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Optical bandgap thermometry in molecular beam epitaxy Johnson, Shane R 1995

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OPTICAL B A N D G A P T H E R M O M E T R Y LN M O L E C U L A R B E A M E P I T A X Y by Shane R. Johnson B.Sc, Simon Fraser University, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF GRADUATE STUDIES (Department of Physics)  We accept this thesis as conforming to the require^ standard  The University of British Columbia December 1995 © Shane R. Johnson, 1995  In presenting this  thesis in partial fulfilment of the requirements  for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted department  or by his or her representatives.  It  by the head of my  is understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  ihy^/C  The University of British Columbia Vancouver, Canada 4 Date  DE-6 (2/88)  S  ll  Abstract The temperature dependence of the optical absorption edge (Urbach edge) is measured in semi-insulating and n-type (n = 2*10 cm- ) GaAs from room 18  3  temperature to 700 °C. The characteristic energy of the exponential absorption edge is found to increase linearly with temperature, from 7.5 meV at room temperature to 12.4 meV at 700 °C, for semi-insulating GaAs. The temperature dependent part of the width of the Urbach edge for semiinsulating GaAs and InP is smaller than predicted by the standard theory where the width of the edge is proportional to the phonon population. The part of the width not characteristic of the phonon occupation number in semi-insulating GaAs and InP, is attributed to static fluctuations in the band edges due to point defects.  The absorption edge in n-type GaAs and InP is broadened by  fluctuations in the band edge caused by the electric fields and the strain fields of the ionized donor impurities. A n optical method for measuring the temperature of a substrate material with a temperature dependent bandgap is presented. In this method the substrate is illuminated with a broad spectrum lamp; the bandgap is determined from the spectrum of the diffusely scattered light. Light with energies below the bandgap is transmitted through the substrate and reflected from the back surface of the substrate while light with energies above the bandgap is absorbed by the substrate. The front surface of the substrate is polished and its back surface is either rough or polished with a scatterer behind the substrate. The reflection of light from the front surface is specular and the light diffusely reflected at the back  in  of the substrate is detected in a non-specular location. Substrate temperature is determined from the wavelength of the onset of the non-specular reflection. A n algorithm is presented that utilizes the position and the width of the knee in the diffuse reflectance spectrum as a reference point for the onset of transparency and hence temperature of the substrate. A model is developed that relates the onset of transparency of the substrate to the optical bandgap and the width of the absorption edge of the substrate. From this model the sensitivity and absolute accuracy of the measurement for differences in substrate thickness, back surface texture, and conductivity is determined. The temperature sensitivity and reproducibility of the diffuse reflectance technique is better than 1°C when using GaAs substrates. Using the diffuse reflectance technique the temperature of GaAs substrates is profiled in a molecular-beam-epitaxy system with a spatial resolution of 3 mm and a thermal resolution of 0.4 °C. The effects of substrate doping, back surface textures, thermal contact to holder, and a pyrolytic boron nitride diffuser plate, on the temperature uniformity of radiatively heated substrates is explored. Both positive and negative curvatures are observed in the temperature profiles.  iv  Table of Contents Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  viii  Acknowledgments  xvi  Chapter 1  Introduction  1  Chapter 2  Urbach Edge  6  §2-1  Absorption Edge of Semiconductors  6  §2-2  Bandgap of GaAs  9  §2-3  The Urbach Parameter  14  §2-4  Diffuse Reflectance Measurement  16  §2-5  Determination of Absorption Coefficient  23  §2-6  Determination of Optical Bandgap and Urbach Parameter  30  §2-7  Urbach Focus for GaAs  39  §2-8  Residual Absorption Below the Urbach Edge in GaAs  42  Band Tails  46  The Urbach Parameter and Band Tails  46  Chapter 3 §3-1  V  §3-2  Width of Band Tails in HI-V Materials  62  Diffuse Reflectance Spectroscopy  79  §4-1  Non Contact Temperature Measurement of Substrates  79  §4-2  The Diffuse Reflectance Technique  83  §4-3  Calibration of Critical Points of the Spectrum to Temperature  85  §4-4  Diffuse Reflectance in Indirect Bandgap Materials  93  §4-5  Modeling Diffuse Reflectance  97  §4-6  Sensitivity to Substrate Characteristics  107  §4-7  Reducing Sensitivity to Variations in Material Properties  127  §4-8  Bandwidth of the Diffuse Reflectance Spectrum  133  Diffuse Reflectance Spectroscopy During Growth  137  §5-1  Oscillations in the Width of the Knee  138  §5-2  Measuring Composition of Small Bandgap Epilayers  147  §5-3  Growth Rate, Index of Refraction, and Composition  150  §5-4  Sensitivity to Surface Morphology  155  §5-5  Measuring Scattering Parameter from Fitting Parameters  157  §5-6  Ex situ Measurements of Scattering from Textured Wafers .... 160  §5-7  Comparing the Model with Calibration Curves  168  Radiatively Heated Substrates  171  Chapter 4  Chapter 5  Chapter 6  vi  §6-1  Temperature Uniformity  171  §6-2  Spatial Temperature Profiling  172  §6-3  Factors Affecting Temperature Uniformity  173  §6-4  Substrate Holder Design  184  Chapter 7  Substrate Temperature During InGaAs Laser Growth  191  Chapter 8  Conclusions  198  Bibliography Appendix I  202 Algorithm Flow Charts  Appendix II Thin Film Interference and the Wavelet Transform  207 214  §LL 1 Wavelets and the Wavelet Transform  214  §11-2 Wavelet Transform of Diffuse Reflectance Spectrum  217  §11-3 Derivatives and the Wavelet Transform  220  §11-4 Higher Derivatives of the Diffuse Reflectance Spectrum  227  §11-5 Correcting for Spurious Temperature Shifts  232  §11-6 Fourier Transform of the Diffuse Reflectance Spectrum  238  §11-7 Correcting for Variations in the Scattering Parameter  240  Appendix EI Wafer Holder Drawings  246  L i s t of T a b l e s  Table 2-1. Cody's constant and the Urbach focus for undoped (semiinsulating) GaAs and Si-doped (n = 2'10 cm- ) GaAs. 18  33  3  c  Table 2-2. Parameter values for the optical bandgap and Urbach parameter 3 6 in undoped and Si-doped GaAs. Table 3-1. Urbach edge parameter values for undoped (semi-insulating) 72 and Si-doped (n = 2-10 cm- ) GaAs and Fe-doped (semi-insulating) and S-doped (n = 6-10 cnr ) InP. 18  3  c  18  3  c  Table 3-2. Comparison of measured and calculated Urbach edge parameters in undoped (semi-insulating) GaAs and Fe-doped (semiinsulating) InP.  73  Table 3-3. Parameters used in calculating the deformation potentials for GaAs and InP.  74  Table 3-4. Comparison of measured and calculated band tail parameters in 75 Si-doped (n = 2-10 cm" ) GaAs and S-doped (n = 6-10 cm" ) InP. 18  3  c  18  3  c  Table 4-1. List of parameters in Eq. 4-26.  110  Table 4-2. Temperature error associated with variations in the physical 121 properties of a GaAs substrate from those of a 500 Lim thick semiinsulating GaAs substrate at 500 °C with scattering parameter 0.1. Table 4-3. List of parameters for Eq. 4-44.  126  Table 5-1. Scattering parameters for several types of textured surfaces, at a 163 wavelength of 633 nm. Table 6 -1. Summary of temperature profiling results.  180  Table II-1. Shift in the position of knee and the associated temperature 236 error for various parameter values in the correction algorithm. Table II-2. Shift in the position of the peak of the fourth-derivative (wavelet-transform) for different widths of the wavelet.  238  viii  L i s t of Figures  Fig. 2-1. Behavior of the optical joint density of states at the band edge.  7  Fig. 2-2. Absorption in n-type GaAs (n = 10 cnr ) at room temperature 8 in the region of the bandgap. 17  3  c  Fig. 2-3. Published values of the bandgap for GaAs.  11  Fig. 2-4. Schematic of the diffuse reflectance spectroscopy setup used to 17 measure the temperature dependence of the absorption spectrum of GaAs. Fig. 2-5. Diffuse reflectance of the pyrolytic boron nitride diffuser plate.  23  Fig. 2-6. Normalized diffuse reflectance spectra for semi-insulating GaAs. 26 The temperature of each curve is on the right-hand side where the lowest temperature corresponds to the upper-most curve. Fig. 2-7. Normalized diffuse reflectance spectra for Si-doped GaAs. The 27 temperature of each curve is on the right-hand side where the lowest temperature corresponds to the upper-most curve. Fig. 2-8. Normalized diffuse reflectance spectra for undoped (semi- 28 insulating) GaAs and Si-doped (n = 2-10 cm- ) GaAs. The temperatures for the absorption curves are as follows: (a) 58 °C and 52 °C, (b) 258 °C and 258 °C, (c) 435 °C and 426 °C, and (d) 631 °C and 632 °C for the undoped and Si-doped GaAs respectively. 18  3  c  Fig. 2-9. The absorption coefficient as a function of photon energy for 29 undoped (semi-insulating) GaAs and Si-doped (n = 2*10 cnr ) GaAs. The temperatures are as follows: (a) 58 °C and 52 °C, (b) 258 °C and 258 °C, (c) 435 °C and 426 °C, and (d) 631 °C and 632 °C for undoped and Si-doped GaAs respectively. 18  3  c  Fig. 2-10. The absorption edge for semi-insulating GaAs, after the removal 31 of the low energy absorption. The temperatures are given at the righthand side, the lowest temperature is for the right-most curve, as the temperature increases the curves shift to the left.  ix  Fig. 2-11. The absorption edge with the low energy interband absorption 32 removed, for Si-doped (n = 2*10 cm- ) GaAs. The temperatures are given at the right-hand side, where the lowest temperature is for the rightmost curve, as the temperature increases the curves shift to the left. 18  3  c  Fig. 2-12. The diffuse reflectance spectrum and the absorption coefficient 33 for semi-insulating GaAs at 398 °C. The solid circles superimposed over the absorption with background subtracted are fit to determine the optical bandgap and the Urbach parameter. The rectangular box shows the section of the diffuse reflectance spectrum used in the fit. Fig. 2-13. The optical bandgap as a function of the Urbach parameter, for 34 undoped and Si-doped GaAs. Fig. 2-14. The temperature dependence of the optical bandgap for 35 undoped (semi-insulating) GaAs and Si-doped (n = 2-10 cm") GaAs. The solid lines are fits to a model where the phonon population is represented by a single oscillator with an Einstein temperature of 263 K. 18  3  c  Fig. 2-15. The temperature dependence of the Urbach slope parameter for 37 undoped (semi-insulating) GaAs and Si-doped (n = 2*10 cnr ) GaAs. The solid lines are fits to a model where the phonon population is represented by a single oscillator with an Einstein temperature of 263 K. 18  3  c  Fig. 2-16. The temperature dependence of the bandgap of the semi- 38 insulating GaAs sample compared with Thurmond's results. Fig. 2-17. The temperature dependence of the Urbach parameter for pure 38 (semi-insulating) GaAs. Fig. 2-18. The Urbach focus (1.8-10 cm , 1.93 eV) and the exponential 40 absorption spectra for undoped (semi-insulating) GaAs. The absorption curves shift to the left with increasing temperature. The temperatures for this series of curves are shown on the left-hand side. 33  11  Fig. 2-19. Exponential absorption spectra for Si-doped (n = 2-10 cm- ) 41 GaAs. The absorption curves shift to the left with increasing temperature. The temperatures for this series of curves are shown on the left-hand side. The Urbach focus is off scale at (5.0-10 cm-i, 2.87 eV). 18  3  c  51  Fig. 2-20. Temperature dependence of the residual subedge absorption at 43 0.92 eV (1350 nm) for undoped and Si-doped GaAs.  Fig. 3-1. Conduction band and valence band density of states for parabolic bands and bands with Urbach tails.  48  Fig. 3-2. The joint density of states for the different band to band absorption processes in semi-insulating GaAs.  51  Fig. 3-3. Conduction band and valence band density of states for n-type material, showing the occupied and unoccupied states.  54  Fig. 3-4. The optical joint density of states for the different band to band absorption processes in Si-doped (n+) GaAs.  57  Fig. 3-5. Optical bandgap shift in the joint density of states of n+ GaAs.  58  Fig. 3-6. Widths of the band tails in a (hypothetical) heavily doped semiconductor.  61  Fig. 3-7. Fluctuations in the conduction and valence bands caused by 63 Coulomb potentials and strain fields. The impurities or defects are represented by: an open circle for compressive strain, a solid circle for tensile strain, a "+" sign for a positive charge, and a "-" sign for a negative charge. Fig. 3-8. Width of the Urbach edge in n-type and semi-insulating GaAs and n-type and semi-insulating InP.  71  Fig. 3-9. Electron-phonon coupling constant for the width of the absorption edge as a function of the ionic character of the energy bond.  76  Fig. 3-10. Electron-phonon coupling constant for the width of the absorption edge as a function of the Frbhlich coupling constant.  78  Fig. 4-1. Thermocouple and substrate temperature response to a 10 to 50% 80 step in the heater power. The power step occurs at time zero. Fig. 4-2. Diffuse reflectance spectra for semi-insulating GaAs, at several different temperatures.  84  Fig. 4-3. Illustration of a fit to the diffuse reflection spectrum using 87 Algorithm A showing the location of the knee in the spectrum.  Fig. 4 4 . Diffuse reflectance spectrum and its second derivative showing 91 the location of the knee using Algorithm B . Fig. 4-5. Diffuse reflectance spectra from an unintentionally doped p-type Si substrate.  94  Fig. 4-6. Normalized diffuse reflectance spectra from an unintentionally 95 doped p-type Si substrate. In the inset the high temperature spectra are scaled by a factor of 25. Fig. 4-7. The residual absorption below the band edge in unintentionally 96 doped p-type Si. Fig. 4-8. Ray tracing diagram for light scattered from a textured surface.  99  Fig. 4-9. The distributions of scattered light inside the substrate (dashed 103 lines) and transmitted through the front surface of the substrate (solid lines), for scattering parameters 0.01, 0.1, and 1.0. Fig. 4-10. The fractions of light trapped and in the escape cone. The ratio 104 of the trapped light and light in the escape cone is shown in the inset. Fig. 4-11. The intensity of the diffuse reflectance signal at the inflection 105 point for four angular positions of the detector, namely 15°, 23°, 30°, and 45°, where the solid angle of the detector is 0.005 sr. The total diffuse reflectance signal at the inflection point is shown in the inset. Fig. 4-12. The shift in the critical points the spectrum from those of the 115 fundamental ray in the diffuse reflectance model, for a weighted average of critical points, A , and numerical simulations of the inflection point, A , and the knee A^. The upper curves are for substrates that are transparent below the edge and the lower curves are for a d = 0.5. c  p  0  Fig. 4-13. The correction parameter, A , using the DRS model for a 116 500 |im thick semi-insulating GaAs substrate. The scattering parameter for each curve is listed next to it. c  W  Fig. 4-14. The correction parameter for 500 Jim thick n+ GaAs substrates, 117 for the DRS model. The scattering parameter for each curve is listed next to it. W  Fig. 4-15. The Fourier transform of the asymptotic fitting function. The 134 integral of the power spectral density is shown in the inset. Fig. 4-16. Spectral resolution for diffuse reflectance measurements on 135 semi-insulating GaAs substrates. The higher curve is for a textured back (y = 0.20) and the lower curve is for a polished back (Y = 0.0). Fig. 5-1. The transmission of light through an epilayer with an index of 139 refraction that differs from that of the underlying material. Fig. 5-2. Oscillations in the width of the knee during the growth of 142 Al.5Ga.5As on GaAs (dashed line). The solid line is a fit to Eq. 5-5. Fig. 5-3. The position of the knee (solid line) and the width of the knee 144 (dashed line) during the growth of Al.sGa 5As on GaAs. Fig. 5-4. Substrate temperature given by, the position of the measured 145 knee (dashed line), the position of the corrected knee (solid line) and the true temperature (broken line), during the growth of Al.5Ga.5As on GaAs. Fig. 5-5. Diffuse reflectance spectra during the growth of InGaAs on GaAs 149 at 530 °C. The thickness of the InGaAs layer for each scan is shown on the left-hand side. As the InGaAs layer thickens the signal decreases. Fig. 5-6. Diffuse reflectance at individual wavelengths; the curves are at 153 10 nm intervals, starting at 1190 nm for the lowest signal, and ending at 1250 nm for the largest signal. Fig. 5-7. Index of refraction of GaAs at 608 °C (solid circles), over the 154 wavelength region 1190 nm to 1250 nm, in steps of 10 nm. The published room temperature values (solid line) are also shown. The energy scale is in units of the Urbach parameter. Fig. 5-8. Diffuse reflectance spectra from a GaAs substrate; one at 600 °C 156 before the oxide removal, and one at 636 °C after the oxide removal. Fig. 5-9. Experimentally measured values of a A for a 430 Lim thick semi- 158 insulating GaAs substrate with nitric acid etched back (solid circles). Calculated values of a A for the DRS model (dashed lines) and the DRSp model (solid lines), for scattering parameters 0.10, 0.20, and 0.30. m  m  c  W  c  Fig. 5-10. The width of the knee and the Urbach parameter as functions of 160 the position of the knee for a 430 fim thick semi-insulating GaAs wafer with a nitric acid etched back surface. Fig. 5-11. Schematic of apparatus used to measure the scattering distributions of textured wafers.  161  Fig. 5-12. The angular distribution of scattered laser light (solid circles) 162 from a nitric acid textured GaAs wafer. The solid line is a fit to a power law cosine distribution. Fig. 5-13. The position of the knee given by a calibration curve (broken 168 line) and a model (solid line), for a 450 p:m thick semi-insulating GaAs substrate with a nitric acid etched back. For comparison the bandgap wavelength of semi-insulating GaAs is also shown. Fig. 6-1. Schematic diagram of the optical system used to collect the 173 diffusely scattered light in the substrate temperature profiling measurements. Fig. 6-2. Temperature at a fixed location on a n+ GaAs substrate at 175 constant heater power. The substrate is allowed to equilibrate for one hour after fixing the heater power. Fig. 6-3. Temperature profile of the Ta foils in the substrate heater. In the 176 top panel the heater is partly covered by an annular M o ring used for holding 50 mm substrates. In the lower panel the heater is uncovered. Fig. 6-4. Selected temperature profiles for GaAs substrates from the 178 summary in Table 6T. The temperature at the center of the wafer is shown in the upper left-hand corner. The letters on the right hand side correspond to the labels in the first column in Table 6-1. Fig. 6-5. Schematics of different substrate mounting systems: standard 185 backside mount (top), standard front side mount (center), and three point mount with diffuser plate (bottom). Fig. 6-6. Schematic of the mounting system for the substrate and the P B N 189 backing plate, at one of the three contact points on the new Mo holders. Fig. 7-1. Substrate temperature and heater power during the growth of an 192 InGaAs laser.  Fig. 7-2. Substrate temperature and heater power during the growth of an 194 InGaAs quantum well. Fig. 7-3. Room temperature photoluminescence from the InGaAs quantum 197 well shown in Fig. 7-2. Fig. II-1. A mexican hat wavelet with width a. The Fourier transform of 216 this wavelet is shown in the inset. Fig. II-2. Diffuse reflectance spectrum from a semi-insulating GaAs 218 substrate at 331 °C (DRS ), and a thin film interference simulation for a 4 Jim thick AlGaAs layer on the surface of the substrate (DRSf). p  Fig. II-3. The wavelet transform of the diffuse reflectance spectrum from a 219 bare semi-insulating GaAs substrate at 331 °C. Fig. II-4. The wavelet transform of the diffuse reflectance spectrum from a 219 semi-insulating GaAs substrate, at 331 °C, with a 4 Lim thick AlGaAs overlay er on the front surface. Fig. II-5. Fourier transform of the wavelets derived from the first four 221 derivatives of the Gaussian. Fig. II-6. The smooth second derivative of the diffuse reflectance spectrum 222 determined by the wavelet transform and conventional methods. Fig. II-7. The peak in the wavelet transform of the diffuse reflectance 224 spectrum, for the Gaussian with widths ranging from 0.6 to 2.2 Urbach parameters, at intervals of 0.2 Urbach parameters. The peak in the second derivative of the diffuse reflectance spectrum is also shown. Fig. II-8. The position of the peak in the wavelet transform as a function of 225 the width of the Gaussian, a. The solid line is a linear fit to the position of the peak for a greater than 1.2 Urbach parameters. Fig. II-9. The peaks of the second derivative and the wavelet transform (a 226 equal 1.4 and 2.0 Urbach parameters) of the diffuse reflectance spectrum from a bare substrate and a substrate with a 4 Lim thick AlGaAs overlayer. Fig. 11-10. The fourth derivative wavelet transform of the diffuse 228 reflectance spectrum from a semi-insulating GaAs substrate at 331 °C. The vertical axis is 8*10~ meV at the bottom and -24-10" meV' at the top. 6  -4  6  4  XV  Fig. I H 1. The position of the sharpest and the highest peak of the first six 229 derivatives of the diffuse reflectance spectrum. Fig. 11-12. Diffuse reflectance spectra from a semi-insulating GaAs 230 substrate at 331 °C; with (solid line) and without (dashed line) an overlayer with index of refraction equal 2.5. :  Fig. II-13. Thin film interference induced shift in the position of the highest 231 and sharpest peak in the derivatives of the diffuse reflectance spectrum. Fig. 11-14. Oscillations in the position and the width of the knee of the 233 diffuse reflectance spectrum from a semi-insulating GaAs substrate at 331 °C during the growth of an epilayer with index of refraction 2.5. Fig. 11-15. Oscillations in the position of the knee of the diffuse reflectance 235 spectrum and the corrected position of the knee using various algorithms. Fig. 11-16. Oscillations in the position of the peak of the fourth-derivative 237 (wavelet-transform) for two widths of the wavelet; namely 1.5 and 2.0 Urbach parameters. The corrected reference point, given by a linear extrapolation from these peaks, is also shown. Fig. 11-17. Fourier transforms of the knee region up to the inflection point 239 of the diffuse reflectance spectrum (solid lines) from a bare substrate and a substrate with an AlGaAs overlayer and the Fourier transform of the asymptotic function fit to the spectrum of the bare substrate (dashed line). Fig. I H 8 . The Fourier transform of the diffuse reflectance spectrum (upper 240 curves) from a bare substrate (dashed line) and a substrate with a 4 \im thick AlGaAs layer (solid line). Fig. I H 9 . Simulated diffuse reflectance spectra for a semi-insulating GaAs 241 substrate at 331 °C for several values of the scattering parameter. Fig. 11-20. The position and the width of the knee in the diffuse 242 reflectance spectrum, as a function of the scattering parameter. Fig. 11-21. The shift in the position of the knee with the scattering 243 parameter and the associated temperature error. The correction in Eq. 11-23 (solid line) and the linear correction in Eq. 11-24 (broken line) are shown.  xvi  Acknowledgments I would like to thank my supervisor and friend Tom Tiedje for his intellectual and financial support during the course of this work. I would also like to thank my friends and colleagues with whom I worked and played over the past four and a half years; namely, Christian Lavoie, Jim Mackenzie, Tom Pinnington, Eric Nodwell, Dave Yarker, Manoj Kanskar, Mark Nissen, Sayuri Ritchie, Steve Patitsas, Stefan Eisebitt, Tony van Buuren, John Smith, Richard Morin, Duncan Rogers, Mario Beaudoin, Robin Coope, Bernard Haveman, Anders Ballestad, Ahmad Kassai, Yuan Gao, and Anton De Vries. I would also like to thank my PhD committee, namely Professors, John Eldridge, Phil Gregory, Tom Tiedje, and Jeff Young, for the time and effort they put in to this project. To my wife and children Heather, Bobbi Jo, and Ryan I would like to give a very special thanks for their emotional support during the past eight years I have spent in University.  1  Chapter 1 Introduction As  critical  dimensions  i n semiconductor  devices  have  become  progressively smaller and device designs progressively more sophisticated, the requirements for control over the fabrication processes have become more stringent. For example, semiconductor lasers for telecommunications w i l l in the near future contain quantum well or multiple quantum well structures in which the allowed tolerances on the individual layer thicknesses w i l l be in the range of a few atomic layers. A t the same time the composition of these layers w i l l need to be controlled to better than 1% to obtain the desired emission wavelengths. Ideally the optimum properties of the layers need to be fixed over the entire surface of a 75 m m (3 inch) diameter wafer in order to achieve a high yield of good devices. Molecular beam epitaxy [1,2] is a thermal beam deposition technique, carried out in ultra high vacuum, where the epilayer growth is controlled at the monolayer level. With this technique, compound semiconductor material with purity levels better than 10"  8  are achievable. The source material is thermally  evaporated from effusion cells with fast action shutters in front of the cell.  A  typical growth rate is 1 |im/hr, which means that growth of a semiconductor laser requires about one day.  Chapter 1  Introduction  2  The fundamental requirements for high quality device fabrication using molecular-beam epitaxy (MBE) are: fanatical cleanliness, precise flux control of source material, and precise substrate temperature control. In M B E cleanliness is required in three areas: ultra-pure source material, ultra-high vacuum (UHV) growth chamber, and substrate surfaces with contamination levels below a few hundredths of a monolayer. In solid source M B E the flux or deposition rate of a given element is controlled through the use of shutters and the temperature of the source material. Pure source material is commercially available. By using effusion cells constructed with U H V compatible refractory materials, which are cleaned and baked, one can maintain source purity during deposition.  The U H V growth  chamber is constructed with stainless steel and contains cryo-shrouds which are filled with liquid nitrogen during growth.  The growth chamber is baked at  200 °C for several days to obtain background pressures below 5-10"  11  mbar  when the cryo-shroud is cooled and the effusion cells are at their standby temperatures.  Maintaining an ultra clean M B E system is necessary for the  growth of high quality epilayers with good electronic properties and surface morphology. Another source of epilayer contamination and poor quality growth is dirty substrates. Wafer manufacturers are responding to the high demand for clean substrates in M B E by introducing "epi-ready" substrates. In general epi-ready is defined as a substrate with a surface particle count that is less than some maximum allowable value. Although epi-ready wafers in general are cleaner than other wafers, the term usually refers to particles, and contaminants such as partial  Chapter 1  Introduction  3  monolayers of carbon or silica are not known and can vary from wafer to wafer. Furthermore, the packaging material can contaminate the wafer.  This  contamination tends to increase with the amount of time the wafer spends in the container. To insure minimal contamination, wafer handling procedures are kept to a minimum. This is achieved in part by using holders that are compatible with one touch mounting. Wafer mounting is discussed in Chapter 6. During the M B E growth of III-V materials, such as AlGaAs, growth rates and composition are controlled by the group III flux while the growth mode, step flow (2D) growth or island (3D) growth, is controlled by substrate temperature and group V over pressure.  Measuring and controlling substrate temperature  during M B E deposition is particularly challenging as a non-contact technique is required. Recently there has been considerable interest in the use of optical bandgap thermometry for measuring substrate temperature.  In optical bandgap  thermometry, the temperature of the substrate is inferred from the absorption edge of the substrate material. The wavelength of the absorption edge is a measure of the bandgap of the substrate material; the bandgap of semiconductors typically shrinks with temperature. One of the most versatile implementations of this type of temperature measurement technique is diffuse reflectance spectroscopy [3]. The physical properties of direct gap semiconductors, such as GaAs, that relate to optical bandgap thermometry are discussed in Chapters 2 and 3. In these materials, the optical absorption edge is exponential in energy. It is the behavior of the absorption edge that determines temperature sensitivity and reproducibility in optical bandgap thermometry. The origin of exponential tails in the densities of states of the bands and broadening of these tails with structural  Chapter 1  Introduction  4  and thermal disorder is discussed in Chapter 3. Partial filling and broadening of these tails in intentionally doped semiconductors is also discussed. The relationship between the temperature dependent parameters of the substrate material, the physical properties of the substrate, and the onset of transparency of the substrate are discussed in Chapter 4. In particular, the effect of light scattering and trapping in substrates with rough back surfaces is modeled. Temperature sensitivity and accuracy, as well as calibration and range, are also discussed in Chapters 4 and 5. Analysis techniques for relating the spectrum to temperature and the sensitivity of these techniques to thin film interference are discussed in Chapter 5 and Appendix II. To maintain uniformity in the properties of the epilayer during M B E , it is essential that the temperature be uniform across the substrate, particularly in the growth of alloys in which one of the components preferentially re-evaporates. For example, in the growth of AlGaAs/GaAs graded index laser structures with InGaAs quantum wells; the sticking coefficients of Ga and In depend critically on temperature above 600 °C and 500 °C respectively. The electrical properties of AlGaAs cladding layers are optimal for growth temperatures above 670 °C, and InGaAs quantum wells are grown below 550 °C to insure In incorporation. This means the substrate temperature must be rapidly decreased from 680 °C to 520 °C during growth of the graded AlGaAs and the GaAs spacer layers and stabilized both temporally and spatially at 520 + 2 °C during growth of the quantum well in order to achieve emission wavelengths that are reproducible and uniform across the substrate.  Chapter 1  Introduction  5  The spatial uniformity of temperature in radiatively heated GaAs substrates is discussed in Chapter 6. The effects of substrate doping, back surface textures, thermal contact to holder, and a pyrolytic boron nitride diffuser plate, on the temperature uniformity is determined.  Substrate holder design for radiatively  heated substrates is also discussed in Chapter 6. Finally in Chapter 7, diffuse reflectance spectroscopy is used as a temperature monitoring tool during the growth of AlGaAs/GaAs graded index separate confinement heterostructure (GRINSCH) laser structures with InGaAs quantum wells. In this work substrate heater power recipes are developed that give reproducible temporal-profiles of substrate temperature during the growth of these laser structures. These heater power recipes allow the substrate temperature to be rapidly decreased from 680 °C to 520 °C during the growth of the first graded index AlGaAs layer, stabilized at 520 °C during the growth of the InGaAs quantum well, and rapidly increased from 520 °C to 680 °C during the growth of the second graded index AlGaAs layer.  6  Chapter 2 Urbach Edge §2*1  Absorption Edge of Semiconductors The optical absorption in a wide variety of semiconductors is found to  increase exponentially with photon energy hv in the region just below the bandgap energy  [4-7].  The optical absorption edge is an important quantity in  the optical properties of semiconductors, in part because it controls the shape of the optical emission spectrum according to the principle of detailed balance. It is also a manifestation of the effect of structural and thermal disorder on the electronic properties of semiconductors  [5,6].  In crystalline semiconductors and  insulators the characteristic slope of the exponential part of the absorption edge, commonly referred to as the Urbach edge [8], is found to be proportional to kT for temperatures above the Debye temperature [4], and is believed to be due to band tailing associated with disorder produced by thermal fluctuations in the crystal lattice  [5-7].  The absorption cross-section (dipole matrix element) in semiconductors is found to be a slowly varying function of photon energy [7].  Therefore the  exponential absorption edge originates in the optical joint density of states. Parabolic band theory predicts a sharp cutoff in the optical joint density of states  Chapter 2  Urbach Edge  7  at the bandgap energy [9], with  A ^hv-E  for hv>E  0  for hv<E  r  p(hv) = where A  b  is a constant.  b  g  g  g  (2-1)  ,  The bandgap energy, E , is the separation of the  conduction band minimum, E , and the valence band maximum, E : c  v  E = E -E . g  c  (2-2)  v  The optical joint density of states predicted by parabolic band theory and the experimentally observed Urbach tail are shown schematically in Fig. 2 - 1 .  10  20  i—I—i—r  T  T  > B  io  19  i  o a  •4—>  < •— >  o  1018 tParabolic Band Theory  -I—1  <u  Q +-> a •  i-H  1017 t-  -3 10 o O  Experimentally Observed  16  • r—I •4—>  OH  O  1015  _i  •100  0  50 hv-E  e  100  i  i  i  i  150  (meV)  F i g . 2-1. B e h a v i o r o f the optical j o i n t density o f states at the b a n d edge.  i_  200  Chapter 2  Urbach Edge  8  The absorption coefficient in the region of the bandgap for the direct bandgap semiconductor GaAs (n = 1 0  cm- ) is shown in Fig. 2-2. At long  17  3  c  wavelengths the absorption is dominated by three types of free carrier absorption [10]: the interaction of electrons with longitudinal optical phonons, the electron scattering by acoustic phonons, and the interaction of free electrons with impurities. Free carrier absorption increases with both carrier concentration and temperature.  1.1  1.2  1.3  1.4  1.5  Photon Energy (eV) F i g . 2-2. A b s o r p t i o n i n n-type G a A s (n = 1 0 c m " ) at r o o m temperature i n the r e g i o n o f the bandgap. T h e b a n d structure o f G a A s is also s h o w n . 1 7  3  c  The flat absorption region below the absorption edge is dominated by indirect inter-conduction band transitions [10].  In this process a photon is  Chapter 2  Urbach Edge  9  absorbed by an electron at the conduction band minimum, T , which makes a 6  phonon assisted transition to the higher X energy state in the conduction band. 6  A schematic of the band structure of GaAs showing this transition is given in the insert of Fig. 2-2.  Inter-conduction band absorption increases with carrier  concentration and temperature [10]. The exponential part of the absorption edge originates from tails in the conduction and valence band densities of states. This exponential behavior rolls off to the parabolic band density of states at the bandgap energy.  Since the  parabolic band approximation is only valid for electrons with small crystal momentum, the square root energy dependence of the absorption coefficient is not observed above the bandgap. The peak in the absorption coefficient just below the bandgap is caused by free exciton absorption. A free exciton can be regarded as an electron-hole pair bound together by the Coulomb interaction, creating a hydrogen-like particle. In GaAs the free exciton binding energy is about 4 meV [11] which places the absorption peak 4 meV below the bandgap. For temperatures above 300 K the lifetime broadening of the free exciton exceeds the binding energy, and the exciton peak is not observed in the absorption spectrum. §2-2  Bandgap of GaAs  The bandgap is difficult to identify in continuously varying absorption data. Pankove [12] therefore defines an "optical energy gap" for GaAs as the photon energy where the absorption coefficient is 8000 cm- . This definition is 1  based on the work of Sturge [13], who measured the absorption coefficient of  Chapter 2  Urbach Edge  10  GaAs in the bandgap region at low temperatures, where an accurate position of the bandgap is given by the free exciton peak.  For measurements of the  absorption edge below the bandgap it is convenient to define the "optical bandgap" as the energy at which the extrapolated absorption coefficient is 8000 cm- : 1  a(hv) = a exp  f hv-E,  (2-3)  where a(hv) is the absorption coefficient, hv is the photon energy, E is the a  inverse slope of the exponential absorption edge, E is the optical bandgap g  energy, and a  g  = 8000 cirr  1  is the bandgap absorption coefficient.  The  characteristic energy of the Urbach edge, E , is referred to as the Urbach a  parameter [14]. The bandgap of GaAs as a function of temperature is shown in Fig. 2-3. There are three distinct features in the bandgap: the bandgap energy at zero Kelvin, the temperature where the curve bends over, and the constant negative slope at high temperatures.  This means that a three parameter curve fit is  sufficient to describe its temperature dependence. In Fig. 2-3, Panish et al. [15] determine the temperature dependence of the bandgap of GaAs by measuring the shift in the absorption edge at an absorption coefficient of 40 c n r , over the temperature range 24 to 700 °C. This is an 1  accurate method for determining the temperature dependence of the bandgap, provided the slope of the absorption edge is independent of temperature. For example, the bandgap absorption coefficient is a factor of 200 larger than 40 cmand the natural logarithm of 200 is roughly 5; this means the measurement occurs  1  Chapter 2  Urbach Edge  11  five Urbach parameters below the bandgap energy. For each 1 meV change in the Urbach parameter with temperature, the absorption edge shift at 40 c n r over 1  estimates the bandgap shift by about 5 meV. 1  1  r  * - Equilibrium Position  1.50 h t  >  1.40  Conductiori^^^ \^band Valence j> band ^ ^ - - ^ ^  - 2p  - 2s  !-l  a i.3o  Atomic Separation  too C  a  Panish Thurmond O'Donnell Lautenschlager Shen  1.20  1.10 _i  0  200  400  600  i  i_  800  1000  Temperature (K) F i g . 2-3. P u b l i s h e d v a l u e s o f the b a n d g a p o f G a A s . T h e r e l a t i o n s h i p between the bandgap and lattice constant is also illustrated.  Panish et al. related their absorption edge measurements to the bandgap energy, by adding the offset between the room temperature (297 K ) position of the absorption edge at 40 cm- and the room temperature (294 K ) bandgap value 1  (1.435 eV) given by Sturge [12]. Panish et al. fit their (297 to 973 K ) bandgap data and Sturge's (21 to 294 K) bandgap data to the Varshni equation [16]: ^2  EAT) = E {p)-a K  T+b  (2-4)  Chapter 2  Urbach Edge  12  where E (0) is the energy gap at 0 K , b is approximately the 0 K Debye g  temperature, and a is the slope of E (T) at high temperatures. Thurmond [17] adjusted the bandgap values obtained by Panish et al. to coincide with the more recent room temperature value E  p  = 1.424 eV [18].  Thurmond set E {0) to 1.519 eV [11] and fit the remaining two parameters of g  Eq. 2-4 to the adjusted data. At high temperatures Thurmond's bandgap curve is 11 meV lower in energy compared to the bandgap curve of Panish et al. Cody [7] noted that an equation based on the phonon occupation number gives a good fit to the bandgap as a function of temperature for semiconductors. This form of E (T) had previously been overlooked by others such as Thurmond g  who used the semi-empirical Varshni equation. In the present work the following bandgap equation, based on the phonon occupation, is used:  where S is a dimensionless coupling constant, [exp((ha))/kT)-  i]  g  Einstein occupation number, and (frco) [exp((fta))/&T)-7]  1  1  is the Bose-  is the average  thermal energy of the phonon population. This average is based on the Einstein single oscillator model for the phonons. E (0) is the zero temperature bandgap. At high temperatures the bandgap goes asymptotically to E (T) = E -S kT g  where E  b  b  g  ; E =±S (ha>)Y  reflects the bond energy.  b  g  ,  (2-6)  The kink or knee in the temperature  dependence of the bandgap can be identified by the intersection of the asymptote (Eq. 2-6) with E (0). This intersection occurs at kT = (hco)/2 which  Chapter 2  Urbach Edge  13  means that the temperature for the onset of bandgap shrinkage from thermally excited phonons is around one half of the Einstein temperature. Shrinkage of the bandgap in semiconductors with temperature is caused by the temperature dependent electron-phonon interaction [19-21] and by thermal expansion [7]. Bandgap shrinkage due to expansion of the crystal lattice is depicted in the inset of Fig. 2-3 where the formation of the conduction and valence bands from the 2s and 2p energy levels of isolated atoms is shown. The electron-phonon contribution to the temperature dependence of the bandgap depends on the phonon population and hence the Bose-Einstein occupation number. Cody [7] showed that the temperature dependence of the shift in the bandgap, from the intrinsic electron-phonon coupling at constant volume and the lattice expansion at constant pressure depend on the same phonon occupation number. This justifies the use of Eq. 2-5 to describe the temperature dependence of  the  bandgap  in  S (ho)) = K = K + K g  v  p  semiconductors.  his  analysis,  Cody  used  where K is the intrinsic contribution to the shift in the v  bandgap at constant volume and K calculated K  In  arises from the thermal expansion. Cody  using the Griineisen model for thermal expansion.  This model  relates the volume change due to thermal expansion to the total thermal energy. Several years later O'Donnell et al. [22] reinvented Cody's bandgap equation, again, showing that the intrinsic and thermal expansion contribution to the bandgap depend on the Bose-Einstein occupation number. They advocate an equation based on the phonon occupation number as a replacement for the Varshni equation and fit the phonon occupation number to the bandgap data of  Chapter 2  Urbach Edge  14  Panish et al. ignoring the adjustments of Thurmond. Small differences between the curves of O'Donnell et al. and Panish et al. are observable in Fig. 2-3. The final two curves in Fig. 2-3 were obtained from reflectivity measurements. Lautenschlager et al. [23] infer the bandgap from the peak in the second derivative of the real part of the dielectric function. Shen et al. [24] infer the bandgap from the peak in the first derivative of the reflectivity.  