LO-plasmon modes in doped GaAs/AL[subcript x]ga[subscript 1-x]As superlattices

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1 Indiana University of Pennsylvania Knowledge IUP Theses and Dissertations (All) LO-plasmon modes in doped GaAs/AL[subcript x]ga[subscript 1-x]As superlattices Robert A. Rodgers Indiana University of Pennsylvania Follow this and additional works at: Recommended Citation Rodgers, Robert A., "LO-plasmon modes in doped GaAs/AL[subcript x]ga[subscript 1-x]As superlattices" (2011). Theses and Dissertations (All) This Thesis is brought to you for free and open access by Knowledge IUP. It has been accepted for inclusion in Theses and Dissertations (All) by an authorized administrator of Knowledge IUP. For more information, please contact

2 LO-PLASMON MODES IN DOPED GAAS/AL X GA 1-X AS SUPERLATTICES A Thesis Submitted to the School of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree Master of Physics Robert A. Rodgers Indiana University of Pennsylvania August 2011

3 Indiana University of Pennsylvania The School of Graduate Studies and Research Department of Physics We hereby approve the thesis of Robert A. Rodgers Candidate for the degree of Master of Physics Dr. Devki Talwar Professor of Physics & Chairman Thesis Advisor Dr. Muhammad Numan Professor of Physics Dr. Ajawad Haija Professor of Physics ACCEPTED Timothy P. Mack, Ph.D Dean School of Graduate Studies and Research

4 Title: LO-PLASMON MODES IN DOPED GAAS/AL X GA 1-X AS SUPERLATTICES Author: Thesis Committee Chair: Thesis Committee Members: Robert A. Rodgers Dr. Devki Talwar Dr. Ajawad Haija Dr. Muhammad Numan A great deal of interest has emerged in recent years to design novel compound semiconductor materials to fulfill the growing societal needs of efficient light sources, powerful solar cells, miniaturized-electronic-circuitry for lab-on-chip equipment, and a plethora of handheld opto-electronic devices. GaAs-based III-V compounds are significant materials with important fundamental characteristics allowing the scientists and engineers to envision their use in a variety of devices including light-emitting diodes (LEDs), laser diodes (LDs), and high-electron mobility transistors (HEMTs). In this regard, there has been a growing interest of studying the far-infrared optical properties in both un-doped and doped bulk III-V compounds, thin films, ternary alloys, and their nano-structured quantum-wells and superlattices. For GaAs/Al x Ga 1-x As materials grown especially by molecular-beam epitaxy (MBE), the optical constants [viz., n, k, N (Charge carrier concentration), R (Reflection), and T (Transmission)] over a broad frequency regime are fundamental inputs that must be known for constructing optoelectronic devices. Despite some success by using Raman Spectroscopy, the influence of free charge carrier concentration on the phonon-plasmon coupled modes (L ± ) by farinfrared (FIR) spectroscopy are still scantily known. Any effort to extract accurate ii

5 information about the charge carrier concentration N in either n- or p-type doped III-V compounds using FIR would be of significant importance to the scientific community. In this thesis, we will use the electromagnetic theory to study the IR reflectivity and transmission at oblique incidence in both undoped and doped GaAs, Al x Ga 1-x As thin films and superlattices in order to correlate the shifts of the L ± mode frequencies with the free charge carrier concentration. The transmission study in compound semiconductors for s-polarization reveals a single minimum at the resonance frequency of the TO mode, while for p-polarization the transmission minima occur at both the resonance frequencies of the TO and LO modes. In doped semiconductor thin films, the transmission in p- polarization exhibits minima at TO and L + with a shift of L + mode to higher frequency as the charge carrier concentration increases. iii

6 ACKNOWLEDGEMENTS I would first like to express my deepest thanks to my thesis advisor, Dr. Devki Talwar. Without his help and guidance, none of this would have been possible. I find his dedication and interest in this subject matter truly inspiring, and I can only hope to one day be as passionate for my work as he is for his. Also, I would like to thank the IUP Physics faculty, particularly Dr. Numan and Dr. Haija for serving on my thesis committee. I d also like to thank Ms. Kala Markel for her assistance in revising and formatting this thesis. All of their patience and time are greatly appreciated. And lastly, but certainly not least, I d like to thank my family and friends: My mother, Jane, and my father, David, for their everlasting love, care, and support, my siblings, Pamela, Kristin, and David for their friendship and encouragement. Lastly, I d like to thank Stephanie Kuhne for her support and inspiration. Without her, this wouldn t have been possible. My most sincere love and gratitude goes out to all of them. iv

7 TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION Optical Properties Growth Methods and Structure Characterization Techniques. 5 II. EXPERIMENTAL DATA Bulk Samples Thin Films Plasmon-Phonon Coupled Modes Superlattices III. THEORETICAL MODELING Theoretical Background Anisotropy for Oblique Incidence Ternary Alloys Iterative Method Plasmon-Phonon Coupled Modes. 33 IV. RESULTS AND DISCUSSION Thin Films Ternary Alloys v

8 4.3 Superlattices Plasmon-Phonon Coupled Modes. 51 V. CONCLUSIONS vi

9 LIST OF FIGURES 1. Direct band gap between valence and conduction bands Transmission spectrum for LiF of thickness 0.20 µm Reflection spectra for LiF of thickness µm Transmission spectra for MnTe/ZnTe/GaAs (a,b,c) and for CdSe/GaAs (d,e,f) Transmission spectra for AlAs (a) and for GaAs (b) Transmission spectra for AlN at 0, 15, 30, and 45 degree incidence (a) Room temperature Raman intensities of phonons in Al x Ga 1-x As showing shifts of optical modes as a function of x (b) the variation of LO-TO phonons... as a function of composition x Raman spectra showing the L + shift for various doping levels Experimental data for frequency shift with free charge carrier concentration. for Al 0.19 Ga 0.81 As 10. Plot of the phonon content for Al 0.19 Ga 0.81As at different free carrier concentrations 11. Infrared reflection spectra for GaAs/AlAs superlattice at 45 degree incidence Diagram of the superlattice structure studied by Z. Ristovski et al Reflection spectrum for GaAs/Al x Ga 1-x As superlattice structure at normal... incidence Transmission spectrum for a 0.2μm CdSe thin film at normal incidence Transmission spectrum for a 0.2μm thin film at 45 degree incidence for.. both s- (solid) and p- (dashed) polarization Reflection spectrum for a 0.2μm CdSe thin film at normal incidence Reflection spectrum for a 0.2μm CdSe thin film at 45 degree incidence for both s- (solid) and p- (dashed) polarization 18. Reflection spectrum for a 0.2μm GaAs thin film at 45 degree incidence for both s- (solid) and p- (dashed) polarization 19. Transmission spectrum for ZnTe of thicknesses 1μm, 5μm, 25μm, and 100μm at 45 degree oblique incidence for p-polarization 20. Transmission spectrum for ZnTe of thickness 1μm at 45 degree oblique.. incidence for p-polarization vii

