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Characterization of GaAs/Ga₁₋ Al As multilayer system by infrared spectroscopy at normal incidence Zeng, An 1992

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CHARACTERIZATION OF GaAs/Ga AlAs MULTILAYER SYSTEM 1 BY INFRARED SPECTROSCOPY AT NORMAL INCIDENCE By An Zeng B. A. Sc. Tsinghua University, China, 1985 M. A. Sc. Institute of Semiconductors, Chinese Academy of Sciences, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  Sept. 1992  ©  An Zeng, 1992  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  is  by the head of my  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department  of  I  (.1  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  Qt.  /s  /  Abstract  The infrared reflectance spectra of GaAs/GaiAlAs multilayer structures at normal incidence from the far-infrared to the near-infrared have been measured using a Bruker IF 113v spectrometer. The main structure in the spectra are due to the phonon reststrahlen bands, the Fabry-Perot interference and the plasma edge of the free carriers. An optical impedance method was successfully used to calculate the reflectance spectra at normal incidence. Through fitting the spectra, we determined both electrical and structural parameters, which are usually determined by several different methods. These parameters include the carrier concentration n, mobility  i,  conductivity u, Al concentration x, and  the thickness d of each layer. The agreement between the theoretical spectra and the experiments is excellent.  11  Table of Contents  Abstract  ii  List of Tables  vi  List of Figures  vii  Acknowledgement  ix  Introduction  1  1.1  Semiconductor Characterization Techniques  1  1.1.1  Photoluminescence  2  1.1.2  X-ray Diffraction  3  1.1.3  Van der Pauw  4  1.1.4  Four-point Probe Method  5  1.1.5  Scanning Electron Microscopy  5  1.1.6  Scanning Tunneling Microscopy  6  1.1.7  Secondary Ion Mass Spectrometry  7  1  1.2  The Need for a Non-Destructive Technique  1.3  Recent Infrared Studies  10  1.3.1  Free Carrier Absorption  10  1.3.2  Theoretical Calculation of Reflection and Transmission Spectra  1.3.3  8  from Multilayer Structures  11  Experimental Study of GaAs Multilayer Structure  11  iii  1.3.4 1.4 2  3  Outline of Thesis  16 18  2.1  18  Fourier Transform Spectroscopy 2.1.1  Interferometer  19  2.1.2  Fourier Transformation  21  2.1.3  Resolution  29  2.1.4  Advantages of FT-JR Instrument  31  2.2  Bruker 113V Spectrometer  32  2.3  Reflectivity Measurement  35  Calculation of Reflectance  38  Reflectance of Bulk Material  38  Reflectivity of a Multiple Layer Structure  38  Experiment Results and Analysis 4.1  4.1.2  4.1.3 4.2  41  Reflectance of Bulk Material 4.1.1  41  Pure GaAs  41  Doped GaAs  43  Pure Gai_AlAs  47  Reflectance of Multiple Layer Structures 4.2.1 4.2.2  5  12  Experiment  3.2 4  Far-infrared Studies of Superlattices  Reflectance of GaAs/AlAs multilayer structure Reflectance of GaiA4 As/GaAs/AlAs multilayer structure. 7  Conclusion  50 51 59 67  Appendices  69  iv  A Interference Fringes  69  Bibliography  71  V  List of Tables  4.1  The best fitting parameters for Gai_AlAs.  4.2  The best fitting parameters for each layer of sample #3  4.3  The electrical parameters of the top GaAs layer of sample #3, calculated from the plasma frequency w,  =  ii  and -y in cm’  53  450cm’ and free-carrier relaxation time  1 7=45cm 4.4  49  53  The best fitting parameters for the top Gai_AlAs layer of sample #4. cm 1 , v and -y in 1 d in itm  63  vi  List of Figures  Li  Geometric structure of a GaAs/A1GaAs multiple quantum well laser diode  1.2  Reflectance of the sample GaAs 262 in reference [16]  13  1.3  Reflectance of the superlattice sample in reference [27]  15  2.1  Schematics of a Michelson Interferometer  20  2.2  Spectra and their corresponding interferograms  23  2.3  Illustration of the convolution theorem of Fourier transform  26  2.4  Apodization functions and their correspondillg FT  30  2.5  Optical path of the Bruker IF 113v  33  2.6  The reflectance module  36  3.1  Illustration of optical impedance methods  40  4.1  The far-infrared spectra of Sample #1  42  4.2  The far-infrared spectrum of sample #2 and its best fitting  44  4.3  The conductivity spectrum of sample #2  46  4.4  The reflectance spectra of 7 A 1 Ga . As 4  48  4.5  The TO and LO phonon energies of Ga _,AlAs vs. the Al concentration x 50 1  4.6  The far-infrared spectrum of sample #3 and its best fitting  52  4.7  The calculated spectra of a sample similar to sample #3  54  4.8  The mid-infrared spectrum of sample #3  55  4.9  The near-infrared spectrum of sample #3 and its best fitting  57  4.10 The calculated near-infrared spectrum of a sample similar to sample #3 vii  2  58  4.11 The geometric structure of sample #4  60  4.12 The far-infrared reflectance spectrum of sample #4 and its best fitting 4.13 The dependence of the far-infrared spectrum on Al concentration  .  .  .  61 62  4.14 The mid-infrared spectrum of sample #4  65  4.15 The calculated mid-infrared spectrum of sample #4  66  4.16 The near-infrared spectrum of sample #4  66  viii  Acknowledgement  I would like to thank my supervisor, Dr. J. E. Eldridge, for his patient and knowledgeable supervision. I am very grateful to Dr. T. Tiedje for his valuable instructions and suggestions. All the samples used in this project were supplied by Dr. T. Tiedje and his co-workers. I am grateful to Christian Lavoie for growing the MBE samples. I would like to thank Dr. Martin Dressel and Dr. Kevin E. Kornelsen for their help in the lab. During this research I have been supported by a University Graduate Fellowship from the University of British Columbia. This work was also supported by the Natural Sciences and Engineering Research Council of Canada.  ix  Chapter 1  Introduction  1.1  Semiconductor Characterization Techniques  1, mo 3 ’ 2j, detectors[ 1 Many optoelectronics devices, such as semiconductor laser diodes[ ’ 7], are based on multilayer structures com 6 ’ 1 and nonlinear optical devices[ 4 dulators[ posed of various thin homo- or heterolayers of different thickness and doping levels. Homolayers are layers made of the same material but with different doping levels. Ret erolayers are layers made of different materials with the same or different doping levels. For example, a semiconductor laser diode is usually composed of five heterolayers of differ ent doping levels as shown in Fig. 1.1. The characterization of the electrical and structural parameters of this complex structure is a challenge. With the fast development of many thin film preparation techniques such as molecular beam epitaxy (MBE), metal organic chemical vapour deposition (MOCVD) and chemical beam epitaxy (CBE) etc.  ,  various  characterization techniques have been developed or used to determine the electrical and structural parameters. These include the free carrier concentration, the layer thickness and the composition concentration. These well developed characterization techniques are photoluminescence(PL), double crystal X-ray diffraction (DCD), Van Der Pauw method, four point probing method, scanning electron microscope (SEM), scanning tunnelling microscope(STM), and secondary ion mass spectrometry (SIMS) etc. These will now be described in some detail.  1  Chapter 1. Introduction  2  .—  p GaAs 027 p- Al  Gao.73  As  C... I X lO (... 7 X 1018  ) 3 cm Cm)  WELLS u—Al A 0 Ga 027 S BARRIERS UNDOPED .—u-GaAs  V  As 073 Ga 027 .—N-AI  cm—) 17 (..3X10  —GGA5 4 .—fl  Figure 1.1: Geometric structure of a GaAs/A1GaAs multiple quantum well laser diode. The active layer is composed of -undoped multiple quantum wells. The numbers in the parentheses are the carrier concentration of the corresponding layers. u, p, and n denote undoped, p-type, and n-type respectively. The “+“ sign denotes the heavy doped layers for Ohmic contacts. 1.1.1  Photoluminescence  The photolurninescence equipment is an excitation laser, a grating monochromator, a V  photon detector connected to a photon counting system and a cryogenic system for installing samples. The photoluminescence is generated by the recombination of the electrons and holes which are created by the excitation light, such as laser light with the photon energy larger than the band gap energy of the subject material. Photolumine scence is a conventional method used to investigate the near-band-gap energy levels[ ], 8 such as the impurity binding energies of both the donors and the acceptors and the binding energy of both the free- and bound-excitons as well as the band gap energy of the subject material. Photoluminescence is also commonly used to determine the  Chapter 1. Introduction  3  composition concentration of a ternary compound, since the composition concentration x of a ternary compound is usually related to the band gap energy Eg by a simple empirical relation. For example AlGai_As has the following relation[ ]: 13 2 E  (eV)  =  l.424+l.247x  =  1.900 + 0.125x + 0.143x 2  0<x<0.45 (eV)  (1.1)  0.45 <x < 1  (1.2)  There are, however, some requirements for the samples used in a photoluminescence measurement. For example, the upper layers must have a larger band gap than the lower ones in a sample with a multilayer structure. Otherwise, the photoluminescence light from the lower layers will be absorbed by the top layer. The AlGa_As layers in Fig. 1.1 cannot be characterized by photoluminescence since the top GaAs layer, which has a smaller band gap than that of A4Gai_As , will absorb all the photoluminescence light irom the lower layers. 1.1.2  -  X-ray Diffraction  The lattice coustant of a crystal can be determined from the Bragg Angle  .B 0  in an X-ray  diffraction measurement. =  2) arcsin--  (1.3)  where ) is the wavelength of the X-ray, and d is the distance between a set of planes in the crystal. The lattice constant a(x) of a ternary compound, e.g. in 1, 13 , 2 Al A 1 Ga s[ is usually a linear function of the composition concentration x. a(x)  =  5.6533 + 0.0078x  (A)  (1.4)  So the composition concentration of a ternary compound can also be determined by X ray diffraction. The resolution of the X-ray diffractioll technique depends on the spectral width and the divergence of the X-ray beam used in the measurement. By placing  4  Chapter 1. Introduction  a high quality crystal right after the X-ray source to monochromate the X-ray beam by diffraction, one can improve the resolution of the X-ray diffraction measurement. This improved method is called double crystal X-ray diffraction (DCD). Because of its high resolution, double crystal diffraction has also been widely used to measure some other structural properties, such as dislocation density and the thickness of the grown ’ 14, 11• 8 layers[  Since the X-ray diffraction peaks are the results of the constructive  interference of the X-ray beams reflected from a set of lattice planes, the thickness of the sample layers must not be too small in order to be detected.  1.1.3  Van der Pauw  17 is the standard method for measuring the resistivity and The Van der Pauw method mobility of an epitaxial semiconductor thin film, particularly for compound semiconduc tors.- (Net- carrier concentration can be calculated from the Hall mobility.) Essentially, it is necessary only to make four small contacts(A, B, C, and D) along the circumference of an arbitrarily shaped flat sample, and to measure the resistances given by and  VBC/IAD.  VDC/IAB  The sample resistivity is then given by p  =  ird VDC + 21n2 (— ‘AB  VBC  —)f ‘AD  where d is the sample thickness and f is a function of  (1.5) VDCIAD/VBCIAB.  The latter  is a geometrical correction factor only. When the sample is a circle or a square and the contacts are spaced at equal distances around the periphery, then Hall mobility is determined by measuring the change of the resistance  f  =  VAC/IBD  1.  The  when a  magnetic field B is applied in a direction perpendicular to the sample. The Hall mobility, ,UH,  is given by d  /VAc/IBD  (1.6)  Chapter 1. Introduction  (Note that  H  5  is independent of d.) Since errors can be introduced by the finite diameter  and position of the contacts on the circumferences, it is common to sandblast the sample in a clover-leaf shape. The conducting film used in Van der Pauw measurement must be grown on a insulating substrate. If there is more than one conducting layer, the measure ment will give an average value of the entire structure. This method will completely fail if the conducting films are grow on a conducting substrate as is the case for most of the optoelectronics devices, where multiple doping layers are grown on a highly conducting substrate. 