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Smoothing of patterned gallium arsenide surfaces during epitaxial growth Ballestad, Anders 1998

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S M O O T H I N G O F P A T T E R N E D G A L L I U M A R S E N I D E S U R F A C E S D U R I N G E P I T A X I A L G R O W T H B y Anders Ballestad Bachelor of Appl ied Science A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A 1998 © Anders Ballestad, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics and Astronomy The University of Br i t i sh Columbia 2075 Wesbrook Place Vancouver, Canada V 6 T 1W5 Date: Abstract Control over the surface structure of semiconductor films during growth is crit ical for devices of recent technological importance. Typical ly the length scales of interest range from nanometers to micrometers. Examples include the size and spacing of quantum dots in quantum dot lasers, and the pitch and amplitude of grating structures for distributed Bragg reflectors. Elastic light scattering has atomic height sensitivity to this surface structure, on lateral length scales as low as half the incident wavelength, and is easily implemented for in-situ monitoring during film growth [1]. For the smooth surfaces of interest here, the distribution of the scattered light intensity as a function of scattering angle directly maps out the power spectral density (PSD) . The P S D gives the 'root mean square' roughness of the surface structure as a function of inverse length scale, or spatial frequency. Here we present in-situ light scattering measurements performed during III-V semi-conductor film growth by molecular beam epitaxy ( M B E ) . We have used the technique to monitor the smoothing of one-dimensional grating structures during regrowth. For the regrowth experiments, the grating pitch was chosen such that the detection angle of the in-situ measurement coincided with the scattering peak associated with a harmonic of the grating periodicity. Because the ini t ia l shape of the patterned surface is known, it is possible to reconstruct the shape of the grating from the P S D as it evolves in time during growth. We find that for homoepitaxy of gall ium arsenide (GaAs) on textured substrates, the time evolution follows the Kardar-Parisi-Zhang ( K P Z ) model [2]. i i Table of Contents Abstract ii List of Tables v List of Figures vi Acknowledgement ix 1 Introduction 1 1.1 Evolut ion of the Semiconductor Interface 2 2 Surface Growth Modelling 8 2.1 Discrete Growth Models 8 2.1.1 Scaling Behaviour of Large Surface Structures 11 2.2 Lateral Growth 13 2.3 Continuous Growth Models 15 2.3.1 Deriving Cont inuum Equations From Symmetry Arguments . . . 16 2.3.2 First Principles Derivation of Continuum Models 22 2.3.3 Mass Conservation and the Kardar-Parisi-Zhang Model 25 3 Elastic Laser Light Scattering 27 3.1 Diffuse Scattering 27 3.2 Scattering From Fla t Surfaces 30 3.3 Scattering From Textured Surfaces 31 i i i 3.4 Mapping the Surface by Moni tor ing its Fourier Coefficients 32 4 Grating Fabrication 35 4.1 Fabrication Steps 35 4.2 Details on the Lithography Process 38 4.3 Wet Etching (100) GaAs 40 5 Experiment 42 5.1 Experimental Setup 42 5.2 Cleaning the Sample 43 5.3 Annealing Pr ior to Regrowth 45 5.4 Regrowth on GaAs Gratings 47 5.5 Discussion 49 6 Conclusions 54 List of Notation 57 Bibliography 60 i v List of Tables 4.1 Parameters for preparation of photoresist films, pattern transfer and strip-ping of photoresist. . . 37 4.2 Parameters for electron beam lithography in the scanning electron micro-scope 38 6.1 Parameters obtained by fitting the Kardar-Parisi-Zhang equation to ex-perimental data 55 v L i s t o f F i g u r e s 1.1 Cross-section T E M micrograph of AIN and NbN multilayer film. (After Mi l le r [3]) 3 1.2 Evolut ion of the interface width by random deposition according to bal-listic deposition on an ini t ial ly flat, one-dimensional surface 6 2.1 Growth rule for ballistic deposition (after Barabasi and Stanley [4]). . . 10 2.2 Simulation of growth by ballistic deposition on grating of pitch 786nm. . 11 2.3 Smoothing of grating structures of different heights according to the bal-listic deposition model. L=128 lattice sites 12 2.4 The first two terms of a Fourier cosine-decomposition of two ini t ial surface shapes 14 2.5 Temporal evolution of the first and second Fourier coefficients of the de-composed grating during ballistic deposition 14 2.6 Decay of two similar shapes with different widths according to the Edwards-Wilk inson model 18 2.7 Diagram showing one possible origin for the slope dependent term in the Kardar-Parisi-Zhang growth equation [2] 19 2.8 Temporal evolution of a grating according to the Kardar-Parisi-Zhang equation 20 2.9 Temporal evolution of the first and second coefficients of the Fourier de-composed grating during growth according to the Kardar-Parisi-Zhang equation 21 v i 3.1 Diagram showing geometry of laser light scattering (LLS) 28 3.2 Simulated surface factor at 1 6 / x m - 1 according to the Fraunhofer far field approximation 29 3.3 Fraunhofer diffraction from a coherently i l luminated random-walk gener-ated rough surface wi th standard deviation lnm (top) and 25nm (bottom). 31 3.4 Fraunhofer diffraction from sinusoidally textured surface of pitch 786nm and 125nm amplitude including random noise 32 3.5 A F M images of gratings before (top) and after (bottom) regrowth . . . . 33 4.1 American standard GaAs (100) wafer with crystal orientations indicated [5, 6] 36 4.2 SEM-image showing the selective growth around the beam dump area of the exposed pattern 39 4.3 S E M image of gratings on (100) GaAs 40 4.4 Simulation of etched surface for relative etchrate ratio of (100) to 111 of a) 2.1 and b) 1.5 41 5.1 Angular dependence of Fraunhofer diffraction from a sinusoidally textured surface of pitch 786nm 42 5.2 SEM-image of gratings after unsuccessful regrowth , 43 5.3 R H E E D patterns from (100) GaAs 44 5.4 Annealing of gratings before growth showing temperature dependence of smoothing parameter 45 5.5 Determining the activation energy of the Edwards-Wilkinson u parameter for GaAs 46 5.6 F i t of Kardar-Paris i-Zhang equation to experimental data: T = 6 0 0 ° C (top, run 1) and T = 592°C (bottom, run 2) 48 v i i 5.7 A F M linescan of 1.3^m gratings from run 1 (top left); top view (top right); 3 D view (bottom) 49 5.8 A F M linescan of 786nm gratings from run 2 (top left); top view (top right); 3 D view (bottom) 50 5.9 Growth of GaAs on GaAs gratings with AlGaAs spacer-layers 52 v i i i A c k n o w l e d g e m e n t A great big thank you goes out to my good friends in the M B E - l a b at U B C , where many rewarding hours have been spent. Thank you, Mar t in , for helping me out with the M B E - r u n s . Hope you learned something, too! Likewise to Tom P. for practical help and good advice along the way. Naturally, none of this would be possible without our very inspiring supervisor, Tom Tiedje. No other person with so many reponsibilites has quite so much time to discuss physics and general affairs as Tom. Thanks for giving me this opportunity! Thanks to J im, for fixing what we break. Other people I appreciate who have added to my life and work in some way: Drs. Johnson, Patitsas, Lavoie, Levy, Beaudoin, Young, Kanskar, Watson and Bergersen; Sayuri, A l , Peng, Dave, Francois, Alex , Vighen, Sune, L ig i a and Dr . Ogryzlo; Ron and Mary A n n , the guys in the machine shops, the staff and members of this department as well as Drs. Ward and Wetton of Mathematics. Personal thanks to Marianne for understanding and being great, my family and friends for endless support, and finally a wish for good health for the dog Lucas. ix Chapter 1 Introduction Semiconductor devices are rapidly becoming smaller and working single electron devices have been reported in experimental settings [7]. W i t h diminishing device dimensions come the demand for more sophisticated nanofabrication processes, along wi th a better understanding of structure evolution during device fabrication. Similar concerns arise during growth of th in films on top of structured surfaces. The intent of growth may be to grow a thin 'cap' or spacer layer without changing the interface shape, or it may be to cover the surface and create a flat interface over the pre-defined shape. We seek to describe the temporal evolution of a patterned semiconductor surface ' during annealing and epitaxial growth. The surface is described by an in i t ia l height distribution, prefabricated or not, and the growth is sensitive to the settings of parameters such as growth rate, system temperature, ambient pressure, growth technique, and so forth. Epi tax ia l growth is the process in which individual atoms are deposited from heated source cells and condense on the growing interface. The deposited particles may stick immediately upon impact, diffuse along the surface, or re-evaporate from the surface. Interface evolution can be simulated on a computer by implementing discrete growth rules and randomly depositing particles on the surface. This describes the growth on an atomistic level, and is valid for short and long time behaviour. Another approach is to try and capture the essential mechanisms of epitaxial growth by various terms in a continuum equation. This approach is l imited to the description of large scale and long time behaviour, however, it is also an elegant approach that greatly simplifies the 1 Chapter 1. Introduction 2 analysis of interface growth. Apar t from the technological importance that the understanding of the growing semi-conductor interface brings us, the growth of solids is of great fundamental interest since it enlightens the behaviour of complex systems in nature. It has been shown that many growth models produce a spatial scale invariance in the surface structure, which can be described by the language of fractal geometry, introduced by Mandelbrot in 1982 [8]. It has been remarked that the observations of Mandelbrot raises fundamental questions as to what the mechanisms are that accounts for this scale invariant behaviour. Interface evolution has been investigated extensively, but mostly theoretically and in simulations. We feel