10 21. Transmission spectrum for ZnTe of thickness 5μm at 45 degree oblique.. incidence for p-polarization 22. Transmission spectrum for ZnTe of thickness 25μm at 45 degree oblique incidence for p-polarization 23. Transmission spectrum for ZnTe of thickness 100μm at 45 degree oblique... incidence for p-polarization 24. Transmission spectrum for GaAs of thickness 1μm at 45 degree oblique. incidence for p-polarization 25. Transmission spectrum for GaAs of thickness 5μm at 45 degree oblique. incidence for p-polarization 26. Transmission spectrum for GaAs of thickness 25μm at 45 degree oblique incidence for p-polarization 27. Transmission spectrum for GaAs of thickness 100μm at 45 degree oblique.. incidence for p-polarization 28. Reflection spectrum for Al Ga As of thickness 1μm at 45 degree.. incidence for p-polarization Calculated phonon content of the three coupled modes in Al 0.2 Ga 0.8 As Reflection spectrum for Al Ga As of thickness 1μm at 45 degree.. incidence for p-polarization Reflection spectrum for the structure shown in figure 12 at normal incidence Reflection spectrum for the structure shown in figure 12 at 45 degree.. incidence for s- (solid) and p-polarization (dashed) 33. Reflection spectrum for the structure shown in figure 12 at normal incidence... as predicted by the iterative method Transmission spectrum for GaN depicting the shift in the L + mode at.. 7.8x10 17 cm -3,1.2x10 18 cm -3, and 3.6x10 18 cm Theoretical model showing the frequency shift of the L + and L - coupled.. modes in GaN with respect to the square root of the free charge carrier concentration 36. Theoretical model showing the frequency shift of the L -, L 0, and L + coupled.. modes for Al 0.2 Ga 0.8 As with respect to the square root of the free charge carrier concentration 37. Three-dimensional plot for InN showing the coupled mode frequency. strength with respect to the frequency shift at different values of ω p viii

11 38. Three-dimensional plot for InN showing the coupled mode frequency. strength with respect to the frequency shift at different values of ω p 39. Three-dimensional plot for GaAs rotated to show only two dimensions,.. frequency versus plasma frequency ix

12 CHAPTER I INTRODUCTION In recent years, both elemental and compound semiconductor based devices have become an important field of study due to their increasing demands for use in smaller size and more efficient personal computers, light-emitting diodes, cell phones, mp3 players, and dozens of other electronic devices. Because of theirsignificance in everyday technology, it is essential to understand the properties and behaviors of semiconductor materials such that these devices can become more effectively marketable. Among others, the most important properties include the perfect and imperfect semiconductor s electronic band structure, lattice dynamical behavior (translational and rotational) vibrational modes, oscillator strengths, and its resonance frequencies. Experimental measurements of these fundamental properties allow scientists and engineers toconstructtheoretical models tocompare and predictthe behaviors of various materials characteristics. In this thesis, the optical properties of both perfect and imperfect GaAs and Al x Ga 1-x As ultrathin films, bulk samples, and superlattices will be studied for both normal and oblique incidence. The effect of free charge carrier concentration on coupled phonon-plasmon modes (L ± ) is analyzed. Theoretical calculations are performedfor many compound semiconductor materials (binary, ternary and superlattice structures) by using Mathematica codes. The related graphs are generated and theoretical results are found to compare favorably well with the existing experimental data. 1

13 1.1 Optical Properties The optical properties of any elemental and compound semiconductor are innate properties unique to each material. They include the high-frequency dielectric constant (ɛ ), the transverse optical and longitudinal optic phonon (ω TO and ω LO, respectively), the oscillator strength (S), and the plasmon and phonon damping constant (γ and Γ, respectively). Knowledge of these properties is important in the characterization of semiconductors. They can be determined experimentally by analysis of the infrared spectrum. From these optical properties, the dielectric function can be determined.the calculation of the dielectric function for semiconductor materials is the foundation of finding the infrared spectrum. 1.2 Growth Methods and Structure The fabrication of semiconductors with precise optical, structural, and electrical properties is an important part of the experimental research of semiconductors. Previously, applications of semiconductor devices were hindered by the lack of availability of quality semiconductors and controlled doping levels. Through the advancement of modern growth techniques, one can obtain a greater flexibility and accuracy in the manufacturing of semiconductor based devices. Recent improvement in Molecular Beam Epitaxy (MBE), Chemical Beam Epitaxy (CBE), Metalorganic Chemical Beam Epitaxy (CBE), Metalorganic Vapor Phase Epitaxy (MOVPE), and several other growth methods has offered higher quality films with more controlled physical properties. 2

14 The term epitaxy refers to the deposition of monolayers upon each other. It is a slow method of crystal growth that is done one monolayer on top of another. For example, Molecular Beam Epitaxy is a type of film growth whose growth rate is generally lower than 1000nm per hour, and is done in a high vacuum. In this case, the desired materials are heated in separate cells until they sublimate. They are they introduced to the substrate, on which they condense to form the aforementioned monolayers. Various semiconductor materials bond in different patterns. For example, GaAs and InSb will condense to form the zincblende structure, whereas the silicon and germanium will condense to form the diamond structure (Hook and Hall, 2008). As these materials condense into solid crystals, they also form band gaps. Band gaps are the difference in energy levels between the valence band and the conduction band. An important property of any semiconductor material is its band gap, which is usually on the order of magnitude of 1eV. Figure 1: Direct band gap between valence and conduction bands 3

15 Here, E G represents the energy gap, which is specific to each type of semiconductor material. Using the top of the valence band as the point of zero potential energy, the total energy of a point on each band can thus be written: (1.1) (1.2) relation: In the case of a crystal, the momentum can be written in terms of k by the (1.3) Thus, the energy of each band is simply the total energy, or kinetic energy plus potential energy. In this thesis, optical properties of several epitaxially grown structures will be analyzed. First, bulk materials and thin films will be studied. The effect of doping will be noted, using the Berreman effect, and comparing normal incidence with oblique incidence by accounting for both the s-polarized and p-polarized components of the incident infrared radiation. In addition, binary and ternary structures will be studied, primarily GaAs, and Al x Ga 1-x As. Finally, various complex superlattice structures will be studied by using two different calculation methods: one primarily using the dielectric function, and another using an iterative calculation via the index of refraction. These two calculation methods will be compared and contrasted. 4