1.1.4  Four-point Probe Method  The four-point probe method is a well established technique for silicon. It does not, however, work well with compound semiconductors, such as GaAs and InP. The fourpoint probe on thin layers gives -an average resistivity of: p=4.5•X  (1.7)  where X is the layer thickness. The problem is that the point-contacts will form electrical barriers to the semi-conductor with a larger band-gap. These barriers are broken down -by alloying contacts or by electrical means. The result, though, is a destructive measure ment. A resistivity measurement can also be made by point-contact-breakdown using two or three probes and biasing one until breakdown occurs. But, all probe methods give unreliable results and are generally not used[ ]. 9 1.1.5  Scanning Electron Microscopy  Scanning Electron Microscope (SEM) is composed of two sub-systems: • the electron optical column which produces a finely focussed beam of electrons that is scanned in a television-type raster over the specimen surface  Chapter 1. Introduction  6  • and a signal detection, processing and display system, which detects one of the six forms of signal emitted by the specimen under electron bombardment. These six forms of signal are 1) emitted electrons, 2)emitted X-ray, 3) cathodolumines cence (CL), 4) electron beam induced current or voltage (EBIC or EBIV), 5) transmitted electrons under continuous electron bombardment and 6) ultrasonic waves under chopped electron beam bombardment. The SEM operates in one of the six modes named after the signal it detects, such CL mode and EBIC mode which are often used for semiconductor analysis. The electron optical column provides the “illumination” and the detection and display system produces the television-like picture. The resolution of SEM is not limited by diffraction as is the ordinary optical microscope. It is determined by the physics of the electron optical column and of the signal detection and display system. It also de pends on the physics of the signal generation process of the particular mode. The SEM techniques provide measurement with a micron scale resolution and so can determine defect microstructures and local values of physical properties. One of the great advan tages of SEM is that it can examine microscopic specimens with little or no preparation, e.g. whole wafers or electronic devices can be inspected. Moreover, SEM offers not only lateral, spatial resolution but also depth resolution, spectral (signal) resolution, time resolution, etc. More details can be found in references [10, 11].  1.1.6  Scanning Tunneling Microscopy  Scanning tunneling microscopy (STM) is a technique which is capable of viewing a con ducting or semiconducting material with atomic resolution. Quantum mechanically, when two materials, such as a probe tip of STM and a semiconducting sample, are brought so close together that the electronic wave functions of the tip and the sample overlap significantly, a tunneling current IT can be induced by an applied bias voltage VT. The  7  Chapter 1. Introduction  tunneling probability depends on the overlap of the wavefunctions of the filled and empty states of the tip and the sample. The exponentially smaller  ‘T  contributions of the longer  paths in the tunnel gap leads fortuitously to the dominance of a single atom to the current from a tip and therefore to the atomic resolution of the STM. In the most usual STM mode  ,  VT is held fixed and  ,  as the tip is piezoelectrically rastered across the sample  (assumed for now to be homogeneous), the gap is held constant by actively holding IT constant. The changes  iii  voltage of the piezoelectric transducer perpendicular to the  surface are proportional to the surface topographic variations. Since STM does not work well for rough surfaces, it is critical to have a very clean and smooth sample surface. The sample preparation is one of the most difficult steps in a STM experiment. The detailed description and applications of STM can be found in reference [18]. 1.1.7  Secondary Ion Mass Spectrometry  The Secondary Ion Mass Spectrometry (SIMS) setup is mainly composed of an ion gun, a mass spectrometer and a data processing system. In a SIMS measurement the sample is bombarded in a vacuum by an energetic beam of primary ions (1 to 2OkeV). As a result, particles are sputtered from the sample surface, some of which are secondary ions. The positive or negative ions are extracted into a mass spectrometer and separated according to their mass to charge (m/e) ratio. With primary beam rastering  ,  the sputter erosion  causes the sample surface to recede in a very controlled way. By monitoring the intensity of one or more mass peaks as a function of bombardment time, an in-depth concentration profile is obtained. SIMS is often used to answer the following questions: (1) what is the form of the dopant profile (i.e. concentration as a function of depth)? (2) does the dopant profile show abrupt step changes between different concentration levels? (3) how sharp is the interface at a hetero-junction? and (4) are spurious electrically active impurity elements present? SIMS is one of the few methods which can characterize both electrical  Chapter 1. Introduction  8  and structural parameters at the same time using only one complex multilayer sample. For example, SIMS was used to determine the free carrier concentration and thickness of each layer of the multilayer structure in reference [16]. The disadvantage of SIMS is that it is destructive, time consuming and difficult to calibrate. For a detailed description of SIMS and its applications the reader is referred to the book by Benninghoven et.  From the above sections one can see that almost all the techniques have at least one specific requirement for the samples and that electrical and structural parameters are usually determined by two or more different methods using different samples. For exam ple, one uses photoluminescence to determine the Al concentration and crystal quality, SEM to determine the layer thickness, and Van der Pauw method to determine the carrier concentration. The only method mentioned above, which can characterize both electrical and structural parameters at the same time using only one complex multilayer sample, is SIMS. But this method is destructive, time consuming and difficult to calibrate, especially if a ternary compound layer like AlGai_As is involved.  [.2  The Need for a Non-Destructive Technique  Recent publications, however, have shown an increasing interest and need for a non destructive and non-contact method to characterize both electrical and structural pa rameters  ill  a multilayer structure ’ 19, 20, 21] 16  There are several immediate needs in  addition to those already mentioned: 1. ZnSe is a good candidate for a blue-light-emitting device because of its suitable direct band gap (2.7 eV at room temperature)[ 1. However, the formation of ZnSe 22 p  —  n junctions has been difficult due to problems in controlling the conductivity  of the bulk crystals.  Much progress has been made, however, with ZnSe thin  9  Chapter 1. Introduction  films grown on GaAs. p-type concentration characterization however still remains 1, 23 a major problem because of the lack of a suitable ohmic contact technology[ which makes the conventional four-point probing method impossible. A non-contact method to determine the p-type carrier concentration is urgently needed. 2. The InAs/GaAs heterostructure has potential in infrared detection or emission as ’ 25j• 24 well as in other optoelectronics applications[  However due to the lattice  mismatch between these two material (7.3%), misfit dislocations propagate into the InAs layer deposited on a GaAs substrate. These structural defects primarily affect the electronic properties of the material. Many efforts have been made to ’ 25, 26]• Hydrogenation of the InAs layer is one of 24 reduce the dislocation density[ them. Atomic hydrogen incorporated into a semiconductor passivates impurities or defects and neutralizes their electrical activity. Because hydrogenation is reversible, any contact measurement may change the hydrogen concentration. This hydrogen concentration is determine by the destructive SIMS method. A non-destructive, non-contact method is needed to determine the free carrier concentration in the study of the effect of hydrogen concentration on the free carrier concentration in . 21 InAs/GaAs heterostructures ’ 20] 16 Infrared spectroscopy has shown promising capabilities in solving these problems[ Besides its conventional use in the determination of the lattice vibrational frequencies, it can also be used to determine the electrical and structural parameters of semiconductor thin films, such as the free carrier coilcentration, the composition concentration, the free carrier scattering time and the layer thickness. Although the technique has not been fully developed, it has received increasing attention since 1985. Several papers on vari ’ 20, 27j In the following section, we will 6 ous material have been published since then[ briefly summarize some of the recent publications on this newly developing method.  10  Chapter 1. Introduction  Recent Infrared Studies  1.3 1.3.1  Free Carrier Absorption  Because of its potential in preparing blue light-emitting devices, doped ZnSe has be ] improved Jensen’s 19 come the topic of many recent publications. In 1987, H. E. Ruda[ 1 of free-carrier absorption by considering all major scattering mechanisms, such 28 theory[ as polar-optical phonon scattering and ionized impurity scattering etc. This made it possible to assess the effect of the impurities on free-carrier absorption. The dependence of the electron concentration n on the free-carrier absorption coefficient a was calculated as a function of the ionized impurity concentration and the compensation ratio 0. The 0 is the ratio of the concentration of ionized acceptors to that of the ionized donors in the material. At A  =  lOfLm, a simple formula which can be used to calculate n-type ZnSe  carrier density by freercarrier absorption was proposed. ) 3 n(cm  =  x (3.5  6.7 x  —  p) x a,(cm’)  (1.8)  where a), and p are the absorption coefficient and the derivative logarithmic absorption coefficient respectively at A  =  10pm. p is defined as:  =  dln(—)  (1.9)  0 is a reference wavelength. Experimental data from the literature are analyzed where A in this scheme and values of n determined by Hall-effect measurements are shown to be in excellent agreement with the theoretical predictions. This non-contact and non-destructive method was used by A. Deneuville and D. B. 1 later in 1991 to determine the carrier concentration in n-type ZnSe thin 20 Tanner et al.[ films grown on a GaAs substrate. Deneuville measured both the transmission and the reflection spectra in the far infrared range and found that they are very sensitive to  11  Chapter 1. Introduction  the carrier concentration. Deneuville et al did not find, however, the correct formula to reproduce the reflection and transmission spectra of a multilayer structure.They used instead an approximation to calculation 1.3.2  from the transmission spectra.  Theoretical Calculation of Reflection and Transmission Spectra from Multilayer Structures  29 reported his theoretical work on a matrix method to calculate In 1986, B. Harbecke the coherent and incoherent reflection and transmission of multilayer structures. The complex-amplitude reflection and transmission coefficients r and t of a multilayer struc ture are represented as a product of matrices. The matrices describe the transformation of two plane waves travelling in opposite directions between the films, and their develop ment within the films. One of his contributions is that he is able to easily suppress the substrate interference fringes, which are sometimes noi observed in the experiment, by adding the absolute squares of the partial waves corresponding to an incoherent treat ment. This procedure is shorter and simpler than the conventional method of averaging over an appropriate interval of frequency or thickness, which, in most cases, leads to the same results. The parameters needed in the calculation are the dielectric function c(w), the magnetic permeability p(), the thickness of layers and the angle of the incident infrared beam. 1.3.3  Experimental Study of GaAs Multilayer Structure  1 used the method of the previous section 16 In 1990, P. Grosse and B. Harbecke et al. to simulate the experimental reflection spectra of a conducting GaAs multilayer struc ture. Both the electrical and geometric properties of the GaAs multilayer structures were determined nondestructively by reproducing the infrared reflectance spectra from 1 (50—5000cm  ).  