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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/Ga1AlAs MULTILAYER SYSTEMBY INFRARED SPECTROSCOPY AT NORMAL INCIDENCEByAn ZengB. A. Sc. Tsinghua University, China, 1985M. A. Sc. Institute of Semiconductors, Chinese Academy of Sciences, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASept. 1992© An Zeng, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of___________________I (.1The University of British ColumbiaVancouver, CanadaDate Qt. /s /DE-6 (2/88)AbstractThe infrared reflectance spectra of GaAs/GaiAlAs multilayer structures at normalincidence from the far-infrared to the near-infrared have been measured using a Bruker IF113v spectrometer. The main structure in the spectra are due to the phonon reststrahlenbands, the Fabry-Perot interference and the plasma edge of the free carriers. An opticalimpedance method was successfully used to calculate the reflectance spectra at normalincidence. Through fitting the spectra, we determined both electrical and structuralparameters, which are usually determined by several different methods. These parametersinclude the carrier concentration n, mobility i, conductivity u, Al concentration x, andthe thickness d of each layer. The agreement between the theoretical spectra and theexperiments is excellent.11Table of ContentsAbstract iiList of Tables viList of Figures viiAcknowledgement ix1 Introduction 11.1 Semiconductor Characterization Techniques 11.1.1 Photoluminescence 21.1.2 X-ray Diffraction 31.1.3 Van der Pauw 41.1.4 Four-point Probe Method 51.1.5 Scanning Electron Microscopy 51.1.6 Scanning Tunneling Microscopy 61.1.7 Secondary Ion Mass Spectrometry 71.2 The Need for a Non-Destructive Technique 81.3 Recent Infrared Studies 101.3.1 Free Carrier Absorption 101.3.2 Theoretical Calculation of Reflection and Transmission Spectrafrom Multilayer Structures 111.3.3 Experimental Study of GaAs Multilayer Structure 11iii1.3.4 Far-infrared Studies of Superlattices 121.4 Outline of Thesis 162 Experiment 182.1 Fourier Transform Spectroscopy 182.1.1 Interferometer 192.1.2 Fourier Transformation 212.1.3 Resolution 292.1.4 Advantages of FT-JR Instrument 312.2 Bruker 113V Spectrometer 322.3 Reflectivity Measurement 353 Calculation of Reflectance 38Reflectance of Bulk Material 383.2 Reflectivity of a Multiple Layer Structure 384 Experiment Results and Analysis 414.1 Reflectance of Bulk Material 414.1.1 Pure GaAs 414.1.2 Doped GaAs 434.1.3 Pure Gai_AlAs 474.2 Reflectance of Multiple Layer Structures 504.2.1 Reflectance of GaAs/AlAs multilayer structure 514.2.2 Reflectance of GaiA47As/GaAs/AlAs multilayer structure. 595 Conclusion 67Appendices 69ivA Interference Fringes 69Bibliography 71VList of Tables4.1 The best fitting parameters for Gai_AlAs. ii and -y in cm’ 494.2 The best fitting parameters for each layer of sample #3 534.3 The electrical parameters of the top GaAs layer of sample #3, calculatedfrom the plasma frequency w, = 450cm’ and free-carrier relaxation time7=45cm1 534.4 The best fitting parameters for the top Gai_AlAs layer of sample #4.v and-y in cm1,d1 in itm 63viList of FiguresLi Geometric structure of a GaAs/A1GaAs multiple quantum well laser diode 21.2 Reflectance of the sample GaAs 262 in reference [16] 131.3 Reflectance of the superlattice sample in reference [27] 152.1 Schematics of a Michelson Interferometer 202.2 Spectra and their corresponding interferograms 232.3 Illustration of the convolution theorem of Fourier transform 262.4 Apodization functions and their correspondillg FT 302.5 Optical path of the Bruker IF 113v 332.6 The reflectance module 363.1 Illustration of optical impedance methods 404.1 The far-infrared spectra of Sample #1 424.2 The far-infrared spectrum of sample #2 and its best fitting 444.3 The conductivity spectrum of sample #2 464.4 The reflectance spectra of Ga1A47.As 484.5 The TO and LO phonon energies of Ga1_,AlAs vs. the Al concentration x 504.6 The far-infrared spectrum of sample #3 and its best fitting 524.7 The calculated spectra of a sample similar to sample #3 544.8 The mid-infrared spectrum of sample #3 554.9 The near-infrared spectrum of sample #3 and its best fitting 574.10 The calculated near-infrared spectrum of a sample similar to sample #3 58vii4.11 The geometric structure of sample #4 604.12 The far-infrared reflectance spectrum of sample #4 and its best fitting . 614.13 The dependence of the far-infrared spectrum on Al concentration . . 624.14 The mid-infrared spectrum of sample #4 654.15 The calculated mid-infrared spectrum of sample #4 664.16 The near-infrared spectrum of sample #4 66viiiAcknowledgementI would like to thank my supervisor, Dr. J. E. Eldridge, for his patient and knowledgeablesupervision.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 helpin the lab.During this research I have been supported by a University Graduate Fellowship fromthe University of British Columbia. This work was also supported by the Natural Sciencesand Engineering Research Council of Canada.ixChapter 1Introduction1.1 Semiconductor Characterization TechniquesMany optoelectronics devices, such as semiconductor laser diodes[1’2j, detectors[31,modulators[4’1 and nonlinear optical devices[6’7], are based on multilayer structures composed 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. Reterolayers 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 different doping levels as shown in Fig. 1.1. The characterization of the electrical and structuralparameters of this complex structure is a challenge. With the fast development of manythin film preparation techniques such as molecular beam epitaxy (MBE), metal organicchemical vapour deposition (MOCVD) and chemical beam epitaxy (CBE) etc. , variouscharacterization techniques have been developed or used to determine the electrical andstructural parameters. These include the free carrier concentration, the layer thicknessand the composition concentration. These well developed characterization techniques arephotoluminescence(PL), double crystal X-ray diffraction (DCD), Van Der Pauw method,four point probing method, scanning electron microscope (SEM), scanning tunnellingmicroscope(STM), and secondary ion mass spectrometry (SIMS) etc. These will now bedescribed in some detail.1Chapter 1. Introduction 2_________________________.—p GaAs C... I X lO cm3)p- Al027 Gao.73 As (... 7 X 1018 Cm)___ ___.—u-GaAs WELLSu—Al027GaAS BARRIERS UNDOPEDV .—N-AI027Ga73As (..3X1017cm—).—fl4—GGA5Figure 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 theparentheses are the carrier concentration of the corresponding layers. u, p, and n denoteundoped, p-type, and n-type respectively. The “+“ sign denotes the heavy doped layersfor Ohmic contacts.1.1.1 PhotoluminescenceThe photolurninescence equipment is anVexcitation laser, a grating monochromator, aphoton detector connected to a photon counting system and a cryogenic system forinstalling samples. The photoluminescence is generated by the recombination of theelectrons and holes which are created by the excitation light, such as laser light with thephoton energy larger than the band gap energy of the subject material. Photoluminescence 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 thebinding energy of both the free- and bound-excitons as well as the band gap energyof the subject material. Photoluminescence is also commonly used to determine theChapter 1. Introduction 3composition concentration of a ternary compound, since the composition concentrationx of a ternary compound is usually related to the band gap energy Eg by a simpleempirical relation. For example AlGai_As has the following relation[13]:E2 = l.424+l.247x (eV) 0<x<0.45 (1.1)= 1.900 + 0.125x + 0.143x2 (eV) 0.45 <x < 1 (1.2)There are, however, some requirements for the samples used in a photoluminescencemeasurement. For example, the upper layers must have a larger band gap than thelower ones in a sample with a multilayer structure. Otherwise, the photoluminescencelight from the lower layers will be absorbed by the top layer. The AlGa_As layers inFig. 1.1 cannot be characterized by photoluminescence since the top GaAs layer, whichhas a smaller band gap than that of A4Gai_As , will absorb all the photoluminescencelight irom the lower layers. -1.1.2 X-ray DiffractionThe lattice coustant of a crystal can be determined from the Bragg Angle 0.B in an X-raydiffraction measurement.2)= arcsin-- (1.3)where ) is the wavelength of the X-ray, and d is the distance between a set of planes inthe crystal. The lattice constant a(x) of a ternary compound, e.g. inAl2,Ga1s[3,isusually 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 Xray diffraction. The resolution of the X-ray diffractioll technique depends on the spectralwidth and the divergence of the X-ray beam used in the measurement. By placingChapter 1. Introduction 4a high quality crystal right after the X-ray source to monochromate the X-ray beamby 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 itshigh resolution, double crystal diffraction has also been widely used to measure someother structural properties, such as dislocation density and the thickness of the grownlayers[8’ 14, 11• Since the X-ray diffraction peaks are the results of the constructiveinterference of the X-ray beams reflected from a set of lattice planes, the thickness of thesample layers must not be too small in order to be detected.1.1.3 Van der PauwThe Van der Pauw method17 is the standard method for measuring the resistivity andmobility of an epitaxial semiconductor thin film, particularly for compound semiconductors.