The  measurements of Lautenschlager et al. cover a temperature range from 10 to 520 K and are close to Thurmond's curve. The 77 to 900 K measurements of Shen et al. are lower than Thurmond's curve at low temperatures and higher at high temperatures. §2*3  The Urbach Parameter The exponential part of the absorption edge in crystalline semiconductors  is believed to be due to fluctuations in the electronic energy bands caused by lattice vibrations which give rise to states below the band edge. In the simplest model for the temperature dependence, the Urbach parameter, E , is proportional 0  to the phonon population; which in the Einstein single oscillator model [4,5] is U T ) - - E A 0 ) ^  &  M  {  ^  k  T  )  _  (2-7)  r  where (hco) is an energy which describes the average phonon frequency for the lattice and S is a dimensionless coupling constant which describes the slope of 0  E as a function of temperature for T —»°°. In this expression, E (0) = S (hco)/2 a  o  0  when the zero-point motion of the lattice is included on an equal basis with the thermally excited lattice vibrations. A more sophisticated theoretical treatment [5]  Chapter 2  Urbach Edge  15  predicts the same temperature dependence of the Urbach slope as in Eq. 2-7. In disordered semiconductors it is convenient to introduce a dimensionless parameter X which describes the contribution of "frozen-in" structural disorder [25] to the width of the Urbach edge, where  E {0) = S (ha>)Zy-  .  o  o  (2-8)  Analogous to the bandgap, at high temperatures the Urbach parameter goes asymptotically to  E (T) = E S kT 0  x+  ; E = - S {hco)X .  0  x  2  (2-9)  Q  Since the temperature dependence of the bandgap and the Urbach parameter have the same functional form, the bandgap can be written in terms of the Urbach parameter, E , where 0  E {T) = E -GE {T) g  f  .  0  (2-10)  Ej is the temperature independent Urbach focus and G (Cody's constant) is a proportionality constant that relates the temperature dependence of the bandgap to the temperature dependence of the Urbach parameter [14].  In terms of the  previously defined parameters, G=^  and E = E {0) + GE {0) = S {hco)^^ f  g  o  g  = E + GE b  x  .  (2-11)  When X is zero, Ej = E , where E is the extrapolated low temperature bandgap, b  b  and the focus energy reflects the bond energy. By substituting Eq. 2-10 into Eq. 2-3, the temperature dependence of the absorption edge is characterized by the parameter, E (T): a  Chapter 2  Urbach Edge  a(hv) -  exp  hv-EA E (T) )  16  ; a = a exp(G) , f  g  (2-12)  0  where « y i s the extrapolated absorption coefficient at the Urbach focus energy. §2*4  Diffuse Reflectance Measurement Extensive measurements have been made on the optical absorption edge  of GaAs at room temperature and below for material with different conductivity types and doping levels [13,26,27].  However, the best existing data on the  absorption edge is not good enough to resolve the temperature dependence of the characteristic width of the exponential tail in the absorption of GaAs above room temperature [15]. In the experiments described below, the exponential part of the absorption edge, in semi-insulating and n-type GaAs from 50 to 700 °C and from 50 to 650 °C respectively, is determined from the diffuse reflectance spectrum. Data is obtained on the optical bandgap, the Urbach parameter, the Urbach focus, Cody's constant, and the temperature dependence of the residual absorption below the band edge, in both semi-insulating and n-type GaAs. The absorption measurements are carried out on GaAs wafers in the ultrahigh vacuum (UHV) growth chamber of a Vacuum Generators V80H molecularbeam epitaxy (MBE) machine using diffuse reflectance spectroscopy (DRS) [2830].  Diffuse reflectance spectroscopy is an optical method for measuring the  temperature of a substrate material with a temperature dependent bandgap. A schematic of the DRS technique is shown in Fig. 2-4. In the DRS technique, the broad spectrum light from the W-halogen lamp is chopped (with a mechanical chopper) and focused (using a lens) through the  Chapter 2  Urbach Edge  17  optical mirror port onto the semiconductor substrate heated from the back by a Ta foil heater. The spectrum of the light from the lamp is sufficiently broad that it covers the spectral range above and below the bandgap of the substrate. Photons with energies less than the bandgap are transmitted through the substrate and reflected from the back surface of the substrate, while photons with energies larger than the bandgap are absorbed by the substrate.  Lens  U H V Chamber Window  w-  Halogen Lamp  MirrorsV A'i Diffuse Ray  _JMono-_ chromator  Cooled InGaAs Detector  Chopper Computer  Lock-in Amplifier F i g . 2 4 . S c h e m a t i c o f the diffuse reflectance s p e c t r o s c o p y setup u s e d to measure the temperature dependence o f the absorption spectrum o f G a A s .  The purpose of the mirror port is to ensure that the optical window is not in the line of sight of the substrate. If the substrate is in the line of sight, some of the deposited material may re-evaporate onto the window and eventually spoil its transmission. The optical throughput is less sensitive to coating on the mirror  Chapter 2  Urbach Edge  18  than on the window. A n alternative commercially available solution that may be effective is a heated window. A long pass filter can be placed at the output of the lamp to eliminate the short wavelength light not close to the bandgap of the semiconductor reducing the heat load of the lamp on the substrate and eliminating short wavelength light which could scatter somewhere inside the growth reactor and find its way into the detector as second order in the monochromator. A long pass filter is usually not used because, second order detection is eliminated by using a InP capped InGaAs detector which has a responsivity over a single order wavelength range, and for substrate temperatures above 50 °C the heat load from the full spectrum of the lamp is negligible. Furthermore, the short wavelength visible light scattered from the polished front surface of the substrate is sensitive to surface morphology, which may also be of interest. The semiconductor substrate is polished on the front and polished or textured on the back. The light from the back of the substrate is detected in a nonspecular location; avoiding the specular reflection from the front of the substrate. The pyrolytic boron nitride (PBN) diffuser plate behind the substrate provides a surface with a reflectivity that generates a strong diffuse signal; this is crucial when substrates with polished backs are used. The P B N diffuser plate is placed immediately behind the substrate, attached to the same refractory metal holder as the substrate. The P B N diffuser plate also conducts heat laterally to minimize thermal gradients in the substrate.  Chapter 2  Urbach Edge  19  Substrates with suitable back surface textures for strong diffuse scattering are saw-cut, lapped, or lightly sand blasted with a pencil type sandblaster. In the case of GaAs, etching the back surface with concentrated nitric acid also works. In practice the morphology of the back surface of the GaAs wafer changes during the time that it spends at the growth temperature because of surface diffusion and evaporation. For example when growth is carried out at 700 °C, the typical growth temperature for AlGaAs, there is some decomposition of the back surface of the substrate as As is lost and small Ga droplets form. Changes in the back surface morphology can affect its optical scattering properties which in turn may affect the diffuse reflectance spectrum. For this reason it is important that the method of analysis of the diffuse reflectance spectrum be as insensitive as possible to the scattering characteristics of the back surface of the substrate. In the DRS technique a fraction Rf of the light incident on the front surface of the wafer is specularly reflected, where Rf is the reflectivity of the front surface. For GaAs, at the wavelength of the onset of substrate transparency, Rf = 0.31. The remaining fraction of the light, 1-Rf,  is transmitted through the  front surface into the substrate. In the infrared region of the spectrum (1 to 5 |im), a semi-insulating GaAs substrate at room temperature is transparent. In this case the transmitted beam will propagate to the back surface of the substrate where it is diffusely reflected. Diffuse reflections commonly occur from surfaces such as paper that are microscopically rough. Some of the diffusely reflected light will be outside the critical angle for total internal reflection from the inside of the front surface and will be trapped inside the semiconductor until a subsequent  Chapter 2  Urbach Edge  20  scattering event at the back surface scatters it back into the escape cone and it can escape out the front of the wafer. The scattered light exiting from the front surface will be spread out over a broad range of angles. In particular, some of the scattered light will be scattered into the solid angle of the nonspecular collection port where it is reflected by a 45° mirror onto a lens and focused into an optical fiber bundle and transmitted to the monochromator. The cross section of the optical fiber bundle is round with a diameter of 3 mm at the input end and rectangular with a width of 1 mm at the exit end in order to maximize the coupling with the entrance slit of the monochromator.  The monochromator is a scanning monochromator with a  spectral resolution around 6 nm and is followed by a liquid nitrogen cooled InGaAs photodiode detector. For short wavelengths in the range above the bandgap of the substrate, which for GaAs includes the visible part of the spectrum, the light from the lamp that is transmitted through the front surface of the wafer is absorbed before it reaches the back of the substrate. Thus only the long wavelength light that is transmitted through the substrate is diffusely scattered into the detection system. The spectrum of the diffusely back scattered light is obtained by scanning the wavelength selective detection system through the wavelength region in the vicinity of the substrate's bandgap.  The monochromator is controlled by a  computer which analyzes the data according to the algorithm described below in order to determine the temperature. The detector contains an internal amplifier with two gain settings; high (10  10  Q feedback resistor), and low (10 Q, feedback resistor). The amplified 9  Chapter 2  Urbach Edge  21  signal from the detector is fed to a Stanford Research Systems SR530 lock-in amplifier whose reference signal comes from the mechanical chopper. The output of the lock-in amplifier and the wavelength setting of the monochromator are recorded by a Macintosh Centris 650 computer. The data acquisition software provides a graphical user interface developed on National Instruments Lab View work bench [31]. In these absorption measurements, the non-specularly reflected light from the wafer is imaged onto an adjustable aperture that blocks all the light except the light from the 1 cm diameter central part of the wafer.  This procedure  minimizes background light scattered from the wafer holder and reduces complications associated with spatial variations in the wafer temperature. The temperature of the wafer is uniform to about ± 1 °C over the central 1 cm diameter area [30]. The wafer is polished on both sides so that scattering from the front and back surfaces of the wafer can be ignored. Both the absorption spectrum and the temperature of the wafer are obtained from the diffuse reflectance spectrum. The wavelength of the onset of transparency of the wafer is given by the absorption edge or bandgap of GaAs which shifts towards longer wavelengths at higher temperatures. The "knee" of the DRS spectrum is a measure of the onset of transparency of the wafer and is related to the temperature of the wafer through a calibration curve.  The  calibration curves for semi-insulating GaAs and Si-doped (n+) GaAs are given in references [28] and [29]. The GaAs wafer and PBN diffuser plate are mounted on a Mo wafer holder which sits in front of a Ta foil heater. The wafer is radiatively heated by the  Chapter 2  Urbach Edge  22  heater foils and by the P B N diffuser plate which is partially transparent to the heater radiation. The heater is supplied with a constant input power and the measurements are performed between 0 and 80% power at 5% intervals. Constant heater power maintains the wafer temperature stable to ± 1 °C in the steady state [30].  Lock-in detection is used to discriminate against heater  radiation that is transmitted through the wafer. At high temperatures, the heater radiation is about an order of magnitude larger than the diffuse reflectance signal. To obtain the optical absorption spectrum the diffuse reflectance signal must be normalized to the optical throughput of the system.  The optical  throughput is measured by removing the wafer and recording the diffuse reflectance spectrum of the P B N diffuser plate at 0, 20, 40, and 60% heater power settings.  Measurements at different temperatures are necessary because the  diffuse reflectivity of P B N is slightly temperature dependent.  The signal  measured with the sample in place is normalized to the optical throughput measured at the same heater power. For temperatures that fall between the calibrated heater powers a linear interpolation between the normalization curves for the two nearest calibrated points is used. The diffuse reflectance (for a single angle 23° to the normal) from the P B N diffuser plate at different heater power settings is shown in Fig. 2-5. The absolute diffuse reflectance of the P B N is determined by comparison to Spectralon [32] a standard diffuse reflector. The wafers used in this study were grown by the vertical gradient freeze technique and supplied by American Xtal Technology [33]. The unintentionallydoped (semi-insulating) GaAs wafer is 117 ± 3 |im thick, 5 cm in diameter, and (100) oriented. The Si-doped (n-type) GaAs sample is 79 ± 3 Lim thick, 5 cm in  Chapter 2  Urbach Edge  23  diameter, and (100) oriented, with a 2-10 cm- carrier concentration. Thin wafers 18  3  are used in order to extend the measurements to the highest possible value of the absorption coefficient. 0.70  i— — — —'—r Heater Power 0% -20% 40% 60% 1  0.65  1  1  O  u  0.60  0.55  0.50 900  1000  1100  1200  1300  Wavelength (nm) F i g . 2-5.  §2*5  Diffuse reflectance o f the p y r o l y t i c b o r o n nitride diffuser plate.  Determination of Absorption Coefficient  The absorption coefficient is determined from the diffuse reflectance spectra as follows.  Of the light incident on the wafer, the fraction T is w  transmitted through the wafer and is diffusely scattered at the surface of the PBN diffuser plate. Of this light the fraction R is scattered into the solid angle of the s  Chapter 2  Urbach Edge  detector and of this fraction the fraction T  24  is transmitted back through the  w  wafer. Hence the leading term of the signal is, F(X)R T S  ,  2 W  (2-13)  where F(X) is the optical throughput of the system at the wavelength, X. In this experiment, the angle of incidence and the angle of detection are both 23° to the wafer normal; the plane of detection is perpendicular to the plane of incidence. Since the index of refraction of GaAs is about 3.5 in the infrared wavelength region of these measurements, the path of the light inside the wafer is about 6° to normal. The increase in path length relative to normal propagation is less than the uncertainty in the thickness of wafer. Therefore the path length of the ray passing through the wafer is assumed equal to the thickness, d. In this case the transmittance of the wafer is, T =(l-R) exp(-ad)[l  + R exp(-2ad)  2  + R exp(-4ccd)+---]  2  w  4  .  (2-14)  The power series on the right hand side of Eq. 2-14 is the contribution to the transmission from the multiple reflections. Summing the power series; (1- R) exp{-ad) l-R zxp(-2ad)  (7 - R) expjad) exp(2ad)-R  2  T  2  = w  =  2  ( 2  .  1 5 )  2  Here R is the reflectivity at the GaAs-vacuum interface, which is assumed to be equal to the normal incidence reflectivity. R = 0.31 in the wavelength region of these measurements. Multiple reflections also occur between the back of the wafer and the P B N diffuser plate, adding the following power series to the diffuse reflectance signal, [l  +  R  p  R  w  +  R Rl ...] 2  +  .  (2-16)  Chapter 2  Urbach Edge  25  Here R is the average reflectivity of the P B N which is assumed to be p  independent of temperature and equal to 0.59 [34]. R^ is the reflectance of the semi-transparent GaAs wafer, which has the same series of multiple internal reflections and hence the same denominator as the transmittance f  \2 / o ^ . j ^ (l-R) exp(-2ad) 2  {  K = R  1-R  2  (7  exp(-2ad)  —  T-  R)  2  (2-17)  exp(2ad)-R'  The reflections between the back of the GaAs wafer and the P B N occur at all angles and polarizations. Nevertheless, for simplicity the average reflectivity is assumed to be the same as the normal incidence reflectivity and the path length through the wafer is assumed to equal the wafer thickness. From Eqs. 2-13 and 2-16 the diffuse reflectance signal is: DRS = r  (2-18) i-RpK  The signal recorded with the wafer removed is, DRS = F(X)R 0  (2-19)  .  S  The normalized signal is obtained by dividing Eq. 2-18 with Eq. 2-19: DRS  (2-20)  r  y=-  DRS  l-R K  0  p  Let x be the denominator of T and R . Then, w  w  {1-R) (x + R ) 4  2  [l-R R)x -R R{l-R) x 2  p  2  with x = exp(2ocd)- R  p  Eq. 2-21 is quadratic in x and is written in the form,  z  (2-21)  Chapter 2  Urbach Edge  26  ax -bx-c = 0 where a = {l-R R)y p  (2-22)  , b = (1 -R) [{1 -R) 2  + (R R)y]  2  p  ,mdc  =  R (l-R) 2  4  The positive root of Eq. 2-22 is (2-23)  x = —(b + ^b + 4ac) , 2a \ I 2  and absorption coefficient in terms of the normalized signal is, (2-24)  a = — \n(x + R ) . 2d ' 2  K  Temperature (°C) i— — — — —r  0.4  1  1  1  1  0.3 o  q c3  o CD  0.2 h  cu  s5 0.1 h  0.0  900  1000  1100 Wavelength (nm)  1200  1300  F i g . 2-6. N o r m a l i z e d diffuse reflectance spectra for s e m i - i n s u l a t i n g G a A s . T h e temperature o f e a c h c u r v e i s o n the r i g h t - h a n d side w h e r e the l o w e s t temperature corresponds to the upper-most c u r v e .  Chapter 2  Urbach Edge  27  The normalized diffuse reflectance spectrum for semi-insulating GaAs and n+ GaAs are shown in Figs. 2-6 and 2-7. The temperature of each curve is shown in the column at the right-hand side of each plot. The lowest temperature at the top of the column corresponds to the highest curve, with the higher temperatures corresponding to the lower curves. The "knee" in the spectra, where the diffuse reflectivity starts to rise, shifts to longer wavelengths as the temperature increases. Temperature r  0.4  0.3 h ii O fi  cd  o ii  -4—>  <u  &  0.2 h  ii  0.1 h  0.0  900  1000  1100 Wavelength (nm)  1200  1300  F i g . 2-7. N o r m a l i z e d diffuse reflectance spectra for S i - d o p e d G a A s . T h e t e m p e r a t u r e o f e a c h c u r v e is o n the r i g h t - h a n d s i d e w h e r e the l o w e s t temperature corresponds to the upper-most curve.  The reduction in the diffuse reflectance signal at long wavelengths and high temperatures, is caused by an increase in the residual subedge absorption at high  Chapter 2  Urbach Edge  28  temperatures. In the short wavelength region where the wafer is opaque, there is a small background signal which comes from residual diffuse scattering from the polished front surface of the wafer and from stray chopped light scattered elsewhere in the growth chamber that impinges onto the detector.  This  background is subtracted from the data to obtain the transmitted portion of the spectrum defined in Eq. 2-20. 0.4  I—i— — — — —i— — — — —i— — — — —i— — — — —r 1  900  1  1  1  1  1  1  1000  1  1  1  1100  1  1  1200  1  1  1  1  1300  Wavelength (nm) F i g . 2-8. N o r m a l i z e d diffuse reflectance spectra f o r u n d o p e d (semiinsulating) G a A s and S i - d o p e d (n = 2 * 1 0 c n r ) G a A s . T h e temperatures for the absorption curves are as f o l l o w s : (a) 58 ° C and 5 2 ° C , (b) 258 ° C and 258 ° C , (c) 4 3 5 ° C and 4 2 6 ° C , and (d) 631 ° C and 6 3 2 ° C for the u n d o p e d and S i - d o p e d G a A s respectively. 18  3  c  The normalized diffuse reflectance spectra for semi-insulating GaAs and Sidoped GaAs are compared in Fig. 2-8. Four curves are shown for each material in  Chapter 2  Urbach Edge  29  the temperature range between 50 and 650 °C. The temperatures for the semiinsulating and Si-doped GaAs are respectively: (a) 58 °C and 52 °C, (b) 258 °C and 258 °C, (c) 435 °C and 426 °C, and (d) 631 °C and 632 °C. The shift in the spectra with temperature, from the shrinkage of the optical bandgap, is larger for Si-doped GaAs. Also the large number of electrons in the conduction band of Sidoped GaAs reduces the long wavelength signal.  — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — f :  (d)  1.0  (c)  (b)  1.1 1.2 Photon Energy (eV)  (ai  1.3  1.4  Fig. 2-9. The absorption coefficient as a function of photon energy for undoped (semi-insulating) GaAs and Si-doped (n = 2-10 cm: ) GaAs. The temperatures are as follows: (a) 58 °C and 52 °C, (b) 258 °C and 258 °C, (c) 435 °C and 426 °C, and (d) 631 °C and 632 °C for undoped and Si-doped GaAs respectively. 18  3  c  The absorption coefficient as a function of photon energy is shown in Fig. 2-9 for semi-insulating and Si-doped GaAs.  The temperatures of the  Chapter 2  Urbach Edge  30  absorption curves are: (a) 58 °C and 52 °C, (b) 258 °C and 258 °C, (c) 435 °C and 426 °C, and (d) 631 °C and 632 °C for semi-insulating and Si-doped GaAs respectively. These absorption curves are calculated from the spectra in Fig. 2-8 using Eqs. 2-22, 2-23, and 2-24. Both GaAs samples show the exponential Urbach edge behavior along with a weakly energy dependent absorption at lower energies. The low energy subedge absorption in the n+ material is due to phonon-assisted inter-conduction band transitions [10]. In the semi-insulating material the residual absorption below the absorption edge is due to deep levels associated with defects or impurities except at the highest temperature where absorption by free electrons and holes begins to be important. §2*6  Determination of Optical Bandgap and Urbach Parameter  In order to find E , the characteristic energy for the exponential part of the 0  absorption edge, the slowly varying low energy absorption must be subtracted from the data. This is accomplished by fitting a straight line to the absorption below the exponential edge and subtracting the extrapolated value from the absorption in the higher energy Urbach region. The resulting absorption curves are shown in Figs. 2-10 and 2-11, for undoped and Si-doped GaAs.  The  temperatures are listed at the right-hand side, where the lowest temperature is for the curve at the right; as the temperature increases the curves shift to the left. For undoped GaAs, the removal of the residual low energy absorption leaves an absorption edge with an exponential behavior over two orders of magnitude. For Si-doped GaAs, the removal of the low energy absorption leaves an absorption edge that is exponential for absorption coefficients above 6 c m . -1  Chapter 2  Urbach Edge  1000  8 o  • 1—1  "T  I  I  I  31  Temperature (°C) i r "H 58 75 99 141 183 217 258 289 331 364 398 435 474 510 550 575 631 683  I  1  ioo  <D O  U  0  1 o  io Ir  < 1.0  1.1 1.2 Photon Energy (eV)  1.3  1.4  F i g . 2-10. T h e absorption edge for s e m i - i n s u l a t i n g G a A s , after the r e m o v a l o f the l o w energy absorption. T h e temperatures are g i v e n at the r i g h t - h a n d side, the l o w e s t temperature is for the right-most c u r v e . A s the temperature increases the curves shift to the left.  The exponential portion of the absorption edge in the 10-100 cm" range, is 1  fit to Eq. 2-3. From this fit the optical bandgap energy, E , and the Urbach g  parameter, E , are obtained. Absorption data below 10 cm- is not included in the 1  0  fit because values below 10 cm are more sensitive to the approximations used in 4  the model, and to uncertainties associated with the subtraction of the low energy absorption. Similarly, absorption values above 100 cm- are also omitted from the 1  fit as they involve low signal levels in the diffuse reflectance spectrum and are therefore sensitive to noise and to the accuracy with which the background signal is subtracted.  Chapter 2  Urbach Edge  32  Temperature (°C)  1.0  1.1  1.2  1.3  1.4  Photon Energy (eV) F i g . 2 1 1 . T h e absorption edge w i t h the l o w energy inter-band a b s o r p t i o n r e m o v e d , for S i - d o p e d (n = 2 - 1 0 c n r ) G a A s . T h e temperatures are g i v e n at the r i g h t - h a n d side, where the l o w e s t temperature is for the rightmost curve. A s the temperature increases the curves shift to the left. 1 8  3  c  The diffuse reflectance spectrum, the absorption curve, and the absorption curve with the low energy background removed are shown in Fig. 2-12 for semiinsulating GaAs at 398 °C. The solid circles, superimposed over the absorption less background curve, show the section of data fit to Eq. 2-3. From this fit, the Urbach parameter is 10.38 ± 0.01 meV and the optical bandgap is 1.23796 ± 0.00005 eV. The uncertainties are given by the standard error [35]. The optical bandgap energy as function of the Urbach parameter is plotted in Fig. 2-13, for undoped GaAs (solid circles) and Si-doped GaAs (open circles). The linear relationship predicted by Eq. 2-10 is observed. The solid lines are least  Chapter 2  Urbach Edge  33  squares fits of Eq. 2-10 to the data. The fitting parameters, namely the Urbach focus energy and Cody's constant, are given in Table 2-1. The Urbach focus absorption coefficient defined in Eq. 2-12 is also given in Table 2-1. 0.4  1—I—I—i—I—i—i—i—r -1—i—i—|—i—i—i—i—r~  1000  -i—i—I—i—r  Diffuse Reflectance Spectrum  >  0.3  cr Vi  CJ  ioo oU O  Q cd  13  o o  5H  & 0-2  CD  Ms  o  CB  0.1  10  s 3.  0.0  1.14  1.16  1.15  1.17 1.18 1.19 Photon Energy (eV)  1 1.21  1.20  F i g . 2-12. T h e diffuse reflectance spectrum a n d the a b s o r p t i o n c o e f f i c i e n t for s e m i - i n s u l a t i n g G a A s at 398 ° C . T h e s o l i d c i r c l e s s u p e r i m p o s e d o v e r the a b s o r p t i o n c u r v e w i t h b a c k g r o u n d subtracted are fit to d e t e r m i n e the o p t i c a l bandgap a n d the U r b a c h parameter. T h e rectangular b o x s h o w s the section o f the diffuse reflectance spectrum used i n the fit.  T a b l e 2-1. C o d y ' s constant a n d the U r b a c h focus for u n d o p e d ( s e m i - i n s u l a t i n g ) G a A s and S i - d o p e d (n = 2 - 1 0 c  18  cnr ) GaAs. 3  Sample  G  E (eV)  a (cm-)  Undoped GaAs  67.6 ±2.1  1.928 ±0.021  1.8-10  Si-doped GaAs  110.0 ±3.5  2.866 ± 0.050  5.0-10  f  1  f  33  51  Chapter 2  Urbach Edge  -i  1  1  1  1  r  1  34  1  '  db  1  '  1  1  r  h  1.4  o\8  > 1  3  t-l  CO  c W O H  *S  Undoped Si Doped  cd  «  o o  l.i  h _l  1  1_  _l  I  1_  _l  I  1_  I I I  10 12 Urbach Parameter (meV)  J  .  i_  14  16  F i g . 2-13. T h e o p t i c a l b a n d g a p as a function o f the U r b a c h parameter, f o r undoped and Si-doped G a A s .  The optical bandgap and Urbach parameter, as functions of temperature are shown in Figs. 2-14 and 2-15.  The optical bandgap as a function of  temperature for semi-insulating GaAs is fit with Eq. 2-5 using the known value for the zero temperature bandgap (1.519 eV) [11], and taking S and (hco) as g  adjustable parameters. The resulting best fit value of 22.7 meV for (hco) gives an Einstein temperature of 9 = 263 K. This is in good agreement with the expected E  value of 270 K = 0.75 0 , where 0 is the Debye temperature of GaAs. This D  D  value for (hco) is then used to fit the temperature dependence of the bandgap of the n+ GaAs. In this case the zero temperature bandgap and the dimensionless  Chapter 2  Urbach Edge  35  constant S are used as fitting parameters. The fits are indicated by the solid lines g  in Fig. 2-14 and the values of the fitting parameters are summarized in Table 2-2. 1.6  -|  1  r  i  i  r  Undoped Si Doped Sell et al.  1.0  0  200  800  400 600 Temperature (K)  1000  F i g . 2-14. T h e temperature dependence o f the o p t i c a l bandgap for u n d o p e d ( s e m i - i n s u l a t i n g ) G a A s a n d S i - d o p e d (n = 2 * 1 0 c m " ) G a A s . T h e s o l i d l i n e s are fits to a m o d e l w h e r e the p h o n o n p o p u l a t i o n is represented b y a single oscillator w i t h an E i n s t e i n temperature o f 263 K . 18  3  c  The optical bandgap for n+ GaAs is larger than the optical bandgap for semi-insulating GaAs due to partial filling of the bottom of the conduction band by free electrons (Moss-Burstein shift) [36]. From the relationship between the Urbach slope parameter and the bandgap in Eq. 2-10, one would expect an opposite contribution to the bandgap from structural disorder associated with the donor impurities. The bandgap of the n-i- material would be expected to be  Chapter 2  Urbach Edge  36  smaller since the Urbach slope parameter is larger. The fact that the optical bandgap is observed to increase with doping means that the band filling effect is larger than the structural disorder effect. The difference in the S values for n+ p  o  and semi-insulating GaAs in Table 2-2 may be due in part to the additional temperature dependence of the bandgap of the n-i- material associated with the temperature dependence of the Moss-Burstein shift. T a b l e 2-2. P a r a m e t e r v a l u e s f o r the o p t i c a l b a n d g a p a n d U r b a c h parameter i n u n d o p e d a n d S i - d o p e d G a A s .  Parameter  Undoped GaAs  Si-doped GaAs  £ (0)(eV)  1.519  1.556 + 0.001  Y  23.41  22.42 ± 0.04  E (eV)  1.587 + 0.027  1.629 ± 0.004  V  5.982 + 0.021  6.414 ± 0.013  (hco) (meV)  22.66 ± 0.38  22.66  262.9 ± 4.4  262.9  £ (0)(meV)  6.133 + 0.117  11.977 ±0.078  X  5.25 + 0.30  17.74 ± 0.69  E (meV)  5.15 + 0.34  11.34 + 0.56  S  0.0865 ± 0.0025  0.0564 ± 0.0017  g  b  o  x  0  The temperature dependence of the Urbach parameter for semi-insulating and n+ GaAs is fit to Eqs. 2-8 and 2-9, where (hco) is fixed at the value determined from the fit to the bandgap and S and X are adjustable parameters. 0  The fits are indicated by the solid lines in Fig. 2-15 and the best-fit parameter  Chapter 2  Urbach Edge  37  values are listed in Table 2-2. S for GaAs, other III-V semiconductors, and more 0  ionic materials are compared and discussed in § 3 - 2 .  Even though the semi-  insulating GaAs sample is good quality single crystal material, a non-zero X or "structural disorder" term is required to fit the temperature dependence of the Urbach slope.  X for GaAs and InP are compared in § 3 - 2 .  The effect of the  electric fields of ionized impurities and the strain fields of point defects, on the width of the band tails is also discussed in § 3 - 2 . T  0  I  I  j  200  I  I  1  [  400  I  I  I  |  I  600  I  I  '  |  I  800  1000  Temperature (K) F i g . 2-15. T h e temperature dependence o f the U r b a c h slope parameter for u n d o p e d ( s e m i - i n s u l a t i n g ) G a A s a n d S i - d o p e d (n = 2 * 1 0 c n r ) G a A s . T h e s o l i d l i n e s are fits to a m o d e l w h e r e the p h o n o n p o p u l a t i o n i s represented b y a single oscillator w i t h an E i n s t e i n temperature o f 263 K . 1 8  c  3  Chapter 2  Urbach Edge  1  L_J  0  1  I  1  I  1  I  I  1  I  1  1  I  100  1  l_l  l  1  I  1  I  1  l  1  I  200  I  1  I  1  I  1  1  I  l _ l I  1  1  I  1  I  1  I  I  I  1  1  I  1  I  1  I  I  I  1  I  300 400 500 Temperature (°C)  1  1  I  1  I  I  I  I  I  I  600  L_J  700  F i g . 2-16. T h e bandgap for semi-insulating G a A s is c o m p a r e d T h u r m o n d ' s results.  F i g . 2-17. T h e U r b a c h parameter for u n d o p e d (semi-insulating) G a A s .  with  Chapter 2  Urbach Edge  39  The bandgap of for undoped GaAs is compared to Thurmond's published curve [17], in Fig. 2-16.  In the present work the shift and broadening of the  absorption edge are both considered in determining the optical bandgap; in previous work [17], only the shift in the absorption edge was considered. The agreement between the curves in Fig. 2-16 is better than expected; this may be due in part to uncertainties in temperature. In Fig. 2-17, the Urbach parameter for undoped GaAs is compared with values extracted from absorption curves published by Panish and Casey [15]. Panish and Casey concluded that within the uncertainties of the measurement, the slope of the absorption edge did not exhibit a temperature dependence.  §2-7  Urbach Focus for GaAs The significance of the Urbach focus is most clearly illustrated in Figs. 2-18  and 2-19 where the focus is shown together with the exponential absorption curves. The temperatures for the absorption curves, which shift to the left with increasing temperature, are shown on the left-hand side of each plot. The focus lines are fits to Eq. 2-12 with E  f  and G fixed at the values given in Table 2-1.  E (T) is the only adjustable parameter. 0  The existence of a focus is a mathematical consequence of the fact that the Urbach slope and the bandgap both depend in the same way on the thermal occupation of the phonon modes. Although the existence of a focus is one of the standard attributes of an Urbach edge [4], in the present case the focus may be due to nothing more profound than the fact that above the Debye temperature both the Urbach slope energy and the bandgap are nearly linear functions of  Chapter 2  Urbach Edge  40  temperature. The large absorption coefficient at the focus reflects the fact that the slope of the Urbach edge is weakly dependent on temperature compared with the bandgap, and apparently has no other physical significance.  1.0  1.2  1.4  1.6  1.8  2.0  Photon Energy (eV) F i g . 2 - 1 8 . T h e U r b a c h focus ( 1 . 8 - 1 0 c m , 1 . 9 3 e V ) a n d the e x p o n e n t i a l a b s o r p t i o n spectra f o r u n d o p e d ( s e m i - i n s u l a t i n g ) G a A s . T h e a b s o r p t i o n c u r v e s shift to the left w i t h i n c r e a s i n g temperature. T h e temperatures for this series o f curves are s h o w n o n the left-hand side. 33  - 1  When measuring substrate temperature using techniques that employ an optical measurement of the bandgap [28-30,37], the temperature dependence of the measurement sensitivity can be related to the position of the Urbach focus. For example, a large absorption coefficient at the focus, and hence a large value for G, means that the temperature sensitivity (the ratio of the width of the Urbach  Chapter 2  Urbach Edge  41  edge to the temperature dependence of the bandgap), is weakly dependent on temperature. Also a focus energy close to E , and hence a small value for X, b  means that the sensitivity for temperature measurements is good at low temperatures. Therefore materials best suited for inferring substrate temperature from optical bandgap measurements are ones that have both a large absorption coefficient at the focus and an Urbach focus energy that is close to the zero temperature bandgap. 10 ^  34  I  10  28  i  e  io  22  c  1  1  1  1  1  1  1  1  | 1  1  1  1  1  |  1  1  1  1  -  - Temperature = (°Q =  82 174 265 348 426 495 567  = = I -  A A A A ^ A A A  /  A  /  /  A S ' A A A ~  A A ' ^ A  ~  A ^ A S ^ A A / ^ / A ^  —  yA^y^A^A^  -  A  1 1 1 1 1 1 1 11 1  •i-H o g 10  1  16  u c  10  111 1  OO  <  10  1  • | 10 o  4  io-  /  \A  ^  1.0  A  A  A  Y  1.2  \A \  i  1  i  i  i  1.4 1.6 Photon Energy (eV)  2.0  1.8  F i g . 2-19. E x p o n e n t i a l a b s o r p t i o n spectra for S i - d o p e d (n = 2 * 1 0 c m ) G a A s . T h e a b s o r p t i o n c u r v e s shift to the left w i t h i n c r e a s i n g temperature. T h e temperatures for this series o f c u r v e s are s h o w n o n the left-hand side. T h e U r b a c h focus is o f f scale at ( 5 . O 1 0 c m " , 2.87 e V ) . 1 8  c  5 1  1  - 3  Chapter 2  §2*8  Urbach Edge  42  Residual Absorption Below the Urbach Edge in GaAs  The temperature dependence of the residual absorption below the band edge, at 0.92 eV (1350 nm), for semi-insulating and n+ GaAs is shown in Fig. 2-20. This absorption increases linearly with temperature up to 500 °C, and then rapidly increases for temperatures above 500 °C. The solid lines in Fig. 2-20 are linear fits to the data up to 500 °C. From these fits the temperature coefficient is 0.0228 c m - / ^ for Si-doped (n = 2-10 cnr ) GaAs and 0.0066 c n r V T for 1  18  3  c  semi-insulating GaAs. As discussed in §2-1, the flat absorption region below the band edge in n-type GaAs is dominated by inter-conduction band absorption. Theoretical calculations predict a linear increase in this absorption with temperature.  For n = 7*10 c n r , the temperature coefficient given by these 17  3  c  calculations is 0.018 c n r ^ C [10]. In semi-insulating GaAs, as discussed in §2-6, for all but the highest temperatures the residual absorption below the band edge is dominated by deep levels associated with defects or impurities. The data in Fig 2-20 indicates that for lower temperatures (up to 530 °C) the residual absorption below the band edge is linear in temperature. This is a consequence of the linear decrease in the bandgap energy with temperature; in this case the density of accessible deep levels increases as the bandgap shrinks. In heavily doped GaAs, at room temperature, the interband absorption is much larger than the free carrier absorption. As well, at room temperature, the free carrier absorption due to scattering by impurities, acoustic phonons, and optical phonons contribute on an equal basis to the absorption. The absorption due to  Chapter 2  Urbach Edge  43  impurity scattering increases as the square of the impurity density, while the absorption due to scattering by phonons increases linearly with impurity concentration [10].  Since the temperature dependence of the free carrier  absorption attributed to acoustic phonons is about three times stronger than that of optical phonons and impurities [10], free carrier absorption due to scattering by acoustic phonons dominates at the highest temperatures.  0  100  200  300  400  500  600  700  Temperature (°C) F i g . 2-20. T e m p e r a t u r e dependence o f the r e s i d u a l a b s o r p t i o n b e l o w the b a n d edge o f u n d o p e d and S i - d o p e d G a A s , at 0.92 e V ( 1 3 5 0 n m ) .  At 600 °C the intrinsic carrier concentration in GaAs is 1 0  16  c n r [17], 3  which is roughly equal to the density of deep levels in semi-insulating GaAs. This  Chapter 2  Urbach Edge  44  means that free carrier absorption due to scattering by acoustic phonons is also important in semi-insulating GaAs above 600 °C. Furthermore, at higher temperatures As preferentially evaporates from the surface of the GaAs wafer causing microscopic Ga droplets to form. This effect becomes important at temperatures above the congruent sublimation point of GaAs, which is around 640 °C. The temperature of bare GaAs wafers is limited to temperatures less than 700 °C in a M B E chamber because the 10~ mbar As over 5  pressures attainable in these systems only partially compensates for As losses. At temperatures above 700 °C the loss of As is so large that the Ga droplets on the surface rapidly grow and in a short time the wafer becomes opaque because of strong absorption in the excess Ga layer. To measure absorption in GaAs at temperatures around 800 °C, without appreciable surface absorption through the formation of Ga droplets, a chamber containing As over pressures around 1 mbar is needed. The sharp increase in the subedge absorption at temperatures above 550 °C for both semi-insulating and n-type GaAs (in Fig. 2-20), is believed to be due to a combination of the scattering of free carriers by acoustic phonons and the onset of formation of Ga droplets on the surface of the wafer.  In these  measurements wafer temperatures did not exceed the temperature where the surface of the wafer is destroyed by the preferential evaporation of As. However, at the highest temperatures the surface of the wafer was likely covered by a few atomic layers of excess Ga. As the wafer cooled the incoming As flux reacts with the excess Ga and the lost GaAs layers are re-grown. The wafers used in this  Chapter 2  Urbach Edge  45  experiment had a clear polished metallic surface after heating, indicating that surface decomposition had not reached a point of no return. The wafers used in these measurements are thin (about 100 Lim thick) causing surface absorption to be interpreted as a large absorption coefficient. Since the subedge absorption is subtracted in determining the exponential part of the absorption curve, surface absorption will have little effect on the results pertaining to the exponential part of the absorption curves.  46  Chapter 3 Band Tails In this chapter, the origin of the exponential tails in the densities of states of the conduction and valence bands and the contribution of the width of these tails to the width of the absorption edge, is explored. In §3-1 the width of the absorption edge is shown to be roughly equal to the width of the broadest band tail. Narrowing of band tails due to partial filling of the tail states by carriers in doped material, is also investigated. In §3-2 broadening of the band edges from the Coulomb potentials of ionized impurities is calculated and compared to the measured values. This analysis is extended to encompass fluctuations in the band edge caused by the strain potentials of point defects in semi-insulating GaAs and InP; good agreement is obtained with the experimentally measured part of the width of the absorption edge attributed to structural disorder.  Finally,  broadening of the band edge by the strain potentials of lattice vibrations is used to explain the temperature dependent part of the width of the absorption edge. §3*1  Urbach Parameter and Band Tail States  The exponential absorption edge originates from exponential tails in the density of states of the conduction and valence bands. These tails extend into the bandgap allowing band to band transitions to occur at energies below the  Chapter 3  bandgap.  Band Tails  47  The Urbach tail density of states for the conduction band (subscript  V ) and the valence band (subscript "v") are written as:  E-E ^  r  Pc = c B  x  P  v  v  V  (3-1)  and p = B exp  c  e  J  °c  °v  V  J  where E is the electron energy, B , and B are constants, and o and o c  v  c  the widths of the Urbach tails. The constants B and B c  v  describe  and the energy  v  E,  where the tail states end and the parabolic band states start, are determined by assuming that the density of states is a smooth function. This means the density of states and its first derivative are continuous at the boundary of the two types of states. In this case for the conduction band  'E-E ^  A  c  V  1  2 ^, J and —  °c  E  E-E ^  f  4  1  cr, B exp V == —  C  (3-2)  c  o  °c  c  J  where A is a constant. Solving for E and B , c  c  E = E +±o e  e  and  B =A ^  (3-3)  and  * =AJ|;  (34)  C  C  Similarly for the valence band  E = E -±o v  v  V  The combined tail and continuum density of states for the conduction band is  4,|f  exp  'E-E ^ e  for  E<E +-oc c  c  c  2  (3-5)  Pc(E) =  K ^ E  C  Similarly for the valence band  for E>E c+-o c  2  c  c  .  