that the experimental part of this investigation has been less emphasized, and believe that we have experimental results indicating non-linear gradient dependent growth terms during growth of nanofabricated structures. Our experimental results verify the Kardar-Parisi-Zhang ( K P Z ) growth equation [2] for length scales longer than 400nm for growth of GaAs on GaAs. 1.1 Evolution of the Semiconductor Interface For an ini t ial ly flat surface at low temperature, most atoms on the surface wi l l remain locked in position, not having enough kinetic energy to break the bonds they form with their nearest neighbors. Upon raising the temperature of the system, the surface atoms wi l l have a higher probability of breaking loose. This diffusion follows an Arrhenius be-haviour where each atom on the surface hops according to a rate of R = Roexp(—Ea/kT). Ea is referred to as the site dependent activation energy of the surface atoms, which can be modelled by Ea = EQ + nE^, where E0 is the activation energy of a free atom with no bonds, Ef, is the binding energy per bond and n is the number of nearest neighbors that the atom has in its ini t ia l site. RQ is an 'attempt frequency' which is approximately Chapter 1. Introduction 3 Figure 1.1: Cross-section T E M micrograph of AIN and M>Af multilayer film. (After Mil ler etal. [3]). equal to a lattice vibration frequency [9]. The diffusing atoms wi l l move around on the surface unti l they settle in the nearest local energy minimum. This is a dynamical pro-cess, in which an overall equilibrium situation wi l l be reached with an average roughness dictated by a trade-off between energy and entropy in the system [2, 10, 11, 12, 13]. The relaxation processes and the random noise compete in smoothing and roughening the surface. The diffusion of surface atoms leads to the formation of mounds, islands and steps [14, 15]. Figure 1.1 shows how the interface width is gradually increasing as more and more layers of atoms are deposited. Growth starts out on an 'epi-ready' (flat) wafer, and alternating films of AIN and NbN are deposited [3]. Sharp, short scale roughness early in the growth is soon overtaken by large scale mounds. It is interesting to notice that the surface in this figure grows in the fashion of the Fresnel-Huygen's principle, where a Chapter 1. Introduction 4 sharp irregularity in the surface grows in a radial way to form large, round shapes. In other words, the lower spatial frequencies survive the larger ones as the surface grows [1, 15, 16]. The evolution of the surface roughness is dependent on spatial frequency, or the size of the irregularities on the surface: the small features in figure 1.1 smooth away whereas the large features grow. O n the other hand, when heating a surface with an ini t ia l roughness higher than the equil ibrium state, the processes described above wi l l tend to smooth the surface from the rougher to a smoother state, thus approaching the min imum free energy state from a higher energy configuration. We define the interface width, W, by W2 = ^ ( h i - h ) 2 (1.1) where h is the average height of the surface and hi is the height of the ith site in a discrete lattice model of size L. This quantity gives us a statistical measurement of the surface roughness. Many studies of thin film growth are based on the dynamic scaling behaviour of the surface statistics that depends on the various surface relaxation and roughening processes [17, 18]. B y plotting the interface width as a function of time, two distinct regions are observed: at times less than some 'crossover' time tx, W increases as a power of time [4]: W(L,t)~tfi (1.2) 8 is called the growth exponent, and characterizes the time-dependence of the growth dynamics. This interface width continues to increase unti l a certain value is reached: W saturates according to: Wsat(L)~La (1.3) where a is called the roughness exponent. The crossover time depends on the system size Chapter 1. Introduction 5 as: tx ~LZ [i ~ tx] (1.4) where z is called the dynamic exponent. The three parameters, a, j3 and z are dependent on one another, and they can be extracted from a given system by normalizing W and t [17]. We obtain the relation: " M - f f f ) (1.5) W,at(L) \t, Substituting in from above we get that for an ini t ia l ly smooth surface, the interface width scales according to the so called Family-Vicsek scaling relation [4, 17, 19]: W(L,t) = Laf(t/La^) (1.6a) f(x < 1) ~ x0 (1.6b) f(x 3> 1) —> constant (1.6c) B y plott ing this relationship, we see that systems of different sizes L w i l l collapse onto one curve. Solving for z around t = tx we get: cx Z = H ( 1 - 7 ) This scaling law links the three exponents, and is valid for any growth system that follows the Family-Vicsek scaling relation. A simulation of a discrete growth rule applied to a surface is shown in figure 1.2. The simulation shows the interface width of a growing surface according to the ballistic deposition (BD) rule on an ini t ial ly flat, one-dimensional surface. This model wi l l be discussed in detail in chapter 2. The saturation of the interface width is related to lateral correlations in the system. The random deposition adds amplitude to roughness at all spatial frequencies, and only Chapter 1. Introduction 6 Figure 1.2: Evolut ion of the interface width by random deposition according to ballistic deposition on an ini t ia l ly flat, one-dimensional surface. Two simulations are shown. short range diffusion is possible before more atoms are deposited. The interface width grows unti l there is a balance between the random deposition noise and this short range surface relaxation. Many physical processes can be incorporated into growth models, and the scaling pa-rameters are signatures for different growth models. Growth models that exhibit similar scaling exponents are said to belong to the same universality class. The scaling exponents can also be obtained experimentally by for instance ex-situ light scattering, and one can thereby link discrete and continuous models to experiments by comparing these param-eters. The surface scientist tries to quantify the scaling parameters through experiments that measure the roughness on different length scales during or after epitaxial growth. The researcher has a multitude of devices available, with which the semiconductor Chapter 1. Introduction 7 surface can be characterized. Several probing techniques are made possible by these de-vices, including ex-situ characterization by scanning electron microscope ( S E M ) , scanning tunneling microscope ( S T M ) , atomic force microscope ( A F M ) , and in-situ methods like elastic laser light scattering (LLS) , reflective high energy electron diffraction ( R H E E D ) and low electron energy diffraction ( L E E D ) . In this thesis we wi l l use some of these techniques in order to add to the continued investigation of surface physics by describing experiments in which we have monitored the evolution of the semiconductor interface during epitaxial growth. The thesis contains six chapters, where in chapter 2 we describe surface evolution and modelling, in chapter 3 we discuss the theory behind elastic light scattering, and chapter 4 is left for the description of the sample preparation. We complete this thesis in chapters 5 and 6 where the experiments we conducted have been described and summarized with interpretations, conclusions and suggestions for further work. Chapter 2 Surface Growth Modell ing 2.1 Discrete Growth Models A n educational and simple approach is to describe surface growth by discrete rules which can be simulated by a computer. Quite simple models display features that enhance our understanding of the microscopic processes in question. The temporal evolution of the thin film surface morphology is a dynamical process due to the combination of deposition noise, desorption, surface relaxation and diffusive processes1.. Several discrete models have been studied [4, 9, 17, 19, 20, 21] and linked to continuum growth equations through dynamical scaling. The simplest of the discrete growth models is that of random deposition (RD) , where an incoming particle falls at a randomly chosen location on the surface. The interface sites are uncorrelated, i.e. one site does not have any effect on the next, so the interface width grows indefinitely. The R D model has for this reason been said to have an infinite roughness exponent (see equation 1.3). This is an unrealistic scenario for M B E growth, as the interaction of nearby surface atoms wi l l cause interface correlations. The scaling exponents for R D are [4]: a = oo, 3 = 1/2 [RD] (2.1) Growth in the M B E shows that the interface width does indeed saturate, a fact that brings us to believe that the surface is correlated. The system knows when to saturate, as 1ln this work, we discuss one-dimensional surfaces, i.e. the surface height distribution h = h(x,t). 8 Chapter 2. Surface Growth Modelling 9 'information' from nearby surface sites w i l l travel a certain distance called the correlation length of the surface. In a discrete model, this travel of information must be incorporated into the growth rules by means of analyzing neighboring sites before depositing a particle at the randomly chosen site. Surface relaxation is a mechanism for creating lateral correlations on the surface, and can be incorporated by letting the landed atom move a finite distance along the surface depending on whether a neighbouring site has a lower height. Note that a 'lower height' is an unphysical description of M B E growth, expecially since the growth is performed with the sample upside-down in the growth chamber. However, since a 'lower height' often corresponds to a higher number of nearest neighbours (NN) , this is a fair description of surface relaxation, and therefore constitutes a plausible physical description of M B E growth. Several versions of R D wi th surface relaxation (SR) have been investigated [9, 22, 23, 24], differing in the distance the atoms are allowed to travel, preference of higher or highest number of N N , etc. The scaling exponents have been found to be [25]: a = 0.48 ± 0 . 0 2 , R = 0.24 ± 0 . 