16 1.3 Characterization Techniques A semiconductor can be roughly modeled as a system of quantum oscillators. In this regard, the various oscillators have a broad range of harmonic frequencies. When radiation of those particular frequencies interact with the semiconductor sample, it will be absorbed. This radiation then becomes part of the absorption spectrum. Alternatively, when the frequency of the radiation is far from any harmonic frequencies, the radiation will be reflected off of the sample. In this case, it becomes a part of the reflection spectrum. Both of these spectrums can be studied, and reveal information about the sample. Particularly, these spectrums allow each of the elementary harmonic excitations to be determined, which are important and unique to each semiconductor sample. Infrared spectroscopy is one of the most popular methods of semiconductor characterization. In this process, infrared radiation (generally between the range of cm -1 ) is incident upon the sample. The radiation is often expressed as a wavenumber, which is related to the wavelength and frequency as: (1.4) Where k is the wavenumber (cm), λ is the wavelength (μm), c is the speed of light, and ω is the angular frequency (Tolstoy, Chernyshova, and Skryshevsky, 2003). In 1963, Berreman discovered an important phenomenon dealing with the polarization of incident infrared radiation. When the radiation is incident on the sample in a manner that it is polarized in the same plane as the sample (s-polarization), only the TO mode of the film will be visible. However, when radiation is incident on the sample in a 5

17 manner that is polarized normal to the plane of the sample, an additional mode becomes visible, the LO mode. This phenomenon has been termed the Berreman Effect. Berreman explains in his publication that it had been previously believed that longitudinal optic phonons could not be observed in infinite crystals because electromagnetic waves are transverse, and therefore cannot interact with longitudinal phonons (Berreman, 1963). However, it was first noticed by Fray et al that due to boundary conditions (particularly in thin flat films), absorption could actually be measured at the longitudinal mode frequency. However, in this case they explained that the vibration spectrum near zero propagation vector was strongly dependent upon the size, shape, and orientation of the specimen. With no further explanation by Fray, Berreman investigated the phenomenon, proposing that films have two perpendicular polarized modes: One parallel to the surface (ω t ), and another normal to the surface (ω l ). Thus, when infrared radiation near the frequency of ω t is incident and polarized such that its electric field is parallel to the surface of the film, a high-amplitude absorption band will occur. Similarly, when infrared radiation near the frequency of ω l is incident and polarized such that its electric field is perpendicular to the surface of the film, a high-amplitude absorption band will occur. Using these arguments, Berreman was able to show that for incident radiation polarized at an angle non-normal to plane, both the transverse optic and longitudinal optic phonon absorption bands can be observed. Another method of characterizing semiconductors is by the use of a technique known as inelastic neutron scattering. Here, instead of infrared radiation being used to 6

18 investigate the dispersion curves of excitations, neutrons of wavelength approximately equal to 2 Angstroms are used. This wavelength is used because it is comparable to the atomic spacing of a sample, which makes it ideal for studying its structure. Using a device known as a neutron spectrometer, the incident and scattered neutron beams can be detected and studied. One means of calculating the energy of the scattered particles is through a technique known as the time-of-flight method. In this case, neutrons are introduced into the system in short bursts. By doing this, the start time for the neutrons is known. By recording the amount of time that the neutrons took to reach the detector, the Bragg Law can be used to calculate the diffraction pattern of the sample. In this case, we know that the time: (1.5) Where the momentum, p, is rewritten using the de Broglie wavelength (λ=h/p), and m n is the mass of a neutron. This equation can be rewritten to solve for the wavelength, λ, and used to solve the Bragg Law: Thus: (1.6) (1.7) 7

19 When the time-of-flight, t, satisfies the Bragg Law, a peak will occur in the spectrum at the associated wavelength. This creates the unique spectrum for that particular sample. 8

20 CHAPTER II EXPERIMENTAL DATA This study focuses mainly on the III-V binary semiconductor GaAs, as well as the ternary compound Al x Ga 1-x As. In order to ensure the calculations done are accurate, parameters were input to match existing experimental data. Once these calculations matched, the parameters were changed to meet the focus of this study. 2.1 Bulk Samples The simplest form of semiconductor characterization is done experimentally on bulk samples. Here, data taken from the spectrums of various materials, thicknesses, and charge carrier densities can be most easily extracted. Materials can be varied, or mixed. In this study, focus will be on the binary semiconductor, GaAs, as well as on the ternary alloy, Al x Ga 1-x As. In the case of the ternary alloy, x is known as the molar fraction. It is a measure of the percentage of that element of which the total material is comprised. The data can be taken from either the reflection spectrum, or the transmission spectrum. The effects of charge carrier density on the optical properties of a semiconductor have been studied widely. Experimental data has been taken for a wide range of n-doped samples, and infrared spectrometry has shown strong features in both the reflection and transmission spectrums. Data taken from IR spectrometry of bulk samples can yield valuable information concerning the transverse optic frequency. However, describing the 9

21 longitudinal optic frequency requires that the sample thickness be much smaller than the wavelength of the incident radiation (Holm, Gibson, and Palik, 1976). 2.2 Thin Films First explained by Berreman, the appearance of the longitudinal optic phonon is a crucial phenomenon for characterizing semiconductors. Termed as the Berreman Effect, the strong peak in both the reflection and transmission spectra at the frequency characteristic of the longitudinal optic mode gives insight to measuring the properties of a sample, particularly the charge carrier concentration. Berreman explained that a thin film has two modes that are normal to one another. On mode is the transverse optic mode, which occurs at frequency ω t and propagates parallel to the surface of the film. The other mode is the longitudinal optic mode, which occurs at frequency ω l and propagates normal to the film surface. Thus, when infrared radiation is incident on the film, it will cause a resonant peak in the spectrum if the frequency is near ω t and the radiation is polarized such that its electric field is parallel to the film surface, or if the frequency is near ω l and the radiation is polarized such that its electric field is perpendicular to the film surface. By this principle, Berreman found that the infrared spectrum for transmission or absorption can show both the transverse and longitudinal optic modes if the incident light is at an oblique angle to the film. Berreman s technique for experimentally studying the longitudinal optic mode is now widely known and accepted as a powerful tool for extracting the phonon frequency in thin films (Berreman, 1963; Yen and Wong, 1989). 10

22 Using this technique, Berreman showed the observed experimental transmission spectra for a LiF film of thickness 0.20µm for both s-polarized and p-polarized radiation. In this case, the radiation was incident between 26 and 34 degrees. Figure 2: Transmission spectrum for LiF of thickness 0.20 µm In figure 2, the longitudinal and transverse optic modes can clearly be seen for p- polarized radiation, whereas only the transverse optic mode can be seen for s-polarized radiation. In addition, Berreman also showed the reflection spectrum for a LiF film of thickness 0.325µm, again distinguishing between the s-polarized and p-polarized infrared radiation. 11