The sample consisted of two layers of GaAs with the same thickness  12  Chapter 1. Introduction  but with different doping levels, sitting on a conducting GaAs substrate. Using oblique incidence and both s- and p-polarizations of the infrared beam, the carrier concentration and the thickness of the epitaxial films as well as the carrier concentration of the GaAs substrate were determined.The main structures in the spectra were due to phonon rest strahien bands, Fabry-Perot interferences and the zeros of the dielectric function leading to dips in the reflectance( Berreman mode  ).  The interference fringes were used to es  timate the first order approximation of the film thicknesses. Fourier transformation of the reflectance spectra was suggested to estimate the first order thickness in those cases where the structure consists of two or more layers of different thickness. The Berreman effect, causing dips in the p-polarization reflectance spectra at zeros of the dielectric function, was used as a first order approximation to find the carrier concentration in the films. The final results of the carrier concentration and layer thickness were determined by fitting the -reflectance spectra, using the first order approximation of thickness and concentrations as input. Fig.l.2 shows that the theoretical spectra fit the experimental ones very well. The parameters for the thin films compare favorably with a depth-resolved SIMS sample analysis. A substantial discrepancy was found, however, in the data per taining to the substrate. The substrate concentration determined by SIMS or electrical measurement was substantially higher than that determined by infrared spectroscopy. m 3 c 8 1O’ m 2.5 x ) ersus c 8 1O’ v (4.15 x 3 1.3.4  Far-infrared Studies of Superlattices  In addition to the characterization of multilayer structures with different doping lev els, infrared spectroscopy has also been developed as a new method to characterize the structural parameters of superlattice structures which are composed of many periods of 1 re 30 alternating thin layers of two or more materials. In 1985, V. M. Agranovich et al.[ ported their important theoretical work on superlattice optical properties, which makes  Chapter 1. Introduction  13  to 0.8  0.2 0.0 0.8 0.6 0.4 0.2 0.0  0  100  200 300400500800 ,/2irc Ecnf’]  Figure 1.2: Reflectance of the sample GaAs 262 in reference [16]. The solid line: mea sured data, dotted line: model fit. a) s-polarization, a = 70°; b) p-polarization, cv = 70°. FP: Fabry-Perot resonances; TO: position of the polar optical phonon; B , B 1 : Berre 2 man-mode.  Chapter 1. Introduction  14  it possible to calculate the dielectric function of a superlattice through the dielectric functions for the bulk materials of the constituent layers. This paper is frequently cited by experimental physicist working in this area now. The calculation method here is different from that which we have introduced pre viously, where bulk dielectric functions were used for each layer in the calculation, as suming there is no change for dielectric functions in thin films. In Agranovich’s method, the whole superlattice structure is treated as a slab of an effective medium, whose di electric response depends on the geometrical layer arrangement. The dielectric function is a diagonal tensor whose components, in the long wavelength limit, can be written as weighted averages of the dielectric functions of the constituent layer materials. For the infrared beam with the electric field E parallel to the layers, the dielectric function can be written as: -  where  (w) 1 e  e1(w)+E2(w)  (110)  and e (w) are the dielectric functions for the bulk materials of the con 2  stituent layers, and  e  is the ratio of the thicknesses of the two alternating layers. in  1 reported the first infrared spectra for AlAs/GaAs superlat 27 1987, R. Sudharsanan[ tices and analyzed the data, using Agranovich’s above formula to calculate the dielectric function for the superlattices, in order to find the superlattice layer thicknesses. Fig.l .3 shows the calculated reflectance spectra and the measured data. Although there is a small systematic difference, the results compare favorably to those obtained from double crystal X-ray diffraction which is considered now to be the most accurate method of de termining the superlattice structure. Sudharsanan suggests that far-infrared reflectance spectroscopy is as precise as that of double crystal diffraction and has advantages over the Raman scattering method which has also been used recently to characterize AlAs/GaAs 1. The far-infrared radiation penetrates much more deeply than the laser 31 superlattices[  Chapter 1. Introduction  15  1.0 .0.8  . 0.6 C-)  a) a)  0.4 0.2 0.:0 1.  >‘  0.  I..  :  -4-A  a.  C-)  a)  0.  C)  0. 0.  Wave Number (cm ) 1 Figure 1.3: Reflectance of the superlattice sample in reference [27]. The dots : measured data, solid lines: model fit. (a) Theoretical curve using nominal layer thicknesses. (b) Theoretical curve using best-fit layer thicknesses. Note the improved fit to positions of the minima at 338 and 396 cm . 1  16  Chapter 1. Introduction  light in Raman scattering and samples the entire microstructure. Moreover, the existence of a simple superlattice dielectric response function allows quantitative analysis. In addition to the characterization of a multilayer structure, infrared spectroscopy has also been used recently to probe layer interdiffusion and alloying in HgTe-CdTe ’ 35] and to examine AlGai_As/GaAs[ 34 superlattices[ 1 and AlAs-GaAs heterostruc 36 32 et al. used it to study the dependence of reststrahlen bands . G. Scamarcio[ 37 tures of multiple quantum well heterostructures and the transverse optical phonon of ultrathin 1 used 40 ’ 39] and A. R. El-Gohary et al.[ 38 1. T. Dumelow et al.[ 33 layer superlattices[ it to study surface, bulk plasmons and phonon-polaritons in superlattices and multiple quantum well structures.  Outline of Thesis  1.4  In Chapter 2 on the experiment, the principles and the advantages of an infrared Fourier transform spectrometer are described, along with the Bruker IFS 113V interferometer and the reflectivity module which were used in the experiment. In Chapter 3 on the theory, the optical impedance method used to reproduce the experiment reflectance spectra is introduced. In Chapter 4 on the results, the measured reflectance spectra of four samples in 1 are presented. the range from 50—15000 cm  The main features of the spectra are  due to the phonon reststrahlen bands, interference fringes and the free carrier plasma edges.  Drude-Lorentz model and optical impedance method were used to reproduce  the far-infrared reflectance spectrum of a multilayer structure. An empirical formula of refractive index and the optical impedance method were used to reproduce the near infrared spectrum of a multilayer structure. The fringes in the different region of the spectra were found to have resulted from the interference of the layers with different  Chapter 1. Introduction  17  thicknesses. The plasma edges were used to estimate the carrier density of the samples. The electrical and structural parameters of each layer, e.g. carrier concentration, Al concentration, and layer thickness etc., were obtained through fitting the experiments. The agreement between the theoretical spectra and the experiments is excellent. In Appendix A, the formula used to calculate layer thickness from unevenly separated Fabry-Perot interference fringes due to dispersion effect is introduced. Finally the conclusions of the thesis are reviewed in Chapter 5, and the limit of the method is discussed.  Chapter 2  Experiment  2.1  Fourier Transform Spectroscopy  Infrared spectroscopy is a powerful tool often used by physicists and chemists. Fourier Transform Infrared Spectroscopy(FT-IR) is the most advanced form of infrared spec troscopy. A FT-JR spectrometer is mainly composed of two parts: • a optical hardware(bench) which consists of optical sources, a Michelson interfer ometer, a sample chamber, and optical detectors covering different frequency ranges • and a computer with special software for data processing. Unlike a conventional grating instrument, a FT-JR instrument is not simply controlled by setting appropriate knobs controlling slit widths, scanning speed, etc. but involves a certain amount of mathematical manipulations such as Fourier transformation, phase correction and apodization. The quality of the final spectra depends on both the settings of the optical bench and the parameters or functions used by the software to calculate the spectra. It is important that the user be familiar with the principles of FT-IR data collection and manipulation. In the following sections we will cover briefly the standard operations of the FT-JR spectroscopy from data acquisition to the final spectrum. The following section is mainly abstracted from references [41] [53] [54].  18  Chapter 2. Experiment  19  Interferometer  2.1.1  The essential piece of the optical hardware in a FT-JR spectrometer is the interferometer. The basic scheme of an idealized Michelson interferometer is shown in Figure 2.1. Infrared light emitted by a source is directed to a device called the beam splitter, because it ideally transmits half of the light while reflects the other half. The reflected part of the beam travels to the fixed mirror M 1 through a distance L, is reflected there and hits the beam splitter again after a total path length of 2L. The same happens to the transmitted part of the beam. Except that the reflecting mirror M 2 for this interferometer arm is not fixed at the same position L but can be moved very precisely back and forth around L by a distance x, The total path length of this beam is accordingly 2(L+x). Thus when the two halves of the beam recombine again on the beam splitter they exhibit a path length difference or optical retardation of 2x. The partial beams are spatially coherent and will interfere when they recombine. The combined JR beams leaving the interferometer pass through the sample chamber and are finally focused on the detector D. The quantity actually measured by the detector is thus the intensity 1(x) of the combined JR beams as a function of the moving mirror displacement x, the so-called interferogram shown in Figure 2.1 B. The interference pat tern as seen by the detector is shown in Figure 2.1 C for the case of one monochromatic spectral line. The partial beams interfere constructively, yielding maximum detector signal, if their optical retardation is an exact multiple of the wavelength ). Minimum detector signal occurs at destructive interference where the optical retardation is an odd multiple of \/2. The complete dependence of 1(x) on x is given by a cosine function: 1(x) where  ii  =  S(v) x [cos(27r v x)  +11  (2.1)  is the wavenlLlmber, and S(v) is the intensity of the monochromatic line at  wavenumber  xi.  For a polychromatic source, the interferogram is the integral of the  Chapter 2. Experiment  20  Figure 2.1: (A) Schematics of a Michelson Interferometer. 5: JR source, M : fixed 1 mirror, M : scanning mirror, D: detector. (B) interferogram of a polychromatic source. 2 (C) interferogram of a monochromatic source. Its zero crossings define the positions where the interferogram is sampled(dash lines)  Chapter 2. Experiment  21  monochromatic interferogram over the spectrum range: fVmax  1(z) =  where  min 11  and  Vma  S(z) x [cos(27r.  j  •  x) + 11th’  (2.2)  ’min 1  are the minimum and the maximum wavenumber generated by the  polychromatic source. Equation 2.1 is very useful for practical measurements, because it allows very precise tracking of the scanning mirror. In fact, all modern FT-JR spectrometer use the interfer ence pattern of the monochromatic light of a He-Ne laser to control the movement of the scanning mirror and the sampling interval of the JR interferogram. Figure 2.1 C demon strates how the JR interferogram is sampled precisely at the zero crossings of the laser interferogram. The accuracy of the sampling interval between two zero crossings is solely determined by the precision of the laser wavelength itself. Thus, FT-JR spectrometers have a built-in wavenumber calibration of high precision(practically about 0.01 cm’). This advantage is know as the Connes advantage[53]. Other advantages of the FT-JR instrument will be discussed later.  