Chapter 3  Band Tails  48  for V  P (E) = \  °v  £>£„--<7„  2  J  (3-6)  V  for  E<E --<J V  V  The conduction and valence band tail states and the parabolic band states are shown in Fig. 3-1. Also shown, is the absorption of a photon with energy hv by an electron with initial energy E in the valence band and final energy E in }  2  the conduction band, where E = Ej+hv.  There are three different absorption  2  processes involving the Urbach tail states: (1) E in the valence band continuum }  and E in the conduction band tail, (2) E in the valence band tail and E in the 2  2  • i 11  1  2  n—TI—i i i i 11  r  1  —i—i i i i i  w CD  c W  Parabolic Bands Bands with Tails  hv  o  t-l < •—>  o  <o W E..  >i < > i  10 17  10  18  i  i_  11111  10  10  19  20  1021  Density of States (cm" eV" ) 3  1  Fig. 3-1. Conduction band and valence band density of states for parabolic bands and bands with Urbach tails.  Chapter 3  Band Tails  conduction band tail, and ( 3 ) E  49  in the valence band tail and E  l  in the  2  conduction band continuum. The band tail states are localized with no well defined momentum which means the usual conservation of crystal momentum rule for band to band transitions is not relevant.  Therefore in determining the portion of the joint  density of states involving the band tails, all transitions at an energy hv are summed over.  Furthermore, by restricting hv to hv< E + (7 /2+ c r / 2 , the g  c  v  probability that an electron from a localized tail state ends up in a state with a large momentum, cuts off exponentially with the width of the tail. The joint density of states for the three absorption processes involving the tail states are as follows: (1) E in the valence band continuum and E in the }  2  conduction band tail, E -o /2+hv v  E -E ^  v  r  Pi(hv)=  JA ^E -E +hvA J^-ex v  v  2  c  2  c  dE?  V  V  °c  J  hv-E  n  = V^-7==tfc exp  (3-7)  2  V2e  V  °c  J  3/2 J/2  K  x exp -^-x  c =  a  V  °c  dx . J  (2) Ej in the valence band tail and E in the conduction band tail, 2  E +o /2 c  c  r  E dC  f  AJ| exp  j  ft(*v)=  E -o /2+hv 1 (cyr )3/2 v  c  + hv^  2  dE,  J  G  (3-8)  v  c  24e  V  E -E v  2  o -o v  exp  r _ hv  )  E  a/2  (hv-E -o /2^ K  - exp V  r  v  J  (3) Ej in the valence band tail and E in the conduction band continuum, 2  Chapter 3  Band Tails  50  E -E v  2e  E +a /2 c  dE,  C7„  c  a. o- exp ( 2e  AA C  + hv\  2  v  V  hv-EA  2  (3-9)  v  3/21/2  C  p e x p  dx  c  v  v  J  The joint density of states in the bandgap region is given by the sum of Eqs. 3-7, 3-8, and 3-9 for absorption involving the tail states, and by Eq. 2-1 for absorption between the parabolic bands:  Pj(hv) + p (hv) + p (hv) 2  p(hv) =  for  3  hv<E +la +l(T g  c  v  (3-10)  \3/2  42  mm m + m c  K n  K  Jhv-E,  v  c  V  vJ  for hv>E„ + -a 8  2 9  8  + -a  r  o2  c  v  v  The constant A (from Eq. 2-1) is rewritten in terms of the effective masses of the b  conduction and valence bands. The constant A joint  density  of  states  for  the  c  \ is determined by equating the  band tails  and  the  continuum at  hv = E +G /2 + G /2: c  42  AA C  v  V  nn  \3/2  mm m +m c  K  c  v  vJ  (a a exp(o- /2o- ) + a CT exp(cT /2cr )) ' 2  v  v  2  c  v  c  c  v  (3-11)  c  The joint density of states for parabolic bands and the three absorption processes involving the tail states are shown in Fig. 3-2. The total joint density of states (Eq. 3-10) is also shown. The total joint density of states makes a smooth transition from the exponential edge to the parabolic band region. The widths of the tails in Fig. 3-2, are cr, = 6.1 meV and o = 7.5 meV. The largest width, o , is v  v  Chapter 3  Band Tails  51  the room temperature value of the Urbach parameter for semi-insulating GaAs and the width of the conduction band tail, rj , is assumed to be 0.8 o . c  v  F i g . 3-2. T h e j o i n t d e n s i t y o f states f o r the d i f f e r e n t b a n d to b a n d absorption processes i n semi-insulating G a A s .  In the Fig 3-2, the total joint density of states has the same characteristic slope as the valence band tail. This is shown by the similarity in the slopes of tail to tail absorption below the bandgap, the valence band tail to conduction band continuum absorption, and the total joint density of states. This means that the Urbach parameter is given by the width of the largest of the two tails in the density of states. The fractional difference in the tail widths is defined as  Chapter 3  Band Tails  52  ; 0<\8 \<l .  (3-12)  o  o CT„ + CJ,  It can be shown, by substitution into Eq. 3-8, that for hv-E  g  less than  -2(<J + o )(i - 8^ the Urbach parameter is c  V  fj  =Uy^Uc  £  °  2  \  +  v  1  S  | )  =  » y t »  c  | ^ v - "  2  0 | y  C  (3-13)  |  2  When rj ^ (7 the Urbach parameter is given by the broadest band tail and when v  C  the widths of the band tails are equal E = G = CJ . 0  V  C  In heavily doped n-type material, the optical absorption is complicated by the occupation of a significant number of the conduction band tail states. The probability that a state in the conduction band with energy E is occupied, is given by the Fermi function exp[[E-/u.)/kT)  +1  where fl is the chemical potential and kT is the thermal energy. The chemical potential is function of carrier concentration and temperature and is not easily calculated when the semiconductor material does not obey Boltzmann statistics, as is the case in degenerate semiconductors. However, Joyce and Dixon [38] derived a series expansion for the chemical potential, where the first two terms accurately determine the chemical potential up to 4kT above the conduction band minimum,  E. C  Therefore in n-type material with  n <5N , C  C  the position of the  chemical potential relative to E , in units of kT, is c  kT  In  (3-15)  Chapter 3  Band Tails  53  where N is the conduction band effective density of states. The first term of c  E q . 3-15 is the Boltzmann approximation and the second term is the leading term of a n /N c  c  power series where oo  Aj = ^  (-rn^X  f  and g = m  \  i  si  { E ) d E  kT  (3-16)  J  Here p (E) is the conduction band density of states, g is the Laplace transform c  m  of the density of states, and N =gj.  For parabolic bands A = l/48  c  }  =  0.35355.  The density of states for the parabolic conduction band with effective mass, m ,is c  42 P (E) = C  m*  2  •^E-E  for E>E  c  (3-17)  C  The fraction of empty states in the conduction band tail at energy E , is 2  l~f(E ) 2  = exp = exp  kT  J  (V~E ) kT C  +1  kT  V  (3-18)  ( 2-E ) E  exp  c  {  kT  )  This approximation is valid for the tail states provided the chemical potential is well into the conduction band (fi-E  c  >3kT) and the band tails cut off sharply  compared to the Fermi function (a <kT and o~ < kT). c  v  The conduction and valence band densities of states and the unoccupied density of states in the conduction band are shown, in Fig. 3-3. Also shown is the absorption of a photon with energy hv; in this process, an electron in the valence band tail is excited to the unoccupied portion of the conduction band tail. The occupation of the conduction band tail modifies the absorption in two ways: ( 1 ) the photon energy threshold for optical absorption is shifted up [36], and (2) the  Chapter 3  Band Tails  54  unoccupied density of states in the conduction band tail is narrower than the band tail and has an additional kT dependence due to the temperature dependence of the band filling. i  i  i  i  i  1  i i i  i  i  i i i  i  i i I  i  i  i  i  -i  i i 11  1—I—if  I  i l l  a o  t-i  Unoccupied Occupied  < -—>  o  hv  10 16  10  17  10  10  18  19  10,20  Density of States (cm eV" ) 3  1  F i g . 3-3. C o n d u c t i o n b a n d a n d v a l e n c e b a n d d e n s i t y o f states for n-type material, s h o w i n g the o c c u p i e d and u n o c c u p i e d states.  In n-type material, the fraction of empty states in the conduction band is included in the calculations of the joint density of states. The joint density of states for n-type material for the three absorption processes involving the tail states (see Eq. 3-7, 3-8 and 3-9), are: (1) Ej in the valence band continuum and E  2  in the conduction band tail,  Chapter 3  Band Tails  (hv) = A^a^^o '  e x p f - ^ - ^ exp V kT J  3 2  Pl  2e  \3/2l/2  K  I* exp  f)  nc =  a  55  fhv-EA 'n  )  (3-19)  \  V  °n  dx • — = — + — o o kT J n  c  (2) Ej in the valence band tail and E in the conduction band tail, 2  P (hv) = A 2  c  x exp  A  2Ve  ^  ( exp  cr - CJ  n  v  hv-E-(J /2  V~E kT J C  f  c  -exp  (3-20)  hv-E  n  oJ2 'c  J  J  (3) Ej in the valence band tail and E in the conduction band continuum, 2  P 5 ( M = AAa c  7T f nv =  a  v  nv  J^a  /  2  exp  kT J  exp  hv-EA (3-21)  3/21/2  dx ;  exp V  °a  J  The reduced width of the conduction band tail through partial filling is given by cr„, which is defined in Eq. 3-19. Furthermore, the tail widths o and cr c  for doped material are larger than those for undoped material.  v  The above  calculations use the exponential approximation for the unoccupied states in the conduction band (see Eq. 3-18). This approximation is only valid if the valence band tail cuts off more sharply than the exponential in 1/kT increases. Therefore C7 must be less than 0.7'/kT which is the case for n+ GaAs. V  The joint density of states for absorption involving parabolic bands, where the conduction band is partially filled, is  Chapter 3  Band Tails  56  \3/2  4~2  mm 2*3 n h m +m c  v  ,  z  K  exp  \  kT )  c  exp  vJ  (hv-E ^ g  v m +m { c  v  kT  JJ  +1  ^  •  1 1 hv>E +-o +-o  (3-22)  n  g  c  v  Here the Fermi function describes the probability that a conduction band state is unoccupied.  The effective mass comes into the Fermi function through the  conservation of momentum for transitions between parabolic bands where the energy of the final state in the conduction band, E , is restricted to 2  E = E +-^—(hv-E ) 2  The constant A \ c  c  .  g  (3-23)  is determined by equating the joint density of states involving  the band tails and Eq. 3-22 at hv = E + <J /2+ <J /2. G  c  v  The joint density of states for n+ GaAs is shown in Fig. 34. The widths of the tails in the density of states are a = 10.5 meV, the room temperature value of c  the Urbach parameter for p+ GaAs [26], and <7 = 12.9 meV, the room temperature V  value of the Urbach parameter for n+ GaAs obtained in §2-6. For n+ GaAs the absorption edge is dominated by electron transitions from the valence band tail to parabolic conduction band. In comparison, the absorption for undoped GaAs (Fig. 3-2) is dominated by tail to tail transitions for energies well below the bandgap and by valence band tail to conduction band continuum transitions for energies closer to the bandgap. The calculated absorption edges for undoped and n+ GaAs at 300 K are compared in Fig. 3-5. The impurity shift in the optical bandgap is clearly shown. The width of the absorption edge is larger in n-type material primarily because of  Chapter 3  Band Tails  57  impurity broadening of the tail in the valence band density of states. In n-type material the conduction band states are partially occupied, causing the absorption edge to roll off more slowly at the bandgap. This means the optical bandgap is not as sharply defined in doped material. 10  19  > 1018  O  '<— Band to Band  oo  10" oo e-t—i  O  > » •<—>  10  16  t  +— Tail to Tail  C CD  Q  Valence Band to Conduction Band Tail  •4—I  C3  •i-H .io  io  14  Valence Band Tail to Conduction Band  O  i_  .J  hv-E.  50  -50 (meV)  F i g . 3-4. T h e o p t i c a l j o i n t density o f states for the different b a n d to b a n d a b s o r p t i o n processes i n S i - d o p e d (n+) G a A s .  Similar to the analysis for undoped material, the Urbach parameter for ntype material, E , is expressed in terms of the width of the valence band tail, o~, n  v  and the reduced width of the conduction band tail, o~„, (see Eq. 3-19): <7 + CT,  (3-24)  V  En =  k  ) ( ' + l*.l) =  0- + (T v  n  Chapter 3  where 8  n  Band Tails  58  is the fractional difference in the width of the valence band tail and the  reduced width of the conduction band tail.  E  1  1  1  ' I •  >  i  i  |  i  i  i  i  hv-E  |  g  i  I I  i  |  i  i  i  i  |  i  i  i  r  (meV)  Fig. 3-5. Optical bandgap shift in the joint density of states of n+ GaAs. In p-type material, the valence band is partially filled with holes. Therefore, in a manner similar to n-type material, the effective width of the valence band tail is reduced through the thermal distribution of the holes. In this case the reduced width of the valence band tail, o , p  = l/o  is given by l/o  p  v  +  1/k.T. Similar to the  analysis for n-type material, the Urbach parameter for p-type material is  E= p  r-[i+8 )  —  {  2  )\  =—  D  >  p  2  £• + — 2  - ; 8=—  p  p  ,  L  cr + a c  p  (3-25)  Chapter 3  Band Tails  59  where 8 is the fractional difference in the width of the conduction band tail and p  the reduced width of the valence band tail. The widths of the band tails in terms of their fractional differences are CT + <T,  (3-26) °c + °p )(  P)  1 + S  ;  °P  =  From Eqs. 3-24, 3-25, and 3-26 the Urbach parameters are written as l+8„ ^  1  +  = CJ,  8. n  n  1+ '  'T s.•  =  (3-27)  1+ = CT„  B  l-8„  a  1-S - ~ r  +  P  P  The introduction of dopant impurities to a pure crystalline material add disorder in the following ways: (1) the ionized impurities add electric fields to the material and (2) the mass of, and the size of, the dopant atom differ from that of the atom that normally sits on the host site. When the dopant atom is similar in size to the host atom, the structural disorder and hence the broadening of the band edges is dominated by the effects of the Coulomb potential of the dopant. In the following analysis, the possibility that absorption measurements in both n-type and p-type material probe the same band tail at high temperatures is explored. In this analysis the tail widths are assumed to depend on impurity concentration and not impurity type, that is c-,n-type  G  =  (I) l>8 >0 n  c\p-type  G  V  n  e  n  n  = c  and l>8 >0  E = <J and E n  w  p  p  -o . c  n  cT „_ , v ;  0  / ? e  and  = CT  v  There are three physical solutions to Eq. 3-27:  v  excluding both 8 -0 and 8 = 0, in which case n  (II) l>S >0 n  p  and -l<8 <0, p  in this case E = <7 and n  V  Chapter 3  Band Tails  60  E = <7 ; this solution requires that either o » P  p  T»S X6 /2. 0  (Ill) -l<S <0  E  and l>8 >0,  n  c  c  C  v  c  with  0  in which case E = o  p  E = G \ this solution requires that either o p  <J or o~ > o /(l-S )  v  n  » o  or o >  v  n  o /[l-S ) v  c  and with  0  T»S X0 /2. o  E  For GaAs at 300 K , E = 12.9 meV for n = 2-10 c n r (see §2-6), E = 18  N  10.5 meV for n = 2-10 v  18  3  c  p  cm- [26], and kT = 25.9 meV. From these values 3  heavily doped GaAs at 300 K is a case I material with 5 = 0.266 and S = 0.099; n  p  this means that optical absorption measurements in heavily doped GaAs probe the valence band tail in n-type material and the conduction band tail in p-type material. At low temperatures, all semiconductors are case I materials because the conduction band tail states are full in n-type material and the valence band tail states are full in p-type material. In case II, the optical absorption measurements probe the valence band tail in n-type material and the partially filled valence band tail in p-type material, with ljE = 1/E + 1/kT. In case III, the optical absorption measurements probe the p  n  conduction band tail in p-type material and the partially filled conduction band tail in n-type material, with 1/E = lJE + 1/kT. n  p  For some heavily doped  semiconductors it is possible that at high temperatures the width of the absorption edge of p-type (n-type) material changes from the width of the conduction (valence) band tail to the reduced width of the partially filled valence (conduction) band tail. As an example, the widths of the band tails in a hypothetical heavily doped semiconductor are shown in Fig. 3-6. In this example o~ = E +S kT with v  x  0  E = 12meV, S = 0.06, and o = 0.8 cr . In the n-type material the Urbach x  0  c  v  Chapter 3  Band Tails  61  parameter is given by the width of the valence band tail. In p-type material the Urbach parameter is given by the width of the conduction band tail below 7 3 5 °C and by the reduced width of the valence band tail above 7 3 5 °C. This semiconductor is a case I material below 7 3 5 °C and a case II material above 7 3 5 °C. Furthermore, this material has two Urbach foci: one for temperatures below 7 3 5 °C and one for temperatures above 7 3 5 °C. I  I ™ 'J'"  400  I  I  I  |  I  600  I  I  |  I  I  800  '  |  1000  '  '  I  |  1200  Temperature (°C) F i g . 3-6. T h e w i d t h o f the b a n d tails i n a ( h y p o t h e t i c a l ) h e a v i l y d o p e d semiconductor.  In this section the shift in the optical bandgap and narrowing of the band tail through partial filling of one of the bands in doped material is illustrated. Also explored is the possibility of complications in the temperature behavior of the  Chapter 3  Band Tails  62  width of the absorption edge caused by the temperature dependence of band filling in doped material. Understanding these basic features of band tails is crucial in understanding optical bandgap thermometry. §3-2  Width of Band Tails in III-V Materials  The tails in the densities of states originate from structural and thermal disorder in the material [5-7]. In amorphous material the structural disorder is large (correlation lengths around the atomic spacing) causing very broad band tails; for example amorphous Si has E ~ 50 meV [25]. In high quality crystalline 0  materials such as GaAs substrates used in epitaxial growth, structural disorder is small and the band tails are narrow. In crystalline material the two main types of structural disorder are the strain fields of point defects and the Coulomb potentials of ionized impurities. The effects of these two types of disorder on the band edges are shown in Fig. 3-7. A n impurity that is larger (smaller) than the atom that normally sits at the host site compressively (tensilely) strains the material; this causes the bandgap to increase (decrease) at the host site.  A  positive (negative) charged impurity causes the bands to be bent down (up) at the impurity site. Thermal disorder is caused by thermal vibrations of the crystal; the amplitude of these vibrations increases with temperature.  Thermal vibrations  cause fluctuations in the band edges similar to those caused by strain fields. For example, an electron sees the slow vibrational motion of the atoms as a series of snap shots of a deformed crystal. Fluctuations in the band edges produce localized states in the bandgap which decay in an exponential manner.  Chapter 3  Band Tails  63  +  O  Ev-  F i g . 3-7. F l u c t u a t i o n s i n the c o n d u c t i o n a n d v a l e n c e b a n d s c a u s e d b y C o u l o m b p o t e n t i a l s a n d s t r a i n f i e l d s . T h e i m p u r i t i e s or defects are represented b y : an o p e n c i r c l e for c o m p r e s s i v e strain, a s o l i d c i r c l e for tensile strain, a "+" s i g n for a p o s i t i v e charge, and a "-" s i g n for a negative charge.  In the following analysis the width of the band tails in crystalline material are discussed for: (1) randomly distributed Coulomb potentials of the ionized impurities in intentionally-doped material, (2) randomly distributed strain fields caused by point defects in semi-insulating material, and (3) randomly distributed strain fields caused by lattice vibrations. Following Eqs. 2-7 to 2-9 the widths of the band tails are described by:  (J (T) = a c  xc  + S {hco) • + 2 exp((hco)/kT)-l oc  ; ^xc=^s {hco)x oc  c  (3-28)  ^ {T) = <y + S (hco) 2 exp((hco)/kT)-l v  xv  ov  The first term in these equations, o  xc  or cr^, is the width of the band tail  attributed to structural disorder while the second term is attributed to lattice vibrations.  Chapter 3  Band Tails  <7  XC  64  and C T ^ are calculated for heavily doped material following the work  of Kane [39], Halperin and Lax [40], and Sritrakool et al. [6]. Halperin and Lax give the general form of the density of tail states as f  m\  E  p(E) oc exp  rpm oJ  \  with E<0  and l/2<m<2  ,  (3-29)  where ra depends on impurity concentration and screening length. For n-type GaAs, Halperin and Lax indicate that 0.8 < m < 1.2 when 1 0  14  < n < 10  21  c  This agrees with the results in Chapter 2 where n-type GaAs (n = 2*10 c  18  cnr . 3  cnr ) 3  exhibits the m = 1 Urbach behavior. For the ra = 1 case, Sritrakool et al. [6] give the width of the band tail as: ^•2  °  2  fc  cr  943E,Lc  xc  '  E  „ l  c  E  Lv '  ~  A  n  4 m  c  72 L  '  (3-30)  - - * i -  ~4mL  Lv  2  where E is the energy associated with localizing an electron or hole within a L  correlation length V2L and a  is the variance of a randomly distributed  2  potential. The variance for the screened Coulomb potential of ionized impurities in semiconductors is given by Kane [39]: (r\ = 27i7i C L for V(r) = - e x p  cr =n jv {r)dr 2  2  (3-31)  2  c  c  V(r) is the screened Coulomb potential with  C  =  _£! _ l f e =  a  n  d  L  =  Z  K  \  3 n  \l/3  cj  mme e  c  2  VU,  1/3  (3-32)  Chapter 3  Band Tails  65  where Ry = 13.6 eV, r = 5.29*10~ cm, and e is the dimensionless relative 9  r  0  dielectric constant.  m is the dimensionless effective mass of, and n is the c  c  concentration of electrons in the conduction band. For p-type material, m and c  n become m and n and the sign of C becomes positive. c  v  v  From Eqs. 3-30 through 3-32 the widths of the band tails due to the Coulomb potential in n-type material are:  e  xc x c  =R y  y  — \ ^ 27 \e m r  and c  =* ™  y  m  c  27  5 Z \e m r  .  (3-33)  c  For p-type material the quantities (or subscripts) attached to each band are interchanged.  The width of the band tails increase as the square root of the  carrier concentration. The effective mass of the valence band is usually larger than the effective mass of the conduction band. Therefore the valence band tail is usually broader than the conduction band tail and the tails in n-type material are usually broader than the tails in p-type material. In this work, the above analysis is extended to materials where the disorder due to the potentials of ionized impurities is small and the disorder is dominated by the strain fields of point defects. In this analysis the point defect is assumed to result in a radial point force that creates a strain field in the crystal lattice surrounding the defect.  The strain field for a radial point force in an elastic  material is S(r) = A/r, with A = ±F{1 + O)/2KY [41], where  F  is the magnitude of  the force, cr is Poisson's ratio, and Y is Young's modulus. This field has the same radial dependence as the Coulomb potential. In a material that has a moderate density of point defects, such as semi-insulating GaAs which has about 1 0  16  cnr  3  As anti site defects, the strain field is assumed to have the same functional form as  Chapter 3  Band Tails  66  the screened Coulomb potential, where L is a screening length and V2L is a correlation length that are related to the average volume occupied by a single point defect by  = 1/N, where  4KL J3 3  N  is the density of defects.  Since the magnitude of the point force exerted on the crystal lattice by a point defect is unknown, the leading coefficient of the strain field is determined by equating the average displacement over the strain field to the change in the volume 1/N, caused by the insertion of a point defect into the material: oo  \s{r)4nr dr 2  = 4nL AL 2  J  o  and S(r) = — expf - - ) . r V LJ  (3-34)  AL is the difference in the covalent radius of the atom that normally sits on the host site and the covalent radius of the impurity or defect at that site. This is valid for small values of A L . When the radius of an impurity is very different from the radii of the atoms of the material, the defect will likely be an interstitial. In which case AL will be some fraction of the difference between the atomic radii of the atoms in the material and the atomic radius of the impurity. The strain potential is given by the deformation potential, E , times the strain field: s  C ( 3 y/ V(r) = - e x p — ; C = E AL ; L = AKN) V LJ r  j  (3-35)  S  From Eqs. 3-30, 3-31, and 3-35 the widths of the band tails due to the disorder caused by strain fields are: m„ ElAL 3 V 3 Ryr  2  Q  m „ El,AL 3^3 R r  2  y  Q  Note that the concentration of point defects is not implicit in the width of the tails; the correlation length cubed is inversely proportional to the defect  Chapter 3  Band Tails  67  concentration, canceling the concentration term.  However, the defect  concentration is implied through the validity of Eq. 3-29; the concentration has to be such that ra ~ 1. An additional subscript has been added to the deformation potentials in Eq. 3-36 to distinguish between the deformation potentials of the conduction band and the valence band. The relevant deformation potential for this analysis is the energy shift in the r point of band with lattice constant, which is related to the shift in the bandgap with thermal expansion. The temperature dependence of the bandgap is dE^  (  dT  'V  \  dE„  \  J3 with  /p  (  dE \ g  V  /  P=A p  dT  (3-37)  where P is the coefficient of volume expansion and A is a constant that represents the fractional contribution of thermal expansion to the temperature dependence of the bandgap. At temperatures above the Debye temperature, the change in the bandgap energy with temperature and the coefficient of expansion are approximately constant. Therefore, at high temperatures the deformation of the bandgap with strain is (AV_\ V ,  The term -S k e  (  3A dE  \  dE  n  3AS k  p  3  S  {  p  r  )  g  - fi dT  (3-38)  P  is the slope of the bandgap as a function of temperature and  o  comes from Eq. 2-6. The relationship of the deformation potentials to the deformation of the bandgap are given by, E ~E = SC  SV  3AS k -Je  3AS k , E =-B —^e  sc  c  , ,wiE„  =  n  3AS± B —^v  (3-39)  Chapter 3  Band Tails  68  The constants B and B represent the fractional contributions of the conduction c  v  band and the valence band to the change in the bandgap: B  c  c  dF = ^ dE  ; B =-^v  v  g  dF  ; B + B=l -  dE  c  g  c  v  .  K  (340) }  The sign convention is chosen so that the quantities on the right-hand side of Eq. 3-39 are typically (but not always) positive; these quantities are assumed to be positive for the strain potentials shown in Fig. 3-7.  The sign of the  deformation potential also depends on how AL is defined. In this case AL is defined to be positive when the lattice is under tensile strain. The effects of thermal disorder or phonons is now considered. In the following analysis the amplitude of the lattice vibrations are treated as a strain deformation in the crystal. Consider the vibrations in a diatomic linear lattice, such as the one discussed by Kittel [42], where the lattice vibrations are given by u(x,t) = u cosqxcos cot .  (341)  0  The amplitude of the vibrations, u , is related to the total energy, (hco)N (T), by a  B  C u =2(hco)N (T),  (342)  2  k  0  B  where C is the force constant between the atoms and N (T) is the phonon k  B  occupation number. A convenient value for (hco) is the average of co(q = 0) for the optical and acoustical phonon branches; this gives a expression that is simpler and approximately equal to the average of both branches over all values of the wave vector q. In this case the average phonon energy is given by (hco) = h (co(0)) = h ^ 2  2  2  2  1  1  (343)  Chapter 3  Band Tails  69  where M and M are the two atomic masses. Solving Eqs. 342 and 343 for the }  2  amplitude _ _2h N {T) u = Mf {hco) 2  2  B  B  =  M  +M  I  1  Q  12  _2M  I- -  r  ~  f l 2  f  ]2  (344) Mj-M^  ( 3  M  1  1  Mj  M  2  2 M M K  1+  M  2J  r  2  is a measure of the asymmetry in the mass of the atoms in the diatomic chain. The average deformation of the lattice is obtained by averaging Eq. 341  over space and time:  L  1  AL = (u (x,t)) = ^u = 4 " 2Mf (hco)\ 2, exp((hco)/kT)-l 2  2  2  (345)  12  The term in the brackets on the right-hand side of Eq. 345 is the phonon occupation number. From Eq. 3-28, 3-36, and 345 the dimensionless electronphonon coupling constants for the conduction and valence are respectively g  =  :1  sc  m E c  343 Mf (hco)  oc  2  b o v  ]2  ~ 34~3Mf {n(0)  2  12  '  (346)  where M is in units of the electron mass. The disorder parameter, X, is now determined for a crystalline material with static fluctuation in its band edges due to the strain fields of point defects. In this case the disorder parameters are determined from Eqs. 2-9, 3-36, and 346 and shown in Eq. 347, where AL is defined in Eq. 3-34. The disorder parameters are the same for both bands and do not depend on the deformation potential. =  X c  Mf {hoj)AL S (hco) h 2 0 x v  X  2  =1  v  l2  2  ov  ^ Mf (hco)AL m Ry r  2  2  12  2  e  (347)  Chapter 3  Band Tails  70  John and Grein [5] calculated the Urbach tail widths of the densities of states. For crystalline material in the high temperature limit they give an analytical approximation to the parameter  S  as  a  S = 0.36S y o  where  ac  S  ac  is the  dimensionless electron-longitudinal-acoustic-phonon coupling constant and y is a measure of the nonadiabaticity of this electron-phonon interaction.  The  nonadiabaticity parameter, y, is the ratio of the speed of sound to the speed of an electron at the Brillouin zone boundary. Their equation for S reduces to 0  S = 0.18™ \ M{hco ) E  o  2  ,  (348)  0  where ra* is the effective mass of the hole (or electron), E is the deformation d  potential for the valence band (or conduction band), and hco is the Debye 0  energy. The right-hand side of Eq. 348 is typically larger for the valence band and hence S is determined by the parameters of the valence band [5]. Q  The expression given by John and Grein is similar to the one obtained here except there is no provision for asymmetry in the mass of the atoms in the diatomic unit cell; this is apparently because their analysis does not include optical phonons. The parameters related to the width of the absorption edge are discussed in Chapter 2 for GaAs and in reference [43] for recent work in InP. The widths of the Urbach edge for n-type and semi-insulating GaAs and InP are compared in Fig. 3-8. The carrier concentrations in the n-type material are 2-10  18  cm- for 3  GaAs and 6-10 cm" for InP. The relevant parameter values for these materials 18  3  are compared in Table 3-1. The most striking feature of this comparison is that heavily doped InP does not exhibit an Urbach focus: the width of the Urbach  Chapter 3  Band Tails  71  edge in this material appears to be dominated by the Coulomb potentials and consequently insensitive to the phonon occupation number. In both GaAs and InP, the temperature dependence of the width of the band edge is reduced in heavily doped material. This is likely a consequence of the screening of the thermal fluctuations in the band edge by electrons in the conduction band. 25  -|  1  i  1  1  1  i  r  • • •  > s R CD  20  •  L  15 h  a a S-l  cd  PL.  10 h  c_> cd  Undoped GaAs Si-doped GaAs Fe-doped InP S-doped InP  5 h-  0  -I  I  L.  0  _l  200  I  1_  _!  400 600 Temperature (K)  I  1_  800  1000  Fig. 3-8. Width of the Urbach edge in n-type and semi-insulating GaAs and n-type and semi-insulating InP. The more heavily doped InP shows a larger Moss-Burstein shift [36] and a stronger temperature dependence in this shift when compared to doped GaAs (see E (0) and S in Table 3-1). The temperature dependence of the width of the g  Urbach edge is stronger in semi-insulating InP than in semi-insulating GaAs (see  Chapter 3  Band Tails  72  S ). This along with a smaller S means that the focus absorption coefficient is a  g  much smaller in InP than in GaAs. A smaller S also means that the disorder 0  parameter X is smaller in InP even through E is about the same in both InP and x  GaAs. The Einstein temperature for both materials is about the same. T a b l e 3-1. U r b a c h edge parameter values for u n d o p e d (semi-insulating) a n d S i - d o p e d (n = 2 - 1 0 c n r ) G a A s a n d F e - d o p e d (semi-insulating) a n d S - d o p e d (n = 6 - 1 0 cm" ) I n P .  c  18  3  18  3  c  InP  GaAs Parameter  Undoped  Si-doped  Fe-doped  G  67.6 + 2.1  110.0 ±3.5  32.71±0.85  1.928 ±0.021 2.866 ± 0.050 1.639±0.010  E (eV) f  S-doped —  —  a,f (cm )  1.8-10  5.0-10  1.1-10  a (cm-)  8000  8000  6865  6865  E (0)(eV)  1.519  1.556±0.001  1.424  1.547±0.005  5.982±0.021  6.414±0.013  4.704±0.037  6.562±0.106  (hco) (meV)  22.66±0.38  22.66  23.17±0.78  23.17  e (K)  262.9±4.4  262.9  268.7±9.0  268.7  So  0.087±0.003  0.056±0.002  0.143±.003  -0.021±0.015  E (meV)  5.15±0.34  11.34±0.56  4.97±0.17  22.93±0.82  X  5.25±0.30  17.74±0.69  3.01±0.17  -93.7+61.5  -1  1  g  g  S  33  51  18  —  8  E  x  The dimensionless phonon coupling constants for the widths of the band tails are calculated using Eq. 346. The static part of the width of the band tails for semi-insulating material is given in Eq. 3-36 where the magnitude of the strain, AL, is assumed to equal the difference in the covalent radius (R) of the atom that  Chapter 3  Band Tails  73  normally sits on the lattice site and the covalent radius of the point defect. For semi-insulating GaAs, the dominant point defect (deep level) is an As atom on a Ga site [44], in which case AL = RQ - R . a  As  The covalent radii of the elements are  given in reference [45]. For semi-insulating InP the dominant point defect is probably an Fe-related complex [44]; AL is assumed to be the same as in GaAs. The experimentally measured parameters in Table 3-1 are compared to calculations in Table 3-2. The parameter values used to calculate the deformation potentials (Eq. 3-39) and their references are given in Table 3-3. T a b l e 3-2. C o m p a r i s o n o f measured and c a l c u l a t e d U r b a c h edge parameters i n u n d o p e d (semi-insulating) G a A s and F e - d o p e d (semi-insulating) I n P .  GaAs  InP  Parameter  Theory  E (eV)  -18.9  —  -11.7  —  E„ (eV)  7.7  —  8.2  —  M (amu)  72.32  —  72.90  —  fl2  0.999  —  0.669  —  0.064  —  0.044  —  0.087+0.003  0.122  0.143+0.003  sc  Experiment  Theory  Experiment  Sov  0.084  AL (cm)  0.6-10"  —  0.6-10"  —  (meV)  4.1  —  2.0  —  a„ (meV)  5.4  5.15+0.34  5.5  4.97+0.17  X  5.6  5.25±0.30  3.9  3.01±0.17  G  XC  9  9  Chapter 3  Band Tails  74  For comparison, John and Grein [5] (Eq. 348) give S = 0.053 for GaAs 0  and S = 0.270 for InP; the deformation potentials they use are 7.0 eV for GaAs 0  and 21.0 eV for InP. T a b l e 3-3. Parameters used i n c a l c u l a t i n g the deformation potentials for G a A s and I n P .  Parameter  GaAs  reference  InP  reference  A  0.31  [46]  0.23  [47]  P (K- )  1.8-10"  [48]  1.4-10"  [49]  B  0.71  [50]  0.59  [50]  B  0.29  [50]  0.41  [50]  m  0.063  [51]  0.079  [52]  m  0.50  hh [51]  0.45  This work  4.70  1  c  v  c  v  5  5.98  m  5  hh [52]  m  This work  The widths of the band tails in heavily doped GaAs and InP are compared in Table 3 4 . The widths are calculated using Eq. 3-33 and compared to the measured values. This analysis shows that the valence band tail is much broader than the conduction band tail due to the differences in the magnitude of the localization energy in each band. Because the effective mass of the electrons in the conduction band is small, the localization energy is large, and the effect of the Coulomb potentials on the conduction band edge is reduced. Doped material is also strained by the addition of dopant impurities. In the case of Si-doped GaAs and S-doped InP the dopants are smaller than the host site and will cause tensile strain which will increase (decrease) the depth of the  Chapter 3  Band Tails  75  potential well in the conduction band (valence band).  In heavily doped  (degenerate) semiconductors the screening length of the Coulomb potential is roughly equal to the separation of the dopant impurities; this means that L for the strain and the Coulomb potential are about the same.  Therefore, the relative  change in the depth of the Coulomb potential well due to the addition of a strain potential can be approximated by  c  _  ^Coulomb  +  _j  ^Strain  (3-49)  r ^L  £  E  s  2R r  CCoulomb  y  0  T a b l e 3-4. C o m p a r i s o n o f measured and c a l c u l a t e d b a n d tail parameters i n S i - d o p e d (n = 2 - 1 0 c m ) G a A s a n d S - d o p e d (n = 6 - 1 0 c m ) I n P . c  18  3  18  3  c  InP  GaAs Parameter  Theory  n (cnr )  2-10  3  c  e xc  o  xv  Theory 6-10  —  Experiment  18  —  13.1 [42]  —  12.4 [42]  —  (meV)  3.4  —  5.4  —  (meV)  26.6  sc  1.69  —  1.40  —  sv  0.72  —  0.72  —  —  10.6  —  r  a  18  Experiment  C  c  sc°xc  ( )  9.7  i^xv  ( )  13.8  C  meV  c  meV  11.34+0.56  11.34±0.56  22.93+0.82  30.4  22.93+0.82  15.7  The widths of the band tails in n-type GaAs and InP corrected for the effects of strain are given in the last two rows of Table 3-4, where AL = 0.4-10" cm for both GaAs and InP. 9  For InP AL = R -R . P  S  For GaAs  Chapter 3  Band Tails  A L = R {a -a )/a SI  GaAs  Si  76  where (a -a )/a  Si  GaAs  Si  is the relative difference in the  Si  lattice constants of GaAs and Si. This definition of A L is about four times smaller than RQQ - R  S I  and consequently more realistic.  In general the electron-phonon coupling constant S increases with a  ionicity and can be greater than one in ionic materials [4]. S as a function of 0  fractional ionicity is shown in Fig. 3-9, where the fractional ionicity of these materials is given in Kittel [42] and defined by Phillips [53] who relates ionicity to the relative ionic character of the total bonding energy.  J  0.0  0.2  I  I  1  I  0.4  I  I  I  I  I  0.6  I  I  1  I  L  0.8  Fractional Ionicity F i g . 3-9. E l e c t r o n - p h o n o n c o u p l i n g c o n s t a n t f o r the w i d t h o f the absorption edge as a function o f the i o n i c character o f the energy b o n d .  1.0  Chapter 3  Band Tails  77  The solid line in Fig. 3-9 is an exponential fit to the data that is given by ^ S = 0.0230 exp fi \0.228) f  (3-50)  0  where  is the fractional ionicity. The values of S in Fig. 3-9 are obtained from 0  this work and from Kurik [ 4 ] . The values of S  a  shown here are given by  absorption data that cover a large temperature range.  For all but ZnTe, the  temperature range includes temperatures well above room temperature.  To  accurately determine S the absorption data for several temperatures above room 0  temperature should be considered. The  standard model for coupling of conduction band electrons to  longitudinal-optical phonons is the Frohlich coupling constant [ 5 4 ] : m Ry  2m m co 2hcoLO h e  c  l  c  LO  hcoLO \ oo £  (3-51)  £ J r  For the coupling to valence band holes, m is changed to m . c  v  The Frohlich  coupling constant depends on the ionic polarization of the material, which is related to £ /£ r  oc  = [pwl^Tof  and  e. r  •>  tne  Using the Lyddane-Sachs-Teller relation [ 4 2 ] , Frohlich coupling constant for holes is written as hcoLO m -1 hcoLO yhQ) j  (3-52)  T0  The parameter S is compared to the Frohlich coupling constants for holes 0  in GaAs, InP, ZnTe, AgCl, and NaCI in Fig. 3-10. The parameter values in E q . 3-52 are obtained from references [42] and [ 4 4 ] . The solid line is a linear fit to the data that is constrained to go through the origin: S = 0.23a o  F  .  (3-53)  Chapter 3  Band Tails  i  i  i  i  |  T — i — i — i — | — i — i — r -  1  |  78  1  1  1  1  |  1  1  1  1  |  1  1  1  1  1.4  NaCl*/^  -  1.2  -  AgCl  to  0.4 h_ ZnTe  -InP 0.0  "^.GaAs 0  / . i , , , ,  i .  ,  ,  ,  i ,  .  . ,  3  Fig. 3-10. E l e c t r o n - p h o n o n c o u p l i n g c o n s t a n t f o r the w i d t h o f the absorption edge as a function o f the F r o h l i c h c o u p l i n g constant.  In this section, the temperature dependent part of the width of the Urbach edge in semiconductors is described by equations that depend on the deformation potential and the average phonon energy (see Eq. 3-46). The part of the width of the Urbach edge not described by the phonon occupation number is attributed to static fluctuations in the band edges. These fluctuations are caused by the strain potentials of point defects in semi-insulating material and by a combination of the strain and Coulomb potentials of ionized impurities in n-type material. The results presented here, are the first experimental observations and theoretical calculations (Eq. 3-36) of the part of the width of the Urbach edge not described by the phonon occupation number in undoped crystalline material.  79  Chapter 4 Diffuse Reflectance Spectroscopy §44  Non Contact Temperature Measurement of Substrates  Substrate temperature is an important parameter in epitaxial thin film deposition and semiconductor processing operations. Substrate temperature and its uniformity have a large effect on the quality and composition of the deposited layers. In molecular-beam epitaxy, the substrate is normally heated radiatively and rotated during the thin film growth operation. Physical contact between the substrate and a temperature sensor is neither practical nor desirable because the sensor itself would cause local perturbations in temperature or even contamination of the substrate. Even if the substrate is not rotating, and heating is accomplished by thermal contact with a temperature regulated support, the temperature of the substrate can deviate substantially from the temperature of the support because of thermal contact problems which frequently exist in vacuum processing environments. Therefore a non-contact method for measuring the temperature of the substrate is needed. The simplest non-contact temperature measurement technique is to place a thermocouple in the radiation cavity between the substrate and the substrate heater so that it is in radiative contact with the substrate. While this solution is  Chapter 4  Diffuse Reflectance Spectroscopy  80  simple and cheap its accuracy is not adequate. In fact in molecular-beam epitaxy (MBE) it is not uncommon to have temperature errors of 100 °C with this approach. These errors are difficult to calibrate for because the temperature offset between the thermocouple and the substrate drifts in a non reproducible way. For example, in Fig. 4-1, the temperature response of the thermocouple is initially slow, under estimating the temperature of the substrate because the large thermal mass surrounding the radiation cavity is still cool. The thermocouple then over shoots the substrate temperature by about 200 °C and slowly decreases to within 60 °C of substrate after 150 min.  20Q I 0  i  i  i  i  i  i i 30  i  i  i  i  i i 60  i  i  i  i  I i  90  i i  i  i  i i 120  i  i  i  i i—L  150  Time (min) F i g . 4-1. T h e r m o c o u p l e and substrate temperature response to a 10 to 5 0 % step i n the heater p o w e r . T h e p o w e r step occurs at zero minutes.  