23 As can be seen from figure 3, below, the reflection spectrum for s-polarization only shows a strong mode at the frequency characteristic of the transverse optic mode, whereas the spectrum for p-polarization shows a strong mode at the transverse optic mode, as well as a mode at the longitudinal optic mode (Berreman, 1963). Figure 3: Reflection spectra for LiF of thickness µm Dean Sciacca et al went on, using Berreman s technique, to deduce ω TO and ω LO of a semiconductor from the infrared reflection and transmission spectra. In their experiments, transmission spectra were obtained for thin films at both normal and oblique incidences. Furthermore, the results of s-polarization and p-polarization are expressed. 12

24 The following are the results of an experiment performed by Dean Sciacca et al on two different thin films. Both show normal incidence and oblique, 45 degree incidence, as well as the effects of s-polarization and p-polarization. The left-hand side (Graphs a, b, and c) shows the transmission spectrum for a 1µm MnTeepilayer grown on a GaAs substrate with a 2µm ZnTe buffer. The right-hand side (Graphs d, e, and f) shows a CdSe epilayer grown on a GaAs substrate. By contrasting these two films, the effects of the 2µm ZnTe buffer layer on the left-hand side can be clearly seen. Figure 4: Transmission spectra for MnTe/ZnTe/GaAs (a,b,c) and for CdSe/GaAs (d,e,f) 13

25 These results show the characteristic ZnTe transverse optic mode at 181cm -1 for both normal and oblique incidence in graphs a, b, and c. Furthermore, the ZnTe longitudinal optic mode can be seen at oblique incidence for p-polarized radiation at 210cm -1. Dean Sciacca went on to show the experimental results for thin films of AlAs and GaAs. In this case, the left-hand side shows a 0.5µm AlAs epilayer grown on a GaAs substrate, while the right-hand side shows a 1.04µm GaAs epilayer grown on a Si substrate. In both cases the infrared radiation was unpolarized (both s- and p- polarization). Figure 5: Transmission spectra for AlAs (a) and for GaAs (b) 14

26 Again, from these two graphs the characteristic modes can clearly be seen. For AlAs, the characteristic transverse optic mode occurs near 365cm -1, while its longitudinal optic mode occurs at ~407cm -1. For GaAs, the characteristic transverse optic mode occurs near 271cm -1, while its longitudinal optic mode occurs at ~294cm -1. More recently, Ibáñez et al also utilized the Berreman effect to simultaneously observe the transverse and longitudinal optic phonons in AlN thin films. Using the method of infrared transmission, J. Ibáñez et al reported data for several films. In one, particular case the effects of changing the angle of incident radiation were evaluated. The angle was varied from 0 degrees to 45 degrees. As expected, the resulting infrared transmission spectrum revealed the transverse optic mode in all four cases (0, 15, 30, and 45 degrees). However, the longitudinal optic mode appeared only in the three non-normal incidence spectra (15, 30, and 45 degrees). Figure 6: Transmission spectra for AlN at 0, 15, 30, and 45 degree incidence 15

27 As can be seen from the above figure, the transverse optic mode appears in all four cases at approximately 670 cm -1. However, the longitudinal optic mode is only visible in the three oblique incidence spectra at frequency of approximately 890 cm -1. In each case, the increasing angle also shows an increasing strength of the longitudinal optic mode (Ibáñez et al, 2008). Silva-Castillo and Pérez-Rodríguez have also proposed a method of analyzing the anisotropy of thin film samples. In this method, the reflection spectrum of a sample at 45 degrees should be taken. According to this method, the reflection spectrum with s- polarization squared subtracted from the reflection spectrum with p-polarization (R p -R 2 s ) should yield a result of zero at all frequencies for an isotropic sample. Being that a thin film is an anisotropic medium, this results in modes at the characteristic transverse and longitudinal optic frequencies of the sample (Silva-Castillo and Pérez-Rodríguez, 1999) 2.3 Plasmon-Phonon Coupled Modes In doped polar compound semiconductors there exist a large number of free charge carriers. The collective quantized excitation of the charge carriers is known as a plasmon. As the ions in the lattice are displaced by atomic vibrations, a resultant dipole moment interacts with the electric field of the free charge carriers causing the coupled plasmon-phonon modes. These coupled plasmon-phonon modes have been examined experimentally using Raman scattering spectroscopy by many investigators in a number of binary polar compound semiconductors such as GaAs, GaP, CdTe, and CdS. 16

28 Although many infrared studies have been carried out for the optical phonons in binary and ternary III-V compound semiconductors, there has been only one study known for the plasmon-phonon coupled modes in doped Ga 1-x In x As alloys. The Ga 1-x Al x As system is of particular interest for us since it is an example of the phonon-coupling phenomenon in a material having "two-mode behavior. In a two-mode system one observes two transverse-optical (TO) phonon modes which occur at frequencies close to those of the end members, i. e., at x = 0 (GaAs) and x = 1 (AlAs) with the strength of each mode dependent on the fraction of each component present in the alloy system. In the reststrahlen band region the reflectivity and Raman spectrum of the undoped Ga 1- xal x As shows a clear two-mode behavior(see fig. 7), thusmaking this alloy system quite attractive for studying the behavior of coupled plasmon-lo phonon modes(feng et al, 1993). a) b) Figure 7: (a) Room temperature Raman intensities of phonons in Al x Ga 1-x As showing shifts of optical modes as a function of x (b) the variation of LO-TO phonons as a function of composition x 17

29 In the Raman scattering studies by Yuasa et al. one can clearly notice (c.f. figure 9) three plasmon-phonon coupled modes L -, L 0, and L +. The shift in these modefrequencies with the increase of doping level is undoubtedly visible in the experimental results. For instance, the L - mode (low frequency mode) begins with low amplitude, due to the small number of free charge carriers and roughly follows the square root of the charge carrier concentration,. This mode then bends and begins to asymptotically approach the transverse optic mode of the lowest frequency of the ternary alloy. The L + mode (high frequency mode) on the other hand, begins above the longitudinal optic mode of the highest frequency of the ternary alloy and then asymptotically approaches the plasmon frequency, which varies as the square root of the charger carrier concentration,. The L 0 mode (medium frequency mode) varies as a square root of the charger carrier concentration, but it stays within the optical modes of the two end members (Ibáñez et al, 2004). The Raman scattering results by Yuasa et al, who studied the coupled plasmonphonon mode frequency dependence with free carrier concentration in Al x Ga 1-x As are displayed in figure 8. Clearly, one can notice the shifts of each of the three (L -, L +, L 0 ) plasmon phonon coupled modes for different doping values ranging between 1.2x10 16 cm -3 and 37.8x10 17 cm

