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

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SMOOTHING OF PATTERNED GALLIUM ARSENIDE SURFACES DURING EPITAXIAL  GROWTH  By Anders Ballestad Bachelor of A p p l i e d 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 REQUIREMENTS FOR T H E D E G R E E OF MASTER OF APPLIED  SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES 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 i n partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the L i b r a r y 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 A s t r o n o m y T h e University of B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, C a n a d a V 6 T 1W5  Date:  Abstract  Control over the surface structure of semiconductor films during growth is critical for devices of recent technological importance. T y p i c a l l y 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 ( P S D ) . T h e 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 semiconductor 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 w i t h the scattering peak associated w i t h a harmonic of the grating periodicity. Because the initial 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 i n time during growth. We find that for homoepitaxy of gallium arsenide (GaAs)  on textured  substrates, the time evolution follows the K a r d a r - P a r i s i - Z h a n g ( K P Z ) model [2].  ii  Table of Contents  Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgement  ix  1  2  Introduction  1  1.1  2  E v o l u t i o n of the Semiconductor Interface  Surface Growth Modelling  8  2.1  8  Discrete G r o w t h Models 2.1.1  3  Scaling Behaviour of Large Surface Structures  11  2.2  Lateral G r o w t h  13  2.3  Continuous G r o w t h Models  15  2.3.1  Deriving C o n t i n u u m 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 M o d e l  25  Elastic Laser Light Scattering  27  3.1  Diffuse Scattering  27  3.2  Scattering From F l a t Surfaces  30  3.3  Scattering From Textured Surfaces  31  iii  3.4 4  5  6  M a p p i n g the Surface by M o n i t o r i n g its Fourier Coefficients  32  Grating Fabrication  35  4.1  Fabrication Steps  35  4.2  Details on the Lithography Process  38  4.3  Wet E t c h i n g (100) GaAs  40  Experiment  42  5.1  Experimental Setup  42  5.2  Cleaning the Sample  43  5.3  A n n e a l i n g P r i o r to Regrowth  45  5.4  Regrowth on GaAs  47  5.5  Discussion  Gratings  49  Conclusions  54  List of Notation  57  Bibliography  60  iv  List of Tables  4.1  Parameters for preparation of photoresist films, pattern transfer and stripping of photoresist. . .  4.2  37  Parameters for electron beam lithography i n the scanning electron microscope  6.1  38  Parameters obtained by fitting the K a r d a r - P a r i s i - Z h a n g equation to experimental data  55  v  List of Figures  1.1  Cross-section T E M micrograph of AIN and NbN multilayer film. (After M i l l e r et.al. [3])  1.2  3  E v o l u t i o n of the interface w i d t h by random deposition according to ballistic deposition on an initially flat, one-dimensional surface  6  2.1  G r o w t h 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 ballistic deposition model. L=128 lattice sites  2.4  T h e first two terms of a Fourier cosine-decomposition of two initial surface shapes  2.5  14  Temporal evolution of the first and second Fourier coefficients of the decomposed grating during ballistic deposition  2.6  18  D i a g r a m showing one possible origin for the slope dependent term i n the K a r d a r - P a r i s i - Z h a n g growth equation [2]  2.8  19  Temporal evolution of a grating according to the Kardar-Parisi-Zhang equation  2.9  14  Decay of two similar shapes w i t h different widths according to the EdwardsW i l k i n s o n model  2.7  12  20  Temporal evolution of the first and second coefficients of the Fourier decomposed grating during growth according to the Kardar-Parisi-Zhang equation  21  vi  3.1  Diagram showing geometry of laser light scattering ( L L S )  3.2  Simulated surface factor at 1 6 / x m  -1  28  according to the Fraunhofer far field  approximation 3.3  29  Fraunhofer diffraction from a coherently illuminated random-walk generated rough surface w i t h standard deviation lnm  3.4  (top) and 25nm (bottom).  31  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 w i t h crystal orientations indicated [5, 6]  4.2  36  S E M - i m a g e 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  4.4  Simulation of etched surface for relative etchrate ratio of (100) to 111 of  40  a) 2.1 and b) 1.5 5.1  41  Angular dependence of Fraunhofer diffraction from a sinusoidally textured surface of pitch 786nm  42  5.2  S E M - i m a g e of gratings after unsuccessful regrowth  5.3  R H E E D patterns from (100) GaAs  5.4  Annealing of gratings before growth showing temperature dependence of smoothing parameter  5.5  43 44  45  Determining the activation energy of the Edwards-Wilkinson u parameter  for GaAs 5.6  ,  46  F i t of K a r d a r - P a r i s i - Z h a n g equation to experimental data: T = 6 0 0 ° C (top, run 1) and T = 5 9 2 ° C (bottom, run 2)  v i i  48  5.7  A F M linescan of 1.3^m gratings from run 1 (top left); top view (top right); 3 D view (bottom)  5.8  49  A F M linescan of 786nm gratings from run 2 (top left); top view (top right); 3 D view (bottom)  5.9  50  G r o w t h of GaAs on GaAs gratings w i t h AlGaAs  v i i i  spacer-layers  52  Acknowledgement  A great big thank you goes out to my good friends i n the M B E - l a b at U B C , where many rewarding hours have been spent. T h a n k you, M a r t i n , for helping me out w i t h the M B E - r u n s . Hope you learned something, too! Likewise to T o m P. for practical help and good advice along the way. Naturally, none of this would be possible without our very inspiring supervisor, T o m Tiedje. N o other person w i t h so many reponsibilites has quite so much time to discuss physics and general affairs as T o m . Thanks for giving me this opportunity! Thanks to J i m , for fixing what we break. Other people I appreciate who have added to my life and work i n some way: Drs. Johnson, Patitsas, Lavoie, Levy, Beaudoin, Young, Kanskar, Watson and Bergersen; Sayuri, A l , Peng, Dave, Francois, A l e x , Vighen, Sune, L i g i a and D r . Ogryzlo; R o n and M a r y A n n , the guys in the machine shops, the staff and members of this department as well as Drs. W a r d 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 i n experimental settings [7]. W i t h diminishing device dimensions come the demand for more sophisticated nanofabrication processes, along w i t h a better understanding of structure evolution during device fabrication. Similar concerns arise during growth of t h i n films on top of structured surfaces. T h e intent of growth may be to grow a t h i n '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.  T h e surface is described by an i n i t i a 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.  E p i t a x i a l growth is the process i n 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.  T h i s 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 limited 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. A p a r t from the technological importance that the understanding of the growing semiconductor interface brings us, the growth of solids is of great fundamental interest since it enlightens the behaviour of complex systems i n nature. It has been shown that many growth models produce a spatial scale invariance i n the surface structure, which can be described by the language of fractal geometry, introduced by Mandelbrot i n 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. O u r 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 initially flat surface at low temperature, most atoms on the surface w i l l remain locked i n position, not having enough kinetic energy to break the bonds they form w i t h their nearest neighbors. U p o n raising the temperature of the system, the surface atoms w i l l have a higher probability of breaking loose. T h i s diffusion follows an Arrhenius behaviour where each atom on the surface hops according to a rate of R = E  a  Roexp(—E /kT). a  is referred to as the site dependent activation energy of the surface atoms, which can  be modelled by E  a  = EQ + nE^, where E  0  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 initial 