Chapter 4  Diffuse Reflectance Spectroscopy  81  Optical pyrometry is another method for measuring the temperature of an object without touching it.  However, pyrometry has serious deficiencies for  semiconductor processing applications.  A pyrometer works by detecting the  intensity of the thermal radiation that is emitted by any object that is not at absolute zero. The spectrum of the thermal emission depends on the product of the spectral dependence of the emittance of the object and the emission spectrum of a blackbody at that temperature.  For the temperature range of interest in  semiconductor processing, from about 0 to 1100 °C, the peak in the blackbody spectrum is in the infrared.  Furthermore, the emissivity of semiconductors is  normally low in the infrared because semiconductor wafers are typically transparent in the 1 to 5 Lim region. This means that the radiation which must be detected by the pyrometer is relatively weak which in turn limits the temperature range of the technique for semiconductors to temperatures above 500 °C for standard commercial pyrometers such as the instrument manufactured by IRCON. The transparency of semiconductors in the infrared also means care must be taken not to inadvertently measure the temperature of the heater behind the semiconductor substrate. Another complication with pyrometers has to do with losses in optical elements used to transport the substrate radiation to the detector. In semiconductor processing operations it is not uncommon for optical elements such as windows and mirrors to become coated during the process. This affects the intensity of the thermal radiation from the substrate that reaches the detector which causes temperature errors.  While the pyrometer can be useful for  semiconductor temperature measurements it is not the complete answer.  Chapter 4  Diffuse Reflectance Spectroscopy  82  It has been recognized for some time that the bandgap of a semiconductor is a reliable indicator of its temperature because the bandgap is typically a smooth, almost linear function of temperature, in the 0 to 1000 °C range. Various optical methods have been proposed for measuring the bandgap of the substrate. In the method of Hellman and Harris [55], the radiation from heater filaments behind the substrate is transmitted through the substrate and detected by a detector outside the process chamber. By measuring the spectrum of the transmitted light they are able to infer the bandgap and hence the temperature. This method suffers from the variability in the intensity of the heater radiation as a function of the temperature of the heater. For example at low temperatures the heater produces very little radiation which makes accurate temperature measurements difficult. To solve this problem, Kirillov and Powel [37] put a small lamp behind the substrate as an additional, brighter source of radiation.  This increases the  sensitivity of the measurement but introduces additional complications in the heater design. Because it is not practical to rapidly modulate the intensity of the light behind the substrate, this technique is not compatible with lock-in detection techniques which means that it is not possible to exclude background light from hot filaments or effusion ovens that may also be radiating in the same spectral range. In addition, with a fixed light source internal to the process chamber it is difficult to spatially resolve the temperature across the substrate. Temperature uniformity is a critical problem in growth of reproducible device structures with high yield. These problems were solved by Weilmeier et al. [28] who put the light source outside the process chamber and determined the bandgap from the  Chapter 4  Diffuse Reflectance Spectroscopy  83  spectrum of the back scattered light. In this method the light source is outside the process chamber so it does not interfere with the heater and is relatively easy to chop with a mechanical chopper.  This makes lock-in detection techniques  possible so that stray light from other sources can be rejected. To further enhance the sensitivity the back surface of the substrate is textured and the detector is placed in a nonspecular position. The important optical signal in measuring the bandgap is the signal which is transmitted through the substrate.  The diffuse  reflection technique detects only that part of the back scattered signal which has been transmitted through the substrate; the reflected signal from the front surface is specular and does not reach the detector which is located away from the specular reflection. This eliminates the background signal reflected from the front surface of the substrate and thus reduces the sensitivity of the measurement to the surface properties of the substrate which are irrelevant to the temperature measurement. §4*2  The Diffuse Reflectance Technique  In the present work, the diffuse reflection technique developed by Weilmeier et al. [28] is extended to encompass substrates with polished or textured back surfaces [3], spatial profiles of substrate temperature [3,30], surface morphology [29], composition of small bandgap epilayers, epilayer growth rates, and heavily doped substrates that are not completely transparent in the wavelength region below the bandgap [29]. This more general technique is called diffuse reflectance spectroscopy (DRS) and a commercial system similar to the one presented here is under development at Thermionics [56].  Chapter 4  Diffuse Reflectance Spectroscopy  84  Diffuse reflectance spectroscopy is an optical method for measuring the temperature of a substrate material with a temperature dependent bandgap and is used to measure the absorption edge of GaAs in § 2 4 . A schematic of the diffuse reflectance technique is shown in Fig. 24. A detailed description of the diffuse reflectance apparatus and technique is given in § 2 4 . T  1060  I  I  I  j  I  1070  I  I  I  J  -  I  1080  I  I  I  |  I  1090  I  I  I  I  I  1100  I  I  I  |  I  1110  l  l  l  1120  Wavelength (nm) Fig. 4-2. Diffuse reflectance spectra for semi-insulating GaAs, at several different temperatures. Several diffuse reflectance spectra for semi-insulating GaAs from 421 to 433 °C, at intervals of about 2 °C, are shown in Fig. 4-2. As the temperature of the substrate increases the onset of transparency shifts to longer wavelengths with the decrease in the bandgap energy. There is no clearly defined feature in  Chapter 4  Diffuse Reflectance Spectroscopy  85  the smooth diffuse reflection curves which corresponds to the generally accepted definition of the bandgap.  In fact the bandgap lies in the short wavelength  section of the spectrum where the signal is near zero. The separation in the spectra in Fig. 4-2 indicate the sensitivity of the DRS measurement is around 1 °C. §4*3  Calibration of Critical Points of the Spectrum to Temperature  A n elementary analysis method can be used to obtain a qualitative estimate of the bandgap from the diffuse reflection spectrum, for example by taking the wavelength where the diffuse reflectance is 50% of the peak value. Qualitatively the bandgap is at the wavelength where the diffusely scattered light intensity increases. However to determine the temperature accurately and reproducibly with a minimum of calibrations requires a precise procedure for finding an optical signature of the onset of transparency that can be related to the temperature. The point of inflection in the transmitted or reflected optical signal has been proposed by Kirillov and Powell [37] as a suitable optical signature for specular optical signals. The point of inflection lies below the optical bandgap. For maximum accuracy it is desirable to measure as close to the bandgap as possible because the absorption below the bandgap is variable depending on the quality of the material and the doping density. For accurate measurements of the absolute temperature it is desirable to have a technique which is as insensitive as possible to properties of the material that can vary between specimens. The problems with residual absorption below the band edge are exacerbated in the case of the diffuse reflection method where the back scattered signal experiences multiple  Chapter 4  Diffuse Reflectance Spectroscopy  86  reflections inside the substrate and hence has a longer effective path inside the substrate. The diffuse reflectance spectrum has a sharp bend in the wavelength region at the onset of substrate transparency. This region of the spectrum is shown in Fig. 4-3. Consider the following two straight lines: Linel, determined by linear extrapolation through the background at the wavelengths shorter than the transparency wavelength of the sample and Line2, determined by linear extrapolation through the data points closest to the steepest part of the spectrum. These lines are the asymptotes to the positive curvature section of the spectrum shown by the solid dots in Fig. 4-3. These two linear functions intersect at the wavelength adjacent to the sharpest bend in the spectrum. The intersection of these two lines assigns an analytical definition to the sharpest part of the positive curvature bend in the spectrum. The wavelength of highest positive curvature in the spectrum will henceforth be defined as the "knee" of the spectrum. In a first algorithm (Algorithm A), the knee of the diffuse reflectance spectrum is mathematically determined by fitting an asymptotic function to the section of the spectrum with positive curvature. The section of spectrum with positive curvature is shown by the solid circles in Fig. 4-3 and will be referred to as the DRS + data. In general the form of the asymptotic function is  y = f(X-X )  ; f(X-X ) ^^Linel  k  k x  ; f(X  -4)^  +00  -> Line2  where the knee, X , is a fitting parameter and the asymptotes of k  intersect at X . k  ,  (4-1) f(X-X ) k  The maximum in the second derivative of the asymptotic function  occurs at the knee.  Chapter 4  Diffuse Reflectance Spectroscopy  87  0.10 Line2  0.08 h 0  Ij  0.06  1  Asymptotic function fit to the solid circles  0.04  Asymptotes  Knee  0.02 h  (1035.1 nm) Linel  CpOCK300QO«»  0.00 1000  1020  1040  1060  1080  Wavelength (nm) Fig. 4-3. Illustration of a fit to the diffuse reflection spectrum using Algorithm A showing the location of the knee in the spectrum. In the present work, the following form for the asymptotic function is used:  y = y +m (X-X ) 0  1  where y +m (X -X ) 0  mj+m  2  }  k  k  + m X In 1 + exp 2  a  \X-X ) k  (4-2)  Xa  J  is the linear back ground asymptote Linel with slope m , 7  is the slope of the asymptote Line2, and the parameter X  a  determines  how sharply the spectrum is bent at the knee. This form of the asymptotic fitting function gives an excellent fit to the DRS+ data from GaAs substrates. DRS+ data is selected and fit to the function y = f(X-X ) k  The  using a computer  algorithm. The DRS + data is given by the part of the spectrum around the knee which has a second derivative greater than zero.  Chapter 4  Diffuse Reflectance Spectroscopy  88  A flow chart for determining the exact position of the knee, using Algorithm A , is given in Appendix I. The diffuse reflectance signal versus wavelength is collected from the detector by scanning the monochromator over the wavelength range X to A . To select the DRS + section of data for fitting, Q  w  the derivatives of the spectrum are calculated. Taking derivatives tends to magnify the noise in the spectrum, therefore the data is digitally filtered (smoothed) and the derivatives are taken using the Savitzky-Golay method [57]. The filtered data will be referred to as DRS. The limits of the DRS + data are determined from the second derivative of DRS where the maximum of the second derivative of the DRS data is a good estimate of the knee of the spectrum. The right-hand limit (long wavelength limit) of the DRS+ data is determined by the wavelength of the zero crossing of the second derivative. The left-hand limit (short wavelength limit) of the DRS + data is given by the estimator of the knee less twice the difference between the zero crossing of the second derivative and the estimator of the knee. To ensure quick convergence during the fitting process of the asymptotic function to the DRS + data, the DRS data and its derivatives are used to estimate the parameter values of Eq. 4-2. This is done as follows: Estimate X  k  as the  wavelength at which the second derivative of the DRS data is maximum, m as 2  the maximum value of the first derivative of the DRS data, and y as the value of 0  the data point at the left-hand side of the DRS data.  The parameter X , is a  estimated by m divided by four times the maximum in the second derivative of 2  the DRS data. The term, m , is included in Eq. 4-2 to accommodate any slope in ;  the background of the DRS spectrum. In general the background signal is due to  Chapter 4  Diffuse Reflectance Spectroscopy  89  the very small diffuse scattering of the chopped white light from the polished front surface of the substrate which has a negligible slope as a function of wavelength. Therefore m is usually zero and not included as a fitting parameter 1  in Eq. 4-2. However for robustness, the algorithm accommodates larger amounts of stray chopped white light with substantial slope that might impinge onto the detector in a less than ideal situation by including the parameter m in the more 1  general fit. Therefore if the slope of the DRS data is less than 5 % of ra at the 2  left-hand side of the DRS+ data, m is set to zero and excluded as a fitting 1  parameter in Eq. 4-2. The slope at the left hand limit of the DRS+ data is less than 1% of m when the background has a slope of zero. If the slope of the DRS 2  data is greater than or equal to 5 % of m at the left-hand limit of the DRS+ data, 2  nij is estimated as the value of the first derivative of the DRS data at the left hand limit of the DRS+ data. The exact position the knee is located by fitting Eq. 4 - 2 to the DRS + section of data using the nonlinear least squares Levenberg-Marquardt method [35].  The temperature of the substrate is calculated from a predetermined  calibration curve which gives the temperature, T, as a function of the wavelength of the location of the knee, X , for a substrate of a given material, thickness, k  doping, and back surface texture. Methods for determining the calibration curve are described below. In a second algorithm (Algorithm B), the knee of the diffuse reflectance spectrum is mathematically determined by fitting a polynomial to the peak of the second derivative of the spectrum and determining its maximum. This method follows from Algorithm A because the knee occurs at the maximum in the second  Chapter 4  Diffuse Reflectance Spectroscopy  90  derivative of the asymptotic fitting function. Again the data is digitally filtered before the derivatives are taken to reduce noise. The data is filtered using a least squares moving quadratic fit [57] where the number of data points used in the fit is determined by the noise in the data and the goodness of fit to the data. The diffuse reflectance spectrum from a semi-insulating GaAs substrate, its second derivative, and the knee determined by the peak in the second derivative, are shown in Fig. 4 4 . The flow chart for determining the position of the knee using the maximum of the second derivative (Algorithm B) is given in Appendix I. The diffuse reflectance signal as a function of wavelength is collected from the detector by scanning the monochromator over the wavelength range X to X . 0  n  The data is  filtered creating the DRS data set. The second derivative of the DRS data and its maximum value are determined.  The top 30% of the peak in the second  derivative is selected and referred to as the data set f(X ). (  is fit to the data set f{X ). i  A quadratic equation  Fitting an equation to the second derivative peak  allows a more precise location of the maximum, by interpolating between the discrete data points. For example when the quadratic equation is of the form g = g c(X-X )  ,  2  0+  k  (4-3)  where g and c are constants, X is the location of the maximum of the fitting 0  k  function and in general will be at a location intermediate between the data points. The position of the knee is given by X . k  The temperature of the substrate is  calculated from the predetermined calibration curve, which gives the temperature  Chapter 4  Diffuse Reflectance Spectroscopy  91  T, as a function of the position of the knee, X , for a substrate of a given material, k  thickness, doping, and back surface texture.  1000  1020  1040  1060  1080  1100  1120  Wavelength (nm) Fig. 44. Diffuse reflectance spectrum and its second derivative showing the location of the knee using Algorithm B. Algorithm B is faster than Algorithm A, because there are no nonlinear systems of equations to solve. However, Algorithm B is more sensitive to noise because it requires the second derivative of the spectrum to be fit. Algorithm A on the other hand only uses derivatives to select the data to be fit and to get the initial parameter values for the fit. The actual fit is done to a large section of raw data (30 to 60 data points) making it insensitive to data noise. Using Algorithm A the temperature sensitivity of the DRS technique is about 0.4 °C [30] for average  Chapter 4  Diffuse Reflectance Spectroscopy  92  quality data, where Algorithm B only achieves this sensitivity for data with a good signal to noise ratio (high quality data).  For example, the temperature  analysis of the spectrum from substrates with a low residual subedge absorption, such as semi-insulating GaAs, works equally well using either algorithm, while Algorithm A is superior for analyzing the noisier data from conducting substrates with a high residual subedge absorption, such as n-i- GaAs. In order to obtain an accurate value for the absolute temperature of the substrate as a function of the wavelength of the knee in the spectrum, a temperature calibration curve is required. The simplest method is to look up the temperature dependence of the bandgap for the material of interest in a handbook such as the Landolt-Bornstein Tables, assume that the knee occurs at the bandgap, and from the handbook values for the bandgap and the wavelength of the knee determine the temperature. A standard reference for the temperature dependence of the bandgap of GaAs.is given by Thurmond [17] and more recently in reference [58].  This method of calibrating the temperature will be  adequate for many applications. However for best accuracy a calibration against another temperature sensor is required. The wavelength of the knee is calibrated by measuring the diffuse reflectance spectra from a substrate with a thermocouple temperature sensor clipped to it, for a series of temperatures. The temperature as a function of the position of the knee is given by a polynomial fit to the data series.  The  temperature as a function of the wavelength of the knee for a 450 p:m thick semiinsulating GaAs substrate is given in reference [28].  Chapter 4  §4*4  Diffuse Reflectance Spectroscopy  93  Diffuse Reflectance in Indirect Bandgap Material  It is well known that Si is an indirect bandgap semiconductor in contrast to GaAs which is a direct gap semiconductor. A practical effect of this property of Si is that its optical absorption for photon energies just above the threshold for absorption at the bandgap is not as strong as it is in direct gap materials such as GaAs. In addition the absorption edge is broadened by the phonons that are involved in the indirect optical transitions. In spite of these differences between the optical properties of Si and GaAs the diffuse reflectance temperature measurement technique also works for Si substrates.  A series of diffuse  reflectance spectra for unintentionally-doped p-type (n ~ 1 0 v  16  c n r ) Si are 3  shown in Fig. 4-5. The calibration curve for this material is given in reference [59]. The prominent "dip" in the spectra at a wavelength of 1.42 Lim is due to absorption by O H species in the optical elements in the detection system, believed to be principally in the optical fiber bundle.  The reduction of the overall  amplitude of the signal at high temperatures is caused by residual absorption below the band edge that depends on thermally generated free carriers. The prominent "dip" or absorption notch in the spectra of Fig. 4-5 interferes with the algorithms used to determine the position of the knee, by distorting and shifting the knee. Therefore in order to accurately determine the position of the knee and hence the temperature above 500 °C in Si, the absorption notch must be removed. The absorption notch, is in the most part, removed by dividing the raw data with the optical throughput of the system (see Fig. 4-6). In the normalized spectra the knee is more clearly defined in the region  Chapter 4  Diffuse Reflectance Spectroscopy  1.0  1.1  1.2  1.3  94  1.4  1.5  Wavelength (|Ltm) Fig. 4-5. Diffuse reflectance for a 400 um thick, unintentionally-doped, n ~ 10 cm- , Si wafer.  v  16  3  of the absorption notch, as shown in the inset of Fig. 4-6.  However, the  combination of the small uncertainties in the wavelength and the sharpness of the notch means that it is impossible to completely remove the effects of the OH absorption through normalization. A better solution is eliminating the absorption notch in the raw data by replacing the glass fiber bundle with a fused silica fiber bundle that does not absorb at these wavelengths. Optical bandgap thermometry during Si processing is limited to temperatures where the substrate is not opaque at wavelengths below the band edge. The residual absorption below the band edge in Si (at 1.5 |im) is shown in Fig. 4-7. This absorption is determined from the spectra in Fig. 4-5. The subedge  Chapter 4  Diffuse Reflectance Spectroscopy  95  absorption increases exponentially with temperature above 300 °C with a characteristic temperature of 61 °C. This means the absorption coefficient increases by a factor of ten for every 140 °C increase in temperature. Therefore to extend the temperature range of optical bandgap thermometry in Si to 800 °C, the signal to noise ratio has to be increased by several orders of magnitude.  1.0  1.1  1.2  1.3  1.4  1.5  Wavelength (fim) Fig. 4-6. Normalized diffuse reflectance for an unintentionally doped, n ~ 10 cm , 400 urn thick, Si wafer. The high temperature spectra are scaled by a factor of 25 in the inset. 16  -3  v  As in GaAs, the residual absorption just below the band edge in Si is dominated by interband absorption at room temperature while at longer wavelengths free carrier absorption is important. Free carrier absorption typically increases as the cube of wavelength, A . Since Si has a smaller bandgap than 5  Chapter 4  Diffuse Reflectance Spectroscopy  96  GaAs, 1.1 eV compared to 1.4 eV at room temperature, the intrinsic thermally generated free carrier density is higher in Si than in GaAs and the band edge intersects the free carrier absorption curve at higher absorption values in Si than in GaAs. This means that the effect of the stronger temperature dependent absorption due to scattering of free carriers by acoustic-phonons is seen at lower temperatures in Si than in GaAs.  0  100  200  300 400 500 Temperature (°C)  600  700  F i g . 4-7. T h e r e s i d u a l a b s o r p t i o n b e l o w the b a n d edge i n u n i n t e n t i o n a l l y d o p e d p - t y p e (n ~ 1 0 c m ) S i . 1 6  - 3  v  For example, the residual absorption below the band edge starts to rapidly increase for temperatures above 300 °C in Si (Fig. 4-7) and above 550 °C in GaAs (Fig. 2-20). At 300 °C the intrinsic carrier concentration is about 10 cm16  3  Chapter 4  Diffuse Reflectance Spectroscopy  97  in Si [17], this is about where unintentionally-doped Si becomes intrinsic. For comparison, the intrinsic carrier concentration does not reach 10 cnr until 16  3  600 °C in GaAs [17]. At 700 °C the intrinsic carrier concentration has increased to 10  18  cm- [17] in Si. In addition, the large effective electron mass of the 3  electrons in Si means that absorption due to free carrier scattering is larger in Si than in GaAs. Furthermore, in contrast to GaAs, the surface of Si is stable at temperatures up to and above 1000 °C and surface absorption is therefore not believed to be a contributing factor to the strong temperature dependence of the subedge absorption in Si. §4*5  Modeling Diffuse Reflectance  Optical bandgap thermometry is becoming an accepted way to measure substrate temperature in semiconductor processing. The standard method for dealing with variations in the characteristics of the substrate has been to generate a separate calibration curve for each group of substrates, even when the fundamental properties of the substrate material haven't changed. In the present work, the basic knowledge of the absorption edge in direct gap semiconductors, developed in the previous two chapters, is used to determine the effect of variations in the physical characteristics of the substrate on optical bandgap thermometry. In the following three sections, the effects of varying substrate thickness, doping, scattering at the back surface, and reflectivity of the front surface, on the sensitivity and accuracy of the optical bandgap thermometry, is explored. Also discussed is the effect of variations in, the width of the absorption edge, the  Diffuse Reflectance Spectroscopy  Chapter 4  98  optical throughput of the system, and the configuration of the measurement (reflectance or transmittance), on algorithms that extract temperature from the spectrum. In §4-7 algorithms are developed that use the information given by the width of the knee of the spectrum to reduce the sensitivity of Algorithm A (§4-3) to variations in the properties of the substrate. The material presented here is both new and useful to anyone using a temperature measurement system that uses the bandgap of the substrate to determine temperature. One of the most important substrate characteristics related to optical bandgap thermometry, because it is both difficult to quantify and can change during growth, is the diffuse scattering of light inside substrates that have textured back surfaces. A theoretical model for diffuse reflectance in terms of the thickness, reflectivity, and absorption coefficient of the substrate was developed by Weilmeier et al. [28]. In this model the back surface of the wafer is assumed to be textured in such a way that scattering from the back surface has a Lambertian angular distribution or cosine law. This is the angular distribution one would expect for a perfectly scattering surface. In this case the diffuse reflectance spectrum is given by, R (l-R ) ex (-2cxd)  =  R diff  2  b  f  V  n -R R exp(-2ccd)-(n -l)Qxp(-4ad) 2  2  f  b  '  where the absorption coefficient, a , is a function of wavelength, n is the index of refraction of the substrate, Ry is the front surface reflectivity, R is the back b  surface reflectivity, and d is the substrate thickness. In the present work, this model is extended to include back surfaces that are less than perfect scatterers or even polished. In this model (see Fig. 4-8),  Chapter 4  Diffuse Reflectance Spectroscopy  99  scattering from the rough back surface of the substrate is represented by two rays; one which is scattered into the escape cone and escapes from the substrate and one which is scattered at an angle greater than the critical angle for total internal reflection. The angle of the escape cone (or critical angle) is given by sin(6\.) = — or v  '  cos(d ) = * r  n  v  '  V  n  (4-5)  Transmitted Specular reflection  Textured back F i g . 4-8. R a y tracing d i a g r a m for light scattered f r o m a textured surface.  Of the light incident on the substrate, the fraction R (l-R )ex (-cxd ) b  f  V  (4-6)  s  is scattered from the back surface of the substrate. Of this light the fraction f  e  is scattered into the escape cone, and of this fraction, the fraction (l-R )ex (-ad ) f  V  d  is transmitted out through the front surface.  (4-7) Of the fraction in Eq. 4-6, the  fraction / , (represented by the trapped ray in Fig. 4-8) is trapped inside the  Chapter 4  Diffuse Reflectance Spectroscopy  100  substrate. It is assumed that this trapped ray cannot escape either surface without scattering again. When the trapped ray strikes the textured back, a fraction f is e  scattered into the escape cone while a fraction f remains trapped. Similarly, t  when the portion of the ray left in the escape cone (after reflection from the front surface) strikes the textured back, a fraction f  t  is trapped while a fraction f  e  remains in the escape cone. Since the index of refraction in semiconductors is large, the paths of the incident and transmitted light rays inside the substrate are close to normal. Therefore the path lengths d and d are in general approximately equal to the s  d  thickness of the substrate. For example, in the present M B E diffuse reflectance system the angles of incidence and detection are both 23° to the normal; the diffuse signal is detected in the plane that is perpendicular to the plane of incidence. This gives an internal path 6.4° to the normal and a internal path length 1.006 times the substrate thickness, when the substrate is GaAs. The mathematical terms used to describe the scattering process can be messy. However, noting that the fraction of light scattered from the textured back surface that survives to scatter again is y =f R R ^ (-2ad ) s  e  f  b  V  + f ex (-2cxd )  e  t  V  ,  t  (4-8)  the multiple reflections are given by a power series in y and the explicit handling s  of cross terms is eliminated. Here 2d and 2d are the average round trip path e  t  lengths of the rays in the escape cone and the rays that are trapped. The total diffuse reflectance (given by the product of f , Eq. 4-6, Eq. 4-7, e  and the multiple reflections of y ) is s  Chapter 4  Diffuse Reflectance Spectroscopy  DRS = f R {l-R f W  e  b  Defining ft =f /f , t  e  zx (-ad )exp(-ad )[l  f  V  s  101  + y + y +-• ] . 2  d  s  s  summing the power series, and noting f = l-f , t  e  (4-9) Eq. 4-9  becomes R (l-R f b  w  exp(-ad )exp(-ad )  f  s  d  ~ 1 - R R exp(-2ad ) +ft(i - txp(-2ad )) b  f  e  (4-10)  t  The effect on the diffuse reflectance from the light scattered outside of the escape cone is g =/3 (l-ex (-2ad )) s  s  P  .  t  (4-11)  In general the scattering distribution from a rough surface can be represented by a function that narrows as the scattering from the surface becomes more specular. A cosine function with this property is S(0) = (cos0)  ,  //r  (4-12)  where the limit y = 0 represents a specular surface and the limit y — 1 represents the maximal scattering Lambertian surface. For a two dimensional surface the probability that light is scattered at an angle 6, to the normal, is P(e) = {l/Y + l)(cosd) sm0  ..  1/Y  (4-13)  Of the scattered light, the fraction in the escape cone and the fraction trapped are  f = jp(6)d6 = [l-(cos e,)  1-  1  e  •  0 71/2  /,= jp(o)de=(cos e )1/7+1 c  1-  1 \ J n J  (4-14)  J \(l/Y+l)/2 V  rf J  The path length of a ray traversing the substrate at the angle 0 is 1(0) = d/cos6;  Chapter 4  Diffuse Reflectance Spectroscopy  102  the average path lengths of the rays in the escape cone and the trapped rays are, 0  1/Y  1 - cos QA -'-(<»se )1/Y+T  1 o  c  4n  3y  2  '  for  16n  4  d\l+y-^zxp  for y « l  l  d = \p(e)i(e)deJ-^^^(i y)d( t  i ^)^(i )d  +  J  (4-15)  =d  2n y  V  .02<y<l  cosv  +  V  r  +7  2n J  Hence the diffuse reflectance becomes R [l-R )  exp(-2ad)  2  b  w  f  l-R R exp(-2ad) b  + p (l-exp(-2(l+y)ccd)  f  s  '  (4-16)  where f  VK  \ -1  -{l/Y+l)/2  (  1\  -1  nJ  (4-17)  J  In the Lambertian Umit y = 1, fi =n -1 and Eq. 4-16 reduces to Eq. 4-4. In the 2  s  specular limit y=0,/3 = 0 and Eq. 4-16 reduces to the well known contribution s  of the back surface to the reflectance of a polished semi-transparent slab [60]. The scattering distributions of rays inside and outside the substrate are shown in Fig. 4-9, for scattering parameters, y = 0.01, 0.1, and 1.0. The distribution of the rays inside the substrate is given by Eq. 4-12 while the distribution of the rays outside the substrate is given by Eq. 4-12 corrected for refraction and variations in transmission at the GaAs-vacuum interface. The angle to the normal outside the substrate, 6 , is given by Snell's law: nsin 6 = sin 6 . 0  0  Chapter 4  Diffuse Reflectance Spectroscopy  103  The angular dependence of the transmission through the GaAs-air interface is given by the Fresnel equations for randomly polarized light. For 0.1 <y< 1.0 and angles less than 65° the distribution of light outside the substrate is fairly uniform. For angles greater than 65° the angular dependence of the transmission becomes important. 1.2 h  Outside Substrate Inside Substrate  1.0 c  0  1  0.8  •c  -<—>  Q c  0.6  SuCcdD 0.4 h CJ  0.2 0.0 0  20  40 60 Angle to Normal (°)  80  F i g . 4-9. T h e d i s t r i b u t i o n s o f scattered l i g h t i n s i d e the substrate (dashed lines) a n d transmitted through the front surface o f the substrate ( s o l i d lines), for scattering parameters 0 . 0 1 , 0 . 1 , a n d 1.0.  The fractions of the light inside and outside of the escape cone of a GaAs substrate as a function of the scattering parameter, y, are shown in Fig. 4-10. As well, f3 is shown in the inset. For large values of y the denominator of Eq. 4-16 s  and hence the diffuse reflectance signal is dominated by light trapping.  Chapter 4  Diffuse Reflectance Spectroscopy  1  T  1.0  1  104  1—i—i—r  0.8 J3  CM) i—I  o  •§ cd  0.6  0.4  S-l  0.2  0.0 F 0.01  0.1  Scattering Parameter F i g . 4-10. T h e fractions o f l i g h t trapped and i n the escape cone. T h e ratio o f the trapped l i g h t and the light i n the escape cone is s h o w n i n the inset.  Eq. 4-16 (DRS ) gives the fraction of the incident light that is transmitted W  back out the front of the substrate over the entire solid angle 27t. The light detected at an angle, 6 , and smaller solid angle Q is, 0  DRS = - + 7  0  r 2n\ 1V  V  1  1-  \0/Y+l)/2'  f  DRS  tt  n  j  1-  sin f?  n  &o ,  (4-18)  where the scattering distribution is written in terms of the external angle, 6 . The 0  normalization of Eq. 4-18 is confirmed by 2nO  ^l/7+l)/2\  c  JJ 00  7  + 1 2K 1-  1-A V  (cos e)  1/r  n J A  sin dded(j) = l .  (4-19)  Chapter 4  Diffuse Reflectance Spectroscopy  105  Eq. 4-18 (DRS ) as a function of the scattering parameter is plotted in S  Fig. 4-11 for angular positions of the detector at 15°, 23°, 30°, and 45°, where the solid angle of detection is Q = 0.005 sr. The total angle integrated signal, DRS , 0  W  as a function of scattering parameter is shown in the inset. The intensity shown is the fraction of the incident light detected at the inflection point of the spectrum. The magnitude of the signal is independent of the angular position of the detector for Lambertian scattering. However, for maximum signal the back surface texture should be tailored to the particular angle of the detector, with best overall performance at small angles and small scattering parameters.  0.001  0.01  0.1  Scattering Parameter F i g . 4 - 1 1 . T h e intensity o f the diffuse reflectance s i g n a l at the i n f l e c t i o n p o i n t for four angular p o s i t i o n s o f the detector, n a m e l y 1 5 ° , 2 3 ° , 3 0 ° , a n d 4 5 ° , w h e r e the s o l i d angle o f the detector is 0.005 sr. T h e angle integrated diffuse reflectance signal at the inflection point is s h o w n i n the inset.  1  Chapter 4  Diffuse Reflectance Spectroscopy  106  The statistical ray analysis used in obtaining Eq. 4-16 is extended to include the reflectance and transmittance from both sides of a substrate with a polished front surface and a textured back surface. These models are given in Eq. 4-20 where r is the reflectance of light incident on the front surface, r is f  b  the reflectance of light incident on the back surface, /y is the transmittance of light incident on the front surface, and t is the transmittance of light incident on b  the back surface. R is the reflectivity for light incident on the back surface from b  outside the substrate.  R' is introduced because the reflectivity of a textured b  surface can differ depending on the whether the light is going into or coming out of the substrate. The leading term of rj is the specular reflection and the second term is the diffuse reflectance.  When either y or a are zero, g -0, s  and  Eqs. 4-20 reduce to the well known transmittance and reflectance of a polished semi-transparent slab [60]. \2  R (l-R ) b  r =R + f  f  f  1 - R R exp(-2ad) + g b  _ r = R' + n  b  /  exp(-2ad)  f  s  R (l-R' ){l-R )ex (-2cxd)  |  f  b  b  V  1 - R R exp(-2ad) + g  b  b  f  s  (  t =  (l-R )(l-R )ex (-ad)  f  f  b  V  ^  R R exp(-2ad) b  (4-20)  f  1 - R R exp(-2ad) + g b  f  s  _{l-R' )(l-R )exp(-ad) b  f  l-R R exp(-2ad) b  f  +g  s  g =P (l-exp(-2(l y)ccd)) s  s  +  When a diffuse scatterer such as PBN is placed behind the substrate the total diffuse reflectance becomes  Chapter 4  Diffuse Reflectance Spectroscopy  DRS, = DRS + DRS p  W  ; DRS. =  ^  p f  !-  R  ,  h r  P  107  (4-21)  b  where DRS is the contribution to the diffuse reflectance from the diffuser plate. p  W h e n the scattering parameter of the substrate is zero the contribution to the diffuse signal from DRS is zero and DRS reduces to R times Eq. 2-20, where W  R  p  is the reflectivity of the diffuser plate.  p  When the back of the substrate is  textured, the scattering parameter dependence of the diffuse reflectance signal (at the inflection point of the spectrum) is the same for both DRS and DRS p  W  while  the contribution to the overall diffuse reflectance from DRS (with P B N diffuser p  plate) is slightly larger than the contribution from DRS . W  In conclusion, diffuse reflectance is modeled, for substrates with polished and textured backs, with and without a diffuse scatterer placed behind the substrate.  Light scattering inside the substrate for various back surface textures  is described by the scattering parameter  y.  In diffuse reflectance spectroscopy,  the optimal back surface texture for the substrate has a scattering parameter that is much smaller than that of a Lambertian scatterer.  §4*6  Sensitivity to Substrate Characteristics In this section the sensitivity of optical bandgap thermometry to variations  of the optical throughput and substrate properties is determined. In this analysis the effects of substrate thickness, doping, back surface texture, reflectivity, width of the absorption edge, and the residual absorption below the absorption edge are considered.  Furthermore, this analysis is achieved by modeling the two  prominent implementations of optical bandgap thermometry: diffuse reflectance  Chapter 4  Diffuse Reflectance Spectroscopy  108  and transmittance. The inflection point and the knee are considered as the key features (or critical points) in the spectrum that are related to temperature. These two features are essentially given by the maximums of the first and second derivatives of the spectrum. In this analysis each ray that exits the wafer is considered. The first or fundamental ray has the shortest path length in the substrate and each subsequent ray has a path length that increases with each additional reflection inside the substrate. Each ray has an unique spectrum and hence an unique energy at each critical point; as the path length of each subsequent ray increases the critical points shift to lower energies. The fundamental ray has the highest intensity while the intensity of each subsequent ray decreases with each additional reflection inside the substrate. The total spectrum is given by the sum of the spectra of the individual rays. Furthermore, in this analysis, the position of a critical point of the total spectrum is written in terms of the position of the critical point of the fundamental ray plus a correction term. This correction term describes the effect on the total spectrum from the shifted spectra of the higher order rays. Four models are considered in this analysis: (1) Transmission of light incident on the back of the substrate, where TS =f (l-R) exp(-ccd)[l 2  w  e  + y +y +-] 2  s  s  .  (4-22)  This represents the case where the light source is placed behind the substrate in the vicinity of the substrate heater. (2) Transmission of heater or diffuser plate radiation through the back of the substrate, where  Chapter 4  Diffuse Reflectance Spectroscopy  v2  TS = TS l + p  109  (4-23)  R r +(R r y p  W  b  p  b  In this case the optical feedback between the radiation source and substrate is included. (3) Diffuse reflection, of light incident on the front of the substrate, at the back of the substrate, where DRS =f (l-R) exp(-2ad)[l 2  w  e  + y + y +-} .  (4-24)  2  s  This represents the case where the substrate heater is smooth and does not contribute to the diffuse scattering. (4) Diffuse reflection, of light incident on the front of the substrate, that is transmitted through the substrate, scattered at the diffuser plate, and transmitted back through the substrate, where DRS = DRS R (l-R) (l 2  p  w  p  + f R cx (-2ad)[l 2  e  P  + y +---])[l + R r +--] s  p  b  . (4-25)  In this case, the scattering of light as it passes through the back surface of the substrate is included. For simplicity, in these models Rf, R , and R' are assumed b  b  to equal the normal incidence reflectivity/?. The rational for the notation used in these models is as follows:  DRS  indicates a diffuse reflectance measurement and TS indicates a transmission measurement. The choice of subscripts originates in the diffuse reflectance model, where "w" stands for wafer and indicates the diffuse signal originates at the back of the substrate and "p" stands for PBN diffuser plate and indicates the diffuse signal originates at the diffuser plate.  This notation is extended to the  transmission models by noting the similarities between, TS and DRS , and TS W  and DRS . p  W  p  Chapter 4  Diffuse Reflectance Spectroscopy  110  The above models are written in the general form oo  A f exp(-a ad)\s  m  0 e  0  ; with,  Z£ S = f R (l  (4-26)  + R )exp(-2ocd) + f, exp(-2(i + y)ad)  2  E  0  ,  where the constants A , a , and R are listed in Table 4-1 for each of the four a  0  Q  models. Physically A is the maximum intensity of the fundamental ray, a is the Q  a  T a b l e 4-1. L i s t o f parameters i n E q . 4-26. T h e m o d e l s i n the first c o l u m n are g i v e n i n E q s . 4 - 2 2 t h r o u g h 4 - 2 5 ; the subscript "p" indicates there i s a reflector o r diffuser plate b e h i n d the substrate a n d "w" indicates there i s none.  a  RQ  (.48)  1  0  i-R R  (-58)  1  DRS  R(l-R)  (.15)  DRS  RJi-R)  M  «  4  d e l  TS  (1-R)  2  W  0  ^2  TS  p  2  p  (l.H)  R{i-R R)  p  W  '  -El P  2  0  ~ (.16)  2  7+  l-R R p  RM-R) ^ (2.11) R(1-RR)  path length of the fundamental ray in units of substrate thickness, and R  a  represents the enhancement of the signal through optical feedback between the back of the substrate and the reflector behind the substrate.  