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## Material Information- Title:
- Theoretical Studies of Coherent Optic and Acoustic Phonons in GaN/InGaN Heterostructures
- Creator:
- LIU, RONGLIANG (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Conceptual lattices ( jstor )
Dielectric materials ( jstor ) Electrons ( jstor ) Lasers ( jstor ) Light refraction ( jstor ) Oscillators ( jstor ) Phonons ( jstor ) Pumps ( jstor ) Reflectance ( jstor ) Semiconductors ( jstor )
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- University of Florida
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- University of Florida
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- Copyright Rongliang Liu. 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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- 4/30/2004
- Resource Identifier:
- 55893129 ( OCLC )
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THEORETICAL STUDIES OF COHERENT OPTIC AND ACOUSTIC PHONONS IN GaN/InGaN HETEROSTRUCTURES By RONGLIANG LIU 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 To Linlin and Bryan. ACKNOWLEDGMENTS First and foremost I would like to thank Professor Christopher J. Stanton, my thesis advisor, for his guidance and encouragement throughout the course of my graduate study here at the University of Florida. Among many other things, he taught me how to construct and develop a theory from simple 1.00. ..1. l ideas and models. I also learned a lot from discussions with his postdoes and graduate students, especially from Dr. Gary Sanders and Yongke Sun. I am honored and grateful to have Professor Peter J. Hirschfeld, Professor Jeffrey L. Krause, Professor Mark E. Law, Professor David H. Reitze, and Professor David B. Tanner to serve on my supervisory committee. I want to thank the McLaughlin family for the dissertation fellowship. Many thanks go to the secretaries of the Physics Department, in particular, Darlene Latimer and Donna Balkum, who gave me much indispensable assistance. It is a great pleasure to have such wonderful friends as Bob and Janis Jackson. The times we are together are the happiest ones. I want to thank my parents. They have gone through all kinds of hardships to raise me and my brothers. They always remind me to work hard, to waste nothing, and to be kind to other people. I also want to thank my parents-in-law. They spent a year with me and my wife, which was one of the happiest times. They help me and my wife to take care of our little son Tianyue during our study. Lastly, I want to thank my family, especially my wife for all the love, support, and fun she brings to me. TABLE OF CONTENTS paget ACKNOWLEDGhlENTS ......... .. iii LIST OF FIGURES ......... .. .. vi ABSTR ACT . .. .... .. .. viii 1 INTRODUCTION . ...... ... .. .._ 1 1.1 GaN/InGaN: Structures and Parameters .. .. .. 2 1.2 The Dynamics of Photoexcited Carriers and Phonons .. .. .. 7 1.3 Experiment Setup For Coherent Phonons ... .. .. 9 2 DIPOLE OSCILLATOR MODEL .... .. .. 11 2.1 Optical Processes and Optical Coefficients .. .. .. 11 2.1.1 Classification of Optical Processes ... .. .. .. 12 2.1.2 Quantization of Optical Processes .. .. . .. 13 2.2 The Dipole Oscillator Model ..... .. . 16 2.2.1 The Atomic Oscillator ..... .. . 16 2.2.2 The Free Electron Oscillator ... . .. 18 2.2.3 The Vibrational Oscillator .... .. .. 20 2.2.4 A Series Of Oscillators .... ... .. 22 2.3 Reflectance And Transmission Coefficient ... .. .. .. 23 3 GENERAL THEORY OF COHERENT PHONON .. .. .. .. 29 3.1 Phenomenologfical Model . .... .. 29 3.2 Microscopic Theory . ...... ... .. 30 3.3 Interpretation Of Experimental Data ... .. .. 33 3.4 Three Kinds Of Coherent Phonons .... .. .. 37 4 THE COHERENT ACOUSTIC PHONON ... . .. 40 4.1 Microscopic Theory ....... ... .. 41 4.2 Loaded Stringf Model . ..... .. 47 4.3 Solution of The Stringf Model ..... ... .. 49 4.4 Coherent Control ........ ... .. 60 4.5 Summary ......... . .. 67 5 PROPAGATING COHERENT PHONON ... . .. 70 5.1 Experimental Results . ..... .. 71 5.2 Theory ............ .......... 78 5.3 Simple model ......... .. .. 90 5.4 Summary ......... . .. 102 6 CONCLUSION ......... . .. 103 REFERENCES ............. ..... ..... 106 BIOGRAPHICAL SKETCH ......... .. .. 109 LIST OF FIGURES Figure page 1-1 Band structure of a direct gap III-V semiconductor. .. .. .. 5 1-2 Schematic setup for a two-beam nonlinear experiment. .. .. .. 10 2-1 Spectral dependence of Seraphin coefficients and differential reflectivity. 27 2-2 Spectral dependence of the dielectric function for DT.. .. .. 28 41Schematic diagram of the In,GCal_,.N MQW diode structure. .....4 4-2 Effects of the built-in piezoelectric field to the MQW bandgap. 44 4-3 Forcing function. ......... .. .. 53 4-4 Displacement as a function of position and time.. .. .. .. .. 54 4-5 The image of strain. ......... .. .. 55 4-6 Energy density as a function of position and time. .. .. .. .. 56 4-7 Energy as a function of time for 4 quantum wells. .. .. .. .. 58 4-8 Energy as a function of time for 14 quantum wells.. .. .. .. .. 59 4-9 The temporally in-phase oscillations of potential energies. .. .. .. 62 4-10 The temporally out of phase reduction of oscillations. .. .. .. .. 63 4-11 Coherent control of the change of transmission. .. .. .. .. 64 4-12 Effects of phase xo temporallyy in phase) on coherent controls. .. 66 4-13 Effects of phase xo temporallyy out of phase) on coherent controls. 68 5-1 The diagrams of ClaN/In~laN sample structures. .. .. .. .. 72 5-2 DB for the In,GCla_,.N epilayers with various In composition .. .. 73 5-3 The oscillation traces of a SQW (III) at different probe energies. .. 75 5-4 DB for the blue LED at different external bias. .. .. .. .. 76 5-5 Differential transmission of DQW's (II) at 3.22 eV. .. .. .. .. 77 5-6 The long time-scale differential reflectivity traces. .. .. .. .. 79 5-7 Generation and propagation of coherent acoustic phonons. .. .. 89 5-8 Propagating strained ClaN layer in our simple model. .. .. .. .. 91 5-9 DB for different frequencies of the probe pulse.. .. .. . .. 93 5-10 The numerically calculated differential reflection. .. .. .. .. .. 95 5-11 Calculated DB varying the thickness of the strained ClaN layer. .. 96 5-12 Schematic diagram of the single-reflection approximation. .. .. .. 97 5-13 The absorption coefficient as a function of the probe energy. .. .. 100 5-14 The change of the dielectric function vs. the probe energy. .. .. 101 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 THEORETICAL STUDIES OF COHERENT OPTIC AND ACOUSTIC PHONONS IN GaN/InGaN HETEROSTRUCTURES By Rongfliangf Liu 11. lv~ 2004 Chair: Christopher J. Stanton II. li..r Department: Physics Coherent phonons are collective lattice oscillations which can periodically modulate the phti~-i~... properties of the crystal. Three kinds of experimentally ob- served coherent phonons are discussed in this thesis. They are the coherent optical phonons in bulk semiconductors, the coherent acoustic phonons in GaN/InGaN multiple quantum wells and superlattices, and the propagating coherent phonon wavepackets in GaN/InGaN heterostructures and epilayers. A phenomenologfical oscillator model is presented to explain the oscillating changes in the reflection of bulk semiconductors. The proper definition of the coherent phonon amplitude as the quantum-mechanical average of the phonon creation and annihilation operators constitutes the basis of the microscopic theory of the coherent phonons, which justifies the macroscopic oscillator model. The lattice displacement is related to the coherent phonon amplitude through Fourier transform. Since the laser wavelength is much larger than the lattice constant, the photoexcited carriers distribute uniformly and the the carrier density matrix has only the q m 0 Fourier component. As a result the coupling of the photoexcited carriers to the phonons leads to coherent optical phonons only. The large piezoelectric fields due to the built-in strain in GaN/InGaN semi- conductor superlattices or multiple quantum wells can be partially screened by the photoexcited carriers whose density has the same periodicity of the superlattice. In response the crystal relaxes to a new configuration which tri-i asi- the coherent acoustic phonon oscillations. The microscopically derived equation of motion for the coherent phonon amplitude can be mapped onto a one-dimensional wave equation for the lattice displacement which is called the string model. Based on the string model, the coherent control of the coherent acoustic phonons can be carried out theoretically. The last kind is the coherent acoustic phonon wavepackets generated and detected in InGaN/GaN epilayers and heterostructures. We constructed a theoret- ical model that fits the experiments well and helps to deduce the strength of the coherent phonon wavepackets. This model shows that localized coherent phonon wavepackets are generated by the femtosecond pump laser in the epilayer near the surface. The wavepackets then propagate through a GaN layer changing the local index of refraction, primarily through the Franz-K~eldysh effect, and as a result modulate the reflectivity of the probe beam. CHAPTER 1 INTRODUCTION Coherent phonons are collective lattice oscillations. When we shine a laser pulse with duration much shorter than the inverse of a lattice vibration frequency, the lattice mode will be excited coherently, i.e., there will be a large number of phonons in one mode with distinct phase relation [1]. This coherent phonon mode may behave like macroscopic oscillators. The nonzero time-dependent lattice displacement modulates the optical properties of the material through the dielectric constant, which can then be detected by changes in the reflection or transmission of a probe laser pulse. There are basically two mechanisms for the generation of coherent phonons. One is the impulsive stimulated Raman scattering (ISRS) first observed by Fu- jimoto and Nelson et al. [2] They excited and detected coherent optic phonons by ISRS in a~-perylene crystals in the temperature range 20-30 K(. The impulsive stimulated Raman scattering can occur with no laser intensity threshold and with only one ultrafast laser pulse because the Stokes frequency is contained within the bandwidth of the incoming pulse. ISRS is a ubiquitous process through which coherent excitation lattice will happen whenever an ultrafast laser pulse passes through a Raman-active solid. The other mechanism is the displacive excitation of coherent phonons (DECP). It is first proposed by Dresselhaus group [3]. DECP requires a significant absorp- tion at the pump frequency so that an interband electron excitation occurs, while ISRS does not require absorption in the material [2, 4]. The electronic excitation makes the lattice to relax to a new quasiequilibrium configuration which then tri-i asi- coherent phonon oscillations. With the development of ultra fast laser systems, the generation of coher- ent phonons in semiconductors, metals, and superconductors by femotosecond excitation of these materials has received considerable attention in recent years because of the potential applications such as non-destructive measurement and the theoretical interests. Shah has given a thorough review of the field [5]. This dissertation discusses three kinds of coherent phonons observed in the exp eriments: 1. the coherent optical phonons in the bulk semiconductor, 2. the coherent acoustic phonons in multiple quantum wells and superlattices, 3. and the propagating coherent phonons in ClaN/In~laN heterostructures and epilayers. I will present theories about these three kinds of coherent phonons and focus on the coherent acoustic phonons and the propagating coherent acoustic phonon wavepackets. 1.1 GaN/InGaN: Structures and Parameters In 1995 Shuji Nakamura at Nichia Chemical Industries in Japan reported the successful development of LEDs based on ClaN compounds [6]. Since then research and application interests have been intensified on this material. The distribution of electrons in space and in momentum and their energy levels determine the electronic properties of a material. There are two implicit assumptions with regard to the electronic structure: 1. The electronic motion and nuclear motion are separable, 2. The electrons are independent of each other. In principle neither of the assumptions is correct. The Hamiltonian of the Ii-v-b III includes electron-electron interaction, nucleus-nucleus interaction, and electron-nucleus interaction. It is a complex many-body system. But these assumptions are still practical approximations. They can be justified by first noticing that the ratio of the electron mass to nuclear mass (10-4-10-5 in a typical solids) is so small that the fast moving electrons can adjust to the motion of nuclei almost instantly. And secondly, any one electron experiences an average potential exerted by all the other electrons. In an infinite and perfect crystal, according to Bloch's theorem, the form of the eigfenfunctions of the Schrodingfer equation are the product of a plane wave and a periodic function with the same periodicity of the periodic potential. The energy levels are so close that they become bands. The methods for band structure calculations fall into two categories: Global methods to obtain the bands on the entire Brillouin zone, Local methods describing band structures near some special points (e.g., L point) inside the Brillouin zone. Just a few examples of the first category include the tight binding method, the orthogonalized plane wave method, and the pseudopotential method. The perturbative k p methods belongs to the second category. The methods for the band calculation can also be classified as empirical and ab into. Most of the techniques are empirical which means that they need experiments to provide input parameters. Slater and K~oster [7] were the first to use the tight binding method empirically. The atomic structure of the elements making up GaN can be described as Ga 1S22S2 p63S23r'~ '.dl 4S2 1 N 1S2 2S2 p3 The covalent bond between a gallium atom and a nitrogen atom is made by sharing the electrons, as a result, each atom ends up with four electrons. In the ground state these valence electrons occupies the s and p atomic orbitals. When the gallium and nitrogen atoms come close to form a molecule, the s and p states form bonding and antibondingf molecular orbitals, which are s bonding, p bonding, s antibonding, and p antibonding orbitals in the order of increasing energy. These molecule orbitals then evolve into the conduction and valence bands of the GaN semiconductor. Because of the specific ordering of the molecular orbitals, the bottom of the conduction band of a GaN semiconductor is s-type, while the top of the valence band is p-type, which allows electric-dipole transitions between the two bands according to the selection rules. The band structure of the GaN semiconductor is shown schematically in Fig. 1-1. This four band model was originally developed by K~ane for InSb [8], which is typical of direct gap III-V semiconductors. Near the zone center all four bands have parabolic dispersions. Two of the hole bands that are degenerate at k = 0 are know as the heavy hole (hh) and the light hole (lh) bands. The third split-off hole (so) band gets its name from the split-off to lower energy by the spin-orbit coupling. In bulk semiconductors the energy gaps are temperature dependent which can be fitted to the empirical Varshni function [9] aT 2 E, (T) = E, (T = 0) ,(1. 1) T p' where a~ and p are adjustable and called Varshni parameters. There are other functional forms [10] but the Varshni's is the most widely used one. GaN is a wide-gap semiconductor. It usually appears in two crystal structures, the wurtzite GaN and the Zinc blende GaN. The wurtzite GaN is the more usual one of the two structures. The following parameters we used are from Vurgaftman et al. [11] They gave a very rich set of references in the article. /\E 4 _______0hh lh SO Figure 1-1: Schematic band diagram of a direct gap III-V semiconductor near the Brillouin zone center. The zero energy point is located at the top of the valence band, while E E, corresponds to the bottom of the conduction band. The con- duction band is an electron (e) band, while the valence bands are: the heavy hole (hh) band, the light hole (lh) band, and the split-off hole (so) band. GaN 3.189 5.185 3.507 0.909 830 0.019 0.014 0.20 0.20 14.0 390 145 106 398 105 -0.35 1.27 8.04 7.96 4.13 4.13 6.31 InN 3.545 5.703 1.994 0.245 624 0.041 0.001 0.12 0.12 14.6 223 115 92 224 48 -0.57 0.97 5.17 5.28 1.21 1.21 2.51 GaAs 5.653 1.519(C) 0.5405(C) 204(C) 0.341 0.067(C) 0.067(C) 28.8 1221 566 6;00 4.73 3.35 300 K 300 K m/s) m/s) m/s) m/s) m/s) Table 1-1: Recommended Band Structure Parameters (for comparison we also listed parameters for GaAs.) for wurtzite GaN and InN Parameters al, (A) at T = cic (A) at T = E, (eV) a~ (meV/K) 79 (K) acr (eV) aso (eV) m6 (eV) m~ (eV) EP (eV) cll (GPa) C12 (GPa) c13 (GPa) cas (GPa) c44 (GPa) els (GPa) es (GPa) C, [001] (10" C, [100] (10" C, [001] (10" C, [100] (10" C, [010] (10" For ternary alloys, the dependence of the energy gap on alloy composition is assumed to fit the following quadratic form, E,(A,B,_,) = xE,(A) +t (1 x)E,(B) x(1 x)C, (1.2) where the bowing parameter C represents the deviation from the linear interpo- lation between the two binaries A and B. In the case of In,Gay _,N, C is about 3.0 eV. For other parameters, one usually linearly interpolate between the values of two binaries if there are no generally accepted experimental data. The [001] longitudinal acoustic wave speed of In,Gal_,N For x: = 0.08 is about 7800 m/s. The difference of the [001] LA mode speed between GaN and In,Gai_,N is very small when the In component x: is small. For x: = 0.08 this difference is less than ;'.' Th Irelecivt d.I~~IY ue to ~lthe differently soundI speedsH between GaN andU In,Gai_,N is given by C, C' r = (1.3) C, +t Ci' where C, and C' are sound speeds in Ga an nGlNrsetiey o x: = 0.08, r is approximately 0.014 and the reflection constant R = |r|2 1S Velln smaller on the order of 10-4 1.2 The Dynamics of Photoexcited Carriers and Phonons Ultrafast femtosecond lasers are indispensable powerful tools for studying the dynamical behavior of photoexcited electrons and holes in semiconductors and other condensed matter systems. These lasers are ideal for obtaining a snapshot of the nonequilibrium photoexcited carriers and studying the scattering processes because the scattering time of carriers in semiconductors is about tens to hundreds of femtoseconds. As mentioned in the last section the periodicity of the semiconductor lattice leads to energy band structure, which forms the basis of understanding most of the optical phenomena in a semiconductor. Electrons can be excited into the conduction band creating holes in the valence band. The periodicity of the semiconductor lattice also allows the description of the quantized vibrational modes of the lattice in terms of phonon dispersion relations. The dynamics of electrons, holes, and phonons is influenced by their interaction with each other, as well as with defects and interfaces of the system. Carrier-carrier scattering determines the exchange of energy between carriers and is primarily responsible for the thermalization of photoexcited non-thermal carriers. Carrier-phonon interactions play a major role in the exchange of energy and momentum between carriers and the lattice. Optical phonons interact with carriers through polar coupling and non-polar optical deformation potential, while acoustic phonons interact with carriers through deformation potential and the piezoelectric potential. After a semiconductor in thermodynamic equilibrium is excited by an ultra- short laser pulse, it undergoes several stages of relaxation before it returns once again to the thermodynamic equilibrium. The carrier relaxation can be classified into four temporally overlapping regimes. First there is the coherent stage. The ultrashort laser pulse creates excitation with a well-defined phase relationship within them and with the laser electromagf- netic field. The scattering processes that destroy the coherence are extremely fast so pico- and femtosecond techniques are required to study the coherent regime in semiconductors. The dynamics are described by the semiconductor Bloch equations [12]. Second is the thermalization stage. After the destruction of coherence through dephasing the distribution of the excitation is very likely to be non-thermal; that is, the distribution function cannot be characterized by an effective temperature. This regime provides information about various carrier-carrier scattering processes that bring the non-thermal distribution to a hot, thermalized distribution. This relaxation stage is governed by the Boltzmann transport equation (BTE) [13]. Third is the hot carrier or the carrier cooling stage. In this regime the carrier distribution is characterized by an effective temperature. The temperature is usually higher than the lattice temperature and may be different for different sub--i~--r I nI- Investigation of hot carrier regime focuses on the rate of cooling of carriers to the lattice temperature and leads to information concerning various carrier-phonon and phonon-phonon scattering processes. At the end of the carrier cooling regime, all the carriers and phonons are in equilibrium with each other and can be described by the same temperature. However, there is still an excess of electrons and holes compared to the thermo- dynamic equilibrium. These excess electron-hole pairs recombine and return the semiconductor to the thermodynamic equilibrium. This is the recombination stage. It should be emphasized that many 1.1..~--1...1 processes in the different regimes can overlap. For example, the processes that destroy coherence may also contribute to the thermalization of carrier distribution functions. 1.3 Experiment Setup For Coherent Phonons There are many techniques developed to investigate the optical properties of semiconductors using ultrafast lasers, which include pump-probe spectroscopy, four- wave mixing spectroscopy, luminescence spectroscopy, and terahertz spectroscopy. Figure 1-2 shows a schematic setup for a general two laser beam pump-probe exp eriment. Pump-probe spectroscopy is the most common setup for the generation and detection coherent phonons. The light source is usually a Titanium:sapphire mode-locked ultrafast laser with a pulse width ranging from a few femtoseconds to hundreds of femtoseconds and a wavelength around 800 nm, the frequency of which can be doubled if photons of higher energy are needed. The light beam from the ultrafast laser is splitted into a pump and a probe. The time delay between the pump and the probe is controlled by the optical path of the probe. The pump laser pulse excites carriers and coherent phonons in the semiconductor sample, which results in the change in the dielectric function of the material. By measuring differential reflection or transmission which represent changes in the reflected or transmitted probe pulse energy between the pump on and pump off at different time delays, one obtains the response of the dielectric function to the light due to the carriers and lattice vibrations in the material. Pump Probe Delay Figure 1-2: Schematic setup for a two-beam nonlinear experiment. It can be used for pump-probe transmission or reflection spectroscopy, four-wave-mixing spec- troscopy, or luminescence correlation spectroscopy. Pump-probe reflection signal CHAPTER 2 DIPOLE OSCILLATOR 1\ODEL The description of the interaction between light and matter falls into one of three categories: the classical, the semiclassical, and the fully quantum. In classical models we treat both the light and matter as classical objects that behave according to the laws of classical pkli--h In semiclassical models we apply quantum mechanics to describe the matter, but treat light as a classical electromagnetic wave. The fully quantum approach belongs to the realm of quantum optics, where both light and matter are treated as quantum objects. When we speak of a light beam in terms of photons and draw Feynman diagram to depict the interaction processes, we are using the fully quantum approach implicitly, even though quantitatively we may have treated the light classically. The dipole oscillator is a typical example of classical models. It is the basic starting point for understanding the effects due to carriers and phonons. The proper comprehension of this simple classical model is indispensable in appreciating more complicated models. In the first section of this chapter I will give a general introduction to the optical processes occurring when a light beam is incident on an optical medium and their quantization. The main purpose is to define the notation and collect together the relationships between the optical coefficients. 2.1 Optical Processes and Optical Coefficients The wide range of optical processes observed in a semiconductor can be orgfa- nized into several groups of general phenomena. At the macroscopic level all the optical phenomena can be described quantitatively by a small number of parame- ters or optical coefficients that characterize the properties of the semiconductor. 2. 1.1 Classification of Optical Processes When a light beam is shined on a semiconductor, some of the light is reflected from the front surface, some enters the semiconductor and propagates through it, and some comes out from the back end of the semiconductor. Thus, the intuitive and simplest classification of optical processes is ref7~l.~~, H. propagation, and transmission. The light propagation phenomena can be further classified into refraction, absorption, luminescence, and scattering. Refraction happens when a light beam travels obliquely from one optical medium to another, e.g., from free space into a semiconductor, in which its speed changes, e.g., the speed of the light becomes smaller in the semiconductor than in free space. The direction of the light beam changes according to Snell's law of refraction. Absorption describes the loss of intensity when a beam of light passes through a semiconductor. There are two kinds of absorption processes: scattering, which will be discussed below, and absorption of photons by atoms or molecules in the semiconductor. Absorption and transmission are related because only the unabsorbed light will be transmitted. Luminescence is the spontaneous emission of light by atoms of a semicon- ductor making transitions from an excited state to the ground state or to another excited state of lower energy. Depending on the causes of excitation, luminescence could be photoluminescence if the excitation is caused by a photon, electrolumi- nescence if it is an electron, chemiluminescence if it is a chemical reaction, etc. Phosphorescence or fluorescence depends on whether the luminescence persists sig- nificantly after the exciting causes is removed. There is some arbitrariness in this distinction, usually a persistence of more than 10 ns is treated as phosphorescence. Thus, photoluminescence accompanies the propagation of light in an absorbing medium. The emitted light beams are in all directions and the frequencies are different from the incoming beam. The spontaneous emission takes a characteristic amount of time. The excitation energy can dissipate as heat before the radiative re-emission. As a result the efficiency of the luminescence is closely related to the dynamics of the de-excitation mechanism in semiconductors. Scattering occurs when a light beam is deflected by atoms or molecules in a semiconductor. If the frequency of the scattered light remains unchanged, it is called an elastic scattering; otherwise it is called an inelastic scattering. The total number of photons does not change in either kind of scattering, but the number in the forward direction decreases because some photons are being redirected into other directions. So scattering also has the attenuating effect as absorption does. There are other phenomena such as frequency doubling if the intensity of the propagating light beam is very high. These phenomena are described by nonlinear optics. 2.1.2 Quantization of Optical Processes The Reflection and transmission of a beam of light are described by the Reflectance or re l7eli~, i .. .~ ..~ .: :I R and the transmittance or transmission .. 5, ..~ :A :T respectively. Reflectance R is defined as the ratio of the reflected power to the incident power on the surface, while transmittance is the ratio of the transmitted power to the incident power. If there is no absorption or scattering, we must have R + T = 1 from energy conservation. The propagation of the light through a semiconductor can be characterized by the refractive index: n and the absorption I .. 5, ..~ :t a: The refractive index is the ratio of the speed of light in vacuum c to the speed of light in the semiconductor vI, n = -. (2.1) The frequency dependence of the refractive index is called dispersion. The absorption coefficient a~ is defined as the fraction of power absorbed after a light beam travels a unit length of the medium. Suppose the light beam travels in the z direction, and denote the intensity at position z as I(z), then the decrease of the intensity in a thin slice of thickness dz is given by dl = -a~dz x I(z). After integrating both sides of the above equation, we obtain Beer's law I(z) = Io e-oz, (2.2) where Io is the light intensity at position z = 0. The absorption coefficient is strongly related to the frequency of the incident light, which is why materials may absorb the light of one color but not another. In an absorbing medium the refraction and absorption can also be described by a single quantity, the complex refractive index fi, which is defined as S= n +isc. (2.3) The real part of n, i.e. n, is the same as the refractive index defined in Eq. (2.1). The imaginary part of n, i.e. sc, is called the extinction I .. FI, T7. :al and is related to the absorption coefficient. We can derive the relationship between a~ and ac by considering a plane electromagnetic wave propagating through an absorbing medium. Let the z axis be the direction of propagation and let the electric field be given by S(z, t) = Soei(kz--wt), 2 where k is the wave vector and wc is the angular frequency, which are related in a non-absorbing medium through k =(2.5) A/n c Here A is the vacuum wavelength of the light. Eq. (2.5) can be generalized to the case of an absorbing medium by way of the complex refractive index, k = f- (n +t inc)- (2.6) c c On substituting Eq. (2.6) into Eq. (2.16) and after rewriting Eq. (2.16), we obtain S(z, t) = Soe-siwzl ee gnzlc-wt) (2.7) Comparing Beer's law in Eq. (2.2) with the intensity of the light wave, which is proportional to the square of the electric field, i.e. I oc SE*, we obtain 2Knc 4xx a (2.8) c A which shows that the absorption coefficient a~ at a given wavelength is proportional to the extinction coefficient Kc. We still have one last optical coefficient to introduce, that is the relative dielec- tric constant e. Sometimes it is simply called the dielectric constant or dielectric function. The relationship between the refractive index and the dielectric constant is a standard result derived from Maxwell's equations (cf. any electrodynamics textbook, e.g. Jackon's [14]), n = (2.9) Correspondingf to the complex refractive index fi, we have the complex dielectric constant, 2 = ex +t iE2 = 2, (2.10) where et is the real part of the complex dielectric constant and E2 the imaginary part. The relationships between the real and imaginary parts of it and 2 are not difficult to derive from Eq. (2.10). They are el = n2 ~C2 (2.11a) t2 = 2nic, (2.11b) and m =i2 -e / (2.12b) Thus, we can calculate n and ac from et and E2, and vice versa. In the weakly absorbing case, i.e. ac n = 4,(2.13) x = (2.14) 2n 2.2 The Dipole Oscillator Model The originator of the classical dipole oscillator model is Lorentz, so it is also called the Lorentz model. In this model, the light is treated as electromagnetic waves and the atoms or molecules are treated as classical dipole oscillators. There are different kinds of oscillators. The atomic oscillator at optical frequencies is due to the oscillations of the bound electrons within the atoms. The vibrations of charged atoms within the crystal lattice give rise to the vibrational oscillators in the infrared spectral region. There are also free electron oscillators in metals. Based on the model we can calculate the frequency dependence of the complex dielectric function and obtain the reflection and the transmission coefficient. 