- (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 circumferenceof an arbitrarily shaped flat sample, and to measure the resistances given by VDC/IABand VBC/IAD. The sample resistivity is then given byird VDC VBCp= (— + —)f (1.5)21n2 ‘AB ‘ADwhere d is the sample thickness and f is a function of VDCIAD/VBCIAB. The latteris a geometrical correction factor only. When the sample is a circle or a square andthe contacts are spaced at equal distances around the periphery, then f = 1. TheHall mobility is determined by measuring the change of the resistance VAC/IBD when amagnetic field B is applied in a direction perpendicular to the sample. The Hall mobility,,UH, is given byd /VAc/IBD (1.6)Chapter 1. Introduction 5(Note that H is independent of d.) Since errors can be introduced by the finite diameterand position of the contacts on the circumferences, it is common to sandblast the samplein a clover-leaf shape. The conducting film used in Van der Pauw measurement must begrown on a insulating substrate. If there is more than one conducting layer, the measurement will give an average value of the entire structure. This method will completely failif the conducting films are grow on a conducting substrate as is the case for most of theoptoelectronics devices, where multiple doping layers are grown on a highly conductingsubstrate.1.1.4 Four-point Probe MethodThe 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 four-point 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 electricalbarriers 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 measurement. A resistivity measurement can also be made by point-contact-breakdown usingtwo or three probes and biasing one until breakdown occurs. But, all probe methods giveunreliable results and are generally not used[9].1.1.5 Scanning Electron MicroscopyScanning Electron Microscope (SEM) is composed of two sub-systems:• the electron optical column which produces a finely focussed beam of electrons thatis scanned in a television-type raster over the specimen surfaceChapter 1. Introduction 6• and a signal detection, processing and display system, which detects one of the sixforms of signal emitted by the specimen under electron bombardment.These six forms of signal are 1) emitted electrons, 2)emitted X-ray, 3) cathodoluminescence (CL), 4) electron beam induced current or voltage (EBIC or EBIV), 5) transmittedelectrons under continuous electron bombardment and 6) ultrasonic waves under choppedelectron beam bombardment. The SEM operates in one of the six modes named after thesignal it detects, such CL mode and EBIC mode which are often used for semiconductoranalysis. The electron optical column provides the “illumination” and the detection anddisplay system produces the television-like picture. The resolution of SEM is not limitedby diffraction as is the ordinary optical microscope. It is determined by the physics ofthe electron optical column and of the signal detection and display system. It also depends on the physics of the signal generation process of the particular mode. The SEMtechniques provide measurement with a micron scale resolution and so can determinedefect microstructures and local values of physical properties. One of the great advantages 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 onlylateral, spatial resolution but also depth resolution, spectral (signal) resolution, timeresolution, etc. More details can be found in references [10, 11].1.1.6 Scanning Tunneling MicroscopyScanning tunneling microscopy (STM) is a technique which is capable of viewing a conducting or semiconducting material with atomic resolution. Quantum mechanically, whentwo materials, such as a probe tip of STM and a semiconducting sample, are broughtso close together that the electronic wave functions of the tip and the sample overlapsignificantly, a tunneling current IT can be induced by an applied bias voltage VT. TheChapter 1. Introduction 7tunneling probability depends on the overlap of the wavefunctions of the filled and emptystates of the tip and the sample. The exponentially smaller ‘T contributions of the longerpaths in the tunnel gap leads fortuitously to the dominance of a single atom to the currentfrom a tip and therefore to the atomic resolution of the STM. In the most usual STMmode , 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 ITconstant. The changes iii voltage of the piezoelectric transducer perpendicular to thesurface are proportional to the surface topographic variations. Since STM does not workwell for rough surfaces, it is critical to have a very clean and smooth sample surface. Thesample preparation is one of the most difficult steps in a STM experiment. The detaileddescription and applications of STM can be found in reference [18].1.1.7 Secondary Ion Mass SpectrometryThe Secondary Ion Mass Spectrometry (SIMS) setup is mainly composed of an ion gun, amass spectrometer and a data processing system. In a SIMS measurement the sample isbombarded 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. Thepositive or negative ions are extracted into a mass spectrometer and separated accordingto their mass to charge (m/e) ratio. With primary beam rastering , the sputter erosioncauses the sample surface to recede in a very controlled way. By monitoring the intensityof one or more mass peaks as a function of bombardment time, an in-depth concentrationprofile is obtained. SIMS is often used to answer the following questions: (1) what is theform of the dopant profile (i.e. concentration as a function of depth)? (2) does the dopantprofile show abrupt step changes between different concentration levels? (3) how sharpis the interface at a hetero-junction? and (4) are spurious electrically active impurityelements present? SIMS is one of the few methods which can characterize both electricalChapter 1. Introduction 8and 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 ofeach layer of the multilayer structure in reference [16]. The disadvantage of SIMS is thatit is destructive, time consuming and difficult to calibrate. For a detailed description ofSIMS 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 onespecific requirement for the samples and that electrical and structural parameters areusually determined by two or more different methods using different samples. For example, 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 carrierconcentration. The only method mentioned above, which can characterize both electricaland structural parameters at the same time using only one complex multilayer sample, isSIMS. But this method is destructive, time consuming and difficult to calibrate, especiallyif a ternary compound layer like AlGai_As is involved.[.2 The Need for a Non-Destructive TechniqueRecent publications, however, have shown an increasing interest and need for a nondestructive and non-contact method to characterize both electrical and structural parameters ill a multilayer structure16’19, 20, 21] There are several immediate needs inaddition to those already mentioned:1. ZnSe is a good candidate for a blue-light-emitting device because of its suitabledirect band gap (2.7 eV at room temperature)[221.However, the formation of ZnSep — n junctions has been difficult due to problems in controlling the conductivityof the bulk crystals. Much progress has been made, however, with ZnSe thinChapter 1. Introduction 9films grown on GaAs. p-type concentration characterization however still remainsa major problem because of the lack of a suitable ohmic contact technology[231,which makes the conventional four-point probing method impossible. A non-contactmethod to determine the p-type carrier concentration is urgently needed.2. The InAs/GaAs heterostructure has potential in infrared detection or emission aswell as in other optoelectronics applications[24’25j• However due to the latticemismatch between these two material (7.3%), misfit dislocations propagate intothe InAs layer deposited on a GaAs substrate. These structural defects primarilyaffect the electronic properties of the material. Many efforts have been made toreduce the dislocation density[24’25, 26]• Hydrogenation of the InAs layer is one ofthem. Atomic hydrogen incorporated into a semiconductor passivates impurities ordefects and neutralizes their electrical activity. Because hydrogenation is reversible,any contact measurement may change the hydrogen concentration. This hydrogenconcentration is determine by the destructive SIMS method. A non-destructive,non-contact method is needed to determine the free carrier concentration in thestudy of the effect of hydrogen concentration on the free carrier concentration inInAs/GaAs heterostructures21.Infrared spectroscopy has shown promising capabilities in solving these problems[16’20]Besides its conventional use in the determination of the lattice vibrational frequencies, itcan also be used to determine the electrical and structural parameters of semiconductorthin films, such as the free carrier coilcentration, the composition concentration, the freecarrier scattering time and the layer thickness. Although the technique has not beenfully developed, it has received increasing attention since 1985. Several papers on various material have been published since then[6’20, 27j In the following section, we willbriefly summarize some of the recent publications on this newly developing method.Chapter 1. Introduction 101.3 Recent Infrared Studies1.3.1 Free Carrier AbsorptionBecause of its potential in preparing blue light-emitting devices, doped ZnSe has become the topic of many recent publications. In 1987, H. E. Ruda[19] improved Jensen’stheory[281of free-carrier absorption by considering all major scattering mechanisms, suchas polar-optical phonon scattering and ionized impurity scattering etc. This made itpossible to assess the effect of the impurities on free-carrier absorption. The dependenceof the electron concentration n on the free-carrier absorption coefficient a was calculatedas a function of the ionized impurity concentration and the compensation ratio 0. The 0is the ratio of the concentration of ionized acceptors to that of the ionized donors in thematerial. At A = lOfLm, a simple formula which can be used to calculate n-type ZnSecarrier density by freercarrier absorption was proposed.n(cm3)= 6.7 x x (3.5—p) x a,(cm’) (1.8)where a), and p are the absorption coefficient and the derivative logarithmic absorptioncoefficient respectively at A = 10pm. p is defined as:= dln(—) (1.9)where A0 is a reference wavelength. Experimental data from the literature are analyzedin this scheme and values of n determined by Hall-effect measurements are shown to bein excellent agreement with the theoretical predictions.This non-contact and non-destructive method was used by A. Deneuville and D. B.Tanner et al.