R is zero in the 0  models where a reflector is not present behind the substrate. The enhancement of the signal from a reflector behind the substrate comes from the higher order rays. Therefore, R is also a measure of the broadening of the total spectrum from the a  Chapter 4  Diffuse Reflectance Spectroscopy  111  higher order rays that originate at the reflector behind the substrate. In the case where the substrate is GaAs and the diffuser plate is P B N the numerical values of the constants are given in brackets in Table 4-1. The leading term in Eq. 4-26 is the fraction of the incident ray that exits as the fundamental ray. To a first approximation, this term relates the onset of transparency of the substrate to the absorption edge and the substrate thickness. The first term of S gives the correction due to multiple reflections in the substrate and the second term of S gives the correction due to light trapping in the substrate. The second term of S is zero when the back of the substrate is polished. To determine the position of the knee and the inflection point of the spectrum, relative to the absorption edge and ultimately the bandgap, the following analysis is done. The spectral dependence of an exponential term in Eq. 4-26 is described by / = exp(-jc); where x = aocd is a dimensionless variable, and a is the path length of the ray in units of substrate thickness. The inflection point and the knee of the spectrum of an individual ray (or each term in Eq. 4-26) are given by the zero crossings of the second and third derivatives of / , respectively. The first three derivatives of / in terms of x are:  /' = -*'/; f" = - "f x  /"' = - »'f x  >f  (4-27)  2  +  x  ;  + 3x'x"f - x' f 3  .  The spectral dependence of the absorption coefficient, and hence x, is given in Eq. 2-3 for materials with an Urbach edge.  Chapter 4  Diffuse Reflectance  Spectroscopy  112  In this analysis, the models in Eq. 4-22 through 4-25 are developed for materials that exhibit the Urbach behavior. To better represent the absorption characteristics of a material with an Urbach edge a constant absorption term, a {T), 0  that depends on temperature is added to Eq. 2-3. a (T) 0  represents the  slowly varying absorption below the band edge. The resulting model for the spectral dependence of absorption is a(hv) = a cxp  \  8  g  +a (T)  .  a  (4-28)  The position or energy of a critical point, E , has a corresponding c  absorption coefficient that is given by a = oc(E ). c  c  Furthermore, by introducing  a dimensionless parameter a = aa d — aa d whose value depends on the critical c  c  Q  point, the following relation between the energy of the critical point, the optical bandgap, and the Urbach parameter is obtained: E {T) =  aa d n  EAT)-\n  c  E (T) 0  c  V  •  (4-29)  J  a  In order to determine a for each of the two critical points, / " and / ' " in c  Eq. 4-27 are set to zero. The derivatives of x with respect to energy are, _ x-aa d 0  x -  c  o  t  ,  x  _ x-aa d  .  0  -  ,  2 c  x  _  x-aa d 0  -  3  o  c  .  (/fou;  o  Note that the numerators of the derivatives of x equal a at a critical point. c  Substituting a for the numerators of the derivatives of x and substituting the c  resulting equations into the equations for / " = 0 and  = 0, two equations are  obtained, one for a at the inflection point and one a at the knee. Solving for c  c  a : a = 1 at the inflection point and a = (3 c  c  c  +  45^J2  =  2.62 at the knee. The  Chapter 4  Diffuse Reflectance Spectroscopy  113  absorption coefficient at the critical points is a = a /ad + a (T). c  c  0  The spectral  positions of the inflection point and the knee for each ray depend on the optical bandgap energy, the bandgap absorption coefficient, the Urbach parameter, and the total path length of the ray inside the wafer. It is interesting to note that the critical points of an individual ray are independent of the subedge absorption, cc (T), and the interface reflectivity, R.  Also note that the knee is  0  \n(2.62)E = 0.96E closer to the bandgap than the inflection point, for both 0  0  transmittance and reflectance (see Eq. 4-29). Recall that the total spectrum is made up of the sum of the spectra of the individual rays represented by the terms in Eq. 4-26. The positions of the critical points of total spectrum are assumed to be given by a weighted average of the positions of the critical points of individual rays, where the weighting is proportional to the intensity of the ray. In this approximation the position of the critical points of the total spectrum are,  £w,ln - i ac  i=0  E (T) = E(T)C  J  E (T) ,  (4-31)  0  i=0  where the i ray has weight W and path length a,. A convenient value for W is th  t  t  the intensity of the spectrum of the i ray at the critical point; th  m(f R {l 2  i = Kfe exp(-a - a a d)  w  c  a =a i  0  t  0  e  m  {  0  [i-m(m + l)l 2]\[m{m + 3)/  + 2(m + [i-m(m +l)/2]y)  ze(ra(ra + 7)I2,m(m + 3)I2)  •\(m(m+3)/2-i)^ -^i- {m+l)l2)  + R )]  ;  ; m = 0,1,2,3,---  ,  2-i]\  (4-32)  Chapter 4  Diffuse Reflectance Spectroscopy  114  where there are m + 1 rays for each order m in Eq. 4-26. In this case, Eq. 4-31 becomes E {T) = EJT)- In  ( a a (T 0  n  C  {  A = [l- f R {l 2  e  e  c  a  g  8  E (T) ; 0  J  + K)exp(-2a J) - f exp(-2(i + y)a d)]£M 0  t  0  m  ;  m=l  j  (4-33)  2(m + jy)  ,  The position of a critical point of the total spectrum is given by the position of the critical point of the fundamental ray plus the shift -A E , c  a  due to the contribution  of the higher order rays to the overall spectrum. The correction term A depends c  on the scattering parameter, 7, the subedge absorption, a , the wafer thickness, 0  d, and the reflectivity, R. Also since A is independent of a it takes on the same c  c  value at the inflection point and the knee.  Since the path length of the  fundamental ray, a , is half as long in transmission as in reflection, the critical a  points are about 0.7E closer to the bandgap in transmission than in reflection. 0  The shift in the critical point given by A (Eq. 4-33) is compared to the c  shift in the critical point determined by numerical simulations using the models (Eq. 4-22 to 4-25). In this simulation, the true shift of the inflection point, A , is p  inferred from the separation in the maximum of the first derivative of the spectrum of the fundamental ray and the maximum of the first derivative of the total spectrum. The true shift of the knee, A , is inferred from the separation in the k  Chapter 4  Diffuse Reflectance Spectroscopy  115  knee of the spectrum of the fundamental ray and the knee of the total spectrum, where the position of the knee is determined by fitting the asymptotic function to the spectrum (Algorithm A). A , A , and A for the DRS model are compared c  k  W  for a d = 0 and a d = 0.5, in Fig. 4-12. The correction term, A , falls between 0  A, p  Q  and A  k  c  with roughly the same scattering dependence.  This means the  weighted average of the critical points of the spectra of the rays or terms in the DRS  W  model, is useful for characterizing the shift in the critical points of the  diffuse reflectance spectrum, caused by multiple reflections and light trapping.  0.1  0.01  1  Scattering Parameter F i g . 4-12. T h e shift i n the c r i t i c a l p o i n t s the s p e c t r u m f r o m those o f the fundamental ray i n the diffuse reflectance m o d e l , for a w e i g h t e d average o f c r i t i c a l points, A,, and n u m e r i c a l simulations o f the i n f l e c t i o n point, A , and the knee A . T h e upper curves are for substrates that are transparent b e l o w the edge a n d the l o w e r curves are for a d = 0.5. p  k  0  Chapter 4  Diffuse Reflectance Spectroscopy  116  Comparisons of A , A , and A for the TS , TS , and DRS models are c  p  k  similar to those of the DRS  W  model.  W  p  p  Since the terms A  and  A, k  are  approximately proportional to A , the scalar a , is introduced to relate A to the c  m  c  actual correction given by numerical simulations. In this case A„=a A P  m  with a = 1.1 ±0.1 ;  r  A =aA  m  k  m  c  with a = 0.6 ±0.2 m  (4-34)  ,  where a is an average over the TS , TS , DRS , and DRS , models for y equal m  W  p  W  p  0.0, 0.1, 0.3, and 1.0 with a d equal 0.0 and 0.5. 0  The correction parameter, A , for the DRS model, is plotted in Fig. 4-13, c  0  100  W  200  300  400  Temperature (°C) Fig. 4-13. The correction parameter, A , using the DRS model for a 500 urn thick semi-insulating GaAs substrate. The scattering parameter for each curve is listed next to it. c  W  500  Chapter 4  Diffuse Reflectance Spectroscopy  117  for a 500 |im thick semi-insulating GaAs substrate, where oc (T) is given by the 0  linear fit to the data for undoped GaAs in Fig. 2-20. Each curve is for a different scattering parameter, whose value is listed next to the curve. As the scattering parameter increases, the critical points of the spectrum shift away from those of the fundamental ray, with shifts as large as two Urbach parameters for Lambertian scattering in substrates that are transparent below the bandgap. The correction parameter, A , for the DRS model, is plotted in Fig. 4-14, c  W  for a 500 pm thick n-i- GaAs substrate, where oc (T) is given by the linear fit to a  the data for Si-doped GaAs in Fig. 2-20. The value of the scattering parameter is  0  i  1  1  1  1  i  1  1  1  1  1  1  1  1  1  1  1  1  1  1  i  1  1  r  J  i  i  i  I  i  i  i  i  I  i  i  i  i  I  i  i  i  i  I  i  i  i  L  100  200  300  400  Temperature (°C) Fig. 414. The correction parameter for a 500 um thick n+ GaAs substrate, for the DRS model. The scattering parameter for each curve is listed next to the curve. W  500  Chapter 4  Diffuse Reflectance Spectroscopy  118  listed next to the curve. The behavior of the correction parameter for n+ GaAs is similar to that of semi-insulating GaAs with the exception of the extreme Lambertian scattering, where the combination of a large a (T) 0  and the large path  length of the trapped rays attenuate the contribution of the trapped light, and hence the correction parameter, to the signal. In general the correction parameter is smaller in doped material because the higher subedge absorption reduces the intensity of the higher order rays. However, since the Urbach parameter is larger in doped material the overall spectral shift is about the same. One implementation of optical bandgap thermometry that is not modeled here is, that given by tj in Eq. 4-20, the transmission of light incident on the front of the substrate. In this method optical access to the back of the substrate, where the substrate heater is located, is required. In most existing M B E equipment optical access to the back of the substrate requires modifications to the substrate heater and manipulator; these modifications are costly and difficult to implement. This method may become more feasible in the future as M B E manufactures move towards manipulator designs with optical access to the back of the substrate. Furthermore, this transmission method is not conducive to spatial temperature profiling. However, the redeeming feature of this method is the intensity of the fundamental ray is insensitive to scattering at the back surface and the shift in the critical points due to light scattering, appears one order higher in the higher order rays (see Eq. 4-20).  This means, of all the methods of optical bandgap  thermometry, this one is the least sensitive to variations in the texture of the back surface of the substrate.  Chapter 4  Diffuse Reflectance Spectroscopy  119  The temperature dependence of the positions of the critical points of the spectrum, using the more accurate correction in Eq. 4-34, are  E (T) =  aad 0  EJT)- ln  C  g  + a A {T) E (T) m  V  c  (4-35)  0  j  c  a  Where the temperature shift associated with a shift in the critical point is AE.  AkT =  dE (T) g  In  dkT  ( a a d^\ n  I  0  o g  + ci A (T) m  The contribution of a , a , and A c  c  (4-36)  ,^dA {T) a E (T) — dkT c  m  dkT  J  c  U  0  dE (T)  c  0  in Eq. 4-36 is small (particularly at higher  temperatures) and the temperature dependence of the Urbach parameter is small. The temperature dependence of the optical bandgap and the Urbach parameter for GaAs are given in Eqs. 2-5 and 2-7 and Table 2-2. In the temperature region where the bandgap and Urbach parameter are linear in temperature with slopes S and S (given in Table 2-2), the temperature shift in Eq. 4-36 is approximately g  0  , f AkT = —  \ l-^\n(a d)  (4-37)  AE„  g  From Eq. 4-35 the shift in the position of a critical point due changes in the physical characteristics of the substrate is  In  AE =-E C  0  +a„  dA, dy  JjL + dn^ 3  + a„  d  V  dA  Ay + a  -Aa„+a„ da„  n  dA, dR  o  AR  a  f  —  In  0  V  I J ( a a d^\ °  n  c  j  +  mc  a  + a A AE (4-38) m  An  A  dn„  p  g  c  g  a  Aa„  a  a a d}  In  da„  )  da  0  c  dn  c  .  c  0  Chapter 4  Diffuse Reflectance Spectroscopy  120  The temperature error associated with variations in the physical characteristics of the substrate from those of the calibration substrate, is determined by substitution of Eq. 4-38 into Eq. 4-37. From Eq. 4-37, the temperature error for a 500 pm thick semi-insulating GaAs substrate, is AT = -1.8 °C/meV AE . The temperature error, C  AT, due to changes in the physical properties of GaAs substrates from those of a 500 °C, 500 pm thick, semi-insulating GaAs reference substrate, are given in Table 4-2, where the reference spectrum is obtained in the DRS configuration with a W  scattering parameter of 0.1. In this case a is 0.61 for the knee and 1.12 for the m  inflection point. The Urbach parameter is 11.0 meV and the subedge absorption coefficient is 3 cm- for semi-insulating GaAs at 500 °C. 1  The first two columns of Table 4-2 give the ratio of the physical characteristics of the substrate in question to those of the reference substrate. Columns three and four list the contributions of the correction parameter to the temperature error as a percentage of the total error. Columns five and six list the temperature error caused by using the calibration curve for a 500 pm thick semiinsulating GaAs with scattering parameter 0.1 to determine the temperature of the substrate in question. Column five is the error using the inflection point and column six is the error using the knee, to extract temperature. The first line of Table 4-2 is for a 100 pm thick substrate (instead of 500 (im); the temperature inferred from the spectrum is 26 °C cooler (at the inflection point) and 29 °C cooler (at the knee), than the real temperature of the substrate.  The second line gives the error associated with 5% thickness  variations between substrates, where the temperature deviation is around the sensitivity of the technique.  The third line is for the case where substrate  Chapter 4  Diffuse Reflectance Spectroscopy  121  temperature is inferred from the transmission spectrum and the calibration curve is determined using reflection measurements. contribution of A  c  In these first three examples the  opposes the spectral shift in the fundamental ray decreasing  the temperature error. This partial offset is smaller at the knee (as a  m  is smaller),  hence the knee is slightly more sensitive than the inflection point to changes in substrate thickness or measurement technique. However, since these changes are well defined and can be calibrated for, they should not introduce large errors. Table 4-2. The temperature error associated with variations in the physical properties of a GaAs substrate from those of a 500 pm thick semi-insulating GaAs calibration substrate at 500 °C with scattering parameter 0.1. For semi-insulating GaAs at 500 °C the Urbach parameter is 11.0 meV and the subedge absorption coefficient is 3 cm . -1  Physical Characteristic  Contribution of A  Temperature Error  c  Parameter  Value  d'/d  0.2  -22  -11  -26  -29  d'/d  1.05  -36  -17  0.71  0.83  'ol o  0.5  -58  -25  -8.7  -11.0  10  100  100  5.7  3.1  0.8  100  100  -1.5  -0.9  0.1  100  100  -9.6  -5.2  1.5  100  100  -2.2  -1.2  0.8  100  100  -0.16  -0.09  K/E  1.2  8  5  29  24  KK  1.0  -28  -21  24  18  a  a  fir fir y'/y <X'o/<Xo f/ f  R  R  0  % of AT  p  % of AT  k  AT  p  (°C)  AT ( ° C ) k  Chapter 4  Diffuse Reflectance Spectroscopy  122  Furthermore, these temperature shifts can be estimated from Eq. 4-35 provided the Urbach parameter is known. The remaining substrate characteristics listed in Table 4-2 are more difficult to determine. The temperature errors for variations in the back surface texture of the substrate (or scattering parameter) are shown for y ' / y = 10 where the scattering is increased to that of a Lambertian scatterer, for y ' / y = 0.8 where the scattering is decreased by 20%, and for y ' / y = 0.1 where the scattering is decreased by a factor of ten.  These results show that small changes in the  scattering at the back surface cause errors around the sensitivity of the measurement, while large changes in scattering need to be calibrated for. The effect of increasing the subedge absorption is also shown. This represents the case where additional impurities in the substrate cause the residual subedge absorption to increase. During M B E the reflectivity of the front surface of the substrate changes as the alloy type and composition of the epilayer changes.  The reflectivity  change shown in Table 4-2 is for a case where the index of refraction of the epilayer is 3.0 compared to 3.5 for the substrate, this depicts the growth of high A l content AlGaAs on GaAs. Clearly variations in surface reflectivity have a negligible effect on the absolute accuracy of these temperature measuring techniques. If a large amount of impurities are present in the substrate (such as compensation doped material) the absorption edge will be broader; this case is represented by a 20% increase in the Urbach parameter. The errors listed for this case are determined assuming no impurity shift in the optical bandgap. Broadening of the edge by 20% causes an over estimation of the substrate  Chapter 4  Diffuse Reflectance Spectroscopy  123  temperature by 29 °C using the inflection point and 24 °C using the knee. Finally in the event one uses a semi-insulating GaAs calibration curve for n+ GaAs substrates, the temperature is over estimated by 24 °C using the inflection point and 18 °C using the knee. The shift in the inflection point is larger than the shift in the knee for variations in the back surface texture, subedge absorption, and front surface reflectivity, because it is more sensitive to variations in the correction parameter, A. c  The knee is less sensitive than the inflection point to changes in the width of  the absorption edge and doping levels because it is closer to the bandgap. Therefore the knee is a more suitable reference point for temperature than the inflection point. Another thing that affects the position of the critical points of the spectrum is variations in the optical throughput (or response) of the optical detection system. Since the optical throughput and the signal / are multiplicative, the higher order terms in the spectral dependence of the optical throughput appear with subsequently higher orders of the Urbach parameter at the critical points. This is because each higher order term of the optical throughput appears with a subsequently lower derivative in / at a critical point. These higher order terms can be neglected when there are no sharp kinks in the spectrum of the optical throughput, that are on the order of the Urbach parameter. Therefore, only the contribution of the linear part of a slowly varying optical throughput has a significant effect on the position of a critical point. When the optical throughput, g, has slope 1/E , the functional form of the S  spectra are  Chapter 4  Diffuse Reflectance Spectroscopy  124  (4-39) S  Where the position of the inflection point and the knee are given by f" + 2(l/E )f' s  w  =0  (440)  s J  f'" + 3{l/E )f" = 0 . s  Solving these equations, the shift in the critical point for slope 1/E in the optical S  throughput, to first order in E /E , a  s  (  a=c a  c  is given by E ^ l+ a ^ s  V  E  (441)  sJ  Where a = 2 at the inflection point and a = 0.8 at the knee. Positive slopes in s  s  the optical throughput shift the critical points to higher energies while negative slopes shift them to lower energies.  The inflection point is 2.4 times more  sensitive than the knee to changes in the slope of the optical throughput. Typically \E \ > 0.33 eV for optical throughput variations due to mirror and S  window coating during the M B E growth of GaAs based materials. However, when variations in the optical throughput are sharp, such as O H absorption in the optical elements (see Fig. 4-5) where \E \ ~ 0.02 eV, this analyses is not valid and S  the higher order terms in the optical throughput become important. The shifts in the critical points of spectrum due to variations in the optical throughput, for E = -0.33 eV, result in a temperature error of 1.3 °C at the s  inflection point and 0.5 °C at the knee, for the substrate discussed in Table 4-2. Up to now the analysis is done for spectra plotted in terms of energy, which is the nicest way to relate the critical points of the spectrum to the physical properties of the substrate. However, since transmittance and diffuse reflectance  Chapter 4  Diffuse Reflectance Spectroscopy  125  spectra are commonly measured in terms of wavelength, the shift in the position of the critical points, to first order in the Urbach parameter, is determined for a spectrum plotted in terms of wavelength. The positions of the critical points of a spectrum (in terms of wavelength) are determined by differentiating x in terms of wavelength, X, where he ~EJ?  (x — aa d) 0  ; x" =  2hc E J  3  he  •+  (x — aoc d) ; 0  (442) x'" = -  6he EX  4  +  0  „ he 2hc 3 — - T — - T EX EX 2  0  f  he  ^'  (x-acc d) 0  +  3  .  0  Solving Eq. 4-27 at the critical points, where the derivatives of x are given in Eq. 442, and neglecting higher order terms in E jE , 0  g  the energies of the critical  points are given by r  «c  where a^=2  E  A  (443)  =c a  at the inflection point and a =1.7 k  at the knee.  Plotting the  spectrum in terms of wavelength, shifts the critical points up in energy. If not corrected for, this shift introduces a temperature error of -0.37 °C at the inflection point and -0.31 °C at the knee, for at substrate like the one discussed in Table 4-2. Finally, the positions of the critical points of the spectrum in terms of energy and wavelength are given in Eq. 444.  Where A (T) C  (Eq. 4-33) is a  function of 7, R, a , d, and oc (T) and represents the shift in the critical point a  0  due to multiple reflections and light trapping. The other parameters of Eq. 444 are listed in Table 4-3.  Chapter 4  Diffuse Reflectance Spectroscopy  E (T) =  E (T)- In  c  (  a a d\ o  g  g  j  c  a  V  +  E„(T)  a A (T)-a m  c  i  126  E {T) a  (444)  he EAT)-  a„a d\ n  °  In  v  g  c  a A (T)-a ^P-a ^ E (T)  +  m  c  s  x  0  j  a  The values of the absorption coefficient at the critical points, neglecting the higher order terms in E , are 0  a  c  {  T  )  =  ^ M - " . M T ) )  +  a  A  T  (445)  )  ad 0  T a b l e 4-3. L i s t o f parameters for E q . 4 44. T h e first r o w is the p a t h l e n g t h o f the fundamental ray i n s i d e the substrate i n units o f the substrate thickness. T h e s e c o n d r o w is the d i m e n s i o n l e s s parameter that describes the p o s i t i o n o f the c r i t i c a l point. T h e t h i r d r o w is a n u m e r i c a l l y determined constant that scales the correction parameter to the actual shift i n the p o s i t i o n o f the c r i t i c a l point. T h e fourth r o w is the constant that relates variations i n o p t i c a l throughput to shifts i n the p o s i t i o n o f a c r i t i c a l point. T h e last r o w is the constant that gives the shift i n the c r i t i c a l point w h e n the spectrum is plotted i n terms w a v e l e n g t h .  Reflectance  Transmittance  Parameter  knee  point  knee  point  o  2  2  1  1  c  2.62  1  2.62  1  m  0.6  1.1  0.6  1.1  s  0.8  2  0.8  2  x  1.7  2  1.7  2  a  a  a  a  a  The temperature errors associated with variations in the physical properties of the substrate are determined. Front surface reflectivity and residual subedge  Chapter 4  127  Diffuse Reflectance Spectroscopy  absorption only affect the spectrum through the higher order rays; consequently changes in front surface reflectivity and residual subedge absorption cause temperature errors that are within the sensitivity of the measurement technique and can be ignored. Temperature errors due to small changes in thickness, the scattering parameter, and optical throughput also fall within the sensitivity of the measurement. The positions of the knee and the inflection point of the spectrum are given in terms of the optical bandgap, the bandgap absorption coefficient, the width of the absorption edge, the thickness of the substrate, the measurement technique used (transmittance or reflectance), the slope of the optical throughput, and a correction term due to multiple reflections and light trapping in the substrate (see Eqs. 444 and 4-33 and Table 4-3). §4*7  Reducing Sensitivity to Variations in Material Properties  As a first step in reducing the sensitivity of Algorithm A (§4-3) to variations in the properties of the substrate, the fitting parameters of Eq. 4-2, the position of the knee X or E and the width of the knee X or E , are related to k  k  a  the properties of the substrate.  a  When the background is zero the asymptotic  function (Eq. 4-2), fit to the spectrum, (Algorithm A) becomes 'E -hv" y(hv) = m E In 7 +exp k  2  a  Ea  y(X) = m X In 7 +exp 2  a  J  (446)  Xa J  where the fitting parameters are for a spectrum plotted in terms of energy or wavelength. Furthermore, at the knee y = -2\n2E y' a  in terms of energy and  Chapter 4  Diffuse Reflectance Spectroscopy  y = 2\n2X y'  128  in terms of wavelength, where y' is the derivative of the  a  asymptotic function. Writing the total spectrum as a single ray / = exp(-*) with f' = -x'f,  and writing the position of the knee of the total spectrum as the  position of the knee of a single ray (see Eq. 4-29), E (T) =  E(T)-ln  k  ( a„a„d  exp(-a A (T))  0  c  m  (4-47)  ;  c  J  s  C  E {T)' a {X) = a 1 + a,  1 + a,  0  k  ;  0  E (T) a {hv) = a 1 + a, k  E (T) j  k  a  V  c  J  E (T) EAT)  exp(-a A (T))  0  m  c  where a is determined by comparing Eq. 4-29 with Eq. 4-44. Furthermore at the k  knee, from Eqs. 4-29 and 4-30 E x' = a (hv) and from Eqs. 4-29 and 4-42 0  E x' = -he a (X)lX .  Finally, the width of the knee is related to the Urbach  2  k  0  k  k  parameter by equating the fitting function divided by its derivative to the spectrum divided by its derivative, at the knee: / F —= ° x' a {hv) '  2\n2E= a  k  21n2A  "  = — =  x'  (4-48)  -AM_  hca ( A ) k  Solving Eq. 4-48 for a (hv) and substituting this solution into Eq. 4-47, the knee k  of the total spectrum is E (T) = EAT)- In  ( a a d^ n  °  k  {  c  a  e  8  )  + ln  ( 2\n2a E {T) c  V  a  E (T)  E (T) 0  (4-49)  0  The position of the knee of the total spectrum relative to the knee of the fundamental ray is written in terms of the width of the knee, E . The shift in the a  Chapter 4  Diffuse Reflectance Spectroscopy  129  knee from multiple reflections and light trapping (A ) and the shift in the knee c  due to nonuniform optical throughput (1/E ), are sensed through the width of the S  knee. This is explained as follows: when the position of the knee shifts AE toward lower (higher) energies, due to changes in A and 1/E , the position of c  S  the inflection point shifts about 2AE toward lower (higher) energies, causing the width of the knee to broaden (narrow). Therefore the width of the knee is a measure of spurious shifts in the position of the knee. Similarly, in terms of wavelength, the position of the knee of the total spectrum is written in terms of the position of the knee of the fundamental ray and the width of the knee of the total spectrum;  EAT)-  In  he f 2 In 2a hcX (T) + ln V E {T)X\{T)  'a a d\ n0  I  9  c  8  c  )  a  a  E (T)  (4-50)  0  0  From Eqs. 4-47 and 4-48 and in the case where the optical throughput is flat (this can be achieved by normalizing the data) the correction parameter, A , in c  terms of the width of the knee, E or X , is a  E  f  aA m  c  = In A  a  A  ; A = a 2\n2 = 3.63 ;  a  m  r  EJ  (4-51)  0  aA m  c  = In A V  a 7 J? X  m  k c  2  A  k o J L  +  a  l K  E  o  he  Recall these expressions are a result of comparing the model to the fitting parameters of the asymptotic function at the knee of the total spectrum. When the reflectivity, Urbach parameter, and subedge absorption are known the scattering parameter can be estimated using Eqs. 4-33 and 4-51.  Chapter 4  Diffuse Reflectance Spectroscopy  130  When the substrate has a polished back (y = 0) and the optical throughput is flat, the optical bandgap and Urbach parameter are easily determined from the fitting parameters. For example, from Eqs. 4-44 and 4-51 ( a cc d^  EJT) = E (T) + In  0  n  a  8  k  I c  +  m  E (T) = A oxp(-a A (T))E (T) 0  where a A m  c  m  m  E {T) ;  ci A {T) c  Q  (4-52)  J  a  c  ,  a  is small and a is accurately estimated by numerical simulations of m  the TS and DRS models for y = 0. In this case W  p  A (T) = 1.15R cxp{-2.06a (T)d)  , a ~0.21  for TS  A {T) = 2.62R exp(-2.29a {T)d)  , a ~0.53  for DRS  2  c  o  m  2  c  0  ;  W  m  p  .  The numeric constants in A (T) are determined by a fit to Eq. 4-33 for y = 0 and C  the values for a are determined by numerical simulations. m  The position of the knee, E .  or X ,  k cal  or X , a;cal  and the width of the knee, E .  tcal  a cal  are measured as a function of temperature for a given substrate in a  calibration run. The position of the knee, E or X , and the width of knee, E or k  k  a  X , for a different substrate of the same material with different thickness or a  texture, are related to the calibration values (with extended subscript "cal") as follows (see Eqs. 4-49 and 4-50): fF 'a;calno;cal^cal^ c  k ~ k;cal +  E  E  l n  a  u  Ea d a  he XK  he X  Q  • + ln  k,cal  2  (4-54)  ^  ^a\cal o;cal cal %k E Xad k\cal J a  d  n  a  a  A  The temperature of the substrate is inferred from the calibration curve by iteratively solving Eq. 4-54.  Chapter 4  s i n c e  Diffuse Reflectance Spectroscopy  E  . {T),  E  k cal  . (T),  131  and E0(T) are known, Ea.cal and E0 are  a cal  expressed as functions of Ek.cal. Similarly, in terms of wavelength, Xa.cal, and E0 are expressed as functions of X . tcal  f  From Eqs. 4-54 \  ao  E  £ { ' = 2 # ( r ) + E (25)ta +  a  d  0  a;cal{ k) o;cald al  E  E  a  (4-55)  C  he  Xad n  n  k  )  l  where E (T) or A^(7/) is the position of the knee at temperature T. Finally, the k  temperature of the substrate is T = fcal{Ek)  f  cal  is the inverse of Ek.cal(T)  or Xtcal(T).  or T = fcal(Xnk),  where the function  The equation is iterated until  El - El' < SE or X\ - Xf < 8X. n=3 is usually sufficient. 1  1  In Eq. 4-55 the position of the knee of the spectrum of the substrate relative to the knee of the calibration curve is given by the Urbach parameter times the sum of natural logarithms of the ratios of the thicknesses, the widths of the knees, and the path lengths of the fundamental rays. These equations correct for the shift in the knee due to differences between the substrate in question and the calibration substrate. These differences are substrate thickness, light trapping, optical throughput, and the path length of the fundamental ray. Variations in light trapping, and optical throughput are sensed through the width of the knee. For a material where the absorption edge is exponential over a few orders of magnitude, such as GaAs, the thickness of the substrate can be many times different from that of the calibration wafer without introducing large errors when one extrapolates from a single calibration curve using Eq. 4-55. Absorption  Chapter 4  Diffuse Reflectance Spectroscopy  132  edges are not always exponential, as is the case for Si which is an indirect bandgap material. This limits the thickness range of Eq. 4-55, where the notion of an Urbach parameter is only useful over a small range in absorption coefficient. For a calibration curve from a substrate with both sides polished where the measurement is done in transmission (TS model with y = 0), the correction term W  aA m  c  < 0.023 and can be ignored. In this case the width of the absorption edge,  E , can be approximated by the width of knee, E , using E = A E Q  a  0  m  a  (see Eqs.  4-52 and 4-53). This allows for the following more general implementation of Eq. 4-55 for materials where the width of the absorption edge is not known:  4  + ;  = ^ (r) + A £ 0  m  a;cfl/  (4)ln  E  a o a  d  a;cal{ k)d l  E  E  (4-56)  ca  j ,  A  «;4 *) f K;cal{^)dcal A  m  AV  X„a„d  ( X° {T)  >2\  k  I A'  t  The temperature is obtained from these equations in the same manner as it is for Eq. 4-55.  In E q . 4-56 the thickness of the substrate is restricted to  a d e((l/b)d ,(b)d ), Q  cal  cal  where b is about 2 for an indirect bandgap material  (such as Si) and b is about 10 for a direct bandgap material with a Urbach edge (such as GaAs). To improve the accuracy of the extrapolation scheme, calibration curves could be obtained at each limit of the substrate thickness, with the temperature of substrate obtained using an equation determined by interpolating between the equations for the two thicknesses. A linearized version of Eq. 4-55 is given in Appendix II, Eq. 11-25. Simulations of the position and width of the knee of the diffuse reflectance  Chapter 4  Diffuse Reflectance Spectroscopy  133  spectrum for various scattering parameters is given in Appendix II §11-7. For large values of the scattering parameter, the corrections given by Eq. 11-25 are found to be better than those given by Eq. 4-55. §4*8 Bandwidth of the Diffuse Reflectance Spectrum The calibration of diffuse reflectance spectra to temperature can be complicated by the spectral resolution of the monochromator. For example, if the resolution of the monochromator is below the critical value required to resolve spectrum, the knee region of the spectrum will be broadened and the position of the knee will most likely be shifted. Therefore, in order to maintain universal calibration curves, the resolution of the monochromator must be such that the width of the knee is limited by the fundamental properties of the substrate and not the resolution of the monochromator. In this section, the minimum resolution required to ensure the width and position of the knee of the spectrum is unique to a particular substrate, is determined for both textured and polished substrates. The bandwidth or spectral resolution required to resolve the knee of the diffuse reflectance spectrum is estimated using Fourier analysis of the asymptotic fitting function. The absolute value of the Fourier transform of the asymptotic fitting function in units of one over the width of the knee is shown in Fig. 415. The integral of the power spectral density (PSD) is shown in the inset. The bandwidth is estimated as 0.30/X where the integral of the PSD is 85%. a  From Eq. 4-51 the spectral resolution given by the bandwidth 0.30/Pi is a  A(hv) = - *- = 0.9E exp(a A ) ' 0.30 ¥\ m E  o  m  c  ; A(X) = - ^ - = 0.9^-exp(a  A)  m  cJ >  Q  3  Q  h  c  f\  c  m  c)  . (4-57)  Chapter 4  Diffuse Reflectance  Spectroscopy  134  The resolution depends on the width of knee which depends on the Urbach parameter and A .  A depends primarily on the scattering parameter.  c  c  0.0 —'— — —'— —•—•— —'—'— 0.0 0.5 1.0 1  1  1  1  1  11X  L  1.5  2.0  (I/X ) a  Fig. 4-15. The Fourier transform of the asymptotic fitting function. The integral of the power spectral density is shown in the inset. The minimum resolution required to resolve the knee of the diffuse reflectance spectrum from a 450 pm thick semi-insulating GaAs substrate, A (X),is shown in Fig. 4-16. The DRS model for a substrate with a polished back ( y = p  0.0) is shown by the solid line and the DRS  W  model for a substrate with a  textured back (y = 0.20) is shown by the dashed line. A higher resolution is required for a polished back. At temperatures above 650 °C the substrate is no longer transparent to higher order rays; this means that A  c  is small and the  Chapter 4  Diffuse Reflectance Spectroscopy  135  resolution only depends on the Urbach parameter. The optical bandwidth of the system is determined by the smallest scattering parameter and the lowest temperature that accurate substrate temperatures are needed.  Temperature  measurements as low as 200 °C on polished substrates may be required during growth, therefore the optical bandwidth should be at least 0.15 nm- . 1  14  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  a  D R S Model DRSp Model  ;  W  -  1  0  1  1  1  1  1  100  1  1  1  1  1  200  1  • i  i i i  300  i  i i i  i  400  i i i i i  500  i i  i  i  600  i ii  700  Temperature (°C) Fig. 4-16. Spectral resolution for diffuse reflectance measurements on semi-insulating GaAs substrates. The higher curve is for a textured back (y = 0.20) and the lower curve is for a polished back (y = 0.0). The bandwidth of the electronics, B, is given by the scan rate of the monochromator, S, times the optical bandwidth, If A: B = S/A. The lock-in time constant for this bandwidth is  Chapter 4  Diffuse Reflectance Spectroscopy  T =  1 2npB  _  A  _0.9X\E  27ipS  2itphcS  o  ex  136  (4-58)  p(«m4) .  where p is the number of poles in the lock-in (typically two).  For a semi-  insulating GaAs substrate and a temperature update every minute the monochromator scan rate is 2 nm/s, the bandwidth is 0.29 Hz, and the time constant is 270 ms. When the temperature update rate is increased to 1 Hz the bandwidth is 15 Hz and the time constant is 11 ms (for one pole). The Nyquist frequency, 2B, determines the minimum sampling interval (A/2)  required to determine the spectrum.  However, because of noise  spectroscopists generally sample five times the Nyquist interval (A/10).  In this  case the sampling interval for GaAs is around 1 nm. In the second part of this chapter, an algorithm (Algorithm A and Eq. 4-55) is developed that uses the position and the shape of the knee of the spectrum to determine temperature. This algorithm is a leap forward in terms of using the information contained in the spectrum to extract temperature; consequently this algorithm is insensitive to variations in substrate properties. Furthermore, this algorithm reduces the number of calibrations needed to determine temperature, to one calibration for each material with a given doping level.  In conventional  methods, calibrations for five different thicknesses times three different back surface textures times two measurement configurations may be required; giving a total of thirty different calibrations for each material with a given doping level. This algorithm is also insensitive to thin film interference and variations in optical throughput.  In some situations, this means that the time wasted doing daily  measurements of optical throughput is eliminated.  137  Chapter 5 Diffuse Reflectance Spectroscopy During Growth The sensitivity of the diffuse reflectance spectrum to both surface morphology and the growth of epilayers on the substrate, is explored in the first part of this chapter (see §5-1 to § 5 4 ) . The growth of overlayers which have a different index of refraction than the substrate, cause thin film interference in the overlayer. These interference fringes can play havoc with algorithms that extract substrate temperature from the spectrum. The growth of overlayers that have a smaller bandgap than the substrate absorb at wavelengths where the substrate is transparent.  In this case, the wavelength of the onset of transparency of the  overlayer is a measure of its composition. Thin film interference can be induced in any epilayer by first growing a thin marker layer with a different index of refraction. These interference fringes can be used to measure the growth rate and index of refraction of the epilayer. Interference effects induce small changes in the shape of the diffuse reflectance spectrum which can cause the temperature reading inferred from the spectrum to oscillate during growth. Spurious shifts in the temperature reading of the substrate are undesirable. For example, in the case where a feedback-loop is used to control temperature, interference fringes can induce oscillations in substrate heater power. To avoid these problems, algorithms are needed that can  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  138  both sense changes in the shape of the spectrum and compensate for their effects. In §5-1 the algorithm in Eq. 4-55 is shown to be effective in correcting for spurious shifts in the position of the knee of the spectrum caused by thin film interference during the growth of AlGaAs on GaAs. In the last part of this chapter (§5-5 to §5-7) the fitting parameters of Algorithm A are used to estimate the shift in the position of the knee from multiple reflections and light trapping in GaAs wafers with nitric acid etched backs. From these results, the scattering parameter for nitric acid etched GaAs is estimated and compared to measurements using laser light scattering. Finally, the position of the knee given by the optical bandgap, the bandgap absorption coefficient, the width of the absorption edge, the subedge absorption, the scattering parameter, and wafer thickness (see Eq. 4-44), is compared to the position of the knee given by the calibration curve. §54  Oscillations in the Width of the Knee  The growth of wide bandgap material such as AlGaAs on a smaller bandgap substrate such as GaAs, produces thin film intensity oscillations in the spectrum of the light transmitted through the substrate. The index of refraction of the wide bandgap overlayer is lower than that of the substrate, partially reflecting the light as it travels through the interface. The partially reflected light wave interferes with the incident wave causing thin film interference in the epilayer. A ray diagram depicting multiple reflections and in a wide bandgap epilayer of thickness dj-, is shown in Fig. 5-1. In this figure r and r are the reflection ;  2  coefficients for light at normal incidence to the vacuum-epilayer interface and the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  139  epilayer-substrate interface, respectively. The indices of refraction are, n^ for the overlayer and n for the substrate, with n^ <n . s  The complex amplitude for light  s  transmitted through the overlayer is tj for light incident on the vacuum side of the layer, and t for light incident on the substrate side of the layer: 2  (J-r )(i-r )exp(-ig/2) 7  2  l + r r exp(-i5) 1  2  . '  l-n ^Jj^L  .  f  r r exp(-iS) }  2  f  l '+ n  f  f  X  s  n +n  1  4nnfdf l+  _n -n  f  he  f  s  (5-1)  hv .  F i g . 5-1. T h e t r a n s m i s s i o n o f l i g h t t h r o u g h an e p i l a y e r w i t h an i n d e x o f refraction that differs f r o m that o f the u n d e r l y i n g material.  For optical bandgap measurements done in reflection, the fundamental ray passes through the overlayer two times, once in each direction. Transmission of  Chapter 5  Diffuse Reflectance Spectroscopy During  Growth  140  the fundamental ray including thin film interference in the wide bandgap overlayer is given by , * Tj = t/ t/ = — - i }  (1-R,) (l-R ) \ 2  2  2 2 )  1 }  2  [1 + RjR + 2sjRjR 2  .  