2.2.1 The Atomic Oscillator Let us consider first the atomic oscillator in the context of a light wave interacting with an atom with a single resonant frequency Lco due to a bound electron. We assume the mass of the nucleus m, is much greater than the mass of the electron mo so that the motion of the nucleus can be ignored, then the displacement x: of the electron is governed by the classical equation of motion, d2X: dx mo d2 MTd mo w x =i0 -C;~a eS (2.15) where y is the damping constant and e is the charge of an electron The electric field 8 of a monochromatic light wave of angular frequency Lc is given by S(t) = Soe-ist. (2.16) If we substitute Eq. (2.16) into Eq. (2.15) and look for solutions of the form x:(t) = zoe-iwt, then we have -e~o/mo xo = 2 2) The resonant polarization due to the displacement of the electrons from their equilibrium position is Ne2 ~Pr= -Nex: = 2218 where N is the number of atoms per unit volume. The electric displacement ~D, the electric field 8, and the polarization p are related through ~D = coff = cof +t p. (2.19) We split the total polarization into the resonant term ~Pr we discussed above and a non-resonant background term ~Pb = t0X8, then from Eq. (2.19) we can obtain the complex dielectric constant Ne2 e(lc) = 1+ + (2.20) como ac, 111 squc We can also write this complex dielectric constant in terms of its real and imagfi- nary parts: Ne2 L12 112 ~1(Li)=l~t omo (Lo2 1122 2t(~ia)(.1 Ne2 Yc caLi) omo (Lc0 112 2 -t(~12 (.2 We define two dielectric constant in the low and high frequency limits, Ne2 e(ci= 0) es=1 + tC2 (2.23) and e(Lc = 00) em = 1 +t 3. (2.24) Thus, we have Ne2 Lc2 es emo ', (2.25) where Lc2 Ne2 is called the plasma frequency. Close to resonance, where wei o >ci 7 and wi1, w12 2 0iAsc With Awc = Lc woi as the detuning from Lco, we can rewrite the real and imaginary parts of the complex dielectric constant as ei(Awc) = eo (es em) 2,(2.26) 4Awc2 2' The frequency dependence of et and E2 is called Lorentzian named after the originator of the dipole oscillator model. The imaginary part e2 is StrOngly peaked with a maximum value at Lco and a full width y at half maximum. The real part ex first gradually rises from the low frequency value es when we approach woi from below. After reaching a peak at Lco 7/2, it falls sharply, going through a minimum at Lco +t 7/2, then rising again to the high frequency limit em. The frequency range over which these drastic changes occur is determined by y. 2.2.2 The Free Electron Oscillator Both metals and doped semiconductors contain large densities of free carriers such as electrons or holes. Both of them can be treated as plasmas, i.e. a neutral gas of heavy ions and light electrons, because they contain equal number of positive ions and free electrons. Unlike bound electrons, the free electrons have no restoring force acting on them. The free electron model of metals was first proposed by Drude in 1900. The Drude model treats the valence electrons of the atoms as free electrons. A detailed discussion of the Drude model can be found in e.g. Ashcroft & Mermin [15]. These free electrons accelerate in an electric field and undergo collisions with a characteristic scattering time 7r. The free electron oscillator is actually a combination of the Drude model of free electron conductivity and the Lorentz dipole oscillator model. The equation of motion for electrons in the Drude model is given by dp p -eS. (2.28) dt -r Comparing it with the equation of motion for the displacement x: of a free electron oscillator d2X: dx mo + mo7 -S (2.29) dt2 dt we have a relation between the damping constant and scattering time y = -. (2.30) We can solve Eq. (2.29) in the same way as we did in the atomic oscillator and obtain the complex dielectric constant for the free electron oscillators, Ne2 e(W) = 1 -(2.31) como 2c ~ iYc where we did not take into account the background polarization. If we define Ne2 w = (2.32) Como where Lci is known as the plasma f,. qu, to~ ;I. we can write Eq. (2.31) in a more concise form e(W) = 1- "..(2.33) The AC conductivity amc in Drude model is obtained by solving Eq. (2.28), a(w)i = (2.34) 1 iwr7 where ao = Ne27 rlO is the DC conductivity. Comparing Eq. (2.31) and (2.34), we have e(W) = 1 +iw)(2.35) which tells us that optical measurements of the dielectric constant are equivalent to AC conductivity measurements. Again we can split the dielectric constant into its real and imaginary parts as ex = 1-(.6 1+ wc272 )(.6 62= (2.37) wc(1+ w272"> The plasma frequency upi typically lies in the visible or ultraviolet spectral region, which corresponds to Lci > 1015 sec-1. The mean free collision time for electrons in metals is typically -r 10-14 sec. So for metals in the region of plasma frequency, Lcr > 1. 2.2.3 The Vibrational Oscillator The atoms in a crystal solid vibrate at characteristic frequencies determined by the phonon modes of the aI i<--.l1 The resonant frequencies of phonons usually occur in the infrared spectral region, which results in strong absorption and reflection of light. Since a longitudinal optical (LO) phonon has no effect on a light wave because the longitudinal electric field induced by an LO phonon is perpendicular to that of the light wave, we will consider the interaction between an electromagnetic wave and a transverse optical (TO) phonon, which is easily visualized by a linear chain. The chain is made up of a series of unit cells with each cell containing a positive ion of mass m+ and a negative ion of mass m_. If we assume the waves propagate in the z direction and the displacement of the positive and negative ions in a TO mode is denoted by x+ and x_ respectively, then the equations of motion are given by d22 m -K(x+ x_) +t qS, (2.38a) dt2 m_- = -K(x_ x ) qS, (2.38b3) where K is the restoring constant and +q is the effective charge per ion. Combin- ing equations of motion (2.38a) and (2.38b), we obtain d2X p dt = -Kx: + qS, (2.39) where pL is the reduced mass given by 1/pL = 1/m + 1/ Im_ and x: = x+ x_ is the relative displacement of the positive and negative ions within the same cell. We can introduce a phenomenological damping constant y to account for the finite life time of a phonon mode. Following the same procedures as the forgoing two subsections, we can solve the motion of equation for x: to obtain the dielectric constant Nq2 e(wc) = 1 +X + 2 (2.40) t0- c# "o Lci squi where 3C is the non-resonant background susceptibility and N is the number of unit cells per unit length. Similar to the definitions of es and em in subsection 2.2.1 we can also write the dielectric constant as e(Lc) = em +t (es em) 2 2041 As we said before usually an LO phonon does not interact with light waves, but in special cases it will. In a medium without free charges, Gauss's law gives V ~D = V (cod~) = 0. (2.42) There are two ways to satisfy the equation, one of them is a transverse wave withk 8 = 0. The other is longitudinal waves but with e = 0. In a weakly damped system we can set y = 0. There is a special frequency called wcLO at which the dielectric constant is zero and the longitudinal modes interact with light waves. We can solve Eq. (2.41) to obtain the so called Lyddane-Sachs-Teller (LST) relationship 2cLO t (2.43) TOi r000 The dielectric constant is below zero when Lio < L < LcO, which leads to 100%o reflectance so no light can propagate into the medium. The frequency range between Lio and wcLO is called the restrahlen (German word for I. -i.111..1 rays") band. 2.2.4 A Series Of Oscillators In a semiconductor, the atomic oscillator, the free electron oscillator, and the lattice vibrational oscillator may be all present. For a specific type of oscillator there can be several different resonant frequencies. The dielectric constant due to these multiple oscillators can be written as Ne2 f where myi and yj are the frequency and damping constant of a particular oscillator. The phenomenological parameter fj is called the oscillator strength, which has no explanation in classical 1.1.i~--i. In quantum ]l' i'--i. -, the oscillator strengths satisfy the sum rule fy, = 1. (2.45) In classical 1.1.i~--i. we just take fj = 1 for each oscillator. 2.3 Reflectance And Transmission Coefficient For normal incidence the reflectance is given by 2n _2 2K R t .n12 2 (2.46) The change of reflectance comes from several different sources. It can be due to the change of carrier density N, the scattering time -r, the restoring force Lci, etc. The differential reflection can be expressed in terms of the change of the real and imaginary part of the dielectric constant respectively, aR 1 aR 1 BR R R Bez~1 R Be2 d 6 2 b)a 1 , E 31 So + t6~ 32 621 (2- 8 where the superindex D and O indicate the contribution from the intraband Drude term and interband oscillator term respectively. 31 and 32 are called the Seraphin coefficients and are given by the following equation: 31 =(2.50) [(ei 1)2 -tt221titt JZa(2~ ~2 21 - [(ei 1)2 -ttl21 tt From the Drude formula in Eqs. (2.37) and (2.37) we can obtain the change of the dielectric function, Be~D 6N 3D - SeD 1o ]i(.2 Dy+ D, (2.53) SeDD 2N 3~ DN2 +t D~ (2.55) The coefficients are shown below: 4x~Noe2 r2 DN e (2.56) 1 m 1 Lc2-r2 4x~Noe2 DN2 = eD (2.57) 8xNoe2 -2 D(P-I D (2.58) Sm (lc1 2 r2 2 1122 2 4xNoe2 _r~ 2 Li-2 112 2 D m w(1+ 2 W'2 2 I 2 2 2P(." For most semiconductors Lcr > 1 thus we have, D = 0 (2.60) D -D, (2.61) 1 = w > 1.(2.62) For weak absorbing material the imaginary part of the dielectric function is small ED 0 The Seraphin coefficient can be simplified as P1 +O(e ), (2.63) 2a 3/2 -2 + (e ), (2.64) 2ez pl1/ 2 Lcr > 1. (2.65) 1 3ex Therefore in the Drude model the differential reflectivity is dominated by the real part of the Seraphin coefficient and is more sensitive to the change of the carrier density. Now let us turn to the transmission. Right at the interface the transmission coefficient is To = 1- R 4n (n + 1) 2 -t ~2 (.6 61 - t Inside the media the transmittance decreases as described by Beer's law in Eq. (2.2). For a film of thickness L the differential transmission as defined be- low is AT T T(z, a~) T(z, c~o) T (z, c~o) Z=L -6co L, (2.67) (2.68) where t~o and a~ are the absorption coefficient before and after the excitation respectively and |So~ L| absorption coefficient and the extinction coefficient in Eq. (2.8) we can obtain the change of the absorption coefficient, Sov = 2w~iK. c (2.69) The change of extinction coefficient can be described similarly as the differential reflectivity, 6K = -Lbe + v6E2, (2.70) with (2.72) (2.73) (2.74) +t O(e ), - + ~e ) and 62 2 61 (2.75) The change of the absorption coefficient in terms of the change of carrier density and scattering time is given by the following, 6N 6-r 6co = DN +t Dr (2.76) with ~1 D, ~ (e ,). (2.78) Therefore the differential transmission is dominated by the change of the imaginary part of the dielectric function and the change of carrier density and scattering time have the same order of effect. Following the same procedure we can discuss the differential reflectivity and differential transmission in the oscillator model. Some results are shown in Fig. 2-1 and Fig. 2-2. Figure 2-1: Spectral dependence of Seraphin coefficients and differential reflectiv- -4 0 5 10 Drd:ra Series Ocs: real -- S~eries~Rss.iQs.. my Drude: img --- 101 l100 -1 6101 '1-2 O 0 0.1 O _ 8 I 1 0-3 L 2.1 2.2 1. -0.2 1.8 1.9 2 m (eV) 1.9 2 a (eV) 2.1 2.2 a, 0.5 v I o-r C 0 -1 0 5 10 15 20 to (e V) Figure 2-2: Spectral dependence of the coefficient of real and imaginary part of the dielectric function for differential transmission. CHAPTER 3 GENERAL THEORY OF COHERENT PHONON 3.1 Phenomenological Model The coherent phonon motion can be described in a phenomenological model of a driven harmonic oscillator [3]. The evolution of a coherent phonon amplitude Q in the presence of a driving force exerted by ultrafast laser pulse is governed by the differential equation 2y + 2 +0 Q (3.1) Of2 Of m where Lco is the frequency of the phonon mode, y is the damping parameter, m is the mass of the oscillator, and F is the driving force, which may depend on carrier density, temperature, and other parameters of the system. The damping parameter y is the inverse of the dephasing time T2 of the coherent phonon mode [16]. The dephasing time T2 COmes from a combination of phase-i1. -r .dwl_ processes with relaxation time Ts, and population decreasing processes with relaxation time Ty. Examples of the latter are anharmonic decay processes such as the decay of LO phonons into acousite phonons and electron-phonon interaction processes where a phonon can be absorbed by an electron. Ultrafast generation of carriers by femtosecond lasers causes the driving force to rapidly turn on and tri-i lr the oscillations. Oscillator Eq. (3.1) can be solved formally by using either Green's functions or Laplace transforms with the initial condition that both Q and 8Q/8t are zero before the force F is applied with the result [17], We consider two kinds of forcing functions [17]. The first kind is impulsive forces, which have the form Fi (t) = I 6(t), (3.3) where 6(t) is a Dirac delta function in time. After integrating Eq. (3.2) directly we have The solution shows that an impulsive force starts oscillations about the current equilibrium position, which will damp out exponentially. The other kind of forcing function is displacive with a form given by Fd(t) = D O(t), (3.5) where 0(t) is a Heaveside step function. Again we can integrate Eq. (3.2) directly to get the coherent phonon amplitude Q(t) = 0(t) 1 e-Yt Icos w-7 2Sn 0- (3.6) The solution in the case of a displacive forcing function shows oscillations and exponential damping too, but compared to the impulsive case, a displacive force will move the oscillator to a new equilibrium position with a different initial phase. 3.2 Microscopic Theory The phenomenologfical oscillator model captures the essential pkli--1 However it leaves open the question of exact definition of coherent phonon amplitude. The microscopic quantum mechanical justification of the oscillator model is given by K~uznetsov and Stanton [18]. In a simplified system consisting of two electronic bands interacting with phonon modes, the Hamiltonian consists three parts: free Bloch electrons and holes in a perfect static crystal lattice, the free phonons, and the electron-phonon interaction. H = EakC krak CLqb ,b+ Mg~(,+ by+ t ckkq (3.7) a,k q a,k,q The Lattice displacement operator C(r) is expressed in terms of the phonon creation and annihilation operators: i(r) = C {bgesq.r + bheizq~r}. (3.8) 2pVwc, The coherent phonon amplitude of the qth mode is defined as: DqE(b)+ b, EB +B" (3.9) Therefore the coherent amplitude is proportional to the Fourier components of the displacement in Eq. (3.8) The average of the coherent amplitude will vanish in a phonon oscillator eigen- state. The average displacement of the lattice vanishes, but there are fluctuations: (U2) OC (bbt +t btb). These phonons are incoherent phonons in the mode. A nonzero displacement requires that the wave function of the oscillator be in a coherent superposition of more than one phonon eigenstate. In a general state there can be a number of both coherent and incoherent phonons. The canonical coherent states are defined for each complex number z in terms of eigfenstate of harmonic oscillator. They have two important properties. Firstly, they are eigenvector of the anni- hilation operator with eigenvalue z, therefore in a canonical coherent state, the coherent amplitude defined in Eq. (3.9) is Ben (z~,|z)= z.(3.11) Secondly, canonical coherent states are minimum-uncert .iinri v wave-packets. When the amplitude z is large, they behave like a macroscopic harmonic oscillator. So they are called "-quasi-classical" states. Using the operator form of the Schroidingfer equation 8 OA ih ( ) [, ] ih( ), (3.12) we can obtain the dynamic equation of motion for the coherent phonon amplitude +2D 2D =-Liqn s MC k,k-q. (3.13) a,k Here n(~,kl-q (fkrak-q is there electr-onic density matr-ix. The electronic density matrix n ,k-q in the right hand side of Eq. (3.13) is nonzero only after excitation with an ultrafast laser pulse. This equation is written in momentum space. Because of the Fourier transform relation between the coherent amplitude and the lattice displacement, we have obtained the phenomenological Eq. (3.1). However there is no dampening term because we neglected anharmonic terms in the lattice potential. In bulk materials since the laser wavelength is much larger than the lattice constant, the created carriers are in a macroscopically uniform state, so that the electronic density matrix is diagonal. So the only phonon mode that is coherently driven by the optical excitation is the qm 0 mode. In superlattice and multiple quantum well systems, the carriers are created in the wells only. Because of the periodicity of the superlattice structure, the electronic density matrix has a q f 0 element which can excite the coherent acoustic phonon mode. 3.3 Interpretation Of Experimental Data In the pump-probe experiment the measured quantities are usually the differential reflectivity or differential transmission. This section discusses the relation between the lattice displacement and the measured phli~-i~.. quantity. Assume the only motion is parallel to the z axis and the only nonzero com- ponent of the elastic strain is rj3. The strain is related to the lattice displacement U(z, t) in the z direction through the following equation, dU (z, t) Usa = (3.14) dz The change of reflection or transmission is due to strain induced variation of the optical constants of the material under consideration. In the linear approxima- tion we have On An(z, t) = as, (3.15) where an and Asc are the changes in the real and imaginary part of the complex index of refraction. For normally incident light the Maxwell's equation for electric field gives 8z2 C2 (x 2 = 0, (3.17) where the change of the dielectric constant is related to the change of the index of refraction through e(z, t) = (n +t ~)2, (3. 18) ae(z, t) = 2(nt + m)A(n + in), =2 ~a(n +t in)~. (3. 19) We write the probe electric field as 8 = F(z, t)eiw, (3.20) where F(z, t) is a slowly varying envelope function. Substituting Eq. (3.20) into Eq. (3.17) we obtain the following equation for the envelope function F(z, t), d2 + [e+ eX, t)]X F12, =i 0.I 2F (3.21) 8z C2i d c2 d2 Since we assumed that F is slowly varying we can neglect the derivative of F with respect to time and rewrite the above equation as 8 2FX t) ,,,2 8z2 C2 [e +t ae(z, t)] F = 0, (3.22) which is analogous to Eq. (3.26) in Thomsen's paper [19] where he calculated the change of reflection. The result is AR(t) = f 9(z)113 dz7, (3.23) where f(z) is the --. I-ii ivity f'lll. I b''ll which determines how strain at different depths below the interface contributes to the change in the reflectivity and is given by f(z) =[ fo n sm +n cos -c, O n 1I; (3.24) .fo = 8 (3.25) c [(n 1) 2 ~C2 2 K(n2 t C2 I S= arctan (3.26) n(n22 and A is the laser wavelength, ( is the absorption length defined as the reciprocal of the absorption coefficient. In the following we will carry out the calculation for the differential trans- mission. If we set the probe electric field of a normal incident laser at z = 0 as a Gaussian function S(z- = 0, t) = oer-Y(t-r)2 e'"', (3.27) where -r is the probe delay with respect to the pump, then in the absence of strain we have the wave equation solution of the field as Sozt =Sz -,where v (3.28) vI n = oe:-Y(t-ziv--r)2 _"I't-z~).~ (3.i29) Comparing the above equation with Eq. (3.20) we can obtain the envelope function for the field without the strain Fo(z-, t) = oeu-7(t-7-Zl?1)2 _'""iWZ. (3.30) With strain present the probe electric field will change to S(z, t) = So(z, t) + t 1(z, t), (3.31) =[Fo(z, t) + F,(z, t)] (3.32) where Fi (z, t) is the correction due to the change of optical constants caused by the strain that resulted from the pump field. Substituting Eq. (3.32) into the partial differential Eq. (3.22) for the envelope function we find the equation for the correction to the envelope function 82 Fl ( t c2 Lc2 n(, 8z2 92 F 2)- ~- F O~z, t). (3.33) For the forward propagating wave at the boundary conditions are Fi (z, t = -oo) = 0, (3.34) Fi (z = -oo, t) = 0. (3.35) The Green's function for FI (z, t) satisfies 82G(z) 8z2 w2 ,2 (z) = 6(t) (3.36) The solution with the boundary condition (3.35) taken into consideration is G(z) G(z) 0 (z < 0), (3.37) (3.38) C sin(wz/v~) + D cosW"I' (z > 0). 0. Integrating Eq. (3.36) From the continuity condition at z 0 we have D about z = 0 we have 8G(0+) iz 8G(-0-) iOz (3.39) which gives C v/w~c. Therefore the Green's function is G(z () =- smn -(z (3.40) Using the above Green's function we can obtain the correction to the envelope function C() 'U2 (, t) ( ) d(, GY(z FI (z, t) (3.41) -So- e-iwc/'e-y(t-r-c/w)2 (3.42) Before the pump laser is turned on the transmission intensity is (3.43) After the pump laser excitation the transmission intensity becomes TP(z, Lc) = |SoP(z, Lc) + t 1(z, Lc)|2, (3.44) d(C sini -( () To(z, c) = |So(z, Lc)|12. where, SKP1, ) is ; the probe electric field in the absence of strain but with the pump 011. By definition, the differential transmission is given by (AT ATi (3.48) T Tox Tci ()0(X 1c)1 (3.49) of strain to the differential transmission is 2Be (SK(c w)*El(z:, L)) ( 4 n Buc r" Be~ J + i33- disin -(a (3.50) ()) e-aW!1)-c where the first term of the differential transmission gives the background and the coherent phonon oscillation comes from the strain induced second term. 3.4 Three Kinds Of Coherent Phonons In the experiment three kinds of different coherent phonons have been ob- served. They are the optical coherent phonons in bulk semiconductors, the acoustic coherent phonons in superlattices or 1\IWs, and the propagating coherent phonon wavepackets in epilayer systems. The contribution The coherent optical phonons have been observed in many different materials. In layered or low-symmetry materials such as Sb, Bi, Te, and Ti20s [20, 3] the Al phonon modes can be excited by the deformational phonon coupling mechanism. In cubic materials such as Ge [21], the driving force vanishes for deformational potential coupling. What causes the driving force not to vanish is the finite absorption depth of the ultrafast laser pulse and band structure effects. But the oscillations are weaker than that of the other systems. In polar materials like GaAs [22] the coupling between electrons and phonons can be both deformational and polar with the latter stronger and being the dominant effect. As an example Cho et. al. measured the differential reflection of (100)-oriented bulk GaAs for three different angles 8 between the probe beam polarization and the [010] w.i--L.1l axis. The orthogonally polarized beams are kept close to normal incidence. In the case of 8 = 900 the reflectivity rises during and peaks towards the end of the excitation pulse. After passing through a minimum at a time delay of 200 fs the reflectivity rises again to a quasistationary value on a picosecond time scale, without any periodic oscillations. The temporal evolution of the reflection transient is entirely due to susceptibility changes induced by the optically excited electronic carriers, their thermalization, and relaxation down to the band edges as well as intervalley transfer. Rotating the sample to 8 = 450 results in a shift of the entire reflectivity signature to higher values. The signal response for positive time delays shows a periodic modulation with an oscillation frequency of 8.8 + 0.15 THz, exactly matching the frequency of the Fis LO phonon in GaAs. In the case of 8 = 1350 a complementary shift to lower reflectivity changes together with an additional phase shift of x in the phonon-induced oscillations is observed. The distinct dependence of reflection modulation on probe beam polarization shows the electro-optic nature of this effect, which is well known for cubic zinc-blende group materials with 43m (li--1.11 symmetry. In a bulk semiconductor the laser wavelength is much larger than the lattice -1.. heil~. so the photo-generated carriers are typically excited by the optical pump over spatial areas that are much larger than the lattice unit cell, which means the excited carrier populations are generated in a macroscopic state and the carrier density matrix has only a qm 0 Fourier component. As a result only the qm 0 phonon mode is coupled to the photoexcited carriers. Since the frequency of the qm 0 acoustic phonon is zero, only the coherent optic phonons are excited in bulk semiconductors. Now the case is different for semiconductor multiple quantum wells and superlattices. The pump can preferentially generate electron-hole pairs only in the wells even though the laser pump has a wavelength large compared to the lattice spacing. So the photoexcited carrier distributions have the periodicity of the superlattice. Since the density matrix of the photo-excited carrier populations now have a q f 0 Fourier component, the photo-excited carriers can not only couple to the optical phonon modes, but they can also generate coherent acoustic phonon modes with a nonzero frequency and wavevector q m 2x/L where L is the superlattice period. In the next chapter I will discuss the coherent acoustic phonons in multiple quantum wells. The experimental data and theory of propagating coherent phonon in GaN/InGaN epilayer system will be presented in chapter 5. CHAPTER 4 THE COHERENT ACOUSTIC PHONON The second type of coherent phonon is the acoustic one. The earlier experi- mental observations of zone-folded acoustic phonons are made by Colvard et al. [23] They used a photoelastic continuum model to predict the scattering intensities of the folded acoustic modes. The acoustic branch of semiconductor superlattices folds within the mini- Brillouin zone because of the artificial periodicity of the superlattices. This leads to the observation of coherent oscillation of the zone folded acoustic phonons in A1As/GaAs superlattices [24, 25]. The observed differential reflection oscillation was very small, however, on the order of AR/R ~ 10-5 10-6. The detection mechanism was based on interband transitions due to the acoustic deformation potential [25]. In the year 2000 Sun et al. reported huge coherent acoustic phonon oscillations in wurtzite (0001) InGaN/GaN multi-quantum well samples with strain induced piezoelectric fields [26]. The oscillations were very strong with differential transmis- sion AT/T ~ 10-2 10-3 compared to the usual differential reflection on the order of 10-5-10-6. The oscillation frequency, in the THz range, corresponded to the LA phonon frequency with q m 2x/1, where 1 is the period of MQWs. The experiments of Sun et al. were done at room temperature on samples of 14 period InGaN/GaN MQWs with barrier widths fixed at 43 A~, well widths varying from 12 to 62 A~, and In composition varying between 6-10' The oscillation period for 50 A~ MQW was 1.38 ps, which corresponds to a frequency of 0.72 THz. By changing the period widths of MQWs, Sun et al. obtained a linear relation between the observed oscillation frequency wc and the wavevector q = 2x/1 of photogenerated carriers. The slope of this dispersion, 6820 m/s, agrees well with the sound velocity of GaN. In this chapter, I will discuss briefly a microscopic theory [27] for coherent acoustic phonons in strained wurtzite InGaN/GaN MQWs. This microscopic theory can be simplified and mapped onto a loaded string model instead of a forced oscillator model [1] as in the case for coherent optical phonons in bulk systems. Based on the string model the simulation will show the strain, energy density, and other p~llw--h.l1 quantities. I will also discuss the coherent control of the acoustic coherent phonon. 4. 1 Microscopic Theory The microscopic theory for coherent acoustic phonons in multiple quantum well system has many details and is very convoluted, so I will just give an outline of the theory. The basic approach is the same as that used for coherent optic phonons in bulk semiconductors, i.e. to obtain the total Hamiltonian of the system and then to derive the equations of motion for the coherent phonon amplitudes, which are coupled to the equations of motion for the electron density matrices. First of all one wants to obtain the energy dispersion relation and wavefunc- tions for carriers moving in a multiple quantum well system so that one can write down the electronic part of the Hamiltonian for the independent and free carriers. The GaN/InGaN multiple quantum wells is shown schematically in Fig. 4-1. The intrinsic active region consists of a left GaN buffer region, several pseudo- morphically strained (0001) In,Gay _,N quantum wells sandwiched between GaN barriers, and a right GaN buffer region. The total length of the MQWs between the P and N regions is L, across which a voltage drop, AV, is maintained. Pho- toexcitation of carriers is done using an ultrafast laser pulse incident normally along the (0001) growth direction, taken to coincide with the z-axis. GaN Barrier InGaN WNell GaN Barrier InGaN WNell n-GaN Figure 4-1: Schematic diagram of the In,Gal_,N MQW diode structure. The offsets and effects of the built-in piezoelectric field are shown in Fig. 4-2 from which we can see that the built-in piezoelectric field is responsible for the spatial separation of the photoexcited electrons and holes so that the density distribution of these two kinds of carriers differ from each other which results in a non-zero non-uniform driving force for the coherent phonon oscillations. In bulk systems, the conduction and valence bands in wurtzite crystals including the effects of strain can be treated by effective mass theory. Near the band edge, the conduction band Hamiltonian is a 2 x 2 matrix because of the electron spin, while the Hamiltonian for the valence bands is a 6 x 6 matrix due to the heavy holes, light holes, and split-off holes. [28, 29] In quantum confined systems shown in Fig. 4-1, the bulk Hamiltonian is modified. The finite MQW structure breaks translational symmetry along the z direction but not in the xy plane. The quantum confinement potentials comes from three sources: (i) bandgap discontinuities between well and barrier regions, (ii) the strain-induced piezoelectric field, and (iii) the time-dependent electric field due to photoexcited electrons and holes. Vo (z, t) = Vol,gap(t Vpiezo(t Vphoto(X -) 41) where a~ = {c, v}) refers to conduction or valence subbands. The confinement of carriers in the MQW leads to a set of two-dimensional subbands. In envelope function approximation, the wavefunction consists of factors of a slowly varying real envelope function in the superlattice direction and a rapidly varying Bloch wavefunction. Solving Schroidinger equations, one obtains the eigenvalues as the subband energies, Eg(k), with n the subband index, and the corresponding eigenvectors as the envelope functions. With this energy dispersion, one can write down the second I -30 """ -0.01 1.0 0.5 0.0 -0.5 -3 -20 -10 0 Position (nm) 10 20 30 - ... IIIIIIIII IIIIIIIII ..IIIIIII Electric Field ( MV/cm) ----- Electron Potential (eV) -I - -20 -10 0 10 20 30 Position (nm) 0.6 conduction band 0.4 (shifted 3.15 eV) 0.2 0.0 -0.2 valence band -0.4 -30 -20 -10 0 10 Position (nrn) 20 30 Figure 4-2: Effects of the built-in piezoelectric field to the bandgap of the MQWs. The upper figure shows strain tensor components for pseudomorphically strained InGaN MQW diode as a function of position. The middle figure shows the electric field and potential due to the the above built-in strain field. The lower figure shows the conduction and valence band edges. The applied dc bias has been adjusted so flat-band biasing is achieved, i. e., so that band edges are periodic functions of position. [See Sanders et al., PRB, 64:235316, 2001] quantized Hamiltonian for free electrons and holes in the multiple quantum well, a,n,k where: c~,n,k and ca,n,k are electron creation and dlestruction operators in condulction and valence subbands respectively. The acoustic phonons in the multiple quantum well are taken as plane-wave states with wave vector q. Due to the cylindrical symmetry of the system, one need consider only longitudinal acoustic phonons with q = q ^a because these are the ones coupled by the electron-phonon interaction. The free LA phonon Hamiltonian is given by .FA0l qb b,~. (4.3) wher-e bJ and b, arle creation and destr-uction oper-ator-s for LAi phonons with wavevector q = q ^a and Lci = c |q| is the linear dispersion of the acoustic phonons with c being the LA phonon sound speed. Statistical operators are normally defined in terms of the electron and phonon eigenstates. So one can write down the electron density matrix as where ( ) denotes the statistical average of the non-equilibrium state of the system. The diagonal component of the electron density matrix is the distribution function for electrons in the subband, while other components describe the coher- ence between carriers in different subbands, including both intraband and interband components. The coherent phonon amplitude of the q-th phonon mode, |q), is defined the same as in the last chapter [1] D, (t) E ( b (t) + b _, (t) ) (4.5) Note again that the coherent phonon amplitude is related to the the macro- scopic lattice displacement U(z, t) through Uitz, t) = "82 Bsq D(t) (4.6j) 2po (hLcq) To derive the equations of motion for the electron density matrix and the coherent phonon amplitude, one must know the total Hamiltonian of the system, which in this case is given by 'F = 'Feo +'~,- N Fee + N t + NA0 FeA -n. The first term is just free electrons and holes as in Eq. (4.2). The second term describes the Coulomb interaction between carriers, including screening and ne- glecting the Coulomb-induced interband transitions because they are energetically unfavorable [30]. The third term describes the creation of electron-hole pairs by the pump laser, the electric field of which is treated in the semiclassical dipole approximation. The last two terms are related to the acoustic phonons. One is the free longitudinal acoustic phonons given by Eq. (4.3) and the other is the electron-LA phonon interaction, which describes the scattering of an electron from one subband to another by absorption or emission of an LA phonon. NFeA has the same form as that of the bulk semiconductor. The interaction matrix includes both deformation and screened piezoelectric scattering. In the end one obtains a closed set of coupled partial differential equations for the electron density matrices and coherent phonon amplitudes. The equation of motion for the coherent phonon amplitude D,(t), which is similar to the case of bulk semiconductor, is given by a driven harmonic oscillator equation , 82D, (t) 8t2 +t ef D2,(t) = f (N, w, q, t), (4.8) where N is the electron density matrix defined in Eq. (4.4), and the initial condi- tions are 8D,(t = -oo) D4(t = -oo) 0. (4.9) 4.2 Loaded String Model The microscopic theory is very detailed. However we are more interested in the lattice displacement U(z, t). It is both more insightful and much easier to deal with the lattice displacement directly. Given the linear acoustic phonon dispersion relation, we can do Fourier transformation on the equation of motion for the coherent phonon amplitude D,(t). The result is the following equation for the lattice displacement U(z, t), 82 -C @,, = S(z, t), (4. 