2.1.2  Fourier Transformation  Data acquisition yields the digitized interferogram 1(x), which must be converted into a spectrum by means of a mathematical operation called Fourier transformation (FT). Generally, the Fourier transformation of a continuous interferogram generates a contin uous spectrum. However, if the interferogram is sampled and consists of N discrete, equidistant points, one has to use the discrete version of the Fourier transformation: S(k. v)  =  I(n. Ax) exp(Z2k)  (2.3)  22  Chapter 2. Experiment  where the continuous variables x,  ii  have been replaced by n• Ax and k Au respectively.  The spacing Au in the spectrum is related to Ax by Av =  (2.4)  N Ax  This process is reversible. If S(k. Au) is known, one can easily reconstruct the interfer ogram I(ri. Ax) by the so-called inverse discrete Fourier transformation: I(n. Ax)  1 =  N—i  —  Nk_o  S(k. Au) exp(  i2irnk N )  (2.5)  The above Fourier transformations are best illustrated in the simple case of a spectrum with one or two monochromatic lines, as shown in parts A and B of Figure 2.2. For a limited number of functions like the Lorentzian in part C of Figure 2.2, the correspond ing Fourier transformation is known analytically and can be looked up from a Fourier transformation table-. However, in the general case of measured data, the discrete Fourier transformation must be calculated numerically by a computer. Although the precise shape of a spectrum cannot be determined from the interfero gram without a computer, it may nevertheless be helpful to know two simple trading rules for an approximate description of the correspondence between I(ri. Ax) and S(k. Au). From part C of Figure 2.2 we can extract the general qualitative rule that a finite spectral line width(as is always present for real samples) is due to damping of the amplitudes in the interferogram: the broader the spectrum the stronger the damping. Comparing the widths at half height (WHH) of I(n Ax) and S(k. Au) one reveals another related .  rule: the WHH of a “hump-like” spectrum is reversely proportional to that of its Fourier transformation—the interferogram. The rule explains why in Figure 2.2 D the interfero gram due to the broad spectrum shows a very sharp peak around the zero path difference position, while the wings of the interferogram, which contain most of the useful spectral information, have a very low amplitude.  Chapter 2. Experiment  23  1(x)  S(v)  A  B  C 4-  h. vw  2000 4000 VVENUMBERS CM-I  Tz,  I,  xz  Figure 2.2: Examples of spectra(on the left) and their corresponding interferograms(on the right). (A) one monochromatic line. (B) two monochromatic lines. (C) Lorentzian line. (D) broadband spectrum of polychromatic source.  24  Chapter 2. Experiment  For n  =  0, the exponential in equation (2.5) is equal to unity. For this case expression  (2.5) states that the intensity 1(0) measured at the interferogram centerburst is equal to the sum over all N spectral intensities divided by N. This means the height of the center burst is a measure of the average spectral intensity. In practice, equation (2.3) is seldom used directly because it is highly redundant. Instead a so-called fast Fourier transforms(FFT) is used. The aim of the FFT is to re duce the number of complex multiplications and sine and cosine calculations appreciably, leading to a substantial saving of computer time. The small price paid for the speed in calculation is that the number of interferogram points N can not be chosen freely. For the most commonly used Cooley-Tukey algorithm, N must be a power of two. It should be noted that discrete Fourier transformation approximates the continu ous Fourier transformation only when it is used correctly. Blind use of equation (2.3), however, -can lead- to three well—known spectral artifacts: the picket-fence effect, aliasing, and leakage. In the following paragraphs, we will discuss these three artifacts and their solutions. Picket-fence Effect and Its Solution: Zero Filling The picket-fence effect[54] becomes evident when the interferogram contains frequencies which do not coincide with the discrete frequency points k. Av of the spectrum obtained by discrete Fourier transformation. If, in the worst case, a frequency component lies exactly halfway between two discrete frequency points, one seems to be viewing the true spectrum through a picket-fence, thereby clipping those spectral contributions lying ‘behind the pickets’, i.e. between the sampling positions k Lv. The picket-fence effect can be overcome by adding zeros to the end of the interferogram before performing the discrete Fourier transformation, thereby increasing the number of points per wave number in the spectrum. Thus  ,  zero filling the interferogram has the effect of interpolating the  25  Chapter 2. Experiment  spectrum, reducing the picket-fence effect. As a rule of thumb, one should always at least double the original interferogram size for practical measurements by zero filling it, i.e. one should choose a zero filling factor(ZFF) of two. it should be noted, that zero filling does not introduce any errors because the instrumental line shape is not changed. It is therefore superior to polynomial interpolation procedures working in the spectral domain. Aliasing and Its Solution: filtering Another artifact caused by discrete Fourier transformation is the so-called aliasing effect. To understand aliasing and leakage which we will discuss later, it is convenient to intro duce a theorem called the convolution theorem of Fourier transform. It states that the Fourier transform of the product of two functions, e.g. 1(x) and (x), is the convolution of their individual transform S(ii) and  S(v) 0 A(v)  (i),  =  J  where tha convolution is defined as:  S(u)(v  —  u)du  (2.6)  The coilvolution theorem is used in Figure 2.3 to illustrate how to determine the discrete Fourier transformation of a sampled interferogram. The discrete interferogram is treated as a multiplication of the continuous interferogram 1(x) and a sampling function A(x) which is a set of delta functions with sampling illterval Lxx. From (f) of Figure 2.3 we can see that the FT of the sampled interferogram is a periodic function where one period within a constant  is equal to (c) the FT of the continuous interferogram. The  periodicity can also be seen from equation 2.3 which is not oniy valid for indices k from  o  to N  —  1 but for all integers including negative numbers. In particular, if we replace k  in Equation (1) by k + m N , we get the equation:  S(k+m.N) =S(k)  (2.7)  26  L(x)  1(x)  (b)  (a)  •1  UA  D . J;f\• (e)  0  Figure 2.3: (a) 1(x): continuous interferogram. (b) Ls(x): a set of delta functions; tx: sampling interval (c) S(v): real spectrum(positive part)—FT of 1(x). (d) v): FT of the set of delta functions. (e)I(x). L(x): sampled interferogram. (f) FT of the discrete interferogram which is real spectrum plus the mirror images.  Chapter 2. Experiment  27  which states that the mirror-symmetrical N-point sequence is endlessly and periodically replicated as indicated in Figure 2.3 (f). By picking out one period of (f) we can deter mine the real spectrum of the interferogram 1(x). As only the positive frequency has real physical meaning, we ignore the negative frequency range. By comparing the real spectrum(positive part) Figure 2.3 (c) and the FT of the sampled interferogram (f), one finds that only the first half period from 0 to The other half period from  of (f) represents the real spectrum.  is just the mirror image of the first half period  to  and therefore should be discarded. The folding wavenumber  Uf =  is also called  Nyquist-wavenumber[54}. From Figure 2.3 (f) it is clear that a unique spectrum can only be calculated if the spectrum does not overlap with its mirror-image(alias). No overlap will occur if the spectrum is zero above a maximum wavenumber  Vma  and if  Vma  is  smaller than the folding wavenumber vf: Vmax  2x  (2.8)  If, however, the spectrum contains a non-zero contribution above the folding wavenumber Vf,  e.g. non-zero contribution from  area below  Vf  (from  Vf  —  i’  to  , this will be ‘folded back’ to the 1 + 200cm  ij  1 to vf). This is the possible artifact due to aliasing. In 200cm  previous section we have explained that the sampling positions are derived from the zero crossings of a He-Ne laser with wavelength  = 158OOcm’  the minimum possible sample spacing Axmjn is  1 31600cm  As zero crossing occurs every According to equation 2.8, this  corresponds to a folding wavenumber of 15800cm’, i.e the maximum bandwidth which can be measured without overlap is 15800cm’. Very often, the investigated spectrum has a bandwidth much smaller than 15800cm 1  .  In these cases, one can choose Lx to be an  rn-fold multiple of AXmin. This leads to an rn-fold reduction of the interferogram size and therefore increases the speed of computing. Further reduction of the data size is possible by using the so-call undersampling technique, if the spectrum is non-zero only in a limited  Chapter 2. Experiment  range  (“max  —  “mm)  and  28  (Vmax  vmmn)  lies between two of the folding wavenumbers which  are a natural fraction or integer multiple of the He-Ne laser wavenumber: Pfn =  n 15800cm (n 1  (2.9)  An advanced FT-JR software package will automatically account for proper sampling and undersampling, if the upper and lower limits of the desired spectral range are specified. To avoid aliasing, the user only needs to make sure that the investigated spectrum is really zero outside the two folding wavenumbers by inserting optical or electronic filters. Leakage and its solution: apodization Unlike the picket-fence effect and aliasing, leakage is not due to using a digitized version of a continuous interferogram. Leakage is caused by the truncation of the interferogram at finite optical path difference. Mathematically, an interferogram trimcated at optical  path difference x  L can be obtained by multiplying the infinite interferogram 1(x) by  a ‘boxcar’ or square window function B(x). According to the convolution theorem, the FT of the truncated interferogram can be obtained by convoluting the spectrum S(v) corresponding to infinite optical path difference with the FT of the window function B() which determines the instrument lineshape function. The analytical form of the  ‘instrument lineshape’ function corresponding to boxcar truncation is the well known sinc function. The FFT of a square window function is plotted in Figure 2.4 B. Besides a main peak, one sees many additional side peaks, called lobes or ‘feet’. These side peaks cause a ‘leakage’ of the spectral intensity. The largest side lobe amplitude is 22% of the main lobe amplitude. As the side lobes do not correspond to actually measured spectrum but rather represent an artifact due to the abrupt truncation of the interferogram, it is desirable to reduce their amplitude. The process to attenuate the spurious ‘feet’ in the spectrum is known as ‘apodization’ (originating from the Greek word a7roS, which means  Chapter 2. Experiment  29  ‘removal of the feet’). The solution to the problem of leakage is to truncate the interferogram less abruptly by using an apodization fimction. There are numerous such functions. In Figure 2.4, three such functions and their FFT are plotted together with a square function. One can see that the ‘feet’ of these three functions are much smaller than those of the square function. However, one must realize that this is at the cost of the resolution. 