2  •  (5-2)  cos 5)  2  The diffuse reflectance spectrum from a substrate during the growth of a wide bandgap overlayer is given by the product of T and the spectrum from the t  substrate with no overlayer. This multiplicative term is analogous to the optical throughput of system: see for example Eq. 4-39. The relationship between T , the {  optical throughput, g, and the slope of the optical throughput 1/E is S  = TF\^1  S  V /) 7  ~ ^L  [l  Rl  [1 + RjR + 2  ^ = g' = 48'Jlyi sm5 E 2  s  . ^l-4^R cos8 2  ;  2  RjR cosS)  (5-3)  2  ; $' = — " he  .  Where the interface reflectivities are Rj = r\ and R = r with R 2  f~l  n  Rr = 1  fl  n  +  ; JR-, = 2  An  2nj-+An  An 2n^ V  An  2  2  « l and  ; An = n -n >0 s  f  .  (5-4)  fJ  2n  In this analysis absorption in the overlayer is neglected because the wide bandgap material is transparent at the wavelength region of the onset of transparency of the substrate. Furthermore, interference effects are considered only for the fundamental ray. This approximation is valid for wavelengths around the onset of transparency of the substrate, where the contribution to the intensity from the higher order rays is small. The period of the sinusoidal oscillation in the optical throughput decreases with epilayer thickness. These oscillations cause the position and the width of  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  141  the knee to oscillate with an amplitude that increases as the layer thickens. The position and width of the knee oscillate in phase when the spectrum is plotted in terms of wavelength, and out of phase when the spectrum is plotted in terms of energy: see for example Eqs. 4-47 and 4-48. A n analytical expression for the oscillations in the width of the knee is derived from Eqs. 4-47, 4-48, 4-51, and 5-3, and shown in Eq. 5-5. The amplitude of the oscillations increase with overlayer thickness, the Urbach parameter, and the difference in the index of refraction of the overlayer and the substrate. E  r  A  = A,  he  V  _ E X Qxp(a A )  _E X (l  2  0  An df sin 8 2nfj  A  k  m  2  c  0  k  + a A) m  (5-5)  c  AJic  A hc 1 + m  The oscillations in the width of the knee are observed during the growth of a 4.2 pm thick A l G a A s layer on a 450 pm thick semi-insulating GaAs 5  5  substrate with a polished back surface. The substrate is rotated at 0.5 Hz during growth to improve the deposition uniformity. Variations in the layer thickness cause phase variations across the substrate which reduce the effects of thin film interference. The substrate heater is maintained at constant power to maintain a constant substrate temperature.  The knee region of the diffuse reflectance  spectrum is scanned, recorded, and fit using Algorithm A , at intervals of about 1 min. The width of the knee, X , as a function of layer thickness, from 2.5 pm to a  4.2 pm, is shown in Fig. 5-2. The solid line in Fig. 5-2 is a fit of Eq. 5-5 to the data. The following parameter values are obtained from this fit:  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  X =3.62 + 0.02 nm ,  => a A  a  m  = 0.1733 ± 0 . 0 0 0 1 / m i ,  2n  142  = 0.080 ;  c  =»n =3.32; f  J  (5-6)  f  2)  QI 2.5  i  i  i  i  i  i  i  i  i  3.0  i  i  i  i  i  i  3.5  i  i  i  4.0  i  I 4.5  A l G a A s Layer Thickness (|im) F i g . 5-2. O s c i l l a t i o n s i n the w i d t h o f the k n e e d u r i n g the g r o w t h o f Al.5Ga.5As o n G a A s (dashed line). T h e s o l i d l i n e is a fit to E q . 5-5.  From these values R = 0.288, R = 0.0014, and n = 3.58. The average }  2  s  substrate temperature during growth is 567.3 °C {X = 1150.4 nm). The inferred k  index for AlGaAs at 567 °C and 1.15 urn is 3.32. This suggests the A l content is 44% [61], which is close to the 50 ± 2% concentration determined from the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  143  growth rates determined by calibration of the A l and Ga effusion cells with a scanning electron microscope. The index of refraction for the GaAs substrate at 567 °C and 1.15 pm is 3.58 ± 0.05, where the uncertainty is a combination of the standard error [35] determined from the fit (0.02) and the uncertainty in the growth rate calibration (0.04). This value of the index of refraction for the GaAs substrate is close to the value of 3.68 for GaAs at 579 °C [61]. The experimental value of a A m  c  = 0.080 is in agreement with 0.070 given by the model in Eq. 4-53.  The position and the width of the knee, during the growth of AlGaAs on GaAs, are compared in Fig. 5-3. As predicted, the position and the width of the knee oscillate in phase.  The position of the knee also exhibits larger period  fluctuations due to small variations in the temperature of the substrate during growth.  The width of the knee is insensitive to these small differences in  temperature. However, the width of the knee is sensitive to large rates of change in temperature; in which case rapid shifts in the position of the absorption edge while the knee is scanned, cause fluctuations in the width of the knee. The data shown in Fig. 5-3 is used to determine the effectiveness of the algorithm given in Eq. 4-55, where spurious shifts in the knee are sensed by the width of the knee and corrected for. In this case the corrected position of the knee is iteratively given by (5-7) where X and X are the experimentally measured position and width of the k  a  knee, and E = 11.5 meV at 567 °C. The temperatures given by the measured 0  position of the knee (dashed line) and corrected position of the knee (solid line),  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  144  are shown in Fig. 5-4. Also shown, is a smooth fit to the position of knee (broken line), which is assumed to represent the true temperature of the substrate. The root mean square (RMS) temperature fluctuations about the true temperature are 1.5 °C for the measured knee and 0.7 °C for the corrected knee.  1153  5.0  4.5  a  4.0  3.5  3.0 3.5 AlGaAs Layer Thickness (|im)  3.0  4.0  Fig. 5-3. The position of the knee (solid line) and the width of the knee (dashed line) during the growth of Al.5Ga.5As on GaAs. The correction algorithm reduces the RMS fluctuations in the temperature to 0.7 °C which is within a few tenths of a degree of the R M S background (typically around 0.5 °C). Using the additional information given by the width of knee improves the absolute accuracy of the measurement. However, the cost of this additional information is poorer sensitivity, because adding another experimentally measured parameter adds to the noise. Defining AE and AE as k  a  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  145  the standard errors [35] in the positions and the widths of the knee, it is observed that AEJE  a  = 0.010 and AE /E k  k  = 0.00006 independent of temperature and  doping concentration. The noise in the width of the knee is 160 times larger than the noise in the position of the knee; however, since the energy of the knee is about 100 times larger than the Urbach parameter, the noise in the position and the width of the knee contribute about the same temperature error. For example, for a 500 pm thick semi-insulating GaAs substrate at 500 °C, the measurement uncertainty in the knee contributes AE 1.8 °C/meV = 0.12 °C to the noise and k  the width of the knee contributes AE 1.8 °C/meV = 0.20 °C to the noise. a  572  2.5  Growth Time (hr) 3.0  3.5  570 h  1152  1150  564  Temperature( A,k) Temperature( X^,X ) True Temperature  A 1148  a  562 2.5  3.0 3.5 AlGaAs Layer Thickness (pm)  4.0  Fig. 54. Substrate temperature given by the position of the measured knee (dashed line), the position of the corrected knee (solid line) and the true temperature (broken line), during the growth of Al.5Ga.5As on GaAs.  |  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  146  The effect of thin film interference on optical bandgap thermometry can be problematic during the growth of II-VI material on III-V substrates [62]. In this case the index of refraction of the substrate and the overlayer differ by about one. This means the amplitude of the intensity oscillations are twice as large as those for AlGaAs on GaAs. The effectiveness of the algorithm in Eq. 4-55 and other algorithms in correcting for thin film interference are further explored in Appendix II using data simulations of the growth of ZnTe on GaAs. In these simulations a real diffuse reflectance spectrum from a semi-insulating GaAs substrate is multiplied by a simulated interference term. These simulations explore the effects of material systems that can not be grown in the present M B E system. Both Algorithm A and algorithms that use the derivatives of the spectrum to determine temperature are investigated. In Appendix II, a novel way of obtaining the derivatives of the spectrum using the wavelet transform is developed. The definition of wavelets and the wavelet transform are given in §11-1. The sensitivity of the wavelet transform to the growth of AlGaAs on GaAs is shown in §11-2. The relation between the wavelet transform of the spectrum and the derivatives of the spectrum is discussed in §11-3 and §11-4. The sensitivity of the Fourier transform of the diffuse reflectance spectrum to thin film interference is discussed in §11-6. In §11-5 a detailed analysis of algorithms that correct for fictitious temperatures shifts in optical bandgap thermometry is presented. Thin film interference oscillations are simulated for the case where the index of refraction is 3.5 for the substrate and 2.5 for the overlayer, and where the overlayer thickness is varied from 1.8 to 2.5 Lim. The ability of Eq. 4-55 and variations on Eq. 4-55 to  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  147  correct for spurious shifts in the position of the knee, is examined. The results are as follows: using Algorithm A without correction the RMS temperature error is 2.2 °C, after correcting with Eq. 4.55 the error is reduced to 1.4 °C, by modifying Eq. 4-55 to accommodate the phase difference between the position and width of the knee the error is reduced further to 1.3 °C, and finally by doubling the amplitude of the correction term, in addition to the phase correction, the error is 0.7 °C. These results are summarized in Appendix II, Table II-1. Another algorithm is developed in Appendix II that uses the peak of the fourth-derivative (wavelet-transform) of the spectrum as a reference for temperature. The RMS temperature error is 2.0 °C when the width of the wavelet is  2.0E  o  and 1.6 °C when the width of the wavelet is 1.5E . By linearly Q  extrapolating from these peaks to a width of zero the RMS temperature error is reduced to 0.5 °C. These results are summarized in Table II-2. This algorithm relies on the fact that thin film interference causes an asymmetry in the peak of the fourth derivative of the spectrum that oscillates as the overlayer thickens. Oscillations in the asymmetry of the peak are then used to correct for oscillations in its position. §5*2  Measuring Composition of Small Bandgap Epilayers  When the bandgap of the epilayer is smaller than the bandgap of the substrate, absorption by the epilayer becomes an important factor in diffuse reflectance measurements. In this case the onset of transparency of the epilayer is at longer wavelengths than the onset of transparency of the substrate. This means the knee of the substrate is obscured when the overlayer is thick. The  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  148  position of the knee of the overlayer occurs at energies around its bandgap energy, when the overlayer is around 1 pm thick. See for example Eq. 445. A second knee forms in the diffuse reflectance spectrum in the region of the bandgap of the overlayer. Since the absorption curve is not as steep above the bandgap, as below the bandgap, the knee of the overlayer is broader than the knee of the substrate. A series of diffuse reflectance measurements, during the growth of InGaAs on GaAs, are shown in Fig. 5-5. The back of the substrate is textured with nitric acid and the substrate heater is set at constant power during the growth. This keeps the substrate temperature constant at about 530 °C during the growth of the InGaAs layer. The In concentration of the overlayer is nominally 24%. The smaller bandgap of the InGaAs overlayer produces a second knee in the diffuse reflectance spectra at longer wavelengths. The thickness of the InGaAs overlayer for each scan is shown in the lower left-hand corner of Fig. 5-5. As the layer thickens the signal of each subsequent scan decreases and the second knee becomes more pronounced. Furthermore, as the layer thickens the second knee shifts to shorter wavelengths because the absorption coefficient for the onset of transparency of the overlayer moves to lower values on the absorption edge. When the overlayer is 1.0 pm thick the position of the second knee is at 1299.9 ± 0.3 nm. This means the bandgap of the InGaAs layer is around 1300 nm at 530 °C. When the overlayer is 1.8 pm thick the position of the second knee is at 1332.3 ± 0.3 nm. The width of the second knee (19.8 nm) is larger than the width of the knee of the substrate (4.8 nm).  As the overlayer thickens it  eventually becomes opaque in the wavelength region of the onset of  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  149  transparency of the substrate and the temperature of the substrate is no longer measurable using the diffuse reflectance technique.  1100  1150  1200  1250  1300  1350  1400  Wavelength (nm) F i g . 5-5. D i f f u s e reflectance spectra d u r i n g the g r o w t h o f I n G a A s o n G a A s at 5 3 0 ° C . T h e thickness o f the I n G a A s layer for each scan is s h o w n o n the left-hand side. A s the I n G a A s layer thickens the s i g n a l decreases.  The second knee can be used to measure epilayer composition provided a calibration is done. For example, by choosing a layer thickness around 0.8 pm, the substrate temperature and the onset of transparency of the epilayer are both measurable. By choosing a growth temperature of say 500 °C where the sticking coefficient of In is one and by using a known growth rate calibration for the In and Ga cells, the position of the second knee as a function of In concentration for a fixed temperature of 500 °C, and a fixed layer thickness of 0.8 pm is  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  150  determined. Once this dependence is known, InGaAs layers 0.8 Lim thick can be grown at a higher temperature, say 500 to 600 °C, then cooled down to 500 °C and the composition measured. Once the composition is determined the sticking coefficient of In at the higher temperature can be determined. Furthermore, this curve can be used to calibrate the growth rate for the In cell, provided the substrate temperature is 500 °C or less and the Ga cell is calibrated.  For  temperatures above 500 °C the re-evaporation of In complicates and reduces the accuracy of the calibration. The composition of quantum wells can not be determined using this method because the second knee is not discernible in spectra from substrates with thin, 10 nm thick, overlayers. §5*3  Growth Rate, Index of Refraction, and Composition  When a similar or larger bandgap material is grown on the substrate, thin film interference can be used to determine growth rate. The accuracy of this method is limited in part by how accurately the index of refraction is known as a function of temperature and composition. Conversely, when the growth rate or layer thickness is known the composition is inferred from the index of refraction. In measuring the growth rate of an epilayer that has an index of refraction the same as or similar to the substrate a thin marker layer with a different index of refraction is used to generate thin film interference. For the growth of GaAs on GaAs an AlGaAs marker layer is used. As the light passes through the marker layer it under goes a phase change that depends on the marker layer thickness and index of refraction, and the wavelength of the light. To accommodate this  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  151  phase shift, a phase <> / is added to the cosine term in Eq. 5-2. For the growth of GaAs on AlGaAs, Eq. 5-2 becomes rp _  (l-R ) {l-R ) exp(-2ad ) 2  2  ]  2  f  (; + RjR exp(-2ad )2  = (l-Rj)  2^R R  f  t  exp(-2ad )[l  2  exp(-ad )cos(8  2  f  + 4- ]R R exp(-ad )cos(8  f  s  ]  2  f  + 0))  (5-8)  + 0)) .  Where " / i s the index and R is the reflectivity of the GaAs overlayer. }  The index of refraction of the GaAs overlayer is greater than that of the AlGaAs marker; therefore the sign of the phase shift at the interface is a reversed from that given in Eq. 5-2. This reversal is shown by the reversal of the sign of the ^]R]R term in Eq. 5-8. Interference also occurs in the marker layer causing 2  the reflectivity at the marker layer, R , to be a more complicated function of 2  marker layer thickness and interface reflectivity. Nevertheless, R «  1 and the  2  same simplifications used in §5-1 are used here. In this analysis an attenuation term is added for absorption in the overlayer with thickness dj.  Combining  Eq. 5-8 and the diffuse reflectance spectrum of the substrate, the diffuse reflectance during the growth of GaAs on an AlGaAs marker is given by DRS = DRS (l + 4jR ^exp(-ad )cos(8 [  The  t  1  f  + (l)j) .  (5-9)  technique described above is used to measure the spectral  dependence of the index of refraction of GaAs at 608 °C, in the vicinity of the absorption edge. A 30 nm thick A l G a A s marker layer and a 850 nm thick 5  5  GaAs overlayer are grown on a semi-insulating GaAs substrate with a textured back. The substrate temperature is 608 °C and the growth rate is 14.2 nm/min, during the growth of the overlayer. The diffuse reflectance is continuously  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  152  scanned from the knee region of the spectrum to the region where the spectrum rolls over approaching maximum transparency (upper knee). The interference effects are strongest at long wavelengths where the substrate is transparent. The temperature is determined from the position of the knee. The diffuse reflectance as function of overlayer thickness, for individual wavelengths spanning the region from the onset of transparency of the substrate to the upper knee, is shown in Fig. 5-6. These wavelengths are at intervals of 10 nm, starting with 1190 nm (lowest signal) and ending with 1250 nm (highest signal). For comparison, the wavelengths of the knee, the inflection point, and the upper knee are about 1190 nm, 1210 nm, and 1235 nm respectively. The solid lines through the data in Fig. 5-6 are fits to Eq. 5-9. In Eq. 5-9 all rays are assumed to have the same interference as the fundamental ray, this is approximation as the rays scatter at the back of the substrate. This approximation makes the amplitude of the cosine term unreliable. However, it has little effect on the period of the oscillations which give the most reliable information on the overlayer. From these fits to Eq. 5-9 an absorption coefficient around 10 cm- for 4  1  the overlayer is obtained. The source of damping in these oscillations is due to the a lack of uniformity in the GaAs overlayer thickness, and not absorption in the overlayer. The overlayer is transparent at these wavelengths, that is  ad«l.  Non-uniformity in overlayer thickness causes spatial variations in the phase of the interference fringes. A n independent S E M measurement of the epilayer thickness gives the relative spatial variation in the Ga flux at the surface of the substrate as 0.03 mm" . The substrate is not rotated during this experiment and the diffuse 1  reflectance is sampled over 5 mm; this means the overall spatial variation in the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  153  growth rate during these measurements is 15%. As the overlayer thickens the spatial variation in the phase of the thin film interference increases, causing the amplitude of the oscillations in the diffuse reflectance to decrease with thickness.  -  0  .llQQnm  200  400  600  800  Layer Thickness (nm) Fig. 5-6. Diffuse reflectance at individual wavelengths; the curves are at 10 nm intervals, starting at 1190 nm for the lowest signal, and ending at 1250 nm for the largest signal. The phase angle, 0 , and the interface reflectivity, R , are found to vary 2  from -23° and 0.0058 at 1210 nm to -46° and 0.0044 at 1250 nm. The index of refraction for GaAs at 608 °C (solid circles), determined from the fit to the period of the interference oscillations is shown in Fig. 5-7.  The published room  temperature values [63] are also shown. The error bars are given by the standard error in the fit to the period of the interference oscillations. For comparison  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  154  purposes, the spectral dependence of the index of refraction is given in units of the Urbach parameter shifted so the bandgap energy is at the origin. The spectral dependence of the index of refraction is related to the absorption edge through a Kramers-Kronig relation and to a first approximation is independent of temperature when plotted in this manner.  3.60 h  -i  1  i  r  T  i  1  r  i  1  1  i  -J  I  I  1_  i  i  r  Marple  fl  o o cd  3.55 h  •H -4—>  & 3.50  0.127  x  la HH  This Work  3.45 h  3.40 h •12  -1  -11  I  I  I  L.  I  •10  -9  -8  -7  ( - *)/ o hv  E  E  F i g . 5-7. I n d e x o f refraction o f G a A s at 608 ° C ( s o l i d c i r c l e s ) , o v e r the w a v e l e n g t h r e g i o n 1190 n m to 1250 n m , i n steps o f 10 n m . T h e p u b l i s h e d r o o m temperature values ( s o l i d line) are also s h o w n . T h e energy scale is i n units o f the U r b a c h parameter.  The solid line through the solid circles in Fig. 5-7 is a quadratic fit to the data in the range of 1250 nm to 1210 nm.  The two outliers at shorter  wavelengths are ignored in the fit, because the lower signal to noise ratio in the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  155  knee region of the spectrum makes them unreliable. The slopes of the fit and Marple's curve are about the same. However, there is a 3.7% dc offset between the two curves. Since this difference is within the uncertainty of the growth rate calibration, it is not known if this offset is real. The relative spectral change in the index of refraction of GaAs at the inflection point of the diffuse reflectance spectrum is 0.0022/E , at both 608 °C 0  and room temperature; while the relative change in the absorption coefficient is 2.72/E or 1200 times larger. Q  This result justifies the constant reflectivity  approximation used in inverting the diffuse reflectance spectra in §2-5. §5'4  Sensitivity to Surface Morphology  The diffuse reflectance measurement is sensitive to changes in the front surface morphology of the substrate.  For example, the front surface of the  substrate roughens during the thermal desorption of the oxide from GaAs. Two diffuse reflectance spectra are shown on a log scale in Fig. 5-8: one at 600 °C before the oxide is desorbed and one at 636 °C after the oxide is desorbed. In this case the oxide desorption temperature is 613 °C. The increase in surface roughness is shown by an increase in the background level of the diffuse reflectance at wavelengths where the wafer is opaque. The increase in surface roughness during the thermal oxide desorption is visible to the unaided eye, when intense visible light is incident on the substrate. This means the temperature uniformity of the wafer is discernible during a thermal desorption of the oxide. The average temperature of the substrate is linearly increased from 500 to 640 °C, during thermal desorption of the oxide, where  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  156  640 °C is above the oxide desorption temperature. During this ramp the oxide starts to desorb at the hottest point on the substrate. This point becomes hazy and as the average temperature of the substrate increases this spot expands. The peripheral edge of the hazy area at any given time defines an isotherm on the substrate. The change in the average temperature of the substrate from the time the hazy area first appears until it reaches the edge, determines the overall uniformity of the substrate temperature.  u  I I I  0.1  1  1  ;  I  I  1  1  1  1  I I  OX  ^  600°c,r< T  i  i  i  o c  i  i  i  i  i  i  ,  i  i  ^ — •  /  CD  i  636°C,T>T  0X  cd  o CD  -4—»  53 CD  Pi CD  /  gcz) 0.01 s3  •  1100  i  i  i  i  i  1150  i  i  /  i  i  i  1200  i  i  i  i  i  1250  i  i  i  i  i  1300  Wavelength (nm) F i g . 5-8. D i f f u s e reflectance spectra f r o m a G a A s substrate at 6 0 0 ° C before the o x i d e r e m o v a l , and at 636 ° C after the o x i d e r e m o v a l .  1350  Chapter 5  §5*5  Diffuse Reflectance Spectroscopy During Growth  157  Measuring Scattering Parameter from Fitting Parameters  In the following analysis, the correction term a A m  c  is determined  experimentally from the fitting parameters of Algorithm A. Recall that a A m  is the  c  shift in the position of the knee of the spectrum, from multiple reflections and light trapping. The relation between a A , m  parameter is given in Eq. 4-52.  c  the fitting parameters, and the Urbach  The fitting parameters are determined by  measuring the position and the width of the knee as function of temperature, for a 430 Lim thick, semi-insulating, GaAs wafer with a nitric acid etched back, mounted in a wafer holder with a P B N diffuser plate. In this case the diffuser plate and the textured back of the wafer both contribute to the diffuse reflectance signal. The temperature dependence of a A m  is determined from these measurements. The  c  results are plotted in Fig. 5-9. The temperature dependence of a A m  c  calculated from the DRS  W  and  DRS models, is also shown in Fig. 5-9, for scattering parameters ( 7 ) 0.10, 0.20, p  and 0.30. A is determined using Eq. 4.33 and a is determined from numerical c  m  simulations where the shift in the knee of the model is compared to A .  For the  c  DRS model a is found to be 0.59, 0.64, and 0.69 for 7 = 0.10, 0.20, and 0.30, W  m  respectively. While for the DRS model a p  m  is found to be 0.60, 0.65, and 0.70  for 7 = 0.10, 0.20, and 0.30, respectively. The measured values of a A m  c  and  those given by the model, rapidly decrease with temperature in the region where cc (T) rapidly increases with temperature (see Fig. 2-20). 0  The residual absorption below the band edge, oc (T), used in the 0  calculation of A , is given by the first equation in Eqs. 5-10, where Eqs. 5-10 are c  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  0.6  "T  0.5  h  0.4  b  I\I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  J  I  I  I  I  I  I  1-  f  i  i  i  i  158  i  i  i  r  0.3 h  0.2 h  •  Measurement D R S Model DRSp Model W  0.1 h-  0.0  _i  0  -J  i i i_  100  I  I  200  I  I  L  300  400  500  700  600  Temperature (°C) Fig. 5-9. Experimentally measured values of a A for a 430 pm thick semi-insulating GaAs substrate with nitric acid etched back (solid circles). Calculated values of a A for the DRS model (dashed lines) and the DRSp model (solid lines), for scattering parameters 0.10, 0.20, and 0.30. m  m  c  c  W  determined from the absorption measurements for GaAs in §2-5. As a more accurate estimation of cx (T), than that given in Fig. 2-20, cx (T) is determined 0  0  by the absorption coefficient where, the linear extrapolation of the slowly varying subedge absorption and the extrapolation of the exponential absorption edge, intersect (see Fig. 2-9). Fits to these data points for semi-insulating and n+ GaAs are, respectively, a (T) = ( l cm" )exp(i.042 + 2.183x - 6.279x - 3.105x + 23.633x ) 1  2  3  4  Q  a (T) = (l cm" ) exp(7.248 + 4.148x - 2.659x -16.089x + 32.586x ) 1  Q  jc = r ( ° C ) / 1 0 0 0 ° C .  2  3  4  1 0 )  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  159  It appears from the results in Fig. 5-9 that the scattering parameter for a GaAs substrate with a nitric acid etched back is around 0.2.  In the high  temperature limit, where the substrate becomes opaque to higher order rays, the model indicates that a A m  c  goes to zero and broadening of the knee due to higher  order rays disappears. The measured values of a A m  c  follow this trend, however  they do not decrease as rapidly with temperature as in the model. This is believed to be, in part, caused by an over estimation of oc (T) at high temperatures, by the 0  absorption curves presented in §2-5. The semi-insulating GaAs wafer used to determine the absorption coefficient in §2-5 is one quarter of the thickness of the one used here, therefore any attenuation in a A m  due to surface absorption will  c  be four times larger in the model calculation than in the measurements. The wafer chosen for the above experiment is from the same batch and has the same thickness and back surface texture as the one used in the original calibration [28]. Therefore this data is used to determine the additional calibration curve required for the implementation of the algorithm given in Eq. 4-55, namely ^a\cai{^k\cai)- This data (solid circles) and a quadratic fit to this data (solid line) and Urbach parameter as a function of the position of the knee, E {X }, Q  kval  are  shown in Fig. 5-10. These curves and the calibration curve T[X ) are given in k  Eq. 5-11, where X is in units of nm. k  T(X ) = -4155.8 + 6.5834X -.0021630X  2  k  k  k  (°C) ;  X (X ) = -32.77+. 058748X -. 00002284X\ (nm) ; a  k  k  E (X ) = -22.13+.04634X -.00001497X  2  o  k  k  k  (meV) .  (5-11)  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  6  15 14  5 h  0  160  7  900  1000  1100  1200  Wavelength at Knee (nm) F i g . 5-10. T h e w i d t h o f the knee and the U r b a c h parameter as functions o f the p o s i t i o n o f the knee for a 4 3 0 u m t h i c k semi-insulating G a A s wafer w i t h a nitric a c i d etched b a c k surface.  §5*6  Ex situ Measurements of Scattering from Textured Wafers  In this section variations in the scattering distribution from textured surfaces is explored using laser light scattering (LLS). In this experiment, diffusely scattered light from the textured back surface of a substrate is detected in back scattering with a Si detector. A schematic of the setup is shown in Fig. 5-11. The sample is mounted on a fixed table and the Si detector is mounted to a rotating arm that is attached to the table. The light source is a normal incidence 633 nm HeNe laser beam; the laser is mounted 30 cm from the sample. The scattering distribution is measured by rotating the detector to the angle 6  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  161  and recording the signal. The detector is 15 cm from the sample and the width of the detector is apertured to 2.5 mm, giving an angular resolution of about 1°.  Si Detector  Laser Beam  Sample F i g . 5-11. S c h e m a t i c o f a p p a r a t u s u s e d to m e a s u r e distributions o f textured wafers.  the  scattering  The scattering distribution for a nitric acid etched GaAs wafer is shown in Fig. 5-12. The solid line is a fit of the scattering distribution (Eq. 4-12) to the data. The scattering parameter is given by this fit is y = 0.31. For comparison the scattering distribution of white paper is shown in the inset of Fig. 5-12. The scattering distribution for several textured surfaces is determined using this technique; the results are summarized in Table 5-1. The cosine law characterizes the scattering distribution of paper better than that of nitric acid etched GaAs. Nevertheless, the cosine law is a good measure of the width of the scattering distribution from nitric acid etched GaAs; the width at half maximum of the distribution and the fit are about the same. The scattering distributions of the other textured wafers in Table 5-1 are narrower than those of nitric acid etched GaAs. As the scattering distributions narrow the fit to the cosine law improves.  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  _J  -80  i  i  i  l _i  .  -40  i  l  i  i__i  0 Angle (°)  L  40  i  i  162  i  l_  80  Fig. 5-12. The angular distribution of scattered laser light (solid circles) from a nitric acid etched GaAs wafer. The solid line is a fit to a power law cosine distribution (y = 0.31). For comparison the scattering distribution from white paper is shown in the inset (y = 0.37). The scattering parameter, 7, is a measure of the width of the scattering distribution. From Eq. 4-12 the scattering parameter is related to the half width at half maximum, AO, by 7ln(2) = -ln(coszif9) = zif9 /2; solving for Ad the width 2  of the scattering distribution is given by the following power law of the scattering parameter, Ad = ^2\n(2)^  = 67°^Jy.  Scattering from nitric acid etched GaAs is isotropic, making this surface well suited as a diffuse scatterer for temperature measurement during substrate rotation. Some of the back surface textures supplied by wafer manufactures are  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  163  slightly anisotropic. For example the scattering parameter for the back surface texture on the as received A X T [33] GaAs wafers is about three times larger in the [Oil] direction than in the [011] direction. When viewing this surface at 400x magnification under a Nomarski microscope, an ordered 20 Lim by 10 Lim pattern, in the [011] by [Oil] directions, is observed. It appears that this asymmetric pattern is responsible for the anisotropic behavior in the scattering. In other as received textured wafers, namely GaAs from LDP [64] and InP from A X T [33], no ordered structures were observed under the microscope and no anisotropy is observed in the width of the scattering-distributions. T a b l e 5-1. S c a t t e r i n g parameters at r o o m temperature a n d a w a v e l e n g t h o f 6 3 3 n m for several textured surfaces.  Material  Treatment  Direction  Surface Color  7  GaAs  as received A X T  [Oil]  metallic  0.031  GaAs  as received A X T  [01T]  metallic  0.009  GaAs  H N 0 etch  dark gray  0.31  GaAs  HNO3  etch  light gray  0.17  3  GaAs  as received L D P  metallic  0.011  InP  as received A X T  gray  0.13  PBN  as received  creamy  0.22  Paper  as received  white  0.37  Spectralon  as received  white  0.27  When the substrate is rotated variations in scattering, from the as received A X T GaAs substrates, cause intensity oscillations in the diffuse reflectance scans  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  164  at a frequency twice that of the substrate rotation. If the monochromator scan speed is around 15 nm per substrate revolution, this anisotropy causes observable ripples in the knee region of the spectrum. These ripples increases the R M S fluctuations in the measured temperature by 1 to 2 °C. A fast scan speed compared to the substrate rotation or a fast substrate rotation compared to the scan speed can be used to minimize this effect. For example in the present system, the monochromator scan speed is 1 to 3 nm/s and the lock-in time constant is 1 s, therefore using substrate rotation speeds around 0.5 Hz minimizes this effect. The results in Table 5-1 show that for the textured surfaces considered, the scattering parameter for 633 nm laser light varies by over an order of magnitude. These results are not an exact measure of the scattering parameters for the longer wavelength light that is scattered off the texture surface from inside the wafer. Nevertheless they are good indicators of the variation one might expect in using different textured surfaces as diffuse scatterers.  Furthermore in the case of  semiconductor substrates, these results show that color is a good indication of the width of the scattering distribution; for example, when the backside of the wafer has a gray matte finish the scattering parameter is likely greater than 0.1 and the darker the finish the broader the scattering distribution. For scattering parameters below 0.03 the finish on the wafer appears metallic, like the surface of a polished wafer. From these results it appears that three calibration curves are required: one for wafers with polished backs (y < 0.01 or AO < 7° at visible wavelengths), one for wafers with textured backs that have a metallic finish (0.01 <y< 0.03 or 7° <A6< 12° at visible wavelengths), and one for wafers with gray matte backs (0.1 < y< 0.3 or 21° < AO < 37° at visible wavelengths).  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  165  A better measure of the scattering parameter, for infrared light incident on the textured back from inside the substrate (during diffuse reflectance measurements), is the scattering distribution of back scattered infrared light (incident of the front surface), that exits back out the front of the substrate. The problem with this type of measurement is that for broad scattering distributions only the first 16° of the peak of distribution escapes from the substrate. For broad distributions, measurements covering 16° are not wide enough to characterize the width of the distribution. Nevertheless, this technique works for narrow distributions such as the one given in row five of Table 5-1 for GaAs. In order to compare the scattering results for 633 nm laser light to those of infrared light, the scattering distribution of the GaAs wafer described in row five of Table 5-1 is also measured using a normal incident (1046 nm) Y L F infrared laser beam. At room temperature GaAs is transparent at 1046 nm, therefore narrow scattering distributions for 1046 nm laser light incident on textured surfaces can be compared for the both the GaAs-air interface and the air-GaAs interface. For 1046 nm laser light incident on the backside of the wafer the scattering parameter is about 30% smaller, at y = 0.008, than that for 633 nm laser light. A surface of a given roughness appears smoother to longer wavelength light; therefore one would expect the scattering parameter to decrease as the wavelength increases. It is also found that varying the intensity of the laser beam does not effect the width of the scattering distribution. For 1046 nm laser light incident on the polished front of the wafer, the width of the broadened scattering distribution outside of the wafer is given by y = 0.15. Correcting the broadened distribution, using Snell's law, the width of the scattering distribution inside the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  166  wafer is determined to be 7 = 0.010. The width of this scattering distribution, AO = 7 ° , is small and it is therefore not necessary to corrected for the angular dependence of transmission through the GaAs-air interface. For angles less than 14°, the variation in transmission of randomly polarized light through the GaAsair interface is within 99% of that for normal incident light. The width of the scattering distribution inside the wafer (for 1046 nm laser light) is 10% smaller than that for 633 nm laser light scattered from the backside of the wafer and 20% larger than that for 1046 nm laser light scattered from the backside of the wafer.  From these measurements the scattering parameter  decreases with wavelength and is slightly larger for light scattered from the textured GaAs-air interface than from the textured air-GaAs interface. The scattering parameter (y  = 0.2) determined by comparison of the  IR  models for, and measurements of, the correction term for nitric acid etched GaAs in §5-5, is close to the values obtained using 633 nm laser light scattering (y = 0.31 and 0.17). The smoother y  633nm  633nm  = 0.17 surface was obtained by less  etching than the standard 15 s etch in concentrated nitric acid. The wafer used in the experiment in §5-5 received the standard etch and has a dark gray matte finish similar to that of the y  633nm  represented by the y  633nm  = 0.31 surface, and is therefore more closely  = 0.31 surface. The scattering parameter for nitric acid  etched GaAs given by laser light scattering is about 50% larger than that determined by the experiment in §5-5. This discrepancy is, in part, due to the decrease in the scattering parameter with wavelength and most likely temperature. For example, in Fig. 2-5 the diffuse reflectivity of P B N decreases with both temperature and wavelength. Furthermore, this discrepancy could be,  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  167  in part, an artifact of the DRS and DRS models. In the scattering models the W  p  trapped ray is assumed to remain trapped when it strikes the textured back of the wafer and can not escape until after it scatters. This is most likely the case for lightly textured surfaces that appear metallic in color. However, for rough gray matte surfaces this may not be true and the reflectivity during the scattering event may be less than unity. Inclusion of this reflectivity term in the model would be easy, however, this would mean that the effect of scattering is described by two parameters instead of one. Both of which are difficult to accurately measure and likely depend on wavelength, temperature, and color of the textured surface. Therefore in order to simplify the models, the effect of scattering is described by a single parameter, 7, which is assumed to be independent of wavelength and temperature.  In this  context it appears that for modeling diffuse scattering of infrared light inside substrates with textured backs, the scattering parameters given by scattering of 633 nm light from the backside of the textured wafers should be reduced by about 35%. In Appendix II §11-5, a detailed analysis of the ability of Eq. 4-55 to correct for spurious shifts in the position of the knee caused by variations in scattering at the back of the substrate, is presented. In this analysis, a linearized version of Eq. 4-55 (see Eq. 11-25) gave the best results, with temperature errors less than 2 °C when the scattering parameter is increased from 0.0 to 1.0. When using only the position of the knee to determine temperature, the temperature error is 10 °C when the scattering parameter is increased from 0.0 to 0.3 and over 30 °C when the scattering parameter is increased from 0.0 to 1.0.  Chapter 5  §5-7  Diffuse Reflectance Spectroscopy During Growth  168  Comparing the Model with Calibration Curves  The position of the knee determined from the model in Eq. 444 (solid line) is compared to the position of the knee given by the calibration curve in reference [28] (broken line), in Fig. 5-13. The substrate is a 450 urn thick semiinsulating GaAs wafer with a nitric acid etched back. The bandgap function and the Urbach parameter function used in the model are based on the phonon occupation number and are given in Eqs. 2-5 and 2-7, the parameter values for these equations are given in Table 2-2. The temperature range of the calibration  1200 i 1100  900 800  -200  0  200  400  600  Temperature (°C)  F i g . 5-13. T h e p o s i t i o n o f the k n e e g i v e n b y a c a l i b r a t i o n c u r v e ( b r o k e n l i n e ) a n d a m o d e l ( s o l i d l i n e ) , for a 4 5 0 u m t h i c k s e m i - i n s u l a t i n g G a A s substrate w i t h a nitric etched back. F o r c o m p a r i s o n the b a n d g a p w a v e l e n g t h o f s e m i - i n s u l a t i n g G a A s is also s h o w n .  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  169  curve is 75 to 600 °C. The bandgap wavelength (Eq. 2-5) for semi-insulating GaAs is also shown. The correction term a A m  c  is calculated using Eq. 4-33.  