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 Miller etal. [3]). equal to a lattice vibration frequency [9]. T h e diffusing atoms w i l l move around on the surface until they settle in the nearest local energy m i n i m u m . This is a dynamical process, in which an overall equilibrium situation w i l l be reached w i t h 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 i n 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 w i d t h is gradually increasing as more and more layers of atoms are deposited. G r o w t h 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 i n the fashion of the Fresnel-Huygen's principle, where a  Chapter  1. Introduction  4  sharp irregularity i n the surface grows i n 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]. T h e evolution of the surface roughness is dependent on spatial frequency, or the size of the irregularities on the surface: the small features i n figure 1.1 smooth away whereas the large features grow. O n the other hand, when heating a surface w i t h an initial roughness higher than the equilibrium state, the processes described above w i l l tend to smooth the surface from the rougher to a smoother state, thus approaching the m i n i m u m free energy state from a higher energy configuration. We define the interface w i d t h , W, by W  2  = ^ ( h i - h )  (1.1)  2  where h is the average height of the surface and hi is the height of the i h site i n a discrete t  lattice model of size L. T h i s quantity gives us a statistical measurement of the surface roughness. M a n y studies of t h i n 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 w i d t h as a function of time, two distinct regions are observed: at times less than some 'crossover' time t , W increases as a power x  of time [4]: W(L,t)~t  (1.2)  fi  8 is called the growth exponent, and characterizes the time-dependence of the growth dynamics. T h i s interface w i d t h continues to increase until a certain value is reached: W saturates according to: W (L)~L  a  sat  (1.3)  where a is called the roughness exponent. The crossover time depends on the system size  Chapter  1.  5  Introduction  as: t  ~L  [i ~ t ]  Z  x  (1.4)  x  where z is called the dynamic exponent. T h e 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 )  W, (L)  (1.5)  \t,  at  Substituting i n from above we get that for an initially smooth surface, the interface w i d t h scales according to the so called Family-Vicsek W(L,t) f(x f(x  scaling relation [4, 17, 19]:  = L f(t/L ^) a  (1.6a)  a  < 1) ~ x  (1.6b)  3> 1) —> constant  (1.6c)  0  B y p l o t t i n g this relationship, we see that systems of different sizes L w i l l collapse onto one curve. Solving for z around t = t  x  we get: cx  Z  =  H  ( 1  -  7 )  T h i s 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 i n figure 1.2. The simulation shows the interface w i d t h of a growing surface according to the ballistic deposition ( B D ) rule on an initially flat, one-dimensional surface.  T h i s model w i l l be  discussed in detail i n chapter 2. The saturation of the interface w i d t h 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: E v o l u t i o n of the interface w i d t h by random deposition according to ballistic deposition on an initially flat, one-dimensional surface. T w o simulations are shown. short range diffusion is possible before more atoms are deposited. T h e interface w i d t h grows until there is a balance between the random deposition noise and this short range surface relaxation. M a n y physical processes can be incorporated into growth models, and the scaling parameters are signatures for different growth models. G r o w t h models that exhibit similar scaling exponents are said to belong to the same universality class. T h e 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 parameters. 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 devices, 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 ( L L S ) , reflective high energy electron diffraction ( R H E E D ) and low electron energy diffraction ( L E E D ) . In this thesis we w i l l use some of these techniques i n order to add to the continued investigation of surface physics by describing experiments i n which we have monitored the evolution of the semiconductor interface during epitaxial growth. T h e thesis contains six chapters, where i n 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 i n chapters 5 and 6 where the experiments we conducted have been described and summarized w i t h interpretations, conclusions and suggestions for further work.  Chapter 2  Surface G r o w t h M o d e l l i n g  2.1  Discrete G r o w t h M o d e l s  A n educational and simple approach is to describe surface growth by discrete rules which can be simulated by a computer. Q u i t e simple models display features that enhance our understanding of the microscopic processes in question. T h e temporal evolution of the t h i n film surface morphology is a dynamical process due to the combination of deposition noise, desorption, surface relaxation and diffusive processes .. Several discrete models have been studied [4, 9, 17, 19, 20, 21] and linked to 1  continuum growth equations through dynamical scaling. T h e simplest of the discrete growth models is that of random deposition ( R D ) , where an incoming particle falls at a randomly chosen location on the surface. T h e interface sites are uncorrelated, i.e. one site does not have any effect on the next, so the interface w i d t h grows indefinitely. T h e R D model has for this reason been said to have an infinite roughness exponent (see equation 1.3). T h i s is an unrealistic scenario for M B E growth, as the interaction of nearby surface atoms w i l l cause interface correlations. T h e scaling exponents for R D are [4]:  a = oo, 3 = 1/2  [RD]  (2.1)  G r o w t h 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 ln this work, we discuss one-dimensional surfaces, i.e. the surface height distribution h = h(x,t).  1  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 w i t h the sample upside-down i n the growth chamber.  However, since a 'lower height' often  corresponds to a higher number of nearest neighbours ( N N ) , 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 w i t h 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. T h e 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 ( B D ) model [8, 20, 21, 26] that was introduced in chapter 1. A l t h o u g h deceptively simple, this model exhibits characteristics that are observed in many growth phenomena, such as interface w i d t h saturation and outward lateral growth [4, 17, 27]. T h e 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. T h e 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\ — m a x [ / i j _ i , hi + l,h ] i+1  (2.3)  T h e incoming particle sticks to the first height-level where it has at least one N N . B D  Chapter  2.  Surface Growth  Modelling  • •  A  T  10  • •  • • B  i •  •  •  T  • A' i  •  B' i  Figure 2.1: G r o w t h rule for ballistic deposition (after Barabasi and Stanley [4]). is a model w i t h a unity sticking coefficient and no diffusion from thermal activation. T h i s w i 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 i n all outward directions,' like the flamefront on a burning field. A more detailed discussion of lateral growth w i l l come later. B D growth has been studied extensively [12, 17, 28, 29] and detailed numerical simulations i n 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 coefficients 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 i n figure 2.2. The initial shape used forms an integral part of the experiments conducted i n this work. The actual physical dimensions of this shape is a lateral pitch of 786nm, w i t h a height of about 250nm. To correctly account for these physical dimensions, the atoms i n the growth model have been given a size of 7 8 6 n m / (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). T h e dimensions of this shape are comparable to state-of-the-art processing limits, and hence semiconductor device dimensions today. T h e 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 i n h is broken and that cusps are formed between the bumps. These cusps were also observed i n a different system in figure 1.1. T h e 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.  