2.1.3  Resolution  Figure 2.4 reveals that all apodization functions produce an instrument lineshape function (ILS) with lower sidelobe levels than the sinc function. However, one also sees that the main lobes of all the ILS’s are broader than that of the sinc function. The full width at half maximum (FWHM) of a ILS defines the best resolution achievable with the given apodization function. This is because if two spectral lines are to appear resolved from one another, they must be separated by at least their FWHM, otherwise no ‘dip’ will occur between them. As side lobe suppression always causes main lobe broadening, leakage reduction is only possible at the cost of resolution. The choice of a particular apodization function depends on the experimental require ment. If the optimum resolution is mandatory, the square window function with the resolution of 0.61/L (no apodization) should be chosen. If a resolution loss of 50% com pared to the square truncation can be tolerated, the Parzen(triangle) or even better the Hanning function with resolution of O.9/L (roughly) is recommended. If the interfero gram contains strong low-frequency components, it may show an offset at the end, which would produce ‘wiggles’ in the spectrum. To suppress these wiggles, one should use a function which is close to zero at the boundary, such as triangular or Hanning function. Detailed information about the apodization functions can be found in reference [42]. Until now, we considered only a plain wave incident normally on the mirrors. With  30  Chapter 2. Experiment  II  IIIIIIII  1111111111  IlIlIlIlIllIlIl  square window—” /  Welch window-.-/ /  ,/  .7  /  CO  / /  2  //  0  \  \\  \\ ,“/ -Parzen window \\  \  \\ \\  \\\s  11/  ‘I  /  \  // I’  ,  \ \ S \\  l  ,“Hanning window  ‘\  \ \  \  -  -—  I  I  0  20  ‘I’’’  40  I  I  I  I  I  I  60  80  100  120  140  160  I 160  200  II  jiTIIjIiILJIlIIjEIIPI  220  240  —  .6 —  .5 .  :11! :111 Iii  Ii I ‘/1 I  i’l // /  :111 -  -  n :111 l\\’,  .8-  .7  260  Tr—r 1 rrIr  (B)  .9  .3  \  \\  /1’  1  \  \\%\  I” I’  / /  \\  /  /  .3  ‘  \\\  / 1/  /  / // I.  .4  /  /  /  (A)  \\  \  1,1/  /  5 a  /\  /  ,  IlIlIllIllIlIlI  /1/  —  \‘\‘ \\\\ I  I I\  —  \‘\\  Hanning Parzen \\Welch I\  Figure 2.4: (A) apodization functions (B) the corresponding FT of the window functions.  31  Chapter 2. Experiment  any finite sized aperture, the radiation from the off-axis points will be incident at some angle on the mirrors. This radiation will have a reduced optical path difference between the two arms, which will transform into a smaller wavenumber. Thus the effect of the entire aperture is to spread a monochromatic v into a range from z-’ to  ii  —  ii  cos  This affects both the accuracy and the resolution. It is obvious that the improvement of resolution and accuracy by reducing the size of aperture is actually at the cost of the optical throughput. This is similar to the dispersive instrument. However, unlike the resolution of a dispersive instrument which is oniy controlled by the width of the slits, the resolution of a FT-JR spectrometer depends on the optical path difference L(scanning distance), apodization function, zero filling of interferogram as well as the aperture size of the optical source. Generally speaking, a FT-JR spectrometer has a higher resolution than that of a dispersive spectrometer. However, one should avoid measuring-a spectrum with very high resolution over a large frequency range, as this may result in a poor signal to noise ratio for the calculated spectrum. When measuring a high resolution spectrum, one must realize that the signal to noise ratio of the interferogram (S/N)IFG  is different from that of the calculated spectrum (S/N)spE. They are related  by the following formula[57j: (S/N)spE —  (S/N)IFG  —  V  2 10  where 6v and Lv are the resolution and spectrum range respectively. Thus a measure 1 will result in a spectrum with ment with resolution 0.lcm’ over a range of 1000cm one hundredth of the signal to noise ratio of the interferogram. 2.1.4  Advantages of FT-JR Instrument  Besides its high wavenumber accuracy discussed before, a FT-IR spectrometer has other features which make it superior to a conventional grating IR spectrometer. The so-called  32  Chapter 2. Experiment  Jacquinot or throughput advantage arises from the fact that a FT-JR spectrometer can have a large circular source at the input or entrance aperture of the instrument with no strong limitation on the resolution. Also, it  can  be operated with large solid angles  at both the source and the detector. However, the resolution of a conventional grating spectrometer depends linearly on the width of the input and output slits. Also, for high resolution, a grating spectrometer requires large radii for the collimation mirror, and this condition in turn necessitates small solid angles. Thus, for the same resolution, a FT-JR spectrometer can collect much more energy than a conventional grating spectrometer. In conventional spectrometers the spectrum S(v) is measured directly by recording the intensity at different monochromator settings of wavenumber v. In FT-IR, all frequencies emanating from the JR source impinge simultaneously on the detector. This accounts for the so-called Fellget advantage.  Finally, the Feilget and the Jacquinot advantages permit construction of interferome ters having much higher resolving power than dispersive instruments. Further advantages ]. 41 can be found in the book by Bell[  2.2  Bruker 113V Spectrometer  All the reflectivity measurements were performed on a Bruker 113V spectrometer that is a Genzel-type interferometer[43] designed to operate under vacuum, which helps to remove the unwanted water absorption and preserves the thermal stability. It covers the full range of wavenumber from the far infrared maximum resolution of  ( lOcm’) to the near infrared (15800cm’) with a  . Figure 2.5 shows a ray diagram of the optical bench 1 O.03cm  of a Bruker IFS 113V. The Bruker spectrometer is directly interfaced with a computer performing data collection, fast Fourier transformation, and control of the motor-driven optical components, e.g. optical filters, beam splitters, apertures, and mirrors which are  Chapter 2. Experiment  33  Figure 2.5: Optical path of the Bruker IFS 113V. (I) Source Chamber: a—Tungsten/Halogen/Quartz lamp, glowbar, mercury arc lamp; b—automated aper ture. (II) Interferometer Chamber: c—optical filters; d—beamsplitter; c—double-sided scanning mirror; f—control interferometer; g—reference laser; h—remote control align ment mirror. (III) Sample Chamber: i—sample focus; j—reference focus.(IV) Detector Chamber: k—far-infrared DTGS detector, mid-infrared MCT and InSb detectors and near-infrared Si- diode detectors. used to change sample chambers, sources, and detectors. Measurements covering different spectral regions require specific choices of sources, filters, beamsplitters, and detectors. With the Bruker IFS 113V, there is a choice of 3 sources, 4 apertures, 4 optical filters, 6 beamsplitters, and 4 detectors. In the Genzel interferometer, the radiation from the source chamber is focused on the beamsplitter, which allows the beamsplitters to be very small (about 2 cm in diameter.) In a conventional interferometer, the radiation is not focused at the beam splitter and therefore the beam splitters are quite large (about 12-20 cm). These large beam splitters vibrate slightly, resulting in a diffusion of the infrared beam and increased spectral noise. This is called the ‘drum-head’ effect. The  Chapter 2. Experiment  34  small size beam splitters in the Genzel interferometer greatly reduce the ‘drum-head’ effect. Another advantage of this design is that the angle of the incident beam at the beam splitters is oniy 14°. This small angle of incidence compared with that of other interferometers results in increased light throughput and decreased polarization effects. The two beams from the beamsplitters are incident on opposite sides of a double-sided scanning mirror. The mirror is supported on a dual gas bearing that uses dry nitrogen and is driven by a linear induction motor. This design gives an optical path difference twice as large as that of a conventional interferometer and therefore achieves a given resolution with oniy half of the usual mirror movement. Data is collected in a fast-scanning mode, with the scanning mirror position deter mined by the fringes of another separated interferometer with a white source and a He-Ne laser. The centerburst of the white source interferogram, which occurs at the zero path -  difference as discussed in the previous section, is used to initiate counting of the laser interference fringes. According to the response time of the detector used, one can choose an appropriate scanning speed v out of the 15 choices provided by Bruker. A scanning mirror with velocity v (in cm/sec.) results in alternating electronic signal at the detector with frequency: fL’  where  f.  =  4vv  (2.11)  is the modulation frequency in Hz and v is the wavenumber in cm . 1  To  avoid aliasing, high-pass and low-pass electronic filters are used in the Bruker instead of optical filters used in other spectrometers since each electronic  f,,  corresponds to one  specific optical wavenumber v. This relation is also important in determining the noise features caused by mechanical vibrations or electromagnetic crosstalking. Each scan of the mirror typically takes a few seconds and several hundreds of interferograms are measured and averaged to increase the signal to noise ratio before performing the Fourier  Chapter 2. Experiment  35  transform to derive the final spectrum. The apodization function that we have used is the so-called ‘three-term Blackmann-Harris’ function: W(x) where n  2.3  =  1,2, 3,  ..  =  ,  0.42323 -f- O.49755cos(rn/N) + 0.07922cos(2rn/N)  (2.12)  N and N is the number of points sampled in the interferogram.  Reflectivity Measurement  The reflectivity measurements were performed using a reflectance module placed in one of the Bruker sample chambers. Figure 2.6 is the reflectance module viewed from top. Light from the interferometer chamber is focused on a rectangular aperture F, which is imaged at the sample or reference surface at about 15 degrees. Using the visible light from the tungsten source, the positions of the aperture and the first toroidal mirror are adjustecF to produce a well focused image at the plane of the sample. The aperture is set to a size which is slightly smaller than that of the sample. Light reflected from the sample is collected by the second toroidal mirror and directed to the detector chamber along the original spectrometer path. The sample and the aluminum reference mirror were mounted on a home made sample holder with a very smooth holding surface. The sample and the reference were mounted against the smooth surface so that their surfaces were  iii  the same plane. This makes  sure that the light reflected from the sample is in the same direction as that from the reference. The sample and the reference can be exchanged by moving the sample holder from outside in the direction designated by M in Figure 2.6. The typical size of our sample and reference is about 5 x 10 mm. There are several vacuum feedthroughs to allow adjustment of the toroidal mirrors and other components in order to maximize the detector signal. When maximizing the signal, a chopper is brought into the beam using one of the vacuum feedthroughs to  Chapter 2. Experiment  36  N  Figure 2.6: The reflectance module view from top. (A),(B): reference mirror and sample. (C) radiation shield. (D) Vacuum Shroud. (E) Polarizer. (F) rectangular aperture. (G) plane mirrors. (H) Toroidal mirrors (I) Chopper. (J) vacuum feedthrough to move chopper into the beam (K) sample chamber extension. (L) plexiglass cover. (M) direction of translation exchanging sample and reference mirror.  37  Chapter 2. Experiment  produce AC signal which is required by the detectors. Since our sample and reference mirror are in the same plane, we need only to maximize the signal once by adjusting the second toroidal mirror. A detailed description of the reflectance module can be found in reference [56]. The far-infrared measurements in the 10  —  1 region were made using a mer —500cm  cury arc lamp with a blue polyethylene film optical filter and a 6tm mylar beamsplitter. The filter is intended primarily to block ultraviolet radiation which might otherwise damage the thin mylar beamsplitter. A deuterated triglycerine sulfate (DTGS) detector working at room temperature was used to detect the far infrared signal. infrared measurements in the 550  —  The mid-  —5500cm’ region were made using a glowbar source  without any optical filter and a KBr beamsplitter. A MCT detector working at 77K was used to detect the mid-infrared radiation. The near-infrared measurements in the 9000  —  —l5000cm’ were-made using a tungsten lamp with a red film optical filter and  a quartz beamsplitter. A si-diode working at room temperature was used to detect the near-infrared radiation. For each sample and reference mirror, 200  —  —500 interferograms are measured and  averaged to increase the signal to noise ratio. The velocity of the scanning mirror was cho sen according to the response time of the detectors. They are 0.099cm/sec, 0.166cm/sec, and 0.333cm/sec for the DTGS detector, the si-diode detector, and the MCT detector respectively. It takes about 5 to 10 minutes for the computer to collect data and calculate the final spectrum. The absolute value of the reflectivity was obtained by comparison with the aluminum mirror and the published values of the aluminum reflectivity [47].  