0 1 [RDwSR] (2.2) A model belonging to a different universality class is the ballistic deposition (BD) model [8, 20, 21, 26] that was introduced in chapter 1. Al though deceptively simple, this model exhibits characteristics that are observed in many growth phenomena, such as interface width saturation and outward lateral growth [4, 17, 27]. The characteristics of the B D model prove to be closely linked to the K P Z equation that I w i l l discuss later. The model is depicted in the diagram of figure 2.1: an incoming particle aiming for site i on the surface sticks at a new height h'(i) depending on the previous height distribution h(i) at that site: h\ — max [/i j_ i , hi + l,hi+1] (2.3) The incoming particle sticks to the first height-level where it has at least one N N . B D Chapter 2. Surface Growth Modelling 10 • • • i • A • • B • T T • A' i • • • B' i Figure 2.1: Growth rule for ballistic deposition (after Barabasi and Stanley [4]). is a model wi th a unity sticking coefficient and no diffusion from thermal activation. This wi l l create a porous structure [28], which is unphysical for M B E growth. B D is therefore a first approximation to a more complete model where the landed particles are allowed to relax or diffuse. Lateral growth can loosely be defined as 'similar growth rates in al l outward directions,' like the flamefront on a burning field. A more detailed discussion of lateral growth wi l l come later. B D growth has been studied extensively [12, 17, 28, 29] and detailed numerical simulations in ID show that the scaling exponents are: a = 0.47 ± 0.02, 3 = 0.330 ± 0 . 0 0 6 [BD] (2.4) The B D model thus belongs to a different universality class than R D . The scaling coeffi-cients obtained for B D turn out to closely match predictions made for the K P Z equation. The ballistic deposition rule has been simulated on a periodic stucture of the shape indicated in figure 2.2. The ini t ia l shape used forms an integral part of the experiments conducted in this work. The actual physical dimensions of this shape is a lateral pitch of 786nm, with a height of about 250nm. To correctly account for these physical dimensions, the atoms in the growth model have been given a size of 786nm/ (number of points in Chapter 2. Surface Growth Modelling 11 Figure 2.2: Simulation of growth by ballistic deposition on grating of pitch 786nm. lattice model). The dimensions of this shape are comparable to state-of-the-art processing limits, and hence semiconductor device dimensions today. The simulation in figure 2.2 indicates lateral growth, where the sloped parts of the surface appear to grow laterally along the surface. We also see that the inversion symmetry in h is broken and that cusps are formed between the bumps. These cusps were also observed in a different system in figure 1.1. The simulation used 300 points across in space, and was wrapped to give periodic boundary conditions. 2.1.1 Scaling Behaviour of Large Surface Structures Interesting effects have been observed when applying the B D model to large surface structures. The scaling relations in equation 1.6 showed us that the interface width should grow as t@ unt i l it reaches a saturation level, Wsat, at a characteristic crossover time, tx. This was verified in simulations shown in figure 1.2. For a surface of ini t ia l roughness larger than this saturation level, the interface width wi l l decrease unti l it Chapter 2. Surface Growth Modelling 12 10' 10 10" 10 10 Initial height (*La): 100 — 30 10 3 flat convergence point 10 10 10 Figure 2.3: Smoothing of grating structures of different heights according to the ballistic deposition model. L=128 lattice sites. reaches the equlibrium state. Some simulations were done to verify this behaviour, see figure 2.3. The in i t ia l shape used was the same as that of figure 2.2, where the ini t ia l height was varied. We found that for the ini t ia l ly rough surface shapes, there are two crossover times: one separating W in the in i t ia l stages where it is fairly constant, from the smoothing phase. There is also the usual crossover time occuring when W reaches the saturation interface level. These times are labelled as tx\ and t x 2 in figure 2.3, respectively. Some observations were made when simulating the smoothing of structures with ver-tical dimensions larger than ~ 3 * La. After the first crossover time, tx\, the interface width was found to drop dramatically. Several large ini t ia l heights were simulated, and the resulting curves were found to converge at a common point, indicated in the figure as the 'convergence point'. We believe that this effect is due to the rapid filling of the Chapter 2. Surface Growth Modelling 13 cusps, as seen in figure 2.2. We have seen that the interface width settles at the satura-tion width, even for man-made surface shapes of large vertical dimensions. However, the pre-saturation phase is not explained by the Family- Viscek scaling relation. 2.2 Lateral Growth It is useful to define 'lateral growth' in a quantitative manner. Let us consider the time evolution of the coefficients of the Fourier series decomposition of the surface shape h(x, t). We decompose the ID periodic surface shape of length L at time t' according to: h(x,t') = a0 ^ /2mix\ , . / 2 w r x \ /(O ( 2 - 5 ) n=l where the coefficients an and bn are defined as follows: 2 /2nnx\ , an = — / h(x,t) cos —-— )dx (2.6a) L J-L/2 \ L J 2 fL/2 /2mrx\ , bn — — J ft(x,t) sm —jdx (2.6b) according to Fourier's theorem, and f(t) is the time evolution of h(x,t). Let us consider the decomposition of two shapes: a narrow and a wide 'bump', similar to the ini t ia l shape used in figure 2.2. When Fourier decomposing these two shapes (letting the mean be zero and shifting the shapes so that they are even around x = 0), we find that for the narrow peak, both a\ and are positive, and a 0 , along with all of the 6-terms, is zero. However, for the wide peak, the a\ is positive, whereas 0,2 is negative. We define lateral growth as the transition in which a 'bump' evolves from being narrow to becoming wide. Since the B D model simulated in the previous section appears to widen such bumps, we would expect that the first two Fourier coefficients would evolve as an extrapolation of the shapes in figure 2.4: ax would die away monotonically, whereas a2 would go through zero before decaying to zero. Indeed, in figure 2.5 we see the temporal Chapter 2. Surface Growth Modelling 14 Figure 2.4: The first two terms of a Fourier cosine-decomposition of two ini t ia l surface shapes. Notice that the second harmonic for the wide shape has a negative coefficient. Figure 2.5: Temporal evolution of the first and second Fourier coefficients of the decom-posed grating during ballistic deposition. Chapter 2. Surface Growth Modelling 15 evolution of the first two Fourier coefficients of the grating as the B D growth progresses. The second coefficient, ci2, goes negative and then decays away. This is a sufficient quantitative measure of lateral growth: the 'bumps' w i l l evolve so that their second Fourier coefficient start at a positive value (for a narrow peak) and go into negative values before decaying. A s wi l l be shown later, growth without lateral movement w i l l not display such behaviour of the second Fourier coefficient, no matter what the in i t ia l surface shape looks like. 2.3 Continuous Growth Models However useful on a smaller scale, the discrete models become computer demanding on larger scale structures, and we would like to capture the essence of these models in continuum equations. The derivation of continuum equations can be based on symmetry principles (subsection 2.3.1) or on first principles (subsection 2.3.2) [4, 27, 30]. The equations wi l l have the form of a Langevin [31, 32, 33, 34, 35] continuum equation 2 dh — = G(h,x,t)+rj(x,t) (2.7) Here h = h{x,t) is the single valued interface height distribution, G(h,x,t) is a general function and rj(x, t) is the noise due to deposition. The noise has a Gaussian distribution with: <r]{x,t)>=0 (2.8a) <r](x,t)r](x',t')>=2D5(x-x')S{t-t') (2.8b) The average velocity of the interface growth is subtracted by a change of variable h —>• h — Ft, where F is the growth rate of the surface, in our experiments Ijim/hr. 2Higher order time derivatives are neglected on the left hand side of the equation, as we are interested in the long-time behaviour of the system [4]. Chapter 2. Surface Growth Modelling 16 2.3.1 Deriving Continuum Equations From Symmetry Arguments A systematic method to derive continuum descriptions of growth equations is to consider the 'simplest possible equation compatible with the symmetries of the problem'. A detailed description is found in Barabasi [4]. The general function G(h,x,t) should not depend on where we define our coordinate system, therefore it can depend on x or t only through h. The terms dh/dt and dh/dx are invariant under time and space translations, respectively, and obey this criterion. The growth rule should furthermore not depend on where we define zero height, so the growth equation should be invariant under the translation h —>• h + 5h. This rules out explicit dependence on h, so the equation must be constructed from spatial derivatives of h. The growth equation must obey rotation symmetry about the growth direction. This rules out odd order derivatives in the spatial coordinate, terms like Vh, V ( V 2 / i ) , etc. Note that the terms ( V / i ) 2 and V 2 / i both survive the transformation x —> —x since they have an even number of derivatives in x. The last symmetry to obey is the inversion symmetry in h [36, 37]. This last argument means that it must be possible to write dh/dt as d(f(h,x,t))/'dx, where f(h,x,t) is some general function; this is the mathematical description of mass conservation. The most general equation describing surface growth is therefore: ^- = V2h + V 4 / i + ... + V2nh + V2h(Vhf + ... + V2kh(Vh)2j + n{x, t) (2.9) where n, k and j can take on any positive integer value, and a comoving frame with zero growth velocity is used. In forming equations, it can be shown using dynamic scaling theory that higher order derivatives should be less important for large x and t (the hydrodynamic limit) [4]. The