T h e scaling relations i n equation 1.6 showed us that the interface w i d t h  should grow as t@ until it reaches a saturation level, W , at a characteristic crossover sat  time, t . x  T h i s was verified in simulations shown i n figure 1.2. For a surface of initial  roughness larger than this saturation level, the interface w i d t h w i l l decrease until it  Chapter  2.  Surface Growth  10'  Modelling  12  Initial height (*L ): a  100  —  30  10  convergence point  10 3  10"  10  10  flat  10  10  10  Figure 2.3: Smoothing of grating structures of different heights according to the ballistic deposition model. L = 1 2 8 lattice sites. reaches the equlibrium state. Some simulations were done to verify this behaviour, see figure 2.3. T h e i n i t i a l shape used was the same as that of figure 2.2, where the initial height was varied. We found that for the initially rough surface shapes, there are two crossover times: one separating W i n the i n i t i a 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 t \ and t x  x 2  i n figure 2.3, respectively.  Some observations were made when simulating the smoothing of structures w i t h vertical dimensions larger than ~ 3 * L . a  After the first crossover time, t \, x  the interface  w i d t h was found to drop dramatically. Several large initial 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 i n figure 2.2. We have seen that the interface w i d t h settles at the saturation w i d t h , 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')  =  a  ^  0  /2mix\  ,  .  / 2 w r x \  /(O  (2-5)  n=l  where the coefficients a  n  and b are defined as follows: n  a  n  2 /2nnx\ , = — / h(x,t) cos —-— )dx L J-L/2 \ L J  2 fL/2 b —— J ft(x,t) n  according to Fourier's theorem, and f(t)  sm  /2mrx\ , —jdx  is the time evolution of h(x,t).  (2.6a)  (2.6b) Let us consider  the decomposition of two shapes: a narrow and a wide 'bump', similar to the initial shape used in figure 2.2.  W h e n 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 , along w i t h all of the 6-terms, is zero. 0  However, for the wide peak, the a\ is positive, whereas 0,2 is negative. We define lateral growth as the transition i n which a 'bump' evolves from being narrow to becoming wide. Since the B D model simulated i n 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: a would die away monotonically, whereas a x  2  would go through zero before decaying to zero. Indeed, i n 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 i n i t i a 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 decomposed 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.  T h i s 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 w i 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 i n i t i a 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 i n continuum equations. T h e 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 w i l l have the form of a Langevin [31, 32, 33, 34, 35] continuum equation dh — = G(h,x,t)+rj(x,t)  2  (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, i n our experiments  Ijim/hr.  Higher 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]. 2  Chapter 2.  2.3.1  Surface Growth  Modelling  16  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 i n 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. T h e 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. T h i s rules out explicit dependence on h, so the equation must be constructed from spatial derivatives of h. T h e growth equation must obey rotation symmetry about the growth direction. T h i s rules out odd order derivatives i n the spatial coordinate, terms like Vh, etc.  Note that the terms ( V / i )  2  V(V /i), 2  and V / i both survive the transformation x —> —x 2  since they have an even number of derivatives in x. The last symmetry to obey is the inversion  symmetry in h [36, 37]. T h i s 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: ^-  = Vh 2  + V / i + ... + V h 4  2n  + V h(Vhf 2  + ... + V h(Vh) 2k  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]. T h e simplest continuum equation describing equilibrium  Chapter  2. Surface Growth  Modelling  17  interface fluctuations is: — = V h + (x,t)  [EW]  2  V  (2.10)  T h i s equation was analysed by Edwards and W i l k i n s o n [13] i n 1982, although i t was derived earlier by C h u i et.al. [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 i n surface growth is related to the fourth spatial derivative of the surface, not the second) w i t h conservative noise. T h e scaling exponents for the E W equation can be solved exactly, and i n 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 w i t h 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 / t - t e r m i n the continuum equation describ2  ing surface growth. T h i s term has also been linked to re-evaporation of surface atoms, see section 2.3.2. T h e E W model has been used successfully to interpret experimental results of strained InGaAs  on GaAs [1, 16].  W h e n the noise term is dropped i n E W , the solution takes on the simple form: h(x,t) = J2c e-' ^ Ux) /  (2.12)  )2t  n  n where d> {x) are orthonormal functions that span the basis space for h(x,0). n  T h e func-  tion Ci<f>i(x) is therefore the i h expansion term of the Fourier decomposition of the i n i t i a l t  surface shape, according to Sturm-Liouville theory on a periodic boundary-value problem. E a c h term (f> (x) dies away monotonically cx exp(—An t), 2  n  i n good agreement w i t h  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  -0.2  m)  -0.2  0.2  O  x  (n  m)  Figure 2.6: Decay of two similar shapes w i t h different widths according to the E d wards-Wilkinson model. B o t h 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 i n 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 w i t h the w i d t h of the 'bump' used as initial 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. T h e E W equation is there integrated numerically, and we see that the narrow bump widens, and the wide bump narrows. T h e y both approach the first harmonic, a\ cos (2irx/L),  as growth progresses. T h e '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 initial w i d t h of the surface shape. The numerical integrations performed i n 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: D i a g r a m showing one possible origin for the slope dependent term in the Kardar-Parisi-Zhang growth equation [2]. T h i s 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  + dh )  (2.13)  2  where Hf is the i h space point and n h time point of the surface height distribution on a t  t  discrete grid w i t h space and time-steps dh and dr, respectively. Periodic boundary conditions are implemented, and the Courant stability condition is obeyed in the simulations: 0 < udr/dh  2  = 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 w i l l have a higher density of steps. A s 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. T h i s description is not applicable to M B E . T h i s 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) + (vdtVh) ] / , 2  2  1 2  which  Chapter  2. Surface Growth  Modelling  20  Figure 2.8: Temporal evolution of a grating according to the K a r d a r - P a r i s i - Z h a n g equation, v = 10nm /s and A = O.Snm/s. The interface velocity due to growth is incorporated to show similarity to B D growth. 2  for |V/i| <C 1 can be expanded according to the binomial theorem:  m  ~  2  v +  ( 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 i n K P Z breaks the inversion symmetry i n 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. T h i s problem w i l l be addressed i n section 2.3.3. The K P Z equation is numerically integrated i n figure 2.8. The numerical scheme is similar to that of equation 2.13, and the initial shape is that of a periodic grating of pitch  Chapter  2.  Surface Growth  Modelling  21  0.2  1i  l  0.15  First Fourie r coefficient . a/t)  0.1 g<  0.05 Secor i d Fourier coefficient, a _(t)  0  2  -0.05  i  i  4  W  6  10  time (minutes)  Figure 2.9: Temporal evolution of the first and second coefficients of the Fourier decomposed grating during growth according to the K a r d a r - P a r i s i - Z h a n g equation. v — 10nm /s and A = 0.8nm/s. 2  786nm: Jf"  + 1  1T^  =  V  ^  ^  ~  +  2  2ik  +0(<fr + «fc')  P16)  A s previously mentioned, the K P Z equation i n I D scales w i t h exponents comparable to those of the B D model [2, 41, 42, 43, 44]: a = 1/2, B = 1/3  [KPZ]  (2.17)  T h e y are therefore i n the same universality class. The grating in figure 2.8 develops cusps and exhibits lateral growth [41], w i t h 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 a (t) 2  for B D and K P Z (see figures 2.5 and 2.9).  