Chapter 3  Calculation of Reflectance  3.1  Reflectance of Bulk Material  The optical properties of a solid can be calculated from the dielectric function E(w) or complex refractive index  =  n + ik. In terms of the refractive index, the dielectric  function is given by: = ‘  =  +  (ii  Equating the real and the imaginary parts gives =  =  —  2 + ik)  and  €“  (3.1)  in terms of n and k:  2 k  2nk  (3.2) (3.3)  For normally incident radiation on a surface in a vacuum, the equation relating m and k to the reflectivity R is (n+1)2+k2  (34  For non-normally incident light, the reflectivity in terms of n and k becomes a complicated function of both the angle of incidence and the polarization [44].  3.2  Reflectivity of a Multiple Layer Structure  There are several methods for calculating the reflectance of a multilayer system [45]. We will use the so-called optical impedance method[58] which allows us to calculate the  38  Chapter 3.  Calculation of Reflectance  39  reflectance of a system with any number of layers. The optical impedance propagation coefficient -y in a solid with dielectric function 77-  1-  Voc =  where  17  j  and the  are defined as:  377 n-I-ik  (3.5)  —2iri7ii/  (3.6)  is the wavenumber (cm’). For a single layer system shown in Figure 3.1 (a),  the load impedance seen at the first interface is coshQy + ) d 2 )+ d 2 772 coshQy 773  ZL1  where  773  =  772  772  sinhQy ) d 2 ) 2 sinh(-yd .  (3.7)  is the impedance of the backing region, e.g. a thick substrate holding the film.  Then the reflectance of the system when light is normally incident from a medium with optical impedance  is calculated using the formulas  r  =  ZL1  R  +  3.8)  77i  =  (3.9)  In the case where the backing region is not a thick substrate but another dielectric layer sitting on a substrate as shown in Figure 3.1 (b), the value of 173 in Eq. 3.7 is replaced by an effective load impedance  ZL2,  which is calculated using the same equations.  cosh( + 773 sinh(-y ) d 3 ) d 3 773 cosh( ) + sznh(d 3 7)4 ) 3 d 13 774  ZL2  =  773  (3.10)  This procedure can be repeated for any series of layers with different optical properties and thickness, i.e. the reflectance of a system can be calculated no matter how many layers it has as long as we know the dielectric function or complex refractive index of each layer. However, the relation between dielectric function and complex refractive index must be defined as Eq. 3.1. Changing the sign of extinction function k will result in unreasonable reflectance in some region, e.g. R  >  1 near reststrahlen band. If the  Chapter 3.  definition of  Calculation of Reflectance  =  n  —  40  ik is used, the minus sign in the definition Eq. (3.6) must be  dropped. The above method is only good for normal incidence. The reflectance of a multilayer system at non-normal incidence can be calculated by a matrix method introduced in detail in reference [29].  2 d p  ‘li  773  12  normal incidence  (a)  2 d  3 d  772  ‘13  ‘14  2 ZL  1 ZL  (b) Figure 3.1: (a) a single layer described by the impedance ‘12, propagation coefficient 72, and thickness d 2 is sitting on a thick substrate described by the impedance 773. is the the impedance of the medium from which the light is incident on the sample. (b) a two layer system with the thin layers described by ‘12, 72, d 2 and 3 is sitting on a -ye, d substrate described by the impedance ‘14  Chapter 4  Experiment Results and Analysis  Reflectance measurements were performed on 4 samples at near normal incidence in the wavenumber range from the far-infrared to the near-infrared. We started with samples of single layer structure, i.e. bulk materials. The knowledge and experience gained in analyzing the reflectance spectra of these samples are essential to understand the com plicated spectrum of a multilayer structure. The optical constants obtained in analyzing bulk materials were used to simulate the reflectance spectrum of multilayer structures. We will introduce the measurement results in the order of increasing complexity of the sample structure. All experiments were conducted at room temperature.  4.1 4.1.1  Reflectance of Bulk Material Pure GaAs  Figure 4.1 (a) is the far-infrared reflectance spectrum of sample #1 which is a semiinsulating GaAs chip with polished parallel surfaces. The main features of the spectrum are a phonon reststrahlen band near 270cm 1 and interference fringes. The latter are due to the interference of the light reflected from the top and the bottom surfaces of the chip and therefore can be used to estimate the thickness of the sample. Because of the strong dispersion effect near the reststrahlen band, the separation between closest fringes is not a constant, e.g. Lv  1 in the range (50cm 2.685cm 1  2.474cm’ in the range (380cm 1  —  —  lOOcm’) and  340cm ) 1 . The thicknesses calculated from  41  Chapter 4. Experiment Results and Analysis  42  100  80 >.  60  H  40  20  0  0  100  200  300  400  500  100 (b)  I  50 > H > H L)  60  40  20  0  GaAs Sub. Parameters T wo = 269 cm  r=3cm—’ =  1 360 cm  4... 0  100  200  300 WAVENUMBER (cm-i)  400  500  Figure 4.1: (a) Far-infrared reflectance spectrum of a semi-insulating GaAs substrate with polished parallel surfaces. (b) Calculated spectrum of a pure GaAs substrate with thickness d The parameters listed on the graph are Lorentz oscillator param eters used for calculating the spectrum.(see text)  Chapter 4. Experiment Results and Analysis  43  the two different separations using a simple formula d index n constant are d measured thickness  =  545gm and d  =  =  1/(2nLv) with the refractive  59Om respectively, which are larger than the  51Oim. This is because the simple formula did not not take the  dispersion of n into account. We derived a formula, which includes a dispersion term, in Appendix A to improve the accuracy of the thickness estimate. The phonon reststrahlen band can be very well fitted by an oscillator model of the form: e(w)  (4.1)  = —  where e(w) is the dielectric function, the resonance frequency,  €,  —  is the high frequency dielectric constant, w 0 is  the oscillator strength, and F the damping of the phonon.  Figure 4.1 (b) is the calculated spectrum using formula 4.1 and the impedance method introduced in Chapter 3. We can see that the calculated reflectance fits the measured spectrum very well. Sample #1 is in fact a semi-insulating substrate used for molecular beam epitaxy with carrier concentration  1.65 x . 3 c 7 10 m Another commonly used sllbstrate is conducting  substrate with carrier concentration 4.1.2  2 x 10 . 3 cm 18  Doped GaAs  The far-infrared reflectance spectrum of a conducting substrate (sample #2 ) from Ocm’ to 500cm 1 is shown in Figure 4.2 (a). As expected, the main features are a reststrahlen band and a plasma edge. Because of the high reflectance in this region, the intensity of light reflected from the top sample surface is much larger than that from the bottom surface and therefore no efficient interference occurs and no fringes are observed. The reststrahlen band in Figure 4.2 appears different from that in Figure 4.1. Instead of showing a peak and a dip in the reflectance spectrum as in Figure 4.1, the reststrahlen band of sample #2 causes a hole in the so-called Drude reflectance spectrum. The latter is  Chapter 4. Experiment Results and Analysis  44  100 Sample #2 80 >  H •1  60  -  -  H C)  40  -  20 (a) 00  100  200  300  400  500  400  500  100  80 >  11.1 1350 cm’ 1 =75cm y  60  =  H C)  40 269 cm 1 3 cm 1 = 360 cm 1 =  =  20 (b) 0  0  100  200  300 WAVENUMBER (cm-i)  Figure 4.2: (a) Far-infrared reflectance spectrum of a conducting GaAs substrate with free-carrier concentration n = 1.6 2.4 x 3 cm given by the manufacturer. (b) 8 1O’ , Calculated spectrum of a conducting GaAs substrate with free-carrier concentration n = 1.45 x 10 . The parameters listed on the graph are the parameters for 3 cm 18 Drude-Lorentz model used for calculating the spectrum. —  Chapter 4. Experiment Results and Analysis  45  caused by the plasma formed by the carriers and ion cores. The plasma edge appearing at about 400cm’ in Figure 4.2 can be used to estimate the carrier concentration. However, one must be aware that the plasma frequency  and the plasma edge frequency  7i  are  different:  /  2 ne  ‘/  =  m 0 v e  (in rad/sec)  *  (4.2) (4.3)  where n and m* are the free-carrier concentration and the effective mass of the carriers. For n-type GaAs, the effective mass of the electrons is m*  =  O.071m. The best way  to determine the carrier concentration is to fit the spectrum using the so-called Drude Lorentz model which is obtained by adding a plasma term into equation 4.1: e(w)  2 c =  -  —  ‘.  2 + vyw w  —  2  (4.4)  .  —  + wI’  where the second term is the plasma term; w, is the plasma frequency defined by equation ). Figure 4.2 (b) is the calculated 1 4.2; -y is the carrier relaxation time in wavenumber (cm spectrum of a conducting GaAs substrate, which fits the experiment very well. The carrier concentration determined by equation 4.2 using the fitting parameter w,,  =  1350cm’  . 3 ) is 1.45 x iO cm 1 405.2cm  (wY  From the fitting parameters listed on the graph, we can determine other electrical parameters, e.g. conductivity o, mobility and relaxation time r (r  =  etc., as well as the carrier concentration n  in .sec/rad). The D.C. conductivity can be calculated  using the following formula: D.C.  where  and ‘y are in cm . The 1  OD.C.  =  —i-  607  1 (1cm)  (4.5)  calculated from equation 4.5 is 405(fcm)’ which  agrees very well with the D.C. conductivity obtained from the conductivity spectrum of  Chapter 4. Experiment Results and Analysis  600  •  46  I  Sample #2  11.1 = 1350 cm 1 1 7 =75cm E  600  -  269 cm 1 P = 3 cm 1 = 360 1 cm p 0 =  400  200  00  1O0  200  300  400  500  WAVENUMBER (cm-i) Figure 4.3: The calculated conductivity spectrum of the conducting substrate. parameters used in the calculation are the same as those in Figure 3.2.  The  Figure 4.3, which was calculated using Eq. 4.6 below.  =  (2cm)  60  (4.6)  —  where e” is the imaginary part of the complex dielectric function e(w). The mobility can be calculated using the following formula: e 27rc.ym* The units for ,u, a-, and n are cm /volt 2 mobility  —  (4.7)  en  .sec, (fcm) , and cm 1 3 respectively. The  of sample #2 calculated from the fitting parameters is 2 1745.7cm / volt  which is within the range given by the manufacturer 1650-1992cm 2 /volt  —  —  sec  sec. However,  Chapter 4. Experiment Results and Analysis  47  both the conductivity and the carrier concentration calculated from the fitting parameters are smaller than those given by the manufacturer, 500 1018  625 (1cm) and 1.6  —  —  2.4 x  3 respectively. cm  4.1.3  Pure Gai_AlAs  Although the Gai_A4,As system is technologically one of the most important alloy systems, systematical studies on the optical constants (n and k) over a wide spectral range and over the Al concentration range of x  =  0 to x  =  1 with Lx  =  0.05 have not  been reported. Even in the recently published handbook of optical constants [46), we can find data only at almost random values of x over a narrow far-infrared spectral range. We will use the optical constants reported in reference [48] and interpolate or extrapolate the data to those x values which are not reported in the reference. Ijireference 148], the Qptical constauts im the far-infrared for Ga AlAs were ob 1 tained by fitting the reflectivity data taken at near normal incidence on samples of thickness ranging from 250tm to 300tm.  