[201 later in 1991 to determine the carrier concentration in n-type ZnSe thinfilms grown on a GaAs substrate. Deneuville measured both the transmission and thereflection spectra in the far infrared range and found that they are very sensitive toChapter 1. Introduction 11the carrier concentration. Deneuville et al did not find, however, the correct formulato reproduce the reflection and transmission spectra of a multilayer structure.They usedinstead an approximation to calculation from the transmission spectra.1.3.2 Theoretical Calculation of Reflection and Transmission Spectra fromMultilayer StructuresIn 1986, B. Harbecke29 reported his theoretical work on a matrix method to calculatethe coherent and incoherent reflection and transmission of multilayer structures. Thecomplex-amplitude reflection and transmission coefficients r and t of a multilayer structure are represented as a product of matrices. The matrices describe the transformationof two plane waves travelling in opposite directions between the films, and their development within the films. One of his contributions is that he is able to easily suppress thesubstrate interference fringes, which are sometimes noi observed in the experiment, byadding the absolute squares of the partial waves corresponding to an incoherent treatment. This procedure is shorter and simpler than the conventional method of averagingover an appropriate interval of frequency or thickness, which, in most cases, leads to thesame 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 incidentinfrared beam.1.3.3 Experimental Study of GaAs Multilayer StructureIn 1990, P. Grosse and B. Harbecke et al.16 used the method of the previous sectionto simulate the experimental reflection spectra of a conducting GaAs multilayer structure. Both the electrical and geometric properties of the GaAs multilayer structureswere determined nondestructively by reproducing the infrared reflectance spectra from(50—5000cm1 ). The sample consisted of two layers of GaAs with the same thicknessChapter 1. Introduction 12but with different doping levels, sitting on a conducting GaAs substrate. Using obliqueincidence and both s- and p-polarizations of the infrared beam, the carrier concentrationand the thickness of the epitaxial films as well as the carrier concentration of the GaAssubstrate were determined.The main structures in the spectra were due to phonon reststrahien bands, Fabry-Perot interferences and the zeros of the dielectric function leadingto dips in the reflectance( Berreman mode ). The interference fringes were used to estimate the first order approximation of the film thicknesses. Fourier transformation ofthe reflectance spectra was suggested to estimate the first order thickness in those caseswhere the structure consists of two or more layers of different thickness. The Berremaneffect, causing dips in the p-polarization reflectance spectra at zeros of the dielectricfunction, was used as a first order approximation to find the carrier concentration in thefilms. The final results of the carrier concentration and layer thickness were determinedby fitting the -reflectance spectra, using the first order approximation of thickness andconcentrations as input. Fig.l.2 shows that the theoretical spectra fit the experimentalones very well. The parameters for the thin films compare favorably with a depth-resolvedSIMS sample analysis. A substantial discrepancy was found, however, in the data pertaining to the substrate. The substrate concentration determined by SIMS or electricalmeasurement was substantially higher than that determined by infrared spectroscopy.(4.15 x 1O’8cm3versus 2.5 x 1O’8cm3)1.3.4 Far-infrared Studies of SuperlatticesIn addition to the characterization of multilayer structures with different doping levels, infrared spectroscopy has also been developed as a new method to characterize thestructural parameters of superlattice structures which are composed of many periods ofalternating thin layers of two or more materials. In 1985, V. M. Agranovich et al.[301 reported their important theoretical work on superlattice optical properties, which makesChapter 1. Introduction 13to0.,/2irc Ecnf’]Figure 1.2: Reflectance of the sample GaAs 262 in reference [16]. The solid line: measured 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; B1, B2: Berreman-mode. 100 200Chapter 1. Introduction 14it possible to calculate the dielectric function of a superlattice through the dielectricfunctions for the bulk materials of the constituent layers. This paper is frequently citedby experimental physicist working in this area now.The calculation method here is different from that which we have introduced previously, where bulk dielectric functions were used for each layer in the calculation, assuming 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 dielectric response depends on the geometrical layer arrangement. The dielectric functionis a diagonal tensor whose components, in the long wavelength limit, can be written asweighted averages of the dielectric functions of the constituent layer materials. For theinfrared beam with the electric field E parallel to the layers, the dielectric function canbe written as:- e1(w)+E2(w) (110)where e1(w) and e2(w) are the dielectric functions for the bulk materials of the constituent layers, and e is the ratio of the thicknesses of the two alternating layers. in1987, R. Sudharsanan[271reported the first infrared spectra for AlAs/GaAs superlattices and analyzed the data, using Agranovich’s above formula to calculate the dielectricfunction for the superlattices, in order to find the superlattice layer thicknesses. Fig.l .3shows the calculated reflectance spectra and the measured data. Although there is asmall systematic difference, the results compare favorably to those obtained from doublecrystal X-ray diffraction which is considered now to be the most accurate method of determining the superlattice structure. Sudharsanan suggests that far-infrared reflectancespectroscopy is as precise as that of double crystal diffraction and has advantages over theRaman scattering method which has also been used recently to characterize AlAs/GaAssuperlattices[311. The far-infrared radiation penetrates much more deeply than the laser.C-)a)a)>‘I..:-4-AC-)a)C)Wave Number (cm1)Figure 1.3: Reflectance of the superlattice sample in reference [27]. The dots : measureddata, 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 ofthe minima at 338 and 396 cm1.Chapter 1. Introduction 1. Introduction 16light in Raman scattering and samples the entire microstructure. Moreover, the existenceof a simple superlattice dielectric response function allows quantitative analysis.In addition to the characterization of a multilayer structure, infrared spectroscopyhas also been used recently to probe layer interdiffusion and alloying in HgTe-CdTesuperlattices[34’35] and to examine AlGai_As/GaAs[361and AlAs-GaAs heterostructures37. G. Scamarcio[32 et al. used it to study the dependence of reststrahlen bandsof multiple quantum well heterostructures and the transverse optical phonon of ultrathinlayer superlattices[331. T. Dumelow et al.[38’ 39] and A. R. El-Gohary et al.[401 usedit to study surface, bulk plasmons and phonon-polaritons in superlattices and multiplequantum well structures.1.4 Outline of ThesisIn Chapter 2 on the experiment, the principles and the advantages of an infrared Fouriertransform spectrometer are described, along with the Bruker IFS 113V interferometerand the reflectivity module which were used in the experiment.In Chapter 3 on the theory, the optical impedance method used to reproduce theexperiment reflectance spectra is introduced.In Chapter 4 on the results, the measured reflectance spectra of four samples inthe range from 50—15000 cm1 are presented. The main features of the spectra aredue to the phonon reststrahlen bands, interference fringes and the free carrier plasmaedges. Drude-Lorentz model and optical impedance method were used to reproducethe far-infrared reflectance spectrum of a multilayer structure. An empirical formula ofrefractive index and the optical impedance method were used to reproduce the nearinfrared spectrum of a multilayer structure. The fringes in the different region of thespectra were found to have resulted from the interference of the layers with differentChapter 1. Introduction 17thicknesses. 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, Alconcentration, 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 separatedFabry-Perot interference fringes due to dispersion effect is introduced.Finally the conclusions of the thesis are reviewed in Chapter 5, and the limit of themethod is discussed.Chapter 2Experiment2.1 Fourier Transform SpectroscopyInfrared spectroscopy is a powerful tool often used by physicists and chemists. FourierTransform Infrared Spectroscopy(FT-IR) is the most advanced form of infrared spectroscopy. A FT-JR spectrometer is mainly composed of two parts:• a optical hardware(bench) which consists of optical sources, a Michelson interferometer, 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 controlledby setting appropriate knobs controlling slit widths, scanning speed, etc. but involvesa certain amount of mathematical manipulations such as Fourier transformation, phasecorrection and apodization. The quality of the final spectra depends on both the settingsof the optical bench and the parameters or functions used by the software to calculatethe spectra. It is important that the user be familiar with the principles of FT-IR datacollection and manipulation. In the following sections we will cover briefly the standardoperations of the FT-JR spectroscopy from data acquisition to the final spectrum. Thefollowing section is mainly abstracted from references [41] [53] [54].18Chapter 2. Experiment 192.1.1 InterferometerThe 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. Infraredlight emitted by a source is directed to a device called the beam splitter, because it ideallytransmits half of the light while reflects the other half. The reflected part of the beamtravels to the fixed mirror M1 through a distance L, is reflected there and hits the beamsplitter again after a total path length of 2L. The same happens to the transmitted partof the beam. Except that the reflecting mirror M2 for this interferometer arm is not fixedat the same position L but can be moved very precisely back and forth around L by adistance x, The total path length of this beam is accordingly 2(L+x). Thus when thetwo halves of the beam recombine again on the beam splitter they exhibit a path lengthdifference or optical retardation of 2x. The partial beams are spatially coherent and willinterfere when they recombine.The combined JR beams leaving the interferometer pass through the sample chamberand are finally focused on the detector D. The quantity actually measured by the detectoris thus the intensity 1(x) of the combined JR beams as a function of the moving mirrordisplacement x, the so-called interferogram shown in Figure 2.1 B. The interference pattern as seen by the detector is shown in Figure 2.1 C for the case of one monochromaticspectral line. The partial beams interfere constructively, yielding maximum detectorsignal, if their optical retardation is an exact multiple of the wavelength ). Minimumdetector signal occurs at destructive interference where the optical retardation is an oddmultiple of \/2. The complete dependence of 1(x) on x is given by a cosine function:1(x) = S(v) x [cos(27r v x) +11 (2.1)where ii is the wavenlLlmber, and S(v) is the intensity of the monochromatic line atwavenumber xi. For a polychromatic source, the interferogram is the integral of theChapter 2. Experiment 20Figure 2.1: (A) Schematics of a Michelson Interferometer. 5: JR source, M1: fixedmirror, M2: scanning mirror, D: detector. (B) interferogram of a polychromatic source.