10) subject to the initial conditions dU(z, t =-oo) U(z, t = -oo) 0. (4.11) The forcing function S(z, t) packages all the microscopic details. Since the absorption occurs only in the wells, the forcing function is not uni- form. It normally has the same period as the superlattice. The sound speeds of GaN and InGaN are almost the same as discussed in the introduction chapter. The difference of sound speed for a typical In component of x: = 0.08 is about ''' .. which causes an even smaller reflection on the order of 10-4, SO this difference can be safely neglected. Thus, Eq. 5.17 describes a uniform string with a nonuni- form forcing function. We call this one-dimensional wave equation of the lattice displacement the --1 ang model". Note the difference between a bulk system and a MQW or superlattice. In a bulk semiconductor, both the amplitude U and the Fourier transform of the amplitude D,(t) for an q m 0 optic mode satisfy a forced oscillator equation. For the nonuniform, multiple quantum well case, one can excite acoustic modes with q f 0. The Fourier transform of the amplitude D, obeys a forced oscillator equation, but owing to the linear dependence of Lc(q) on q the amplitude itself U obeys a 1-D wave equation with a forcing term S(z, t). Under ideal conditions the forcing function takes a simple form [27] S(z, t) = S,(z, t), (4.12) where the index, v, is for different kinds of carriers and the forcing function satisfies the sum rule dz S(z, t) = 0, (4.13) /OO which requires that the average force on the string be zero so that the center of mass of the string would have no accelerating motion. The partial driving functions, S,(z, t), are given by [27] S,(z, t)= a +4||esp(z )41) po 8zp(t ea,14 where the plus sign is for conduction electrons and the minus sign for holes. The photogenerated electron or hole number density is p,(z, t) and po is the mass density. The partial forcing function has two terms, the first due to deformation potential scattering and the second to piezoelectric scatteringf. The piezoelectric coupling constant, ess, is the same for all carrier species, while the deformation potential, a,, depends on the species. Another interesting point is that Planck's constant cancels out in the string model. It does not appear in either the string equation (5.17), or in its related forcing function defined in Eqs. (4.12) and (4.14). So essentially, coherent LA phonon oscillations in an MQW is also an classical phenomenon, like the coherent LO phonon oscillations in bulk semiconductors [1]. 4.3 Solution of The String Model Using Green's function method, we can solve the wave equation (5.17) for a given forcing function with the initial conditions in Eq. (4.11). The general solution is given by U(z, t) = dr~x d( G(z (, t 7) S((, 7). (4.15) In performing the above integration of the forcing function S(z, t) over time we differentiate between two cases of the forcing function. One case is for the impulsive force, which is a delta function in time, fi(z, t) = 6(z) 6(t). (4. 16) The Green's function is given by G,(z, t) = 0e(t) 0(z -tct) 0(z ct) ,(4.17) where the two terms in the solution stands for the wave propagating along the positive and negative direction of z axis respectively. The other case is for the displacive force, which is a Heaviside step function in time, fd(z, t) = 6(z) 0(t). (4. 18) The Green's function is readily obtained by integrating Gi in Eq. (4.17), Gd(z, t) = 2 O ) Ig Z Ct) 2g Z) +g(z ct) (4. 19) 2c2 where function g(z) is defined as g (z) = z 0(z). (4.20) In addition to the two propagation terms as in the case of impulsive force, there is also a third term which is constant in time and sits right in the range of the multiple quantum well where the force function is nonzero. In our multiple quantum well system, the photogenerated carriers will persist for a certain amount of time to give a displacive forcing function S(z, t) = f(z)0(t - to). Thus the lattice displacement can be written as U(z, t) =_ d( Gd(z (, t to) f ((). (4.21) The only nonzero component of the elastic strain rjas is obtained from the derivative of the lattice displacement with respect to space coordinate z, dU (z, t) rsa (z, t) = (4.22) The kinetic and potential energy density uk, and u, according to definition can be obtained from the derivatives of the lattice displacement with respect to time and space coordinate respectively, uk Z, t) Oc ,Uxt (4.23) Up(z, t) oc r 3(z, t) = U x, t) 2424 If we substitute the displacive Green's function into the displacement equation and then calculate the kinetic energy density, we will obtain for to = 0 UkZ,1)Oc0() d~l(; [0z t ( 0(z-c -()] f((1) (4.25) where only two propagating term appear because dr and the integral of the first term with a Dirac 6-function is zero. We can calculate the following integral = d(0()f(z -S(s) = ds( f (z ) SF (z), and the kinetic energy density will be Uk Z, t) OC B 1) F& Z CE) -F Z Ce) 2 (.7 Similarly we can obtain the potential energy density n,(z, t) oc 0(t) [F~(z + ct) 2F(z) + F(z ct)]2 ~ (.3 From the above equations we can see that if F(z) is an oscillating function with a certain frequency then the kinetic energy density will oscillate with a doubled frequency because it is the square of two oscillating functions, but the potential energy will have the same frequency as F(z) due to the interference term between the static F(z) and the propagating F(z +t ct) and F(z ct). The total energy density is just the sum of the kinetic and potential part, u(z, t) = Uk~X p)tU(z, t). (4.29) The total energy E as a function of time is obtained by integrating the total energy density over the whole z axis, E(t) = Ek(t) + E,(t) = [uk/L t) pi~z, t)] dz. (4.310) If the forcing function is limited in some region with length L, then the kinetic energy will remain constant after t = L/2c at which time the two oppositely propagating parts does not cover each other anymore. The wave form does not change with the propagation, so the integral over the whole z-axis will remain constant t = L/2c. However the potential energy density will not become a constant until t = L/c which is twice the time of the kinetic energy density because the static part combined with the propagating part will keep changing until the propagating part is totally outside the forcing function region which gives the time L/c. Since the numerical calculation of microscopic theory [27] uses a multiple quantum well of four periods, we will first consider the four-period case for compar- 1SOH. Figure 4-3 shows the forcing function obtained from the microscopic model [27] and a simple sinusoidal forcing function used in the string model. The common features of both forcing function are their oscillations and periodicity in the range of the multiple quantum wells. In Fig. 4-4 we show the displacement of the lattice calculated from the string model using the simple four-period sinusoidal function shown in Fig. 4-3. It shows clearly the propagating parts of the motion and the static part inside the multiple quantum well system that remains behind when the dynamic parts travel outside the quantum wells region. From the displacement we can calculate the strain, which is shown in Fig. 4-5. The image of strain again shows clearly the propagation of the wave motion. Figure 4-6 shows the total energy density calculated from the string model. It has the very similar wave propagating features as shown in the lattice displacement and strain figures. We plot the energy as a function of time from the string model in Fig. 4-7. It has four peaks and four periods of oscillation, which is the same as the number of multiple quantum well periods. It also shows that the kinetic energy oscillates O C O o -2 -15 -10 -5 5 10 15 30 20 10 v E C t n ps Piezo Deforrn -10 -20 20 40 Position (c.u.) Total -40 Position (nm) Figure 4-3: Forcingf functions. The upper one is a sinusoidal function of four pe- riods corresponding to a four-period MQW. The lower one is from the numerical calculation of the microscopic theory, [See Sanders et al., PRB, 64:235316, 2001]. 20 40 60 10 ,10 -1Es i --J OE, Ba ds za O _zc;~-=~g~-aq ,,~,csac- 60 E a, -0 i (b t=4 40 20 =2\//~ S=l 1 -20 Position (c.u.) Figure 4-4: Displacement as a function of position and time. It is the solution of wave equation with the simple four-period sinusoidal force shown in Fig. 4-3 (,) Displacement 0 -60 -40 -20 0 Figure 4-5: The image of strain as a function of position and time from the strings model. Time (arb. unit) t= 2 JUV V y V vj V 9 \ t,=, 1~ .. (a~) EnergYDest R 1.5 o 60.5 C i. 2 a5F=_ RO ,2o ~o ,19"~i~;~-a0 ~,~e5~-40'h O, C (b) -60 -40 -20 20 (a.u.) 40 60 Position Figure 4-6: Energy Density as a function of position and time. After a long time (t=4 here), the energy density divides into a static part and a traveling part at twice the frequency of the potential energy and the time for the oscillation to die out of kinetic energy is half of the potential energy as discussed in Eq. (4.30). The total energy changes with time because we have treated the driving force as an external one. If we treat the driving force as internal, then the system will have a new equilibrium configuration, around which the calculated energy will be conserved. For the case of a multiple quantum well system of fourteen periods, the energy is plotted in Fig. 4-8. The energy plot has fourteen peaks and it dies out after fourteen periods of oscillation. Again the number of peaks and the number of oscillations is the same as the number of periods in the multiple quantum well The energy plot can be viewed in two ways. In the real space the wave will propagate with the acoustic sound speed in both directions of the x axis. After the wave travels totally out of the range of the multiple quantum well, the energy stops changing and remains constant, which explain the die out of the oscillation. The number of wells is like the number of sources of force. The waves they induce superimpose and gives the number of oscillations in the energy. In the wavevector space, because the periodic forcing function is confined in a limited range of multiple quantum wells, the wavevector q will have some uncertainty Aq. Thus the angular frequency Lc of the coherent acoustic phonon oscillation will also have an uncertainty Asc with Aw/w~c = Aq/q. It is this uncertainty Asc that leads to a dephasing of the oscillation. Since the more periods of quantum wells and the greater extent of the system will produce a smaller uncertainty Aq. So the die-out time of oscillation is related to the number of quantum wells, more specifically it is related to the total length of MQWs divided by the sound velocity in the MQWs. 40 (a) a ,.s .-s Poten tialI O l!i ! c Kinetic II O 10 20 30 Time (a.u.) 2- E Pulse Shape a 031 ~ n~_Total /! / Potential \\ 1I i K inetic 0 -20 2 4 6 8 Time (ps) Figure 4-7: Energy as a function of time for 4 quantum wells. The number of peaks of total energy equals the number of wells. The kinetic energy oscillates twice fast as the potential energy. (b) is from calculation of microscopic theory, see [See Sanders et al., PRB, 64:235316, 2001]. 1 50 100 50 0 20 ,- -'il'' Potential K inetic LL 0 100 120 40 60 8 Time (a.u.) Figure 4-8: Energy as a function of time for 14 quantum wells. This figure shows the same features as that of figure 4-7(a). 4.4 Coherent Control In recent years there are quite a few coherent control experiments with multi- ple quantum wells from several different groups such as Sun [31], Ojzgiir [32], and Nelson [33]. In InGaN/GaN multiple quantum wells the effect is more prominent because of the stronger piezoelectric field. By using double-pulse pump excitation Dekorsy et al. showed the controlling of the amplitude of the coherent optical phonons in GaAs [34]. Hase et al. conducted this kind of optical control of coherent optical phonons in bismuth films [35]. Bartels et al. reported coherent control of acoustic phonons in semiconductor superlattices [36]. In these experiments, the pulse train of pump laser was splited into two or more beams that travels different optical path distances by means such as Michelson interferometer. The sample will excited twice or more at different times. The changes in reflectivity were measured the same as the pump-probe setup. The results from these experiments show that the coherent phonons can interfere with each other and can be coherently controlled by using multiple optical pump trains. Depending on different phase delays of pump pulses, the coherent oscillation can be enhanced or annihilated. In this section I will use the string model to analyze the coherent control of coherent acoustic phonons. In the generation of coherent acoustic phonons, the ultrafast laser pump generates electron-hole pairs, which is the source of the forcing function. In the string model we use a simplified sinusoidal forcing function to replace the microscopic one. The coherent control uses two pump pulses, which mean we will have two forcing functions. In experiments, one can control the time interval between the two pump pulses. Theoretically, we can control not only the time interval between the start time of the two forcing function but also the phase difference between them. Based on the string model, we can try coherent control theoretically by using a forcing function with two terms. The first forcing term takes effect at time t = 0 and the second forcing term appears at a later time t = to with a relative spatial phase zo and amplitude A with respect to the first term. Note that while the first forcing term corresponds directly to the excitation pump, the relation between the second forcing term and the control pulse is not so direct and clear because the second forcing term will depend on both the excitation pump and the control pulse. The forcing functions have the form A sin(z zo) 0(t to), describing displacive forces starting at time to and having an initial phase of zo. The amplitude A is taken as unit for the first force. But the second force may have a relative larger or smaller amplitude. The phase effects have two aspects. One is the temporal phase effect due to the time interval between the two forcing terms. The other is the spatial effect caused by the relative initial phase difference between the spatial parts of the two forcing terms. First let us consider the effects of the temporal phase and the relative ampli- tude. The total forcing function can be written down as S(z, t) = sin(z) 0(t) + A sin(z) 0(t to). (4.31) When the time intervals to are even multiples of x, the two terms in the forcing function will be temporally in phase. In the cases of to = (2n +t 1)x with n being an integer, the two terms will be temporally out of phase. To save numerical computation time without losing any features we calculate the potential energies only which are proportional to the square of the strain. Figure 4-9 shows the temporally in-phase constructive enhancement of the oscillations. The time interval in this case is 4x, the relative amplitudes of the second forcing term are 1.5, 1, and 0 respectively. The lowest curve with zero relative amplitude means that there is only one forcing term. We see that no matter what the relative amplitudes are the constructive enhancement effects are always there. 600 500 400 300 100 r. I\I 20 40 60 80 Time (arb. unit) 100 Figure 4-9: Potential energy as a function of time for 14 quantum wells with two forcing terms delayed by an in-phase time interval of 4x. This figure shows the constructive enhancement of the oscillations as a result of coherent effect of the two forcingf terms. The destructive effects of two temporally out of phase forcing terms are plotted in Fig. 4-10. The lowest curve is for one forcing term only. The other three curves have the relative amplitudes 0.7, 1, and 1.5 respectively. The destructive effect is most obvious when the relative amplitude of the second force is smaller than the first, while for larger amplitudes the oscillations reduces but does not disappear. Since the coherent phonon oscillations have a tendency to dephase, the oscillations will become smaller with time. To further reduce the oscillation amplitudes the second pump pulse does not need to be stronger. It does not need even to reach the same strength as the first one. This is why the case of a force with relative amplitude A = 0.7 has the strongest destructive effect. 500 400 300 200 1 0 0 V ,\ ' I V I II lilalal 20 4060 80 Tim (arb unit)\ 100 Figure 4-10: Potential energy as a function of time for 14 quantum wells with two forcing terms delayed by an out of phase time interval of 5x. This figure shows the destructive effect of the oscillations as a result of coherent control by two forcing terms. Using the formalism discussed in Chapter 3 we can calculate the differential transmission. To further simplify the matter, the sensitivity function is taken to be a constant in the wells. Outside of the wells it is zero. Fig. 4-11 shows the constructive and destructive effects of the two forcing terms as discussed above. I'I'I'I'I'I :: - in phase 500 400 300 200 100 0 ~ out of phase ;:force: cos z) : -.force: sin(z) '-' 0 20 100 120 t (arb. unit) Figure 4-11: The change of transmission as a function of time for 14 quantum wells with two forcing terms delayed by an in phase time interval of 4x and an out of phase time interval of 5x. The upper and lower two sets of curves are displaced with regard to each other for a better view. The temporal phases, i.e. the time intervals between the two forces, are easy to control experimentally by varying the optical path of the two pump pulses, but the control of the spatial phases zo is not so obvious. In theory, however, it is not hard to observe the effects of spatial phases zo on the coherent control. We will take the amplitudes of the two forces to be the same. If we keep the temporally in phase time interval as to = 4x, then we can have four typical cases of the two forcing term. Case Il: S(z, t) = cos(z) 0(t) + cos(z ~) 0(t 4x). Case 12: S(z, t) = cos(z) 0(t) +t cos(z x) 0(t 4x). Case IS: S(z, t) = cos(z) 0(t) +t cos(z ~) 0(t 4x). Case 14: S(z, t) = cos(z) 0(t) + cos(z) 0(t 4x). These cases are plotted in Fig. 4-12. We have already discussed case (I4) in the above, which is the constructive temporally in phase enhancement of the oscillations. For case (I2) at time t > 4x, the sum of the two forcing terms will give a zero total forcing function. As a result, the oscillation will totally disappear. In cases (I1) and (IS) the spatial part of the total forcing function is given by cos(z) + sin(z). The amplitudes of the oscillations for these two cases are not as big as the coherent case (I4) where the phases lock in both temporally and spatially. Thus for temporally in phase time interval, if the spatial phases are between the two extremes of in phase and out of phase as shown in case (I4) and (I2) respectively, the enhancement of the oscillations will also vary between the totally coherent and the totally destructive. There are four similar cases for the temporally out of phase time interval. Again we take to = 5x. The four cases are listed below. Case 01: S(z, t) = cos(z) 0(t) +t cos(z () 0(t 5x). Case 02: S(z, t) = cos(z) 0(t) +t cos(z x) 0(t 5x). 300 \ 14: cos + cos - -- 11,3: cos+/-sin 250 IIII -------- 12: cos cos - = 200-- S 150-- 100- - 50-- 0 20 40 60 80 100 Time (arb. unit) Figure 4-12: Effects of phase zo on the change of transmission as a function of time for 14 quantum wells. The two forcing terms are delayed by an in phase time interval of 4x. The different curves are for different combination of phases zo as discussed in the text. Case 03: S(z, t) = cos(z) 0(t) +t cos(z ~) 0(t 5x). Case 04: S(z, t) = cos(z) 0(t) +t cos(z) 0(t 5x). Figure 4-13 shows the changes of transmissions for these four temporally out of phase cases. In cases (01), O(3), and (04) the spatial part of the total forcing function is the same as the corresponding temporally in phase cases, i.e., cos(z) + sin(z) and we see the expected destructive effects of the oscillations, although the reduction is not drastic because the relative amplitude of the second forcing term is as big as the first. For the spatially out of phase case (02) the total forcing function is identically zero at times t > to = 5x, but the oscillation persists, which is a distinct contrast to the corresponding case (I2). In case (I2) the oscillation almost totally disappear after the second temporally in phase force comes in because at time t = to = 4x the oscillation reaches one of its lowest points the value of which is nearly zero. However, in case (02) at the starting time t = to = 5x the oscillation is at one of its highest point. Although from then on there is no forces any more, the oscillation will keep on like a free oscillator. Note that we have not taken into consideration of any damping force, but the oscillation still dies out. The dephasing time is about a half of that of the one forcing term case. This dephasingf effect, as we mentioned before, is due to the limited range of the multiple quantum well system. 4.5 Summary We have discussed briefly a microscopic theory for the generation and detec- tionl of cohlerent acoustic phonons in G7aN/InGaN multiple quantum well system. Following the outlined procedures we arrive at a set of coupled differential equa- tions of motion for the electron density matrices and coherent phonon amplitude. The latter ones can be mapped onto a one-dimensional wave equation called the 150 S01,3: cos+/-sin --- 02: cos cos 320- 0 -------- 0:cs o 60 20I 40 6 8 0 Tie(rb nt Figure~~~~~~~ ~ ~~~~~~ 4-3 fet fpaez ntecag ftasiso safnto ftm for~~~~~~~~~~~ ~ ~~~~~~~ 14qatmwls h w ocn em r eae ya u fpaetm interval ~ ~ I: of 5x Th ifrn uvsaefrdfern obnto fpae oa discused n thetext string model, which describes the motion of a uniform string under a non-uniform forcingf function. We can solve the string model using Green's function method to obtain the lattice displacement and other related pkli~-i~... quantities such as strain, energy density, energy, and the differential transmission. All these quantities have the oscillating property. The strain and energy density propagate in both directions along the superlattice growth axis z. The oscillations of energy and differential transmission have the same number of peaks as the number of the wells and they dies out because of the limited range of the multiple quantum well system. Another interesting application of the string model is coherent control of these coherent acoustic phonons. By using two forcing terms corresponding to two pump pulses, we can change three elements of the second forcing term with respect to the first, the temporal phase, the spatial phase, and the relative amplitude. The temporal phase is the most effective control. We have shown both the temporally in phase constructive and out of phase destructive enhancement. To get better destructive effect the relative amplitude of the second forcing term should be small than the first. In spatially out of phase cases the total forcing function will be zero after the starting time of the second forcing term, but there is a distinctive contrast between the temporally in phase and out of phase cases. In the former case the oscillation almost disappear, while in the latter case the oscillation persists with a dephasing time a half of the case when there is only one forcing term. CHAPTER 5 PROPAGATING COHERENT PHONON Heterostructures of GaN and InGaN are important materials owing to their applications to blue laser diodes and high power electronics [6]. Strong coherent acoustic phonon oscillations have recently been detected in InGaN/GaN multiple quantum wells [26, 37]. These phonon oscillations were much stronger than folded acoustic phonon oscillations observed in other semiconductor superlattices [24, 25, 38]. InGaN/GaN heterostructures are highly strained at high In concentrations giving rise to large built-in piezoelectric fields [39, 40, 41, 42], and the large strength of the coherent acoustic phonon oscillations was attributed to the large strain and piezoelectric fields [26]. In this chapter, we discuss the generation of strong localized coherent phonon wave-packets in strained layer In,Gal_,N/GaN epilayers and heterostructures grown on GaN and Sapphire substrates [43]. This work was done in cooperation with Gary D. Sanders, Christopher J. Stanton, and Chang Sub K~im. The experi- ments were performed by J. S. Yahng, Y. D. Jho, Kt. J. Yee, E. Oh and D. S. K~im. (See cond-mat/0310654.) By focusing high repetition rate, frequency-doubled femtosecond Ti:Sapphire laser pulses onto strongly strained InGaN/GaN heterostructures, we can, through the carrier-phonon interaction, generate coherent phonon wavepackets which are initially localized near the epilayer/surface but then propagate away from the surface/epilayer and through a GaN layer. As the wavepackets propagate, they modulate the local index of refraction and can be observed in the time-dependent differential reflectivity of the probe pulse. There is a sudden drop in the amplitude of the reflectivity oscillation of the probe pulse when the phonon wave-packet reaches a GaN-sapphire interface below the surface. Theoretical calculations as well as experimental evidences support this picture; the sudden drop of amplitude when the wave encounters the GaN-sapphire interface cannot be explained if the wave-packet had large spatial extent. When the wave-packet encounters the GaN- sapphire interface, part of the wave gets reflected while most of it gets transmitted into the sapphire substrate, depending on the interface properties and the excess energy of the exciting photons. This experiment illustrates a non-destructive way of generating high pressure tensile waves in strained heterostructures and using them to probe semiconductor structure below the surface of the sample. Since the strength of this non-destructive wave is de~~~llltermne by ~lthe str~ain be~ttween GaN and InGaN, it is likely that even stronger coherent phonons can be generated in InGaN/GaN digital alloys grown on a GaN substrate. 5.1 Experimental Results In the experiments, frequency-doubled pulses of mode-locked Ti:sapphire lasers are used to perform reflective pump-probe measurements on four different sample types which are shown in Fig. 5-1: Type (I) InGaN epilayers; Type (II) In~7aN/GaN double quantum wells (DQ vvs); Type (III) In~7aN/GaN single quantum well (SQW); and Type (IV) InGaN/GaN light-emitting diode (LED) structures. The peak pump power is estimated to be 400 MW/cm2, COrresponding to a carrier density of 1019 cm-3 and the doubled pulse width is 250 fs. All samples were grown on a c-plane sapphire substrate by metal organic chemical vapor deposition. The InGaN epilayers shown in Fig. 5-1(a) consist of 1 pLm GaN grown on a sapphire substrate and capped with 30 nm of In,Gay _,N with In composition, x, varying from 0.04 to 0.12. The DQW sample shown in Fig. 5-1(b) consists of GaN (1 pLm), double quantum wells of In0.12Gao.ssN (1-16 nm) and barrier of GaN (1-16 nmr), and a G7aN cap layer (0.1 pLm). The SQW sample in Fig. 5-1(c) consists of G7aN (2 pLm), In0.12Gao.ssNu (24 nm), and GaN cap layer (0.1 pLm). The blue LED structure shown in Fig. 5-1(d) consists of n-GaN (4 pLm), 5 quantum wells of Ino.1sGao.ssN (2 nm) and 4 barriers of GaN (10 nm), p-Alo.1Ga0.9N (20 nm), and p-GaN (0.2 pLm). p-GaN 0.2pm p-Alo IGa0 9N 20nm GaN 0.1pm GaN 0.1pm Ino isGaosxsN 2nm x GaN 10nm/ In0 12Gao ssN (1~16nm) In0 12Gao ssN 24nm GaN (1~16nm) InxGal-xN 30nm 180 12Gao ssN (1~16nm) GN2mn-GaN 4pm GaN l ym GaN l ym Sapphire substrate Sapphire substrate Sapphire substrate Sapphire substrate (a) (b) (c) (d) Figure 5-1: The simple diagram of the sample structures used in these experi- ments. (a)InGaN epilayer (type I), (b)InGaN/GaN double quantum wells (type II), (c)GaN single quantum well (type III), (d)InGaN/GaN blue light-emitting diode structure (type IV). Differential reflection pump-probe measurements for the In,Gal_,N epilayers are shown in Fig. 5-1(a). Fig. 5-2 shows the oscillatory component of the measured probe differential reflectivity for the InGaN epilayers (type I) with various In con- centrations, x. For comparison purposes, differential reflectivity was measured on a pure GaN HVPE grown sample in order to show that no differential reflectivity oscillations are present in the absence of strain and an epilayer. The energy of the pump laser was varied between 3.22 and 3.35 eV, to keep the excess carrier energy above the InGaN band gap but below the GaN band gap. We note that if the laser energ-y was below the In~7aN band gap, no signal was detected.TI..foe carrier generation is essential to observing the oscillations, unlike a recent coherent optical phonon experiment in GaN [44]. The inset shows the pump-probe signal prior to the background subtraction for x = 0.12. The background results from the 0.06 0.08 0.05 1- 0.04- 0.00- 0.04 O 40 80 12C GaN (HVPE) 0.03- x =0.0 x = 0.10 0.01 x = 0.12 0.00 20 40 60 80 100 120 Time Delay (ps) Figure 5-2: The oscillatory component of the differential reflection pump-probe data for the In,.Gal_,.N epilayers with various In composition (:r=0.04, 0.08, 0.10, and 0.12). For comparison, differential reflection in a pure GaN HPVE grown sam- ple is shown. The reflection signal prior to the background subtraction for x: = 0.12 is shown in the inset. relaxation of the photoexcited electrons and holes. The oscillations are quite large, on the order of 10-2-10-3 and the period is 8-9 ps, independent of the In compo- sition but dependent on the probe photon energy. The amplitude of the oscillation is approximately proportional to the In concentration indicating that the strain at the InGaN/GaN interface is important. The observed period is approximately r = A/2Csn, where A is the probe beam wavelength, Cs the longitudinal acoustic sound velocity, and n the refractive index of GaN [19]. Two-color pump-probe experiments were performed for a type III InGaN SQW sample as shown in Fig. 5-1(b). Fig. 5-3 shows the differential reflectivity oscillations for different probe energies. Note that the period of the oscillation changes and is proportional to the probe wavelength. In addition, the amplitude of the differential reflectivity oscillation decreases as the detuning (with respect to the pump) becomes larger. The inset shows the oscillation amplitude as a function of the probe energy in a logarithmic scale and there is an ~-2-order-of magnitude decrease in differential reflectivity when the probe energy changes from 3.26 eV to 1.63 eV. Interestingly, Fig. 5-4 shows that the amplitude of the oscillatory component of the differential reflectivity is independent of the bias voltage, even though the carrier lifetime changes dramatically with voltage bias. Fig. 5-4 shows the bias dependent acoustic phonon differential reflectivity oscillations in a type IV blue LED structure (see Fig. 5-1(d)) at a pump energy of 3.17 eV. The lifetime of the background signal drastically decreases as the bias increases as shown in Fig. 5-4(a). This is due to the carrier recombination time and the decrease in the tunneling escape time in the strong external bias regime [45]. On the other hand, the amplitude and frequency of the oscillatory component of the differential reflectivity doesn't change much with bias voltage [Fig. 5-4(b)]. Since the observed reflectivity oscillation is independent of the carrier lifetime for lifetimes as short as 0.030 d & 10-4 10 * 0.025 -1 2 3 prb=3.26 eV Photon Energy (eV) 0.020 0; .015 -Ap b 3.18 eV 0.010 prb= 3.10 eV 0.005- a =1.63 eV x 30 probe 0.000 20 40 60 80 100 Time Delay (ps) Figure 5-3: The oscillation traces of a SQW (III) at different probe energies. The pump energy is centered at 3.26 eV. The bottom curve has been magnified 30 times. The inset shows the oscillation amplitude as a function of probe photon energy on a logarithmic scale. 