Chapter 2. Surface Growth Modelling  2.3.2  22  First Principles Derivation of Continuum Models  The continuum equations for surface growth are rooted i n the principle of mass conservation, where the number of atoms in the system is balanced i n a general continuity equation:  ^  ^  + V-j(x,t) =  0  (2.18)  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)  (2.19)  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]. T h i s is the continuum description of 'number of nearest neighbours', as the curvature is inversely proportional to the local radius: fi(x,t) oc -\7 h(x,t) 2  (2.20)  T h i s w i l l give us an equation that describes relaxation by surface diffusion [37, 45]: dh  ^-  dt  = -  F T  V /i 4  (2.21)  We include random deposition by adding a constant interface growth, F, as well as a random term, r)(x,t) [9, 2 2 ] :  dh  —  =  - K V  4  H  F + -q{x,t)  (2.22)  C/t-  Similar considerations can give rise to more terms i n 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 . T h e equation that incorporates both surface diffusion and relaxation is the so-called 'linear M B E equation': —  = vV h 2  - KV /I + 4  n(x, t)  (2.24)  B y Fourier transforming the deterministic part of this equation w i t h respect to the spatial coordinate, we get: rlhin  A  = u(-q )h(q ,  t) - K(q )h(q ,  2  x  dt where h(q ,t) x  4  x  x  is the Fourier transform of h(x,t),  spatial frequency). T h i s can easily be solved for  +K  The  two linear decay-terms (uV h 2  t)  (2.25)  and q is the transform variable (the x  h(q ,t): x  h(q ,t)^h(q ,0)e-^ ^ x  x  t  x  (2.26)  and K V / J ) generate a characteristic critical length 4  scale L it given by: cr  Lor* = 2vr ( - J  (2.27)  For spatial frequencies corresponding to smaller length scales, the fourth derivative term will be dominant, and for larger length scales, the second derivative w i l l be dominant, giving us the familiar E W equation. T h i s gives the familiar faster decay of small scale roughness, but even faster here than i n the E W equation. The increase of large scale roughness (like that observed i n figure 1.1) can be accounted for by considering diffusion bias. T h i s effect arises due to the potential barrier present at the edge of a descending step, also called the Schwoebel barrier [47, 48]. T h e barrier at  Chapter 2. Surface Growth  Modelling  24  the step-edge reflects an approaching atom w i t h some probability instead of letting it fall off the edge. T h i s results i n a current of surface atoms away from the descending step, proportional to the local slope of the surface: j oc Vh.  T h i s term gives us a different  version of the linear M B E equation [15]: = _ | z , | V / i - KV h + r)(x,t)  —  (2.28)  4  2  T h i s equation is unstable, and for spatial frequencies smaller than q i = 27r/L cr  t  crit  it w i l l  diverge. T h i s problem was solved for by Johnson et.al. [27, 49, 50] when they proposed a surface current which peaks at some intermediate slope: ] = FS a  J 1 + (am)  2  (2.29)  h  a  2  where F is the incident particle flux, S incorporates a Schwoebel parameter proportional a  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. T h e uphill current w i l l for small slopes be limited by the diffusion length of the atoms. T h e diffusion length is the average distance a surface atom w i l l travel before it nucleates w i t h another atom to form an island, or simply reevaporates off the surface. T h e high-slope l i m i t a t i o n is due to the decreased length of the terrace, as well as re-evaporation of the edge-atoms which w i l l eventually lead to diffusion over the unfavourable Schwoebel barrier. T h i s process becomes more probable as the steps get closer together [51]. T h e full growth equation overcomes the unstability problem of equation 2.28:  ¥ = - °° • (TT^W) " " '<*• FS  2V  KV4 +  )(  '  (2 30)  Finally, we mention an equation that is similar to the K P Z equation, but obeys mass conservation [10, 24]. T h e equation reads Bh  —  = -KV h 4  + X V (Vh) 2  1  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 conservation. In our experiments, we verify q dependence of the surface evolution, indicating 2  the presence of a second spatial derivative i n the growth equation. A m o n g the non-linear equations, this would favour K P Z over this last equation. The continued analysis of continuum equations for epitaxial surface growth w i 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 i n 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]: (2.32) (2.33) In discrete modelling, this could be dealt w i t h by simply subtracting the mean of the interface height at any time t [52]. Notice that a moving interface w i 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. Analytically, we can write the 'true' surface height, ri(x,t)  as [51, 52]:' (2.34)  where v(t) is the excess growth rate due to lateral growth. Substituting into the K P Z equation, we get: &H{x,t) dt  vV rl{x,t) 2  +  -(Vrl{x,t))  2  v(t)  (2.35)  Chapter 2.  Surface Growth  Modelling  26  T h i s corrected equation satisfies mass conservation globally (although not locally), and the growth mechanisms described by the equation are not changed. T h e 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 i n 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 ) of the Fourier series decomposition 0  of the grating shape. . T h e K P Z equation is a derivation based on mathematical simplicity [27]. U n 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 i n the specular direction is insensitive to structure on the reflecting surface [53], and limited information can be achieved by monitoring it. T h e nonspecular scattered light, however, contains information about the surface morphology. The intensity of the scattered signal at some non-specular angle w i 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 approximation). In that figure, we get constructive interference from the two incoming beams when \segmentA  — segmentB\  = m\[ , aser  w i t h m an integer. W i t h the notation i n that figure,  this means that at an angle 9 we are observing the scattered signal off a surface feature S  of spatial frequency: 27T  qx,obsewed = T (sin Oi - sin 9 ) "laser S  (3.1)  The details of the three-dimensional scattering from a surface was described by Lavoie [54], but we w i l l settle for the two-dimensional scattering geometry i n the plane of incidence. 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: D i a g r a m showing geometry of laser light scattering ( L L S ) .  Q, is given by [56, 57]: 1  dP  16TT cos Bi cos 9 \Qab{0i,O ,6)\ 2  2  A  where P  0  g(q)  2  S  s  (3.2)  4  and dP are the incident and scattered power, respectively, and a metallic  interface is assumed w i t h 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 i g h t [58]. 1  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 Q , ab  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 S k i n depth, S, derived from l / | f c | , where k = n w /c and n As - 4.392 + jO.476 at E = 2.54eV 1  2  z  Ga  laser  2  2  2  - k , where k = 16/xm , 2  1  x  A;  o s e  r  = 488nm  Chapter 3.  Elastic Laser Light  Scattering  29  1  o o  0.4 0.6 G r a t i n g d e p t h (\x m)  0.2  Figure 3.2: Simulated surface factor at 16yum approximation.  