simplest continuum equation describing equilibrium Chapter 2. Surface Growth Modelling 17 interface fluctuations is: — = V2h + V(x,t) [EW] (2.10) This equation was analysed by Edwards and Wilk inson [13] in 1982, although it was derived earlier by Chui [38]. Ignoring the noise-term, we see that it takes the form of a simple heat equation (sometimes misleadingly referred to as the 'diffusion equation', since diffusion in surface growth is related to the fourth spatial derivative of the surface, not the second) wi th conservative noise. The scaling exponents for the E W equation can be solved exactly, and in I D they are [4, 13]: a = 1/2, 8 = 1/4 [EW] (2.11) These exponents are the same as those derived for R D with surface relaxation [25]. Therefore, the two models belong to the same universality class, and this indicates that surface relaxation can be implemented by a V 2 / t - t e r m in the continuum equation describ-ing surface growth. This term has also been linked to re-evaporation of surface atoms, see section 2.3.2. The E W model has been used successfully to interpret experimental results of strained InGaAs on GaAs [1, 16]. When the noise term is dropped in E W , the solution takes on the simple form: h(x,t) = J2cne-'/^)2tUx) (2.12) n where d>n{x) are orthonormal functions that span the basis space for h(x,0). The func-tion Ci<f>i(x) is therefore the ith expansion term of the Fourier decomposition of the ini t ia l surface shape, according to Sturm-Liouvil le theory on a periodic boundary-value prob-lem. Each term (f>n(x) dies away monotonically cx exp(—An2t), in good agreement with experiments that show how smaller surface features die away quicker than the large features. Chapter 2. Surface Growth Modelling 18 0.2 0.2 -0.2 -0.2 -0.2 0 X (±1 m ) 0.2 -0.2 O x (n m ) 0.2 Figure 2.6: Decay of two similar shapes wi th different widths according to the E d -wards-Wilkinson model. Both shapes decay towards the first harmonic. Comoving frame is used to more clearly show lateral growth behaviour. The E W model lacks lateral growth as defined in section 2.2. In that section, lateral growth was defined as a process that takes the second Fourier coefficient through zero and into negative values before decaying. One might wonder if that process would change with the width of the 'bump' used as ini t ial condition. According to the solution of the deterministic E W , this should not happen. Every Fourier coefficient should die away monotonically, as can be seen in figure 2.6. The E W equation is there integrated numerically, and we see that the narrow bump widens, and the wide bump narrows. They both approach the first harmonic, a\ cos (2irx/L), as growth progresses. The 'bumps' w i l l travel laterally, to approach this harmonic, however, the lateral mass transport can go both ways, inward and outward, depending on the ini t ia l width of the surface shape. The numerical integrations performed in this work were done using a second order accurate finite-difference scheme for the space-derivatives, and a first order accurate Chapter 2. Surface Growth Modelling 19 Figure 2.7: Diagram showing one possible origin for the slope dependent term in the Kardar-Parisi-Zhang growth equation [2]. This mechanism is not applicable in M B E . forward Euler approximation for the time-derivative [39]. For the E W equation this becomes: H ? + l ~ H ? = " H ? + 1 ~ 2 ^ H t l + 0(dr + dh2) (2.13) where Hf is the ith space point and nth time point of the surface height distribution on a discrete grid with space and time-steps dh and dr, respectively. Periodic boundary condi-tions are implemented, and the Courant stability condition is obeyed in the simulations: 0 < udr/dh2 = 0.25 < 1/2. Lateral growth can be incorporated into continuum equations by adding terms that depend on the local slope of the surface ('Eden model, ' [40]). A sloped surface wi l l have a higher density of steps. As a result, atoms are more likely to stick to regions of higher local slope due to the presence of the steps. Figure 2.7 explains the justification of the slope dependent term in the growth equation in another way. This description is not applicable to M B E . This mechanism is relevant when the growth occurs normal to the interface at every point, so the growth projected onto the h axis is 5h = [(v5t)2 + (vdtVh)2]1/2, which Chapter 2. Surface Growth Modelling 20 Figure 2.8: Temporal evolution of a grating according to the Kardar-Parisi-Zhang equa-tion, v = 10nm2/s and A = O.Snm/s. The interface velocity due to growth is incorpo-rated to show similarity to B D growth. for |V/i| <C 1 can be expanded according to the binomial theorem: m ~ v + 2 ( v f c ) + -(2.14) The steady interface growth v can be excluded by a variable transformation, and after including the surface relaxation and the random deposition noise terms from E W , we get [2]: f t o = u V 2 h + * ( y h ) 2 + v , X i t ) [ K p Z ] . ( 2 1 5 ) The non-linear term in K P Z breaks the inversion symmetry in h, and excess mass is being created on sloped parts of the surface. K P Z can therefore not be the final, correct description for M B E growth. This problem wi l l be addressed in section 2.3.3. The K P Z equation is numerically integrated in figure 2.8. The numerical scheme is similar to that of equation 2.13, and the ini t ia l shape is that of a periodic grating of pitch Chapter 2. Surface Growth Modelling 21 0.2 0.15 0.1 g- 0.05 < 0 -0.05 l 1 i First Fourie r coefficient . a/t) Secor id Fourier c oefficient, a _(t) 2 W i i 4 6 time (minutes) 10 Figure 2.9: Temporal evolution of the first and second coefficients of the Fourier decomposed grating during growth according to the Kardar-Paris i-Zhang equation. v — 10nm2/s and A = 0.8nm/s. 786nm: J f " + 1 1 T ^ = V ^ ^ ~ + 2 2ik +0(<fr + «fc') P 1 6 ) A s previously mentioned, the K P Z equation in I D scales wi th exponents comparable to those of the B D model [2, 41, 42, 43, 44]: a = 1/2, B = 1/3 [KPZ] (2.17) They are therefore in the same universality class. The grating in figure 2.8 develops cusps and exhibits lateral growth [41], wi th a remarkable resemblance to the B D evolution of the same shape in figure 2.2. Also notice the similar behaviour of the Fourier coefficients ai(t) and a2(t) for B D and K P Z (see figures 2.5 and 2.9). Chapter 2. Surface Growth Modelling 2 2 2.3.2 First Principles Derivation of Continuum Models The continuum equations for surface growth are rooted in the principle of mass conser-vation, where the number of atoms in the system is balanced in a general continuity equation: ^ ^ + V - j ( x , t ) = 0 ( 2 . 1 8 ) where j is the current of atoms along the surface, and deposition noise has been neglected. The surface current can be linked to the chemical potential of the surface, n(x,t) [37]: ]{x,t) oc -Vp,(x,t) ( 2 . 1 9 ) To describe surface diffusion by thermal activation, we can assume that the chemical potential describing the surface is proportional to the local curvature of the surface [37]. This is the continuum description of 'number of nearest neighbours', as the curvature is inversely proportional to the local radius: fi(x,t) oc -\72h(x,t) ( 2 . 2 0 ) This w i l l give us an equation that describes relaxation by surface diffusion [37, 45]: dh ^ - = - F T V 4 / i ( 2 . 2 1 ) dt We include random deposition by adding a constant interface growth, F, as well as a random term, r)(x,t) [9, 22] : dh — = - K V 4 H F + -q{x,t) ( 2 . 2 2 ) C/t-Similar considerations can give rise to more terms in the equations above, for instance by including relaxation by evaporation [46]. B y assuming that the deposition-desorption growth processes are dominated by the difference between the average chemical potential Chapter 2. Surface Growth Modelling 23 of the vapor, //, and the local chemical potential on the surface, p(x, t) [10], we can write: dh . . -^<x-v(x,t) (2.23) Substituting [J.{x,t) from equation 2.20, we get the familiar second space derivative from E W and K P Z . The equation that incorporates both surface diffusion and relaxation is the so-called 'linear M B E equation': — = vV2h - K V 4 / I + n(x, t) (2.24) B y Fourier transforming the deterministic part of this equation with respect to the spatial coordinate, we get: rlhin A = u(-q2x)h(qx, t) - K(q4x)h(qx, t) (2.25) dt where h(qx,t) is the Fourier transform of h(x,t), and qx is the transform variable (the spatial frequency). This can easily be solved for h(qx,t): h(qx,t)^h(qx,0)e-^+K^t (2.26) The two linear decay-terms (uV2h and K V 4 / J ) generate a characteristic cri t ical length scale Lcrit given by: Lor* = 2vr ( - J (2.27) For spatial frequencies corresponding to smaller length scales, the fourth derivative term wi l l be dominant, and for larger length scales, the second derivative wi l l be dominant, giving us the familiar E W equation. This gives the familiar faster decay of small scale roughness, but even faster here than in the E W equation. The increase of large scale roughness (like that observed in figure 1.1) can be accounted for by considering diffusion bias. This effect arises due to the potential barrier present at the edge of a descending step, also called the Schwoebel barrier [47, 48]. The barrier at Chapter 2. Surface Growth Modelling 24 the step-edge reflects an approaching atom with some probability instead of letting it fall off the edge. This results in a current of surface atoms away from the descending step, proportional to the local slope of the surface: j oc Vh. This term gives us a different version of the linear M B E equation [15]: — = _ | z , | V 2 / i- KV4h + r)(x,t) (2.28) This equation is unstable, and for spatial frequencies smaller than qcrit = 27r/Lcrit it wi l l diverge. This problem was solved for by Johnson [27, 49, 50] when they proposed a surface current which peaks at some intermediate slope: ] = FSaa2 J h (2.29) 1 + (am)2 where F is the incident particle flux, Sa incorporates a Schwoebel parameter proportional to the reflection probability at the step edge, a is the diffusion length on a terrace and m is the local slope of the surface. The uphil l current wi l l for small slopes be l imited by the diffusion length of the atoms. The diffusion length is the average distance a surface atom wi l l travel before it nucleates with another atom to form an island, or simply re-evaporates off the surface. The high-slope l imitat ion is due to the decreased length of the terrace, as well as re-evaporation of the edge-atoms which wi l l eventually lead to diffusion over the unfavourable Schwoebel barrier. This process becomes more probable as the steps get closer together [51]. The full growth equation overcomes the unstability problem of equation 2.28: ¥ = -FS°°2 V • (TT^W) " KV4" + '<*•() (2'30) Finally, we mention an equation that is similar to the K P Z equation, but obeys mass conservation [10, 24]. The equation reads Bh — = -KV4h + X1V2(Vh)2 + rj{x,t) (2.31) Chapter 2. Surface Growth Modelling 25 which can be written as the divergence of a current, and therefore obeys mass conser-vation. In our experiments, we verify q2 dependence of the surface evolution, indicating the presence of a second spatial derivative in the growth equation. Among the non-linear equations, this would favour K P Z over this last equation. The continued analysis of continuum equations for epitaxial surface growth wi l l no doubt result in improved models for the surface evolution, and experimental work should supplement and verify these models. 