The Gai_AlAs samples were grown by  an isothermal liquid-phase-epitaxy (LPE) technique for x values ranging from 0 to 0.54. All material had unintentionally doped n-type carrier concentrations of less than 5 x . The free-carrier plasma edge was not observed in the frequency range investi 3 cm 16 10 gated (50— 500cm’). The measured far-infrared spectra were assumed to be the same as those of pure Gai_AlAs. The reflectance spectra of the 1 _ Ga , As Al shown in Figure 4.4 were generated using the following dielectric function: (  —  ‘  1 2 iw 11  2 —  2  1 (w  where  wj,  w,  and  yj  \f  —  —  2  2 —  2  ’Y11A’°l2 1  —  2 iLc)71)(Wt  —  W  2  W2  —  —  2 1W71 ZW7t2)  denote TO phonon frequency, LO phonon frequency, TO  damping constant, and LO damping constant respectively and are obtained by fitting the far-infrared experiments. The best fitting values are listed in table 4.1. The index  Chapter 4. Experiment Results and Analysis  48  1  0.8  H > H 0 -I  0.6  0.4  0.2  0 100  300  200  -  400  500  WAVENUMBER (cm-i) Figure 4.4: The reflectance spectra of GaAs, Gai_AlAs,.and AlAs reported in reference [47]. i  =  1 and i  =  2 identify the parameters for the GaAs-like and the AlAs-like modes  respectively. Unlike the phonons of GaAs and AlAs, the optical phonons of the Gai_AlAs have two modes, a GaAs-like mode and an AlAs-like mode [49] [50], [51]. From Figure 4.4, we see that the phonon energy of the two modes changes as the Al concentration x changes. Adachi [13] reported the following empirical formulas for the LO and TO phonon energies of the Gai_AlAs material. LO GaAs LO AlAs  —  —  like mode:  11 E  =  36.25  like mode:  12 E  =  44.63 + 8.78x  —  6.55x + 1.79x 2 (meV) —  2 (meV) 3.32x  (4.9) (4.10)  Chapter 4. Experiment Results and Analysis  49  _AlAs. 1 Table 4.1: The best fitting parameters for Ga X  l”li  l”i  268.8 267.1 266.9 265.2 264.5 262.9 261.8 0  0 0.14 0.18 0.30 0.36 0.44 0.54 1.00  TO GaAs TO AlAs  292.8 285.7 283.4 278.3 276.5 273.7 269.8 0  —  —  7ti  2.65 5.67 8.76 8.64 10.69 10.05 12.43 0  Vt2  711  2.85 4.85 4.24 6.15 5.58 6.44 7.97 0  2 i  0 358.8 360.1 360.2 360.4 360.2 361.5 361.8  0 369.0 372.4 379.9 381.3 385.4 390.1 400.0  0.64x  like mode : E 1  =  33.29  like mode :  =  44.63 + 0.55x  2 E  —  ii  7t2  0 10.56 12.20 12.10 12.23 9.55 8.75 8.00  —  —  and  in cm.  712  c  0 11.31 10.24 9.42 8.08 7.90 8.68 8.68  10.9 10.57 10.47 10.16 10.04 9.84 9.60 8.20  2 (meV) 1.16x  (4.11)  2 (meV) 0.30x  (4.12)  The formulae for the TM phonons will be used to determine the Al concentration after ob taining the LO phonon energies through fitting the far infrared reflectance measurement, since the LO phonon energies are more sensitive to the change of the Al concentration than are the TO phonon energies. Figure 4.5 shows the phonon energies as functions of the Al concentration x. GaAs and AlAs are the two limiting cases of Gai_A4As where the Al concentration 0 and x  =  1 respectively. Figure 4.4 also shows the reflectance of GaAs and AlAs.  By equating the parameters of one mode to zero, Eq. 4.8 can also be used to describe the dielectric function of GaAs and AlAs. The parameters for GaAs and AlAs are also listed in Table 4.1. To our knowledge, the far-infrared optical constants of highly conducting Ga _AlAs 1 are not available. Eq. (4.8), however, can be used to calculate the reflectance spectrum of a highly-doped Ga _AlAs by adding a Drude-term to it as introduced in section 1 4.1.2.  Chapter 4. Experiment Results and Analysis  450  50  --  430—-•-•-  a  410  390 370 :...--  350 r  —  -  330 310  2500:20406081 Al Concentration x Figure 4.5: The TO and LU phonon energies of Gai_AlAs are functions of the Al concentration x.The LU phonon energies E 11 and E 12 are more sensitive to the change of x than the TO phonon energies E . 2 1 and E 4.2  Reflectance of Multiple Layer Structures  In section (4.1), we introduced experiments and calculations on the far-infrared re flectance spectra of bulk GaAs, AlAs, and Gai_AlAs materials and obtained the dielectric functions describing these materials in the far-infrared region.  In this sec  tion, we will introduce the reflectance spectra of two multilayer samples made of these materials and use the dielectric functions obtained in the previous sections to analyze the spectra. Structural and electrical parameters for each layer, e.g. layer thickness, free-carrier concentration, and composition concentration, were obtained through fitting the experiments.  Chapter 4. Experiment Results and Analysis  4.2.1  51  Reflectance of GaAs/AlAs multilayer structure  Far-infrared Measurement A sketch of sample #3 which consists of a Si-doped GaAs layer and an undoped AlAs marker sitting on a semi-insulating substrate is shown in the upper right corner of Figure 4.6 (b). It was grown by Molecular Beam Epitaxy system in Dr. T. Tiedje’s group. The reflectance spectrum is shown in Figure 4.6 (a). The main features in the spectrum are a GaAs reststrahlen band at about 270 cm , an AlAs reststrahlen band at about 360 1 1 cm  ,  and a free-carrier plasma Drude response at low wavenumbers. The spikes under  80 cm’ are mainly noise. The width of the GaAs reststrahlen peak is about the same as that shown in Figure 4.1 for bulk GaAs. However, the AlAs reststrahlen band associated with the thin AlAs marker is much sharper than its GaAs counterpart. This is probably because the GaAs reststrahlen band shown in Figure 4.6 is the contribution of both the top doped GaAs layer and the semi-insulating GaAs substrate. The total thickness of these two layers are about 643tm which is much larger than the thickness of the AlAs marker of 0.14tm. Figure 4.6 (b) is the calculated spectrum of sample #3, which is obtained using the dielectric functions in section 4.1 and the impedance method in Chapter 3. The structural and electrical parameters for each layer were obtained. Table 4.2 lists the best fitting parameters for each layer of sample #3. The electrical parameters of the top GaAs layer are listed in table 4.3. The calculated spectrum fits the experiment very well. The Berreman Modes[16] which are caused by zeros in the dielectric function and occur only in a p-polarization measurement were not observed in our spectrum measured at near normal incidence. According to reference [16], the two Berreman modes B 1 and B 2 of sample #3 are at 130cm and 290cm respectively. B 2 coincides with the frequency of the LO phonon mode. Neither were observed.  Chapter 4. Experiment Results and Analysis  52  100  80 > H > H  60  40 12  20  0  0  100  200  300  400  500  0  100  200  300  400  500  100  80 > H  60  H C)  40  20  0  WAVENUMBER (cm-i) Figure 4.6: (a) The far-infrared reflectance spectrum of sample #3 measured with reso lution 4cm . (b) The calculated far-infrared spectrum of sample #3 with the top GaAs 1 layer d 1 = 3pm and AlAs marker d 2 = O.l4pm. Table 3.2 lists other parameters used in the calculation.  Chapter 4. Experiment Results and Analysis  53  Table 4.2: The best fitting parameters for each layer of sample #3. Layers top GaAs layer AlAs marker SI-GaAs Sub.  Drude-term y (cm’) 450 45  wp (cm—’)  e  / /  11.1 8.2 11.1  / /  Lorentz-term T (cm’) 269 3 362 8 269 3  0 (cm’) w  (cm’)  360 500 360  Thickness d (sm) 3 0.14 643  Table 4.3: The electrical parameters of the top GaAs layer of sample #3, calculated from the plasma frequency = 450cm 1 and free-carrier relaxation time -y = 45cm’.  We know that the position of the free-carrier plasma edge in a reflectance spectrum is determined by the free-carrier concentration. The plasma edge shifts towards smaller wavenumber as the free-carrier concentration decreases (Eq.4.2).  All solids with the  same free-carrier concentration and effective mass have the same plasma frequency, but their plasma-edge-frequencies, which are related to the refractive index of each solid and defined by Eq. 4.3, may be different and therefore appear in different places in their reflectance spectra. The slop of the plasma edge is determined by the free-carrier relax ation time as shown in Figure 4.7. The plasma edge becomes steeper as the relaxation time increases or the damping constant decreases. Although the AlAs marker in sample #3 is much thinner than the other two layers, it cannot be ignored. Besides the AlAs reststrahlen band, the shoulder of the GaAs reststrahlen peak at about 250 cm 1 is also caused by the AlAs marker. Figure 4.7 showing a GaAs reststrahlen peak without the shoulder was calculated from a sample  Chapter 4. Experiment Results and Analysis  54  Calculated Reflectivity of Si—doped GaAs on Pure GaAs Substrate Thin film thickness d 3 jm  1 0.8  >  0.6  I— C) bJ -J  Li. fl LI-i  4  0.2 0 0  100  200 300 WAVENUMOER (cm-1)  400  500  Figure 4.7: The calculated far-infrared spectra of a sample with almost the same structure as sample #3 except fo not having the AlAs marker. The only difference between the dotted line and the solid line is the free-carrier relaxation time. which has the same structure as sample #3 except for not having the AlAs marker. The interference fringes resulting from the substrate were not observed in Figure 4.6. The reasons for this are that the bottom substrate surface of sample #3 was not polished and the resolution used in the measurement was 4 cm 1 which is larger than the fringe separation of 2.29cm’. However, small fringes of the substrate were observed when the resolution was increased to 0.5cm’. These may be seen in the spectral range from 600 cm’ to 1000 cm 1  ill  Figure 4.8. The thickness of the substrate, calculated  from the fringes, is 4 642.5 u m which agrees very well with the digital caliper measurement 643gm. The accuracy of the digital caliper is +1im. The detailed calculation method will be discussed in the Appendix A. The large fringes in Fig. 4.8 (a) are the results of the interference in the two top thin layers. These fringes can also be seen in the near-infrared region. We will fit the near-infrared spectrum to determine the thickness  Chapter 4. Experiment Results and Analysis  55  100  80  I,  > C)  40  20  0 500  1500  2500  3500  4500  I  I  60 (b)  5500  I  48  36  VVVVVJWAMMAMWf\MMNvwAAJv\ fringes of the substrate  24  12  -  0 600  I  620  •  I  640  I  660 Wavenumber (cm—i)  I  680  700  Figure 4.8: (a) Mid-infrared measurement of sample #3 with resolution O.5cm’. The interference fringes of both the substrate and the top layer are seen. (b) The fringes of the substrate are shown in detail.  Chapter 4. Experiment Results and Analysis  56  of the two layers instead of fitting the spectrum in the mid-infrared region, because the refractive index n and extinction function k in the near-infrared region can be obtained more accurately by using an empirical formula, which will be introduced in the following section. Near-infrared Measurement In order to determine the thickness of the two epitaxial layers in sample #3 more accu rately, we measured the reflectance spectra of sample #3 from 9000cm’ to lS000cm’, and this is shown in Figure 4.9 (a). The main features in the spectrum are the interfer ence fringes resulting from the top GaAs layer. The separation of the interference fringes resulting from the AlAs marker is about 12000cm’ which is larger than the spectral range measured and therefore is difficult to be noticed. However, the AlAs marker do have a contribution to the appearance of the fringes resulting from the top OaAs layer. Through calculating the near-infrared spectrum, we found that the thickness of the top GaAs layer determined the fringe separation while the thickness of the AlAs marker de termined the fringe envelope. The damping of the fringe amplitudes in the spectral range from 9O00cm to 11400cm 1 is not caused by the increasing absorption near the GaAs ) but by the interference of the AlAs layer. Figure 4.10 1 band gap (1.42eV or 11452cm shows the calculated reflectance spectra from 5500cm 1 to 11500cm’ with the same top layer thickness but different thickness of AlAs marker. It is obvious that the fringes re sulting from the top layer are modulated by the much larger AlAs fringes. The dielectric function introduced in section 4.1 cannot be used in the calculation of the near-infrared reflectance spectrum, since the dispersion in this region is caused by the energy gap. The extinction coefficient in the wavenumber range under the band gap is so small that it can actually be taken as zero in the reflectance calculation. In this region, the reflectance is mainly determined by the refractive index which can be calculated using an empirical  Chapter 4. Experiment Results and Analysis  57  50  40  H  30  > H C)  20  10  000  13000  14000  15000  13000 WAVENUMBER (cm-i)  14000  15000  10000  11000  12000  10000  11000  12000  50  40 >  H  30  H  C)  20  10  0 9000  Figure 4.9: (a) Near-infrared reflectance spectrum of sample #3. The fringes were re sulted from the top GaAs layer. The damping of the fringe amplitudes under the GaAs band gap is caused by the thin AlAs marker. (b) The calculated near-infrared spectrum of sample #3 with the top layer thickness d 1 = 3.O3um and the AlAs marker thickness 2 = O.14,um. d  Chapter 4. Experiment Results and Analysis  I  —-  ——  2 d  r  =  O.O5im  58  .sc C-  -————---r——-——  i  2 d  —  O.1m  C  r  JWJV V \/ A/ \! Doped GaAo  3m  AlAs Meeker SI-GaAs Sabatrato  .00 5500.0  I  —  Wavenumber (cm’)  Wavenumber (cm—’)  .0  I  11500.0  30.0  Wavenumber (cm’)  ) 1 Wavenumber (cm  Figure 4.10: The calculated near-infrared spectrum of a sample similar to sample #3 but with a variable thickness d 2 of the AlAs marker.  