(C) interferogram of a monochromatic source. Its zero crossings define the positionswhere the interferogram is sampled(dash lines)Chapter 2. Experiment 21monochromatic interferogram over the spectrum range:fVmax1(z)= j S(z) x [cos(27r. • x) + 11th’ (2.2)1’minwhere 11min and Vma are the minimum and the maximum wavenumber generated by thepolychromatic source.Equation 2.1 is very useful for practical measurements, because it allows very precisetracking of the scanning mirror. In fact, all modern FT-JR spectrometer use the interference pattern of the monochromatic light of a He-Ne laser to control the movement of thescanning mirror and the sampling interval of the JR interferogram. Figure 2.1 C demonstrates how the JR interferogram is sampled precisely at the zero crossings of the laserinterferogram. The accuracy of the sampling interval between two zero crossings is solelydetermined by the precision of the laser wavelength itself. Thus, FT-JR spectrometershave 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-JRinstrument will be discussed later.2.1.2 Fourier TransformationData acquisition yields the digitized interferogram 1(x), which must be converted intoa spectrum by means of a mathematical operation called Fourier transformation (FT).Generally, the Fourier transformation of a continuous interferogram generates a continuous 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)Chapter 2. Experiment 22where 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 byAv= N Ax(2.4)This process is reversible. If S(k. Au) is known, one can easily reconstruct the interferogram I(ri. Ax) by the so-called inverse discrete Fourier transformation:1 N—i i2irnkI(n. Ax) = — S(k. Au) exp( ) (2.5)Nk_o NThe above Fourier transformations are best illustrated in the simple case of a spectrumwith one or two monochromatic lines, as shown in parts A and B of Figure 2.2. For alimited number of functions like the Lorentzian in part C of Figure 2.2, the corresponding Fourier transformation is known analytically and can be looked up from a Fouriertransformation table-. However, in the general case of measured data, the discrete Fouriertransformation must be calculated numerically by a computer.Although the precise shape of a spectrum cannot be determined from the interferogram without a computer, it may nevertheless be helpful to know two simple trading rulesfor 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 spectralline width(as is always present for real samples) is due to damping of the amplitudes inthe interferogram: the broader the spectrum the stronger the damping. Comparing thewidths at half height (WHH) of I(n . Ax) and S(k. Au) one reveals another relatedrule: the WHH of a “hump-like” spectrum is reversely proportional to that of its Fouriertransformation—the interferogram. The rule explains why in Figure 2.2 D the interferogram due to the broad spectrum shows a very sharp peak around the zero path differenceposition, while the wings of the interferogram, which contain most of the useful spectralinformation, have a very low amplitude.Chapter 2. Experiment 234000 2000VVENUMBERS CM-IFigure 2.2: Examples of spectra(on the left) and their corresponding interferograms(onthe right). (A) one monochromatic line. (B) two monochromatic lines. (C) Lorentzianline. (D) broadband spectrum of polychromatic source.S(v)A1(x)BC4-h.vwTz, I,xzChapter 2. Experiment 24For 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 tothe sum over all N spectral intensities divided by N. This means the height of the centerburst 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 reduce 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 incalculation is that the number of interferogram points N can not be chosen freely. Forthe most commonly used Cooley-Tukey algorithm, N must be a power of two.It should be noted that discrete Fourier transformation approximates the continuous 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 theirsolutions.Picket-fence Effect and Its Solution: Zero FillingThe picket-fence effect[54] becomes evident when the interferogram contains frequencieswhich do not coincide with the discrete frequency points k. Av of the spectrum obtainedby discrete Fourier transformation. If, in the worst case, a frequency component liesexactly halfway between two discrete frequency points, one seems to be viewing thetrue 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 effectcan be overcome by adding zeros to the end of the interferogram before performing thediscrete Fourier transformation, thereby increasing the number of points per wave numberin the spectrum. Thus , zero filling the interferogram has the effect of interpolating theChapter 2. Experiment 25spectrum, reducing the picket-fence effect. As a rule of thumb, one should always atleast 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 zerofilling does not introduce any errors because the instrumental line shape is not changed.It is therefore superior to polynomial interpolation procedures working in the spectraldomain.Aliasing and Its Solution: filteringAnother 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 introduce a theorem called the convolution theorem of Fourier transform. It states that theFourier transform of the product of two functions, e.g. 1(x) and (x), is the convolutionof their individual transform S(ii) and (i), where tha convolution is defined as:S(v) 0 A(v)= J S(u)(v — u)du (2.6)The coilvolution theorem is used in Figure 2.3 to illustrate how to determine the discreteFourier transformation of a sampled interferogram. The discrete interferogram is treatedas 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 2.3we can see that the FT of the sampled interferogram is a periodic function where oneperiod within a constant is equal to (c) the FT of the continuous interferogram. Theperiodicity can also be seen from equation 2.3 which is not oniy valid for indices k fromo to N — 1 but for all integers including negative numbers. In particular, if we replace kin Equation (1) by k + m N , we get the equation:S(k+m.N) =S(k) (2.7)A J;f\•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 ofthe set of delta functions. (e)I(x). L(x): sampled interferogram. (f) FT of the discreteinterferogram which is real spectrum plus the mirror images.261(x) L(x)(a) (b)(e)•1U0 D.Chapter 2. Experiment 27which states that the mirror-symmetrical N-point sequence is endlessly and periodicallyreplicated as indicated in Figure 2.3 (f). By picking out one period of (f) we can determine the real spectrum of the interferogram 1(x). As only the positive frequency hasreal physical meaning, we ignore the negative frequency range. By comparing the realspectrum(positive part) Figure 2.3 (c) and the FT of the sampled interferogram (f), onefinds that only the first half period from 0 to of (f) represents the real spectrum.The other half period from to is just the mirror image of the first half periodand therefore should be discarded. The folding wavenumber Uf = is also calledNyquist-wavenumber[54}. From Figure 2.3 (f) it is clear that a unique spectrum can onlybe calculated if the spectrum does not overlap with its mirror-image(alias). No overlapwill occur if the spectrum is zero above a maximum wavenumber Vma and if Vma issmaller than the folding wavenumber vf:Vmax 2x(2.8)If, however, the spectrum contains a non-zero contribution above the folding wavenumberVf, e.g. non-zero contribution from i’ to ij + 200cm1,this will be ‘folded back’ to thearea below Vf (from Vf — 200cm1 to vf). This is the possible artifact due to aliasing. Inprevious section we have explained that the sampling positions are derived from the zerocrossings of a He-Ne laser with wavelength= 158OOcm’ As zero crossing occurs everythe minimum possible sample spacing Axmjn is 31600cm According to equation 2.8, thiscorresponds to a folding wavenumber of 15800cm’, i.e the maximum bandwidth whichcan be measured without overlap is 15800cm’. Very often, the investigated spectrum hasa bandwidth much smaller than 15800cm . In these cases, one can choose Lx to be anrn-fold multiple of AXmin. This leads to an rn-fold reduction of the interferogram size andtherefore increases the speed of computing. Further reduction of the data size is possibleby using the so-call undersampling technique, if the spectrum is non-zero only in a limitedChapter 2. Experiment 28range (“max — “mm) and (Vmax vmmn) lies between two of the folding wavenumbers whichare a natural fraction or integer multiple of the He-Ne laser wavenumber:Pfn = n 15800cm(n (2.9)An advanced FT-JR software package will automatically account for proper sampling andundersampling, 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 isreally zero outside the two folding wavenumbers by inserting optical or electronic filters.Leakage and its solution: apodizationUnlike the picket-fence effect and aliasing, leakage is not due to using a digitized versionof a continuous interferogram. Leakage is caused by the truncation of the interferogramat finite optical path difference. Mathematically, an interferogram trimcated at opticalpath difference x L can be obtained by multiplying the infinite interferogram 1(x) bya ‘boxcar’ or square window function B(x). According to the convolution theorem, theFT 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 functionB() which determines the instrument lineshape function. The analytical form of the‘instrument lineshape’ function corresponding to boxcar truncation is the well knownsinc function. The FFT of a square window function is plotted in Figure 2.4 B. Besidesa main peak, one sees many additional side peaks, called lobes or ‘feet’. These side peakscause a ‘leakage’ of the spectral intensity. The largest side lobe amplitude is 22% of themain lobe amplitude. As the side lobes do not correspond to actually measured spectrumbut rather represent an artifact due to the abrupt truncation of the interferogram, it isdesirable to reduce their amplitude. The process to attenuate the spurious ‘feet’ in thespectrum is known as ‘apodization’ (originating from the Greek word a7roS, which meansChapter 2. Experiment 29‘removal of the feet’).The solution to the problem of leakage is to truncate the interferogram less abruptlyby 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. Onecan see that the ‘feet’ of these three functions are much smaller than those of the squarefunction. However, one must realize that this is at the cost of the resolution.2.1.3 ResolutionFigure 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 themain lobes of all the ILS’s are broader than that of the sinc function. The full width athalf maximum (FWHM) of a ILS defines the best resolution achievable with the givenapodization function. This is because if two spectral lines are to appear resolved from oneanother, they must be separated by at least their FWHM, otherwise no ‘dip’ will occurbetween them. As side lobe suppression always causes main lobe broadening, leakagereduction is only possible at the cost of resolution.The choice of a particular apodization function depends on the experimental requirement. If the optimum resolution is mandatory, the square window function with theresolution of 0.61/L (no apodization) should be chosen. If a resolution loss of 50% compared to the square truncation can be tolerated, the Parzen(triangle) or even better theHanning function with resolution of O.9/L (roughly) is recommended. If the interferogram contains strong low-frequency components, it may show an offset at the end, whichwould produce ‘wiggles’ in the spectrum. To suppress these wiggles, one should use afunction 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. WithChapter 2. Experiment 30II IIIIIIII 1111111111 IlIlIlIlIllIlIl IlIlIllIllIlIlIsquare window—” / \ \\ (A)/, / \ ‘Welch window-.