2.3.3 Mass Conservation and the Kardar-Parisi-Zhang Model The K P Z equation in explicitly non-conservative, i.e. it is non-physical in the sense that it does not obey the mass conservation dictated by the general continuity equation for surface atoms [36]. In the comoving reference frame where the deposition growth rate has been subtracted, we find an excess velocity of the interface given by [4]: In discrete modelling, this could be dealt wi th by simply subtracting the mean of the interface height at any time t [52]. Notice that a moving interface wi l l affect the D C term of the Fourier series decomposition, and therefore not affect the second Fourier coefficient as discussed in section 2.2. Analytical ly, we can write the 'true' surface height, ri(x,t) as [51, 52]:' where v(t) is the excess growth rate due to lateral growth. Substituting into the K P Z equation, we get: (2.33) (2.32) (2.34) &H{x,t) dt vV2rl{x,t) + -(Vrl{x,t))2 v(t) (2.35) Chapter 2. Surface Growth Modelling 26 This corrected equation satisfies mass conservation globally (although not locally), and the growth mechanisms described by the equation are not changed. The apparent excess mass that occurs due to the lateral growth is simply mass that accumulates on the steeper sections of the surface, and can be compensated for by the new term, v(t). Since M B E growth forms non-porous structures, the inherent non-conserving nature of the K P Z equation means that it cannot be an exact description of M B E growth. Nevertheless, we have found that it describes the behaviour seen in our experiments rather well. The breach of inversion symmetry has been experimentally observed (see chapter 5), and the excess growth rate arising from the non-linear term is not affecting our simulations, as it only affects the D C term (a 0) of the Fourier series decomposition of the grating shape. . The K P Z equation is a derivation based on mathematical simplicity [27]. Un t i l we can describe M B E growth in a better way, it seems unjustifiable to add terms ad-hoc to clean up the shortcomings of this equation. Chapter 3 Elastic Laser Light Scattering 3.1 Diffuse Scattering The intensity of reflected light in the specular direction is insensitive to structure on the reflecting surface [53], and l imited information can be achieved by monitoring it. The non-specular scattered light, however, contains information about the surface morphology. The intensity of the scattered signal at some non-specular angle wi l l be proportional to the roughness of the surface at the corresponding spatial frequency. Figure 3.1 shows diagramatically how the intensity of scattered light from a rough surface can be estimated. We assume that the roughness of the surface is small enough so that we can ignore shadowing effects and second order scattering (Born approxima-tion). In that figure, we get constructive interference from the two incoming beams when \segmentA — segmentB\ = m\[aser, wi th m an integer. W i t h the notation in that figure, this means that at an angle 9S we are observing the scattered signal off a surface feature of spatial frequency: 27T qx,obsewed = T (sin Oi - sin 9S) (3.1) "laser The details of the three-dimensional scattering from a surface was described by Lavoie [54], but we wi l l settle for the two-dimensional scattering geometry in the plane of inci-dence. More rigorous treatment using Kirchhoff theory [55] or perturbation theory can give the relationship between the surface morphology and the scattered light intensity. In the Fraunhofer far-field approximation, the power scattered elastically into a solid angle 27 Chapter 3. Elastic Laser Light Scattering 28 light out Pitch of surface feature, L Figure 3.1: Diagram showing geometry of laser light scattering (LLS) . Q, is given by [56, 57]: 1 dP 16TT2 A 4 cos Bi cos 2 9S \Qab{0i,Os,6)\2 g(q) (3.2) where P 0 and dP are the incident and scattered power, respectively, and a metallic interface is assumed with a zero electric field boundary condition. We estimate a 20nm skin depth for the evanescent component of the electric field, and believe that the metallic interface approximation does not qualitatively change the behaviour of the scattered l ight 1 [58]. The function g(q) is often referred to as the 'surface factor', and the terms preceding it are called the 'optical factor', including the polarization dependent Qab, where a and b can take the values of s or p indicating the polarization of the light. The optical factor constitutes the physics of the material, as well as the angular dependence arising from the 1 S k i n depth, S, derived from l / | f c z | , where k2 = n2w2/c2 - k2, where kx = 16/xm 1 , A ; o s e r = 488nm and nGaAs - 4.392 + jO.476 at Elaser = 2.54eV Chapter 3. Elastic Laser Light Scattering 29 1 o o 0.2 0.4 0.6 Grating depth (\x m) 0.8 1 Figure 3.2: Simulated surface factor at 16yum 1 according to the Fraunhofer far field approximation. density of scatterers. The surface factor is often taken to be the power spectral density (PSD) of the surface height distribution, h(x,t), and is thus the only time-dependent term in this equation. This approximation is valid for a surface of small height variations and the wavelength of the light source must be much greater than the interface width. A more accurate description in the Fraunhofer far field approximation can be given by adding up the phase information rather than the height, s t i l l assuming that there is no where the incident light is assumed to be laterally coherent, qx is the in-plane component of the spatial frequency and qz is the vertical component. In our experiment, qx is given by equation 3.1 and, shadowing effect, as follows [51, 59]: (3-3) (cos 6i + cos 0S) (3.4) aser Chapter 3. Elastic Laser Light Scattering 30 2TT A laser COS 9; + COS a r c s i n s i n & — - — (3.5) In this expression, 9S was substituted in from equation 3.1. For a given observation angle, the experiment is sensitive to surface structure of a length scale L = 2n/qx. In the Fraunhofer approximation, an incoming plane-wave is assumed, as well as a point receiver for the scattered light. For incoherent i l lumination, an ensemble average must be calculated. When the surface height is small, equation 3.3 reduces to: |2 9(q) J elQxX (cos (qzh(x, t)) + i sin [qzh(x, t)) dx j eiq*x(l + iqzh{x,t))dx |2 KQx) + iqzh(qx,t) ${qx) + q\ h(qx,t) (3-6) (3.7) (3.8) (3.9) where we recognize | /z(g x ,£) | as the P S D of the surface. A specular component is also present in this equation. The intensity response from a sinusoidal grating is simulated in figure 3.2 as a function of depth of the grating. The response oscillates, reflecting the fact that at some grating depths, the top and the bottom of the gratings wi l l be out of phase by an amount appropriate for destructive interference. 3.2 Scattering From Flat Surfaces The surface factor, g(q), is simulated in two plots in figure 3.3 for spatial frequencies in the scattering range with Oi =25 degrees and A ; a s e r = 488nm. The numbers chosen here reflect those used in our experiments. The roughness on the surfaces in the two plots is approximated by a random walk of interface width Inm and 25nm, respectively. The intensity profile of the top figure is that of the lower roughness and has the shape of a sinc-squared function, as expected from the Fourier transform of a rectangular aperture. Chapter 3. Elastic Laser Light Scattering 31 q x (V- m 1) Figure 3.3: Fraunhofer diffraction from a coherently il luminated random-walk generated rough surface wi th standard deviation lnm (top) and 25nm (bottom). The strong peak at qx = 0/j,m~1 corresponds to the specular reflection. The effect of the lnm rms surface roughness is barely visible. In the bottom figure, the surface roughness is increased to 25nm, and the effect on the scattered signal is obvious: the specular intensity is unchanged, but the diffuse scattering is affected by the increased noise. 3.3 Scattering From Textured Surfaces If we texture the surface wi th a periodic grating, the reflected signal wi l l have peaks at corresponding spatial frequencies. A sinusoidal grating of pitch 786nm wi l l have intensity peaks at qx = 2ir/786nm = 8 . 0 / i m _ 1 in the forward and backward directions, as well as harmonic frequencies at integer multiples of this spatial frequency. W i t h this grating pitch, we were able to detect the diffracted intensity peaks in the optical ports of our M B E . We have repeated the exercise from figure 3.3 in figure 3.4. The increased surface Chapter 3. Elastic Laser Light Scattering 32 Qx m ) Figure 3.4: Fraunhofer diffraction from sinusoidally textured surface of pitch 786nm and 125nm amplitude including random noise of roughness lnm (top) and 2bnm (bottom). roughness of the bottom plot st i l l has an impact on the signal; however, it is the 250nm deep grating (125nm amplitude) that dominates the response in the scattered signal at the corresponding spatial frequencies. This means that although the increased surface roughness affects the diffuse scattering, the presence of the large surface gratings is so strong that the added roughness has l i t t le effect on the intensity peaks the gratings cause. This justifies the exclusion of random noise when numerically integrating growth equations in previous chapters. 