30 Figure 8: Raman spectra showing the L + shift for various doping levels As can be seen from figure 8, the L + mode clearly shifts from lower to higher frequencies depending on the doping level. In case (d), the sample with the lowest doping, 7.0x10 17 cm -3, the L + frequency occurs at ~450cm -1. In the next case, (b), the doping is 2.6x10 18 cm -3, and the L + frequency occurs at ~550cm -3. In case (a) the doping is 2.7x10 18 cm -3, and the L + frequency occurs at ~600cm -3. And finally in case (c) the doping is 3.7x10 18 cm -3, and the L + frequency occurs at ~625cm

31 Figure 9: Experimental data for frequency shift with free charge carrier concentration for Al 0.19 Ga 0.81 As Yuasa et al went on to graphically show the experimental results for the free charge carrier concentration s effect on the coupled mode frequencies: As can be seen from the above figure, the L -, L 0, and L + modes behave as previously described. Here, the solid curves represent theoretical calculation, while the open circles represent the experimental data points. The characteristic modes (LO and TO) of AlAs and GaAs are shown for reference, and do not actually appear in the experimental results. The diagonal line extending from zero towards the top right corner is representative of the plasma frequency. Yuasa also showed the experimental results and fitted curves for the phonon content of the three plasmon-phonon coupled modes. In this case, Yuasa measured the coupled-mode half-width in order to find its inverse, which is proportionate to the phonon 20

32 content of that mode. Thus, by measuring the half-width, a plot of the phonon content can be created. (Yuasa et al, 1986). Figure 10: Plot of the phonon content for Al 0.19 Ga 0.81As at different free carrier concentrations The L - mode tends to have a strong frequency shift for lower doping concentrations (between 1x10 16 cm -3 and 5x10 17 cm -3 for Al 0.19 Ga 0.81 As) and very little shift for the higher > 5x10 17 cm -3 charge carrier concentrations. On the other hand, the L + mode tends to have a strong frequency shift for higher doping concentrations (above 5x10 17 cm -3 for Al 0.19 Ga 0.81 As) and very little shift for lower charge carrier concentrations. The L 0 mode shifts relatively linearly in the mid-doping ranges, and asymptotically 21

33 approaches the LO-phonon frequencies for each of the binary compounds (GaAs LO phonon for low free carrier concentrations, and AlAs LO phonon for high free carrier concentrations in the case of Al x Ga 1-x As). In other words, for low free carrier concentrations the L + and L 0 modes are phonon-like, while the L - mode is plasmon-like and for high free carrier concentrations the L + and L 0 modes are plasmon-like, and the L - mode is phonon-like. 2.4 Superlattices A semiconductor superlattice is a layered structure in which at least two different materials are periodically configured. The idea of the superlattice is to layer two materials with differing band gaps together such that the dielectric function of the system is manipulated to a desired value. A study done by B. Lou et al shows the experimental data for a GaAs/AlAs superlattice. In this study, infrared radiation was incident on the superlattice at an angle of 45 degrees in order to observe the characteristic optical phonon modes of their particular sample. The purpose of the study was to describe the anisotropic behavior of the optical phonons, which occurs only when the incident radiation has a wavelength much smaller than the periodicity of the superlattice. The following results are the experimental (and theoretical) results obtained by B. Lou et al for a GaAs/AlAs superlattice. The results are split into two graphs: one displaying the infrared reflection spectrum for s-polarization at 45 degree incidence, and the other displaying the infrared reflection spectrum for p-polarization at 45 degree incidence. 22

34 Figure 11: Infrared reflection spectra for GaAs/AlAs superlattice at 45 degree incidence As can be seen from the results, the characteristic GaAs TO mode can be seen at 270cm -1. However, in this case there is an interference fringe at the characteristic LO mode frequency for GaAs, which appears at about 290cm -1. Here, the anisotropy of the system can be seen in the splitting of the second peak from R s to R p. In the spectrum for R s the second peak is broad. However, in the spectrum for R p the second peak has split into two separate peaks. This is a consequence of the dielectric function taking on a different value for p-polarized radiation, or in other words, the anisotropy of the superlattice system (Lou, Sudharsanan, and Perkowitz, 1988). 23

35 In another study done by Z. Ristovski et al, the infrared spectrum was studied for more complicated GaAs/Al x Ga 1-x As superlattice structures (see Figure 12). In this case a new, iterative method of calculation was proposed, which will be further discussed in chapter 3. The sample that was studied by Z. Ristovski et al is a multilayered structure, consisting of a GaAs top layer, followed by the layered GaAs/Al x Ga 1-x As superlattice. Beneath the superlattice lies an Al x Ga 1-x As buffer layer, which rests on the GaAs substrate. The following diagram depicts the sample structure more clearly. Figure 12: Diagram of the superlattice structure studied by Z. Ristovski et al Here, the top, GaAs layer is 500 Angstroms thick. The superlattice consists of alternating layers of GaAs and Al x Ga 1-x As of thicknesses Angstroms and

36 Angstroms, respectively. The molar fraction, x, is The following shows the experimental results obtained from infrared spectroscopy. Figure 13: Reflection spectrum for GaAs/Al x Ga 1-x As superlattice structure at normal incidence As shown in figure 13, there are two strong bands defining the spectrum. The first appears near the characteristic GaAs transverse optic mode frequency of 270 cm -1. The second appears near 375cm -1, which is the characteristic frequency of the longitudinal optic mode for AlAs(Ristovski et al, 1997). 25

37 CHAPTER III THEORETICAL MODELING Understanding the properties of semiconductor materials and their effects on their behaviors is of particular interest in the development of optoelectronic devices. Such materials, not available in nature, must be synthesized by modern epitaxial growth techniques. In order to overcome the challenge of controlling the behavior of the material, the theory must be understood such that the desired parameters can be achieved from these growth techniques. 3.1 Theoretical Background The process of modeling far infrared reflection and transmission for semiconductors requires the use of the complex dielectric function, ε. The contributing factors to the dielectric function are the free charge carrier effects, ε e (ω,q), which originates from either the electrons in the conduction band or the holes in the valence band, and the lattice effects, ε i (ω,q), which originate from the lattice vibrations, or optical phonons. In the limiting case that q goes to zero, the dielectric function can be written as:, (3.1) or, rewriting the complex dielectric function in more general terms: 26