0.15 ((8) 0b oV- 0.O10 0.10- 0.05 -10 V0.5 0.00 -0.000 0 40 80 120 40 80 120 Time Delay (ps) Time Delay (ps) Figure 5-4: Pump-probe differential reflectivity for the blue LED structure (type IV). (a) External bias varies at a pump energy of 3.17 eV. The decay time of the background signal is drastically reduced as the bias increases. (b) The oscillatory amplitude does not change much with bias. 1 ps due to ultrafast tunneling [bottom curve of Fig. 5-4(a)], it implies that once the source that modulates the experimentally observed reflectivity is launched by the sub-picosecond generation of carriers, the remaining carriers do little to affect the source. This -II__I that the reflectivity oscillation is due to the strain pulse which is generated at short times once the pump excites the carriers and modulates the lattice constant. 0.010 0.008 0.004 0.002 0.000 -0.002 0.004 0.003 0.002 0.001 0.000 20 40 60 80 100 20 40 60 80 100 Time Delay (ps) Time Delay (ps) Figure 5-5: The oscillatory component of the pump-probe differential transmission traces of DQW's (II) at 3.22 eV. The left figure shows the well width dependence and the right figure shows the barrier width dependence. Fig. 5-5 shows the well- and barrier-dependent acoustic phonon differential reflectivity oscillations in the type II DQW samples (see Fig. 5-1(b)) at a pump energy of 3.22 eV. The amplitude increases as the well width increases. However, the oscillation amplitude of the differential reflectivity doesn't change much with the barrier width. This means that the generation of the acoustic phonons is due to the InGaN layer (well) and not GaN layer (barrier). This also verifies that the oscillation is due to the strain in InGaN layer. Interesting results are seen in the long time behavior of the reflectivity oscillations shown in Fig. 5-6(a). The long time scale reflectivity oscillation is plotted for the epilayer (I), DQW (II), SQW (III), and the blue LED structure (IV) at 3.29 eV (below the GaN band gap). Astonishingly, the oscillation amplitude abruptly decreases within one cycle of an oscillation at a critical time which appears to scale with the thickness of the GaN layer in each sample. In addition, the slope of the GaN thickness vs. the critical time is very close to the known value of the sound velocity in GaN [inset of Fig. 5-6(a)] [26]. Fig. 5-6(b) shows results when the probe laser energy is changed to 3.44 eV which is above the GaN band gap. Then the laser probe is sensitive to coherent phonon oscillations only within an absorption depth of the surface. We see that the amplitude of the reflectivity oscillation exponentially decays with a decay time of 24.2 ps corresponding to a penetration depth in GaN of about 0.17 micron (=24.2 psx 7000 m/s). The oscillation reappears at 260 ps for epilayer (I) and 340 ps for DQW (II). This is twice the critical time for the oscillations to disappear when the photon energy is 3.29 eV. This further shows that the probe pulse is sensitive to the coherent acoustic wave. The "echo" in the probe signal results from the partial reflection of the coherent phonon off the GaN/sapphire interface. 5.2 Theory To explain the experimental results discussed in the last section, we have developed a theoretical model of the generation, propagation and detection of coherent acoustic phonons in strained GaN/InGaN heterostructures. The pump laser pulse generates a strain field that propagates through the sample which, in turn, causes a spatio-temporal change in the index of refraction. This change is responsible for the oscillatory behavior seen in the probe-field reflectivity in various -(b) a = 3.44 eV 260 ps 340 ps 0.020 -(a) a = 3.29 eV 0.015- 0.010 0.005 0 100 200 300 Time Delay 400 500 600 0.008 0.004~ 0.000 0 100 200 300 Tim~e Delay 400 (ps) 500 600 Figure 5-6: The long time-scale differential reflectivity traces. (a) Differential re- flectivity oscillation traces of an epilayer (I), a DQW (II), a SQW (III), and blue LED structure (IV). The inset shows the GaN thickness between the sapphire sub- strate and InGaN active layer of the sample as a function of the die out time of the oscillations. The solid line indicates that the velocity of the wavepacket in the GaN medium is about 7000 m/s. (b) The oscillation traces of epilayer and DQW [top two curves in (a)]at 3.44 eV which corresponds to a probe laser energy above the band gap of GaN. Experimental data are from the group of D. S. K~im. [APL, 80 4723, 2002] semiconductor heterostructures. An approximate method of solving Maxwell's equations in the presence of spatio-temporal disturbances in the optical properties and obtaining the reflectivity of the probe field in thin films excited by picosecond pump pulses can be found in Thompsen [19]. The spatio-temporal disturbance of the refractive index is caused by the propagating coherent phonon wavepackets. Thus, an essential ingredient in the understanding the probe reflectivity is a model for the generation and propagation of the very short strain pulse in the sample. Recently, a microscopic theory explaining the generation and propagation of such a strain pulse was reported by Sanders et al. [27, 46] where it was shown that propagating coherent acoustic phonon wavepackets are created by the nonequilibrium carriers excited by the ultrafast pump pulse. The acoustic phonon oscillations arise through the electron- phonon interaction with the photoexcited carriers. Both acoustic deformation potential and piezoelectric scattering were considered in the microscopic model. It was found that under typical experimental conditions, the microscopic theory could be simplified and mapped onto a loaded string model. Here, we use the string model of coherent phonon pulse generation to obtain the strain field seen by the probe pulse. First, we solve Maxwell's equations to obtain the probe reflectivity in the presence of a generalized spatio-temporal disturbance of the index of refraction. Let ab be the index of refraction without the strain which is real because initially the absorption can be neglected and let 6ft = Sn +t isc be the propagating change in the index of refraction due to the strain. When the effect of the change of the index of refraction is taken into account, the probe field with energy Lc can be described by the following generalized wave equation 82E(z, t) 112 8z2 C2 [nb nx l2E(z, t) = 0 (5.1) where E(z, t) is the probe field in the slowly varying envelope function approxima- tion and wc is the central frequency of the probe pulse. Eq. (5.1) is obtained from Maxwell's equations assuming that the polarization response is instantaneous and that the probe pulse obeys the slowly varying envelope function approximation. Since |Sh| < nb under typical conditions, Eq. (5.1) can be cast into 82E(z, t) 8z2 bk)2E(z, t) = 2nbk26n(z, t)E(z, t) (5.2) where k = w/cl is the probe wavevector. To relate the change of the index of refraction to the strain field, rl(z, t) dU(z, t)/8z, we assume |Sh| < nb and adopt the linear approximation [19] Sn(z, t) = q~,t.(5.3) We view Eq. (5.2) as an inhomogeneous Helmholtz equation and obtain the solution using the Green's function technique. The desired Green's function is determined by solving 82G(z, z') 8z2 Obk)2G(z, z') = 6(z z') (5.4) and the result is G(z, z') =exp (inbklz z'|I) (5.5) 2nbk Then, the solution to Eq. (5.2) can be written as E(z, t) = Eh(z, t)t + d~'Cz'Gz z') (-2nbk2 1, tl) Ei(z' t) } (5.6) SEh(z, t)t + d~'Cz'Gz z') (-2nbk2 1, tl) EhL(zi' t) } (5.7) where we have chosen the lowest Born series in the last line. In Eq. (5.7), Eh is the homogeneous solution which takes the form Eh(z, t) = Eh(z, t) exp {i(nbkz Lct)} for the probe pulse moving to the right in the sample without optical distortion. We now apply this approximate solution to our structure where the interface between air and the sample is chosen at z = 0. In the air, where z < 0, there is an incident probe pulse traveling toward the sample as well as a reflected pulse. The electric field in the air, E, (z, t), can thus be written as the sum E, (z, t) = Ei(z, t) ei(k~z-wt)t r Z, t) 6-i(kz+wt) (5.8) where Ei(z, t) and Er(z, t) are the slowly varying envelope functions of the incident and reflected probe fields, respectively. Inside the sample, z > 0, the solution is given as E>(z, t) = Eh(z, t)t dz'/ exp (i~nbk(z' z)) ik Sh(z', t) Eh(Xt = t(z, ig cr .. -"t) + A(nbk, t) Et(z, t)e-i(nbkz+wt) (5.10) where we used Eh(z, t) = Et(z, t) exp(i(nbkz Lct)) in the second step, assum- ing that Et(z, t) is nearly constant within the slowly varying envelope function approximation, and we define a reflected amplitude function A(Hk, ) E dz' exp (2inbkz') ik Sh(z', t). (5.11) The expression for the reflected amplitude function, A(nbk, t), in Eq. (5.9) says there is a frequency-dependent modulation of the amplitude in the reflected wave in the sample due to the propagating strain. Having determined the waves on both sides of the interface, we can now calculate the reflectivity. We apply the usual boundary conditions to the slowly varying envelope functions and the results are written compactly as -1 +A Er Ei ) j (5.12) We solve this equation to obtain -~~ = o +t A (5.13) Ei 1 + roA where ro = (1 ab)l ~ -tb). TO the same order, we find that Et/Ei a to(1 roA) where to = 2/(1 +t nb). It is HOw straightforward to calculate the differential reflectivity as AR |ro +arl2- 70 2 ~ ReA. (5.14) R |rol2 r0 Finally, by substituting the linear law Eq. (5.3) into Eq. (5.11) and using Eq. (5.14) we get AR ~",, U (z,t) dz F~, w)(5.15) R o, dz where the sensitivity function, FT(z, Lc), is defined as 2k On Onc F(z, w) = --sin(2nbkz) + co(nb,,nkz) (5.16) ro drl dr S "-""" In Eq. (5.15), the differential reflectivity is expressed in terms of the lattice displacement, U(z, t), due to propagating coherent phonons. Sanders et al. [27, 46] developed a microscopic theory showing that the coherent phonon lattice displacement satisfies a driven string equation, 8t s ,,2 S(z, t), (5.17) where Cs is the LA sound speed in the medium and S(z, t) is a driving term which depends on the photogenerated carrier density. The LA sound speed is related to the mnass density, p, and thle elastic stif'fnessh constant, Cas, by Cs, = .C~ Th'e LA~ soundU speed is taken to be different in the G7aN/InGaN heterostructure and Sapphire substrate. For simplicity, we neglect the sound speed mismatch between the GaN and In,Gai_,N layers. The driving function, S(z, t), is nonuniform and is given by S(z, t) = S,,(z, t), (5.18) where the summation index, v, runs over carrier species, i.e., conduction electrons, heavy holes, light holes, and crystal field split holes, that are created by the pump pulse. Each carrier species makes a separate contribution to the driving function. The partial driving functions, S,(z, t), in piezoelectric wurtzite crystals depend on the density of the photoexcited carriers. Thus, 1 8;ii |e| eSS (.9 S,(z, t) = s 4x ,z ),(.9 where the plus sign is used for conduction electrons and the minus sign is used for holes. Here p,(z, t) is the photogenerated electron or hole number density, which is real and positive, p is the mass density, a, are the deformation potentials, ess is the piezoelectric constant, and em is the high frequency dielectric constant. To be more specific, we will consider a SQW sample of the type shown in Fig. 5-1(c). We adopt a simple model for photogeneration of electrons and holes in which the photogfenerated electron and hole number densities are proportional to the squared ground state particle in a box wavefunctions. The exact shape of the electron and hole number density profile is not critical in the present calculation since all that really matters is that the electrons and holes be strongly localized. The carriers are assumed to be instantaneously generated by the pump pulse and are localized in the In,Gay _,N quantum well. Having obtained a model expression for p,(z, t), it is straightforward to obtain S(z, t) using Eqs. (5.18) and (5.19). To obtain U(z, t), we solve driven string equations in the GaN epilayer and the Sapphire substrate, namely 8 2 U(,, t 2 U(, , -' CJ = S(z, t) (0 < z < L) (5.20a) 8t2 ,, ,2 and 8 2 U(,, t)~ 2 U(, d C/12 X = 0 (L < z < Zs) (5.20b) where Co and C1 are LA sound speeds in the GaN and Sapphire substrate, re- spectively. In Eq. (5.20b), the Sapphire substrate has finite thickness. To simulate coherent phonon propagation in an infinite Sapphire substrate, Zs in Eq. (5.20b) is chosen large enough so that the propagating sound pulse generated in the GaN epilayer does not have sufficient time to reach z = Zs during the simulation. If Tsim is the duration of the simulation, this implies Zs > L +t C1 Tsim. Equations (5.20a) and (5.20b) are solved subject to initial and boundary conditions. The initial conditions are dU(z, 0) U(z, 0) 0. (5.21) At the GaN-air interface at z = 0, we assume the free surface boundary condition aU(0, t) = 0 (5.22a) since the air exerts no force on the GaN epilayer. The phonon displacement and the force per unit area are continuous at the GaN-Sapphire interface so that U(L e, t) = U(L +t e, t) (5.22b3) and 2n 8U(L e, t) 2 8, U(L+ ,) polo io= l 01 (5.22c) The boundary condition at z = Zs can be chosen arbitrarily since the propagating sound pulse never reaches this interface. For mathematical convenience, we choose the rigid boundary condition U(Zs, t) = 0. (5.22d) To obtain U(z, t) for general S(z, t), we first need to find the harmonic solutions in the absence of strain, i.e. S(z, t) = 0. The harmonic solutions are taken to be Un(z, t) = Wn(z) e "wn ( wei > 0 ) (5.23) and it is easy to show that the mode functions, W,(z), satisfy d2 n,,,) L2 +t "W,(z) = 0 ( 0 < z < L) (5.24a) dz2 C:2 and + W,(z) = ( L < z Zs ) (5.24b3) d z2 1-(2 Applying the boundary conditions from Eq. (5.22) we obtain the mode functions Wn(z) = cos (Lc,z/Co) if 0 with cos (wc,L/Co) B, =.(5.25b3) sin (wc,(Zs L)/C1) The mode frequencies, ci,, are solutions of the transcendental equation 1 o AL 1o L,(Zs L) (.6 cot tan (.6 poCo Po pi1 C1 which we solve numerically to obtain the mode frequencies, wei (n = 0, 1, 2, ...). The index, n, is equal to the number of nodes in the mode functions, W,(z). A general displacement can be expanded in terms of the harmonic modes as U(z, t) =) q,(t) W,(z). (5.27) n= o Substituting Eq. (5.27) for U(z, t) into Eq. (5.20) and taking the initial conditions from Eq. (5.21) into account, we find that the expansion coefficients, q,(t), satisfy a driven harmonic oscillator equation 4n(t)t + iw ,(t) = Q,(t), (5.28) subject to the initial conditions q,(0) = q,(0) = 0. The harmonic oscillator driving term Q,(t) is given by fo^ dz W,:(z) S(z, t) Q.(t) = z(5.29) In our simple displacive model for photogeneration of carriers, S(z, t)= S(z) 8(t) where 8(t) is the Heaviside step function. In this case, the lattice displacement is explicitly given by U~~z,, t)= cos(wc, t) ) Wn(z) (5.30) with S, defined as fo^ dz Wv(z) S'(z) Sn (5.31) So^" d z W, n(z) 2 Using the lattice displacement from Eq.(5.30), we obtain the time-dependent differential reflectivity at the probe frequency, Lc, from Eq.(5.15). The result is R, 2 COs(Lci t) ) Rn(w) (5.32) where /zs dWn(z) R,() dz F(z,Y iL) (5.33) Jo dz can be evaluated analytically. With the above formalism, we solve for the lattice displacement, U(z, t), for a coherent LA phonon pulse propagating in a multilayer structure consisting of a 1.124 pLm thick GaN epilayer grown on top of an infinitely thick Sapphire substrate with1 lthe growthI direction alonlg z. Wet atak ~rthe olgrigin utob at rthe G7aN- air interface and the GaN-Sapphire interface is taken to be at z = L = 1.124 pLm. We assume that carriers are photogenerated in a single 240 A~ thick In,Gal_,N quantum well embeddedut~ inl ~the GaN layer 0.1 pLm below the GaN-air interface and 1 pLm above the Sapphire substrate. Our structure thus resembles the SQW sample shown in Fig. 5-1(c). In the GaN epilayer, we take Cas = 379 GPa and po = 6.139 gm/cm3 [47 from whUich weII obta ~ in C/o= 7857 m/s. For the Sapphire substrate, we take Cas = 500 GPa and pr = 3.986 gm/cm3 [48] from which we find C = 11200 m/s. The results of our simulation are shown in Fig. 5-7. A contour map of the strain, dU(z, t)/8z, is shown in Fig. 5-7(a). We plot the strain as a function of the depth below the surface and the probe delay time. Photoexcitation of electrons and holes in the InGaN quantum well generates two coherent LA sound pulses traveling in opposite directions. The pulses are totally reflected off the GaN/air interface at z = 0 and are partially reflected at the Sapphire substrate at z = 1.124 pLm. Approximately 95% of the pulse energy is transmitted and only 5% is reflected at the substrate. The speed of the LA phonon pulses is just the slope of the propagating wave trains seen in Fig. 5-7(a) and one can clearly see that the LA sound speed is greater in the Sapphire substrate. From the strain the differential reflectivity can be obtained from Eq. (5.15). From Fig. 5-7(a), the strains, dU(z, t)/8z, associated with the propagating pulses are highly localized and travel at the LA sound speed. Each pulse contributes a term to the differential reflectivity that goes like AR Lc nb~c (i, t) ~ FT(Cot, W) oc smn Cot +) (5.34) R c c: The period of the oscillations of FT depends on the probe wavelength, A = 2xIc/w, with the result that the observed differential reflectivity oscillates in time with period, T = xc/(nboci 0 ""b 0) whr ab'-- = 2.4 is the index of refraction, and Co = 7857 m/s is the LA sound speed in GaN. For A = 377 nm (hac = 3.29 eV), this gives us T = 10 ps. The sensitivity function, FT(z, Lc), defined in Eq. (5.16) 1.5 (a) Sapphire S1.0 ..'... G N. In' Ga aNW 1 Ii I, .~ O -1 05 ~ (~~ eratedn~a inasnl nla elebd ed ina re tadn114mCaNeiar grw on to faSphr usrae n()acnorplto h tanfed 8Uz /2,i sow asafnto ofd thblwteCa-iitraeadth prb ea.I btersligdfeetilrfetvt nue ytesri il in(a s honasa uctono teprobe delay.ps is an oscillating function in the GaN/InGaN epilayer and is assumed to vanish in the Sapphire substrate. Our computed differential reflectivity is shown in Fig. 5-7(b) for a probe wavelength of A = 377 nm. We find that the reflectivity abruptly attenuates when the strain pulse enters the Sapphire substrate at t = 170 ps. The reflected strain pulses give rise to the weaker oscillations seen for t > 170 ps. These oscillations are predicted to continue until the reflected pulses are again partially reflected off the Sapphire substrate at t = 430 ps. 5.3 Simple model Since the coherent oscillation observed in the differential reflectivity stems essentially from the strain pulse propagating into the layers, most phenomena can be understood by a simple macroscopic model that is presented in this section. Instead of solving the loaded string equations for the strain to obtain a propagating disturbance in the refractive index, the propagating strain pulse at a given moment can be viewed as a thin strained layer in the sample, where the index of refraction is assumed to be slightly different from the rest of the sample. This situation is schematically depicted in Fig. 5-8 where a fictitious, thin GaN strained layer is located at z in the thick host GaN layer. The thickness of the strained layer, d, is approximately the width of the traveling coherent phonon strain field, dU(z, t)/8z, and is to be determined from the microscopic theory. From the last section, it was seen that the propagating strain field is strongly localized so that d is small. In the example of the last section, d is approximately equal to the quantum well width. Here we assume the strain pulse has been already created near the air/GaN interface and do not consider its generation procedure. We treat d as a phenomenological constant and also assume that the change in the index of refraction is constant. This strained GaN layer travels into the structure with the speed of the acoustic phonon wavepacket Co = 7 x 103 m/s, so the location of the stained layer is given as z = Co-r where -r is the pump-probe delay time. Air z z+d Figure 5-8: Propagating strained GaN layer in our simple model. The pump laser pulse creates a coherent acoustic phonon wavepacket in the InGaN layer near the air/GaN surface, which is modelled as a thin strained layer. The strained GaN layer propagates into the host GaN layer. The index of refraction in the strained layer is perturbed relative to the background GaN due to the strong strain induced piezoelectric field (Franz-K~eldysh effect). Strained GaN layer Cs Sapphire GaN GaN |

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PAGE 1 THEORETICAL STUDIES OF COHERENT OPTIC AND A COUSTIC PHONONS IN GaN/InGaN HETER OSTR UCTURES By R ONGLIANG LIU A DISSER T A TION PRESENTED TO THE GRADUA TE SCHOOL OF THE UNIVERSITY OF FLORID A IN P AR TIAL FULFILLMENT OF THE REQUIREMENTS F OR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORID A 2004 PAGE 2 T o Linlin and Bry an. PAGE 3 A CKNO WLEDGMENTS First and foremost I w ould lik e to thank Professor Christopher J. Stan ton, m y thesis advisor, for his guidance and encouragemen t throughout the course of m y graduate study here at the Univ ersit y of Florida. Among man y other things, he taugh t me ho w to construct and dev elop a theory from simple ph ysical ideas and mo dels. I also learned a lot from discussions with his p ostdo cs and graduate studen ts, esp ecially from Dr. Gary Sanders and Y ongk e Sun. I am honored and grateful to ha v e Professor P eter J. Hirsc hfeld, Professor Jerey L. Krause, Professor Mark E. La w, Professor Da vid H. Reitze, and Professor Da vid B. T anner to serv e on m y sup ervisory committee. I w an t to thank the McLaughlin family for the dissertation fello wship. Man y thanks go to the secretaries of the Ph ysics Departmen t, in particular, Darlene Latimer and Donna Balkum, who ga v e me m uc h indisp ensable assistance. It is a great pleasure to ha v e suc h w onderful friends as Bob and Janis Jac kson. The times w e are together are the happiest ones. I w an t to thank m y paren ts. They ha v e gone through all kinds of hardships to raise me and m y brothers. They alw a ys remind me to w ork hard, to w aste nothing, and to b e kind to other p eople. I also w an t to thank m y paren ts-in-la w. They sp en t a y ear with me and m y wife, whic h w as one of the happiest times. They help me and m y wife to tak e care of our little son Tian yue during our study Lastly I w an t to thank m y family esp ecially m y wife for all the lo v e, supp ort, and fun she brings to me. iii PAGE 4 T ABLE OF CONTENTS page A CKNO WLEDGMENTS . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . vi ABSTRA CT . . . . . . . . . . . . . . . . . . viii 1 INTR ODUCTION . . . . . . . . . . . . . . . 1 1.1 GaN/InGaN: Structures and P arameters . . . . . . . 2 1.2 The Dynamics of Photo excited Carriers and Phonons . . . 7 1.3 Exp erimen t Setup F or Coheren t Phonons . . . . . . . 9 2 DIPOLE OSCILLA TOR MODEL . . . . . . . . . . . 11 2.1 Optical Pro cesses and Optical Co ecien ts . . . . . . 11 2.1.1 Classication of Optical Pro cesses . . . . . . . 12 2.1.2 Quan tization of Optical Pro cesses . . . . . . . 13 2.2 The Dip ole Oscillator Mo del . . . . . . . . . . 16 2.2.1 The A tomic Oscillator . . . . . . . . . . 16 2.2.2 The F ree Electron Oscillator . . . . . . . . 18 2.2.3 The Vibrational Oscillator . . . . . . . . . 20 2.2.4 A Series Of Oscillators . . . . . . . . . . 22 2.3 Rerectance And T ransmission Co ecien t . . . . . . . 23 3 GENERAL THEOR Y OF COHERENT PHONON . . . . . . 29 3.1 Phenomenological Mo del . . . . . . . . . . . 29 3.2 Microscopic Theory . . . . . . . . . . . . . 30 3.3 In terpretation Of Exp erimen tal Data . . . . . . . . 33 3.4 Three Kinds Of Coheren t Phonons . . . . . . . . 37 4 THE COHERENT A COUSTIC PHONON . . . . . . . . 40 4.1 Microscopic Theory . . . . . . . . . . . . . 41 4.2 Loaded String Mo del . . . . . . . . . . . . 47 4.3 Solution of The String Mo del . . . . . . . . . . 49 4.4 Coheren t Con trol . . . . . . . . . . . . . 60 4.5 Summary . . . . . . . . . . . . . . . 67 iv PAGE 5 5 PR OP A GA TING COHERENT PHONON . . . . . . . . 70 5.1 Exp erimen tal Results . . . . . . . . . . . . 71 5.2 Theory . . . . . . . . . . . . . . . . 78 5.3 Simple mo del . . . . . . . . . . . . . . 90 5.4 Summary . . . . . . . . . . . . . . . 102 6 CONCLUSION . . . . . . . . . . . . . . . . 103 REFERENCES . . . . . . . . . . . . . . . . . 106 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . 109 v PAGE 6 LIST OF FIGURES Figure page 1{1 Band structure of a direct gap I I I-V semiconductor. . . . . . 5 1{2 Sc hematic setup for a t w o-b eam nonlinear exp erimen t. . . . . 10 2{1 Sp ectral dep endence of Seraphin co ecien ts and dieren tial rerectivit y 27 2{2 Sp ectral dep endence of the dielectric function for DT. . . . . 28 4{1 Sc hematic diagram of the In x Ga 1 x N MQW dio de structure. . . 42 4{2 Eects of the built-in piezo electric eld to the MQW bandgap. . . 44 4{3 F orcing function. . . . . . . . . . . . . . . 53 4{4 Displacemen t as a function of p osition and time. . . . . . . 54 4{5 The image of strain. . . . . . . . . . . . . . . 55 4{6 Energy densit y as a function of p osition and time. . . . . . 56 4{7 Energy as a function of time for 4 quan tum w ells. . . . . . 58 4{8 Energy as a function of time for 14 quan tum w ells. . . . . . 59 4{9 The temp orally in-phase oscillations of p oten tial energies. . . . 62 4{10 The temp orally out of phase reduction of oscillations. . . . . 63 4{11 Coheren t con trol of the c hange of transmission. . . . . . . 64 4{12 Eects of phase z 0 (temp orally in phase) on coheren t con trols. . . 66 4{13 Eects of phase z 0 (temp orally out of phase) on coheren t con trols. . 68 5{1 The diagrams of GaN/InGaN sample structures. . . . . . . 72 5{2 DR for the In x Ga 1 x N epila y ers with v arious In comp osition . . 73 5{3 The oscillation traces of a SQW (I I I) at dieren t prob e energies. . 75 5{4 DR for the blue LED at dieren t external bias. . . . . . . 76 5{5 Dieren tial transmission of DQW's (I I) at 3.22 eV. . . . . . 77 5{6 The long time-scale dieren tial rerectivit y traces. . . . . . 79 vi PAGE 7 5{7 Generation and propagation of coheren t acoustic phonons. . . . 89 5{8 Propagating strained GaN la y er in our simple mo del. . . . . 91 5{9 DR for dieren t frequencies of the prob e pulse. . . . . . . 93 5{10 The n umerically calculated dieren tial rerection. . . . . . . 95 5{11 Calculated DR v arying the thic kness of the strained GaN la y er. . 96 5{12 Sc hematic diagram of the single-rerection appro ximation. . . . 97 5{13 The absorption co ecien t as a function of the prob e energy . . . 100 5{14 The c hange of the dielectric function vs. the prob e energy . . . 101 vii PAGE 8 Abstract of Dissertation Presen ted to the Graduate Sc ho ol of the Univ ersit y of Florida in P artial F ulllmen t of the Requiremen ts for the Degree of Do ctor of Philosoph y THEORETICAL STUDIES OF COHERENT OPTIC AND A COUSTIC PHONONS IN GaN/InGaN HETER OSTR UCTURES By Rongliang Liu Ma y 2004 Chair: Christopher J. Stan ton Ma jor Departmen t: Ph ysics Coheren t phonons are collectiv e lattice oscillations whic h can p erio dically mo dulate the ph ysical prop erties of the crystal. Three kinds of exp erimen tally observ ed coheren t phonons are discussed in this thesis. They are the coheren t optical phonons in bulk semiconductors, the coheren t acoustic phonons in GaN/InGaN m ultiple quan tum w ells and sup erlattices, and the propagating coheren t phonon w a v epac k ets in GaN/InGaN heterostructures and epila y ers. A phenomenological oscillator mo del is presen ted to explain the oscillating c hanges in the rerection of bulk semiconductors. The prop er denition of the coheren t phonon amplitude as the quan tum-mec hanical a v erage of the phonon creation and annihilation op erators constitutes the basis of the microscopic theory of the coheren t phonons, whic h justies the macroscopic oscillator mo del. The lattice displacemen t is related to the coheren t phonon amplitude through F ourier transform. Since the laser w a v elength is m uc h larger than the lattice constan t, the photo excited carriers distribute uniformly and the the carrier densit y matrix has only the q 0 F ourier comp onen t. As a result the coupling of the photo excited carriers to the phonons leads to coheren t optical phonons only viii PAGE 9 The large piezo electric elds due to the built-in strain in GaN/InGaN semiconductor sup erlattices or m ultiple quan tum w ells can b e partially screened b y the photo excited carriers whose densit y has the same p erio dicit y of the sup erlattice. In resp onse the crystal relaxes to a new conguration whic h triggers the coheren t acoustic phonon oscillations. The microscopically deriv ed equation of motion for the coheren t phonon amplitude can b e mapp ed on to a one-dimensional w a v e equation for the lattice displacemen t whic h is called the string mo del. Based on the string mo del, the coheren t con trol of the coheren t acoustic phonons can b e carried out theoretically The last kind is the coheren t acoustic phonon w a v epac k ets generated and detected in InGaN/GaN epila y ers and heterostructures. W e constructed a theoretical mo del that ts the exp erimen ts w ell and helps to deduce the strength of the coheren t phonon w a v epac k ets. This mo del sho ws that lo calized coheren t phonon w a v epac k ets are generated b y the fem tosecond pump laser in the epila y er near the surface. The w a v epac k ets then propagate through a GaN la y er c hanging the lo cal index of refraction, primarily through the F ranz-Keldysh eect, and as a result mo dulate the rerectivit y of the prob e b eam. ix PAGE 10 CHAPTER 1 INTR ODUCTION Coheren t phonons are collectiv e lattice oscillations. When w e shine a laser pulse with duration m uc h shorter than the in v erse of a lattice vibration frequency the lattice mo de will b e excited coheren tly i.e., there will b e a large n um b er of phonons in one mo de with distinct phase relation [ 1 ]. This coheren t phonon mo de ma y b eha v e lik e macroscopic oscillators. The nonzero time-dep enden t lattice displacemen t mo dulates the optical prop erties of the material through the dielectric constan t, whic h can then b e detected b y c hanges in the rerection or transmission of a prob e laser pulse. There are basically t w o mec hanisms for the generation of coheren t phonons. One is the impulsiv e stim ulated Raman scattering (ISRS) rst observ ed b y F ujimoto and Nelson et al. [ 2 ] They excited and detected coheren t optic phonons b y ISRS in -p erylene crystals in the temp erature range 20{30 K. The impulsiv e stim ulated Raman scattering can o ccur with no laser in tensit y threshold and with only one ultrafast laser pulse b ecause the Stok es frequency is con tained within the bandwidth of the incoming pulse. ISRS is a ubiquitous pro cess through whic h coheren t excitation lattice will happ en whenev er an ultrafast laser pulse passes through a Raman-activ e solid. The other mec hanism is the displaciv e excitation of coheren t phonons (DECP). It is rst prop osed b y Dresselhaus group [ 3 ]. DECP requires a signican t absorption at the pump frequency so that an in terband electron excitation o ccurs, while ISRS do es not require absorption in the material [ 2 4 ]. The electronic excitation mak es the lattice to relax to a new quasiequilibrium conguration whic h then triggers coheren t phonon oscillations. 