-/ / / \ \ \\/ / \ \/ 1/ \\ \.7,/,“/ \\ \\/ 1,1/ -Parzen window \\/ 11/ \\ \\/ ‘I \s/ /a5 / / \ \/ I” \\ \CO I. I’ %\ \.4 / /// /1’ \\ \/ I’ \.3 /1 lS \\ \/ ,2 //,“Hanning window ‘\ \0 - -—I I I I I I I I I0 20 40 60 80 100 120 140 160 160 200 220 240 260‘I’’’ jiTIIjIiILJIlIIjEIIPI II rrIr1T —r.9 (B) -n —.8-:111 l\\’,:11! \‘\‘.7— :111Iii \\\\Ii I I.6— ‘/1 I I I\ —i’l \‘\\.5. // / Hanning:111 Parzen- /1/ \\Welch.3 I\Figure 2.4: (A) apodization functions (B) the corresponding FT of the window functions.Chapter 2. Experiment 31any finite sized aperture, the radiation from the off-axis points will be incident at someangle on the mirrors. This radiation will have a reduced optical path difference betweenthe two arms, which will transform into a smaller wavenumber. Thus the effect of theentire aperture is to spread a monochromatic v into a range from z-’ to ii — ii cosThis affects both the accuracy and the resolution. It is obvious that the improvementof resolution and accuracy by reducing the size of aperture is actually at the cost of theoptical throughput. This is similar to the dispersive instrument.However, unlike the resolution of a dispersive instrument which is oniy controlled bythe width of the slits, the resolution of a FT-JR spectrometer depends on the optical pathdifference L(scanning distance), apodization function, zero filling of interferogram as wellas the aperture size of the optical source. Generally speaking, a FT-JR spectrometer hasa higher resolution than that of a dispersive spectrometer. However, one should avoidmeasuring-a spectrum with very high resolution over a large frequency range, as this mayresult in a poor signal to noise ratio for the calculated spectrum. When measuring a highresolution 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 relatedby the following formula[57j:(S/N)spE— 2 10(S/N)IFG— Vwhere 6v and Lv are the resolution and spectrum range respectively. Thus a measurement with resolution 0.lcm’ over a range of 1000cm will result in a spectrum withone hundredth of the signal to noise ratio of the interferogram.2.1.4 Advantages of FT-JR InstrumentBesides its high wavenumber accuracy discussed before, a FT-IR spectrometer has otherfeatures which make it superior to a conventional grating IR spectrometer. The so-calledChapter 2. Experiment 32Jacquinot or throughput advantage arises from the fact that a FT-JR spectrometer canhave a large circular source at the input or entrance aperture of the instrument withno strong limitation on the resolution. Also, it can be operated with large solid anglesat both the source and the detector. However, the resolution of a conventional gratingspectrometer depends linearly on the width of the input and output slits. Also, for highresolution, a grating spectrometer requires large radii for the collimation mirror, and thiscondition in turn necessitates small solid angles. Thus, for the same resolution, a FT-JRspectrometer can collect much more energy than a conventional grating spectrometer.In conventional spectrometers the spectrum S(v) is measured directly by recording theintensity at different monochromator settings of wavenumber v. In FT-IR, all frequenciesemanating from the JR source impinge simultaneously on the detector. This accounts forthe so-called Fellget advantage.Finally, the Feilget and the Jacquinot advantages permit construction of interferometers having much higher resolving power than dispersive instruments. Further advantagescan be found in the book by Bell[41].2.2 Bruker 113V SpectrometerAll the reflectivity measurements were performed on a Bruker 113V spectrometer that is aGenzel-type interferometer[43] designed to operate under vacuum, which helps to removethe unwanted water absorption and preserves the thermal stability. It covers the full rangeof wavenumber from the far infrared ( lOcm’) to the near infrared (15800cm’) with amaximum resolution of O.03cm1 Figure 2.5 shows a ray diagram of the optical benchof a Bruker IFS 113V. The Bruker spectrometer is directly interfaced with a computerperforming data collection, fast Fourier transformation, and control of the motor-drivenoptical components, e.g. optical filters, beam splitters, apertures, and mirrors which areChapter 2. Experiment 33Figure 2.5: Optical path of the Bruker IFS 113V. (I) Source Chamber:a—Tungsten/Halogen/Quartz lamp, glowbar, mercury arc lamp; b—automated aperture. (II) Interferometer Chamber: c—optical filters; d—beamsplitter; c—double-sidedscanning mirror; f—control interferometer; g—reference laser; h—remote control alignment mirror. (III) Sample Chamber: i—sample focus; j—reference focus.(IV) DetectorChamber: k—far-infrared DTGS detector, mid-infrared MCT and InSb detectors andnear-infrared Si- diode detectors.used to change sample chambers, sources, and detectors. Measurements covering differentspectral 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 thesource chamber is focused on the beamsplitter, which allows the beamsplitters to bevery small (about 2 cm in diameter.) In a conventional interferometer, the radiation isnot focused at the beam splitter and therefore the beam splitters are quite large (about12-20 cm). These large beam splitters vibrate slightly, resulting in a diffusion of theinfrared beam and increased spectral noise. This is called the ‘drum-head’ effect. TheChapter 2. Experiment 34small 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 thebeam splitters is oniy 14°. This small angle of incidence compared with that of otherinterferometers results in increased light throughput and decreased polarization effects.The two beams from the beamsplitters are incident on opposite sides of a double-sidedscanning mirror. The mirror is supported on a dual gas bearing that uses dry nitrogenand is driven by a linear induction motor. This design gives an optical path differencetwice as large as that of a conventional interferometer and therefore achieves a givenresolution with oniy half of the usual mirror movement.Data is collected in a fast-scanning mode, with the scanning mirror position determined by the fringes of another separated interferometer with a white source and a He-Nelaser. 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 laserinterference fringes. According to the response time of the detector used, one can choosean appropriate scanning speed v out of the 15 choices provided by Bruker. A scanningmirror with velocity v (in cm/sec.) results in alternating electronic signal at the detectorwith frequency:fL’ = 4vv (2.11)where f. is the modulation frequency in Hz and v is the wavenumber in cm1. Toavoid aliasing, high-pass and low-pass electronic filters are used in the Bruker insteadof optical filters used in other spectrometers since each electronic f,, corresponds to onespecific optical wavenumber v. This relation is also important in determining the noisefeatures caused by mechanical vibrations or electromagnetic crosstalking. Each scanof the mirror typically takes a few seconds and several hundreds of interferograms aremeasured and averaged to increase the signal to noise ratio before performing the FourierChapter 2. Experiment 35transform to derive the final spectrum. The apodization function that we have used isthe so-called ‘three-term Blackmann-Harris’ function:W(x) = 0.42323-f- O.49755cos(rn/N) + 0.07922cos(2rn/N) (2.12)where n = 1,2, 3, .. , N and N is the number of points sampled in the interferogram.2.3 Reflectivity MeasurementThe reflectivity measurements were performed using a reflectance module placed in oneof 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 isimaged at the sample or reference surface at about 15 degrees. Using the visible lightfrom the tungsten source, the positions of the aperture and the first toroidal mirror areadjustecF to produce a well focused image at the plane of the sample. The aperture isset to a size which is slightly smaller than that of the sample. Light reflected from thesample is collected by the second toroidal mirror and directed to the detector chamberalong the original spectrometer path.The sample and the aluminum reference mirror were mounted on a home made sampleholder with a very smooth holding surface. The sample and the reference were mountedagainst the smooth surface so that their surfaces were iii the same plane. This makessure that the light reflected from the sample is in the same direction as that from thereference. The sample and the reference can be exchanged by moving the sample holderfrom outside in the direction designated by M in Figure 2.6. The typical size of oursample and reference is about 5 x 10 mm.There are several vacuum feedthroughs to allow adjustment of the toroidal mirrorsand other components in order to maximize the detector signal. When maximizing thesignal, a chopper is brought into the beam using one of the vacuum feedthroughs toChapter 2. Experiment 36NFigure 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 movechopper into the beam (K) sample chamber extension. (L) plexiglass cover. (M) directionof translation exchanging sample and reference mirror.Chapter 2. Experiment 37produce AC signal which is required by the detectors. Since our sample and referencemirror are in the same plane, we need only to maximize the signal once by adjusting thesecond toroidal mirror. A detailed description of the reflectance module can be found inreference [56].The far-infrared measurements in the 10 — —500cm1 region were made using a mercury 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 otherwisedamage the thin mylar beamsplitter. A deuterated triglycerine sulfate (DTGS) detectorworking at room temperature was used to detect the far infrared signal. The mid-infrared measurements in the 550 — —5500cm’ region were made using a glowbar sourcewithout any optical filter and a KBr beamsplitter. A MCT detector working at 77Kwas used to detect the mid-infrared radiation. The near-infrared measurements in the9000 — —l5000cm’ were-made using a tungsten lamp with a red film optical filter anda quartz beamsplitter. A si-diode working at room temperature was used to detect thenear-infrared radiation.For each sample and reference mirror, 200 — —500 interferograms are measured andaveraged to increase the signal to noise ratio. The velocity of the scanning mirror was chosen 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 detectorrespectively. It