Chapter 4. Experiment Results and Analysis  59  formula. Afromowitz [52] introduced a semi-empirical method for calculating the room temperature refractive index of Gai_AlAs at energies below the direct band gap. 1  =  M_, + 3 M. + 1 2 E -E x ln[(E  M_,  =  —(E  =  -(E  E),  (4.15)  Ef  =  (2E—E),  (4.16)  ri  =  —  where  and  —  —  —  E ) 2 /(E  —  E ) 2 ]  E),  (4.13) (4.14)  ir Ed Eo 2 Er)  (4.17)  E is the photon energy in eV and E , Ed, E are functions of Al concentration x and 0 are determined by following formulas. Ga,_,AlAs :  0 E  =  3.65 -f- 0.871x + 0.179x 2  (4.18)  Ed  =  36.1  (4.19)  E  =  1.424 + 1.266x + 0.26x 2  —  2.45x  (4.20)  where the x rims from 0 to 1. The above formulae cover the two ends of Gai_AlAs, i.e. GaAs and AlAs and can be used for GaAsi_P and GaIniP material as well. This method reproduces the experimental index of refraction for GaAs within 0.004 from 0.895tm (11173.2cm’) to 1.7um (5882.3cm’). The fit to the data for AlAs is within 0.004 for energies up to 1.5 eV, and within 0.014 up to 2 eV. Figure 4.9 (b) is the best fit of sample &3 in the near-infrared region, which gives the same layer thicknesses as found in the far-infrared fitting. However, these values are more accurate than those obtained in far-infrared fitting, having an accuracy of ±0.01,um. 4.2.2  Reflectance of Ga,_,A4,As /GaAs/A1As multilayer structure  Sample #4 with a Gal_TA1XAS/GaAs/A1A5 multilayer structure as shown in Figure 4.11 was also grown by molecular beam epitaxy. All three layers are undoped. The thicknesses  Chapter 4. Experiment Results and Analysis  Gai—A1As GaAs AlAs  60  T2.51m I1.8,Lrn um 1 0.l  GaAs Substrate  Figure 4.11: The geometric structure of sample #4. The thicknesses shown on the graph are nominal values. on Figure 4.11 are nominal values calculated according to the growth conditions. The far-infrared reflectance spectrum of sample #4 from 0 to 500cm 1 with resolution 2cm 1 is shown in Figure 4.12 (a). The main features of the spectrum are reststrahlen bands of GaAs and Gai_AlAs. The two peaks appearing between 220cm’ and 300cm’ are the combined effect of the GaAs reststrahlen band and the Gai_AlAs reststrahlen band of the GaAs-like mode. The peak at about 370cm 1 is mainly the contribution of the GaiAlAs reststrahlen band of the AlAs-like mode. The reflectance spectrum of a three-layer structure can be calculated using the same method as in section 4.2.1 and the best fit is shown in Figure 4.12 (b). The dependence of the spectrum on Al concentration x is illustrated in Figure 4.13, here it is calculated for four x values. The spectra show that the wavenumbers of the three peaks change only slightly as the Al concentration increases from x  =  0.3 to x  =  0.54, but the maximum reflectance of both peaks #2 and  #3 increases as Al concentration increases. However, the maximum reflectance of peak #1 decreases as x increases. The best fit of the experiment is given in Figure 4.12 (b)  Chapter 4. Experiment Results and Analysis  61  100  80 > H  60  H  40 r:i lE  20  0  0  100  200  300  400  500  100  200  300  400  500  100  80 > H  •1  > H  40 cz  20  0  WAVENUMBER (cm-i) Figure 4.12: (a) The far-infrared reflectance spectrum of sample #4. (b) The calculated far-infrared reflectance spectrum of sample #4  Chapter 4. Experiment Results and Analysis  5pm  GA. AlAs Gaae  =  62  0.3  Jl. m 8 . D.lPm  T  #1 #3  (a) .0  1 cm  500.0  Figure 4.13: The calculated spectra of four structures similar to sample #4 but with different Al concentration for the top Gai_AlAs layer. The thickness of each layer are shown at the upper left corner.  Chapter 4. Experiment Results and Analysis  with x  =  63  0.66. Table 4.4 shows the best fitting parameters for the top Gai_rAlAs layer  of sample #4. The LO phonon energies of the GaAs-like mode and the A1As-ilke mode  Table 4.4: The best fitting parameters for the top GaiAlAs layer of sample #4. v and ‘y in cm, d 1 in ,um. 1 d  r2.78  1 are vj  ii.  259.5.  263.8cm’ and  263.8  2 = Vj  -yn. 12.54  711  Vt2  2 Vt  7t2  712  c  8.12  361.7  395  8.4  8.68  9.17  395cm’ respectively. The Al concentrations calculated  from Eq. 4.9 and Eq. 4.10 on page 48 agree with the nominal value x x  =  0.66 and x  0.66 very well,  0.66 respectively. The error on x is estimated to be 0.05 which is  obtained from the uncertainty of the phonon energy equation in 4.9 and 4.10. To determine the layer thickness more accurately, we- measured the mid-infrared re flectance spectrum from 500cm’ to 5500cm with resolution 0.5cm. This is shown in Figure 4.14 (a). The interference fringes resulting from the substrate were observed in the region from 500cm 1 up to about 1300 cm showing that the sample surfaces are fairly smooth and parallel. The substrate thickness calculated from the fringes is 652.8umwhich agrees very well with the digital caliper measurement 654 + l urn. The fringes resulting 1 from the two top layers were also observed. Figure 4.15 is the best fit of the mid-infrared spectrum. It fits the experiment well, except for the range 5000—5500cm where the measured spectrum was deformed because of exceeding the measurement range of the detector(MCT). For clarity, the fitting did not take the substrate interference into ac count. The thicknesses obtained from the fitting agree with the nominal values reasonably well except for the AlAs marker (0.O3itm compared with 0.lpm). All the values of the thicknesses shown in Fig. 4.15 are accurate to 0.01jm. Because of different thicknesses of the the two top layers, the interference fringes from the two layers canceled out in the  Chapter 4. Experiment Results and Analysis  64  near infrared region as shown in Figure 4.16. We will not fit the spectrum in this region.  Chapter 4. Experiment Results and Analysis  65  60  45  30 0  15  0 500  1500  2500  620  640  3500  4500  5500  680  700  60  —  45  j30  15  0 600  660 Wavenumber (em*1)  Figure 4.14: (a) The mid-infrared spectrum of sample #4. The fringes of the substrate and the top two layers were observed. (b) The fringes of the substrate are shown in detail.  Chapter 4. Experiment Results and Analysis  66  60  48  I:  36  24  12  0  1000  2000  3000  WAVENUMBER  4000  5000  (cm—’)  Figure 4.15: The best fitting of the mid-infrared measurement of sample #4. The thick nesses shown outhegraph are the best fitting values. •1  I  I  ——  I  Sample #4 i3. c4  t3ØØ  12Ø3ø 11ØØ VENUME CM-I  løeøs  Figure 4.16: (a) The near-infrared spectrum of sample #4. The fringes of the top two layers are canceled out due to different layer thickness.  Chapter 5  Conclusion  In conclusion we have measured the infrared reflectance spectra of GaAs/Gai_AlAs multilayer structures from the far-infrared to the near-infrared using a Bruker IF 113v spectrometer. The main structures in the spectra are due to the phonon reststrahlen bands, the Fabry-Perot interference and the plasma edge of the free carriers. Through fitting the spectra, we determined both electrical and structural parameters. include the carrier concentration n, mobility  t,  These  conductivity u, Al concentration x, and  the thickness d of each layer. The agreement between the theoretical spectra and the experiments is excellent. Comparing with a recent publication [16] on the same topic, our multilayer samples consisting of GaAs/AlAs/Gai_Alc,As materials are more complicated than theirs which were made of GaAs layers with different doping levels. The optical impedance method we used to fit our normal incident experiments is also different from theirs where a matrix method was used to fit their oblique incident experiments. We also determine the Al concentration x which did not exist in their samples. The limit of this method to determine the free carrier concentration is determined by the measurable wavenumber range of the Fourier spectrometer. The plasma fre quency will be too small to be determined accurately if the concentration is smaller than 2x. 3 c 5 1O’ m The plasma frequency of a n-type GaAs with 2 x 10 3 carrier concen cm 15 tration is about 15 cm’ which is close to the limit of the Bruker IFS 113v spectrometer we used. Although the samples we used were grown on semi-insulating substrates, this 67  Chapter 5.  Conclusion  68  method can also be used to characterize a multilayer structure on a conducting substrate. In this thesis, we have shown that infrared reflectance spectroscopy(IRS) is an excel lent alternative to many existing conventional methods for characterization of electrical and structural parameters. It has the advantage of being non-destructive, non-contact, and being able to characterize both electrical and structural parameters, which are usu ally determined by several different methods. Another advantage of this method is that it has no specific structural requirement for the sample as many other conventional methods have. Since both electrical and structural parameters can be determined by using only one sample, IRS method may change the conventional way of calibrating the thin film preparing systems. Instead of growing several single layer samples and using different measuring methods to measure them, now IRS method makes it possible to grow only one sample with multilayer structure and to calibrate several, if not all, parameters in every single layer at the sarne time. This will save time jn the measurement and reduce the number of samples needed to calibrate a thin film growing system.  Appendix A  Interference Fringes  The interference fringes resulting from a layer can be used to estimate the layer thickness. The fringe separation Lv and the layer thickness d are related by the following simple formula. (A.l) where zz’ is in wavenumber cm’ and n is the refractive index. However, the thickness calculated from Eq. A.l is often larger than the real value, especially for the fringes near the phonon reststrahlen band or the direct band gap where the dispersion of n is quite large. Because of this dispersion, the fringe separation Au is wavenumber dependent. To improve the accuracy of the estimate, we derived a formula to calculate the thickness from unevenly separated fringes. The refractive index of a dispersive material is a function of wavenumber  ii  and can  be written as n(v). The wavenumbers of the (m +1) th and m th Fabry—Perot peaks can be written as: =  m+l  (A.2)  2dfl(Ym+i)  (A.3)  = 2dfl(Vm) The wavenumber separation between two adjacent fringes is =  ‘rn+1  urn  =  1 2dfl(m+i)  69  +  m 2dfl(m+i)  m —  2d(m)  (A.4)  Appendix A. Interference Fringes  Substitute for  = iim(m)  m1  from Eq. A.3 in Eq. A.4:  =  —  2d(m+i) 1 2d fl(Vm+i) where  70  n+1 —  72(11m)  Vmfl(Vm)(__1 fl(Zim) —  ‘m  rn+l  1 ) 1 n(v,+ (A.5)  fl(i’m+i)  and is usually larger then zero. Eq. A.5 is the same as  Eq. A.l except that there is one more term in Eq. A.5. The second term in Eq. A.5 is a dispersion term. As  zmrn+l  is usually larger than zero, the fringe separation calculated  from Eq. A.5 is smaller than that calculated from Eq. A.1. This is why the separations of those fringes near the band gap or the reststrahlen band, where Zri+l is very large, are smaller than those far away from these bands. For example, in Fig. 3.9, the separation around 11000cm , which is near the GaAs direct band gap 11452cm’, is 312cm’, 1 while the separation around 9500cm 1 is 395cm . Also, for the same dispersion, the 1 of the near-infrared fringes are smaller than those of the far-infrared ones because of the  i’m  in the dispersion term.  The substrate thicknesses of sample #3 calculated from Eq. A.5 and Eq. 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