0.8  1  according to the Fraunhofer far field  1  density of scatterers. T h e surface factor is often taken to be the power spectral density ( P S D ) of the surface height distribution, h(x,t),  and is thus the only time-dependent  term i n this equation. T h i s 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, still assuming that there is no shadowing effect, as follows [51, 59]: (3-3) where the incident light is assumed to be laterally coherent, q is the in-plane component x  of the spatial frequency and q is the vertical component. In our experiment, q is given z  x  by equation 3.1 and,  aser  (cos 6i +  cos  0) S  (3.4)  Chapter 3. Elastic Laser Light  Scattering  2TT  9; + COS a r c s i n  COS  A laser  30  s i n& —  -  (3.5)  —  In this expression, 9 was substituted i n from equation 3.1. For a given observation S  angle, the experiment is sensitive to surface structure of a length scale L = 2n/q . x  In  the Fraunhofer approximation, an incoming plane-wave is assumed, as well as a point receiver for the scattered light. For incoherent i l l u m i n a t i o n , an ensemble average must be calculated. W h e n the surface height is small, equation 3.3 reduces to: |2  9(q)  J e  lQxX  (cos (q h(x, t)) + i sin [q h(x, t)) dx z  j e * (l  +  KQx) +  iqzh(q ,t)  iq  x  ${qx) + q\  where we recognize |/z(g ,£)| x  (3-6)  z  (3.7)  iq h{x,t))dx z  |2  (3.8)  x  (3.9)  h(q ,t) x  as the P S D of the surface. A specular component is also  present i n this equation. T h e intensity response from a sinusoidal grating is simulated in figure 3.2 as a function of depth of the grating. T h e response oscillates, reflecting the fact that at some grating depths, the top and the b o t t o m of the gratings w i l l be out of phase by an amount appropriate for destructive interference.  3.2  Scattering From Flat Surfaces  T h e surface factor, g(q), is simulated i n two plots i n figure 3.3 for spatial frequencies i n the scattering range w i t h Oi =25 degrees and A ;  a s e r  = 488nm. T h e numbers chosen here  reflect those used i n our experiments. T h e roughness on the surfaces i n the two plots is approximated by a random walk of interface w i d t h Inm and 25nm, respectively. T h e 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 (V- m ) x  1  Figure 3.3: Fraunhofer diffraction from a coherently illuminated random-walk generated rough surface w i t h standard deviation lnm (top) and 25nm (bottom). The strong peak at q = 0/j,m~ corresponds to the specular reflection. T h e effect of the 1  x  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 w i t h a periodic grating, the reflected signal w i l l have peaks at corresponding spatial frequencies. A sinusoidal grating of pitch 786nm w i l l have intensity peaks at q = 2ir/786nm x  = 8.0/im  _ 1  i n 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 i n figure 3.4. T h e increased surface  Chapter 3. Elastic Laser Light  Scattering  Q  32  x  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 b o t t o m plot still 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. T h i s 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 little effect on the intensity peaks the gratings cause. T h i s justifies the exclusion of random noise when numerically integrating growth equations i n 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. T h e surface shape can easily be verified by ex-situ methods such as A F M ; gratings before and after growth are shown i n figure  Chapter 3. Elastic Laser Light  Scattering  33  Figure 3.5: A F M images of gratings before (top) and after (bottom) regrowth. gratings have a pitch of 1.2pm, and the aspect-ratio is 1:1.  The  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. T h e fabrication of these w i l l be discussed in detail in chapter 4. A s mentioned i n chapter 2, this periodic shape can be decomposed into a Fourier series: h(x) = ao/2 + a i cos (2%x/L) + a cos (4TTX/L) + .... T h e shape is centered at the origin so 2  there are no odd expansion terms. The zero'th term is the mean of the surface and is lost in the specular beam. T h e first term, a\ cos (2nx/L) in section 3.3. The second term, a cos (4irx/L), 2  is detected at 8 . 0 / x m , as discussed -1  is seen at 1 6 . 0 / / m . A s discussed i n _1  chapter 2, the second Fourier coefficient of this grating w i 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 w i l l not show such a transition, and w i 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 m i n d that a signal that crosses zero and goes negative would show up as a m i n i m u m 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 i n 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 ° ) gallium arsenide (GaAs) by electron beam (e-beam) lithography i n conjunction w i t h a wet chemical etching system. T h e GaAs wafers used were supplied by Crystar of V i c t o r i a , B . C . [61]. T h e fabrication requires quite a few steps and exposure to several solvents and chemicals. A l l fabrication was performed i n a Class 1000 cleanroom, w i t h the exception of the e-beam lithography. We w i 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 i n four pieces by scribing the edge of the wafer w i t h a diamond-tip scriber. T h e cleaved edges follow the crystal planes, in this case the ( O i l ) and the ( O i l ) crystal planes to form four (equally large) quarter wafers. Each of these quarter wafers becomes an individual sample. T h i s rather large sample size is dictated by the temperature measurement system used i n the M B E during growth. A larger sample gives a signal that is easier to interpret for the diffuse reflectance spectroscopy ( D R S ) system. T h i s method w i l l be explained in chapter 5.  35  Chapter 4.  Grating  Fabrication  36  Figure 4.1: A m e r i c a n standard GaAs (100) wafer w i t h crystal orientations indicated [5, 6]. T h e cross indicates how the wafer was cleaved. A b o u t seven drops of 950K molecular weight polymethylmethacrylate ( P M M A ) positive photoresist is put on a quarter wafer w i t h a disposable pipette, and immediately spun at 9000 r p m for one minute. T h e spin creates a uniform thickness film of 150nm [62]. T h e uniformity can be inspected by the color of the film, and for this large sample the edge-effects are insignificant. T h e 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. T h e cleanroom is intended for lithography, and is illuminated by lamps w i t h yellow filters to block the U V . T h e sample is now removed from the cleanroom to do electron beam lithography. T h i s 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 Bake  9,000rpm More than 3 hours, ~ 180°C, hotplate  Developer Rinse  M I B K : I P A , 1:3, ~ 2 5 ° C , 90-120 seconds I P A , 20 seconds  Photoresist stripper  Acetone, ~ 5 0 ° C , 20 minutes  Table 4.1: Parameters for preparation of photoresist films, pattern transfer and stripping of photoresist. w i t h DesignCad [65] graphical software to define the patterns.  Operating parameters  for the S E M and the e-beam lithography are summarized i n 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 i n methyl-iso-butyl-ketone  ( M I B K ) and  iso-propyl-  alcohol (IPA) i n a ratio of 1:3 for two minutes followed by a stop solution of pure I P A for 20 seconds.  M e t h a n o l 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 sample. The pattern transfer is done using the so-called piranha-etch H S0 2  4  : # 02 : H0 2  2  (it etches 'anything'):  (1:1:18) that etches GaAs at a rate of 480nm/mm  [62]. More de-  tails on this etch w i l l be explained in section 4.3. T h e remaining resist is finally removed by dissolving it i n 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 )-blow. T h e sample is 2  now ready for M B E growth. T h e hot acetone treatment seemed crucial to the success of regrowth on the sample. W i t h o u t the hot acetone treatment, every attempt on regrowth failed as the surface got coated w i t h a white film as soon as the semiconductor growth was started.  Chapter 4. Grating Fabrication  38  Acceleration voltage Emission current Aperture opening W r i t i n g time after flashing  30kV 20/iA #4 from 2 hours to one day  Centre-to-centre distance Magnification Condenser lens W o r k i n g distance Dose  300A 1000 12 and above 17mm 12.0 f C / p o i n t  Table 4.2: Parameters for electron beam lithography i n 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 l e x 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 w i d t h , which N P G S interprets as a single e-beam pass when it comes to writing 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 i n 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]. W h e n the  same calibration was attempted by the author, w i t h all other parameters equal, at least  Chapter 4.  Grating  Fabrication  39  Figure 4.2: S E M - i m a g e showing the selective growth around the beam dump area of the exposed pattern. T h e 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 . W h e n 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. W h e n attempting to regrow on this hardened carbon i n 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 i n this work used a ~ 2mm diameter laserbeam incident on a GaAs  substrate w i t h uniform gratings, so the larger the area with gratings, the  better (up to 2 m m 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 * 90fim . 