takes about 5 to 10 minutes for the computer to collect data and calculatethe final spectrum. The absolute value of the reflectivity was obtained by comparisonwith the aluminum mirror and the published values of the aluminum reflectivity [47].Chapter 3Calculation of Reflectance3.1 Reflectance of Bulk MaterialThe optical properties of a solid can be calculated from the dielectric function E(w) orcomplex refractive index = n + ik. In terms of the refractive index, the dielectricfunction is given by:= ‘ += (ii + ik)2 (3.1)Equating the real and the imaginary parts gives and €“ in terms of n and k:= — k2 (3.2)= 2nk (3.3)For normally incident radiation on a surface in a vacuum, the equation relating m and kto the reflectivity R is(34(n+1)2+k2For non-normally incident light, the reflectivity in terms of n and k becomes a complicatedfunction of both the angle of incidence and the polarization [44].3.2 Reflectivity of a Multiple Layer StructureThere 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 the38Chapter 3. Calculation of Reflectance 39reflectance of a system with any number of layers. The optical impedance j and thepropagation coefficient-y in a solid with dielectric function are defined as:1- 37777- Voc n-I-ik (3.5)= —2iri7ii/ (3.6)where 17 is the wavenumber (cm’). For a single layer system shown in Figure 3.1 (a),the load impedance seen at the first interface is773 coshQy2d)+ 772 sinhQyd)ZL1 = 772 . (3.7)772 coshQy + sinh(-ydwhere 773 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 withoptical impedance is calculated using the formulasr = 3.8)ZL1 + 77iR = (3.9)In the case where the backing region is not a thick substrate but another dielectriclayer sitting on a substrate as shown in Figure 3.1 (b), the value of 173 in Eq. 3.7 is replacedby an effective load impedance ZL2, which is calculated using the same equations.774 cosh(3d)+ 773 sinh(-yd)ZL2 = 773 (3.10)773 cosh( + 7)4sznh(-13This procedure can be repeated for any series of layers with different optical propertiesand thickness, i.e. the reflectance of a system can be calculated no matter how manylayers it has as long as we know the dielectric function or complex refractive index ofeach layer. However, the relation between dielectric function and complex refractiveindex must be defined as Eq. 3.1. Changing the sign of extinction function k will resultin unreasonable reflectance in some region, e.g. R > 1 near reststrahlen band. If theChapter 3. Calculation of Reflectance 40definition of = n — ik is used, the minus sign in the definition Eq. (3.6) must bedropped.The above method is only good for normal incidence. The reflectance of a multilayersystem at non-normal incidence can be calculated by a matrix method introduced indetail in reference [29].d2p‘li 12 773normal incidence(a)d2 d3772 ‘13 ‘14ZL1 ZL2(b)Figure 3.1: (a) a single layer described by the impedance‘12, propagation coefficient 72,and thickness d2 is sitting on a thick substrate described by the impedance 773. is thethe impedance of the medium from which the light is incident on the sample. (b) a twolayer system with the thin layers described by‘12, 72, d2 and -ye, d3 is sitting on asubstrate described by the impedance ‘14Chapter 4Experiment Results and AnalysisReflectance measurements were performed on 4 samples at near normal incidence in thewavenumber range from the far-infrared to the near-infrared. We started with samplesof single layer structure, i.e. bulk materials. The knowledge and experience gained inanalyzing the reflectance spectra of these samples are essential to understand the complicated spectrum of a multilayer structure. The optical constants obtained in analyzingbulk materials were used to simulate the reflectance spectrum of multilayer structures.We will introduce the measurement results in the order of increasing complexity of thesample structure. All experiments were conducted at room temperature.4.1 Reflectance of Bulk Material4.1.1 Pure GaAsFigure 4.1 (a) is the far-infrared reflectance spectrum of sample #1 which is a semi-insulating GaAs chip with polished parallel surfaces. The main features of the spectrumare a phonon reststrahlen band near 270cm1 and interference fringes. The latter aredue to the interference of the light reflected from the top and the bottom surfaces ofthe chip and therefore can be used to estimate the thickness of the sample. Because ofthe strong dispersion effect near the reststrahlen band, the separation between closestfringes is not a constant, e.g. Lv 2.685cm1 in the range (50cm1—lOOcm’) and2.474cm’ in the range (380cm1— 340cm1). The thicknesses calculated from41Chapter 4. Experiment Results and Analysis 4210080>. 60H40200>H>HL)404...300WAVENUMBER (cm-i)Figure 4.1: (a) Far-infrared reflectance spectrum of a semi-insulating GaAs substratewith polished parallel surfaces. (b) Calculated spectrum of a pure GaAs substrate withthickness d The parameters listed on the graph are Lorentz oscillator parameters used for calculating the spectrum.(see text)0 100 200100(b)300 400 5005060GaAs Sub. Parameterswo = 269 cmTr=3cm—’= 360 cm1I2000 100 200 400 500Chapter 4. Experiment Results and Analysis 43the two different separations using a simple formula d = 1/(2nLv) with the refractiveindex n constant are d = 545gm and d = 59Om respectively, which are larger than themeasured thickness 51Oim. This is because the simple formula did not not take thedispersion of n into account. We derived a formula, which includes a dispersion term, inAppendix A to improve the accuracy of the thickness estimate. The phonon reststrahlenband can be very well fitted by an oscillator model of the form:e(w) =——(4.1)where e(w) is the dielectric function, €, is the high frequency dielectric constant, w0 isthe resonance frequency, 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 methodintroduced in Chapter 3. We can see that the calculated reflectance fits the measuredspectrum very well.Sample #1 is in fact a semi-insulating substrate used for molecular beam epitaxy withcarrier concentration 1.65 x 107cm3.Another commonly used sllbstrate is conductingsubstrate with carrier concentration 2 x 108cm3.4.1.2 Doped GaAsThe far-infrared reflectance spectrum of a conducting substrate (sample #2 ) from Ocm’to 500cm1 is shown in Figure 4.2 (a). As expected, the main features are a reststrahlenband and a plasma edge. Because of the high reflectance in this region, the intensityof light reflected from the top sample surface is much larger than that from the bottomsurface and therefore no efficient interference occurs and no fringes are observed. Thereststrahlen band in Figure 4.2 appears different from that in Figure 4.1. Instead ofshowing a peak and a dip in the reflectance spectrum as in Figure 4.1, the reststrahlenband of sample #2 causes a hole in the so-called Drude reflectance spectrum. The latter isChapter 4. Experiment Results and Analysis 44>H•1HC)Sample #2(a)0 100 200 300 400 50010080 -_60 -40-200-10080> 60HC)402000 100 200 300 500WAVENUMBER (cm-i)Figure 4.2: (a) Far-infrared reflectance spectrum of a conducting GaAs substrate withfree-carrier concentration n = 1.6— 2.4 x 1O’8cm3,given by the manufacturer. (b)Calculated spectrum of a conducting GaAs substrate with free-carrier concentrationn = 1.45 x 108cm3. The parameters listed on the graph are the parameters forDrude-Lorentz model used for calculating the spectrum.11.1= 1350 cm’y =75cm= 269 cm1= 3 cm1= 360 cm1(b)400Chapter 4. Experiment Results and Analysis 45caused by the plasma formed by the carriers and ion cores. The plasma edge appearing atabout 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 aredifferent:/ ne2= ‘/ * (in rad/sec) (4.2)v e0m(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 wayto determine the carrier concentration is to fit the spectrum using the so-called DrudeLorentz model which is obtained by adding a plasma term into equation 4.1:c2e(w)= — ‘. —. (4.4)- w2 + vyw 2 — + wI’where the second term is the plasma term; w, is the plasma frequency defined by equation4.2;-y is the carrier relaxation time in wavenumber (cm1). Figure 4.2 (b) is the calculatedspectrum of a conducting GaAs substrate, which fits the experiment very well. The carrierconcentration determined by equation 4.2 using the fitting parameter w,, = 1350cm’(wY 405.2cm1)is 1.45 x iO cm3.From the fitting parameters listed on the graph, we can determine other electricalparameters, e.g. conductivity o, mobility etc., as well as the carrier concentration nand relaxation time r (r = in .sec/rad). The D.C. conductivity can be calculatedusing the following formula:D.C. = —i- (1cm) (4.5)607where and ‘y are in cm1. The OD.C. calculated from equation 4.5 is 405(fcm)’ whichagrees very well with the D.C. conductivity obtained from the conductivity spectrum ofChapter 4. Experiment Results and Analysis 46600600 -4002000-0 1O0 200 300 400 500WAVENUMBER (cm-i)Figure 4.3: The calculated conductivity spectrum of the conducting substrate. Theparameters used in the calculation are the same as those in Figure 3.2.Figure 4.3, which was calculated using Eq. 4.6 below.____—= 60(2cm) (4.6)where e” is the imaginary part of the complex dielectric function e(w). The mobility canbe calculated using the following formula:e (4.7)27rc.ym* enThe units for ,u, a-, and n are cm2/volt— .sec, (fcm)1 and cm3 respectively. Themobility of sample #2 calculated from the fitting parameters is 1745.7cm2/volt — secwhich is within the range given by the manufacturer 1650-1992cm2/volt — sec. However,Sample #2• IE 11.1= 1350 cm17 =75cm= 269 cm1P = 3 cm10p = 360 cm1Chapter 4. Experiment Results and Analysis 47both the conductivity and the carrier concentration calculated from the fitting parametersare smaller than those given by the manufacturer, 500 — 625 (1cm) and 1.6 — 2.4 x1018 cm3 respectively.4.1.3 Pure Gai_AlAsAlthough the Gai_A4,As system is technologically one of the most important alloysystems, systematical studies on the optical constants (n and k) over a wide spectralrange and over the Al concentration range of x = 0 to x = 1 with Lx = 0.05 have notbeen reported. Even in the recently published handbook of optical constants [46), we canfind 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 extrapolatethe data to those x values which are not reported in the reference.Ijireference 148], the Qptical constauts im the far-infrared for Ga1AlAs were obtained by fitting the reflectivity data taken at near normal incidence on samples ofthickness ranging from 250tm to 300tm. The Gai_AlAs samples were grown byan 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 x106cm3.The free-carrier plasma edge was not observed in the frequency range investigated (50— 500cm’). The measured far-infrared spectra were assumed to be the same asthose of pure Gai_AlAs. The reflectance spectra of the Ga1_Al, s shown in Figure4.4 were generated using the following dielectric function:1 2 2 \f 2 2( ‘ — iw11 — —1’Y11A’°l2 — W — 1W7122 2 2 2(w1 ——iLc)71)(Wt2— W — ZW7t2)where wj, w, and yj denote TO phonon frequency, LO phonon frequency, TOdamping constant, and LO damping constant respectively and are obtained by fittingthe far-infrared experiments. The best fitting values are listed in table 4.1. The indexH-I>H0Chapter 4. Experiment Results and Analysis 4810010. 