38 . (3.2) Here, the real part of the dielectric function is represented by ε 1, and ε 2 represents the imaginary part, which arises due to the absorption of energy as a function of frequency. This absorption yields direct results in the reflectivity spectrum of the material (Talwar et al). Simultaneously, the complex dielectric function can be calculated using experimental data obtained from the specific semiconductor sample. (3.3) Here, ε represents the high-frequency dielectric constant, S is the oscillator strength, γ is the plasmon damping coefficient, Γ is the phonon damping coefficient, ω TO is the transverse optic phonon frequency, and ω p represents the plasma frequency, which only appears in doped materials. That is, ω p is a function of the free charge carrier concentration (n) (Lockwood, Yu, and Rowell, 2005). (3.4) In the above equation, represents the effective mass, e is the elementary charge in statcoulombs, n is the free-carrier density, and n is the free charge carrier concentration in cm -3. Using the complex dielectric function, one can also solve for the optical constants n and κ, which represent the index of refraction and the extinction coefficient, respectively. 27

39 (3.5) and, (3.6) 3.2 Anisotropy for Oblique Incidence Far Infrared theory must also account for the Berreman Effect, that is, the discrepancy between s-polarization and p-polarization. Due to the anisotropy of a uniaxial crystal, the dielectric function is variant, depending on its polarization with the plane of the surface. Using long-wavelength theory, the dielectric tensor for a semiconductor is: (3.7) According to this tensor, the dielectric function for a semiconductor is the same when polarized along either the x- or y-axis, ε (columns one and two). However, the dielectric function becomes different when the incident radiation is polarized into the plane, along the z-axis, ε (column three). The difference in the dielectric function is as follows: (3.8) and, 28

40 (3.9) Here, d 1 and d 2 are the layer thicknesses of the alternating superlattice materials, and ε 1 and ε 2 are the complex dielectric functions for those materials, respectively. Using these values for the isotropy of the superlattice, and following the methods set out by Piro, the reflectivity for oblique incidence can be calculated for both s- and p- polarization. (3.10) Where, for s-polarization: (3.11) (3.12) (3.13) (3.14) and, for p-polarization: 29

41 (3.15) (3.16) (3.17) (3.18) Here, ε s represents the complex dielectric function for the substrate, θ i is the incident angle, and h represents the total thickness of the superlattice. The frequency, ω, is measured in units of cm -1. It can be seen that these coefficients, in the case of s-polarization, do not contain any dependence on ε. However, in the case of p-polarization both ε and ε exist (Lou, Sudharsanan, and Perkowitz, 1988). Consequentially, the resulting infrared spectrum will exhibit new features in comparison to when only ε is used. 3.3 Ternary Alloys The calculation of the dielectric function for a ternary alloy (consisting of three elements) is slightly more complicated. A new property dependence arises due to the molar fraction of the ternary alloy, or the ratio of each of the elements. Because of this, a weighted calculation must be made in order to determine the high-frequency dielectric constant, as well as for the complex dielectric function which arises from the ternary 30

42 composition. In the case of the high-frequency dielectric constant, it is simply a ratio between the dielectric functions of each material with their respective molar fraction. The complex dielectric function, on the other hand, can be calculated using the following: (3.19) Here, the term ε jx represents the weighted average of the high frequency dielectric function between the pure binary compounds, and is calculated as follows: (3.20) Also, S jx is the oscillator strength, ω TO is the transverse optic resonant frequency of the j th TO phonon. And finally, Γ jx represents the damping parameter for the j th TO phonon. All of these parameters are representative for a particular composition, x. Yuasa also shows an expression that can be used to calculate the oscillator strength of a coupled mode, or phonon content, as a function of the frequency. Using this calculation, Yuasa was able to produce a theoretical plot of the phonon content which, with only small disagreement, showed similar features to the experimental plot. (3.21) In this case, the subscripts, j, are representative of the L - and L + modes, respectively. When the strength of the coupled mode is strong, or phonon-like, S should be nearly one. On the other hand, when the strength of the coupled mode is weak, or plasmon-like, S will be nearly zero. 31

43 3.4 Iterative Method Using another, iterative method for calculating the FIR reflection spectrum, a complex semiconductor structure can be accurately characterized. Following the work of Z. Ristovski et al, who applied this method to the structure shown in Figure 12, the reflection spectrum can be calculated for a semiconductor structure of n layers by looking at the boundary conditions at each interface. First, the electric field in a medium, n, must be defined: (3.22) Here, represents the amplitude of the electric field in the +z direction, while represents the amplitude of the electric field in the z direction for a medium, n. Also, k n can be represented as the following: (3.23) Using the electric field interface method, the reflection contribution due to each interface can be found by solving for the complex ratio of the electric field s amplitude in the z direction to the +z direction. (3.24) Solving for the ratio yields: (3.25) Here, r n-1,n is the complex reflection coefficient between the n-1 and n interface, and can be evaluated as: 32

44 (3.26) In the case of the last layer, the substrate, the reflection contribution, R n-1,n (in this case R 3,4 ) is simply: (3.27) This is due to the fact that the substrate layer is considered infinite, so the wave propagates only in the +z direction. Furthermore, the uppermost interface, r 0,1, is between air and the GaAs top layer of the structure. Accordingly, since the dielectric function of air is approximately equal to 1: (3.28) Using the above formulae, the total reflection for the semiconductor structure can be calculated. The total reflection spectrum of the total system will be a function of R 0,1, that is: (3.29) According to the structure shown in Figure 12, the calculation would refer to the GaAs top layer as ε 1, the GaAs/Al x Ga 1-x As superlattice structure as ε 2, the Al x Ga 1-x As buffer layer as ε 3, and the GaAs substrate as ε Plasmon-Phonon Coupled Modes In order to calculate the dependence of the plasmon-phonon coupled mode frequencies, the dielectric functions and polarizations of the lattice and the electron 33

45 system must be evaluated. Following the work of Yuasa et al, the polarizations can be calculated as follows: (3.30a) (3.30b) (3.30c) Where P L is the polarization of the lattice, P e is the polarization of the electrons, and P T is the total polarization of the system. ε L and ε e are the dielectric functions of the lattice and electrons, respectively, and E is the electric field. In using this method, it is also assumed that the polarizations of the lattice and electrons are additive i.e., Which yields the total polarization as: (3.31) From this total polarization equation, the total dielectric function can be calculated, and is equal to: (3.32) (3.33) 34