In the above calculations the substrate characteristics are: 450 pm for the substrate thickness, 0.20 for the scattering parameter, and 0.64 for a . The model m  over-estimates the substrate temperature by 14 °C at 75 °C and by 27 °C at 600 °C. Increasing the temperature dependence of the bandgap (S ) by 2% makes the model match the calibration curve. Increasing the wafer thickness by a factor of 2.5 also makes the model fit the calibration curve.  Increasing the  scattering parameter shifts the model toward the calibration curve at lower temperatures where the substrate is transparent. However, at high temperatures, where the substrate is opaque to higher order rays, increasing the scattering parameter does not move the model toward the calibration curve. The absolute accuracy required in optical bandgap  thermometry  establishes the precision needed in determining the calibration curve.  For  example, when the bandgap as a function of temperature [58] is used as the calibration curve for the position of the knee, the absolute accuracy of the measurement is within 100 °C. Adding the difference between the position of the knee and the bandgap at room temperature, to the bandgap as a function of temperature, improves the accuracy to within 50 °C. For an absolute accuracy within 20 °C the calibration curve given by the model in Eq. 4-44 is sufficient. This requires a fairly accurate knowledge of the physical properties of the substrate. The best absolute accuracies are obtained by calibrating the position of the knee against another temperature sensor, such as a thermocouple. In this case the absolute accuracy is limited by, the absolute accuracy of the  Chapter 5  Diffuse Reflectance Spectroscopy During Growth  170  thermocouple and the quality of the thermal contact between the thermocouple and the substrate.  The sensitivity and reproducibility of the measurement  technique are about 1 °C for GaAs substrates [59].  171  Chapter 6 Radiatively Heated Substrates §64  Temperature Uniformity  Semiconductor lasers are manufactured from compound semiconductor alloys containing two to four different alloying elements from group II, III, V or V I in the periodic table. Thin epitaxial films of these materials are fabricated by chemical or physical vapor deposition techniques such as metal organic chemical vapor deposition or molecular beam epitaxy (MBE).  In these deposition  processes the temperature of the substrate during the epitaxial film growth affects the electronic properties of the film, such as the non-radiative recombination lifetime of electrons and holes. The optimum growth temperature is about 600 °C for GaAs and about 700 °C for AlGaAs.  In addition the temperature of the  substrate affects the composition of the film.  For example at the high  temperatures needed to grow high quality AlGaAs the vapor pressure of the Ga is high enough that some of the deposited Ga evaporates. The fraction of the Ga which evaporates depends critically on temperature so that even a small change in the substrate temperature will change the composition of the film. Thus for high uniformity in the composition and electronic properties of the epitaxial layer it is important to have a controlled and uniform temperature during growth. To  Chapter 6  Radiatively Heated Substrates  172  achieve this objective it is important to be able to measure the wafer temperature at any location on the wafer quickly and accurately. §6*2  Spatial Temperature Profiling  As a first step in minimizing temperature gradients in the substrate it is necessary to measure the temperature profile across the surface of the substrate. Diffuse reflectance spectroscopy is well-suited to temperature profiling.  The  substrate is illuminated in the region to be profiled by the broad spectrum chopped light source shown in Fig. 6-2. The optical detection system is modified to profile the substrate temperature as shown in Fig. 6-1. The illuminated substrate is imaged with the collection lens onto an adjustable aperture in front of the optical fiber bundle. The area of the wafer of interest is selected by locating its position in the image of the brightly lit substrate and substrate holder and moving the appropriate location on the image so that it lines up with the hole in the aperture. The diameter of the aperture is typically 0.7 mm which for a 7.5 cm focal length collection lens corresponds to a spatial resolution of about 5 mm at the usual working distances in a commercial Vacuum Generators M B E system. The temperature profile of a scan line across the substrate is obtained by scanning the aperture across the image of the substrate, while measuring the diffuse reflection spectrum at each location. The aperture and fiber bundle are scanned together with a micrometer driven stage to avoid any possible complications associated with changes in the optical path of the light as it passes through the fiber bundle and into the monochromator.  Chapter 6  Radiatively Heated Substrates  173  Optical fiber Fig. 6-1. Schematic diagram of the optical system used to collect the diffusely scattered light in the substrate temperature profiling measurements.  §6*3  Factors Affecting Temperature Uniformity The diffuse reflectance spectroscopy temperature measurement technique  is used to spatially profile the temperature of GaAs substrates in-situ. The effect of different In-free [65-68] wafer mounting techniques on the thermal uniformity of the substrate are explored. Aperture diameters between 0.4 mm and 0.7 mm are used in this experiment, which for the 7.5 cm focal length collection lens gives a spatial resolution between 3 mm and 5 mm on the substrate.  Chapter 6  Radiatively Heated Substrates  174  A manual temperature profile takes about 20 min, which means that the wafer temperature must be stable as a function of time in order for the measured temperature profile to be reliable. To check this stability the steady state temperature at the center of an n+ GaAs wafer is measured every minute over a 1.5 hour period, while the substrate heater was held at constant power. The results are shown in Fig. 6-2. The substrate temperature is stable for 0.7 hour then drops by about 1.2 °C in less than 10 minutes. Small random shifts in temperature were observed in every wafer that was measured. It is therefore necessary to verify that the measured temperature profile is reproducible in order not to confuse temporal fluctuations with spatial variations. The root mean square (RMS) fluctuation in the measured temperature in Fig. 6-2 is about 0.4 °C neglecting the long term drift. In other experiments presented here this R M S fluctuation ranged from 0.2 °C to 0.7 °C depending on the measurement conditions. The spatial resolution is limited by the signal-to-noise ratio required in the diffuse reflectance spectrum to obtain an accurate temperature measurement. Since semi-insulating GaAs wafers are more transparent in the infrared than n+ wafers, the diffuse reflection signal for the semi-insulating wafers is larger, particularly at high temperatures. Therefore a smaller aperture is used with semiinsulating GaAs, which makes higher spatial resolution possible. Temperature profiles are performed on 50 mm semi-insulating GaAs substrates with spatial resolutions of 3 mm and 5 mm, with similar results. A lower resolution of 5 mm is used in measuring the temperature profile in n+ wafers.  Chapter 6  Radiatively Heated Substrates  175  595  I I-I  590  • •  S H  585  1.0  1.5  2.0  2.5  Time (hours) F i g . 6-2. T e m p e r a t u r e at a f i x e d l o c a t i o n o n a n+ G a A s substrate at constant heater p o w e r . T h e substrate is a l l o w e d to equilibrate for one h o u r after f i x i n g the heater p o w e r .  In order to better understand the origin of the measured wafer temperature profiles, the temperature profile of the substrate heater is first measured with a pyrometer (Minolta-Land Cyclops 152). The substrate heater consists of 6.5 mm wide, Ta foil strips 0.025 mm thick, spaced 2 mm apart and arranged to cover a circular area 91 mm in diameter. Since the heater is completely enclosed by radiation shields and the substrate in normal operation, it was necessary to remove the substrate in order to be able to see the heater with the pyrometer. Two pyrometer profiles of the substrate heater are shown in Fig. 6-3. In the lower profile the heater is completely open on the substrate side and in the upper profile the heater is covered around its outer perimeter with an annular-shaped M o wafer  Chapter 6  Radiatively Heated Substrates  176  holder designed to hold a 50 mm wafer. The temperature profiles of the heater are obtained by measuring the temperature along the four longest Ta foils that cross the center of the heater. The pyrometer views the heater through an ultra high vacuum (UHV) viewport and a 45° gold-coated Si mirror, placed in an effusion cell port [69]. The emissivity setting on the pyrometer is at 0.09, corresponding to the emissivity at 1.0 pm for Ta (0.1) [70] corrected for the transmission of the viewport (0.92) and the reflectivity of the gold mirror (0.98). — i  U o Ul  to  C tu  U  40 30 20 10 0  tu IH  +->  a u  cu CH  CU  H  .  |  i  |  i  |  i  |  i  |  0 -10 -20 -30 -40 -50 -60 -70  "1  r  t  •  I  i  I  I  i  "  J  I  i  -i  i— —i— —i— —i— —i—•—i—•—i— —r  1  r1  1  i  •  j  <+H  U  i  • •  o a  i  933°C  B tu  |  J  i_  1  1  x  L  1  U  887°C  J  -5  -4  i  -3  L  -2  J  -1  0  .  L  1  2  3  4  5  Position (cm) F i g . 6-3. Temperature p r o f i l e o f the T a foils i n the substrate heater. I n the top p a n e l the heater is p a r t l y c o v e r e d b y an a n n u l a r M o r i n g u s e d for h o l d i n g 5 0 m m substrates. In the l o w e r panel the heater is u n c o v e r e d .  In Fig. 6-3 the open heater is hotter in the middle while the partially covered heater is cooler in the middle. The heater foils are not perfectly flat and  Chapter 6  Radiatively Heated Substrates  177  there is some radiative transfer between the foils. Thus when the heater is open, the foils near the edge of the heater receive less radiation from neighboring foils and are cooler. The annular sample holder on the other hand acts like a radiation shield around the outside of the heater, which reduces heat loss and increases the average temperature of the foils for the same power input. The shield inverts the temperature profile in the center of the heater, making it cooler in the center compared with the area around the center. In Fig. 6-4 and in Table 6 1 , the results of temperature profile measurements on substrates are summarized for different carrier concentrations, back surface textures, and thermal contacts to the holder: Also substrates are mounted with and without a pyrolytic boron nitride (PBN) diffuser plate. Some of the profiles for the 50 mm (2 inch) GaAs wafers, labeled with a letter in the first column in the summary in Table 6-1, are also shown in Fig. 6-4. The profile of a 75 mm (3 inch) n+ GaAs wafer is shown for comparison with the smaller diameter wafers, at the bottom of Fig. 6-4. In Fig. 6-4, the temperature at the center of the wafer is given in the upper left-hand corner of each plot and the temperature profile is shown as a deviation from the center temperature. A l l of the temperature profiles are for a scan line that crosses the central heater foils at an angle of about 20°: This angle is large enough to detect non-uniformities associated with the different heater foils and small enough to stay in the area for which the temperature profile of the foils has been established.  Chapter 6  Radiatively Heated Substrates  178  5 550°C 0 -5 h-SI —  0 -5 -10  b  H  d  1  _L  r  1  •  571°C n+  0 -5 -10  589°C  5 0 -5  603°C  0 -5 -10  589°C  0 -5 -10 -15 -20  588°C  i  i  i  8  n+ i  L  SI i  L  n  • •••  o  n+ i —  0 -5 -10 -15  1  L 1  r  • • ••  552°C n+  -4  _L  _L  -1  0 1 Position (cm)  -i—  F i g . 6-4. S e l e c t e d temperature p r o f i l e s for G a A s substrates f r o m the s u m m a r y i n T a b l e 6-1. T h e temperature at the center o f the wafer is s h o w n i n the upper left-hand c o m e r . T h e letters o n the right h a n d side c o r r e s p o n d to the labels i n the first c o l u m n i n T a b l e 6-1.  The letter in the first column in Table 6-1 is a label for the measurement in that row and the second column indicates whether the wafer is semi-insulating (SI) or n+ GaAs. The third column gives the temperature of the center of the  Chapter 6  Radiatively Heated Substrates  179  wafer at the heater power indicated in the fourth column where power is expressed as a percentage of the maximum heater output. The "+" sign on some of the entries in the power column indicates that the As cracker was on at 900 °C, which increases the heat load on the substrate slightly. The next column labeled "Back" indicates whether the back surface of the wafer is polished (P), saw cut (S), or textured by etching in concentrated nitric acid (A ). The saw cut surface is 7  lightly textured compared with the acid etched surface which has a dark gray, matte finish.  Five different substrate mounting configurations were tested as  indicated by the letters A - E in the second last column of Table 6-1 labeled "Mounting". The mounting configurations are as follows: In A the annular M o ring used to hold the GaAs substrate is clean with no GaAs coating. It is in contact with the GaAs wafer at only three points around the perimeter, and is backed with a 0.64 mm thick P B N diffuser plate separated by 1 mm from the back surface of the wafer. Configuration B is the same as A without the P B N diffuser plate. C corresponds to a Mo holder that is coated on the front with GaAs and makes contact with the wafer all the way around the outside edge of the wafer, with no diffuser plate. D is the same as C except that the wafer is bonded to the M o holder around the perimeter with Ga, and E is also the same as C except that the front surface of the wafer holder has been etched to clean off the GaAs coating from earlier growth runs. The last column in the table is the second derivative of the temperature profile found by fitting a quadratic equation to the temperature profile. This parameter is a measure of the uniformity of the wafer temperature. A positive curvature means that the wafer is cooler in the center.  Chapter 6  Radiatively Heated Substrates  180  T a b l e 6-1. S u m m a r y o f temperature p r o f i l i n g results.  Label  Type  Temp. (°C) Power (%)  Back  Mounting  (°C/cm2)  2" GaAs wafers a  SI  403  40  P  A  0.6  b  SI  550  66  P  A  0.3  c  n+  442  40  S  A  -1.3  d  n+  571  60  S  A  -2.5  e  n+  570  60+  s  A  -2.6  f  n+  466  40  s  B  -2.8  8  n+  589  56  s  B  -2.2  h  n+  590  56+  s  B  -3.0  i  n+  469  40  N  B  -2.2  j  n+  588  56  N  B  -.1.8  k  n+  591  56+  N  B  -.2.0  l  SI  567  65  P  C  2.2  m  SI  603  62  .N  C  -0.1  n  n+  589  60  N  C  -2.4  o  n+  588  60  s  D  -6.8  P  n+  468  40  N  E  -1.0  q  n+  585  56  N  E  -1.9  r  n+  595  56+  N  E  -1.9  552  60  P  A  -2.0  3" GaAs wafer s  n-i-  Substrate heater t  —  933  60  —  —  7.6  u  —  887  60  —  —  -6.0  Chapter 6  Radiatively Heated Substrates  181  The results in Fig. 6 4 and Table 6-1 are explained as follows. First the semi-insulating GaAs wafer polished on both sides in row / has a positive curvature temperature profile similar to the heater with the empty sample holder in Fig. 6-3. The polished semi-insulating GaAs has a total transmittance of 0.5 for the heater radiation at 600 °C [10] and hence acts to first approximation as a window in front of the heater. When the same wafer is textured with nitric acid (ra), the wafer profile changes sign to a slightly negative curvature. Texturing causes the heater radiation to scatter inside the substrate so that the average path length of the heater radiation is longer inside the substrate compared with the polished substrate. This increases the absorption of the heater radiation in the substrate, so the substrate now acts as a partial radiation shield in front of the heater making the center of the heater hotter, but not hot enough to generate a large negative curvature.  The P B N diffuser plate has the same effect on the  polished wafer (b) as texturing the wafer for the same reason; namely it is also not transparent to the heater radiation making the heater hotter in the center. The diffuser plate has very little effect on the temperature profiles for the n+ wafers, as can be seen by the fact that d (with a diffuser plate) and g (without a diffuser plate) have almost the same temperature profile. This is because the n+ wafers are nearly opaque to the heater radiation to begin with, so the diffuser plate has a small effect on the radiation balance. In fact all of the mounting systems for n+ wafers in Table 6-1 have temperature profiles that are hotter in the center. Even though the layered P B N structure is known for its high lateral thermal conductivity (0.63 w/cm/K) [34], it is not large enough to significantly alter the relatively small temperature gradients that are observed in these  Chapter 6  Radiatively Heated Substrates  182  experiments (a few degrees per cm). Nevertheless the diffuser plate has several other attributes which could be significant in particular applications. First because the surface of the P B N is a good diffuse scatterer, it is useful in enhancing the diffuse reflectance signal for wafers that are polished on both sides. This is particularly important in the case of a polished n+ wafer at high temperatures where the signal is weak. Furthermore the P B N diffuser plate has the disadvantage of reducing the heater efficiency by shielding the sample from the heater. This means more heater power is required to achieve the same substrate temperature with the diffuser plate in place. This can be seen by comparing c and/, or d and g in Table 6-1. The diffuser plate also seems to reduce the stability of the substrate temperature as a function of time. Thermal contact between the substrate and the substrate holder can also cause temperature non-uniformity if the substrate holder and the wafer are at different temperatures. In o an n-i- sample is thermally bonded to the holder around its periphery with Ga. This wafer is hottest in the middle and has the largest temperature gradient of any of the samples. The same sample held in the same holder with metal clips without the Ga bond (n) has a much smaller temperature gradient at the same heater power. This means the holder is cooler than the substrate and the standard, light-pressure, mechanical contact between the holder and the substrate makes a poor thermal contact. The coated substrate holder is expected to be cooler than the sample because it has the low emissivity characteristic of Mo on the back where it faces the heater, and the relatively high emissivity, characteristic of polycrystalline GaAs, on the front.  Chapter 6  Radiatively Heated Substrates  183  A comparison of n and q shows that the residual thermal contact between the sample holder and the wafer does have a measurable affect on the temperature uniformity. In n the substrate holder plate is coated with GaAs on the front while in q the front of the wafer holder is clean. The clean sample holder has a lower emissivity on the front than the coated sample holder and is thus expected to be hotter. This brings the holder closer to the temperature of the wafer which should reduce the temperature gradient, as observed. Table 6-1 contains measurements of a number of different configurations at different power levels. There is no clear trend in the temperature gradient with heater power: in some cases the gradient increases and in other cases it decreases with increasing power. Finally a comparison of d and e, or g and h, or j and k, or q and r shows that the heat load from the As cracker raises the substrate temperature a few degrees but has very little effect on the thermal gradient. The effect of opening and closing source shutters is expected to be similar. A number of factors are found to affect the temperature uniformity of semiconductor substrates in MBE including the In-free substrate mounting configuration, the optical properties of the substrate and the holder, and the heater design. Both positive and negative curvature temperature profiles are observed. While the heater and sample mounting configuration could be designed to give a uniform temperature for a given type of substrate at a particular operating temperature, the design would need to be modified if the substrate or growth conditions were changed. Accordingly the most promising general solution to the temperature uniformity problem may be a dual element substrate heater, in which the temperature of the center of the heater can be  Chapter 6  Radiatively Heated Substrates  184  controlled independently of the temperature of the outer part of the heater. The temperature gradient across the heater would then need to be tuned for the particular sample and growth conditions of interest.  Apart from this general  solution, temperature gradients are minimized by reducing the thermal contact between substrate and holder in the In-free mounts, and in the special case of semi-insulating substrates, by texturing the back surface or using a diffuser plate.  §6*4  Substrate Holder Design In conventional In-free mounts the substrate is in contact with holder  around its periphery. In this type of design the holder either shades the edge of substrate from heater radiation or acts as a radiation shield to the emission of radiation from the edge of the substrate.  Furthermore, the thermal contact  between the substrate and holder varies randomly around the periphery of the substrate.  In the present work, the effects of shading and thermal contact are  minimized by redesigning the In-free mounting system. The new holder design uses a three point contact system for attaching the substrate to the holder, where the periphery of the substrate is separated from the holder everywhere except at the contact points. In this system the effusion cell flux can pass between the holder and substrate and onto the substrate heater.  The group III flux will  damage the Ta heater foils, therefore a shield such as a PBN diffuser plate is placed between the substrate and the heater to protect the heater foils. Schematics of two conventional substrate holder designs and the new substrate holder design are shown in Fig. 6-5.  In the upper schematic the  substrate is mounted from the backside of the holder. In this system the holder  Chapter 6  Radiatively Heated Substrates  shields the front of the substrate.  185  In the center schematic the substrate is  mounted from the front side of the holder. In this arrangement the holder shades the substrate. In the lower schematic the new three point mounting system is shown.  Mo Holder  Substrate  Mo Holder  Ta Heater Foils  Substrate Mo Holder  Mo Holder  Ta Heater Foils  Mo Holder  Substrate  Mo Holder  PBN Diffuser Plate t Three Point Contact  Ta Heater Foils  F i g . 6-5. S c h e m a t i c s o f different substrate m o u n t i n g systems: standard b a c k s i d e m o u n t (top), standard front side m o u n t (center), a n d three p o i n t m o u n t w i t h diffuser plate (bottom).  In the new three point holder the effects of thermal contact between the substrate and holder are minimal and reproducible. This design requires a diffuser  Chapter 6  Radiatively Heated Substrates  186  plate behind the substrate, which also has advantages. The diffuser plate protects the heater foils from the effusion cell flux even when a substrate is not in the holder. This means any odd shaped wafer can be used as a substrate, such as wafers with non-standard flats.  In conventional holders the heater foils are  protected by notches built into the holder that accommodate the standard flats on a wafer. In this type of design broken wafers or wafers with non-standard flats can not be used. The diffuser plate also enhances the diffuse reflectance signal used in measuring substrate temperature, particularly for substrates with a polished back surface. A cavity is created between the back of the substrate and diffuser plate. As the group V flux evaporates from the back of the substrate it is partially trapped in this cavity, creating a group V over pressure at the back of the substrate. This over pressure is reduced as some of the group V flux escapes from the cavity around the periphery of the substrate. During growth group V flux enters the cavity around the periphery of the substrate offsetting this loss. The group V over pressure behind the substrate reduces the decomposition of the back surface of the substrate. Controlling back surface decomposition is important, particularly when the optimal growth temperature is above the congruent sublimation point of the substrate.  For temperatures above the congruent sublimation point the  evaporation rate of the group V element is greater than the evaporation rate of group III element; this leads to the formation of group III droplets on the surface of the substrate. For example, GaAs substrates are at about 700 °C for at least three hours during the growth of AlGaAs/GaAs lasers; this temperature is well  Chapter 6  Radiatively Heated Substrates  187  above the congruent sublimation point of GaAs (640 °C). In viewing the backs of GaAs substrates after the growth of these lasers structures, using a Nomarski microscope, Ga droplets are observed when conventional holders are used. Ga droplets are not observed when the growth is done on GaAs substrates with polished or textured backs in holders with PBN backing plates. Using a backing plate to reduce back surface decomposition is even more crucial when growing on InP substrates where the congruent sublimation point is 360 °C. When a InP substrate is heated to 500 °C in the growth chamber under an As over pressure of 10 mbar at the front surface, the In droplets that form on -5  2  the front surface are much larger than the ones that form on the back. This indicates that the P over pressure in the cavity behind the substrate at 500 °C is larger than 10" mbar. 5  During growth on substrates mounted in the new wafer holders with backing plates, group III materials passing between the substrate and holder are deposited on the PBN plate. Some of this material re-evaporates and is deposited around the periphery of the backside of the substrate. This deposition is shown by a hazy ring around the edge of the backside of the wafer after growth. The disadvantages of using a diffuser plate are: increased heater power, poorer radiation coupling between the substrate and heater, and decreased temperature stability. About a 10% increase in heater power is required to attain the same substrate temperature when a PBN diffuser plate is used. Higher power substrate heaters will compensate for this effect. Poor heater-substrate coupling increases the response time of the substrate to heater power changes and increases the size of the fluctuations in substrate temperature at constant power.  Chapter 6  Radiatively Heated Substrates  188  These both increase by about a factor of two when using a PBN backing plate; the RMS fluctuations are around 1 °C and the maximum ramp rate is about 20 °C/min, at the standard growth temperatures for III-V material. These values are within the limits required to grow III-V devices. Another important requirement of substrate holder design in MBE is one touch substrate mounting. Mounting systems with fasteners that require multiple operations, such as nut tightening, violate the cleanliness rule of MBE. In the new three point mounting system the substrates are held in place with W clips. During mounting these clips are lifted, rotated, and lowered into place over the substrate in a single operation. A schematic of the W clip mounting system is shown in Fig. 6-6. This mounting system consists of three M2 threaded holes, at intervals of 120° around the edge of the holder. These holes accommodate fasteners that hold both the PBN diffuser plate and the W clips. Three M2 Mo screws with 0.13 mm (5 mils) thick Mo flat washers are threaded into the holes, from the backside of the holder, fastening the 0.3 mm (12 mils) thick PBN plate to three contact pads. The M2 screws stick up 2 mm (80 mils) above the front surface of the holder to accommodate the 2 mm (80 mils) high M2 Mo nuts which hold the W clips. The W clips are formed from 0.5 mm (20 mils) diameter W wire and have a 3.2 mm (120 mils) diameter eye at one end for mounting. The Mo nuts are 3.1 mm (180 mils) in diameter at the base with a 0.6 mm (25 mils) high shoulder and have a 4.6 mm (180 mils) diameter top. The shoulder of the nut goes through the eye in the clip holding it securely to the holder while the clip rotates freely on the nut. A ball of W is formed on the straight end of W clip by liquefying the end of the W wire with the electric arc from a tungsten inert gas (TIG) welder. The  Chapter 6  Radiatively Heated Substrates  189  purpose of the is ball is insure uniform contact to the surface of the substrate. Sharp features on the end of the clip that holds the substrate are undesirable as they can damage or cleave the substrate. The holding pressure maintained on the substrate by the clip is set by bending the clip so that it just touches the contact pad with no substrate in place. This setting, gently holds the substrate with a force that increases with the substrate thickness and hence substrate mass.  Substrate Mo Holder  PBN Diffuser Plate Mo Screw Mo Washer Fig. 6-6. Schematic of the mounting system for the substrate and the PBN backing plate, at one of the three contact points on the new Mo holders. Complete drawings for the three point mounting holders with P B N backing plates are shown in Appendix III, for mounting 50 mm (2 inch) and 1/4 segments of (100) III-V 50 mm (2 inch) wafers. Also shown are drawings for a holder to mount 4 sections of a 1/4 segment of a 50 mm (2 inch) (100) III-V wafer. These segments are cleaved 11 mm from the vertex of the 1/4 segment along both cleaving planes. The P B N backing plates used on the full 50 mm wafer (and the 4 sections of a 1/4 segment) mounting systems are 0.35 mm (12 mils) thick and 63.5  Chapter 6  Radiatively Heated Substrates  190  mm (2.5 inches) in diameter. For the 1/4 segment of a (100) 50 mm (2 inch) wafer holder, the diffuser plate is a 1/4 segment of a 0.35 mm (12 mils) thick and 76.2 mm (3 inch) diameter P B N disk. The diameters of several 50 mm (2 inch) GaAs wafers from various manufactures vary from 50.4 to 50.8 mm (1.986 to 2.000 inches) with an average value of 50.678 ± 0.015 mm (1.995 ± 0.006 inches). The full 50 mm (2 inch) and the 1/4 segment of a (100) 50 mm (2 inch) wafer holders are designed to have a 0.25 mm (10 mils) clearance around the largest wafers.  191  Chapter 7 Substrate Temperature During InGaAs Laser Growth Temperature control is crucial in growing AlGaAs/GaAs graded index separate confinement heterostructure (GRINSCH) lasers with InGaAs quantum wells, that have reproducible Al concentrations and emission wavelengths. For example: the sticking coefficients of Ga and In depend critically on temperature above 600 °C and 500 °C respectively; the electrical properties of AlGaAs cladding layers are optimal for growth temperatures above 670 °C; and InGaAs quantum wells are grown around 520 °C to insure uniform In incorporation. This means the substrate temperature must be rapidly decreased from about 680 to 520 °C during the growth of the graded AlGaAs and GaAs spacer layers and stabilized at 520 ± 3 °C during the growth of the quantum well. Using diffuse reflectance spectroscopy as a temperature monitoring tool, a substrate heater power recipe is developed that gives reproducible temporalprofiles of substrate temperature during the growth of AlGaAs/GaAs GRINSCH lasers with InGaAs quantum wells. Substrate temperature and substrate heater power, during the growth of an InGaAs laser, are shown in Fig. 7-1. The heater power recipe consists of constant power settings and linear power ramps. The InGaAs laser is grown on a, (100) 4° off toward (111), n+ GaAs wafer. The as received wafer is exposed to an ozone atmosphere under an UV lamp for  Chapter 7  700  Substrate Temperature During InGaAs laser Growth  192  I—'—i—i—I—i—i—i—i—i—"—"—i—i—i—i—i—i—i—i—i—i—"—r  0  40  80  120  160  200  240  Time (min) F i g . 7-1. Substrate temperature and heater p o w e r d u r i n g the g r o w t h o f an I n G a A s laser.  20 min to remove residual organics from the wafer. This process leaves a thick oxide on the substrate. Before starting growth this oxide is thermally removed under an A s flux, by ramping the temperature of the substrate up to about 2  650 °C. As soon as the oxide is removed the substrate is cooled to the growth temperature of the first GaAs layer, which is about 600 °C. The oxide removal is monitored using laser light scattering (LLS) [69] and reflection high energy electron diffraction (RHEED) [1]. Thermal desorption of a thick oxide roughens the surface of the substrate [69]. The first layer grown on the substrate is a 600 nm GaAs buffer layer, this layer smoothes the surface of the substrate and buries any residual impurities left  Chapter 7  Substrate Temperature During InGaAs laser Growth  193  on the surface of the substrate after the oxide is removed. Smoothing of the epilayer is monitored by L L S and RHEED. In laser light scattering the intensity of diffusely reflected laser light is a measure of the R M S roughness of the epilayer. In R H E E D a rough (3-D) surface is indicated by a spotty diffraction pattern and a smooth (2-D) surface is indicated by a streaked diffraction pattern. During the growth of the buffer layer the substrate heater power is held constant. At the beginning of the growth of this layer the temperature of the substrate increases by about 7 °C. This increase in temperature is caused primarily by the increased heat load on the substrate, from the hot (1070 °C) Ga and (1120 °C) Si effusion cells, when the Ga and Si shutters are opened. The decrease in the emittance of the substrate as the buffer layer smoothes also contributes to the increase in temperature. The first layers of the laser including the buffer layer are Si-doped (n = 2*10 c  18  cm- ). The layers adjacent to the 3  quantum well (300 nm on each side) are not doped. The layers grown after the quantum well are C-doped (n = 2*10 v  18  c n r ) except for the 140 nm GaAs 3  capping layer, where n is linearly graded from 2*10 cm" to 1 0 c m . The C18  3  19  -3  v  doping source is a resistively heated carbon fiber bundle [71]; the fiber bundle contains 10,000 carbon fibers. Growth temperatures greater than 670 °C are required to produce step flow growth and good surface morphology during the growth of AlGaAs layers with large A l concentrations. Again the monitoring techniques used to determine this are R H E E D and L L S . The AlGaAs cladding layers in the InGaAs laser are 1000 nm thick with a 60% A l concentration. These layers are grown at 680 °C using constant heater power. To ensure uniform In incorporation, the quantum  Chapter 7  Substrate Temperature During InGaAs laser Growth  194  well is grown at 520 °C. This means the substrate temperature must be rapidly decreased before (and rapidly increased after) the growth of the quantum well to insure both the cladding layers and the quantum well are grown at their optimal growth temperatures. These rapid changes in temperature are shown by a sharp dip in the substrate temperature in Fig. 7-1. The quantum well region of the laser is expanded in Fig. 7-2 to better show the rapid substrate temperature and heater power changes. The laser structure is superimposed over the data in Fig. 7-2.  700 90  80 CZ)  CP  < cd o  CD  70 Z o CD  Temperature Power GRIN _1  135  140  145  150  I  60  3  50  L.  155  Time (min) F i g . 7-2. Substrate temperature and heater p o w e r d u r i n g the g r o w t h o f an I n G a A s quantum well.  In the first part of the growth shown in Fig. 7-2 the A l concentration is graded from 60% to 15% over 110 nm, after which the A l shutter is closed. Next  Chapter 7  Substrate Temperature During InGaAs laser Growth  195  a 30 nm GaAs spacer layer is grown, and after that the In shutter is opened and a 6.3 nm I n G a A s quantum well is grown. During the growth of the graded 2  8  index (GRIN) AlGaAs layer, the temperature of the A l effusion cell is linearly ramped (in time) from 1066 °C to 972 °C. This roughly produces an A l concentration that decreases linearly, in units of layer thickness, from 60% to 15%. The substrate heater power is held constant until the A l concentration drops below 40%, at which time the power is linearly decreased from 84% to 56%.  In order to maintain step flow growth throughout the structure, the  substrate temperature is held above 670 °C until the A l concentration drops below 40%. The heater power is reduced another 6% during the growth of the GaAs spacer. The above recipe results in a substrate temperature of 520 ± 3 °C during the growth of the quantum well. The laser structure is symmetric about the quantum well. However, because of the slow response of substrate temperature to a heater power changes, an asymmetric power ramp is used in the region of the quantum well to achieve a temperature profile that matches the symmetry of the quantum well region. The slow response of the substrate is due to the relatively large thermal mass of the substrate heater and holder and the manipulator to which they are mounted. The position of the minimum temperature during the growth of the quantum well typically varies from the left-hand edge to the righthand edge of the quantum well using this heater power recipe. In the growth shown in Fig. 7-2 the minimum happens to occur at the left-hand edge of the quantum well. Furthermore the temperature of the minimum typically varies from about 516 °C to 520 °C.  Chapter 7  Substrate Temperature During InGaAs laser Growth  196  This method of temperature control improves the reproducibility of temperature profiles during the grown of quantum well lasers, when compared to the conventional method where a thermocouple located behind the substrate senses the substrate temperature and a proportional-integral-derivative (PID) control loop determines the heater power settings. Failure of the conventional method is two fold: (1), as shown in Fig. 4-1, the thermocouple behind the substrate is a poor measure of substrate temperature, particularly during rapid changes in substrate power (or temperature) and (2) PID control loops tend to over shoot the end points of rapid temperature ramps and require a finite time to stabilize.  Using the conventional method of temperature control during the  growth a InGaAs laser structure like the one illustrated in Fig. 7-2, means that during the growth of the quantum well the substrate temperature is no where near the set point. Another method used to grow adjacent epilayers at different temperatures is growth interruption. In this method the growth is stopped at the end of one layer, the temperature of the substrate changed and allowed stabilize, and finally the growth of the next layer is started. In general this procedure is undesirable because during the growth interruption the interface is exposed to impurities in the growth chamber for several minutes. The room temperature photoluminescence spectra [72] of the InGaAs quantum well of the laser discussed above, is shown in Fig. 7-3.  The  photoluminescence peak is at a wavelength of 952 nm and has a 12.7 nm full width at half maximum. The excitation source used in these measurements is a 633 nm HeNe laser.  Chapter 7  Substrate Temperature During InGaAs laser Growth  _  6  §  ,  .  ,  1  1  1  1  1  1  I  1  1  •<i  F-  —  1  1  1  1  1  1  197  1  1  1  952 nm  ;  5  -8 a  <u o fi  •/  '_  CO  o  <u  c  j3  o 0 :  1  850  ,  900  ,  ,  ,  1  .  ,  .  950 Wavelength (nm)  ,  i  1000  ,  .  .  ,  i  -  1050  F i g . 7-3. R o o m temperature p h o t o l u m i n e s c e n c e f r o m the I n G a A s q u a n t u m w e l l s h o w n i n F i g . 7-2.  In order to obtain room temperature photoluminescence from these lasers, the M B E chamber must be very clean so that the contamination of the epilayers by impurities such as oxygen is minimized. The photoluminescence signal from the InGaAs quantum wells also improved when the substrates were switched from on axis (100) wafers to off axis (100) wafers (4° off toward (111)).  198  Chapter 8 Conclusions The width of the Urbach edge in GaAs is measured over the temperature range from 300 to 950 K. These are the first measurements of the temperature dependence of the width of the Urbach tail in GaAs above room temperature. Over this temperature range the width of the Urbach edge is found to be a linear function of temperature. The slope of the temperature dependence is weaker than predicted by standard models, from the room temperature width. The position of the absorption edge in GaAs is found to be a linear function of the width of the absorption edge. This is observed in other materials that exhibit the Urbach edge. The temperature dependent part the width of the Urbach edge is described by equations that depend on the deformation potentials and the average phonon energy. The part of the width of the of the Urbach edge in semi-insulating GaAs and InP not described by the phonon occupation number, is attributed to static fluctuations in the band edges due to point defects. The width of the absorption edge is found to broaden with doping in n-type GaAs and InP. This broadening is attributed to fluctuations in the band edge caused by the electric fields and the strain fields of the ionized donor impurities. A n optical method for measuring the temperature of a substrate material with a temperature dependent bandgap is presented. In this method the spectrum  Chapter 8  Conclusions  199  of the diffuse reflectance of the substrate is used to sense the temperature of the substrate.  Algorithms that locate the knee of the spectrum and use it as a  reference for the temperature, are presented. In the most versatile algorithm, a hyperbolic function is fit to the spectrum in the wavelength region of the onset of transparency of the substrate.  Of the fitting parameters, the position and the  width of the knee are insensitive to the absolute intensity of the spectrum and are used to determine the temperature of the substrate.  