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 modesrespectively.Unlike the phonons of GaAs and AlAs, the optical phonons of the Gai_AlAs havetwo modes, a GaAs-like mode and an AlAs-like mode [49] [50], [51]. From Figure 4.4, wesee 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 energiesof the Gai_AlAs material.LO GaAs— like mode: E11 = 36.25— 6.55x + 1.79x2 (meV) (4.9)200 300WAVENUMBER (cm-i)- 400 500LO AlAs — like mode: E12 = 44.63 + 8.78x — 3.32x2 (meV) (4.10)Chapter 4. Experiment Results and Analysis 49Table 4.1: The best fitting parameters for Ga1_AlAs. ii and in cm.X l”i l”li 7ti 711 Vt2 i2 7t2 712 c0 268.8 292.8 2.65 2.85 0 0 0 0 10.90.14 267.1 285.7 5.67 4.85 358.8 369.0 10.56 11.31 10.570.18 266.9 283.4 8.76 4.24 360.1 372.4 12.20 10.24 10.470.30 265.2 278.3 8.64 6.15 360.2 379.9 12.10 9.42 10.160.36 264.5 276.5 10.69 5.58 360.4 381.3 12.23 8.08 10.040.44 262.9 273.7 10.05 6.44 360.2 385.4 9.55 7.90 9.840.54 261.8 269.8 12.43 7.97 361.5 390.1 8.75 8.68 9.601.00 0 0 0 0 361.8 400.0 8.00 8.68 8.20TO GaAs — like mode : E1 = 33.29 — 0.64x — 1.16x2 (meV) (4.11)TO AlAs — like mode : E2 = 44.63 + 0.55x — 0.30x2 (meV) (4.12)The formulae for the TM phonons will be used to determine the Al concentration after obtaining 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 concentrationthan are the TO phonon energies. Figure 4.5 shows the phonon energies as functions ofthe Al concentration x.GaAs and AlAs are the two limiting cases of Gai_A4As where the Al concentration0 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 describethe dielectric function of GaAs and AlAs. The parameters for GaAs and AlAs are alsolisted in Table 4.1.To our knowledge, the far-infrared optical constants of highly conducting Ga1_AlAsare not available. Eq. (4.8), however, can be used to calculate the reflectance spectrumof a highly-doped Ga1_AlAs by adding a Drude-term to it as introduced in section4.1.2.Chapter 4. Experiment Results and Analysis 50450--430-410—-•-•-a 390370:...--350 — -r 3303102500:20406081Al Concentration xFigure 4.5: The TO and LU phonon energies of Gai_AlAs are functions of the Alconcentration x.The LU phonon energies E11 and E12 are more sensitive to the change ofx than the TO phonon energies E1 and E2.4.2 Reflectance of Multiple Layer StructuresIn section (4.1), we introduced experiments and calculations on the far-infrared reflectance spectra of bulk GaAs, AlAs, and Gai_AlAs materials and obtained thedielectric functions describing these materials in the far-infrared region. In this section, we will introduce the reflectance spectra of two multilayer samples made of thesematerials and use the dielectric functions obtained in the previous sections to analyzethe spectra. Structural and electrical parameters for each layer, e.g. layer thickness,free-carrier concentration, and composition concentration, were obtained through fittingthe experiments.Chapter 4. Experiment Results and Analysis 514.2.1 Reflectance of GaAs/AlAs multilayer structureFar-infrared MeasurementA sketch of sample #3 which consists of a Si-doped GaAs layer and an undoped AlAsmarker sitting on a semi-insulating substrate is shown in the upper right corner of Figure4.6 (b). It was grown by Molecular Beam Epitaxy system in Dr. T. Tiedje’s group. Thereflectance spectrum is shown in Figure 4.6 (a). The main features in the spectrum area GaAs reststrahlen band at about 270 cm1, an AlAs reststrahlen band at about 360cm1 , and a free-carrier plasma Drude response at low wavenumbers. The spikes under80 cm’ are mainly noise. The width of the GaAs reststrahlen peak is about the same asthat shown in Figure 4.1 for bulk GaAs. However, the AlAs reststrahlen band associatedwith the thin AlAs marker is much sharper than its GaAs counterpart. This is probablybecause the GaAs reststrahlen band shown in Figure 4.6 is the contribution of both thetop doped GaAs layer and the semi-insulating GaAs substrate. The total thickness ofthese two layers are about 643tm which is much larger than the thickness of the AlAsmarker of 0.14tm.Figure 4.6 (b) is the calculated spectrum of sample #3, which is obtained using thedielectric functions in section 4.1 and the impedance method in Chapter 3. The structuraland electrical parameters for each layer were obtained. Table 4.2 lists the best fittingparameters for each layer of sample #3. The electrical parameters of the top GaAs layerare 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 andoccur only in a p-polarization measurement were not observed in our spectrum measuredat near normal incidence. According to reference [16], the two Berreman modes B1and B2 of sample #3 are at 130cm and 290cm respectively. B2 coincides with thefrequency of the LO phonon mode. Neither were observed.Chapter 4. Experiment Results and Analysis 52>H>H12>HHC)0 100 200 300 400 50010080_6040200100806040200Figure 4.6: (a) The far-infrared reflectance spectrum of sample #3 measured with resolution 4cm1. (b) The calculated far-infrared spectrum of sample #3 with the top GaAslayer d1 = 3pm and AlAs marker d2 = O.l4pm. Table 3.2 lists other parameters used inthe calculation.0 100 200 300 400 500WAVENUMBER (cm-i)Chapter 4. Experiment Results and Analysis 53Table 4.2: The best fitting parameters for each layer of sample #3.Layers Drude-term Lorentz-term Thicknesswp (cm—’) y (cm’) e w0 (cm’) T (cm’) (cm’) d (sm)top GaAs layer 450 45 11.1 269 3 360 3AlAs marker / / 8.2 362 8 500 0.14SI-GaAs Sub. / / 11.1 269 3 360 643Table 4.3: The electrical parameters of the top GaAs layer of sample #3, calculated fromthe plasma frequency = 450cm1 and free-carrier relaxation time-y = 45cm’.We know that the position of the free-carrier plasma edge in a reflectance spectrumis determined by the free-carrier concentration. The plasma edge shifts towards smallerwavenumber as the free-carrier concentration decreases (Eq.4.2). All solids with thesame free-carrier concentration and effective mass have the same plasma frequency, buttheir plasma-edge-frequencies, which are related to the refractive index of each solid anddefined by Eq. 4.3, may be different and therefore appear in different places in theirreflectance spectra. The slop of the plasma edge is determined by the free-carrier relaxation time as shown in Figure 4.7. The plasma edge becomes steeper as the relaxationtime 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 GaAsreststrahlen peak at about 250 cm1 is also caused by the AlAs marker. Figure 4.7showing a GaAs reststrahlen peak without the shoulder was calculated from a sampleChapter 4. Experiment Results and Analysis 54Calculated Reflectivity of Si—doped GaAs on Pure GaAs SubstrateThin film thickness d 3 jm10.8> 0.6I—C)bJ-JLi. fl 4LI-i0.200 100 200 300 400 500WAVENUMOER (cm-1)Figure 4.7: The calculated far-infrared spectra of a sample with almost the same structureas sample #3 except fo not having the AlAs marker. The only difference between thedotted 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 Figure4.6. The reasons for this are that the bottom substrate surface of sample #3 was notpolished and the resolution used in the measurement was 4 cm1 which is larger thanthe fringe separation of 2.29cm’. However, small fringes of the substrate were observedwhen the resolution was increased to 0.5cm’. These may be seen in the spectral rangefrom 600 cm’ to 1000 cm1 ill Figure 4.8. The thickness of the substrate, calculatedfrom the fringes, is 642.5um which agrees very well with the digital caliper measurement643gm. The accuracy of the digital caliper is +1im. The detailed calculation methodwill be discussed in the Appendix A. The large fringes in Fig. 4.8 (a) are the resultsof the interference in the two top thin layers. These fringes can also be seen in thenear-infrared region. We will fit the near-infrared spectrum to determine the thicknessChapter 4. Experiment Results and Analysis 5510080I,>C) 40200500 55006048362412-0600 620 640 660 680 700Wavenumber (cm—i)Figure 4.8: (a) Mid-infrared measurement of sample #3 with resolution O.5cm’. Theinterference fringes of both the substrate and the top layer are seen. (b) The fringes ofthe substrate are shown in detail.1500 2500 3500 4500(b) II IVVVVVJWAMMAMWf\MMNvwAAJv\fringes of the substrateI • I I IChapter 4. Experiment Results and Analysis 56of the two layers instead of fitting the spectrum in the mid-infrared region, because therefractive index n and extinction function k in the near-infrared region can be obtainedmore accurately by using an empirical formula, which will be introduced in the followingsection.Near-infrared MeasurementIn order to determine the thickness of the two epitaxial layers in sample #3 more accurately, 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 interference fringes resulting from the top GaAs layer. The separation of the interference fringesresulting from the AlAs marker is about 12000cm’ which is larger than the spectralrange measured and therefore is difficult to be noticed. However, the AlAs marker dohave 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 topGaAs layer determined the fringe separation while the thickness of the AlAs marker determined the fringe envelope. The damping of the fringe amplitudes in the spectral rangefrom 9O00cm to 11400cm is not caused by the increasing absorption near the GaAsband gap (1.42eV or 11452cm)but by the interference of the AlAs layer. Figure 4.10shows the calculated reflectance spectra from 5500cm1 to 11500cm’ with the same toplayer thickness but different thickness of AlAs marker. It is obvious that the fringes resulting from the top layer are modulated by the much larger AlAs fringes. The dielectricfunction introduced in section 4.1 cannot be used in the calculation of the near-infraredreflectance spectrum, since the dispersion in this region is caused by the energy gap. Theextinction coefficient in the wavenumber range under the band gap is so small that it canactually be taken as zero in the reflectance calculation. In this region, the reflectance ismainly determined by the refractive index which can be calculated using an empiricalChapter 4. Experiment Results and Analysis 57H>HC)>HHC)5040302010000 10000 11000 12000 13000 14000 15000504030201009000 10000 11000 12000 13000 14000 15000WAVENUMBER (cm-i)Figure 4.9: (a) Near-infrared reflectance spectrum of sample #3. The fringes were resulted from the top GaAs layer. The damping of the fringe amplitudes under the GaAsband gap is caused by the thin AlAs marker. (b) The calculated near-infrared spectrumof sample #3 with the top layer thickness d1 = 3.O3um and the AlAs marker thicknessd2 = O.14,um.Chapter 4. Experiment Results and Analysis 58C- -————---r——-—— i—d2 O.1m.scI —-_________——d2 = O.O5imr r CJWJVV \/ A/ \!Doped GaAo 3mAlAs MeekerSI-GaAs Sabatrato.00 I— I .05500.0 11500.0Wavenumber (cm’) 30.0 Wavenumber (cm’)Wavenumber (cm—’) Wavenumber (cm1)Figure 4.10: The calculated near-infrared spectrum of a sample similar to sample #3 butwith a variable thickness d2 of the AlAs marker.Chapter 4. Experiment Results and Analysis 59formula. Afromowitz [52] introduced a semi-empirical method for calculating the roomtemperature refractive index of Gai_AlAs at energies below the direct band gap.