46 The dielectric functions of the lattice and of the electron system are explained to be: (3.34) (3.35) Where ω p is the plasmon frequency, and j represents the j th material in the ternary alloy. Thus S j is the j th oscillator strength, and ω tj is the characteristic transverse optic phonon frequency. The new, coupled mode frequencies occur at the locations where ε T (ω,q) is equal to zero. Thus, we must solve for the total dielectric function in terms of known constants. Using the above equations: (3.36) As can be seen from the above formula, ε T is a sixth order equation, and as a result yields six different solutions. When traced over different levels of doping (and hence plasma frequencies), this equation accurately predicts the L -, L 0, and L + frequencies (Yuasa et al, 1986). 35

47 CHAPTER IV RESULTS AND DISCUSSION By employing the above theory and comparing with experimental results, the behavior of various semiconductor materials of any thickness, free charge carrier concentration, and structure can be predicted. The following results have been produced using programs developed with Mathematica. 4.1 Thin Films The infrared transmission spectrum for a CdSe thin film is specified by Sciacca et al for normal incidence and oblique incidence for both s- and p-polarization. Using the methods presented above, a theoretical plot of the CdSe thin film has been made. Figure 14: Transmission spectrum for a 0.2μm CdSe thin film at normal incidence 36

48 Figure 15: Transmission spectrum for a 0.2μm thin film at 45 degree incidence for both s- (solid) and p- (dashed) polarization As can be seen from figures 14 and 15, the theory is in very good agreement with the experimental data. At normal incidence the theory shows only a strong mode at the characteristic transverse optic frequency of 172cm -1. When done at an oblique angle (in this case 45 degrees), the s- polarization also shows only a strong mode at the characteristic transverse optic frequency, while the p-polarization shows a strong transverse optic mode as well as a weak longitudinal optic mode at a frequency of 213cm -1. Furthermore, the reflection spectra can also be evaluated for normal and oblique incidences, and should show similar features to the transmission curves. Thus, for the same CdSe thin film as shown above, the reflection curves are as follows: 37

49 Figure 16: Reflection spectrum for a 0.2μm CdSe thin film at normal incidence Figure 17: Reflection spectrum for a 0.2μm CdSe thin film at 45 degree incidence for both s- (solid) and p- (dashed) polarization 38

50 As shown in figures 16 and 17, only one strong peak occurs in the reflection spectrum at the transverse optic frequency of 171cm -1 for normal incidence. In the case of oblique, 45 degree incidence, s-polarization again shows only the transverse optic mode, whereas p-polarization shows both the transverse optic mode and the weaker longitudinal optic mode at 213cm -1. Using the same simulation that accurately represents both the reflection and transmissions graphs for a CdSe thin film, the spectra can be calculated for any binary semiconductor thin film. Here, we will evaluate the reflection spectrum for a GaAs film. The characteristic transverse optic frequency of GaAs is 268cm -1, while the longitudinal optic frequency is 291cm -1. Figure 18: Reflection spectrum for a 0.2μm GaAs thin film at 45 degree incidence for both s- (solid) and p- (dashed) polarization Figure 18 clearly shows a strong peak at the frequency characteristic of the transverse optic mode for both s- and p-polarization. Because the two characteristic 39

51 frequencies (LO and TO) are so close together in GaAs, it is difficult to distinguish the LO mode in the p-polarization. Another observation Sciacca et al made showed the transmission spectrum for ZnTe over various thicknesses. In this case, samples were measured at oblique, 45 degree incidence. The samples were ZnTe of thicknesses 1μm, 5μm, 25μm, and 100μm. The spectrums shown are for p-polarization Figure 19: Transmission spectrum for ZnTe of thicknesses 1μm, 5μm, 25μm, and 100μm at 45 degree oblique incidence for p-polarization Here it can be seen that the thin film of 1μm clearly depicts the transverse optic mode at 181cm -1, as well as the longitudinal optic mode at 209cm -1. However, as the 40

52 sample s thickness grows, interference fringes appear below the TO mode frequency (Sciacca et al, 1995).Using the simulation developed on Mathematica in order to reproduce these results: Figure 20: Transmission spectrum for ZnTe of thicknesses 1μm at 45 degree oblique incidence for p-polarization Figure 11: Transmission spectrum for ZnTe of thicknesses 5μm at 45 degree oblique incidence for p-polarization 41

53 Figure 22: Transmission spectrum for ZnTe of thicknesses 25μm at 45 degree oblique incidence for p-polarization Figure 23: Transmission spectrum for ZnTe of thicknesses 100μm at 45 degree oblique incidence for p-polarization 42

54 The above figures are in excellent agreement with the theory introduced by Sciacca et al. This shows the accuracy of the theoretical models for thin films. It can now be used to show the transmission spectra for many thin films of various thicknesses. In this case it will be used to show the spectrum evolution of GaAs over several thicknesses. Figure 24: Transmission spectrum for GaAs of thicknesses 1μm at 45 degree oblique incidence for p-polarization 43

55 Figure 25: Transmission spectrum for GaAs of thicknesses 5μm at 45 degree oblique incidence for p-polarization Figure 26: Transmission spectrum for GaAs of thicknesses 25μm at 45 degree oblique incidence for p-polarization 44

56 Figure 27: Transmission spectrum for GaAs of thicknesses 100μm at 45 degree oblique incidence for p-polarization As expected, the spectrum shows two strong restrahlen bands at each of the characteristic modes of GaAs: A strong dip at the transverse optic frequency of 268cm -1, and a weak dip at the longitudinal optic frequency of 291cm -1. The spectrum then evolves to show some signs of interference fringes, though the two characteristic modes are still noticeable. As the film thickness grows beyond 5μm, however, only the strong transverse optic mode remains visible, as the interference fringes begin to characterize the spectrum. 4.2 Ternary Alloys The theory behind the study of ternary alloys is still mildly lacking in terms of the determination of the longitudinal optic and transverse optic modes with respect to the composition, x. Many studies have been done fitting experimental data to linear or 45

57 quadratic expressions. Here, we compare our model to that obtained by Feng. Here, we attempt to reproduce the results for x = 0.108: Figure 28: Reflection spectrum for Al Ga As of thickness 1μm at 45 degree incidence for p-polarization From the above figure, it is apparent that the mode frequencies match up relatively well. However, the strengths of the peaks are not accurate. This is due to the model not taking into account the small composition of aluminum atoms. The aluminumlike mode at 360cm -1 is therefore too strongly represented. Although this is a downside to the ternary model, the results for compounds near x = 0.50 should be more accurately represented. Here, we will show x = 0.492: 46

58 Figure 29: Reflection spectrum for Al Ga As of thickness 1μm at 45 degree incidence for p-polarization The above figure shows good agreement with the results obtained by Feng et al. It is important to note that the mode found at 290cm -1 in Feng s results are due to a thin GaAs cap layer of the sample, and therefore should not, and does not, appear in our results (Feng et al, 1993). Talwar also calculated a theoretical plot of the phonon content for a ternary alloy similar to that shown by Yuasa. In this case, his calculation was done for Al 0.2 Ga 0.8 As, and shows the phonon content with respect to the free carrier concentration. 47