2  Each of these squares takes on the order of 10-  20 minutes with a 786nm pitch grating under ideal writing conditions. The m a x i m u m 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. T h e image is tilted, but one can still resolve the total lateral etch-distance immediately under the P M M A , as well as the w i d t h of the bottom flat to estimate the vertical to lateral etchrate ratio. E t c h 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  E t c h i n g rates using wet etching systems are crystal orientation dependent. For instance, when etching parallel to the [011] direction, one achieves characteristic w i t h overhangs.  undercuttings  However, when etching parallel to the [011] direction, one obtains the  shapes in figure 4.3. T h e {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  30sed  60sed  90sed  -0.8  -0.2  0 x  (JLI  0.2 m)  -0.8  -0.2  0 x  0.2 m)  Figure 4.4: Simulation of etched surface for relative etchrate ratio of (100) to 111 of a) 2.1 and b) 1.5. critical relative etchrate ratio of (100) to the {111} planes is 1.73 . For a larger ratio, the 1  long time etched gratings will become triangular, whereas for a small ratio, the gratings w i l l eventually annihilate, see figure 4.4. T h e S E M - i m a g e i n figure 4.3 indicate that this etchrate ratio is about 1.84. T h i s number has been verified elsewhere [62, 66, 67].  T h e 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. 1  Chapter 5  Experiment  5.1  Experimental Setup  We w i l l perform regrowth on top of the gratings discussed i n chapter 4 to investigate the temporal evolution of their shape. T h e M B E machine is a single growth chamber V G Semicon V 8 0 H w i t h effusion-cells. A 488nm A r g o n 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 i n the back-scattering direction, and the signal is captured by a standard lock-in amplifier. T h i s setup enables us to monitor surface roughness  5.9672  270  Figure 5.1: A n g u l a r dependence of Fraunhofer diffraction from a sinusoidally textured surface of pitch 78Qnm. 42  Chapter 5.  Experiment  1 03227'  43  X 2 . 8 0 K  Figure 5.2: S E M - i m a g e of gratings after unsuccessful regrowth. Notice beam-dump in top left corner as discussed i n 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 16//m  _ 1  signal emerges at 305° (in the notation of the figure), which corresponds to  backscattering at 9 = 55°. S  5.2  Cleaning the Sample  Sample cleanliness is crucial for regrowth experiments, as growth i n M B E is highly selective and sensitive to surface contamination by for example carbon. T h e sample preparation requires many steps, i n which the sample has been exposed to several solvents, acid and temperature treatments. A n example of failed regrowth is shown i n the S E M - i m a g e in figure 5.2. T h e little dots on the gratings have a diameter of less than a micrometer. Several of these little 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.  44  Experiment  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 w i t h a thick oxide and volatilizes carbon to C0 . 2  T h e sample is transferred to the M B E and put in position for growth.  The oxide just added is removed by a hydrogen etch, i n which the surface oxide reacts w i t h cracked i^-molecules, which are highly reactive and etch away the arsenic and the gallium oxides. T h e /^-molecules are cracked by a hot filament (1800°C), and the substrate reaches a temperature of approximately 200°C7 i n this process. pressure of the hydrogen is I0~ torr, 5  T h e partial  and the i f - e t c h 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  Growth starts : here  1I  1  1  1  1  1  ^600  T3 Q.  580  E , 540  o  O  o 52J5B0SD.: i  10  20  30 40 time (min)  50  60  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 i f - e t c h the oxide is  removed, and the R H E E D pattern develops into diffraction spots when it reflects off the (100) facet. T h e oxide is now removed, and the sample is ready for regrowth [54, 69].  5.3  Annealing Prior to Regrowth  T h e temperature is measured using the diffuse reflectance spectrosopy ( D R S ) technique that was invented i n our lab [70, 71]. T h i s technique applies a broadband light source whose signal transmits through the sample, and reflects off the back of the substrate. T h e 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  12.5  13  46  13.5  14  14.5  1/kT (eV )  15  15.5  16  16.5  _1  Figure 5.5: Determining the activation energy of the E d w a r d s - W i l k i n s o n u parameter for GaAs. ±2°C. We start our experiment by ramping the temperature towards the growth temperature, which is usually set at 6 0 0 ° C . A s the temperature approaches the growth temperature, the L L S signal starts to decrease, as shown i n figure 5.4. T h e Edwards-Wilkinson parameter v can be calculated from the slope of a log-plot, as the slope is equal to —2vq\. A s 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 —E . a  Such a plot is shown in figure 5.5  [69]. T h e results obtained during several regrowth runs are also included there; however, as a l l the regrowth experiments were done at temperatures very close to 6 0 0 ° C , there was  Chapter 5.  Experiment  47  not enough spread i n the data to achieve an activation energy for the surface smoothing during growth. Note the p r o x i m i t y of the points from the regrowth-experiments w i t h the two different q values. T h e E W parameters i n this plot were derived assuming q\ x  behaviour. A fourth order spatial derivative i n the growth equation would spread the two sets of points apart by about a factor of three. D u r i n g pre-annealing, the activation energy was found to be lAeV. includes data obtained by T o m P i n n i n g t o n by growing InGaAs  on GaAs.  Figure 5.5 also A n activation  energy of 2.6eV was obtained i n that study [69].  5.4  Regrowth on GaAs  Gratings  W h e n 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.  1004°C, the As-ceW is at 4 0 0 ° C , and the V : III pressure is about 10~ torr 9  T h e Ga-cell temperature is  ratio is between 15 and 35. T h e base  and the partial pressure of As is on the order of  10~ torr. 5  G r o w t h was started when the sample temperature reached 6 0 0 ° C ('run 1'). A s the shutter to the Ga-cell was opened and the growth started, the L L S signal dropped drastically. T h e signal continued to decrease, until it very abruptly came up again a couple of minutes into the growth. W e interpret this as the change i n sign of the second Fourier coefficient, that was discussed i n chapters 2 and 3. A second r u n was performed w i t h the temperature lowered to 5 9 2 ° C , but otherwise identical to the first r u n ('run 2'). T h e results can be seen i n figure 5.6.  Chapter  5.  Experiment  48  ,  ,  r  time (min) Figure 5.6: F i t of K a r d a r - P a r i s i - Z h a n g equation to experimental data: T = 6 0 0 ° C (top, run 1) and T = 592°C (bottom, run 2).  Chapter 5.  49  Experiment  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  Discussion  W h e n simulating the second coefficient of the Fourier decomposition of the grating according to growth by the K P Z equation, we achieve the graphs in figure 5.6. The simulations are superimposed on the experimental data, and the parameters used i n the simulations were: v = 10nm /s, 2  A = lAnm/s  for the experiment at 600°C and v =  oAnm fs, 2  A = A.Qnm/s for the experiment at 592°C. For the first run at 6 0 0 ° C , we see an excellent fit as the L L S - s i g n a l 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 5 9 2 ° C , w i t h qualitatively similar results. There is no evidence of a second bump like the one we saw at 78 minutes into the first run. T h e 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 w i t h temperature, and that the parameter for the non-linear term decreases w i t h increasing temperature. T h i s is reasonable: the surface atoms w i l l have more kinetic energy at higher temperatures, and w i l l therefore have a higher probability of escaping the stronger bonding at an inclined surface w i t h 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 uV h 2  term might be expected to increase  w i t h 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 r u n 1. Note that this is not the 786nm gratings used for the L L S . T h e gratings of pitch 1.3pm should have a decay time of more than an hour . T h e gratings have therefore barely had time to fill i n , as the growth only lasted 1  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 r u n 2. These gratings have pitch 786nm, giving them a decay time of about 700 seconds or 12 minutes . T h e 30 minute growth 2  of this run has had time to fill i n the gratings quite well, and the formation of cusps has occurred. These cusps were also observed i n simulations of the B D and K P Z models for surface growth i n chapter 2. Note that the y-axis is expanded compared to figure 5.7. T h e 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 x  at even time intervals. T h e objective was to cleave the sample  Decaytime obtained from r =  l/(vq ). Here, q = 27r/1.3/zm = 4 . 