1 PAGE 11 2 With the dev elopmen t of ultra fast laser systems, the generation of coheren t phonons in semiconductors, metals, and sup erconductors b y femotosecond excitation of these materials has receiv ed considerable atten tion in recen t y ears b ecause of the p oten tial applications suc h as non-destructiv e measuremen t and the theoretical in terests. Shah has giv en a thorough review of the eld [ 5 ]. This dissertation discusses three kinds of coheren t phonons observ ed in the exp erimen ts: 1. the coheren t optical phonons in the bulk semiconductor, 2. the coheren t acoustic phonons in m ultiple quan tum w ells and sup erlattices, 3. and the propagating coheren t phonons in GaN/InGaN heterostructures and epila y ers. I will presen t theories ab out these three kinds of coheren t phonons and fo cus on the coheren t acoustic phonons and the propagating coheren t acoustic phonon w a v epac k ets. 1.1 GaN/InGaN: Structures and P arameters In 1995 Sh uji Nak am ura at Nic hia Chemical Industries in Japan rep orted the successful dev elopmen t of LEDs based on GaN comp ounds [ 6 ]. Since then researc h and application in terests ha v e b een in tensied on this material. The distribution of electrons in space and in momen tum and their energy lev els determine the electronic prop erties of a material. There are t w o implicit assumptions with regard to the electronic structure: 1. The electronic motion and n uclear motion are separable, 2. The electrons are indep enden t of eac h other. In principle neither of the assumptions is correct. The Hamiltonian of the system includes electron-electron in teraction, n ucleus-n ucleus in teraction, and electron-n ucleus in teraction. It is a complex man y-b o dy system. PAGE 12 3 But these assumptions are still practical appro ximations. They can b e justied b y rst noticing that the ratio of the electron mass to n uclear mass (10 4 {10 5 in a t ypical solids) is so small that the fast mo ving electrons can adjust to the motion of n uclei almost instan tly And secondly an y one electron exp eriences an a v erage p oten tial exerted b y all the other electrons. In an innite and p erfect crystal, according to Blo c h's theorem, the form of the eigenfunctions of the Sc hr odinger equation are the pro duct of a plane w a v e and a p erio dic function with the same p erio dicit y of the p erio dic p oten tial. The energy lev els are so close that they b ecome bands. The metho ds for band structure calculations fall in to t w o categories: Global metho ds to obtain the bands on the en tire Brillouin zone, Lo cal metho ds describing band structures near some sp ecial p oin ts (e.g., p oin t) inside the Brillouin zone. Just a few examples of the rst category include the tigh t binding metho d, the orthogonalized plane w a v e metho d, and the pseudop oten tial metho d. The p erturbativ e k p metho ds b elongs to the second category The metho ds for the band calculation can also b e classied as empirical and ab in tio. Most of the tec hniques are empirical whic h means that they need exp erimen ts to pro vide input parameters. Slater and Koster [ 7 ] w ere the rst to use the tigh t binding metho d empirically The atomic structure of the elemen ts making up GaN can b e describ ed as Ga 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 3 d 1 4 s 2 4 p 1 | {z } ; N 1 s 2 2 s 2 2 p 3 | {z } : The co v alen t b ond b et w een a gallium atom and a nitrogen atom is made b y sharing the electrons, as a result, eac h atom ends up with four electrons. In the ground state these v alence electrons o ccupies the s and p atomic orbitals. When the PAGE 13 4 gallium and nitrogen atoms come close to form a molecule, the s and p states form b onding and an tib onding molecular orbitals, whic h are s b onding, p b onding, s an tib onding, and p an tib onding orbitals in the order of increasing energy These molecule orbitals then ev olv e in to the conduction and v alence bands of the GaN semiconductor. Because of the sp ecic ordering of the molecular orbitals, the b ottom of the conduction band of a GaN semiconductor is s -t yp e, while the top of the v alence band is p -t yp e, whic h allo ws electric-dip ole transitions b et w een the t w o bands according to the selection rules. The band structure of the GaN semiconductor is sho wn sc hematically in Fig. 1{1 This four band mo del w as originally dev elop ed b y Kane for InSb [ 8 ], whic h is t ypical of direct gap I I I-V semiconductors. Near the zone cen ter all four bands ha v e parab olic disp ersions. Tw o of the hole bands that are degenerate at k = 0 are kno w as the hea vy hole (hh) and the ligh t hole (lh) bands. The third split-o hole (so) band gets its name from the split-o to lo w er energy b y the spin-orbit coupling. In bulk semiconductors the energy gaps are temp erature dep enden t whic h can b e tted to the empirical V arshni function [ 9 ] E g ( T ) = E g ( T = 0) T 2 T + ; (1.1) where and are adjustable and called V arshni parameters. There are other functional forms [ 10 ] but the V arshni's is the most widely used one. GaN is a wide-gap semiconductor. It usually app ears in t w o crystal structures, the wurtzite GaN and the Zinc blende GaN. The wurtzite GaN is the more usual one of the t w o structures. The follo wing parameters w e used are from V urgaftman et al. [ 11 ] They ga v e a v ery ric h set of references in the article. PAGE 14 5 Figure 1{1: Sc hematic band diagram of a direct gap I I I-V semiconductor near the Brillouin zone cen ter. The zero energy p oin t is lo cated at the top of the v alence band, while E = E g corresp onds to the b ottom of the conduction band. The conduction band is an electron (e) band, while the v alence bands are: the hea vy hole (hh) band, the ligh t hole (lh) band, and the split-o hole (so) band. PAGE 15 6 T able 1{1: Recommended Band Structure P arameters for wurtzite GaN and InN (for comparison w e also listed parameters for GaAs.) P arameters GaN InN GaAs a lc ( A) at T = 300 K 3.189 3.545 5.653 c lc ( A) at T = 300 K 5.185 5.703 E g (eV) 3.507 1.994 1.519() (meV/K) 0.909 0.245 0.5405() (K) 830 624 204() cr (eV) 0.019 0.041 so (eV) 0.014 0.001 0.341 m ke (eV) 0.20 0.12 0.067() m ?e (eV) 0.20 0.12 0.067() E P (eV) 14.0 14.6 28.8 c 11 (GP a) 390 223 1221 c 12 (GP a) 145 115 566 c 13 (GP a) 106 92 c 33 (GP a) 398 224 c 44 (GP a) 105 48 600 e 13 (GP a) -0.35 -0.57 e 33 (GP a) 1.27 0.97 C L [001] (10 3 m/s) 8.04 5.17 C L [100] (10 3 m/s) 7.96 5.28 4.73 C T [001] (10 3 m/s) 4.13 1.21 C T [100] (10 3 m/s) 4.13 1.21 3.35 C T [010] (10 3 m/s) 6.31 2.51 PAGE 16 7 F or ternary allo ys, the dep endence of the energy gap on allo y comp osition is assumed to t the follo wing quadratic form, E g ( A x B 1 x ) = xE g ( A ) + (1 x ) E g ( B ) x (1 x ) C ; (1.2) where the b o wing parameter C represen ts the deviation from the linear in terp olation b et w een the t w o binaries A and B In the case of In x Ga 1 x N, C is ab out 3.0 eV. F or other parameters, one usually linearly in terp olate b et w een the v alues of t w o binaries if there are no generally accepted exp erimen tal data. The [001] longitudinal acoustic w a v e sp eed of In x Ga 1 x N F or x = 0 : 08 is ab out 7800 m/s. The dierence of the [001] LA mo de sp eed b et w een GaN and In x Ga 1 x N is v ery small when the In comp onen t x is small. F or x = 0 : 08 this dierence is less than 3%. The rerectivit y due to the dieren t sound sp eeds b et w een GaN and In x Ga 1 x N is giv en b y r = C s C 0 s C s + C 0 s ; (1.3) where C s and C 0 s are sound sp eeds in GaN and In x Ga 1 x N resp ectiv ely F or x = 0 : 08, r is appro ximately 0.014 and the rerection constan t R = j r j 2 is ev en smaller on the order of 10 4 1.2 The Dynamics of Photo excited Carriers and Phonons Ultrafast fem tosecond lasers are indisp ensable p o w erful to ols for studying the dynamical b eha vior of photo excited electrons and holes in semiconductors and other condensed matter systems. These lasers are ideal for obtaining a snapshot of the nonequilibrium photo excited carriers and studying the scattering pro cesses b ecause the scattering time of carriers in semiconductors is ab out tens to h undreds of fem toseconds. As men tioned in the last section the p erio dicit y of the semiconductor lattice leads to energy band structure, whic h forms the basis of understanding most of the optical phenomena in a semiconductor. PAGE 17 8 Electrons can b e excited in to the conduction band creating holes in the v alence band. The p erio dicit y of the semiconductor lattice also allo ws the description of the quan tized vibrational mo des of the lattice in terms of phonon disp ersion relations. The dynamics of electrons, holes, and phonons is inruenced b y their in teraction with eac h other, as w ell as with defects and in terfaces of the system. Carrier-carrier scattering determines the exc hange of energy b et w een carriers and is primarily resp onsible for the thermalization of photo excited non-thermal carriers. Carrier-phonon in teractions pla y a ma jor role in the exc hange of energy and momen tum b et w een carriers and the lattice. Optical phonons in teract with carriers through p olar coupling and non-p olar optical deformation p oten tial, while acoustic phonons in teract with carriers through deformation p oten tial and the piezo electric p oten tial. After a semiconductor in thermo dynamic equilibrium is excited b y an ultrashort laser pulse, it undergo es sev eral stages of relaxation b efore it returns once again to the thermo dynamic equilibrium. The carrier relaxation can b e classied in to four temp orally o v erlapping regimes. First there is the c oher ent stage The ultrashort laser pulse creates excitation with a w ell-dened phase relationship within them and with the laser electromagnetic eld. The scattering pro cesses that destro y the coherence are extremely fast so picoand fem tosecond tec hniques are required to study the coheren t regime in semiconductors. The dynamics are describ ed b y the semiconductor Blo c h equations [ 12 ]. Second is the thermalization stage After the destruction of coherence through dephasing the distribution of the excitation is v ery lik ely to b e non-thermal; that is, the distribution function cannot b e c haracterized b y an eectiv e temp erature. This regime pro vides information ab out v arious carrier-carrier scattering pro cesses PAGE 18 9 that bring the non-thermal distribution to a hot, thermalized distribution. This relaxation stage is go v erned b y the Boltzmann transp ort equation (BTE) [ 13 ]. Third is the hot c arrier or the c arrier c o oling stage In this regime the carrier distribution is c haracterized b y an eectiv e temp erature. The temp erature is usually higher than the lattice temp erature and ma y b e dieren t for dieren t sub-systems. In v estigation of hot carrier regime fo cuses on the rate of co oling of carriers to the lattice temp erature and leads to information concerning v arious carrier-phonon and phonon-phonon scattering pro cesses. A t the end of the carrier co oling regime, all the carriers and phonons are in equilibrium with eac h other and can b e describ ed b y the same temp erature. Ho w ev er, there is still an excess of electrons and holes compared to the thermodynamic equilibrium. These excess electron-hole pairs recom bine and return the semiconductor to the thermo dynamic equilibrium. This is the r e c ombination stage It should b e emphasized that man y ph ysical pro cesses in the dieren t regimes can o v erlap. F or example, the pro cesses that destro y coherence ma y also con tribute to the thermalization of carrier distribution functions. 1.3 Exp erimen t Setup F or Coheren t Phonons There are man y tec hniques dev elop ed to in v estigate the optical prop erties of semiconductors using ultrafast lasers, whic h include pump-prob e sp ectroscop y fourw a v e mixing sp ectroscop y luminescence sp ectroscop y and terahertz sp ectroscop y Figure 1{2 sho ws a sc hematic setup for a general t w o laser b eam pump-prob e exp erimen t. Pump-prob e sp ectroscop y is the most common setup for the generation and detection coheren t phonons. The ligh t source is usually a Titanium:sapphire mo de-lo c k ed ultrafast laser with a pulse width ranging from a few fem toseconds to h undreds of fem toseconds and a w a v elength around 800 nm, the frequency of whic h can b e doubled if photons of higher energy are needed. The ligh t b eam from PAGE 19 10 the ultrafast laser is splitted in to a pump and a prob e. The time dela y b et w een the pump and the prob e is con trolled b y the optical path of the prob e. The pump laser pulse excites carriers and coheren t phonons in the semiconductor sample, whic h results in the c hange in the dielectric function of the material. By measuring dieren tial rerection or transmission whic h represen t c hanges in the rerected or transmitted prob e pulse energy b et w een the pump on and pump o at dieren t time dela ys, one obtains the resp onse of the dielectric function to the ligh t due to the carriers and lattice vibrations in the material. Delay FWM-signal transmissionsignal Pump-probe correlationsignal Luminescence signal reflection Pump-probe 1 t Pump 2 t Probe Sample Ultrafast laser Delay Figure 1{2: Sc hematic setup for a t w o-b eam nonlinear exp erimen t. It can b e used for pump-prob e transmission or rerection sp ectroscop y four-w a v e-mixing sp ectroscop y or luminescence correlation sp ectroscop y PAGE 20 CHAPTER 2 DIPOLE OSCILLA TOR MODEL The description of the in teraction b et w een ligh t and matter falls in to one of three categories: the classical, the semiclassical, and the fully quan tum. In classical mo dels w e treat b oth the ligh t and matter as classical ob jects that b eha v e according to the la ws of classical ph ysics. In semiclassical mo dels w e apply quan tum mec hanics to describ e the matter, but treat ligh t as a classical electromagnetic w a v e. The fully quan tum approac h b elongs to the realm of quan tum optics, where b oth ligh t and matter are treated as quan tum ob jects. When w e sp eak of a ligh t b eam in terms of photons and dra w F eynman diagram to depict the in teraction pro cesses, w e are using the fully quan tum approac h implicitly ev en though quan titativ ely w e ma y ha v e treated the ligh t classically The dip ole oscillator is a t ypical example of classical mo dels. It is the basic starting p oin t for understanding the eects due to carriers and phonons. The prop er comprehension of this simple classical mo del is indisp ensable in appreciating more complicated mo dels. In the rst section of this c hapter I will giv e a general in tro duction to the optical pro cesses o ccurring when a ligh t b eam is inciden t on an optical medium and their quan tization. The main purp ose is to dene the notation and collect together the relationships b et w een the optical co ecien ts. 2.1 Optical Pro cesses and Optical Co ecien ts The wide range of optical pro cesses observ ed in a semiconductor can b e organized in to sev eral groups of general phenomena. A t the macroscopic lev el all the optical phenomena can b e describ ed quan titativ ely b y a small n um b er of parameters or optical co ecien ts that c haracterize the prop erties of the semiconductor. 11 PAGE 21 12 2.1.1 Classication of Optical Pro cesses When a ligh t b eam is shined on a semiconductor, some of the ligh t is rerected from the fron t surface, some en ters the semiconductor and propagates through it, and some comes out from the bac k end of the semiconductor. Th us, the in tuitiv e and simplest classication of optical pro cesses is r ere ction pr op agation and tr ansmission The ligh t propagation phenomena can b e further classied in to r efr action absorption luminesc enc e and sc attering R efr action happ ens when a ligh t b eam tra v els obliquely from one optical medium to another, e.g., from free space in to a semiconductor, in whic h its sp eed c hanges, e.g., the sp eed of the ligh t b ecomes smaller in the semiconductor than in free space. The direction of the ligh t b eam c hanges according to Snell's la w of refraction. A bsorption describ es the loss of in tensit y when a b eam of ligh t passes through a semiconductor. There are t w o kinds of absorption pro cesses: sc attering whic h will b e discussed b elo w, and absorption of photons b y atoms or molecules in the semiconductor. Absorption and transmission are related b ecause only the unabsorb ed ligh t will b e transmitted. Luminesc enc e is the sp on taneous emission of ligh t b y atoms of a semiconductor making transitions from an excited state to the ground state or to another excited state of lo w er energy Dep ending on the causes of excitation, luminescence could b e photoluminesc enc e if the excitation is caused b y a photon, ele ctr oluminesc enc e if it is an electron, chemiluminesc enc e if it is a c hemical reaction, etc. Phosphor esc enc e or ruor esc enc e dep ends on whether the luminescence p ersists signican tly after the exciting causes is remo v ed. There is some arbitrariness in this distinction, usually a p ersistence of more than 10 ns is treated as phosphorescence. Th us, photoluminescence accompanies the propagation of ligh t in an absorbing medium. The emitted ligh t b eams are in all directions and the frequencies are PAGE 22 13 dieren t from the incoming b eam. The sp on taneous emission tak es a c haracteristic amoun t of time. The excitation energy can dissipate as heat b efore the radiativ e re-emission. As a result the eciency of the luminescence is closely related to the dynamics of the de-excitation mec hanism in semiconductors. Sc attering o ccurs when a ligh t b eam is derected b y atoms or molecules in a semiconductor. If the frequency of the scattered ligh t remains unc hanged, it is called an elastic scattering; otherwise it is called an inelastic scattering. The total n um b er of photons do es not c hange in either kind of scattering, but the n um b er in the forw ard direction decreases b ecause some photons are b eing redirected in to other directions. So scattering also has the atten uating eect as absorption do es. There are other phenomena suc h as frequency doubling if the in tensit y of the propagating ligh t b eam is v ery high. These phenomena are describ ed b y nonlinear optics.2.1.2 Quan tization of Optical Pro cesses The Rerection and transmission of a b eam of ligh t are describ ed b y the R ere ctanc e or r ere ction c o ecient R and the tr ansmittanc e or tr ansmission c o ecient T resp ectiv ely Rerectance R is dened as the ratio of the rerected p o w er to the inciden t p o w er on the surface, while transmittance is the ratio of the transmitted p o w er to the inciden t p o w er. If there is no absorption or scattering, w e m ust ha v e R + T = 1 from energy conserv ation. The propagation of the ligh t through a semiconductor can b e c haracterized b y the r efr active index n and the absorption c o ecient The refractiv e index is the ratio of the sp eed of ligh t in v acuum c to the sp eed of ligh t in the semiconductor v n = c v : (2.1) The frequency dep endence of the refractiv e index is called disp ersion PAGE 23 14 The absorption co ecien t is dened as the fraction of p o w er absorb ed after a ligh t b eam tra v els a unit length of the medium. Supp ose the ligh t b eam tra v els in the z direction, and denote the in tensit y at p osition z as I ( z ), then the decrease of the in tensit y in a thin slice of thic kness d z is giv en b y d I = d z I ( z ). After in tegrating b oth sides of the ab o v e equation, w e obtain Beer's la w I ( z ) = I 0 e z ; (2.2) where I 0 is the ligh t in tensit y at p osition z = 0. The absorption co ecien t is strongly related to the frequency of the inciden t ligh t, whic h is wh y materials ma y absorb the ligh t of one color but not another. In an absorbing medium the refraction and absorption can also b e describ ed b y a single quan tit y the complex refractiv e index ~ n whic h is dened as ~ n = n + i: (2.3) The real part of ~ n i.e. n, is the same as the refractiv e index dened in Eq. ( 2.1 ). The imaginary part of ~ n i.e. is called the extinction c o ecient and is related to the absorption co ecien t. W e can deriv e the relationship b et w een and b y considering a plane electromagnetic w a v e propagating through an absorbing medium. Let the z axis b e the direction of propagation and let the electric eld b e giv en b y E ( z ; t ) = E 0 e i ( k z t ) ; (2.4) where k is the w a v e v ector and is the angular frequency whic h are related in a non-absorbing medium through k = 2 =n = n! c : (2.5) PAGE 24 15 Here is the v acuum w a v elength of the ligh t. Eq. ( 2.5 ) can b e generalized to the case of an absorbing medium b y w a y of the complex refractiv e index, k = ~ n c = ( n + i ) c : (2.6) On substituting Eq. ( 2.6 ) in to Eq. ( 2.16 ) and after rewriting Eq. ( 2.16 ), w e obtain E ( z ; t ) = E 0 e z =c e i ( nz =c t ) : (2.7) Comparing Beer's la w in Eq. ( 2.2 ) with the in tensit y of the ligh t w a v e, whic h is prop ortional to the square of the electric eld, i.e. I / E E w e obtain = 2 c = 4 ; (2.8) whic h sho ws that the absorption co ecien t at a giv en w a v elength is prop ortional to the extinction co ecien t W e still ha v e one last optical co ecien t to in tro duce, that is the relativ e dielectric constan t Sometimes it is simply called the dielectric constan t or dielectric function. The relationship b et w een the refractiv e index and the dielectric constan t is a standard result deriv ed from Maxw ell's equations (cf. an y electro dynamics textb o ok, e.g. Jac k on's [ 14 ]), n = p : (2.9) Corresp onding to the complex refractiv e index ~ n w e ha v e the complex dielectric constan t, ~ = 1 + i 2 = ~ n 2 ; (2.10) where 1 is the real part of the complex dielectric constan t and 2 the imaginary part. The relationships b et w een the real and imaginary parts of ~ n and ~ are not dicult to deriv e from Eq. ( 2.10 ). They are 1 = n 2 2 ; (2.11a) PAGE 25 16 2 = 2 n; (2.11b) and n = 1 p 2 1 + q 21 + 22 1 = 2 ; (2.12a) = 1 p 2 1 + q 21 + 22 1 = 2 : (2.12b) Th us, w e can calculate n and from 1 and 2 and vic e versa In the w eakly absorbing case, i.e. n Equation ( 2.12 ) can b e simplied to n = p 1 ; (2.13) = 2 2 n : (2.14) 2.2 The Dip ole Oscillator Mo del The originator of the classical dip ole oscillator mo del is Loren tz, so it is also called the L or entz mo del In this mo del, the ligh t is treated as electromagnetic w a v es and the atoms or molecules are treated as classical dip ole oscillators. There are dieren t kinds of oscillators. The atomic oscillator at optical frequencies is due to the oscillations of the b ound electrons within the atoms. The vibrations of c harged atoms within the crystal lattice giv e rise to the vibrational oscillators in the infrared sp ectral region. There are also free electron oscillators in metals. Based on the mo del w e can calculate the frequency dep endence of the complex dielectric function and obtain the rerection and the transmission co ecien t. 2.2.1 The A tomic Oscillator Let us consider rst the atomic oscillator in the con text of a ligh t w a v e in teracting with an atom with a single resonan t frequency 0 due to a b ound electron. W e assume the mass of the n ucleus m n is m uc h greater than the mass of the electron m 0 so that the motion of the n ucleus can b e ignored, then the PAGE 26 17 displacemen t x of the electron is go v erned b y the classical equation of motion, m 0 d 2 x d t 2 + m 0 r d x d t + m 0 2 0 x = e E ; (2.15) where r is the damping constan t and e is the c harge of an electron The electric eld E of a mono c hromatic ligh t w a v e of angular frequency is giv en b y E ( t ) = E 0 e i! t : (2.16) If w e substitute Eq. ( 2.16 ) in to Eq. ( 2.15 ) and lo ok for solutions of the form x ( t ) = x 0 e i! t then w e ha v e x 0 = e E 0 =m 0 2 0 2 ir : (2.17) The resonan t p olarization due to the displacemen t of the electrons from their equilibrium p osition is P r = N ex = N e 2 m 0 1 2 0 2 ir E ; (2.18) where N is the n um b er of atoms p er unit v olume. The electric displacemen t D the electric eld E and the p olarization P are related through D = 0 E = 0 E + P : (2.19) W e split the total p olarization in to the resonan t term P r w e discussed ab o v e and a non-resonan t bac kground term P b = 0 E then from Eq. ( 2.19 ) w e can obtain the complex dielectric constan t ( ) = 1 + + N e 2 0 m 0 1 2 0 2 ir : (2.20) W e can also write this complex dielectric constan t in terms of its real and imaginary parts: 1 ( ) = 1 + + N e 2 0 m 0 2 0 2 ( 2 0 2 ) 2 + ( r ) 2 ; (2.21) PAGE 27 18 2 ( ) = N e 2 0 m 0 r ( 2 0 2 ) 2 + ( r ) 2 : (2.22) W e dene t w o dielectric constan t in the lo w and high frequency limits, ( = 0) s = 1 + + N e 2 0 m 0 2 0 ; (2.23) and ( = 1 ) 1 = 1 + : (2.24) Th us, w e ha v e s 1 = N e 2 0 m 0 2 0 = 2 p 2 0 ; (2.25) where 2 p = N e 2 0 m 0 is called the plasma frequency Close to resonance, where ! 0 r and 2 0 2 2 0 with = ! 0 as the detuning from 0 w e can rewrite the real and imaginary parts of the complex dielectric constan t as 1 ( ) = 1 ( s 1 ) 2 0 4 2 + r 2 ; (2.26) 2 ( ) = ( s 1 ) 0 r 4 2 + r 2 : (2.27) The frequency dep endence of 1 and 2 is called Loren tzian named after the originator of the dip ole oscillator mo del. The imaginary part 2 is strongly p eak ed with a maxim um v alue at 0 and a full width r at half maxim um. The real part 1 rst gradually rises from the lo w frequency v alue s when w e approac h 0 from b elo w. After reac hing a p eak at 0 r = 2, it falls sharply going through a minim um at 0 + r = 2, then rising again to the high frequency limit 1 The frequency range o v er whic h these drastic c hanges o ccur is determined b y r 2.2.2 The F ree Electron Oscillator Both metals and dop ed semiconductors con tain large densities of free carriers suc h as electrons or holes. Both of them can b e treated as plasmas, i.e. a neutral gas of hea vy ions and ligh t electrons, b ecause they con tain equal n um b er of p ositiv e PAGE 28 19 ions and free electrons. Unlik e b ound electrons, the free electrons ha v e no restoring force acting on them. The free electron mo del of metals w as rst prop osed b y Drude in 1900. The Drude mo del treats the v alence electrons of the atoms as free electrons. A detailed discussion of the Drude mo del can b e found in e.g. Ashcroft & Mermin [ 15 ]. These free electrons accelerate in an electric eld and undergo collisions with a c haracteristic scattering time The free electron oscillator is actually a com bination of the Drude mo del of free electron conductivit y and the Loren tz dip ole oscillator mo del. The equation of motion for electrons in the Drude mo del is giv en b y d p d t = p e E : (2.28) Comparing it with the equation of motion for the displacemen t x of a free electron oscillator m 0 d 2 x d t 2 + m 0 r d x d t = e E ; (2.29) w e ha v e a relation b et w een the damping constan t and scattering time r = 1 : (2.30) W e can solv e Eq. ( 2.29 ) in the same w a y as w e did in the atomic oscillator and obtain the complex dielectric constan t for the free electron oscillators, ( ) = 1 N e 2 0 m 0 1 2 + ir ; (2.31) where w e did not tak e in to accoun t the bac kground p olarization. If w e dene 2 p = N e 2 0 m 0 ; (2.32) where p is kno wn as the plasma fr e quency w e can write Eq. ( 2.31 ) in a more concise form ( ) = 1 2 p 2 + ir : (2.33) PAGE 29 20 The A C conductivit y in Drude mo del is obtained b y solving Eq. ( 2.28 ), ( ) = 0 1 i! ; (2.34) where 0 = N e 2 =m 0 is the DC conductivit y Comparing Eq. ( 2.31 ) and ( 2.34 ), w e ha v e ( ) = 1 + i ( ) 0 ; (2.35) whic h tells us that optical measuremen ts of the dielectric constan t are equiv alen t to A C conductivit y measuremen ts. Again w e can split the dielectric constan t in to its real and imaginary parts as 1 = 1 2 p 2 1 + 2 2 ; (2.36) 2 = 2 p (1 + 2 2 ) : (2.37) The plasma frequency p t ypically lies in the visible or ultra violet sp ectral region, whic h corresp onds to p > 10 15 sec 1 The mean free collision time for electrons in metals is t ypically 10 14 sec. So for metals in the region of plasma frequency 1. 2.2.3 The Vibrational Oscillator The atoms in a crystal solid vibrate at c haracteristic frequencies determined b y the phonon mo des of the crystal. The resonan t frequencies of phonons usually o ccur in the infrared sp ectral region, whic h results in strong absorption and rerection of ligh t. Since a longitudinal optical (LO) phonon has no eect on a ligh t w a v e b ecause the longitudinal electric eld induced b y an LO phonon is p erp endicular to that of the ligh t w a v e, w e will consider the in teraction b et w een an electromagnetic w a v e and a transv erse optical (TO) phonon, whic h is easily visualized b y a linear c hain. The c hain is made up of a series of unit cells with eac h cell con taining a p ositiv e ion of mass m + and a negativ e ion of mass m If w e PAGE 30 21 assume the w a v es propagate in the z direction and the displacemen t of the p ositiv e and negativ e ions in a TO mo de is denoted b y x + and x resp ectiv ely then the equations of motion are giv en b y m + d 2 x + d t 2 = K ( x + x ) + q E ; (2.38a) m d 2 x d t 2 = K ( x x + ) q E ; (2.38b) where K is the restoring constan t and q is the eectiv e c harge p er ion. Com bining equations of motion ( 2.38a ) and ( 2.38b ), w e obtain d 2 x d t 2 = K x + q E ; (2.39) where is the reduced mass giv en b y 1 = = 1 =m + + 1 =m and x = x + x is the relativ e displacemen t of the p ositiv e and negativ e ions within the same cell. W e can in tro duce a phenomenological damping constan t r to accoun t for the nite life time of a phonon mo de. F ollo wing the same pro cedures as the forgoing t w o subsections, w e can solv e the motion of equation for x to obtain the dielectric constan t ( ) = 1 + + N q 2 0 1 2 TO 2 ir ; (2.40) where is the non-resonan t bac kground susceptibilit y and N is the n um b er of unit cells p er unit length. Similar to the denitions of s and 1 in subsection 2.2.1 w e can also write the dielectric constan t as ( ) = 1 + ( s 1 ) 2 TO 2 TO 2 ir : (2.41) As w e said b efore usually an LO phonon do es not in teract with ligh t w a v es, but in sp ecial cases it will. In a medium without free c harges, Gauss's la w giv es r D = r ( 0 E ) = 0 : (2.42) PAGE 31 22 There are t w o w a ys to satisfy the equation, one of them is a transv erse w a v e with k E = 0. The other is longitudinal w a v es but with = 0. In a w eakly damp ed system w e can set r = 0. There is a sp ecial frequency called LO at whic h the dielectric constan t is zero and the longitudinal mo des in teract with ligh t w a v es. W e can solv e Eq. ( 2.41 ) to obtain the so called Lyddane-Sac hs-T eller (LST) relationship 2 LO 2 TO = s 1 : (2.43) The dielectric constan t is b elo w zero when TO < < LO whic h leads to 100% rerectance so no ligh t can propagate in to the medium. The frequency range b et w een TO and LO is called the r estr ahlen (German w ord for \residual ra ys") band.2.2.4 A Series Of Oscillators In a semiconductor, the atomic oscillator, the free electron oscillator, and the lattice vibrational oscillator ma y b e all presen t. F or a sp ecic t yp e of oscillator there can b e sev eral dieren t resonan t frequencies. The dielectric constan t due to these m ultiple oscillators can b e written as ( ) = 1 + N e 2 0 m 0 X j f j 2 j 2 ir j ; (2.44) where j and r j are the frequency and damping constan t of a particular oscillator. The phenomenological parameter f j is called the oscil lator str ength whic h has no explanation in classical ph ysics. In quan tum ph ysics the oscillator strengths satisfy the sum rule X j f j = 1 : (2.45) In classical ph ysics w e just tak e f j = 1 for eac h oscillator. PAGE 32 23 2.3 Rerectance And T ransmission Co ecien t F or normal incidence the rerectance is giv en b y R = ~ n 1 ~ n + 1 2 = ( n 1) 2 + 2 ( n + 1) 2 + 2 : (2.46) The c hange of rerectance comes from sev eral dieren t sources. It can b e due to the c hange of carrier densit y N the scattering time the restoring force 0 etc. The dieren tial rerection can b e expressed in terms of the c hange of the real and imaginary part of the dielectric constan t resp ectiv ely R R = 1 R @ R @ 1 2 1 + 1 R @ R @ 2 1 2 ; (2.47) 1 1 + 2 2 ; (2.48) 1 = D1 + O1 ; 2 = D2 + O2 ; (2.49) where the sup erindex D and O indicate the con tribution from the in traband Drude term and in terband oscillator term resp ectiv ely 1 and 2 are called the Seraphin co ecien ts and are giv en b y the follo wing equation: 1 = p 2 h ( 1 1)( p 21 + 22 + 1 ) 22 i [( 1 1) 2 + 22 ] p 21 + 22 q p 21 + 22 + 1 ; (2.50) 2 = p 2 2 ( p 21 + 22 + 2 1 1) [( 1 1) 2 + 22 ] p 21 + 22 q p 21 + 22 + 1 : (2.51) F rom the Drude form ula in Eqs. ( 2.37 ) and ( 2.37 ) w e can obtain the c hange of the dielectric function, D1 = N 0 @ D1 @ N 0 N 0 N 0 + @ D1 @ ; (2.52) D N 1 N 0 N 0 + D 1 ; (2.53) D2 = N 0 @ D2 @ N 0 N 0 N 0 + @ D2 @ ; (2.54) D N 2 N 0 N 0 + D 2 : (2.55) PAGE 33 24 The co ecien ts are sho wn b elo w: D N 1 = 4 N 0 e 2 m 2 1 + 2 2 = D1 1 ; (2.56) D N 2 = 4 N 0 e 2 m (1 + 2 2 ) = D2 ; (2.57) D 1 = 8 N 0 e 2 m 2 (1 + 2 2 ) 2 = 2( D1 1) 1 + 2 2 ; (2.58) D 2 = 4 N 0 e 2 m (1 2 2 ) (1 + 2 2 ) 2 = 1 2 2 1 + 2 2 D2 : (2.59) F or most semiconductors 1 th us w e ha v e, D 1 0 ; (2.60) D 2 D2 ; (2.61) j D1 1 j j D2 j = 1 : (2.62) F or w eak absorbing material the imaginary part of the dielectric function is small D2 0. The Seraphin co ecien t can b e simplied as 1 2 ( 1 1) p 1 + O ( 22 ) ; (2.63) 2 3 1 1 ( 1 1) 2 3 = 2 1 2 + O ( 32 ) ; (2.64) 1 = 2 ! 