3.4 Mapping the Surface by Monitoring its Fourier Coefficients We have developed tools that allow us to monitor the temporal evolution of textured semiconductor interfaces during epitaxial growth. The surface shape can easily be verified by ex-situ methods such as A F M ; gratings before and after growth are shown in figure Chapter 3. Elastic Laser Light Scattering 33 Figure 3.5: A F M images of gratings before (top) and after (bottom) regrowth. The gratings have a pitch of 1.2pm, and the aspect-ratio is 1:1. 3.5. W i t h the in-situ laser light scattering we can now learn about the transient evolution of textured surfaces. The surface structure used in our experiments consists of wet-etched gratings of pitch 786nm. The shape of the gratings is almost rectangular, with walls sloped at 54.7° relative to the surface plane. The fabrication of these wi l l be discussed in detail in chapter 4. A s mentioned in chapter 2, this periodic shape can be decomposed into a Fourier series: h(x) = ao/2 + a i cos (2%x/L) + a2 cos (4TTX/L) + .... The shape is centered at the origin so there are no odd expansion terms. The zero'th term is the mean of the surface and is lost in the specular beam. The first term, a\ cos (2nx/L) is detected at 8 .0 /xm - 1 , as discussed in section 3.3. The second term, a2 cos (4irx/L), is seen at 16 .0 / /m _ 1 . As discussed in chapter 2, the second Fourier coefficient of this grating wi l l exhibit a change of sign if there is lateral growth present in this system. A system that simply smooths following Chapter 3. Elastic Laser Light Scattering 34 E W wi l l not show such a transition, and wi l l simply decay monotonically. Remembering that we do not see the Fourier coefficients when we monitor the L L S , but rather the P S D , we must keep in mind that a signal that crosses zero and goes negative would show up as a minimum followed by a positive bump [60]. In other words, the phase information of the surface decomposition is lost. Chapter 4 Grating Fabrication The experiments performed in this work required fabrication of periodic gratings on the surface of the sample. These gratings were made on semi-insulating (SI) (lOO)-oriented, on-axis (±0.1°) gall ium arsenide (GaAs) by electron beam (e-beam) lithography in con-junction with a wet chemical etching system. The GaAs wafers used were supplied by Crystar of Vic tor ia , B . C . [61]. The fabrication requires quite a few steps and exposure to several solvents and chemicals. A l l fabrication was performed in a Class 1000 cleanroom, with the exception of the e-beam lithography. We wi l l learn that the cleanliness of the sample is crucial as the experiments involve regrowth using the molecular beam epitaxy ( M B E ) process which is sensitive to surface contamination. 4.1 Fabrication Steps A two inch diameter, 450yum thick GaAs wafer was cleaved in four pieces by scribing the edge of the wafer with a diamond-tip scriber. The cleaved edges follow the crystal planes, in this case the (Oi l ) and the (Oi l ) crystal planes to form four (equally large) quarter wafers. Each of these quarter wafers becomes an individual sample. This rather large sample size is dictated by the temperature measurement system used in the M B E during growth. A larger sample gives a signal that is easier to interpret for the diffuse reflectance spectroscopy (DRS) system. This method wi l l be explained in chapter 5. 35 Chapter 4. Grating Fabrication 36 Figure 4.1: American standard GaAs (100) wafer with crystal orientations indicated [5, 6]. The cross indicates how the wafer was cleaved. About seven drops of 950K molecular weight polymethylmethacrylate ( P M M A ) pos-itive photoresist is put on a quarter wafer wi th a disposable pipette, and immediately spun at 9000 rpm for one minute. The spin creates a uniform thickness film of 150nm [62]. The uniformity can be inspected by the color of the film, and for this large sample the edge-effects are insignificant. The sample is then put on a hotplate at 180°C for at least three hours to harden the photoresist and make it adhere to the sample surface. The film is sensitive to ultraviolet light, so the sample is for all subsequent processing kept in a dark container when transported. The cleanroom is intended for lithography, and is i l luminated by lamps with yellow filters to block the U V . The sample is now removed from the cleanroom to do electron beam lithography. This is done in a Hitachi S-4100 field emission scanning electron microscope ( F E - S E M ) [63] driven by the Nanometer Pattern Generation System ( N P G S , version 7.4 [64]) along Chapter 4. Grating Fabrication 37 Spinning speed 9,000rpm Bake More than 3 hours, ~ 180°C, hotplate Developer M I B K : I P A , 1:3, ~ 25°C, 90-120 seconds Rinse IPA, 20 seconds Photoresist stripper Acetone, ~ 50°C, 20 minutes Table 4.1: Parameters for preparation of photoresist films, pattern transfer and stripping of photoresist. with DesignCad [65] graphical software to define the patterns. Operating parameters for the S E M and the e-beam lithography are summarized in tables 4.1 and 4.2. Once the lithography is done, the sample is returned to the cleanroom for development and pattern transfer. We develop the exposed resist in methyl-iso-butyl-ketone ( M I B K ) and iso-propyl-alcohol (IPA) in a ratio of 1:3 for two minutes followed by a stop solution of pure I P A for 20 seconds. Methanol is squirted on the sample for some 10 seconds, and this is blown away by dry nitrogen gas to avoid evaporation residue on the surface of the sam-ple. The pattern transfer is done using the so-called piranha-etch (it etches 'anything'): H2S04 : # 2 0 2 : H20 (1:1:18) that etches GaAs at a rate of 480nm/mm [62]. More de-tails on this etch wi l l be explained in section 4.3. The remaining resist is finally removed by dissolving it in hot acetone for about 20 minutes, followed by a cold 20 second squirt of acetone, then 10 second squirt of methanol and dry nitrogen (7V 2 )-blow. The sample is now ready for M B E growth. The hot acetone treatment seemed crucial to the success of regrowth on the sample. Without the hot acetone treatment, every attempt on regrowth failed as the surface got coated with a white film as soon as the semiconductor growth was started. Chapter 4. Grating Fabrication 38 Acceleration voltage 30kV Emission current 20/iA Aperture opening #4 Wri t ing time after flashing from 2 hours to one day Centre-to-centre distance 300A Magnification 1000 Condenser lens 12 and above Working distance 17mm Dose 12.0 fC/point Table 4.2: Parameters for electron beam lithography in the scanning electron microscope. 4.2 Details on the Lithography Process The lithography process is perceived by many as somewhat of a black art. There are many v operating parameters that must be fixed, and this art is often taught by an oral tradition from those who knew before you. A knowledgable person on e-beam lithography in our department is A lex Busch. His M . A . S c . thesis from U . B . C . describes e-beam lithography using the F E - S E M [62, 66]. The desired pattern to be written on the photoresist is defined in DesignCad [65], a graphical software package. In the case of gratings, single lines are written with zero width, which N P G S interprets as a single e-beam pass when it comes to wri t ing the pattern on the sample. Peng Chen has defined gratings using a multiple-pass method that greatly enhanced the uniformity of the gratings [66]. The high energy electrons break the polymer chains in the P M M A , and leave them soluble to the developing agent, in our case M I B K and I P A . Now, enough electrons must hit the resist and penetrate through it to make the entire layer of resist soluble. Peng Chen did a careful calibration and found that for a P M M A - layer of 150nm, an incident point dose of at least 2.8fC/point was necessary to expose the P M M A [66]. When the same calibration was attempted by the author, with al l other parameters equal, at least Chapter 4. Grating Fabrication 39 Figure 4.2: SEM-image showing the selective growth around the beam dump area of the exposed pattern. The raised plateau is believed to be inorganic carbon, and the two lumps on top of it GaAs. 8fC/point was needed to penetrate the P M M A . When the beam current is too high, on the other hand, hydrogen is released producing inorganic carbon directly on the GaAs substrate, which is insoluble to the developing agent. When attempting to regrow on this hardened carbon in the M B E , we have verified that there is no growth around an area where this has occurred, i.e. at a beam dump (see figure 4.2). The experiments performed in this work used a ~ 2mm diameter laserbeam incident on a GaAs substrate with uniform gratings, so the larger the area with gratings, the better (up to 2mm diameter). For the kind of resolution we are interested in (gratings of pitch down to 393nm), the largest area that can be exposed at a time without manually moving the beam is ~ 90 * 90fim2. Each of these squares takes on the order of 10-20 minutes with a 786nm pitch grating under ideal wri t ing conditions. The maximum number of squares written in close proximity was 25 (in a square grid), which is about as much as one can write before the e-beam current becomes unstable and the tip needs to Chapter 4. Grating Fabrication 40 Figure 4.3: S E M image of gratings on (100) GaAs. The image is t i l ted, but one can st i l l resolve the total lateral etch-distance immediately under the P M M A , as well as the width of the bottom flat to estimate the vertical to lateral etchrate ratio. Etch time was 25sec, giving an etch depth of 200nm. be flashed. Most of the experiments had only 16 such squares, as the e-beam was only stable for about 4 hours. 