— 1 = M_, + M.3E2+ 1-E x ln[(E —E2)/(E — E2)] (4.13)where M_, = —(E — E), (4.14)= -(E — E), (4.15)Ef = (2E—E), (4.16)ir Edand ri= E E2 (4.17)o r)E is the photon energy in eV and E0, Ed, E are functions of Al concentration x andare determined by following formulas.Ga,_,AlAs : E0 = 3.65 -f- 0.871x + 0.179x2 (4.18)Ed = 36.1 — 2.45x (4.19)E = 1.424 + 1.266x + 0.26x2 (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 from0.895tm (11173.2cm’) to 1.7um (5882.3cm’). The fit to the data for AlAs is within0.004 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 thesame layer thicknesses as found in the far-infrared fitting. However, these values are moreaccurate 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 structureSample #4 with a Gal_TA1XAS/GaAs/A1A5 multilayer structure as shown in Figure 4.11was also grown by molecular beam epitaxy. All three layers are undoped. The thicknessesChapter 4. Experiment Results and Analysis 60Gai—A1As T2.51mGaAs I1.8,LrnAlAs 0.l1umGaAs SubstrateFigure 4.11: The geometric structure of sample #4. The thicknesses shown on the graphare nominal values.on Figure 4.11 are nominal values calculated according to the growth conditions. Thefar-infrared reflectance spectrum of sample #4 from 0 to 500cm1with resolution 2cm1is shown in Figure 4.12 (a). The main features of the spectrum are reststrahlen bandsof 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 reststrahlenband of the GaAs-like mode. The peak at about 370cm1 is mainly the contribution ofthe GaiAlAs reststrahlen band of the AlAs-like mode. The reflectance spectrum of athree-layer structure can be calculated using the same method as in section 4.2.1 and thebest fit is shown in Figure 4.12 (b). The dependence of the spectrum on Al concentrationx is illustrated in Figure 4.13, here it is calculated for four x values. The spectra showthat the wavenumbers of the three peaks change only slightly as the Al concentrationincreases 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>HHr:ilE>H•1>Hcz610 100 200 300 400 500100806040200100806040200500Figure 4.12: (a) The far-infrared reflectance spectrum of sample #4. (b) The calculatedfar-infrared reflectance spectrum of sample #4100 200 300 400WAVENUMBER (cm-i)Chapter 4. Experiment Results and Analysis 62Figure 4.13: The calculated spectra of four structures similar to sample #4 but withdifferent Al concentration for the top Gai_AlAs layer. The thickness of each layer areshown at the upper left corner.AlAs D.lPmGA. Jl.8m5pm= 0.3T #1#3Gaae(a).0cm1 500.0Chapter 4. Experiment Results and Analysis 63with x = 0.66. Table 4.4 shows the best fitting parameters for the top Gai_rAlAs layerof sample #4. The LO phonon energies of the GaAs-like mode and the A1As-ilke modeTable 4.4: The best fitting parameters for the top GaiAlAs layer of sample #4. vand ‘y in cm, d1 in ,um.d1 ii.-yn. 711 Vt2 Vt2 7t2 712 cr2.78 259.5. 263.8 12.54 8.12 361.7 395 8.4 8.68 9.17are vj1 263.8cm’ and Vj2 = 395cm’ respectively. The Al concentrations calculatedfrom Eq. 4.9 and Eq. 4.10 on page 48 agree with the nominal value x 0.66 very well,x = 0.66 and x 0.66 respectively. The error on x is estimated to be 0.05 which isobtained 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 reflectance spectrum from 500cm’ to 5500cm with resolution 0.5cm. This is shown inFigure 4.14 (a). The interference fringes resulting from the substrate were observed in theregion from 500cm1 up to about 1300 cm showing that the sample surfaces are fairlysmooth and parallel. The substrate thickness calculated from the fringes is 652.8umwhichagrees very well with the digital caliper measurement 654 +l1urn. The fringes resultingfrom the two top layers were also observed. Figure 4.15 is the best fit of the mid-infraredspectrum. It fits the experiment well, except for the range 5000—5500cm where themeasured spectrum was deformed because of exceeding the measurement range of thedetector(MCT). For clarity, the fitting did not take the substrate interference into account. The thicknesses obtained from the fitting agree with the nominal values reasonablywell except for the AlAs marker (0.O3itm compared with 0.lpm). All the values of thethicknesses shown in Fig. 4.15 are accurate to 0.01jm. Because of different thicknessesof the the two top layers, the interference fringes from the two layers canceled out in theChapter 4. Experiment Results and Analysis 64near infrared region as shown in Figure 4.16. We will not fit the spectrum in this region.Chapter 4. Experiment Results and Analysis 6501500 2500 3500 4500 550060__453015050060— 45j30150600 700Wavenumber (em*1)Figure 4.14: (a) The mid-infrared spectrum of sample #4. The fringes of the substrateand the top two layers were observed. (b) The fringes of the substrate are shown indetail.620 640 660 680Chapter 4. Experiment Results and AnalysisI:60483624120WAVENUMBER (cm—’)66Figure 4.15: The best fitting of the mid-infrared measurement of sample #4. The thicknesses shown outhegraph are the best fitting values.i3. c4t3ØØ 12Ø3ø 11ØØVENUME CM-IløeøsFigure 4.16: (a) The near-infrared spectrum of sample #4. The fringes of the top twolayers are canceled out due to different layer thickness.1000 2000 3000 4000 5000•1 I I—— ISample #4Chapter 5ConclusionIn conclusion we have measured the infrared reflectance spectra of GaAs/Gai_AlAsmultilayer structures from the far-infrared to the near-infrared using a Bruker IF 113vspectrometer. The main structures in the spectra are due to the phonon reststrahlenbands, the Fabry-Perot interference and the plasma edge of the free carriers. Throughfitting the spectra, we determined both electrical and structural parameters. Theseinclude the carrier concentration n, mobility t, conductivity u, Al concentration x, andthe thickness d of each layer. The agreement between the theoretical spectra and theexperiments is excellent.Comparing with a recent publication [16] on the same topic, our multilayer samplesconsisting of GaAs/AlAs/Gai_Alc,As materials are more complicated than theirs whichwere made of GaAs layers with different doping levels. The optical impedance method weused to fit our normal incident experiments is also different from theirs where a matrixmethod was used to fit their oblique incident experiments. We also determine the Alconcentration x which did not exist in their samples.The limit of this method to determine the free carrier concentration is determinedby the measurable wavenumber range of the Fourier spectrometer. The plasma frequency will be too small to be determined accurately if the concentration is smaller than2 x 1O’5cm3.The plasma frequency of a n-type GaAs with 2 x 105cm3carrier concentration is about 15 cm’ which is close to the limit of the Bruker IFS 113v spectrometerwe used. Although the samples we used were grown on semi-insulating substrates, this67Chapter 5. Conclusion 68method 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 excellent alternative to many existing conventional methods for characterization of electricaland structural parameters. It has the advantage of being non-destructive, non-contact,and being able to characterize both electrical and structural parameters, which are usually determined by several different methods. Another advantage of this method is that ithas no specific structural requirement for the sample as many other conventional methodshave. Since both electrical and structural parameters can be determined by using onlyone sample, IRS method may change the conventional way of calibrating the thin filmpreparing systems. Instead of growing several single layer samples and using differentmeasuring methods to measure them, now IRS method makes it possible to grow onlyone sample with multilayer structure and to calibrate several, if not all, parameters inevery single layer at the sarne time. This will save time jn the measurement and reducethe number of samples needed to calibrate a thin film growing system.Appendix AInterference FringesThe 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 simpleformula.(A.l)where zz’ is in wavenumber cm’ and n is the refractive index. However, the thicknesscalculated from Eq. A.l is often larger than the real value, especially for the fringes nearthe phonon reststrahlen band or the direct band gap where the dispersion of n is quitelarge. 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 thicknessfrom unevenly separated fringes.The refractive index of a dispersive material is a function of wavenumber ii and canbe written as n(v). The wavenumbers of the (m +1) th and m th Fabry—Perot peaks canbe written as:m+l= (A.2)2dfl(Ym+i)= 2dfl(Vm) (A.3)The wavenumber separation between two adjacent fringes is1 m m= ‘rn+1 urn = +— (A.4)2dfl(m+i) 2dfl(m+i) 2d(m)69Appendix A. Interference Fringes 70Substitute for = iim(m) from Eq. A.3 in Eq. A.4:m1 =Vmfl(Vm)(__1— 12d(m+i) fl(Zim) n(v,+1)1— ‘m rn+l (A.5)2d fl(Vm+i) fl(i’m+i)where n+1—72(11m) and is usually larger then zero. Eq. A.5 is the same asEq. A.l except that there is one more term in Eq. A.5. The second term in Eq. A.5 is adispersion term. As zmrn+l is usually larger than zero, the fringe separation calculatedfrom Eq. A.5 is smaller than that calculated from Eq. A.1. This is why the separations ofthose fringes near the band gap or the reststrahlen band, where Zri+l is very large, aresmaller than those far away from these bands. For example, in Fig. 3.9, the separationaround 11000cm, which is near the GaAs direct band gap 11452cm’, is 312cm’,while the separation around 9500cm1 is 395cm1. Also, for the same dispersion, theseparationsof the near-infrared fringes are smaller than those of the far-infrared onesbecause of the i’m in the dispersion term.The substrate thicknesses of sample #3 calculated from Eq. A.5 and Eq. A.1 are642.5km and 671.1pm respectively. The former agrees with a digital caliper measurementvery well 643 ± lm. The dispersion ZXn1 used in the calculation was obtained fromreference [47].Bibliography[1] Y. J. Yang, T. G. Dziura, R. Fernandez, S. C. Wang, G. Du and S. Wang; Appi.Phys. Lett.58, 1780 (1991).[2] T. Sogawa, and Y. Arakawa; Appi. Phys. Lett. 58, 1709 (1991).[3] G. Hasnain, B. F. Levne, D. L. Sivco, and A. Y. Cho; Appi. 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