2 1 1 3 1 1 : (2.65) Therefore in the Drude mo del the dieren tial rerectivit y is dominated b y the real part of the Seraphin co ecien t and is more sensitiv e to the c hange of the carrier densit y No w let us turn to the transmission. Righ t at the in terface the transmission co ecien t is T 0 = 1 R = 4 n ( n + 1) 2 + 2 : (2.66) PAGE 34 25 Inside the media the transmittance decreases as describ ed b y Beer's la w in Eq. ( 2.2 ). F or a lm of thic kness L the dieren tial transmission as dened b elo w is T T T ( z ; ) T ( z ; 0 ) T ( z ; 0 ) z = L (2.67) = L; (2.68) where 0 and are the absorption co ecien t b efore and after the excitation resp ectiv ely and j L j 1 is required. F rom the relationship b et w een the absorption co ecien t and the extinction co ecien t in Eq. ( 2.8 ) w e can obtain the c hange of the absorption co ecien t, = 2 c : (2.69) The c hange of extinction co ecien t can b e describ ed similarly as the dieren tial rerectivit y = 1 + 2 ; (2.70) with @ @ 1 2 (2.71) = 1 2 1 p 21 + 22 p 21 + 22 q 2( p 21 + 22 1 ) 2 4 3 = 2 1 + O ( 32 ) ; (2.72) @ @ 2 1 (2.73) = 1 2 2 p 21 + 22 q 2( p 21 + 22 1 ) 1 2 p 1 + O ( 22 ) ; (2.74) and = = 1 p 21 + 22 2 1 2 2 1 1 : (2.75) PAGE 35 26 The c hange of the absorption co ecien t in terms of the c hange of carrier densit y and scattering time is giv en b y the follo wing, = D N N N + D ; (2.76) with D N 1 + 1 4 3 = 2 1 2 + O ( 32 ) ; (2.77) D 1 2 p 1 + O ( 32 ) : (2.78) Therefore the dieren tial transmission is dominated b y the c hange of the imaginary part of the dielectric function and the c hange of carrier densit y and scattering time ha v e the same order of eect. F ollo wing the same pro cedure w e can discuss the dieren tial rerectivit y and dieren tial transmission in the oscillator mo del. Some results are sho wn in Fig. 2{1 and Fig. 2{2 PAGE 36 27 1.8 1.9 2 2.1 2.2 -0.2 -0.1 0 0.1 w (eV)Seraphin Coefficientsb 1 b 2 0 5 10 -8 -4 0 1.8 1.9 2 2.1 2.2 10 -3 10 -2 10 -1 10 0 10 1 w (eV)Contributions of terms to DRS Drude: realSeries Ocs: realSeries Ocs: imgDrude: img Figure 2{1: Sp ectral dep endence of Seraphin co ecien ts and dieren tial rerectivit y PAGE 37 28 0 5 10 15 20 -1 -0.5 0 0.5 1 w (eV)Differential extinct Coefficientsn m Figure 2{2: Sp ectral dep endence of the co ecien t of real and imaginary part of the dielectric function for dieren tial transmission. PAGE 38 CHAPTER 3 GENERAL THEOR Y OF COHERENT PHONON 3.1 Phenomenological Mo del The coheren t phonon motion can b e describ ed in a phenomenological mo del of a driv en harmonic oscillator [ 3 ]. The ev olution of a coheren t phonon amplitude Q in the presence of a driving force exerted b y ultrafast laser pulse is go v erned b y the dieren tial equation @ 2 Q @ t 2 + 2 r @ Q @ t + 2 0 Q = F ( t ) m ; (3.1) where 0 is the frequency of the phonon mo de, r is the damping parameter, m is the mass of the oscillator, and F is the driving force, whic h ma y dep end on carrier densit y temp erature, and other parameters of the system. The damping parameter r is the in v erse of the dephasing time T 2 of the coheren t phonon mo de [ 16 ]. The dephasing time T 2 comes from a com bination of phase-destro ying pro cesses with relaxation time T p and p opulation decreasing pro cesses with relaxation time T 1 Examples of the latter are anharmonic deca y pro cesses suc h as the deca y of LO phonons in to acousitc phonons and electron-phonon in teraction pro cesses where a phonon can b e absorb ed b y an electron. Ultrafast generation of carriers b y fem tosecond lasers causes the driving force to rapidly turn on and trigger the oscillations. Oscillator Eq. ( 3.1 ) can b e solv ed formally b y using either Green's functions or Laplace transforms with the initial condition that b oth Q and @ Q=@ t are zero b efore the force F is applied with the result [ 17 ], Q ( t ) = Z t 1 F ( ) m e r ( t ) sin p 2 0 r 2 ( t ) p 2 0 r 2 d : (3.2) 29 PAGE 39 30 W e consider t w o kinds of forcing functions [ 17 ]. The rst kind is impulsiv e forces, whic h ha v e the form F i ( t ) = I ( t ) ; (3.3) where ( t ) is a Dirac delta function in time. After in tegrating Eq. ( 3.2 ) directly w e ha v e Q ( t ) = I m p 2 0 r 2 e r t sin q 2 0 r 2 t ( t ) : (3.4) The solution sho ws that an impulsiv e force starts oscillations ab out the curren t equilibrium p osition, whic h will damp out exp onen tially The other kind of forcing function is displaciv e with a form giv en b y F d ( t ) = D ( t ) ; (3.5) where ( t ) is a Hea v eside step function. Again w e can in tegrate Eq. ( 3.2 ) directly to get the coheren t phonon amplitude Q ( t ) = D m! 2 0 ( t ) ( 1 e r t cos q 2 0 r 2 t + r p 2 0 r 2 sin q 2 0 r 2 t # ) : (3.6) The solution in the case of a displaciv e forcing function sho ws oscillations and exp onen tial damping to o, but compared to the impulsiv e case, a displaciv e force will mo v e the oscillator to a new equilibrium p osition with a dieren t initial phase. 3.2 Microscopic Theory The phenomenological oscillator mo del captures the essen tial ph ysics. Ho w ev er it lea v es op en the question of exact denition of coheren t phonon amplitude. The microscopic quan tum mec hanical justication of the oscillator mo del is giv en b y Kuznetso v and Stan ton [ 18 ]. In a simplied system consisting of t w o electronic bands in teracting with phonon mo des, the Hamiltonian consists three parts: free Blo c h electrons and PAGE 40 31 holes in a p erfect static crystal lattice, the free phonons, and the electron-phonon in teraction. H = X ; k k c y k c k + X q h! q b yq b q + X ; k ; q M kq b q + b y q c y k c k + q (3.7) The Lattice displacemen t op erator ^ u ( r ) is expressed in terms of the phonon creation and annihilation op erators: ^ u ( r ) = X q s h 2 V q f b q e i q r + b yq e i q r g : (3.8) The coheren t phonon amplitude of the q th mo de is dened as: D q h b q i + h b y q i B q + B q : (3.9) Therefore the coheren t amplitude is prop ortional to the F ourier comp onen ts of the displacemen t in Eq. ( 3.8 ) The a v erage of the coheren t amplitude will v anish in a phonon oscillator eigenstate. The a v erage displacemen t of the lattice v anishes, but there are ructuations: h u 2 i / h bb y + b y b i These phonons are incoheren t phonons in the mo de. A nonzero displacemen t requires that the w a v e function of the oscillator b e in a coheren t sup erp osition of more than one phonon eigenstate. In a general state there can b e a n um b er of b oth coheren t and incoheren t phonons. The canonical coheren t states are dened for eac h complex n um b er z in terms of eigenstate of harmonic oscillator. coh = j z i e ( z b y z b ) j 0 i = X n z n p n e z 2 j n i (3.10) They ha v e t w o imp ortan t prop erties. Firstly they are eigen v ector of the annihilation op erator with eigen v alue z, therefore in a canonical coheren t state, the PAGE 41 32 coheren t amplitude dened in Eq. ( 3.9 ) is B coh q h z j b q j z i = z : (3.11) Secondly canonical coheren t states are minim um-uncertain t y w a v e-pac k ets. When the amplitude z is large, they b eha v e lik e a macroscopic harmonic oscillator. So they are called \quasi-classical" states. Using the op erator form of the Sc hr odinger equation i h @ @ t h ^ A i = h [ ^ A; ^ H ] i + i h h @ ^ A @ t i ; (3.12) w e can obtain the dynamic equation of motion for the coheren t phonon amplitude @ 2 D q @ t 2 + 2 q D q = 2 q h X ; k M k q n k ; k q : (3.13) Here n k ; k q h c y k c k q i is the electronic densit y matrix. The electronic densit y matrix n k ; k q in the righ t hand side of Eq. ( 3.13 ) is nonzero only after excitation with an ultrafast laser pulse. This equation is written in momen tum space. Because of the F ourier transform relation b et w een the coheren t amplitude and the lattice displacemen t, w e ha v e obtained the phenomenological Eq. ( 3.1 ). Ho w ev er there is no damp ening term b ecause w e neglected anharmonic terms in the lattice p oten tial. In bulk materials since the laser w a v elength is m uc h larger than the lattice constan t, the created carriers are in a macroscopically uniform state, so that the electronic densit y matrix is diagonal. So the only phonon mo de that is coheren tly driv en b y the optical excitation is the q 0 mo de. In sup erlattice and m ultiple quan tum w ell systems, the carriers are created in the w ells only Because of the p erio dicit y of the sup erlattice structure, the electronic densit y matrix has a q 6 = 0 elemen t whic h can excite the coheren t acoustic phonon mo de. PAGE 42 33 3.3 In terpretation Of Exp erimen tal Data In the pump-prob e exp erimen t the measured quan tities are usually the dieren tial rerectivit y or dieren tial transmission. This section discusses the relation b et w een the lattice displacemen t and the measured ph ysical quan tit y Assume the only motion is parallel to the z axis and the only nonzero comp onen t of the elastic strain is 33 The strain is related to the lattice displacemen t U ( z ; t ) in the z direction through the follo wing equation, 33 = @ U ( z ; t ) @ z : (3.14) The c hange of rerection or transmission is due to strain induced v ariation of the optical constan ts of the material under consideration. In the linear appro ximation w e ha v e n ( z ; t ) = @ n @ 33 33 ; (3.15) ( z ; t ) = @ @ 33 33 ; (3.16) where n and are the c hanges in the real and imaginary part of the complex index of refraction. F or normally inciden t ligh t the Maxw ell's equation for electric eld giv es @ 2 E ( z ; t ) @ z 2 1 c 2 [ + ( z ; t )] @ 2 E ( z ; t ) @ t 2 = 0 ; (3.17) where the c hange of the dielectric constan t is related to the c hange of the index of refraction through ( z ; t ) = ( n + ) 2 ; (3.18) ( z ; t ) = 2( n + )( n + i ) ; = 2 p ( n + i ) : (3.19) PAGE 43 34 W e write the prob e electric eld as E = F ( z ; t ) e i! t ; (3.20) where F ( z ; t ) is a slo wly v arying en v elop e function. Substituting Eq. ( 3.20 ) in to Eq. ( 3.17 ) w e obtain the follo wing equation for the en v elop e function F ( z ; t ), @ 2 F ( z ; t ) @ z 2 + 2 c 2 [ + ( z ; t )] F 2 i @ F @ t 1 2 @ 2 F @ t 2 = 0 : (3.21) Since w e assumed that F is slo wly v arying w e can neglect the deriv ativ e of F with resp ect to time and rewrite the ab o v e equation as @ 2 F ( z ; t ) @ z 2 + 2 c 2 [ + ( z ; t )] F = 0 ; (3.22) whic h is analogous to Eq. (3.26) in Thomsen's pap er [ 19 ] where he calculated the c hange of rerection. The result is R ( t ) = Z 1 0 f ( z ) 33 d z ; (3.23) where f ( z ) is the \sensitivit y function", whic h determines ho w strain at dieren t depths b elo w the in terface con tributes to the c hange in the rerectivit y and is giv en b y f ( z ) = f 0 @ n @ 33 sin 4 nz + @ @ 33 cos 4 nz e z = ; (3.24) f 0 = 8 p n 2 ( n 2 + 2 1) 2 + 2 ( n 2 + 2 + 11) 2 c [( n + 1) 2 + 2 ] 2 ; (3.25) = arctan ( n 2 + 2 + 1) n ( n 2 + 2 1) ; (3.26) and is the laser w a v elength, is the absorption length dened as the recipro cal of the absorption co ecien t. PAGE 44 35 In the follo wing w e will carry out the calculation for the dieren tial transmission. If w e set the prob e electric eld of a normal inciden t laser at z = 0 as a Gaussian function E ( z = 0 ; t ) = E 0 e r ( t ) 2 e i! t ; (3.27) where is the prob e dela y with resp ect to the pump, then in the absence of strain w e ha v e the w a v e equation solution of the eld as E 0 ( z ; t ) = E ( z = 0 ; t z v ) ; where v = c n ; (3.28) = E 0 e r ( t z =v ) 2 e i! ( t z =v ) : (3.29) Comparing the ab o v e equation with Eq. ( 3.20 ) w e can obtain the en v elop e function for the eld without the strain F 0 ( z ; t ) = E 0 e r ( t z =v ) 2 e i! z =v : (3.30) With strain presen t the prob e electric eld will c hange to E ( z ; t ) = E 0 ( z ; t ) + E 1 ( z ; t ) ; (3.31) = [ F 0 ( z ; t ) + F 1 ( z ; t )] e i! t ; (3.32) where F 1 ( z ; t ) is the correction due to the c hange of optical constan ts caused b y the strain that resulted from the pump eld. Substituting Eq. ( 3.32 ) in to the partial dieren tial Eq. ( 3.22 ) for the en v elop e function w e nd the equation for the correction to the en v elop e function @ 2 F 1 ( z ; t ) @ z 2 + 2 v 2 F 1 ( z ; t ) = 2 v 2 ( z ; t ) F 0 ( z ; t ) : (3.33) F or the forw ard propagating w a v e at the b oundary conditions are F 1 ( z ; t = 1 ) = 0 ; (3.34) F 1 ( z = 1 ; t ) = 0 : (3.35) PAGE 45 36 The Green's function for F 1 ( z ; t ) satises @ 2 G ( z ) @ z 2 + 2 v 2 G ( z ) = ( t ) : (3.36) The solution with the b oundary condition ( 3.35 ) tak en in to consideration is G ( z ) = 0 ( z < 0) ; (3.37) G ( z ) = C sin ( z =v ) + D cos z =v ( z > 0) : (3.38) F rom the con tin uit y condition at z = 0 w e ha v e D = 0. In tegrating Eq. ( 3.36 ) ab out z = 0 w e ha v e @ G (0 + ) @ z @ G ( 0 ) @ z = 1 ; (3.39) whic h giv es C = v =! Therefore the Green's function is G ( z ) = v sin v ( z ) : (3.40) Using the ab o v e Green's function w e can obtain the correction to the en v elop e function F 1 ( z ; t ) = Z 1 1 G ( z ) 2 v 2 ( ; t ) F 0 ( ; t ) d ; (3.41) = E 0 v Z z 1 d sin h v ( z ) i ( ; t ) e i! =v e r ( t =v ) 2 : (3.42) Before the pump laser is turned on the transmission in tensit y is T 0 ( z ; ) = jE 0 0 ( z ; ) j 2 : (3.43) After the pump laser excitation the transmission in tensit y b ecomes T P ( z ; ) = jE P 0 ( z ; ) + E 1 ( z ; ) j 2 ; (3.44) PAGE 46 37 where E P 0 ( z ; ) is the prob e electric eld in the absence of strain but with the pump on. By denition, the dieren tial transmission is giv en b y T T ( z ; ) = T P ( z ; ) T 0 ( z ; ) T 0 ( z ; ) ; (3.45) = jE P 0 ( z ; ) + E 1 ( z ; ) j 2 jE 0 0 ( z ; ) j 2 jE 0 0 ( z ; ) j 2 ; (3.46) jE P 0 ( z ; ) j 2 jE 0 0 ( z ; ) j 2 jE 0 0 ( z ; ) j 2 + 2Re E P 0 ( z ; ) E 1 ( z ; ) jE 0 0 ( z ; ) j 2 ; (3.47) = T T 0 + T T 1 ; (3.48) where in the last line w e neglected the second order term of E 1 The rst term is that part of the dieren tial transmission without the strain whic h can b e calculated using F ermi's golden rule T T 0 = 4 2 e 2 n r c X nn 0 1 A X k (1 + f c n ( k ; ) f v n 0 ( k ; )) j ^ d cvnn 0 ( k ) j 2 ( E c n ( k ) E v n 0 ( k ) h! ) : (3.49) The con tribution of strain to the dieren tial transmission is T T 1 = 2Re E P 0 ( z ; ) E 1 ( z ; ) jE 0 0 ( z ; ) j 2 ; (3.50) = Re 4 p r v p @ n @ 33 + i @ @ 33 Z z 1 d sin v ( z ) e i! ( z ) =v Z 1 1 d te r ( t =v ) 2 @ u @ z ( ; t ) ; (3.51) where the rst term of the dieren tial transmission giv es the bac kground and the coheren t phonon oscillation comes from the strain induced second term. 3.4 Three Kinds Of Coheren t Phonons In the exp erimen t three kinds of dieren t coheren t phonons ha v e b een observ ed. They are the optical coheren t phonons in bulk semiconductors, the acoustic coheren t phonons in sup erlattices or MQWs, and the propagating coheren t phonon w a v epac k ets in epila y er systems. PAGE 47 38 The coheren t optical phonons ha v e b een observ ed in man y dieren t materials. In la y ered or lo w-symmetry materials suc h as Sb, Bi, T e, and Ti 2 O 3 [ 20 3 ] the A1 phonon mo des can b e excited b y the deformational phonon coupling mec hanism. In cubic materials suc h as Ge [ 21 ], the driving force v anishes for deformational p oten tial coupling. What causes the driving force not to v anish is the nite absorption depth of the ultrafast laser pulse and band structure eects. But the oscillations are w eak er than that of the other systems. In p olar materials lik e GaAs [ 22 ] the coupling b et w een electrons and phonons can b e b oth deformational and p olar with the latter stronger and b eing the dominan t eect. As an example Cho et. al. measured the dieren tial rerection of (100)-orien ted bulk GaAs for three dieren t angles b et w een the prob e b eam p olarization and the [010] crystal axis. The orthogonally p olarized b eams are k ept close to normal incidence. In the case of = 90 the rerectivit y rises during and p eaks to w ards the end of the excitation pulse. After passing through a minim um at a time dela y of 200 fs the rerectivit y rises again to a quasistationary v alue on a picosecond time scale, without an y p erio dic oscillations. The temp oral ev olution of the rerection transien t is en tirely due to susceptibilit y c hanges induced b y the optically excited electronic carriers, their thermalization, and relaxation do wn to the band edges as w ell as in terv alley transfer. Rotating the sample to = 45 results in a shift of the en tire rerectivit y signature to higher v alues. The signal resp onse for p ositiv e time dela ys sho ws a p erio dic mo dulation with an oscillation frequency of 8 : 8 0 : 15 THz, exactly matc hing the frequency of the 15 LO phonon in GaAs. In the case of = 135 a complemen tary shift to lo w er rerectivit y c hanges together with an additional phase shift of in the phonon-induced oscillations is observ ed. The distinct dep endence of rerection mo dulation on prob e b eam p olarization sho ws the electro-optic nature of this eect, whic h is w ell kno wn for cubic zinc-blende group materials with 43m crystal symmetry PAGE 48 39 In a bulk semiconductor the laser w a v elength is m uc h larger than the lattice spacing, so the photo-generated carriers are t ypically excited b y the optical pump o v er spatial areas that are m uc h larger than the lattice unit cell, whic h means the excited carrier p opulations are generated in a macroscopic state and the carrier densit y matrix has only a q 0 F ourier comp onen t. As a result only the q 0 phonon mo de is coupled to the photo excited carriers. Since the frequency of the q 0 acoustic phonon is zero, only the coheren t optic phonons are excited in bulk semiconductors. No w the case is dieren t for semiconductor m ultiple quan tum w ells and sup erlattices. The pump can preferen tially generate electron-hole pairs only in the w ells ev en though the laser pump has a w a v elength large compared to the lattice spacing. So the photo excited carrier distributions ha v e the p erio dicit y of the sup erlattice. Since the densit y matrix of the photo-excited carrier p opulations no w ha v e a q 6 = 0 F ourier comp onen t, the photo-excited carriers can not only couple to the optical phonon mo des, but they can also generate coheren t acoustic phonon mo des with a nonzero frequency and w a v ev ector q 2 =L where L is the sup erlattice p erio d. In the next c hapter I will discuss the coheren t acoustic phonons in m ultiple quan tum w ells. The exp erimen tal data and theory of propagating coheren t phonon in GaN/InGaN epila y er system will b e presen ted in c hapter 5 PAGE 49 CHAPTER 4 THE COHERENT A COUSTIC PHONON The second t yp e of coheren t phonon is the acoustic one. The earlier exp erimen tal observ ations of zone-folded acoustic phonons are made b y Colv ard et al. [ 23 ] They used a photo elastic con tin uum mo del to predict the scattering in tensities of the folded acoustic mo des. The acoustic branc h of semiconductor sup erlattices folds within the miniBrillouin zone b ecause of the articial p erio dicit y of the sup erlattices. This leads to the observ ation of coheren t oscillation of the zone folded acoustic phonons in AlAs/GaAs sup erlattices [ 24 25 ]. The observ ed dieren tial rerection oscillation w as v ery small, ho w ev er, on the order of R =R 10 5 10 6 The detection mec hanism w as based on in terband transitions due to the acoustic deformation p oten tial [ 25 ]. In the y ear 2000 Sun et al. rep orted h uge coheren t acoustic phonon oscillations in wurtzite (0001) InGaN/GaN m ulti-quan tum w ell samples with strain induced piezo electric elds [ 26 ]. The oscillations w ere v ery strong with dieren tial transmission T =T 10 2 10 3 compared to the usual dieren tial rerection on the order of 10 5 {10 6 The oscillation frequency in the THz range, corresp onded to the LA phonon frequency with q 2 =l where l is the p erio d of MQWs. The exp erimen ts of Sun et al. w ere done at ro om temp erature on samples of 14 p erio d InGaN/GaN MQWs with barrier widths xed at 43 A, w ell widths v arying from 12 to 62 A, and In comp osition v arying b et w een 6{10%. The oscillation p erio d for 50 A MQW w as 1.38 ps, whic h corresp onds to a frequency of 0.72 THz. By c hanging the p erio d widths of MQWs, Sun et al. obtained a linear relation b et w een the observ ed oscillation frequency and the w a v ev ector q = 2 =l of 40 PAGE 50 41 photogenerated carriers. The slop e of this disp ersion, 6820 m/s, agrees w ell with the sound v elo cit y of GaN. In this c hapter, I will discuss briery a microscopic theory [ 27 ] for coheren t acoustic phonons in strained wurtzite InGaN/GaN MQWs. This microscopic theory can b e simplied and mapp ed on to a lo ade d string mo del instead of a forced oscillator mo del [ 1 ] as in the case for coheren t optical phonons in bulk systems. Based on the string mo del the sim ulation will sho w the strain, energy densit y and other ph ysical quan tities. I will also discuss the coheren t con trol of the acoustic coheren t phonon. 4.1 Microscopic Theory The microscopic theory for coheren t acoustic phonons in m ultiple quan tum w ell system has man y details and is v ery con v oluted, so I will just giv e an outline of the theory The basic approac h is the same as that used for coheren t optic phonons in bulk semiconductors, i.e. to obtain the total Hamiltonian of the system and then to deriv e the equations of motion for the coheren t phonon amplitudes, whic h are coupled to the equations of motion for the electron densit y matrices. First of all one w an ts to obtain the energy disp ersion relation and w a v efunctions for carriers mo ving in a m ultiple quan tum w ell system so that one can write do wn the electronic part of the Hamiltonian for the indep enden t and free carriers. The GaN/InGaN m ultiple quan tum w ells is sho wn sc hematically in Fig. 4{1 The in trinsic activ e region consists of a left GaN buer region, sev eral pseudomorphically strained (0001) In x Ga 1 x N quan tum w ells sandwic hed b et w een GaN barriers, and a righ t GaN buer region. The total length of the MQWs b et w een the P and N regions is L across whic h a v oltage drop, V is main tained. Photo excitation of carriers is done using an ultrafast laser pulse inciden t normally along the (0001) gro wth direction, tak en to coincide with the z -axis. PAGE 51 42 InGaN WellGaN BarrierInGaN Well GaN Barrier p-GaNn-GaN Figure 4{1: Sc hematic diagram of the In x Ga 1 x N MQW dio de structure. PAGE 52 43 The osets and eects of the built-in piezo electric eld are sho wn in Fig. 4{2 from whic h w e can see that the built-in piezo electric eld is resp onsible for the spatial separation of the photo excited electrons and holes so that the densit y distribution of these t w o kinds of carriers dier from eac h other whic h results in a non-zero non-uniform driving force for the coheren t phonon oscillations. In bulk systems, the conduction and v alence bands in wurtzite crystals including the eects of strain can b e treated b y eectiv e mass theory Near the band edge, the conduction band Hamiltonian is a 2 2 matrix b ecause of the electron spin, while the Hamiltonian for the v alence bands is a 6 6 matrix due to the hea vy holes, ligh t holes, and split-o holes. [ 28 29 ] In quan tum conned systems sho wn in Fig. 4{1 the bulk Hamiltonian is mo died. The nite MQW structure breaks translational symmetry along the z direction but not in the xy plane. The quan tum connemen t p oten tials comes from three sources: (i) bandgap discon tin uities b et w een w ell and barrier regions, (ii) the strain-induced piezo electric eld, and (iii) the time-dep enden t electric eld due to photo excited electrons and holes. V ( z ; t ) = V ; gap ( z ) + V piezo ( z ) + V photo ( z ; t ) ; (4.1) where = f c; v g refers to conduction or v alence subbands. The connemen t of carriers in the MQW leads to a set of t w o-dimensional subbands. In en v elop e function appro ximation, the w a v efunction consists of factors of a slo wly v arying real en v elop e function in the sup erlattice direction and a rapidly v arying Blo c h w a v efunction. Solving Sc hr odinger equations, one obtains the eigen v alues as the subband energies, E n ( k ), with n the subband index, and the corresp onding eigen v ectors as the en v elop e functions. With this energy disp ersion, one can write do wn the second PAGE 53 44 -30 -20 -10 0 10 20 30 Position (nm) -0.01 0 0.01Strain Tensor Component e xx e yy e zz Pseudomorphic Strain -30 -20 -10 0 10 20 30 Position (nm) -0.5 0.0 0.5 1.0 Electric Field ( MV/cm ) Electron Potential (eV) Figure 4{2: Eects of the built-in piezo electric eld to the bandgap of the MQWs. The upp er gure sho ws strain tensor comp onen ts for pseudomorphically strained InGaN MQW dio de as a function of p osition. The middle gure sho ws the electric eld and p oten tial due to the the ab o v e built-in strain eld. The lo w er gure sho ws the conduction and v alence band edges. The applied dc bias has b een adjusted so rat-band biasing is ac hiev ed, i. e., so that band edges are p erio dic functions of p osition. [See Sanders et al., PRB, 64:235316, 2001] PAGE 54 45 quan tized Hamiltonian for free electrons and holes in the m ultiple quan tum w ell, H e 0 = X ;n; k E n ( k ) c y;n; k c ;n; k ; (4.2) where c y;n; k and c ;n; k are electron creation and destruction op erators in conduction and v alence subbands resp ectiv ely The acoustic phonons in the m ultiple quan tum w ell are tak en as plane-w a v e states with w a v e v ector q Due to the cylindrical symmetry of the system, one need consider only longitudinal acoustic phonons with q = q ^ z b ecause these are the ones coupled b y the electron-phonon in teraction. The free LA phonon Hamiltonian is giv en b y H A 0 = X q h! q b yq b q : (4.3) where b yq and b q are creation and destruction op erators for LA phonons with w a v ev ector q = q ^ z and q = c j q j is the linear disp ersion of the acoustic phonons with c b eing the LA phonon sound sp eed. Statistical op erators are normally dened in terms of the electron and phonon eigenstates. So one can write do wn the electron densit y matrix as N ; 0 n;n 0 ( k ; t ) D c y;n; k ( t ) c 0 ;n 0 ; k ( t ) E (4.4) where h i denotes the statistical a v erage of the non-equilibrium state of the system. The diagonal comp onen t of the electron densit y matrix is the distribution function for electrons in the subband, while other comp onen ts describ e the coherence b et w een carriers in dieren t subbands, including b oth in traband and in terband comp onen ts. The coheren t phonon amplitude of the q -th phonon mo de, j q i is dened the same as in the last c hapter [ 1 ] D q ( t ) n b yq ( t ) + b q ( t ) (4.5) PAGE 55 46 Note again that the coheren t phonon amplitude is related to the the macroscopic lattice displacemen t U ( z ; t ) through U ( z ; t ) = X q s h 2 2 0 ( h! q ) V e iq z D q ( t ) (4.6) T o deriv e the equations of motion for the electron densit y matrix and the coheren t phonon amplitude, one m ust kno w the total Hamiltonian of the system, whic h in this case is giv en b y H = H e 0 + H ee + H eL + H A 0 + H eA : (4.7) The rst term is just free electrons and holes as in Eq. ( 4.2 ). The second term describ es the Coulom b in teraction b et w een carriers, including screening and neglecting the Coulom b-induced in terband transitions b ecause they are energetically unfa v orable [ 30 ]. The third term describ es the creation of electron-hole pairs b y the pump laser, the electric eld of whic h is treated in the semiclassical dip ole appro ximation. The last t w o terms are related to the acoustic phonons. One is the free longitudinal acoustic phonons giv en b y Eq. ( 4.3 ) and the other is the electron-LA phonon in teraction, whic h describ es the scattering of an electron from one subband to another b y absorption or emission of an LA phonon. H eA has the same form as that of the bulk semiconductor. The in teraction matrix includes b oth deformation and screened piezo electric scattering. In the end one obtains a closed set of coupled partial dieren tial equations for the electron densit y matrices and coheren t phonon amplitudes. The equation of motion for the coheren t phonon amplitude D q ( t ), whic h is similar to the case of bulk semiconductor, is giv en b y a driv en harmonic oscillator equation @ 2 D q ( t ) @ t 2 + 2 q D q ( t ) = f ( N ; q ; q ; t ) ; (4.8) PAGE 56 47 where N is the electron densit y matrix dened in Eq. ( 4.4 ), and the initial conditions are D q ( t = 1 ) = @ D q ( t = 1 ) @ t = 0 : (4.9) 4.2 Loaded String Mo del The microscopic theory is v ery detailed. Ho w ev er w e are more in terested in the lattice displacemen t U ( z ; t ). It is b oth more insigh tful and m uc h easier to deal with the lattice displacemen t directly Giv en the linear acoustic phonon disp ersion relation, w e can do F ourier transformation on the equation of motion for the coheren t phonon amplitude D q ( t ). The result is the follo wing equation for the lattice displacemen t U ( z ; t ), @ 2 U ( z ; t ) @ t 2 c 2 @ 2 U ( z ; t ) @ z 2 = S ( z ; t ) ; (4.10) sub ject to the initial conditions U ( z ; t = 1 ) = @ U ( z ; t = 1 ) @ t = 0 : (4.11) The forcing function S ( z ; t ) pac k ages all the microscopic details. Since the absorption o ccurs only in the w ells, the forcing function is not uniform. It normally has the same p erio d as the sup erlattice. The sound sp eeds of GaN and InGaN are almost the same as discussed in the in tro duction c hapter. The dierence of sound sp eed for a t ypical In comp onen t of x = 0 : 08 is ab out 2%, whic h causes an ev en smaller rerection on the order of 10 4 so this dierence can b e safely neglected. Th us, Eq. 5.17 describ es a uniform string with a non uniform forcing function. W e call this one-dimensional w a v e equation of the lattice displacemen t the \string mo del". Note the dierence b et w een a bulk system and a MQW or sup erlattice. In a bulk semiconductor, b oth the amplitude U and the F ourier transform of the amplitude D q ( t ) for an q 0 optic mo de satisfy a forced oscillator equation. PAGE 57 48 F or the non uniform, m ultiple quan tum w ell case, one can excite acoustic mo des with q 6 = 0. The F ourier transform of the amplitude D q ob eys a forced oscillator equation, but o wing to the linear dep endence of ( q ) on q the amplitude itself U ob eys a 1-D w a v e equation with a forcing term S ( z ; t ). Under ideal conditions the forcing function tak es a simple form [ 27 ] S ( z ; t ) = X S ( z ; t ) ; (4.12) where the index, is for dieren t kinds of carriers and the forcing function satises the sum rule Z 1 1 d z S ( z ; t ) = 0 ; (4.13) whic h requires that the a v erage force on the string b e zero so that the cen ter of mass of the string w ould ha v e no accelerating motion. The partial driving functions, S ( z ; t ), are giv en b y [ 27 ] S ( z ; t ) = 1 0 a @ @ z + 4 j e j e 33 1 ( z ; t ) (4.14) where the plus sign is for conduction electrons and the min us sign for holes. The photogenerated electron or hole n um b er densit y is ( z ; t ) and 0 is the mass densit y The partial forcing function has t w o terms, the rst due to deformation p oten tial scattering and the second to piezo electric scattering. The piezo electric coupling constan t, e 33 is the same for all carrier sp ecies, while the deformation p oten tial, a dep ends on the sp ecies. Another in teresting p oin t is that Planc k's constan t cancels out in the string mo del. It do es not app ear in either the string equation ( 5.17 ) or in its related forcing function dened in Eqs. ( 4.12 ) and ( 4.14 ). So essen tially coheren t LA phonon oscillations in an MQW is also an classical phenomenon, lik e the coheren t LO phonon oscillations in bulk semiconductors [ 1 ]. PAGE 58 49 4.3 Solution of The String Mo del Using Green's function metho d, w e can solv e the w a v e equation ( 5.17 ) for a giv en forcing function with the initial conditions in Eq. ( 4.11 ). The general solution is giv en b y U ( z ; t ) = Z 1 1 d Z 1 1 d G ( z ; t ) S ( ; ) : (4.15) In p erforming the ab o v e in tegration of the forcing function S ( z ; t ) o v er time w e dieren tiate b et w een t w o cases of the forcing function. One case is for the impulsiv e force, whic h is a delta function in time, f i ( z ; t ) = ( z ) ( t ) : (4.16) The Green's function is giv en b y G i ( z ; t ) = 1 2 c ( t ) 24 ( z + ct ) | {z } ( = ( z ct ) | {z } = ) 35 ; (4.17) where the t w o terms in the solution stands for the w a v e propagating along the p ositiv e and negativ e direction of z axis resp ectiv ely The other case is for the displaciv e force, whic h is a Hea viside step function in time, f d ( z ; t ) = ( z ) ( t ) : (4.18) The Green's function is readily obtained b y in tegrating G i in Eq. ( 4.17 ), G d ( z ; t ) = 1 2 c 2 ( t ) 24 g ( z + ct ) | {z } ( = 2 g ( z ) | {z } + + g ( z ct ) | {z } = ) 35 ; (4.19) where function g ( z ) is dened as g ( z ) = z ( z ) : (4.20) PAGE 59 50 In addition to the t w o propagation terms as in the case of impulsiv e force, there is also a third term whic h is constan t in time and sits righ t in the range of the m ultiple quan tum w ell where the force function is nonzero. In our m ultiple quan tum w ell system, the photogenerated carriers will p ersist for a certain amoun t of time to giv e a displaciv e forcing function S ( z ; t ) = f ( z ) ( t t 0 ). Th us the lattice displacemen t can b e written as U ( z ; t ) = Z 1 1 d G d ( z ; t t 0 ) f ( ) : (4.21) The only nonzero comp onen t of the elastic strain 33 is obtained from the deriv ativ e of the lattice displacemen t with resp ect to space co ordinate z 33 ( z ; t ) = @ U ( z ; t ) @ z : (4.22) The kinetic and p oten tial energy densit y u k and u p according to denition can b e obtained from the deriv ativ es of the lattice displacemen t with resp ect to time and space co ordinate resp ectiv ely u k ( z ; t ) / @ U ( z ; t ) @ t 2 ; (4.23) u p ( z ; t ) / 2 33 ( z ; t ) = @ U ( z ; t ) @ z 2 : (4.24) If w e substitute the displaciv e Green's function in to the displacemen t equation and then calculate the kinetic energy densit y w e will obtain for t 0 = 0 u k ( z ; t ) / ( t ) Z 1 1 d [ ( z + ct ) ( z ct )] f ( ) 2 ; (4.25) where only t w o propagating term app ear b ecause d d z g ( z ) = z ( z ) + ( z ) (4.26) and the in tegral of the rst term with a Dirac -function is zero. PAGE 60 51 W e can calculate the follo wing in tegral Z 1 1 d ( z ) f ( ) = Z 1 1 d ( ) f ( z ) = Z 1 0 d f ( z ) F ( z ) ; and the kinetic energy densit y will b e u k ( z ; t ) / ( t ) [ F ( z + ct ) F ( z ct )] 2 : (4.27) Similarly w e can obtain the p oten tial energy densit y u p ( z ; t ) / ( t ) [ F ( z + ct ) 2 F ( z ) + F ( z ct )] 2 : (4.28) F rom the ab o v e equations w e can see that if F ( z ) is an oscillating function with a certain frequency then the kinetic energy densit y will oscillate with a doubled frequency b ecause it is the square of t w o oscillating functions, but the p oten tial energy will ha v e the same frequency as F ( z ) due to the in terferenc term b et w een the