4.3 Wet Etching (100) GaAs Etching rates using wet etching systems are crystal orientation dependent. For instance, when etching parallel to the [011] direction, one achieves characteristic undercuttings with overhangs. However, when etching parallel to the [011] direction, one obtains the shapes in figure 4.3. The {111} facets etch slower than (100), and <111> oriented walls at 54.7° off the horizontal are formed. The relative etchrate ratio of the (100) plane and {111} can be deduced by observing that the two bottom corners of the grating in figure 4.3 are closer together than the edges of the P M M A . Quick calculations show that a Chapter 4. Grating Fabrication 41 0.1 0.1 Or Or 3 0 s e d 6 0 s e d 9 0 s e d -0.8 -0.8 -0.2 0 x (JLI m ) 0.2 -0.2 0 x m ) 0.2 Figure 4.4: Simulation of etched surface for relative etchrate ratio of (100) to 111 of a) 2.1 and b) 1.5. cri t ical relative etchrate ratio of (100) to the {111} planes is 1.731. For a larger ratio, the long time etched gratings wi l l become triangular, whereas for a small ratio, the gratings wi l l eventually annihilate, see figure 4.4. The SEM-image in figure 4.3 indicate that this etchrate ratio is about 1.84. This number has been verified elsewhere [62, 66, 67]. 1 The critical ratio is obtained from tan 54.7°/ cos (90° - 54.7°) = 1/ cos 54.7°, where 54.7° is the angle between the two planes. Chapter 5 Experiment 5.1 Experimental Setup We wi l l perform regrowth on top of the gratings discussed in chapter 4 to investigate the temporal evolution of their shape. The M B E machine is a single growth chamber V G Semicon V 8 0 H with effusion-cells. A 488nm Argon laser is pointed through an optical port at 25° off the sample normal. A Si-detector or a photomultiplier tube ( P M T ) is located at 55° off the normal in the back-scattering direction, and the signal is captured by a standard lock-in amplifier. This setup enables us to monitor surface roughness 5.9672 270 Figure 5.1: Angular dependence of Fraunhofer diffraction from a sinusoidally textured surface of pitch 78Qnm. 42 Chapter 5. Experiment 43 1 0 3 2 2 7 ' X 2 . 8 0 K Figure 5.2: SEM-image of gratings after unsuccessful regrowth. Notice beam-dump in top left corner as discussed in chapter 4. of lateral wavelengths of about 400nra, equivalent to a spatial frequency of 1 6 / / m _ 1 . Figure 5.1 shows the reflected signal from a grating of pitch 786nm. The peak of the 1 6 / / m _ 1 signal emerges at 305° (in the notation of the figure), which corresponds to backscattering at 9S = 55°. 5.2 Cleaning the Sample Sample cleanliness is crucial for regrowth experiments, as growth in M B E is highly selec-tive and sensitive to surface contamination by for example carbon. The sample prepara-tion requires many steps, in which the sample has been exposed to several solvents, acid and temperature treatments. A n example of failed regrowth is shown in the SEM-image in figure 5.2. The little dots on the gratings have a diameter of less than a micrometer. Several of these litt le dots are clustered in the same area, and we believe they are caused by residual P M M A . These holes create scattering centres that the L L S is very sensitive Chapter 5. Experiment 44 Figure 5.3: R H E E D patterns from (100) GaAs. to. In fact, when regrowth fails, the scattering signal immediately increases by orders of magnitude, and as a result is a good method for detecting whether the regrowth is successful. Immediately before inserting the sample into the M B E , it is oxidized under a U V -lamp for 10 minutes, which terminates the surface with a thick oxide and volatilizes carbon to C02. The sample is transferred to the M B E and put in position for growth. The oxide just added is removed by a hydrogen etch, in which the surface oxide reacts with cracked i^-molecules, which are highly reactive and etch away the arsenic and the gall ium oxides. The /^-molecules are cracked by a hot filament (1800°C), and the substrate reaches a temperature of approximately 200°C7 in this process. The partial pressure of the hydrogen is I0~5torr, and the i f-etch is done for 40 minutes [68]. The desorption of the oxide is monitored using R H E E D (see figure 5.3). A t low temperatures, the reflection of the 12keV electrons off the surface oxide create incoherent scattering, Chapter 5. Experiment 45 ^600 T3 580 Q. E , 540 10 20 30 40 time (min) 50 Growth starts : here 60 1 I 1 1 1 1 1 o O o 52J5B0SD.: i 70 Figure 5.4: Annealing of gratings before growth showing temperature dependence of smoothing parameter. Increased temperature negatively increases the slope of the smoothing. showing up as a uniformly glowing screen. Some time into the if-etch the oxide is removed, and the R H E E D pattern develops into diffraction spots when it reflects off the (100) facet. The oxide is now removed, and the sample is ready for regrowth [54, 69]. 5.3 Annealing Prior to Regrowth The temperature is measured using the diffuse reflectance spectrosopy (DRS) technique that was invented in our lab [70, 71]. This technique applies a broadband light source whose signal transmits through the sample, and reflects off the back of the substrate. The transmission is temperature and wavelength dependent, and a monochromator is used to scan appropriate wavelengths at a given temperature. The system is accurate to Chapter 5. Experiment 46 12.5 13 13.5 14 14.5 15 15.5 16 16.5 1/kT (eV_1) Figure 5.5: Determining the activation energy of the Edwards-Wilkinson u parameter for GaAs. ±2°C. We start our experiment by ramping the temperature towards the growth tempera-ture, which is usually set at 600°C. As the temperature approaches the growth temper-ature, the L L S signal starts to decrease, as shown in figure 5.4. The Edwards-Wilkinson parameter v can be calculated from the slope of a log-plot, as the slope is equal to —2vq\. As v is temperature dependent, we can deduce the activation energy of the surface atoms: a log-plot of v versus 1/kT w i l l have a slope of —Ea. Such a plot is shown in figure 5.5 [69]. The results obtained during several regrowth runs are also included there; however, as al l the regrowth experiments were done at temperatures very close to 600°C, there was Chapter 5. Experiment 47 not enough spread in the data to achieve an activation energy for the surface smoothing during growth. Note the proximity of the points from the regrowth-experiments wi th the two different qx values. The E W parameters in this plot were derived assuming q\ behaviour. A fourth order spatial derivative in the growth equation would spread the two sets of points apart by about a factor of three. Dur ing pre-annealing, the activation energy was found to be lAeV. Figure 5.5 also includes data obtained by Tom Pinnington by growing InGaAs on GaAs. A n activation energy of 2.6eV was obtained in that study [69]. 5.4 Regrowth on GaAs Gratings When the sample is ready for regrowth, and the cells as well as the sample are heated to appropriate temperatures, we grow GaAs at l/im/hr. The Ga-cell temperature is 1004°C, the As-ceW is at 400°C, and the V : III ratio is between 15 and 35. The base pressure is about 10~9torr and the partial pressure of As is on the order of 10~5torr. Growth was started when the sample temperature reached 600°C ('run 1'). As the shutter to the Ga-cell was opened and the growth started, the L L S signal dropped dras-tically. The signal continued to decrease, unti l it very abruptly came up again a couple of minutes into the growth. We interpret this as the change in sign of the second Fourier coefficient, that was discussed in chapters 2 and 3. A second run was performed with the temperature lowered to 592°C, but otherwise identical to the first run ('run 2'). The results can be seen in figure 5.6. Chapter 5. Experiment 48 , , r time (min) Figure 5.6: F i t of Kardar-Parisi-Zhang equation to experimental data: T = 6 0 0 ° C (top, run 1) and T = 592°C (bottom, run 2). Chapter 5. Experiment 49 Figure 5.7: A F M linescan of 1.3/xm gratings from run 1 (top left); top view (top right); 3D view (bottom). 5.5 D i s c u s s i o n When simulating the second coefficient of the Fourier decomposition of the grating ac-cording to growth by the K P Z equation, we achieve the graphs in figure 5.6. The simula-tions are superimposed on the experimental data, and the parameters used in the simula-tions were: v = 10nm2/s, A = lAnm/s for the experiment at 600°C and v = oAnm2fs, A = A.Qnm/s for the experiment at 592°C. For the first run at 600°C, we see an excellent fit as the LLS-signal goes towards zero, and comes back up as the square of the negative signal. There is furthermore a good fit on the secondary bump at 75 minutes, and as the signal dies away towards the Chapter 5. Experiment 50 Figure 5.8: A F M linescan of 786nm gratings from run 2 (top left); top view (top right); 3 D view (bottom). equilibrium state, the noise-less fit of the K P Z equation drops towards zero, as expected. A third peak occurred around 78 minutes into the growth. We have not analyzed this peak, but we believe it is a random phenomenon. A second run was performed at 592°C, with qualitatively similar results. There is no evidence of a second bump like the one we saw at 78 minutes into the first run. The transition from positive to negative values for the second Fourier coefficient is seen to be sharper in run 1 than in run 2. We believe that this is due to a better uniformity of the gratings produced for the first run. For instance, the uniformity of the photoresist can cause the gratings on one part of the wafer to be etched deeper than on Chapter 5. Experiment 51 another part. The parameters obtained from the fitted equations tell us that the smoothing rate increases with temperature, and that the parameter for the non-linear term decreases wi th increasing temperature. This is reasonable: the surface atoms wi l l have more kinetic energy at higher temperatures, and wi l l therefore have a higher probability of escaping the stronger bonding at an inclined surface with more nearest neighbours. A s a result one would expect the enhanced growth rate on inclined surfaces to be reduced. A t the same time the surface relaxation described by the uV2h term might be expected to increase with temperature, as