8 / i m 2  x  T = 4340s  l/{vql) = l / ( 5 . 4 n m / s * (16/um- ) ) = 723s  2  2  1  2  -1  and  v = 10nm /s, giving 2  Chapter 5.  Experiment  52  time (min) Figure 5.9: G r o w t h of GaAs on GaAs gratings w i t h AlGaAs  spacer-layers.  and selectively etch it to show growth structure i n the S E M or A F M . A n example of this technique was shown i n figure 1.1. A l t h o u g h we were not successful i n obtaining an image of the regrown material, we d i d see an interesting development i n the L L S signal (see figure 5.9). Every time the Al-cell was opened, there was a momentary decrease i n the L L S - s i g n a l , probably caused by lower reflectivity i n the AlGaAs  film.  We monitored the surface w i t h two lasers, the usual 488nra Ar laser and a 325nm HeCd  laser. T h e new laser adds a signal at 24u.mr  l  laser gives us the lQ/j-m^ . 1  through the same port that the Ar  T h e two signals were separated by filters before detection.  T h e spatial frequency monitored w i t h the HeCd  laser is exactly at the t h i r d 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  3  T h i s indicates w i t h good accuracy  Exponent obtained from log 1.92/log ( 2 4 ^ m / 1 6 / u m ) = 1.61 _1  _1  3  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. T h i s is approxi-  mately the time between the beginning of the growth and where our fit superimposes the experimental data. There is no change i n sign of the L L S signals this time. We believe that the initial shape decomposed w i t h a negative second Fourier coefficient for this experiment.  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 = L /(\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]. 4  2  Chapter 6 Conclusions  In this work, we have described experiments that were performed i n 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]: dh  (6-1)  ~dt  T h i s indicated a breach of inversion symmetry i n 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 K P Z - e q u a t i o n was dealt  w i t h 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 ( L L S ) , and  showed that for small surface heights, the intensity obtained from the L L S was comparable to the power spectral density ( P S D ) of the evolving surface structure. T h i s 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 i n accordance with the Kardar-Parisi-Zhang model, and then Fourier transformed and compared to the P S D  54  Chapter  6.  Conclusions  55  obtained from the L L S experiments. G o o d agreement was found between the two, and lateral growth was observed, quantified by a change i n sign of the second harmonic Fourier coefficient of the decomposed grating during growth. We also obtained A F M - i m a g e s from the post-growth gratings. The gratings were found to develop cusps, similar to behaviour seen i n simulations of the B D and the K P Z models. We found that the parameters involved i n the K P Z - e q u a t i o n were temperature dependent, where the surface relaxation term parametrized by v increased w i t h increasing temperature, and the parameter for the non-linear term, A, decreased w i t h increasing temperature.  T h i s agrees well w i t h 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 = A=  10nm /s lAnm/s  T = 592°C  v = A=  5Anm /s 4.0nm/s  2  2  Table 6.1: Parameters obtained by fitting the K a r d a r - P a r i s i - Z h a n g equation to experimental data. v i n the absence of growth, we concluded that activation energy of the surface atoms was E  a  — lAeV  during annealing of GaAs.  A s far as we know this work represents the first quantitative analysis of the smoothing of a textured semiconductor surface i n terms of a continuum growth equation. T h i s is also the first measurement of the non-linear, slope-dependent term in the K a r d a r - P a r i s i Zhang equation for GaAs epitaxial film growth. E a r l y on i n the growth, we verified a dependence on the spatial frequency as q ., 2  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 i n the growth. T h i s is i n agreement w i t h studies that indicate that the E W term vV h 2  is the dominant term early i n the growth, and K P Z behaviour is dominant later  on, giving a q]. dependence i n the growth equation [52]. b  We have i n this work verified the validity for the K a r d a r - P a r i s i - Z h a n g growth equation for spatial frequencies less than 16/zm~ . T h i s corresponds to length scales of about x  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  AFM  atomic force microscope  AlGaAs  A l u m i n u m G a l l i u m Arsenide  AIN  A l u m i n u m Nitride  a  coefficient of n h Fourier cosine-series decomposition  BD  ballistic deposition  b  coefficient of n h Fourier sine-series decomposition  DRS  diffuse reflectance spectroscopy, temperature measurement  n  t  n  t  system  EQ  activation energy of a free surface atom w i t h no bonds  E  A  surface atom activation energy  E  N  bonding energy  EW F FE-SEM  Edwards-Wilkinson flux  term or growth rate of surface field  emission scanning electron microscope  GaAs  G a l l i u m 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 i h space point on the n h time step t  t  H(x,t)  'true' surface height: H(x,t)  h  Planck's constant  h(x, t)  height distribution of semiconductor interface  57  = h(x,t) — v(i)t  List of  Notation  h(q ,t)  spatial Fourier transform of h(x,t)  IPA  iso-propyl-alcohol  j(x,t)  current of surface atoms  KPZ  Kardar-Parisi-Zhang  k  Boltzmann's constant  L  lateral system size; also: size of surface feature  x  L  critical length scale:  LEED  low energy electron diffraction  LLS  laser light scattering  MBE  molecular beam epitaxy  MIBK  methyl-iso-butyl-ketone  m  local slope of surface  NbN  Niobium Nitride  NN  nearest neighbours  PMMA  ploymethylmethacrylate  PMT  photo multiplier tube  PSD  power spectral density  q it  critical spatial frequency:  q  in-plane component of spatial frequency  q.  component of spatial frequency perpendicular  crit  cr  x  z  L  c r  i  t  2TT(K/U) I  =  1  q i cr  t  =  2  (V/K) I 1  2  to plane  R  surface atom hopping rate  RQ  surface atom hopping rate at 0 K e l v i n  RD  random deposition  RDwSR  random deposition w i t h surface relaxation  List of  Notation  RHEED  reflectance high energy electron diffraction  SEM  scanning electorn microscope  Si  Silicon  STM  scanning tunneling microscope  S  Schwoebel barrier parameter  TEM  transmission electron microscope  t  transition time when surface w i d t h saturates  v(t)  growth velocity of surface  W  interface width given by rms fluctuation  z  dynamic exponent: z =  a  roughness exponent: W  3  growth exponent: W ~ t&, t small  •n(x, t)  stochastic, random Gaussian deposition noise  4> (x)  orthonormal basis functions  K  diffusion constant  A  electromagnetic wavelength; also: coefficient of  a  x  a/3 sat  n  ~  L  a  non-linear term i n 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  0  angle between scattered light and surface normal  Q  solid angle  S  Bibliography  [1] T . Pinnington, C . Lavoie, and T . Tiedje. Surface morphology dynamics i n strained epitaxial I n G a A s . Phys. Rev. Lett, 79(9):1698-1701, 1997. [2] M . K a r d a r , G . Parisi, and Y . - C . Zhang. Phys. Rev. Lett, 56:889-892, 1986.  D y n a m i c scaling of growing interfaces.  [3] D . J . Miller, K . E . Gray, R . T . K a m p w i r t h , and J . M . M u r d u c k . Studies of growth instabilities and roughening i n sputtered N b N films using a multilayer decoration technique. Europhys. Lett, 19:27-32, 1992. [4] A . - L . Barabasi and H . E . Stanley. Fractal Concepts in Surface University Press, 1995. [5] N . W . Ashcroft and N . D . M e r m i n . Solid State Physics. New York, 1976.  Growth. Cambridge  Holt, Rinehart and W i n s t o n ,  [6] A m e r i c a n X t a l Technology, D u b l i n , C A , U S A . Sheet indicating for X-tal orientations for (100) wafers.  US-wafer  standard  [7] Y . Okada, S. A r m a n o , and M . Kawabe. Nanoscale oxidation of GaAs-based semiconductors using atomic force microscope. J. Appl. Physics, 83(4): 1844-1847, 1998. [8] B . B . Mandelbrot. The Fractal Geometry of Nature.  Freeman, San Francisco, 1982.  [9] S. Das Sarma and P. Tamborenea. A new universality class for kinetic growth: one-dimensional molecular-beam epitaxy. Phys. Rev. Lett, 66:325-328, 1991. [10] J . V i l l a i n . C o n t i n u u m models of crystal growth from atomic beams w i t h and without desorption. J. Phys., I 1:19, 1991. [11] M . Siegert and M . Plischke. Phys. Rev. Lett, 73:1517, 1994. [12] F . Family. D y n a m i c a l scaling and phase transitions i n interface growth. J. Phys., A19:L441-L446, 1990. [13] S. F . Edwards and D . R . W i l k i n s o n . T h e surface statistics of a granular aggregate. Proc. Roy. Soc. London, A381:17-31, 1982. [14] J . Lapujoulada. T h e roughening of metal surfaces. Surf. Science Rep., 20:191-249, 1994. 60  Bibliography  61  [15] M . F . Gyure, J . J . Zinck, C . Ratsch, and D . D . Vvedensky. Unstable growth on rough surfaces. Preprint received at M R S conference i n San Francisco, spring 1998. [16] T . P i n n i n g t o n , C . Lavoie, and T . Tiedje. roughening of relaxed I n G a A s on G a A s . 1997.  Effect of growth conditions on surface J. Vac. Sci. Tech., B15(4):1265-1269,  [17] F . F a m i l y and T . Viscek. Scaling of the active zone i n the E d e n process on percolation networks and the ballistic deposition model. J. Phys., A 1 8 : L 7 5 - L 8 1 , 1985. [18] F . F a m i l y and T . Viscek. Dynamics 1991.  of Fractal Surfaces. W o r l d Scientific, Singapore,  [19] F . F a m i l y and J . G . A m a r . The morphology and evolution of the surface in epitaxial and think film growth: a continuum model with surface diffusion. W o r l d Scientific, Singapore, 1991. [20] T . Viscek. Fractal Growth Phenomena.  W o r l d Scientific, Singapore, 1992.  [21] J . Feder. Fractals. P l e n u m Press, New York, 1988. [22] D . E . Wolf and J . V i l l a i n . G r o w t h w i t h surface diffusion. Europhys. 394, 1990.  Lett,  13:389-  [23] H . Park, A . Provate, and S. Redner. Interface growth w i t h competing surface currents. J. Phys., A24:L1391-L1397, 1991. [24] Z . - W . L a i and S. Das Sarma. K i n e t i c roughening w i t h surface relaxation: Continuum versus atomistic models. Phys. Rev. Lett., 66:2348-2351, 1991. [25] F . Family. Scaling of rough surfaces: Effects of surface diffusion. J. Phys., A 1 9 : L 4 4 1 L446, 1986. [26] H . E . Stanley. Introduction to Phase Transitions University Press, New York, 1971.  and Critical  Phenomena.  Oxford  [27] T . Halpin-Healy and Y . - C . Zhang. K i n e t i c roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Physics Reports, 254:215-414, 1995. [28] P. Meakin, P. R a m a n l a l , L . M . Sander, and R . C . B a l l . surfaces. Phys. Rev. A, A34:5091-5103, 1986.  Ballistic deposition on  [29] R . B a i o d , D . Kessler, P. R a m a n l a l , L . M . Sander, and R . Savit. D y n a m i c a l scaling of the surface of finite-density ballistic aggregation. Phys. Rev. A, A38:3672-3678, 1986.  Bibliography  62  H w a T . and M . K a r d a r . Avalanches, hydrodynamics and discharge events i n models of sandpiles. Phys. Rev. A, 45:7002-7021, 1992. N . G . van K a m p e n . Stochastic Processes in Chemistry A m s t e r d a m , 1981.  and Physics. N o r t h - H o l l a n d ,  T . H w a and M . K a r d a r . Phys. Rev. Lett, 62:1813, 1989. S. K a t z , J . L . Lebowitz, and H . Spohn. J, Stat. Phys., 34:497, 1984. R . B r u i n s m a and G . A e p p l i . Phys. Rev. Lett, 52:1547, 1984. J . K o p l i k and H . Levine. Phys. Rev. B, 32:280, 1985. C . Herring. Structure and Properties of Solid Surfaces. University of Chicago, Chicago, 1953. i n Structure and Properties of Solod Surfaces, edited by R . Gomer and C . S. S m i t h , pp. 5-72, see discussion on pg. 64. W . W . M u l l i n s . J. Appl. Physics, 28:333, 1957. S. T . C h u i and J . D . Weeks. Dynamics of the roughening transition. Phys. Rev. Lett, 40:733-736, 1978. B r i a n Wetton. Private communication, 1997. M . E d e n . . In F . Neyman, editor, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume I V . University of California, Berkeley, 1961. H . G u o , B . Grossmann, and M . Grant. Kinetics of interface growth i n driven systems. Phys. Rev. Lett, 64(11):1262-1265, 1990. K . Moser, D . E . Wolf, and J . Kertesz. Numerical solution of the K a r d a r - P a r i s i - Z h a n g equation i n one, two and three dimensions. Physica, A l 7 8 : 2 1 5 - 2 2 6 , 1991. A . C h a k r a b a r k i and R . Toral. N u m e r i c a l study of a model for interface growth. Phys. Rev. B, 40:11419-11421, 1989. P. M e a k i n . T h e growth of rough surfaces and interfaces. Physics 289, 1993.  Reports, 235:189-  C . Herring. Effect of change of scale on sintering phenomena. 21:301-303, 1950.  J. Appl.  Physics,  L . G o l u b o v i c and R . Bruinsma. Surface diffusion and fluctuations of growing interfaces. Phys. Rev. Lett, 66:321-324, 1991.  Bibliography  63  R . L . Schwoebel and E . J . Shipsey. Step motion on crystal surfaces. J. Appl. 37:3682-3686, 1966. R . L . Schwoebel. Step motion on crystal surfaces i i . J. Appl. Physics, 1968.  Physics,  40:614-619,  A . W . Hunt, C . Orme, D . R . M . W i l l i a m s , B . G . Orr, and L . M . Sander. Instabilities in M B E growth. Europhys. Lett, 27:611-616, 1994. M . D . Johnson, C . Orme, A . W . Hunt, D . Graff, J . Sudojino, L . M . Sander, and B . G . O r r . Stable and unstable growth i n molecular beam epitaxy. Phys. Rev. Lett, 72:116-119, 1994. T o m Tiedje. Private communication, 1998. Michael Plischke. Private communication, 1998. M . B o r n and E . Wolf. Principles  of Optics. Pergamon, New York, 1975.  C . Lavoie. Light scattering measurements of surface morphology during molecular beam epitaxy growth of GaAs-based semiconductors. P h D thesis, U B C , 1994. J . A . Ogilvy. Theory of Wave Scattering A d a m Hilger, 1991.  from Statistically  Rough Surfaces. B r i s t o l .  F . U . Hillebrecht. J. Phys., D13:1625-1631, 1980. E . L . Church, H . A . Jenkinson, and J . M . Zavada. Optical Eng., 18:125, 1979. Vighen Pacradouni. Private communication, 1998. C o r i Pistonesi. Private communication, 1998. W . B . Gauster. Private communication, 1998. Nanostructure and semiconductor physics department, Sandia National Laboratories, Albuquerque, N M , U S A . Contact persons Fran Seeds or D a v i d R e i d , telephone +(250) 479 9922, fax +(250) 479 2734, postal address: 721 V a n a l m a n A v e , V i c t o r i a , B C , Canada, V 8 Z 3B6. Web: http://www.jmcrystar.com. A . Busch. A n electron beam lithography system: Setup and characterization. Master's thesis, University of B r i t i s h C o l u m b i a , 1994. Hitachi, L t d . Instruction tron Microscope, 1991.  Manual for Model S-J^lOO Field Emission  Scanning  Elec-  Bibliography  [64] N a b i t y Lithography Systems, Bozeman, M o n t a n a , U S A . Nanometer eration System, Version 7.4, 1994. [65] A m e r i c a n S m a l l Business Computers, Pryor, O K , U S A . DesignCad Manuals, Version 6, 1992.  64  Pattern  2-D  Gen-  Reference  [66] Peng Chen. L a t e r a l wet oxidation of A l G a A s and its applications i n high index contrast distributed feedback ridge waveguides. Master's thesis, University of B r i t i s h Columbia, 1998. [67] Manoj Kanskar. Private communication, 1996. [68] Sayuri Ritchie. Private communication, 1997. [69] Measurement of I n G a A s smoothing lead by T o m P i n n i n g t o n . [70] S. R . Johnson, C . Lavoie, M . K . Nissen, and T . Tiedje. O p t i c a l method for measuring temperature of a substrate material w i t h a temperature dependent bandgap. U.S. patent 5568978, 1996. [71] S. R . Johnson and T . Tiedje. M e t h o d for determining the temperature of semiconductor substrates from bandgap spectra. F i l e d for patent M a r c h , 1997.  

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