static F ( z ) and the propagating F ( z + ct ) and F ( z ct ). The total energy densit y is just the sum of the kinetic and p oten tial part, u ( z ; t ) = u k ( z ; t ) + u p ( z ; t ) : (4.29) The total energy E as a function of time is obtained b y in tegrating the total energy densit y o v er the whole z axis, E ( t ) = E k ( t ) + E p ( t ) = Z 1 1 [ u k ( z ; t ) + u p ( z ; t )] d z : (4.30) If the forcing function is limited in some region with length L then the kinetic energy will remain constan t after t = L= 2 c at whic h time the t w o opp ositely PAGE 61 52 propagating parts do es not co v er eac h other an ymore. The w a v e form do es not c hange with the propagation, so the in tegral o v er the whole z -axis will remain constan t t = L= 2 c Ho w ev er the p oten tial energy densit y will not b ecome a constan t un til t = L=c whic h is t wice the time of the kinetic energy densit y b ecause the static part com bined with the propagating part will k eep c hanging un til the propagating part is totally outside the forcing function region whic h giv es the time L=c Since the n umerical calculation of microscopic theory [ 27 ] uses a m ultiple quan tum w ell of four p erio ds, w e will rst consider the four-p erio d case for comparison. Figure 4{3 sho ws the forcing function obtained from the microscopic mo del [ 27 ] and a simple sin usoidal forcing function used in the string mo del. The common features of b oth forcing function are their oscillations and p erio dicit y in the range of the m ultiple quan tum w ells. In Fig. 4{4 w e sho w the displacemen t of the lattice calculated from the string mo del using the simple four-p erio d sin usoidal function sho wn in Fig. 4{3 It sho ws clearly the propagating parts of the motion and the static part inside the m ultiple quan tum w ell system that remains b ehind when the dynamic parts tra v el outside the quan tum w ells region. F rom the displacemen t w e can calculate the strain, whic h is sho wn in Fig. 4{5 The image of strain again sho ws clearly the propagation of the w a v e motion. Figure 4{6 sho ws the total energy densit y calculated from the string mo del. It has the v ery similar w a v e propagating features as sho wn in the lattice displacemen t and strain gures. W e plot the energy as a function of time from the string mo del in Fig. 4{7 It has four p eaks and four p erio ds of oscillation, whic h is the same as the n um b er of m ultiple quan tum w ell p erio ds. It also sho ws that the kinetic energy oscillates PAGE 62 53 Figure 4{3: F orcing functions. The upp er one is a sin usoidal function of four p erio ds corresp onding to a four-p erio d MQW. The lo w er one is from the n umerical calculation of the microscopic theory [See Sanders et al., PRB, 64:235316, 2001]. PAGE 63 54 Figure 4{4: Displacemen t as a function of p osition and time. It is the solution of w a v e equation with the simple four-p erio d sin usoidal force sho wn in Fig. 4{3 PAGE 64 55 0 2 0 4 0 6 0 8 0 1 0 0 6 0 4 0 2 00 2 0 4 0 6 0 P o s i t i o n ( a r b u n i t ) T i m e ( a r b u n i t ) Figure 4{5: The image of strain as a function of p osition and time from the string mo del. PAGE 65 56 Figure 4{6: Energy Densit y as a function of p osition and time. After a long time (t=4 here), the energy densit y divides in to a static part and a tra v eling part PAGE 66 57 at t wice the frequency of the p oten tial energy and the time for the oscillation to die out of kinetic energy is half of the p oten tial energy as discussed in Eq. ( 4.30 ). The total energy c hanges with time b ecause w e ha v e treated the driving force as an external one. If w e treat the driving force as in ternal, then the system will ha v e a new equilibrium conguration, around whic h the calculated energy will b e conserv ed. F or the case of a m ultiple quan tum w ell system of fourteen p erio ds, the energy is plotted in Fig. 4{8 The energy plot has fourteen p eaks and it dies out after fourteen p erio ds of oscillation. Again the n um b er of p eaks and the n um b er of oscillations is the same as the n um b er of p erio ds in the m ultiple quan tum w ell system. The energy plot can b e view ed in t w o w a ys. In the real space the w a v e will propagate with the acoustic sound sp eed in b oth directions of the z axis. After the w a v e tra v els totally out of the range of the m ultiple quan tum w ell, the energy stops c hanging and remains constan t, whic h explain the die out of the oscillation. The n um b er of w ells is lik e the n um b er of sources of force. The w a v es they induce sup erimp ose and giv es the n um b er of oscillations in the energy In the w a v ev ector space, b ecause the p erio dic forcing function is conned in a limited range of m ultiple quan tum w ells, the w a v ev ector q will ha v e some uncertain t y q Th us the angular frequency of the coheren t acoustic phonon oscillation will also ha v e an uncertain t y with =! = q =q It is this uncertain t y that leads to a dephasing of the oscillation. Since the more p erio ds of quan tum w ells and the greater exten t of the system will pro duce a smaller uncertain t y q So the die-out time of oscillation is related to the n um b er of quan tum w ells, more sp ecically it is related to the total length of MQWs divided b y the sound v elo cit y in the MQWs. PAGE 67 58 Figure 4{7: Energy as a function of time for 4 quan tum w ells. The n um b er of p eaks of total energy equals the n um b er of w ells. The kinetic energy oscillates t wice fast as the p oten tial energy (b) is from calculation of microscopic theory see [See Sanders et al., PRB, 64:235316, 2001]. PAGE 68 59 Figure 4{8: Energy as a function of time for 14 quan tum w ells. This gure sho ws the same features as that of gure 4{7 (a). PAGE 69 60 4.4 Coheren t Con trol In recen t y ears there are quite a few coheren t con trol exp erimen ts with m ultiple quan tum w ells from sev eral dieren t groups suc h as Sun [ 31 ], Ozg ur [ 32 ], and Nelson [ 33 ]. In InGaN/GaN m ultiple quan tum w ells the eect is more prominen t b ecause of the stronger piezo electric eld. By using double-pulse pump excitation Dek orsy et al. sho w ed the con trolling of the amplitude of the coheren t optical phonons in GaAs [ 34 ]. Hase et al. conducted this kind of optical con trol of coheren t optical phonons in bism uth lms [ 35 ]. Bartels et al. rep orted coheren t con trol of acoustic phonons in semiconductor sup erlattices [ 36 ]. In these exp erimen ts, the pulse train of pump laser w as splited in to t w o or more b eams that tra v els dieren t optical path distances b y means suc h as Mic helson in terferometer. The sample will excited t wice or more at dieren t times. The c hanges in rerectivit y w ere measured the same as the pump-prob e setup. The results from these exp erimen ts sho w that the coheren t phonons can in terfere with eac h other and can b e coheren tly con trolled b y using m ultiple optical pump trains. Dep ending on dieren t phase dela ys of pump pulses, the coheren t oscillation can b e enhanced or annihilated. In this section I will use the string mo del to analyze the coheren t con trol of coheren t acoustic phonons. In the generation of coheren t acoustic phonons, the ultrafast laser pump generates electron-hole pairs, whic h is the source of the forcing function. In the string mo del w e use a simplied sin usoidal forcing function to replace the microscopic one. The coheren t con trol uses t w o pump pulses, whic h mean w e will ha v e t w o forcing functions. In exp erimen ts, one can con trol the time in terv al b et w een the t w o pump pulses. Theoretically w e can con trol not only the time in terv al b et w een the start time of the t w o forcing function but also the phase dierence b et w een them. Based on the string mo del, w e can try coheren t con trol PAGE 70 61 theoretically b y using a forcing function with t w o terms. The rst forcing term tak es eect at time t = 0 and the second forcing term app ears at a later time t = t 0 with a relativ e spatial phase z 0 and amplitude A with resp ect to the rst term. Note that while the rst forcing term corresp onds directly to the excitation pump, the relation b et w een the second forcing term and the con trol pulse is not so direct and clear b ecause the second forcing term will dep end on b oth the excitation pump and the con trol pulse. The forcing functions ha v e the form A sin ( z z 0 ) ( t t 0 ), describing displaciv e forces starting at time t 0 and ha ving an initial phase of z 0 The amplitude A is tak en as unit for the rst force. But the second force ma y ha v e a relativ e larger or smaller amplitude. The phase eects ha v e t w o asp ects. One is the temp oral phase eect due to the time in terv al b et w een the t w o forcing terms. The other is the spatial eect caused b y the relativ e initial phase dierence b et w een the spatial parts of the t w o forcing terms. First let us consider the eects of the temp oral phase and the relativ e amplitude. The total forcing function can b e written do wn as S ( z ; t ) = sin ( z ) ( t ) + A sin( z ) ( t t 0 ) : (4.31) When the time in terv als t 0 are ev en m ultiples of the t w o terms in the forcing function will b e temp orally in phase. In the cases of t 0 = (2 n + 1) with n b eing an in teger, the t w o terms will b e temp orally out of phase. T o sa v e n umerical computation time without losing an y features w e calculate the p oten tial energies only whic h are prop ortional to the square of the strain. Figure 4{9 sho ws the temp orally in-phase c onstructive enhancemen t of the oscillations. The time in terv al in this case is 4 the relativ e amplitudes of the second forcing term are 1.5, 1, and 0 resp ectiv ely The lo w est curv e with zero PAGE 71 62 relativ e amplitude means that there is only one forcing term. W e see that no matter what the relativ e amplitudes are the constructiv e enhancemen t eects are alw a ys there. 0 2 0 4 0 6 0 8 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 A = 1 5 A = 1 0 A = 0 0 P o t e n t i a l E n e r g y ( a r b u n i t )T i m e ( a r b u n i t ) Figure 4{9: P oten tial energy as a function of time for 14 quan tum w ells with t w o forcing terms dela y ed b y an in-phase time in terv al of 4 This gure sho ws the constructiv e enhancemen t of the oscillations as a result of coheren t eect of the t w o forcing terms. The destructive eects of t w o temp orally out of phase forcing terms are plotted in Fig. 4{10 The lo w est curv e is for one forcing term only The other three curv es ha v e the relativ e amplitudes 0.7, 1, and 1.5 resp ectiv ely The destructiv e eect is most ob vious when the relativ e amplitude of the second force is smaller than the rst, while for larger amplitudes the oscillations reduces but do es not disapp ear. Since the coheren t phonon oscillations ha v e a tendency to dephase, the oscillations PAGE 72 63 will b ecome smaller with time. T o further reduce the oscillation amplitudes the second pump pulse do es not need to b e stronger. It do es not need ev en to reac h the same strength as the rst one. This is wh y the case of a force with relativ e amplitude A = 0 : 7 has the strongest destructiv e eect. 0 2 0 4 0 6 0 8 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 A = 1 5 A = 1 0 A = 0 7 A = 0 0 P o t e n t i a l E n e r g y ( a r b u n i t )T i m e ( a r b u n i t ) Figure 4{10: P oten tial energy as a function of time for 14 quan tum w ells with t w o forcing terms dela y ed b y an out of phase time in terv al of 5 This gure sho ws the destructiv e eect of the oscillations as a result of coheren t con trol b y t w o forcing terms. Using the formalism discussed in Chapter 3 w e can calculate the dieren tial transmission. T o further simplify the matter, the sensitivit y function is tak en to b e a constan t in the w ells. Outside of the w ells it is zero. Fig. 4{11 sho ws the constructiv e and destructiv e eects of the t w o forcing terms as discussed ab o v e. PAGE 73 64 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 f o r c e : s i n ( z )DT ( n o t n o r m a l i z e d )t ( a r b u n i t ) i n p h a s e o u t o f p h a s e f o r c e : c o s ( z ) Figure 4{11: The c hange of transmission as a function of time for 14 quan tum w ells with t w o forcing terms dela y ed b y an in phase time in terv al of 4 and an out of phase time in terv al of 5 The upp er and lo w er t w o sets of curv es are displaced with regard to eac h other for a b etter view. PAGE 74 65 The temp oral phases, i.e. the time in terv als b et w een the t w o forces, are easy to con trol exp erimen tally b y v arying the optical path of the t w o pump pulses, but the con trol of the spatial phases z 0 is not so ob vious. In theory ho w ev er, it is not hard to observ e the eects of spatial phases z 0 on the coheren t con trol. W e will tak e the amplitudes of the t w o forces to b e the same. If w e k eep the temp orally in phase time in terv al as t 0 = 4 then w e can ha v e four t ypical cases of the t w o forcing term. Case I1: S ( z ; t ) = cos ( z ) ( t ) + cos ( z 2 ) ( t 4 ). Case I2: S ( z ; t ) = cos ( z ) ( t ) + cos ( z ) ( t 4 ). Case I3: S ( z ; t ) = cos ( z ) ( t ) + cos ( z 3 2 ) ( t 4 ). Case I4: S ( z ; t ) = cos ( z ) ( t ) + cos ( z ) ( t 4 ). These cases are plotted in Fig. 4{12 W e ha v e already discussed case (I4) in the ab o v e, whic h is the constructiv e temp orally in phase enhancemen t of the oscillations. F or case (I2) at time t > 4 the sum of the t w o forcing terms will giv e a zero total forcing function. As a result, the oscillation will totally disapp ear. In cases (I1) and (I3) the spatial part of the total forcing function is giv en b y cos ( z ) sin ( z ). The amplitudes of the oscillations for these t w o cases are not as big as the coheren t case (I4) where the phases lo c k in b oth temp orally and spatially Th us for temp orally in phase time in terv al, if the spatial phases are b et w een the t w o extremes of in phase and out of phase as sho wn in case (I4) and (I2) resp ectiv ely the enhancemen t of the oscillations will also v ary b et w een the totally coheren t and the totally destructiv e. There are four similar cases for the temp orally out of phase time in terv al. Again w e tak e t 0 = 5 The four cases are listed b elo w. Case O1: S ( z ; t ) = cos( z ) ( t ) + cos ( z 2 ) ( t 5 ). Case O2: S ( z ; t ) = cos( z ) ( t ) + cos ( z ) ( t 5 ). PAGE 75 66 0 2 0 4 0 6 0 8 0 1 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 I 4 : c o s + c o s I 1 3 : c o s + / s i n I 2 : c o s c o s DT ( a r b u n i t )T i m e ( a r b u n i t ) Figure 4{12: Eects of phase z 0 on the c hange of transmission as a function of time for 14 quan tum w ells. The t w o forcing terms are dela y ed b y an in phase time in terv al of 4 The dieren t curv es are for dieren t com bination of phases z 0 as discussed in the text. PAGE 76 67 Case O3: S ( z ; t ) = cos( z ) ( t ) + cos ( z 3 2 ) ( t 5 ). Case O4: S ( z ; t ) = cos( z ) ( t ) + cos ( z ) ( t 5 ). Figure 4{13 sho ws the c hanges of transmissions for these four temp orally out of phase cases. In cases (O1), O(3), and (O4) the spatial part of the total forcing function is the same as the corresp onding temp orally in phase cases, i.e., cos( z ) sin ( z ) and w e see the exp ected destructiv e eects of the oscillations, although the reduction is not drastic b ecause the relativ e amplitude of the second forcing term is as big as the rst. F or the spatially out of phase case (O2) the total forcing function is iden tically zero at times t > t 0 = 5 but the oscillation p ersists, whic h is a distinct con trast to the corresp onding case (I2). In case (I2) the oscillation almost totally disapp ear after the second temp orally in phase force comes in b ecause at time t = t 0 = 4 the oscillation reac hes one of its lo w est p oin ts the v alue of whic h is nearly zero. Ho w ev er, in case (O2) at the starting time t = t 0 = 5 the oscillation is at one of its highest p oin t. Although from then on there is no forces an y more, the oscillation will k eep on lik e a free oscillator. Note that w e ha v e not tak en in to consideration of an y damping force, but the oscillation still dies out. The dephasing time is ab out a half of that of the one forcing term case. This dephasing eect, as w e men tioned b efore, is due to the limited range of the m ultiple quan tum w ell system. 4.5 Summary W e ha v e discussed briery a microscopic theory for the generation and detection of coheren t acoustic phonons in GaN/InGaN m ultiple quan tum w ell system. F ollo wing the outlined pro cedures w e arriv e at a set of coupled dieren tial equations of motion for the electron densit y matrices and coheren t phonon amplitude. The latter ones can b e mapp ed on to a one-dimensional w a v e equation called the PAGE 77 68 0 2 0 4 0 6 0 8 0 1 0 0 0 3 0 6 0 9 0 1 2 0 1 5 0 O 1 3 : c o s + / s i n O 2 : c o s c o s O 4 : c o s + c o s DT ( a r b u n i t )T i m e ( a r b u n i t ) Figure 4{13: Eects of phase z 0 on the c hange of transmission as a function of time for 14 quan tum w ells. The t w o forcing terms are dela y ed b y an out of phase time in terv al of 5 The dieren t curv es are for dieren t com bination of phases z 0 as discussed in the text. PAGE 78 69 string mo del, whic h describ es the motion of a uniform string under a non-uniform forcing function. W e can solv e the string mo del using Green's function metho d to obtain the lattice displacemen t and other related ph ysical quan tities suc h as strain, energy densit y energy and the dieren tial transmission. All these quan tities ha v e the oscillating prop ert y The strain and energy densit y propagate in b oth directions along the sup erlattice gro wth axis z The oscillations of energy and dieren tial transmission ha v e the same n um b er of p eaks as the n um b er of the w ells and they dies out b ecause of the limited range of the m ultiple quan tum w ell system. Another in teresting application of the string mo del is coheren t con trol of these coheren t acoustic phonons. By using t w o forcing terms corresp onding to t w o pump pulses, w e can c hange three elemen ts of the second forcing term with resp ect to the rst, the temp oral phase, the spatial phase, and the relativ e amplitude. The temp oral phase is the most eectiv e con trol. W e ha v e sho wn b oth the temp orally in phase constructiv e and out of phase destructiv e enhancemen t. T o get b etter destructiv e eect the relativ e amplitude of the second forcing term should b e small than the rst. In spatially out of phase cases the total forcing function will b e zero after the starting time of the second forcing term, but there is a distinctiv e con trast b et w een the temp orally in phase and out of phase cases. In the former case the oscillation almost disapp ear, while in the latter case the oscillation p ersists with a dephasing time a half of the case when there is only one forcing term. PAGE 79 CHAPTER 5 PR OP A GA TING COHERENT PHONON Heterostructures of GaN and InGaN are imp ortan t materials o wing to their applications to blue laser dio des and high p o w er electronics [ 6 ]. Strong coheren t acoustic phonon oscillations ha v e recen tly b een detected in InGaN/GaN m ultiple quan tum w ells [ 26 37 ]. These phonon oscillations w ere m uc h stronger than folded acoustic phonon oscillations observ ed in other semiconductor sup erlattices [ 24 25 38 ]. InGaN/GaN heterostructures are highly strained at high In concen trations giving rise to large built-in piezo electric elds [ 39 40 41 42 ], and the large strength of the coheren t acoustic phonon oscillations w as attributed to the large strain and piezo electric elds [ 26 ]. In this c hapter, w e discuss the generation of strong lo calized coheren t phonon w a v e-pac k ets in strained la y er In x Ga 1 x N/GaN epila y ers and heterostructures gro wn on GaN and Sapphire substrates [ 43 ]. This w ork w as done in co op eration with Gary D. Sanders, Christopher J. Stan ton, and Chang Sub Kim. The exp erimen ts w ere p erformed b y J. S. Y ahng, Y. D. Jho, K. J. Y ee, E. Oh and D. S. Kim. (See cond-mat/0310654.) By fo cusing high rep etition rate, frequency-doubled fem tosecond Ti:Sapphire laser pulses on to strongly strained InGaN/GaN heterostructures, w e can, through the carrier-phonon in teraction, generate coheren t phonon w a v epac k ets whic h are initially lo calized near the epila y er/surface but then propagate a w a y from the surface/epila y er and through a GaN la y er. As the w a v epac k ets propagate, they mo dulate the lo cal index of refraction and can b e observ ed in the time-dep enden t dieren tial rerectivit y of the prob e pulse. There is a sudden drop in the amplitude 70 PAGE 80 71 of the rerectivit y oscillation of the prob e pulse when the phonon w a v e-pac k et reac hes a GaN-sapphire in terface b elo w the surface. Theoretical calculations as w ell as exp erimen tal evidences supp ort this picture; the sudden drop of amplitude when the w a v e encoun ters the GaN-sapphire in terface cannot b e explained if the w a v e-pac k et had large spatial exten t. When the w a v e-pac k et encoun ters the GaNsapphire in terface, part of the w a v e gets rerected while most of it gets transmitted in to the sapphire substrate, dep ending on the in terface prop erties and the excess energy of the exciting photons. This exp erimen t illustrates a non-destructiv e w a y of generating high pressure tensile w a v es in strained heterostructures and using them to prob e semiconductor structure b elo w the surface of the sample. Since the strength of this non-destructiv e w a v e is determined b y the strain b et w een GaN and InGaN, it is lik ely that ev en stronger coheren t phonons can b e generated in InGaN/GaN digital allo ys gro wn on a GaN substrate. 5.1 Exp erimen tal Results In the exp erimen ts, frequency-doubled pulses of mo de-lo c k ed Ti:sapphire lasers are used to p erform rerectiv e pump-prob e measuremen ts on four dieren t sample t yp es whic h are sho wn in Fig. 5{1 : T yp e (I) InGaN epila y ers; T yp e (I I) InGaN/GaN double quan tum w ells (DQWs); T yp e (I I I) InGaN/GaN single quan tum w ell (SQW); and T yp e (IV) InGaN/GaN ligh t-emitting dio de (LED) structures. The p eak pump p o w er is estimated to b e 400 MW/cm 2 corresp onding to a carrier densit y of 10 19 cm 3 and the doubled pulse width is 250 fs. All samples w ere gro wn on a c -plane sapphire substrate b y metal organic c hemical v ap or dep osition. The InGaN epila y ers sho wn in Fig. 5{1 (a) consist of 1 m GaN gro wn on a sapphire substrate and capp ed with 30 nm of In x Ga 1 x N with In comp osition, x v arying from 0.04 to 0.12. The DQW sample sho wn in Fig. 5{1 (b) consists of GaN (1 m), double quan tum w ells of In 0 : 12 Ga 0 : 88 N (1{16 nm) and barrier of GaN (1{16 nm), and a GaN cap la y er (0.1 m). The SQW sample in Fig. 5{1 (c) consists PAGE 81 72 of GaN (2 m), In 0 : 12 Ga 0 : 88 N (24 nm), and GaN cap la y er (0.1 m). The blue LED structure sho wn in Fig. 5{1 (d) consists of n -GaN (4 m), 5 quan tum w ells of In 0 : 15 Ga 0 : 85 N (2 nm) and 4 barriers of GaN (10 nm), p -Al 0 : 1 Ga 0 : 9 N (20 nm), and p -GaN (0.2 m). 0.88 N 24nm m m 4 Ga m m m n-GaN m m GaN 0.1 m m GaN 0.12 m In 1 1 m GaN m GaN m 0.1 m Sapphire substrate Sapphire substrate Sapphire substrate GaN GaN (1~16nm) In 0.12 Ga 0.88 N (1~16nm) In 0.12 Ga 0.88 N (1~16nm) x5 In Ga 0.85 N 2nm p-Al p-GaN 0.2 2 0.15 GaN 10nm In Ga N 20nm 0.1 0.9 Ga N 30nm x 1-x (a) (b) (c) (d) Sapphire substrate Figure 5{1: The simple diagram of the sample structures used in these exp erimen ts. (a)InGaN epila y er (t yp e I), (b)InGaN/GaN double quan tum w ells (t yp e I I), (c)GaN single quan tum w ell (t yp e I I I), (d)InGaN/GaN blue ligh t-emitting dio de structure (t yp e IV). Dieren tial rerection pump-prob e measuremen ts for the In x Ga 1 x N epila y ers are sho wn in Fig. 5{1 (a). Fig. 5{2 sho ws the oscillatory comp onen t of the measured prob e dieren tial rerectivit y for the InGaN epila y ers (t yp e I) with v arious In concen trations, x. F or comparison purp oses, dieren tial rerectivit y w as measured on a pure GaN HVPE gro wn sample in order to sho w that no dieren tial rerectivit y oscillations are presen t in the absence of strain and an epila y er. The energy of the pump laser w as v aried b et w een 3.22 and 3.35 eV, to k eep the excess carrier energy ab o v e the InGaN band gap but b elo w the GaN band gap. W e note that if the laser energy w as b elo w the InGaN band gap, no signal w as detected. Ther efor e, c arrier gener ation is essential to observing the oscil lations, unlike a r e c ent c oher ent optic al phonon exp eriment in GaN [ 44 ]. The inset sho ws the pump-prob e signal prior to the bac kground subtraction for x = 0 : 12. The bac kground results from the PAGE 82 73 Figure 5{2: The oscillatory comp onen t of the dieren tial rerection pump-prob e data for the In x Ga 1 x N epila y ers with v arious In comp osition ( x =0.04, 0.08, 0.10, and 0.12). F or comparison, dieren tial rerection in a pure GaN HPVE gro wn sample is sho wn. The rerection signal prior to the bac kground subtraction for x = 0 : 12 is sho wn in the inset. PAGE 83 74 relaxation of the photo excited electrons and holes. The oscillations are quite large, on the order of 10 2 {10 3 and the p erio d is 8{9 ps, indep endent of the In comp osition but dep endent on the prob e photon energy The amplitude of the oscillation is appro ximately prop ortional to the In concen tration indicating that the strain at the InGaN/GaN in terface is imp ortan t. The observ ed p erio d is appro ximately = = 2 C s n where is the prob e b eam w a v elength, C s the longitudinal acoustic sound v elo cit y and n the refractiv e index of GaN [ 19 ]. Tw o-color pump-prob e exp erimen ts w ere p erformed for a t yp e I I I InGaN SQW sample as sho wn in Fig. 5{1 (b). Fig. 5{3 sho ws the dieren tial rerectivit y oscillations for dieren t prob e energies. Note that the p erio d of the oscillation c hanges and is pr op ortional to the pr ob e wavelength In addition, the amplitude of the dieren tial rerectivit y oscillation decreases as the detuning (with resp ect to the pump) b ecomes larger. The inset sho ws the oscillation amplitude as a function of the prob e energy in a logarithmic scale and there is an 2-order-of magnitude decrease in dieren tial rerectivit y when the prob e energy c hanges from 3.26 eV to 1.63 eV. In terestingly Fig. 5{4 sho ws that the amplitude of the oscillatory comp onen t of the dieren tial rerectivit y is indep enden t of the bias v oltage, ev en though the carrier lifetime c hanges dramatically with v oltage bias. Fig. 5{4 sho ws the bias dep enden t acoustic phonon dieren tial rerectivit y oscillations in a t yp e IV blue LED structure (see Fig. 5{1 (d)) at a pump energy of 3.17 eV. The lifetime of the bac kground signal drastically decreases as the bias increases as sho wn in Fig. 5{4 (a). This is due to the carrier recom bination time and the decrease in the tunneling escap e time in the strong external bias regime [ 45 ]. On the other hand, the amplitude and frequency of the oscillatory comp onen t of the dieren tial rerectivit y do esn't c hange m uc h with bias v oltage [Fig. 5{4 (b)]. Since the observ ed rerectivit y oscillation is indep enden t of the carrier lifetime for lifetimes as short as PAGE 84 75 Figure 5{3: The oscillation traces of a SQW (I I I) at dieren t prob e energies. The pump energy is cen tered at 3.26 eV. The b ottom curv e has b een magnied 30 times. The inset sho ws the oscillation amplitude as a function of prob e photon energy on a logarithmic scale. PAGE 85 76 Figure 5{4: Pump-prob e dieren tial rerectivit y for the blue LED structure (t yp e IV). (a) External bias v aries at a pump energy of 3.17 eV. The deca y time of the bac kground signal is drastically reduced as the bias increases. (b) The oscillatory amplitude do es not c hange m uc h with bias. PAGE 86 77 1 ps due to ultrafast tunneling [b ottom curv e of Fig. 5{4 (a)], it implies that once the source that mo dulates the exp erimen tally observ ed rerectivit y is launc hed b y the sub-picosecond generation of carriers, the remaining carriers do little to aect the source. This suggests that the rerectivit y oscillation is due to the strain pulse whic h is generated at short times once the pump excites the carriers and mo dulates the lattice constan t. Figure 5{5: The oscillatory comp onen t of the pump-prob e dieren tial transmission traces of DQW's (I I) at 3.22 eV. The left gure sho ws the w ell width dep endence and the righ t gure sho ws the barrier width dep endence. Fig. 5{5 sho ws the w elland barrier-dep enden t acoustic phonon dieren tial rerectivit y oscillations in the t yp e I I DQW samples (see Fig. 5{1 (b)) at a pump energy of 3.22 eV. The amplitude increases as the w ell width increases. Ho w ev er, the oscillation amplitude of the dieren tial rerectivit y do esn't c hange m uc h with the barrier width. This means that the generation of the acoustic phonons is due PAGE 87 78 to the InGaN la y er (w ell) and not GaN la y er (barrier). This also v eries that the oscillation is due to the strain in InGaN la y er. In teresting results are seen in the long time b eha vior of the rerectivit y oscillations sho wn in Fig. 5{6 (a). The long time scale rerectivit y oscillation is plotted for the epila y er (I), DQW (I I), SQW (I I I), and the blue LED structure (IV) at 3.29 eV (b elo w the GaN band gap). Astonishingly the oscillation amplitude abruptly decreases within one cycle of an oscillation at a critical time whic h app ears to scale with the thic kness of the GaN la y er in eac h sample. In addition, the slop e of the GaN thic kness vs. the critical time is v ery close to the kno wn v alue of the sound v elo cit y in GaN [inset of Fig. 5{6 (a)] [ 26 ]. Fig. 5{6 (b) sho ws results when the prob e laser energy is c hanged to 3.44 eV whic h is ab o v e the GaN band gap. Then the laser prob e is sensitiv e to coheren t phonon oscillations only within an absorption depth of the surface. W e see that the amplitude of the rerectivit y oscillation exp onen tially deca ys with a deca y time of 24.2 ps corresp onding to a p enetration depth in GaN of ab out 0.17 micron (=24.2 ps 7000 m/s). The oscillation reapp ears at 260 ps for epila y er (I) and 340 ps for DQW (I I). This is t wice the critical time for the oscillations to disapp ear when the photon energy is 3.29 eV. This further sho ws that the prob e pulse is sensitiv e to the coheren t acoustic w a v e. The "ec ho" in the prob e signal results from the partial rerection of the coheren t phonon o the GaN/sapphire in terface. 