observed. A F M images were obtained from both runs of the gratings after the growth. In figure 5.7, we see gratings of pitch 1.3pm from run 1. Note that this is not the 786nm gratings used for the L L S . The gratings of pitch 1.3pm should have a decay time of more than an hour 1 . The gratings have therefore barely had time to fill in , as the growth only lasted for about 22 minutes. We see that the gratings have started to move 'laterally', and cusps are about to form. Figure 5.8 shows similar A F M images from run 2. These gratings have pitch 786nm, giving them a decay time of about 700 seconds or 12 minutes 2 . The 30 minute growth of this run has had time to fill in the gratings quite well, and the formation of cusps has occurred. These cusps were also observed in simulations of the B D and K P Z models for surface growth in chapter 2. Note that the y-axis is expanded compared to figure 5.7. The A F M profiles confirm the interpretation of the in-situ light scattering presented above. We have also attempted to perform regrowth on top of gratings while depositing spacer-layers of AlGaAs at even time intervals. The objective was to cleave the sample xDecaytime obtained from r = l/(vq2). Here, qx = 27r/1.3/zm = 4.8/ im - 1 and v = 10nm2/s, giving T = 4340s 2l/{vql) = l / (5 .4nm 2 / s * (16/um- 1) 2) = 723s Chapter 5. Experiment 52 time (min) Figure 5.9: Growth of GaAs on GaAs gratings with AlGaAs spacer-layers. and selectively etch it to show growth structure in the S E M or A F M . A n example of this technique was shown in figure 1.1. Al though we were not successful in obtaining an image of the regrown material, we did see an interesting development in the L L S signal (see figure 5.9). Every time the Al-cell was opened, there was a momentary decrease in the LLS-s ignal , probably caused by lower reflectivity in the AlGaAs film. We monitored the surface with two lasers, the usual 488nra Ar laser and a 325nm HeCd laser. The new laser adds a signal at 24u.mrl through the same port that the Ar laser gives us the lQ/j-m^1. The two signals were separated by filters before detection. The spatial frequency monitored wi th the HeCd laser is exactly at the third Fourier coefficient of the 78Qnm grating. We calculated the ratio of the slopes of the two decaying LLS-signals, and got 1.92, meaning that the smoothing is dependent on q^6. This indicates with good accuracy 3 3Exponent obtained from log 1.92/log (24^m _ 1 /16/um _ 1 ) = 1.61 Chapter 5. Experiment 53 that K P Z is the applicable growth equation to our system. In systems that follow K P Z , the early time behaviour is believed to follow q\ ( E W ) , whereas long time follows a q\b dependence [52]. A transition time from E W to K P Z behaviour was calculated to be about seven minutes 4 for run 1 and about three minutes for run 2. This is approxi-mately the time between the beginning of the growth and where our fit superimposes the experimental data. There is no change in sign of the L L S signals this time. We believe that the ini t ia l shape decomposed with a negative second Fourier coefficient for this experiment. 4 T h e analytical solution to the deterministic K P Z equation gives us a transition time of ~ 7min, where t is calculated from t = L2/(\ho), where ho is the initial surface amplitude. Here, ho was ~ 200nm, L — 393nm and A = 1.0nm/s. See chapters 25 and 26 of reference [4]. Chapter 6 Conclusions In this work, we have described experiments that were performed in order to understand the kinetics of epitaxial semiconductor film growth on patterned substrates. Several discrete and continuous models of surface evolution were introduced. Sim-ulations showed a similarity between the discrete ballistic aggregation model and the continuous Kardar-Parisi-Zhang ( K P Z ) model, as given by [2]: This indicated a breach of inversion symmetry in the surface height distribution, as well as lateral mass transport. We discovered that lateral growth necessitated a change in sign of the coefficient of the second term in the Fourier decomposed grating used in the regrowth experiments. The inherent non-conservation of the KPZ-equa t ion was dealt wi th by subtracting a time-dependent growth velocity term, however, it was found not to affect the experiments at hand and therefore left out of the simulated equation. We discussed current theories for in-situ elastic laser light scattering (LLS) , and showed that for small surface heights, the intensity obtained from the L L S was compa-rable to the power spectral density (PSD) of the evolving surface structure. This enabled us to experimentally monitor the evolution of the coefficients of the different terms of the Fourier decomposition of the surface shape. Simulations of the evolution of surface gratings were done in accordance with the Kardar-Parisi-Zhang model, and then Fourier transformed and compared to the P S D dh ~dt (6-1) 54 Chapter 6. Conclusions 55 obtained from the L L S experiments. Good agreement was found between the two, and lateral growth was observed, quantified by a change in sign of the second harmonic Fourier coefficient of the decomposed grating during growth. We also obtained AFM- images from the post-growth gratings. The gratings were found to develop cusps, similar to behaviour seen in simulations of the B D and the K P Z models. We found that the parameters involved in the KPZ-equat ion were temperature de-pendent, where the surface relaxation term parametrized by v increased wi th increasing temperature, and the parameter for the non-linear term, A, decreased wi th increasing temperature. This agrees well wi th the interpretation that the deposited atoms have a higher probability of escaping the stronger binding sites at the inclined parts of the surface when the temperature is higher. B y measuring the surface relaxation parameter T = 600°C v = 10nm2/s A = lAnm/s T = 592°C v = 5Anm2/s A = 4.0nm/s Table 6.1: Parameters obtained by fitting the Kardar-Parisi-Zhang equation to experi-mental data. v in the absence of growth, we concluded that activation energy of the surface atoms was Ea — lAeV during annealing of GaAs. As far as we know this work represents the first quantitative analysis of the smoothing of a textured semiconductor surface in terms of a continuum growth equation. This is also the first measurement of the non-linear, slope-dependent term in the Kardar-Paris i -Zhang equation for GaAs epitaxial film growth. Ear ly on in the growth, we verified a dependence on the spatial frequency as q2., indicating Edwards-Wilkinson type growth. B y monitoring the smoothing of different Chapter 6. Conclusions 56 spatial frequencies, we were able to verify a dependence on the spatial frequency as g 1 6 later in the growth. This is in agreement wi th studies that indicate that the E W term vV2h is the dominant term early in the growth, and K P Z behaviour is dominant later on, giving a q].b dependence in the growth equation [52]. We have in this work verified the validity for the Kardar-Parisi-Zhang growth equation for spatial frequencies less than 16/zm~ x . This corresponds to length scales of about 400nm, and is comparable to semiconductor device dimensions today. For smaller length scales, higher order derivatives i n the growth equation might become significant, and other experiments must be done to verify this behaviour. Further experiments can explore the full temperature and growth rate dependence on the fitting parameters, v and A, and other material compositions can be investigated. List of Notation A F M atomic force microscope AlGaAs A luminum Gal l ium Arsenide AIN A luminum Nitr ide an coefficient of nth Fourier cosine-series decomposition B D ballistic deposition bn coefficient of nth Fourier sine-series decomposition D R S diffuse reflectance spectroscopy, temperature measurement system EQ activation energy of a free surface atom with no bonds EA surface atom activation energy EN bonding energy E W Edwards-Wilkinson F flux term or growth rate of surface F E - S E M field emission scanning electron microscope GaAs Ga l l i um Arsenide G(h,x,t) general surface growth function g(q) surface factor function in light scattering Hf discrete approximation of the surface height at the ith space point on the nth time step H(x,t) 'true' surface height: H(x,t) = h(x,t) — v(i)t h Planck's constant h(x, t) height distribution of semiconductor interface 57 List of Notation h(qx,t) spatial Fourier transform of h(x,t) I P A iso-propyl-alcohol j(x,t) current of surface atoms K P Z Kardar-Parisi-Zhang k Boltzmann's constant L lateral system size; also: size of surface feature Lcrit critical length scale: L c r i t = 2TT(K/U)1I2 L E E D low energy electron diffraction L L S laser light scattering M B E molecular beam epitaxy M I B K methyl-iso-butyl-ketone m local slope of surface NbN Niobium Nitr ide N N nearest neighbours P M M A ploymethylmethacrylate P M T photo multiplier tube P S D power spectral density qcrit critical spatial frequency: qcrit = (V/K)1I2 qx in-plane component of spatial frequency qz. component of spatial frequency perpendicular to plane R surface atom hopping rate RQ surface atom hopping rate at 0 Ke lv in R D random deposition R D w S R random deposition wi th surface relaxation List of Notation R H E E D reflectance high energy electron diffraction S E M scanning electorn microscope Si Silicon S T M scanning tunneling microscope Sa Schwoebel barrier parameter T E M transmission electron microscope tx transition time when surface width saturates v(t) growth velocity of surface W interface width given by rms fluctuation z dynamic exponent: z = a/3 a roughness exponent: Wsat ~ La 3 growth exponent: W ~ t&, t small •n(x, t) stochastic, random Gaussian deposition noise 4>n(x) orthonormal basis functions K diffusion constant A electromagnetic wavelength; also: coefficient of non-linear term in K P Z equation n(x, t) surface chemical potential v coefficient for surface relaxation term in E W a average separation of terrace islands Oi angle; between incoming laser and surface normal 0S angle between scattered light and surface normal Q solid angle Bibliography [1] T . 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