To a first approximation  substrate temperature is given by the position of the knee, while the width of the knee is used to correct for spurious shifts in the position of the knee. Of the algorithms known to date, this one is the most sophisticated and the least sensitive to variations in the properties of the substrate. The sensitivity of optical bandgap thermometry to substrate thickness, back surface texture, and doping is modeled for GaAs substrates. In this model, key points of the spectrum are related to the absorption edge of GaAs. Small changes in substrate thickness, light scattering, and background absorption cause temperature errors that are within the sensitivity of the measurement (1 °C). Large changes in substrate thickness, back surface texture, and doping levels must be calibrated for, if absolute accuracies close to the sensitivity of the measurement are to be maintained. For materials (such as GaAs) where the temperature dependence of the absorption edge is well known, the shift in the position of the knee, from thickness variations between substrates, is given by a simple relation between the width of the absorption edge and the thicknesses of the substrates. Temperature measurement techniques that infer temperature from the optical bandgap of the substrate material are insensitive to changes in the reflectivity of the epilayer, during growth. However, they are sensitive to thin  Chapter 8  Conclusions  200  film interference oscillations in the spectrum. It is found that spurious temperature shifts caused by thin film interference are reduced to the noise level of the measurement technique during the growth of AlGaAs on GaAs, by using both the position and the width of the knee of the spectrum to determine temperature. The effects of light trapping and scattering in textured substrates during optical bandgap thermometry are studied for GaAs substrates. It is found that at temperatures below 500 °C the Lambertian scattering model over estimates the effects of scattering by about a factor of four. The diffuse reflectance technique also gives composition and growth rate information for the epilayer. The composition of small bandgap epilayers is obtained from the onset of transparency of the epilayer. The growth rate and composition of large bandgap epilayers are obtained from thin film interference oscillations at wavelengths where the substrate is transparent. A number of factors are found to affect the temperature uniformity of semiconductor substrates in molecular-beam epitaxy; these include the In-free mounting configuration, the optical properties of the substrate and the holder, and the heater design. Temperature profiles with both positive and negative curvature are observed. A general solution to substrate temperature uniformity, may be a dual element substrate heater where the temperature at the center of the heater is controlled independently of the temperature of the outer part of the heater. The temperature gradient across the heater would then be tuned for a particular sample and growth conditions. 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B 1 2 , 1407 (1994).  [72]  Measurements by Richard Morin and Jeff Young, Department of Physics, University of British Columbia.  [73]  I. Daubechies, Ten Lectures on wavelets, Society for Industrial and Applied Mathematics, Philadelphia, (1992), p. 1-33.  Appendix I Algorithm Flow Charts  Algorithm A  Appendix I  Algorithm Flow Charts  (  208  START )  COLLECT DRS DATA ^ SCAN MONOCHROMATOR OVER WAVELENGTH RANGE Xo TO Xn AND COLLECT DATA FROM THE DETECTOR AT INTERVALS OF AX. A TYPICAL AA, IS 1 nm. STORE THE DRS DATA AS DRS ,X\ = Xo + i AX,, (i=0,1,2,3, ... ,n). FIND FIRST AND SECOND DERIVATIVE AND THEIR MAXIMA i = 0, MAX1 = 0, MAX2 = 0, N1 = 0, N2 = 0  YES  COMPUTE AND STORE FILTERED DATA:  DRS  FIRST DERIVATIVE:  dDRS  SECOND DERIVATIVE:  cftDRS  X = X\ X = Xi  «—  X = X)  Appendix  Algorithm  I  Flow  MAX1 =  d D  209  Charts  *  s  A = Ai  N1 =i  MAX2 =  dA  2  N2 = i  c  SELECT DRS+ SECTION OF DATA  NO = N2 - (N1 - N2) i = N0 DRS+  X = Xi  X. =  Ai  yo  = DRS A = ANO  i = i +1 NO ESTIMATE PARAMETERS OF EQUATION 3-2  -N2 m = 2  nri2  dQBS •A  YES  X=  XNI  4  cfegsi dXf  X =  XN2  Appendix I  Algorithm Flow Charts  \  1  mi = dDBS dA  A =  rm = 0 AND IS EXCLUDED AS A FITTING PARAMETER IN EQUATION 3 2  ANO  DETERMINE KNEE FIT E Q U A T I O N 3-2 T O D R S + D A T A U S I N G T H E N O N L I N E A R LEAST SQUARES LEVENBERG-MARQUARDT METHOD A N D O B T A I N E X A C T L O C A T I O N O F K N E E , XK. CALCULATE TEMPERATURE  ,  <  COMPUTE TEMPERATURE FROM C A L I B R A T I O N C U R V E T(A*)  /OUTPUT  (  TEMPERATURE/  STOP  )  210  Appendix I  Algorithm  Algorithm Flow Charts  B  211  Appendix I  Algorithm Flow Charts  (  START  C O L L E C T D R S DATA  212  )  ^  S C A N M O N O C H R O M A T O R O V E R W A V E L E N G T H R A N G E Xo TO A N D C O L L E C T DATA F R O M THE D E T E C T O R AT I N T E R V A L S O F AX. A T Y P I C A L AX. IS 1 nm. S T O R E T H E D R S | DATA A S DRS ,X\ = Xo+ \ AX, (i=0,1,2,3, ... ,n). FIND S E C O N D D E R I V A T I V E A N D ITS M A X I M U M i = 0, M A X 2 = 0, N2 = 0  i= i+ 1 YES  COMPUTE AND STORE FILTERED DATA:  DRS  S E C O N D DERIVATIVE:  cfPRS cttf  NO  MAX2 =  <%s A = Ai  N2 = i  f  A = Ai  A = Ai  Appendix I  Algorithm Flow Charts  SELECT SECTION OF SECOND DERIVATIVE PEAK TO BE FIT  i =  N2,  j=  213  N2 A = Ai  j=j_"1 m  =  ofDRS A = Aj  A = Ai  N1  N0  = i  = j  DETERMINE KNEE COMPUTE THE EXACT PEAK OF THE SECOND DERIVATIVE OF THE DRS DATA, Apeak, BY LEAST SQUARES FITTING EQUATION 3.3 TO THE DATA SET f(Aj), (i=N0,N0+1,N0+2, ... ,N0-2, N1-1, N1). A,K = W CALCULATE TEMPERATURE COMPUTE TEMPERATURE FROM CALIBRATION CURVE T(A,k)  I  -OUTPUT TEMPERATURE, IPDATE .TEMPERATURE. ? NO  (  S T Q P  )  YES  214  Appendix II Thin Film Interference and the Wavelet Transform In this appendix a new method to analyze the diffuse reflectance spectrum is developed, where the wavelet transform is used to obtain the derivatives of the spectrum. The ability of the wavelet transform method and Algorithm A (and the correction algorithm Eq. 4-55) to deal with spurious fluctuations in the spectrum caused by thin film interference during the growth of epilayers, is explored. In this analysis the thin film interference effects are simulated for the growth of ZnTe on GaAs substrates. The ability of Eq. 4-55 to correct for shifts in the knee caused by changes in back surface texture is examined in §11-7.  In this work the algorithm in  Eq. 4-55 is improved by linearizing the logarithm of the width of the knee term. The new algorithm is given in Eq. 11-25. §IM  Wavelets and the Wavelet Transform Wavelets provide a tool for time-frequency localization. A common form  of the wavelet is the second derivative of the Gaussian sometimes called the "mexican hat function", because of the resemblance. The "mother wavelet" [73] is  vM  = -^ex (-t /2) 2  P  = (l-t )exp(-t /2) 2  2  .  (HM)  Appendix II  Thin Film Interference and the Wavelet Transform  215  The conventional form of the wavelet [73] is given as ft-b^  1  w  K a )  (t-b\  •\ja  \  (II-2)  a )  In this case the normalization is given by 2  oo  °°  dt  (IL3)  To relate the width a directly to frequency, a scaling factor 42n is added to the conventional wavelet. In this case the form of the wavelet is ft-y\_ \  42n_y  {42n{t-b)^  a j  a N27T f a  V  2\  1-2  I  exp a  )  (114)  2\  I  J  a  J J  The Fourier transform of the mother wavelet is \fr (v)= j\f/ {t)exp(i2KVt)dt m  m  = 427t{2)Tv) exp(-(27i:v) /2) 2  2  ,  (II-5)  —oo  and the Fourier transform of the wavelet in Eq. 114 is oo  ¥ {av)= w  .  J^w(V«)exp(/2^vO^ = J ^ ^ = 2^42a(av) exp(-(av) ) 2  2  m  ^ - ^  (H-6)  .  The wavelet in Eq. 114 and its Fourier transform are shown in Fig. II-1. The Fourier transform has a center frequency 1/a and a width around 1/a.  Appendix II  Thin Film Interference and the Wavelet Transform  i • , , • i , , • , i  -1.0  -0.5  0.0  0.5  1.0  J  1.5  i  i  i  i  216  i i_  i  2.0  2.5  t/a F i g . I L L A m e x i c a n hat w a v e l e t w i t h w i d t h a. T h e F o u r i e r t r a n s f o r m o f this w a v e l e t is s h o w n i n the inset.  The wavelet transform [73] is given by  F(a b)=]f(t) (t-±y t  ¥w  t  (H-7)  .  This transform gives frequency information about f(t)  at time b.  As the  parameter a decreases the wavelet narrows and is thus able to zero in on high frequency information. Eq. II-7 is a convolution of the wavelet with f(t), and is therefore expressed in terms of the Fourier transforms of f(t) and \i/ (t/a), using w  the convolution theorem [35]  F{a,b)= Jf(v)\i/ {av)exp{-i27ivb)dv w  .  (II-8)  Appendix II  Thin Film Interference and the Wavelet Transform  217  In the Fourier transform picture, the wavelet transform samples f(t) over the frequency range, 0.5 to 1.5 1/a, at time b. See Fig. DM. §H'2  Wavelet Transform of Diffuse Reflectance Spectrum  The sensitivity of the wavelet transform of the diffuse reflectance spectrum, to thin film interference is now explored. The effect of thin film interference on the spectrum during the growth of a 4 pm thick AlGaAs overlayer on a GaAs substrate is simulated using the multiplicative term T in Eq. 5-2. The growth of {  such a layer is described in §5-1. The phase of the interference term is chosen so the shift in the knee is at a maxima, namely sin 8 = -1. The simulated spectrum DRSj in terms of the measured spectrum, DRS , is p  "  D  ™ i = [l-K) Tr^  D  R  S  P>  ^  where the values Rj, R , and R are experimentally determined in §5-1. The initial 2  condition DRS^df = d) = DRS is verified by substitution of these values into p  Eq. II-9 with cos8 = 1. The diffuse reflectance spectrum, DRS , and the p  simulation, DRS , for a 100 pm thick, semi-insulating GaAs substrate, at 331 °C, {  are shown in Fig. II-2. The wavelet transforms (Eq. II-7) of the two spectra in Fig. II-2 are shown in Figs. IL3 and II-4, for a bare substrate and a substrate with a 4 pm thick AlGaAs layer on the front surface, respectively. In these plots the width a is varied from 2 to 122 meV; for comparison the Urbach parameter, E , is 9.73 meV (at 331 °C) for 0  GaAs. The shape of the transform in Fig. II-3 broadens as a increases, and is similar to the shape of the second derivative of the spectrum. For small values of  Appendix II  r  Thin Film Interference and the Wavelet Transform  218  M-51.7 meV-H DRS  P  0 1100  1150  1200  1250  1300  Energy (meV) Fig. IL2. Diffuse reflectance spectrum from a semi-insulating GaAs substrate at 331 °C (DRS ), and a thin film interference simulation for a 4 urn thick AlGaAs layer on the surface of the substrate (DRSj). p  a, say a less than E , the noise in the spectrum is discernible in the transform. In 0  Fig. II-4 the interference effects are most strongly shown at a equal 52 meV, the period of the interference oscillations. For values of a larger than 104 meV (two periods) the effect of the interference has almost disappeared from the transform. As a goes to zero the structure in Figs. II-3 and II-4 disappears, this is an artifact of the conventional normalization in Eq. II-3. For this normalization, the wavelet transform of a constant, goes to zero as 4a. This reduces the amplitude of structure and noise at the higher frequencies where noise dominates the transform. In the following section at more relevant normalization is obtained in developing a wavelet related to the derivatives of the spectrum.  Appendix II  Thin Film Interference and the Wavelet Transform  1300 F i g . II-3. T h e w a v e l e t transform o f the diffuse reflectance spectrum f r o m a bare semi-insulating G a A s substrate at 331 ° C .  1300 F i g . 114. T h e w a v e l e t transform o f the diffuse reflectance s p e c t r u m f r o m a s e m i - i n s u l a t i n g G a A s substrate, at 331 ° C , w i t h a 4 | i m t h i c k A l G a A s overlayer o n the front surface.  219  Appendix II  Thin Film Interference and the Wavelet Transform  220  §n*3 Derivatives and the Wavelet Transform Extending the notion of a wavelet defined by the second derivative of the Gaussian to a wavelet defined by the n derivative of a Gaussian, G (t), the n th  th  n  wavelet transform is F (a,b)= n  j \t)f(t)dt n  yf (t) = (-l) ^G (t) n)  ;  Wd  n  d  .  n  (H-IO)  —oo  Integrating Eq. 11-10 by parts n times the n wavelet transform becomes th  F {a,b)=\G (t)^f{t)dt n  .  n  (n-11)  — oo  Furthermore, if the normalization of the Gaussian, with width a, is given by JG (t)dt  =l ,  n  (n-12)  — oo  then in the limit that a goes to zero (n-13)  Again choosing the width of the wavelet so the peak of Fourier transform of the wavelet occurs at 1/a; f2K(t-bA  2  G (t) =, —a exp V n n  \  a  ^ I2n 1  )  ; n = 1,2,3,-  (II. 14)  In this case the wavelet is (2n+l)/2 ,  n  (aVn)  y  d  _  2it(t-b) a  n  ^  n  (H-15)  Appendix II  Thin Film Interference and the Wavelet Transform  221  The Fourier transform (with b = 0) of the wavelets derived from the derivatives of the Gaussian are given by \fi(av) = {-i)  n  {—Y (a v)  n  V  aJ  exp(-n (a v )  2  (II-16)  J2\  The first four (n = 1, 2, 3, 4) Fourier transforms in Eq. II-16 are shown in Fig. IL5. The frequency range sampled by these wavelets decreases as the order of the derivative increases. 1.0 \0.8 h 0.6  0.4  0.2  0.0  J  0.0  0.5  1.0  I  I  !_  1.5  2.0  _1  2.5  I  I  1-  3.0  av F i g . II-5. F o u r i e r t r a n s f o r m o f the w a v e l e t s d e r i v e d f r o m the first four derivatives o f the G a u s s i a n .  The picture of the wavelet transform, given by Eq. 11-11, is one of a convolution of a Gaussian of width a, with the n derivative of the spectrum. In th  the case of the second derivative the wavelet transform is  Appendix II  Thin Film Interference and the Wavelet Transform  222  oo  F {a,b)=\w?{t)f{t)dt  ;  2  K 5/2  a'  1-2  2\ (n(t-b)} \  a  J J  (n-n)  2\  f  exp —  1{  a  ) J  In the following analysis the second derivative of the diffuse reflectance spectrum is calculated using the wavelet transform and the wavelet in Eq. II-17 for a on the order of the Urbach parameter. This calculation is compared to the conventional second derivative determined directly from the spectrum. The wavelet transform for a = 2E and the numerical derivative are shown in 0  Fig. E-6. — i — i — . — i — | — . — . — . — i — | -  I  0.0008 I i | i i i | i • i | i  "I'  I "  Wavelet Derivative  0.0008 CN  0.0006 h •'  > CD  a  ^  0.0004  a  0.0004 if i i i i i i 1224 1228  CZ)  d u  1232  ^ 0.0000 CD  I—I  CD  >  cd  -0.0004 J  1100  i  i  i_  _i  1150  i  i  i_  _i  i  1200  i  i_  1250  Energy (meV) F i g . II-6. T h e s m o o t h s e c o n d derivative o f the diffuse reflectance s p e c t r u m determined b y a wavelet transform and c o n v e n t i o n a l methods.  1300  Appendix II  Thin Film Interference and the Wavelet Transform  223  Calculating derivatives of data sets, magnifies noise in the data. Therefore in calculating the derivatives of the spectrum the following smoothing is done: the raw data is smoothed using a window one half the width of the Urbach parameter and then the first derivative is calculated. The first derivative is smoothed using the same window. The second derivative is calculated from the smooth first derivative and smoothed using the same window as before. In the case of the wavelet transform, smoothing is determined by the size of a. The height of and the noise in the peak of the second derivative is about the same for both methods of calculation. However, the peak of the wavelet transform is shifted to higher energies. The peak in the second derivative is asymmetric and when convoluted with the symmetric Gaussian this asymmetry shifts the peak. The effect of noise, in the spectrum, on the wavelet transform is shown in Fig. II-7, where the peak in the wavelet transform is compared for several values in a, starting at 0.6 E and ending with 2.2 E at intervals of 0.2 0  0  E . E = 9.73 meV for this spectrum. The peak in the second derivative is also 0  0  shown. As the width of the wavelet transform decreases, the noise and the amplitude of the peak increase, and the peak moves towards the true peak of the second derivative. The position of the peak in the wavelet transform as a function of a is shown in Fig. II-8. The position of the peak is linear in a for a greater than one Urbach parameter. For a less than one Urbach parameter the position of the peak is sensitive to noise. The solid line in Fig. II-8 is a linear fit to the positions of the peak for a greater than 1.2 E . The y intercept of this fit is 1226.90 ± 0.02 meV 0  which coincides with the position of the peak in the second derivative, namely  Appendix II  Thin Film Interference and the Wavelet Transform  224  1221 Al meV. This is predicted by Eq. 11-13 where the true second derivative of the spectrum is given by the wavelet transform in the limit that a goes to zero.  1224  1226  1228 Energy (meV)  1230  1232  Fig. II-7. The peak in the wavelet transform of the diffuse reflectance spectrum, for widths of the Gaussian ranging from 0.6 to 2.2 Urbach parameters, at intervals of 0.2 Urbach parameters. The peak in the second derivative of the diffuse reflectance spectrum is also shown. Similarly the extrapolated amplitude of the series of wavelet transform peaks in Fig II-7, predicts a true amplitude of 0.00108 meV" for the peak in the 2  second derivative.  In comparison the amplitude of the smoothed second  derivative is 69% of the true amplitude. This shows that the three step data smoothing system used in determining the second derivative decreases the  Appendix II  Thin Film Interference and the Wavelet Transform  225  amplitude of the peak, by broadening the knee of the spectrum, without significantly shifting the position of the peak.  0.5  1.0  1.5  2.0  2.5  "i o) E  F i g . II-8. T h e p o s i t i o n o f the peak i n the w a v e l e t transform as a function o f the w i d t h a, o f the G a u s s i a n . T h e s o l i d l i n e is a linear fit to the p o s i t i o n o f the p e a k for a greater than 1.2 U r b a c h parameters.  The rate of shift in the peak of the wavelet transform as a function of a is 0.745 ± 0.013 meV per Urbach parameter. The position of the knee given by algorithm A (the asymptotic fit) is 1228.46 ± .06 meV which coincides with the position in peak of the wavelet transform when a is about 2.1 E . The width of 0  the second derivative of the asymptotic fitting function is about 7E or 2.3E , a  0  E = 3.25 + 0.03 meV. The width of the Gaussian (a = 2.1 E ) is 1.1 E . a  0  0  Appendix II  Thin Film Interference and the Wavelet Transform  0.0011  226  - 1 — I — I — I — I — I — I — I —  Thin Film Interference 0.0010 CM i  >8  0.0009  | 0.0008 a  SH H S > c3  0.0007 0.0006 r  0.0005 1224  1226  1228  1230  1232  Energy (meV) Fig. II-9. The peak of the wavelet transform of the diffuse reflectance spectrum with a equals 1.4 and 2.0 Urbach parameters and the peak of second derivative of the diffuse reflectance spectrum, for a bare substrate and a substrate with a 4 urn thick AlGaAs overlayer on the front. The sensitivity of the peak in the second derivative and the peak in the wavelet transform to thin film interference, is examined in Fig. II-9. The results shown here are for the two spectra discussed in §11-2. The width of wavelet transforms in Fig. II-9 are given by a equal \AE and 2.0E . The peaks for the 0  0  spectrum perturbed by thin film interference have higher amplitudes and are shifted by about 0.4 meV to lower energies, compared to the peaks for the spectrum from the bare substrate. The peak in the wavelet transform and the peak in the second derivative have the same sensitivity to thin film interference. For comparison, the position of the knee, given by Algorithm A, shifts about  Appendix II  Thin Film Interference and the Wavelet Transform  227  0.6 meV toward lower energies when the effect of thin film interference is added to the spectrum. Using the correction algorithm given in Eq. 3-54, the shift in the knee is reduced to about 0.3 meV. §11*4 H i g h e r Derivatives of the Diffuse Reflectance Spectrum  The higher order derivatives of the diffuse reflectance spectrum are accessible using the wavelet transform. For example, from Eq. 11-15 the fourth derivative wavelet is K  9/2  f  •42a' V  2  (7l(t-b)"\  3-6  \  a  + ,  (n{t-b)^ y  a  )  4\  ) >] / )  exp J  v  I  a  (HIS)  The fourth-derivative (wavelet-transform) of the diffuse reflectance spectrum from a bare, semi-insulating, GaAs substrate at 331 °C, is shown in Fig. 11-10. The vertical axis is 8-10" meV" at the bottom and -24-IO" meV" at the top. Of the 6  4  6  4  peaks in this wavelet transform the sharpest and highest peak occurs about 3 meV below the sharpest and highest peak in the second derivative. The noise in the spectrum is visible in the wavelet transform when, a, the width of wavelet is less than 20 meV or two Urbach parameters. The peak in the fourth derivative is symmetric and does not suffer from the broadening shift observed in the peak of the second derivative. In order to assess the optimal derivative of the spectrum to use as a measure of the onset of transparency of the substrate, the first six derivatives are calculated for the diffuse reflectance spectrum from a bare, semi-insulating, GaAs substrate at 331 °C. The spectrum is smoothed, then the first and each subsequent derivative, is numerically calculated and smoothed. All smoothing  Appendix II  Thin Film Interference and the Wavelet Transform  228  steps are done with a window one half the width of the Urbach parameter. The position of the narrowest and the highest peak of each derivative is shown in Fig. II11. For comparison the position of the knee given by algorithm A is also shown. The positions of the peaks for the third and higher derivatives lie between the peaks of the first and second derivative. Consequently the peak in the second derivative is closest to the bandgap.  F i g . 11-10. T h e f o u r t h d e r i v a t i v e w a v e l e t t r a n s f o r m o f the d i f f u s e reflectance spectrum f r o m a semi-insulating G a A s substrate at 331 ° C . T h e v e r t i c a l a x i s is 8 - 1 0 m e V at the b o t t o m and -24* 1 0 m e V " at the top. -6  - 4  - 6  4  The simulations above show that during the growth of AlGaAs on GaAs thin film interference does not have a large effect on optical bandgap thermometry. However, recently there has been considerable interest on the effect of thin film interference optical bandgap thermometry during the growth of II-VI material on III-V substrates [62]. Such a system is the growth of ZnTe on  Appendix II  Thin Film Interference and the Wavelet Transform  229  GaAs, where the difference in index of refraction between the substrate and overlayer is about one. In these systems the amplitude of the interference oscillations are about twice as large as those in AlGaAs on GaAs. 1  >  CD  1  1  r  I  I  I  I  I  I  I  I  1  I  I  I  I  1  I  1  I  I  Algorithm A  1228  CD CM CD 03  oo C 03 -i—> CZ)  CD  Jl  O  d o cz)  O PH  1216  I  |  I  I  [  I  3 4 nth Derivative F i g . II-11. T h e p o s i t i o n o f the sharpest a n d the highest p e a k o f the first s i x derivatives o f the diffuse reflectance spectrum.  As before the thin film oscillations are simulated using Eq. IL9 where T is t  given by Eqs. 5-1 and 5-2 with n = 3.5, n = 2.5, and dj- = 2 pm. In this s  f  simulation the diffuse reflectance spectrum from a bare semi-insulating GaAs substrate at 331 °C is multiplied by the interference term for a 2 pm thick overlayer. This spectrum is a good choice for these simulations because II-VI material is typically grown at temperatures between 200 and 350 °C on semiinsulating substrates. The phase of the oscillations in the spectrum is chosen for  Appendix II  Thin Film Interference and the Wavelet Transform  230  the maximum effect at the knee, namely sin 8 = -1. The spectrum from the bare substrate (dashed line) and the simulation (solid line) are shown in Fig. 11-12. 0.6 h 0.5 h CD  B 0.4 Ca  CD CD  q=! CD  0.3 h  CD  .=3  ^  a  02  o.i o1100  1150  1200 Energy (meV)  1250  1300  F i g . 11-12. D i f f u s e r e f l e c t a n c e s p e c t r a f r o m a s e m i - i n s u l a t i n g G a A s substrate at 331 ° C , w i t h ( s o l i d line) and without (dashed line) an overlayer.  To determine the sensitivity of the derivatives of the spectrum to thin film interference, the first six derivatives of simulated spectrum are calculated and the positions of their peaks compared to those of the derivatives in Fig. 11-11. The shift in the positions of the peaks due to thin film interference are shown in Fig. 11-13. For comparison, the shift in knee determined using algorithm A, and the shift in knee after applying the correction algorithm in Eq. 4-55, are also shown. Interference induced shifts in the derivative peaks are lowest for the second,  Appendix II  Thin Film Interference and the Wavelet Transform  231  fourth, fifth, and sixth derivatives. The least sensitive method is Algorithm A plus the correction given in Eq. 4-55; recall in this correction, spurious shifts in the knee as sensed by the change in the width of the knee and partially corrected for. T  1  1  1  1  1  2  1  1  1  1  1  1  1  1  1  1  3 4 nth Derivative  1  1  1  1  1  5  1  1  1  r  6  Fig. II-13. Thin film interference induced shift in the position of the highest and sharpest peak in the derivatives of the diffuse reflectance spectrum. From this simulation it appears that the peak in the second and fourth derivatives of the spectrum are best suited of any of derivatives to determine the onset of transparency and hence the temperature of the substrate. Algorithm A and the correction given by Eq. 4-55 is less sensitive to thin film interference than the derivative methods.  Appendix II  Thin Film Interference and the Wavelet Transform  232  §IT5 Correcting for spurious temperature shifts  A detailed analysis of algorithms that correct for fictitious temperatures shifts in optical bandgap thermometry is given in the last part of this Appendix. This analysis includes the ability of Eq. 4-55 to correct for spurious shifts in the position of the knee given by Algorithm A, caused by both thin film interference and variations in scattering at the back of the substrate. A correction algorithm for spurious shifts in the position of the peak of the fourth-derivative wavelettransform, during thin film interference is also developed. As in the pervious section, thin film oscillations are simulated using Eq. 11*9 where T is given by }  Eqs. 5-1 and 5-2 with n = 3.5 and n^ = 2.5. Furthermore, in this case the s  overlayer thickness df is varied from 1.8 to 2.5 pm. Recall, that this represents the stronger interference case where II-VI material is grown on a III-V substrates. The position and the width of the knee of the diffuse reflectance spectrum from a semi-insulating substrate at 331 °C, are shown in Fig. 11-14, as a function of overlayer thickness. As predicted by Eq. 4-54 the position and the width of the knee oscillate out of phase. For comparison, the position and the width of knee for the bare substrate, are 1228.45 ± 0.06 meV and 3.28 ± 0.03 meV, respectively. Using the data shown in Fig. 11-14 the performance of the correction algorithm in Eq. 4-55 and variations on it are compared. Here spurious shifts in the knee are sensed by a change in the width of the knee and corrected for. From Eq. 4-55, the corrected position of the knee is  V  Appendix II  Thin Film Interference and the Wavelet Transform  233  where E is the position of, and E is the width of, the knee given by Algorithm k  a  A, and E' is the width of the knee for a bare substrate. In order to improve the a  effectiveness of the correction given in Eq. 11-19, the constant A is introduced as a  a parameter that scales the amount of correction given by the logarithm term.  1.8  1.9  2.0 2.1 2.2 2.3 Overlayer Thickness (Lim)  2.4  2.5  Fig. 11-14. Oscillations in the position and the width of the knee, of the diffuse reflectance spectrum from a semi-insulating GaAs substrate at 331 °C during the growth of an epilayer with index of refraction equal 2.5. A close look at the curves in Fig. 11-14 shows that the position and the width of the knee are not exactly 180° out of phase, as predicted by Eq. 4-54. The maximum in the width of the knee occurs roughly 10° after the minimum in the position of the knee. This small phase shift A, is due to the fact that the positions of the knee and the inflection point do not oscillate exactly in phase.  Appendix II  Thin Film Interference and the Wavelet Transform  234  The width of the knee is given mainly by the shift in the inflection point and hence has roughly the same phase as the inflection point. Therefore the phase shift in units of overlayer thickness is F  f  A  = zr{h- k)  =d  = f  T  d  \  he , (H-20) 2njhv  A  P  E  J  \ E  K  - E  -1  =  d^ f  0  where T is the period between peaks in the interference fringe as a function of overlayer thickness, see Eq. 5-1. The subscripts " k" and " p" on r refer to the period of the fringe at the knee and the inflection point respectively. djjx  k  number of oscillations since the start of growth.  is the  The phase difference  accumulates with each oscillation in the position of the knee; as the overlayer thickens the interference fringes get closer together in energy and the phase difference between the knee and the inflection point increases. This increase is linear in overlayer thickness. From Eq. 11-20 the dimensionless phase shift is (j) = A/df = E /E A  K  =  0.0079. For comparison, the measured phase shift in Fig.  E-14is 0 = 0.0069. Using the phase difference correction <p and the weighting parameter A , a  the correction algorithm in Eq. 11-19 becomes E' = k  E +A E \n k  a  0  A f)  E  d  ; d' =d {l f  f  + <p) .  (H-21)  E:  The corrected position of the knee given by Eq. 11-21 for A equal 1.0 and 2.0 a  and for (j) equal 0 and 0.0079 are shown in Fig. 11-15. For comparison, the position of the uncorrected knee is also shown.  The RMS deviation in  temperature for each method is summarized in Table II-1. The RMS deviation for  Appendix II  Thin Film Interference and the Wavelet Transform  235  A = 2.0 and <>/ = 0.0079 is within 10% of the lowest value obtained, namely for a  A = 2.0 and 0 = 0.0069. Correcting for the phase difference between the knee a  and the inflection point further reduces the temperature error caused by thin film interference. T — i — i — i — | — i — i — i — i — | — i — i — i — I — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — i — r  I  1.8  ,  1.9  ,  ,  •  I  2.0 2.1 2.2 2.3 Overlayer Thickness (Lim)  I  , , , I  2.4  F i g . 11-15. O s c i l l a t i o n s i n the p o s i t i o n o f the knee o f the diffuse reflectance spectrum and the corrected position o f the knee u s i n g different algorithms.  The position of the peak of the fourth-derivative (wavelet-transform) is sensitive to thin film interference. Interference oscillations in the spectrum cause an asymmetry in the peak of the fourth derivative that oscillates with the overlayer thickness. The fourth-derivative (wavelet-transform) is used to correct for interference oscillations in the peak of the fourth derivative by exploiting this  Appendix II  Thin Film Interference and the Wavelet Transform  236  asymmetry. A simple method of doing this is a linear extrapolation from the position of two peaks at high values of the a, to a point at a lower value of a, such as a equal zero. The shift in the peak, due to any asymmetry in the peak, increases with a (see Fig. II-3), therefore the position of the peak for a transform with a larger a is more sensitive to interference effects. The information given by the sensitivity of the peak to interference fringes as a function of the width of the wavelet, is used to correct for shifts in the peak. T a b l e II-1. Shift i n the p o s i t i o n o f knee a n d the associated temperature error f o r v a r i o u s parameter values i n correction algorithm.  A  A  RMS Shift (meV)  RMS Error (°C)  0.0  0.0  1.16  2.2  1.0  0.0  0.75  1.4  2.0  0.0  0.49  0.9  1.0  0.0079^  0.68  1.3  2.0  0.0079<L  0.37  0.7  a  In the algorithm used as a correction to the shift in the position of the peak in the fourth-derivative wavelet-transform, the position of the peak, P , is a  determined for a = \.5E and a = 2.0E . The reference point used to determine 0  0  temperature is given by the following extrapolation from these peaks; Po.o=4Pj. -3P , 5  For the bare substrate P  20  2 0  .  = 1227.45 meV, P  (n-22) ]5  = 1227.19 meV, and P  00  =  1226.41 meV. The positions of these peaks as a function of overlayer thickness is  Appendix II  237  Thin Film Interference and the Wavelet Transform  determined. The position of these peaks relative to those of the bare substrate are shown in Fig. 11-16.  The RMS temperature deviations for each peak is  summarized in Table II-2. The temperature deviation for a given shift in the peak is determined using Eq. 4 - 3 7 .  1.8  T  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  r  J  I  I  .  I  I  I  I  I  I  1  .  I  I  I  .  I  .  I  I  I  .  L  1.9  2.0 2.1 Overlayer Thickness (lim)  2.2  F i g . 11-16. O s c i l l a t i o n s i n the p o s i t i o n o f the peak o f the f o u r t h - d e r i v a t i v e ( w a v e l e t - t r a n s f o r m ) for t w o w i d t h s o f the w a v e l e t ; n a m e l y 1.5 a n d 2 . 0 U r b a c h p a r a m e t e r s . T h e c o r r e c t e d reference p o i n t , g i v e n b y a l i n e a r e x t r a p o l a t i o n f r o m these peaks, is also s h o w n .  In this case the algorithm in Eq. 11-22 works well. However, it has little theoretical justification and is not tested on other layer thicknesses or for different indices of refraction, and therefore might not work in all situations. In contrast  Appendix II  Thin Film Interference and the Wavelet Transform  238  Algorithm A and its correction have a good theoretical foundation and have been shown to work in many situations. T a b l e II-2. Shift i n the p o s i t i o n o f the fourth-derivative w a v e l e t - t r a n s f o r m p e a k for t w o w i d t h s o f the w a v e l e t , a n d the corrected reference p o i n t u s i n g a l i n e a r e x t r a p o l a t i o n f r o m these peaks. T h e associated temperature error is also g i v e n .  Peak  RMS Shift (meV)  RMS Error (°C)  P  2 . 0 E0  1.06  2.0  P  1.5  E  0.82  1.6  0.25  0.5  20  15  P  00  §n*6  a  —  0  Fourier Transform of the Diffuse Reflectance Spectrum  The Fourier transforms of the knee region, up to the inflection point, of the diffuse reflectance spectrum from semi-insulating GaAs, at 331 °C, for a bare substrate and a substrate with a 4 urn thick AlGaAs overlayer, are shown in Fig. 11-17 by the solid lines. For comparison, the Fourier transform of the asymptotic function in algorithm A, fit to the spectrum of the bare substrate, is shown by the dashed line. The intensity of the curve for the spectrum from the substrate with the AlGaAs overlayer, is higher. The average signal from this substrate is larger because the reflectivity of AlGaAs overlayer is lower than the reflectivity of the bare substrate. The Fourier transforms of the spectrum and the fit for the bare substrate are almost identical. The fact the shape of the Fourier transforms of the spectra from the bare substrate and the substrate with the  Appendix II  Thin Film Interference and the Wavelet Transform  239  overlayer are the same, means the effect of thin film interference is not distinguishable in this region of the spectrum using the Fourier transform. -I  1  0.04  1  1  r  Transform of Data Transform of Fit  With AlGaAs Overlayer  0.03 a lH  SH  CD  3 0.02 HH  0.01 0.00  0.20  0.40 0.60 Frequency (1/EJ  0.80  1.00  F i g . 11-17. F o u r i e r transforms o f the knee r e g i o n u p to the i n f l e c t i o n p o i n t o f the diffuse reflectance s p e c t r u m ( s o l i d lines) f r o m a bare substrate a n d a substrate w i t h an A l G a A s o v e r l a y e r a n d the F o u r i e r t r a n s f o r m o f the asymptotic function fit to the spectrum o f the bare substrate (dashed line).  The Fourier transform of the diffuse reflectance spectrum from a bare substrate (dashed line) and a substrate with a 4 pm AlGaAs overlayer (solid line), are shown in Fig. 11-18. For comparison, the Fourier transform of the knee region of the spectra are also shown by the lower curves. The frequency range of this plot is half the frequency range of Fig. 11-17. Again the magnitude of the curve for the substrate with the AlGaAs overlayer is larger. The interference effects are  Appendix II  Thin Film Interference and the Wavelet Transform  240  clearly shown in the transforms of the total spectrum of the substrate with the AlGaAs overlayer. Since interference occurs for first order and higher order reflections in the overlayer, the interference effects are seen at the frequency of, and the higher order frequencies of, the cosine term in Eq. 5-2. i  " i — i — i — r -  1.00  i  i  i  |  i  1  i  r  Bare Substrate AlGaAs Overlayer b  Total Spectrum  6 J-l  a  Interference oscillation  c  cS  .53 0.10 =s o  fe  Knee Region of Spectrum i i i i i i i i i i_ 0.01 0.00 0.10 0.20 0.30 Frequency (1/E ) _i  0.40  0.50  0  F i g . II-18. T h e F o u r i e r t r a n s f o r m o f the diffuse r e f l e c t a n c e s p e c t r u m (upper c u r v e s ) f r o m a bare substrate (dashed l i n e ) a n d a substrate w i t h a 4 [ i m t h i c k A l G a A s l a y e r ( s o l i d line). F o r c o m p a r i s o n the transform o f the knee r e g i o n o f these spectra are s h o w n b y the l o w e r curves.  §11*7 Correcting for Variations in the Scattering Parameter  A detailed analysis of the ability of Eq. 4-55 to correct for spurious shifts in the position of the knee caused by variations in scattering at the back of the  Appendix II  Thin Film Interference and the Wavelet Transform  241  substrate is provided in this section. Algorithm A and its correction algorithm are tested in a simulation where the scattering parameter is varied from zero to one. In this simulation, the diffuse reflectance spectra for a 100 pm thick semiinsulating GaAs substrate at 331 °C, are calculated using Eq. 4-16, where the absorption coefficient is given by Eq. 4-28, with E - 9.73 meV, E = 1.271 eV, 0  g  and a = 1.78 cm- . Some of the spectra for this simulation are plotted in 1  a  Fig. 11-19 for several values of the scattering parameter, namely 0.0, 0.1, 0.2, 0.4, 0.7, and 1.0. As the scattering increases the knee broadens and the overall intensity of the spectrum decreases. T"~  1100  i  i  |  i  i  i  i  |  i  i  i  I  i  i  \  i  I  i  i  i  1150  1200 Energy (meV)  ~i  i  |  i  i  i  r  1  i  i  i  i  1250  F i g . I I - l 9 . S i m u l a t e d diffuse reflectance spectra for a s e m i - i n s u l a t i n g G a A s substrate at 331 ° C for several v a l u e s o f the scattering parameter. T h e s e values are listed o n the left-hand side; where the smallest value corresponds to the highest curve and as the values increases the intensity decreases.  1300  Appendix II  Thin Film Interference and the Wavelet Transform  242  Using Algorithm A , the position and width of the knee as a function of the scattering parameter for the simulated spectra are plotted in Fig. 11-20. The width of the knee is given by the solid line and the position of the knee is given by the dashed line. The position of the knee and the width of the knee have roughly the same scattering dependence.  0.0  T  I  l  |  I  I  I  |  I  I  I  |  I  I  1  |  I  I  I  1  .  I  I  I  I  .  I  I  I  .  I  I  I  I  I  .  .  L  0.2  0.4  0.6  0.8  1.0  Scattering Parameter F i g . 11-20. T h e p o s i t i o n a n d the w i d t h o f the k n e e , o f the reflectance spectrum, as function o f the scattering parameter.  diffuse  In the following analysis the information in the width of the knee is used to correct for spurious shifts in the position of the knee caused by variations in back surface scattering. In a first method a scalar A is added to Eq. 4-55 to vary y  the weight of the logarithmic term:  Appendix II  Thin Film Interference and the Wavelet Transform  EM E (0)  E' =E +A E ln k  k  r  243  0  (11-23)  a  where E is the position and E (7) is the width of the knee of a spectrum with k  a  scattering parameter 7 . E (0) is the width of the knee of a spectrum from a a  substrate with a polished back. Since the curve for the shift in the position of the knee and the curve for the width of the knee have a similar shape, a second method given by the linearization of Eq. 11-23, with A = 1, is explored, namely y  f  E'k = E + k  t..\  j  Eg(r) E  \ (n-24)  0  yE (0) a  1  I  T  0  1  r  1  r  0  >  -4 H CD  B  CD CD  Uncorrected  Linearized  V 1  CD  0  1-1  -8  —1— —1— —1— —r 1  a «o -6 o  1  1  1-!  CD  tn i-i o 11  CM  *  43  -8  o  -10 h  n  -15  OO  B  0.0  -10  0.2  0.4  _L  0.6  _L 0.8  H-16  -30 1.0  _L  0.00  0.05  0.10 0.15 Scattering Parameter  0.20  Fig. 11-21. The shift in the position of the knee with scattering and the associated temperature error. The correction in Eq. 11-23 is given by the solid lines and the linear correction in Eq. 11-24 is given by the broken line.  Appendix II  Thin Film Interference and the Wavelet Transform  244  The shift in the position of the knee and the associated temperature error, are shown in Fig. 11-21, as a function of the scattering parameter over the range 0.0 to 0.22. This range covers the scattering from most textured wafers, see §5-6. The larger 0.0 to 1.0 range is shown in the inset. In Fig. 11-21 the dashed line gives the uncorrected shift in the position of the knee. The solid lines are for the correction given in Eq. 11-23, and the broken line is for the linear correction given by Eq. 11-24. The linearization of Eq. 11-23 is an improvement over Eq. 11-23. Values of A slightly above one also improve the correction. For thin film interference, the y  linearization of Eq. 11-21 is not an improvement over Eq. 11-21, because these methods are equivalent for small variations in the width of the knee. Since the linearization of the correction equation improves the results for scattering and does not degrade the results of thin film interference an improvement on the correction algorithm given by Eq. 4-55 is:  El  +1  = E (T) k  ad  E (Ei) In  0  +  a  o  y o;cald al J  +  a  C  j  ^k o(^k) E  he  a  In  o d  a;cal( k) E  \  +  d  \ o\cal cal a  rr-1 E  J  (f  j  (11-25)  Al  4(T))  A,  A  a;caZ  (4)  -7  Algorithm A and Eq. 11-25 form a complete algorithm that can extract substrate temperature from both transmission and reflection spectra. This is a powerful algorithm because it can maintain the accuracy of the measurement close to that of the calibration curve, in spite of changes in back surface texture, subedge absorption, optical throughput, and substrate thickness. This algorithm  Appendix II  Thin Film Interference and the Wavelet Transform  245  can also accommodate thin film interference and variations in the measurement technique.  For direct gap semiconductors this algorithm is valid for substrate  thicknesses from 100 to 1000 Lim and back surface textures from polished to Lambertian. Furthermore, this algorithm only requires a single calibration for each semiconductor material with given doping level. However, this algorithm requires knowledge of the width of the absorption edge as a function of temperature.  246  Appendix III Wafer Holder Drawings Tungsten Clip  , 0.02  L + 0.10  - 0.01 r  0.02 R  0.063 R  T h e tungsten b a l l i s f o r m e d o n the e n d o f the w i r e b y heating it w i t h the electric arc f r o m a T I G w e l d e r . L = 0.360 inches f o r the f u l l 2 i n c h wafer h o l d e r a n d the 4 sections o f a 1/4 o f a 2 i n c h wafer holder. L = 0.500 inches for the 1/4 segment o f a 2 i n c h wafer holder.  Appendix III  Round Molybdenum Nut  Wafer Holder Drawings  247  Appendix III  Wafer Holder Drawings  Molybdenum Holder for Two inch Wafer  248  Appendix III  Wafer Holder Drawings  Molybdenum Holder for 1/4 Segment of Two inch Wafer  249  Appendix III  Wafer Holder Drawings  250  Molybdenum Holder for Four sections of 1/4 Segment of Wafer  2  3.ovo p  .S 1 o 0  Appendix III  Wafer Holder Drawings  251  Molybdenum Flat Washer  U»  0.082  »4  L = 0 . 2 5 0 i n c h e s for the f u l l 2 i n c h w a f e r h o l d e r a n d the 4 sections o f a 1/4 o f a 2 i n c h wafer holder. L = 0.500 inches for the 1/4 segment o f 2 i n c h wafer holder.  

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