5.2 Theory T o explain the exp erimen tal results discussed in the last section, w e ha v e dev elop ed a theoretical mo del of the generation, propagation and detection of coheren t acoustic phonons in strained GaN/InGaN heterostructures. The pump laser pulse generates a strain eld that propagates through the sample whic h, in turn, causes a spatio-temp oral c hange in the index of refraction. This c hange is resp onsible for the oscillatory b eha vior seen in the prob e-eld rerectivit y in v arious PAGE 88 79 Figure 5{6: The long time-scale dieren tial rerectivit y traces. (a) Dieren tial rerectivit y oscillation traces of an epila y er (I), a DQW (I I), a SQW (I I I), and blue LED structure (IV). The inset sho ws the GaN thic kness b et w een the sapphire substrate and InGaN activ e la y er of the sample as a function of the die out time of the oscillations. The solid line indicates that the v elo cit y of the w a v epac k et in the GaN medium is ab out 7000 m/s. (b) The oscillation traces of epila y er and DQW [top t w o curv es in (a)]at 3.44 eV whic h corresp onds to a prob e laser energy ab o v e the band gap of GaN. Exp erimen tal data are from the group of D. S. Kim. [APL, 80 4723, 2002] PAGE 89 80 semiconductor heterostructures. An appro ximate metho d of solving Maxw ell's equations in the presence of spatio-temp oral disturbances in the optical prop erties and obtaining the rerectivit y of the prob e eld in thin lms excited b y picosecond pump pulses can b e found in Thompsen [ 19 ]. The spatio-temp oral disturbance of the refractiv e index is caused b y the propagating coheren t phonon w a v epac k ets. Th us, an essen tial ingredien t in the understanding the prob e rerectivit y is a mo del for the generation and propagation of the v ery short strain pulse in the sample. Recen tly a microscopic theory explaining the generation and propagation of suc h a strain pulse w as rep orted b y Sanders et al. [ 27 46 ] where it w as sho wn that propagating coheren t acoustic phonon w a v epac k ets are created b y the nonequilibrium carriers excited b y the ultrafast pump pulse. The acoustic phonon oscillations arise through the electronphonon in teraction with the photo excited carriers. Both acoustic deformation p oten tial and piezo electric scattering w ere considered in the microscopic mo del. It w as found that under t ypical exp erimen tal conditions, the microscopic theory could b e simplied and mapp ed on to a loaded string mo del. Here, w e use the string mo del of coheren t phonon pulse generation to obtain the strain eld seen b y the prob e pulse. First, w e solv e Maxw ell's equations to obtain the prob e rerectivit y in the presence of a generalized spatio-temp oral disturbance of the index of refraction. Let n b b e the index of refraction without the strain whic h is real b ecause initially the absorption can b e neglected and let ~ n = n + i b e the propagating c hange in the index of refraction due to the strain. When the eect of the c hange of the index of refraction is tak en in to accoun t, the prob e eld with energy can b e describ ed b y the follo wing generalized w a v e equation @ 2 E ( z ; t ) @ z 2 + 2 c 2 [ n b + ~ n ( z ; t )] 2 E ( z ; t ) = 0 (5.1) PAGE 90 81 where E ( z ; t ) is the prob e eld in the slo wly v arying en v elop e function appro ximation and is the cen tral frequency of the prob e pulse. Eq. ( 5.1 ) is obtained from Maxw ell's equations assuming that the p olarization resp onse is instan taneous and that the prob e pulse ob eys the slo wly v arying en v elop e function appro ximation. Since j ~ n j n b under t ypical conditions, Eq. ( 5.1 ) can b e cast in to @ 2 E ( z ; t ) @ z 2 + ( n b k ) 2 E ( z ; t ) = 2 n b k 2 ~ n ( z ; t ) E ( z ; t ) (5.2) where k = =c is the prob e w a v ev ector. T o relate the c hange of the index of refraction to the strain eld, ( z ; t ) @ U ( z ; t ) =@ z w e assume j ~ n j n b and adopt the linear appro ximation [ 19 ] ~ n ( z ; t ) = @ ~ n @ ( z ; t ) : (5.3) W e view Eq. ( 5.2 ) as an inhomogeneous Helmholtz equation and obtain the solution using the Green's function tec hnique. The desired Green's function is determined b y solving @ 2 G ( z ; z 0 ) @ z 2 + ( n b k ) 2 G ( z ; z 0 ) = ( z z 0 ) (5.4) and the result is G ( z ; z 0 ) = i 2 n b k exp ( in b k j z z 0 j ) : (5.5) Then, the solution to Eq. ( 5.2 ) can b e written as E ( z ; t ) = E h ( z ; t ) + Z 1 1 dz 0 G ( z ; z 0 ) 2 n b k 2 ~ n ( z 0 ; t ) E ( z 0 ; t ) (5.6) E h ( z ; t ) + Z 1 1 dz 0 G ( z ; z 0 ) 2 n b k 2 ~ n ( z 0 ; t ) E h ( z 0 ; t ) (5.7) where w e ha v e c hosen the lo w est Born series in the last line. In Eq. ( 5.7 ), E h is the homogeneous solution whic h tak es the form E h ( z ; t ) = ~ E h ( z ; t ) exp f i ( n b k z t ) g for the prob e pulse mo ving to the righ t in the sample without optical distortion. PAGE 91 82 W e no w apply this appro ximate solution to our structure where the in terface b et w een air and the sample is c hosen at z = 0. In the air, where z 0, there is an inciden t prob e pulse tra v eling to w ard the sample as w ell as a rerected pulse. The electric eld in the air, E < ( z ; t ), can th us b e written as the sum E < ( z ; t ) = ~ E i ( z ; t ) e i ( k z t ) + ~ E r ( z ; t ) e i ( k z + t ) (5.8) where E i ( z ; t ) and E r ( z ; t ) are the slo wly v arying en v elop e functions of the inciden t and rerected prob e elds, resp ectiv ely Inside the sample, z 0, the solution is giv en as E > ( z ; t ) = E h ( z ; t ) + Z 1 0 dz 0 exp ( in b k ( z 0 z )) ik ~ n ( z 0 ; t ) E h ( z 0 ; t ) = ~ E t ( z ; t ) e i ( n b k z t ) + Z 1 0 dz 0 exp (2 in b k z 0 ) ik ~ n ( z 0 ; t ) ~ E t ( z ; t ) e i ( n b k z + t ) (5.9) = ~ E t ( z ; t ) e i ( n b k z t ) + A ( n b k ; t ) ~ E t ( z ; t ) e i ( n b k z + t ) (5.10) where w e used E h ( z ; t ) = ~ E t ( z ; t ) exp ( i ( n b k z t )) in the second step, assuming that ~ E t ( z ; t ) is nearly constan t within the slo wly v arying en v elop e function appro ximation, and w e dene a rerected amplitude function A ( n b k ; t ) Z 1 0 dz 0 exp (2 in b k z 0 ) ik ~ n ( z 0 ; t ) : (5.11) The expression for the rerected amplitude function, A ( n b k ; t ), in Eq. ( 5.9 ) sa ys there is a frequency-dep enden t mo dulation of the amplitude in the rerected w a v e in the sample due to the propagating strain. Ha ving determined the w a v es on b oth sides of the in terface, w e can no w calculate the rerectivit y W e apply the usual b oundary conditions to the slo wly v arying en v elop e functions and the results are written compactly as 0B@ 1 1 + A 1 n b (1 A ) 1CA 0B@ ~ E r ~ E t 1CA = 0B@ ~ E i ~ E i 1CA : (5.12) PAGE 92 83 W e solv e this equation to obtain ~ E r ~ E i = r 0 + A 1 + r 0 A r 0 + A (5.13) where r 0 = (1 n b ) = (1 + n b ). T o the same order, w e nd that ~ E t = ~ E i t 0 (1 r 0 A ) where t 0 = 2 = (1 + n b ). It is no w straigh tforw ard to calculate the dieren tial rerectivit y as R R = j r 0 + r j 2 j r 0 j 2 j r 0 j 2 2 r 0 Re A : (5.14) Finally b y substituting the linear la w Eq. ( 5.3 ) in to Eq. ( 5.11 ) and using Eq. ( 5.14 ) w e get R R = Z 1 0 dz F ( z ; ) @ U ( z ; t ) @ z (5.15) where the sensitivit y function, F ( z ; ), is dened as F ( z ; ) = 2 k r 0 @ n @ sin(2 n b k z ) + @ @ cos (2 n b k z ) (5.16) In Eq. ( 5.15 ), the dieren tial rerectivit y is expressed in terms of the lattice displacemen t, U ( z ; t ), due to propagating coheren t phonons. Sanders et al. [ 27 46 ] dev elop ed a microscopic theory sho wing that the coheren t phonon lattice displacemen t satises a driv en string equation, @ 2 U ( z ; t ) @ t 2 C 2 s @ 2 U ( z ; t ) @ z 2 = S ( z ; t ) ; (5.17) where C s is the LA sound sp eed in the medium and S ( z ; t ) is a driving term whic h dep ends on the photogenerated carrier densit y The LA sound sp eed is related to the mass densit y , and the elastic stiness constan t, C 33 b y C s = p C 33 = The LA sound sp eed is tak en to b e dieren t in the GaN/InGaN heterostructure and Sapphire substrate. F or simplicit y w e neglect the sound sp eed mismatc h b et w een the GaN and In x Ga 1 x N la y ers. PAGE 93 84 The driving function, S ( z ; t ), is non uniform and is giv en b y S ( z ; t ) = X S ( z ; t ) ; (5.18) where the summation index, runs o v er carrier sp ecies, i.e., conduction electrons, hea vy holes, ligh t holes, and crystal eld split holes, that are created b y the pump pulse. Eac h carrier sp ecies mak es a separate con tribution to the driving function. The partial driving functions, S ( z ; t ), in piezo electric wurtzite crystals dep end on the densit y of the photo excited carriers. Th us, S ( z ; t ) = 1 a @ @ z + 4 j e j e 33 1 ( z ; t ) ; (5.19) where the plus sign is used for conduction electrons and the min us sign is used for holes. Here ( z ; t ) is the photogenerated electron or hole n um b er densit y whic h is real and p ositiv e, is the mass densit y a are the deformation p oten tials, e 33 is the piezo electric constan t, and 1 is the high frequency dielectric constan t. T o b e more sp ecic, w e will consider a SQW sample of the t yp e sho wn in Fig. 5{1 (c). W e adopt a simple mo del for photogeneration of electrons and holes in whic h the photogenerated electron and hole n um b er densities are prop ortional to the squared ground state particle in a b o x w a v efunctions. The exact shap e of the electron and hole n um b er densit y prole is not critical in the presen t calculation since all that really matters is that the electrons and holes b e strongly lo calized. The carriers are assumed to b e instan taneously generated b y the pump pulse and are lo calized in the In x Ga 1 x N quan tum w ell. Ha ving obtained a mo del expression for ( z ; t ), it is straigh tforw ard to obtain S ( z ; t ) using Eqs. ( 5.18 ) and ( 5.19 ). T o obtain U ( z ; t ), w e solv e driv en string equations in the GaN epila y er and the Sapphire substrate, namely @ 2 U ( z ; t ) @ t 2 C 2 0 @ 2 U ( z ; t ) @ z 2 = S ( z ; t ) (0 z L ) (5.20a) PAGE 94 85 and @ 2 U ( z ; t ) @ t 2 C 2 1 @ 2 U ( z ; t ) @ z 2 = 0 ( L z Z s ) (5.20b) where C 0 and C 1 are LA sound sp eeds in the GaN and Sapphire substrate, resp ectiv ely In Eq. ( 5.20b ), the Sapphire substrate has nite thic kness. T o sim ulate coheren t phonon propagation in an innite Sapphire substrate, Z s in Eq. ( 5.20b ) is c hosen large enough so that the propagating sound pulse generated in the GaN epila y er do es not ha v e sucien t time to reac h z = Z s during the sim ulation. If T sim is the duration of the sim ulation, this implies Z s L + C 1 T sim Equations ( 5.20a ) and ( 5.20b ) are solv ed sub ject to initial and b oundary conditions. The initial conditions are U ( z ; 0) = @ U ( z ; 0) @ t = 0 : (5.21) A t the GaN-air in terface at z = 0, w e assume the free surface b oundary condition @ U (0 ; t ) @ z = 0 (5.22a) since the air exerts no force on the GaN epila y er. The phonon displacemen t and the force p er unit area are con tin uous at the GaN-Sapphire in terface so that U ( L ; t ) = U ( L + ; t ) (5.22b) and 0 C 2 0 @ U ( L ; t ) @ z = 1 C 2 1 @ U ( L + ; t ) @ z : (5.22c) The b oundary condition at z = Z s can b e c hosen arbitrarily since the propagating sound pulse nev er reac hes this in terface. F or mathematical con v enience, w e c ho ose the rigid b oundary condition U ( Z s ; t ) = 0 : (5.22d) PAGE 95 86 T o obtain U ( z ; t ) for general S ( z ; t ), w e rst need to nd the harmonic solutions in the absence of strain, i.e. S ( z ; t ) = 0. The harmonic solutions are tak en to b e U n ( z ; t ) = W n ( z ) e i! n t ( n 0 ) (5.23) and it is easy to sho w that the mo de functions, W n ( z ), satisfy d 2 W n ( z ) dz 2 + 2 n C 2 0 W n ( z ) = 0 ( 0 z L ) (5.24a) and d 2 W n ( z ) dz 2 + 2 n C 2 1 W n ( z ) = 0 ( L z Z s ) (5.24b) Applying the b oundary conditions from Eq. ( 5.22 ) w e obtain the mo de functions W n ( z ) = 8><>: cos ( n z =C 0 ) if 0 z L B n sin ( n ( Z s z ) =C 1 ) if L z Z s (5.25a) with B n = cos ( n L=C 0 ) sin ( n ( Z s L ) =C 1 ) : (5.25b) The mo de frequencies, n are solutions of the transcenden tal equation 1 0 C 0 cot n L C 0 = 1 1 C 1 tan n ( Z s L ) C 1 (5.26) whic h w e solv e n umerically to obtain the mo de frequencies, n ( n = 0 ; 1 ; 2 ; ::: ). The index, n is equal to the n um b er of no des in the mo de functions, W n ( z ). A general displacemen t can b e expanded in terms of the harmonic mo des as U ( z ; t ) = 1 X n =0 q n ( t ) W n ( z ) : (5.27) Substituting Eq. ( 5.27 ) for U ( z ; t ) in to Eq. ( 5.20 ) and taking the initial conditions from Eq. ( 5.21 ) in to accoun t, w e nd that the expansion co ecien ts, q n ( t ), satisfy a PAGE 96 87 driv en harmonic oscillator equation q n ( t ) + 2 n q n ( t ) = Q n ( t ) ; (5.28) sub ject to the initial conditions q n (0) = q n (0) = 0. The harmonic oscillator driving term Q n ( t ) is giv en b y Q n ( t ) = R Z s 0 dz W n ( z ) S ( z ; t ) R Z s 0 dz W n ( z ) 2 : (5.29) In our simple displaciv e mo del for photogeneration of carriers, S ( z ; t ) = S ( z ) ( t ) where ( t ) is the Hea viside step function. In this case, the lattice displacemen t is explicitly giv en b y U ( z ; t ) = 1 X n =0 S n 2 n ( 1 cos ( n t ) ) W n ( z ) (5.30) with S n dened as S n = R Z s 0 dz W n ( z ) S ( z ) R Z s 0 dz W n ( z ) 2 : (5.31) Using the lattice displacemen t from Eq.( 5.30 ), w e obtain the time-dep enden t dieren tial rerectivit y at the prob e frequency from Eq.( 5.15 ). The result is R R ( ; t ) = 1 X n =0 S n 2 n ( 1 cos ( n t ) ) R n ( ) (5.32) where R n ( ) = Z Z s 0 dz F ( z ; ) dW n ( z ) dz : (5.33) can b e ev aluated analytically With the ab o v e formalism, w e solv e for the lattice displacemen t, U ( z ; t ), for a coheren t LA phonon pulse propagating in a m ultila y er structure consisting of a 1.124 m thic k GaN epila y er gro wn on top of an innitely thic k Sapphire substrate with the gro wth direction along z W e tak e the origin to b e at the GaNair in terface and the GaN-Sapphire in terface is tak en to b e at z = L = 1 : 124 m. W e assume that carriers are photogenerated in a single 240 A thic k In x Ga 1 x N PAGE 97 88 quan tum w ell em b edded in the GaN la y er 0.1 m b elo w the GaN-air in terface and 1 m ab o v e the Sapphire substrate. Our structure th us resem bles the SQW sample sho wn in Fig. 5{1 (c). In the GaN epila y er, w e tak e C 33 = 379 GP a and 0 = 6 : 139 gm/cm 3 [ 47 ] from whic h w e obtain C 0 = 7857 m/s. F or the Sapphire substrate, w e tak e C 33 = 500 GP a and 1 = 3 : 986 gm/cm 3 [ 48 ] from whic h w e nd C 1 = 11200 m/s. The results of our sim ulation are sho wn in Fig. 5{7 A con tour map of the strain, @ U ( z ; t ) =@ z is sho wn in Fig. 5{7 (a). W e plot the strain as a function of the depth b elo w the surface and the prob e dela y time. Photo excitation of electrons and holes in the InGaN quan tum w ell generates t w o coheren t LA sound pulses tra v eling in opp osite directions. The pulses are totally rerected o the GaN/air in terface at z = 0 and are partially rerected at the Sapphire substrate at z = 1 : 124 m. Appro ximately 95% of the pulse energy is transmitted and only 5% is rerected at the substrate. The sp eed of the LA phonon pulses is just the slop e of the propagating w a v e trains seen in Fig. 5{7 (a) and one can clearly see that the LA sound sp eed is greater in the Sapphire substrate. F rom the strain the dieren tial rerectivit y can b e obtained from Eq. ( 5.15 ) F rom Fig. 5{7 (a), the strains, @ U ( z ; t ) =@ z asso ciated with the propagating pulses are highly lo calized and tra v el at the LA sound sp eed. Eac h pulse con tributes a term to the dieren tial rerectivit y that go es lik e R R ( ; t ) F ( C 0 t; ) / c sin 2 n b c C 0 t + (5.34) The p erio d of the oscillations of F dep ends on the prob e w a v elength, = 2 c=! with the result that the observ ed dieren tial rerectivit y oscillates in time with p erio d, T = c= ( n b C 0 ) = = (2 n b C 0 ), where n b = 2 : 4 is the index of refraction, and C 0 = 7857 m/s is the LA sound sp eed in GaN. F or = 377 nm ( h! = 3 : 29 eV), this giv es us T = 10 ps. The sensitivit y function, F ( z ; ), dened in Eq. ( 5.16 ) PAGE 98 89 0.5 1.0 1.5 Depth ( m m) Sapphire GaN (a) InGaN Well 0 100 200 300 400 500 Probe Delay (ps) -15 -10 -5 0 5 10 Reflectivity (a.u.) (b) Figure 5{7: Generation and propagation of coheren t acoustic phonons photogenerated in a single InGaN w ell em b edded in a free standing 1 : 124 m GaN epila y er gro wn on top of a Sapphire substrate. In (a) a con tour plot of the strain eld, @ U ( z ; t ) =@ z is sho wn as a function of depth b elo w the GaN-air in terface and the prob e dela y In (b), the resulting dieren tial rerectivit y induced b y the strain eld in (a) is sho wn as a function of the prob e dela y PAGE 99 90 is an oscillating function in the GaN/InGaN epila y er and is assumed to v anish in the Sapphire substrate. Our computed dieren tial rerectivit y is sho wn in Fig. 5{7 (b) for a prob e w a v elength of = 377 nm W e nd that the rerectivit y abruptly atten uates when the strain pulse en ters the Sapphire substrate at t = 170 ps. The rerected strain pulses giv e rise to the w eak er oscillations seen for t > 170 ps. These oscillations are predicted to con tin ue un til the rerected pulses are again partially rerected o the Sapphire substrate at t = 430 ps. 5.3 Simple mo del Since the coheren t oscillation observ ed in the dieren tial rerectivit y stems essen tially from the strain pulse propagating in to the la y ers, most phenomena can b e understo o d b y a simple macroscopic mo del that is presen ted in this section. Instead of solving the loaded string equations for the strain to obtain a propagating disturbance in the refractiv e index, the propagating strain pulse at a giv en momen t can b e view ed as a thin strained la y er in the sample, where the index of refraction is assumed to b e sligh tly dieren t from the rest of the sample. This situation is sc hematically depicted in Fig. 5{8 where a ctitious, thin GaN strained la y er is lo cated at z in the thic k host GaN la y er. The thic kness of the strained la y er, d is appro ximately the width of the tra v eling coheren t phonon strain eld, @ U ( z ; t ) =@ z and is to b e determined from the microscopic theory F rom the last section, it w as seen that the propagating strain eld is strongly lo calized so that d is small. In the example of the last section, d is appro ximately equal to the quan tum w ell width. Here w e assume the strain pulse has b een already created near the air/GaN in terface and do not consider its generation pro cedure. W e treat d as a phenomenological constan t and also assume that the c hange in the index of refraction is constan t. This strained GaN la y er tra v els in to the structure with the sp eed of the acoustic phonon w a v epac k et C 0 = 7 10 3 m=s so the lo cation of the stained la y er is giv en as z = C 0 where is the pump-prob e dela y time. PAGE 100 91 z L z+d 0 Strained GaN layer Sapphire GaN GaN Air Cs Figure 5{8: Propagating strained GaN la y er in our simple mo del. The pump laser pulse creates a coheren t acoustic phonon w a v epac k et in the InGaN la y er near the air/GaN surface, whic h is mo delled as a thin strained la y er. The strained GaN la y er propagates in to the host GaN la y er. The index of refraction in the strained la y er is p erturb ed relativ e to the bac kground GaN due to the strong strain induced piezo electric eld (F ranz-Keldysh eect). PAGE 101 92 Within the slo wly v arying en v elop e function appro ximation, the solutions to the Maxw ell equation can b e written as plane w a v es in eac h region, E i ( z ; t ) = a i e ik i z i i! t + b i e ik i z i i! t (5.35) where E i is the electric eld in the left la y er of i th in terface, and a i and b i are the slo wly v arying amplitudes. The magnetic eld is giv en b y B i @ E i =@ z F or a normally inciden t prob e eld, w e apply the usual matc hing conditions on E and B to obtain 0B@ a i b i 1CA = M i 0B@ a i +1 b i +1 1CA ; (5.36) where the transfer matrix, M i is giv en explicitly as M i = (5.37) 1 2 0B@ (1 + k i +1 k i ) e i ( k i +1 k i ) z i (1 k i +1 k i ) e i ( k i +1 + k i ) z i (1 k i +1 k i ) e i ( k i +1 + k i ) z i (1 + k i +1 k i ) e i ( k i +1 k i ) z i 1CA : T o apply this form ula to our conguration, w e normalize the inciden t amplitude to 1, let r b e the rerected amplitude in the air, and imp ose the b oundary condition that there is only a transmitted w a v e in to the Sapphire substrate with amplitude t and no rerected w a v e from the GaN-Sapphire in terface bac k in to the GaN epila y er. The latter assumption is reasonable since the microscopic theory of the previous section sho ws that only 5% of the pulse energy is rerected from the in terface b et w een the GaN and the Sapphire substrate. The total rerection and transmission amplitudes r and t for the GaN epila y er structure are determined b y 0B@ 1 r 1CA = 0B@ M 11 M 12 M 21 M 22 1CA 0B@ t 0 1CA (5.38) PAGE 102 93 where M = M 1 M 2 M n The rerection amplitude is readily found to b e r = M 21 = M 11 = r 0 + r ; (5.39) where r 0 is the bac kground con tribution without the strained la y er. The total transmission amplitude is giv en as t = 1 = M 11 W e can no w n umerically calculate the dieren tial rerectivit y b y substituting r in to Eq. ( 5.14 ). D Figure 5{9: Dieren tial rerection for dieren t frequencies of the prob e pulse. Plot (a) is exp erimen tal data and (b) comes from our calculations describ ed in the text. PAGE 103 94 In Fig. 5{9 w e compare our theoretical mo del with exp erimen tal results. In Fig. 5{9 (a), the oscillatory part of the dieren tial rerectivit y is plotted as a function of prob e dela y for three v alues of the prob e photon energy This is the same data that w as sho wn in Fig. 5{3 where it w as seen that the oscillation p erio d of the coheren t phonon rerectivit y oscillations are prop ortional to the prob e w a v elength. In Fig. 5{9 (b), w e plot the corresp onding theoretical dieren tial rerectivities obtained from Eq. ( 5.14 ). Comparing Figs. 5{9 (a) and 5{9 (b), w e see that our geometrical optics mo del successfully explains the observ ed relation b et w een the coheren t phonon oscillation p erio d and the prob e w a v elength. Ho w ev er, the calculated amplitudes of oscillation is inconsisten t with the exp erimen tal data b ecause in this case w e ha v e not tak en in to accoun t of the c hange of index of refraction with resp ect to the prob e w a v elength whic h w e will do later in the c hapter. In Fig. 5{10 w e plot the dieren tial rerectivit y as a function of the prob e dela y for three dieren t v alues of the c hange in the index of refraction in the strained la y er, ~ n As exp ected, the greater the c hange in index of refraction, the greater the amplitude of the dieren tial rerectivit y oscillations. A larger c hange in the index of refraction implies that more electron-hole pairs are excited near the air-GaN in terface, whic h acts as a stronger source for the coheren t acoustic phonon rerectivit y oscillations. These results are qualitativ ely consisten t with the exp erimen tal results sho wn in Fig. 5{2 In Fig. 5{11 w e x the c hange in the index of refraction, n = 0 : 01, and v aried the thic kness of the strained la y er, d The result sho ws a larger amplitude for the dieren tial rerection in wider strained la y ers, whic h is consisten t with what w as exp erimen tally observ ed in Fig. 5{5 Both Fig. 5{10 and Fig. 5{11 suggest that the amplitude of the dieren tial rerectivit y seems to increase monotonically with n and d All these features can b e understo o d more easily using the single-rerection appro ximation to the full form ula for the rerectivit y in Eq. ( 5.39 ). Assuming the c hange in the index of refraction is PAGE 104 95 0 20 40 60 80100Time Delay (ps) -0.0100.01 0.02 0.03 0.04D R/R d n = 0.01d n = 0.005 d n = 0.001Figure 5{10: The dieren tial rerection calculated n umerically using Eq. ( 5.14 ) for three dieren t v alues in the c hange in index of refraction. The parameters used are n b = 2 : 65, = 3 : 29 eV d = 30 nm L = 1 m and C s = 7000 m=s [Note that the curv es are sifted to a v oid o v erlapping.] PAGE 105 96 0 20 40 60 80100Delay Time (ps) 00.02 0.04 0.06D R/R d = 30 nm d = 15 nmd = 3 nmFigure 5{11: Calculated dieren tial rerection for dieren t v alues of the thic kness of the strained GaN la y er. The parameters are n b = 2 : 65, = 3 : 29 eV L = 1 m and C s = 7000 m=s The c hange in the index of refraction in the strained GaN la y er w as assumed to b e n = 0 : 01. PAGE 106 97 small, j ~ n j 1, w e ma y select con tributions from only terms prop ortional to ~ n in the innite F abry-P erot series for the total rerection amplitude. The relev an t rerection pro cesses selected are sc hematically depicted in Fig. 5{12 In this case the total rerection amplitude is giv en b y the leading terms in the F abry-P erot series r = r 0 + r 1 + r 2 + O ~ n 2 (5.40) where r 0 is the bac kground rerection amplitude, and the rst order terms in ~ n are r 1 / e 2 ik z ~ n; r 2 / e i 2 k ( z + d ) ~ n : (5.41) z z+d 0 r 1 r 2 r 0 n n n L Cs Figure 5{12: Sc hematic diagram of the single-rerection appro ximation in the F abry-P erot rerection; where w e ha v e selected the pro cesses only prop ortional to ~ n (= n 0 n ). PAGE 107 98 T o linear order in ~ n the dieren tial rerectivit y Eq. ( 5.14 ) b ecomes R R = 8 sin ( k d ) n 2 1 [ n sin (2 k z + k d ) + cos (2 k z + k d )] = 8 sin ( k d ) n 2 1 j ~ n j sin( k d ) sin(2 k z + ) ; (5.42) where the phase is giv en b y = k d + arctan n : (5.43) Note that for small v alues of k d w e can expand Eq. ( 5.42 ) in a T a ylor series to obtain R R = 8 k d n 2 1 [ n sin (2 k z ) + cos(2 k z )] (5.44) to rst order in k d F rom Eq. ( 5.42 ) one can see that the amplitude of the oscillation scales linearly with the c hange of the real part and imaginary part of the index of refraction resp ectiv ely Since the strain-pulse-fron t mo v es with the sp eed of sound, z = C 0 t one can rewrite Eq. ( 5.42 ) as R R ( ; t ) / j ~ n j sin( k d ) sin 2 t T + (5.45) where T = c= ( n b C 0 ) = = (2 C 0 n b ). Note that in the limit k d 0, Eq. ( 5.45 ) is equiv alen t to Eq. ( 5.34 ) of the microscopic theory describ ed in the last section. Equation ( 5.45 ) sho ws that the p erio d of oscillations in the dieren tial rerectivit y is giv en b y T whic h is consisten t with what w as seen in the exp erimen ts. It also explains that the amplitude of the oscillation is prop ortional to the c hange in the index of refraction and that the amplitude mo dulates as sin ( k d ), so increasing linearly with the thic kness of the strained la y er as w ell as the prob e frequency for k d = n b ( =c ) d 1. PAGE 108 99 The c hange of index of refraction in Eq. ( 5.45 ) comes from t w o eects. The rst one whic h is small is the band gap c hange due to the deformational p oten tial. The other one is the built-in piezo electric eld whic h leads to non-zero absorption b elo w the band gap b ecause of the F ranz-Keldysh eect as sho wn in Fig. 5{13 In general, it is dicult to determine the F ranz-Keldysh eld exp erimen tally Here w e pro vide a w a y to estimate the order-of-magnitude of the built-in piezoelectric eld from the dieren tial rerectivit y measuremen ts. F or a prob e energy b elo w band-gap, w e read o the amplitude of the dieren tial rerectivit y from the exp erimen tal data. F rom this v alue, and an estimate for d based on information ab out the sample geometry w e obtain the c hange in the index of refraction whic h, in turn, giv es an estimate for the absorption co ecien t ( ) at the prob e photon energy In the F ranz-Keldysh mec hanism, ( ) is related to the piezo electric eld, F via [ 12 ] ( ) 1 n b c f E g h! exp ( 4 3 f ( E g h ) E 0 a 20 3 = 2 ) : (5.46) Here f = eF = ( E 0 a 20 ) where E 0 = 3 : 435 eV and a 0 = 39 : 68 A are the excitonic energy and length scales in GaN. F rom this form ula, for instance, w e estimate that a piezo electric eld on the order of F 0 : 93 MeV/cm is resp onsible for a 1% c hange in the index of refraction. In Fig. 5{13 w e displa y the bulk absorption co ecien t as a function of photon energy h! in bulk GaN (dotted line) at a xed piezo electric eld, F = 0 : 93 M V =cm (solid line). The inset sho ws the absorption co ecien t at a xed photon energy h! = 3 : 29 eV as a function of the piezo electric eld, F The c hange in the absorption at h! = 3 : 29 eV (b elo w the band-gap E g = 3 : 43 eV ), can b e used to estimate the piezo electric eld. The corresp onding c hanges in the real and imaginary parts of the dielectric function are sho wn in Fig. 5{14 PAGE 109 100 3.2 3.4 3.6 3.84w (eV) 0 20 40 60a (103cm-1 ) 00.511.52F (106 V/cm) 00.511.5a (103cm-1) w = 3.29 eVF = 0.93 MV/cmFigure 5{13: The absorption co ecien t as a function of the prob e energy The absorption tail b elo w the band gap E g = 3 : 43 eV is due to the built-in F ranz-Keldysh eld. The dotted curv e is the free-carrier absorption. The inset sho ws the absorption co ecien t as a function of the piezo electric eld. The parameters used in this mo del calculation are: d =30 nm, n b =2.65, a 0 =4 nm, and E 0 =15 meV. PAGE 110 101 3 2 3 4 3 6 3 8 4 0 0 1 5 0 1 0 0 0 5 0 0 0 0 0 5 0 1 0 R e (D e) I m (D e) D ew ( e V ) Figure 5{14: The c hange in the real and imaginary parts of the dielectric function in GaN as a function of the prob e energy for the situation describ ed in Fig. 5{13 The v ertical dotted line at 3.43 eV is the band gap energy Just b elo w the band gap the c hange in the dielectric function is dominated b y the imaginary part as indicated b y the dotted o v al. PAGE 111 102 5.4 Summary W e ha v e presen ted a theory for the detection of a new class of large amplitude, coheren t acoustic phonon w a v epac k ets in fem tosecond pump-prob e exp erimen ts on In x Ga 1 x N/GaN epila y ers and heterostructures. The InGaN/GaN structures are highly strained and at high In concen trations ha v e large built in piezo electric elds whic h accoun t for the large amplitude of the observ ed rerectivit y oscillations. This new class of coheren t acoustic phonons is generated near the surface and propagates in to the structure. The frequency of the rerectivit y oscillations is found to b e prop ortional to the frequency of the prob e. These coheren t phonon w a v epac k ets can b e used as a p o w erful prob e of the structure of the sample. W e are able to mo del the generation and propagation of these acoustic phonon w a v epac k ets using a simple string mo del whic h is deriv ed from a microscopic mo del for the photogeneration and propagation of coheren t acoustic phonon w a v epac k ets in InGaN/GaN m ultiple quan tum w ells. Our mo del successfully predicts the observ ed dep endence of the coheren t phonon rerectivit y oscillations on prob e w a v elength and epila y er thic kness. PAGE 112 CHAPTER 6 CONCLUSION I ha v e dev elop ed a string mo del for coheren t acoustic phonons in GaN/InGaN in GaN/InGaN m ultiple quan tum w ells whic h ha v e v ery large built-in piezo electric elds on the order of MeV/cm. It is a one dimensional w a v e equation for the lattice displacemen t, whic h describ es the motion of a uniform string under a nonuniform forcing function. I also calculated dieren tial transmission using the strain calculated from the string mo del. This mo del can b e justied b y a microscopic theory Using this mo del, I can explain and predict man y c haracteristics of exp erimen tally observ ed oscaillations in dieren tial rerection and transmission. I ha v e solv ed the string mo del using Green's function metho d to obtain the lattice displacemen t and other related ph ysical quan tities suc h as strain, energy densit y energy and the dieren tial transmission. All these quan tities ha v e the oscillating prop ert y The strain and energy densit y propagate in b oth directions along the sup erlattice gro wth axis z The oscillations of energy and dieren tial transmission ha v e the same n um b er of p eaks as the n um b er of the w ells and they dies out b ecause of the limited range of the m ultiple quan tum w ell system. I ha v e applied the string mo del to the coheren t con trol of the coheren t acoustic phonons in GaN/InGaN MQWs. By using t w o forcing terms corresp onding to t w o pump pulses, w e can c hange three con trollable parameters of the second forcing term with resp ect to the rst one, the temp oral phase, the spatial phase, and the relativ e amplitude. The temp oral phase is the most eectiv e con trol. W e ha v e sho wn b oth the temp orally in phase constructiv e enhancemen t and out of phase destructiv e cancelation of the coheren t phonon oscillations. T o get b etter destructiv e eect the relativ e amplitude of the second forcing term should b e small 103 PAGE 113 104 than the rst. In spatially out of phase cases the total forcing function will b e zero after the starting time of the second forcing term, but there is a distinctiv e con trast b et w een the temp orally in phase and out of phase cases. In the former case the oscillation almost disapp ear, while in the latter case the oscillation p ersists with a dephasing time a half of the case when there is only one forcing term. I also dev elop ed a theory for the detection of a new class of large amplitude, coheren t acoustic phonon w a v epac k ets in fem tosecond pump-prob e exp erimen ts on GaN/InGaN epila y ers and heterostructures. The InGaN/GaN structures are highly strained and at high In concen trations ha v e large built in piezo electric elds whic h accoun t for the large amplitude of the observ ed rerectivit y oscillations. This new class of coheren t acoustic phonons is generated near the surface and propagates in to the structure. The frequency of the rerectivit y oscillations is found to b e prop ortional to the frequency of the prob e. I mo delled the generation and propagation of these acoustic phonon w a v epac k ets using the simple string mo del dev elop ed for the coheren t acoustic phonon in InGaN/GaN MQWs. By solving Maxw ell's equations in the presence of spatio-temp oral disturbances in the refractiv e index caused b y the propagating coheren t phonon w a v e pac k ets, I can explain the sudden die out of the oscillations in the rerection. A simple macroscopic mo del treating the propagating strain pulse as a thin strained la y er in the host material is dev elop ed, whic h successfully predicts the observ ed dep endence of the coheren t phonon rerectivit y oscillations on prob e w a v elength and epila y er thic kness. I also pro vided a w a y to estimate the order-of-magnitude of the bulit-in piezo electric eld from the dieren tial rerection measuremen ts. Man y sp eculations on p ossible applications of the coheren t phonons ha v e b een discussed in the literature. I will giv e a few here with regards to the coheren t acoustic phonons and w a v epac k ets. PAGE 114 105 Coheren t acoustic phonon disp ersion relation in MQW/sup erlattice system can b e used to obtain tunable oscillation frequencies ranging from lo w frequency up to sev eral THz with one material system. The coheren t con trol of coheren t phonons ma y nd application in the m ultiplexed generation of tailored THz signals. The energy dissipation b y con v en tional diusiv e heat transp ort could b e impro v ed or replaced b y propagating non thermal lattice excitation, whic h has p oten tial application in future high p o w er nitride eld-eect transistors. 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F rom 1989 to 1996 he studied in Shanghai Jiaotong Univ ersit y and receiv ed his B.S. and M.S. degree in summer of 1993 and spring of 1996 resp ectiv ely Then he w ork ed as a programmer and system in tegration engineer in Shanghai T riman Compan y for t w o and a half y ears from 1996 to 1998. In the fall of 1998 he came to the United States and b ecame a Gator. F rom then on he has b een studying for a Ph.D. degree in the Departmen t of Ph ysics at the Univ ersit y of Florida. 109 |