59 Phonon content S m (q) 1.2 n - Al 0.2 Ga 0.8 As 0.8 S + S o S o 0.4 S [N (cm -3 )] 1/2 Figure 30: Calculated phonon content of the three coupled modes in Al 0.2 Ga 0.8 As 4.3 Superlattices Following the classical method in order to reproduce the data acquired by Z. Ristovski has also provided theoretical results which agree very well with the experimental data. In this case, the structure studied is the same as that shown in figure 12. Using the theory provided in chapter 3, a theoretical reflection plot of this structure has been made. 48

60 Figure 31: Reflection spectrum for the structure shown in figure 12 at normal incidence Upon comparison between figure 13 and figure 31, one can immediately see that theory is in nearly perfect agreement with experiment. As also shown by Ristovski, the spectrum shows two strong restrahlen bands at 270cm -1 (GaAs) and 370cm -1 (AlAs). Now that it has been shown that this program can accurately reproduce the reflection spectrum for a complicated semiconductor structure, it can be used to predict new spectra for various semiconductor structures. For example, if the same structure were to have been analyzed at oblique incidence rather than normal incidence, the data obtained would show similar features as the following: 49

61 Figure 32: Reflection spectrum for the structure shown in figure 12 at 45 degree incidence for s- (solid) and p-polarization (dashed) From figure 32 it can be seen that s-polarization yields similar results as normal incidence, which is expected. However, the p-polarization shows a splitting of the complex mode at 290cm -1. Similarly, the iterative method yields a spectrum almost exactly the same as that acquired by the classical method. Here, the iterative method followed by Z. Ristovski et al has been employed to produce the reflection spectrum for the same semiconductor structure. 50

62 Figure 33: Reflection spectrum for the structure shown in figure 12 at normal incidence as predicted by the iterative method As can be seen from figure 33, the spectrum nearly exactly agrees with the previous theoretical prediction shown in figure 31. Thus the two models are equally valid, and modeling may be done using either of the two. 4.4 Plasmon-Phonon Coupled Modes The study of plasmon-phonon coupled modes is an important part of characterizing semiconductor materials. Theoretical knowledge of the frequency shift of these modes can lead to easily determining the free charge carrier concentration of a particular sample. It is important to note that the study of plasmon-phonon coupled modes is dependent on a large number of parameters, though the coupled modes themselves tend to follow the same patterns. 51

63 Transmission (arb. units) Here we present the theory done by Dr. Devki Talwar for the shift of the L + mode in doped GaN at different free charge carrier concentrations ranging from 7.8x10 17 to 3.8x10 18 cm -3. LO phonon plasmon coupled modes L + in n - GaN TO (a) (b) (c) L + L + L + 7.8x x x Frequency in wavenumber Figure 34: Transmission spectrum for GaN depicting the shift in the L + mode at 7.8x10 17 cm -3,1.2x10 18 cm -3, and 3.6x10 18 cm -3 As can be seen from figure 34, the L + mode shifts only slightly (about 15cm -1 ) over the lower free charge carrier concentrations, while it shifts greatly (about 100cm -1 ) over the higher free charge carrier concentrations. Using this data, Talwar went on to 52

64 Frequency in wavenumber produce a theoretical model showing the shift of both the L + and L - modes in GaN over a number of free charge carrier concentrations n - GaN o Kozawa et al. o Harima et al. L LO p TO 400 L [N cm -3 ] 1/2 Figure 35: Theoretical model showing the frequency shift of the L + and L - coupled modes in GaN with respect to the square root of the free charge carrier concentration Figure 35 showsthe L + mode and L - mode behave exactly as described in section 2.3. These results have also been compared to existing data produced by Kozawa et al and Harima et al, and are in excellent agreement (Harima et al, 1998; Kozawa et al, 1994). 53

65 Frequency (cm -1 ) Moving forward, Talwar went on to use this model to predict the frequency shift of the L +, L 0, and L - modes in the ternary alloy, Al 0.2 Ga 0.8 As. Here, ω LO1 and ω TO1 are representative of the longitudinal and transverse optical frequencies characteristic of GaAs, while ω LO2 and ω TO2 are representative of the longitudinal and transverse optical frequencies characteristic of AlAs. 600 n-al 0.2 Ga 0.8 As L LO2 P TO2 L o LO1 200 L - TO [N (cm -3 )] 1/2 Figure 36: Theoretical model showing the frequency shift of the L -, L 0, and L + coupled modes for Al 0.2 Ga 0.8 As with respect to the square root of the free charge carrier concentration As can be seen from figure 36, the L - mode frequency shifts greatly for low charge carrier concentrations, and only slightly at higher charge carrier concentrations. Meanwhile, the L + mode shifts only slightly at low charge carrier concentrations, and 54

66 greatly for higher charge carrier concentrations. Thus showing that for low free carrier concentrations the L + and L 0 modes are phonon-like, while the L - mode is plasmon-like and for high free carrier concentrations the L + and L 0 modes are plasmon-like, and the L - mode is phonon-like. Using a similar method, we have produced a three-dimensional plot which depicts the L - and L + mode frequency shifts for binary compounds with respect to the plasmon frequency, ω p, and hence the square root of the free charge carrier concentration. In addition, this three-dimensional plot shows the strength of each of the two coupled modes, and is portrayed as the height of the peak. Here, we present the three-dimensional plot for InN in hopes of reproducing the results obtained by Thakur et al in a similar simulation (Thakur et al, 2004). 55

67 Figure 37: Three-dimensional plot for InN showing the coupled mode frequency strength with respect to the frequency shift at different values of ω p This theoretical model is in very good agreement with Thakur s predictions. This program can be used to predict the behaviors of other materials. Here, the parameters have been changed in order to represent GaAs. 56

68 Figure 38: Three-dimensional plot for InN showing the coupled mode frequency strength with respect to the frequency shift at different values of ω p When rotated to only show a two-dimensional plot of the frequency with respect to the plasmon frequency: 57

69 Figure 39: Three-dimensional plot for GaAs rotated to show only two dimensions, frequency versus plasma frequency From these two three-dimensional plots, one can see that the L - mode shifts strongly for lower values of ω p, and hence lower free charge carrier concentrations, and approaches the GaAs transverse optic frequency for higher ω p. The L + mode shifts only slightly for lower values of ω p, and strongly for higher values of ω p. These theoretical models all display good agreement with known experimental data. This shows that the models accurately depict the features of real samples, and as such can be used to either determine the specific parameters of a particular 58