lithography, metallization and lift-off procedures described above, we use ion b eam etching to remove the pre-deposited bismuth layer and open up the blank areas.

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93 Figure 8-1. Some examples of the sub-micr on sized bismuth patterns. (a)-(c) shows AFM amplitude images of two reservoirs, and a A-B ring patterns, respectively. (d) shows the SEM picture of a cross pattern. (a) (b) (c) (d)

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94 Preliminary data taken from some of thes e patterns shows some typical phenomena for the transport in the diffusive and ballistic region. Figure 8-2 shows the magnetoresistance measurement on the reservoi r pattern with two films connected by a narrow and short wire shown in Figure 8-1 (a). The resistance is generally linear with field. On top of the linear field depend ence, we can see the small, random, but reproducible fluctuations. These are universal conductance fluctuations (UCF), originated from the elastic impurity scattering in the sample in the magnetic field. Figure 8-2. Magnetic field depende nce of the resistivity for the reservoir pattern shown in Figure 8-1 (a). Inset: differential conduc tance vs. bias voltage in different magnetic fields. -20-15-10-50510152 0 1000 1050 1100 1150 1200 1250 B (T)R ()-0.020.000.02 dI/dV (Arb. unit)Vbias (V)

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95 The universal conductance fl uctuations are also observed in the differential conductance vs. bias voltage sweep, as shown in the inset of Figure 8-2 in which the curves are taken from the same sample. Here we also observed random fluctuations as a function of bias voltage in diffe rent fields. This is just another way of probing the UCF in that, instead of changing the phase by the ch anging magnetic field, we change the phase by changing the energy of the electrons. Fabrication of the sub-micr on or nanometer sized bismuth patterns also provide an opportunity studying the ballistic transport behavior in bismuth. For example, by making films with size comparable to the grain size, we can study th e transport inside a single grain or across a single boundary. Figure 83 shows the magnetoresistance measurements on a bismuth nano-cavity pattern. From the AFM image we can clearly see a grain boundary inside the cavity. By changing the geometry of the measurements and by changing the input signals, we can possibly study the transport across a single grain boundary. Note, for example, the more than ten times higher resistance scale for configuration A compared to configuration B. Figure 8-3. Measurements of a nano-cav ity with a single grain-boundary in it. -10-50510 55 60 65 70 75 R ()B (T) V V I I DC AC -8-6-4-202468 3.0 3.1 3.2 3.3 3.4 3.5 R ()B (T) V V I I DC AC A B

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96 There are still many pending problems in the study of bismuth nano-structures. First of all, as the bismuth structures are made smaller and smalle r, the effect of the surface states become more and more signifi cant. The surface states are more metallic than the bulk bismuth and will smear out the intrinsic properties (e.g., long Fermi wavelength, long phonon mean free path, quantum size effect) in the bulk part of the bismuth patterns. Hence it is very important to control the metallic surface states. Second, its difficult to make the bismuth patterns sma ll because they tend to have large crystals. We are still pushing the limit of our lithogra phy to make the sizes comparable to the Fermi wavelength. Another interesting future direction of bismuth study is about the strong spin-orbit coupling in Bi. Recently, Koroteev et.al. observed the strong spin -orbit splitting on Bi surfaces by angular resolved photo emission.53 Bi can be a very interesting material in the spintronics study. We are working on ma king magnetic tunnel junctions on bismuth surface, and studying the possibility of obser ving the spin-hall effect in bismuth.

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97 LIST OF REFERENCES 1. Kempa, H., Kopelevich, Y., Mrowka, F ., Setzer, A., Torres, J.H.S., Hohne, R., and Esquinazi, P., Solid St ate Communications, 2000. 115(10): p. 539. 2. Khveshchenko, D.V., Physical Review Letters, 2001. 87(24): p. 246802/1. 3. Kopelevich, Y., Torres, J.H.S., da S ilva, R.R., Mrowka, F., Kempa, H., and Esquinazi, P., Physical Review Letters, 2003. 90(15): p. 156402/1. 4. Kopelevich, Y., Lemanov, V.V., Moehleck e, S., and Torres, J.H.S., Physics of the Solid State, 1999. 41(12): p. Numbers: 1959. 5. Ashcroft, N.W. and Mermin, N.D., Solid Sate Physics 1976: Holt, Rinehart and Winston. 6. Abrikosov, A.A., Fundamentals of the Theory of Metals 1988: North-Holland. 7. Gantmakher, V.F. and Levinson, Y.B., Carrier scattering in metals and semiconductors 1987: North-Holland. 8. McClure, J.W. and Spry, W.J., Physical Review, 1968. 165(3): p. 809. 9. Smith, G.E., Baraff, G.A., and Rowell, J.M., Physical Review, 1964. 135(4A): p. A1118. 10. Brown, R.D., Physical Review B (Solid State), 1970. 2(4): p. 928. 11. Khveshchenko, D.V., Phys. Rev. Lett., 2001. 87(20): p. 206401. 12. Brandt, N.B., Schudinov, S.M., and Ponomarev, Y.G., Semimetals 1: Graphite and its compounds 1988: North-Holland. 13. Khveshchenko, D.V., Physical Review Letters, 2001. 87(20): p. 206401. 14. Matsui, T., Kambara, H., Niimi, Y., Tagami, K., Tsukada, M., and Fukuyama, H., cond-mat/0405011, 2004. 15. Biagini, C., Maslov, D.L., Reizer, M. Y., and Glazman, L.I., Europhysics Letters, 2001. 55(3): p. 383. 16. Murzin, S.S., Physics-Uspekhi, 2000. 43(4).

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101 BIOGRAPHICAL SKETCH Xu Du was born on July 16, 1974, in Chengdu, Sichuan province, P. R. China. He spent a happy childhood with hi s grandparents in Chengdu, a nd entered school in 1980. In 1982, he moved to a rural area of Sichuan an d lived with his parents. There he spent 10 years enjoying the beauty of nature and fini shing his pre-college education. As a child, Xu Du became deeply interested in math, physics, and engineering, because of the influence of his father. He spent a lot of time working on self-proposed math and physics problems, and on making airplane models a nd electronics. He was also addicted to classical guitar, which became a major pastime. In 1992, Xu Du entered Beijing University of Aeronautics and Astronautics to study mechanical engineering. Driven by his intere st, he studied the main courses for a physics major by himself, and entered the physics depa rtment at Beijing University as a graduate student in 1996. There he worked on th e structural influe nce of the high Tc superconductors and GaN, under the supe rvision of Professor Han Zhang. In 1999, Xu Du received his M..S degree in physics. He received the Alumni Fellowship and entered the University of Flor ida (UF) for his Ph.D. study. There he met his future wife, Zhihong Chen, who was also a physics graduate student. He started working under the supervision of Professor Arth ur F. Hebard in 2000. From then on, he spent 5 years enjoying the freedom of res earch. This dissertation represents the culmination of his research work during the past 5 years.