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Kinetically determined surface morphology in epitaxial growth Jones, Aleksy K. 2008

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Kinetically Determined Surface Morphology in Epitaxial Growth by Aleksy K. Jones B.Sc., Laurentian University, 2006 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics) The University of British Columbia (Vancouver) October, 2008 © Aleksy K. Jones 2008 Abstract Molecular beam epitaxy has recently been applied to the growth and self assembly of nanostructures on crystal substrates. This highlights the impor- tance of understanding how microscopic rules of atomic motion and assembly lead to macroscopic surface shapes. In this thesis, we present results from two computational studies of these mechanisms. We identify a kinetic mechanism responsible for the emergence of low- angle facets in recent epitaxial regrowth experiments on patterned surfaces. Kinetic Monte Carlo simulations of vicinal surfaces show that the preferred slope of the facets matches the threshold slope for the transition between step flow and growth by island nucleation. At this crossover slope, the surface step density is minimized and the adatom density is maximized, respectively. A model is developed that predicts the temperature dependence of the crossover slope and hence the facet slope. We also examine the “step bunching” instability thought to be present in step flow growth on surfaces with a downhill diffusion bias. One mechanism thought to produce the necessary bias is the inverse Ehrlich Schwoebel (ES) barrier. Using continuum, stochastic, and hybrid models of one dimensional step flow, we show that an inverse ES barrier to adatom migration is an insufficient condition to destabilize a surface against step bunching. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Theory of Epitaxial Crystal Growth . . . . . . . . . . . . . . 4 2.1 Continuum theory in the sub-monolayer regime . . . . . . . . 4 2.1.1 Nucleation length . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Critical nucleus size . . . . . . . . . . . . . . . . . . . 5 2.1.3 Scaling with temperature and flux . . . . . . . . . . . 6 2.2 Evolution of macroscopic shape . . . . . . . . . . . . . . . . . 9 2.2.1 Kinetic step edge barriers . . . . . . . . . . . . . . . . 10 2.2.2 Faceting in and out of equilibrium . . . . . . . . . . . 11 2.3 Stochastic modeling and simulation . . . . . . . . . . . . . . . 12 2.3.1 Solid-on-solid model . . . . . . . . . . . . . . . . . . . 12 2.3.2 Kinetic Monte Carlo method . . . . . . . . . . . . . . 14 iii Table of Contents 3 Diffusion Bias and Step Flow Growth . . . . . . . . . . . . . 18 3.1 Analytical step flow model . . . . . . . . . . . . . . . . . . . 20 3.2 Stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Hybrid model . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Step bunching in 2D . . . . . . . . . . . . . . . . . . . . . . . 28 4 Kinetic Faceting During Patterned Regrowth . . . . . . . . 32 4.1 Step density as a function of vicinal angle . . . . . . . . . . . 32 4.1.1 Estimating θmin via the nucleation length . . . . . . . 35 4.1.2 Improving the estimate of θmin . . . . . . . . . . . . . 38 4.2 Slope selection . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Appendices A Surface Characterization . . . . . . . . . . . . . . . . . . . . . 55 A.1 Edge D/F ratio . . . . . . . . . . . . . . . . . . . . . . . . . 55 A.2 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . 58 B Altered Growth Conditions . . . . . . . . . . . . . . . . . . . . 60 B.1 Straightened step edges . . . . . . . . . . . . . . . . . . . . . 60 B.2 Change of binding energies . . . . . . . . . . . . . . . . . . . 61 iv List of Figures 2.1 Dimer survival rate vs. temperature . . . . . . . . . . . . . . . 7 2.2 Cluster density nI and nucleation length ln vs. Γ. . . . . . . . 8 2.3 Exponent χ and critical nucleus size n∗ . . . . . . . . . . . . . 9 2.4 Kinetic energy landscape of a growing crystal surface. . . . . . 11 2.5 The solid-on-solid model . . . . . . . . . . . . . . . . . . . . . 13 2.6 Periodic boundary conditions on a vicinal surface. . . . . . . . 14 3.1 Adatom density profiles . . . . . . . . . . . . . . . . . . . . . 22 3.2 Schematic representation of a step train. . . . . . . . . . . . . 23 3.3 Evolution of step bunching from an analytical model . . . . . 25 3.4 Evolution of step bunching via a stochastic model. . . . . . . . 27 3.5 Evolution of step bunching via a hybrid and a stochastic model. 28 3.6 Step unbunching with a negative ES barrier. . . . . . . . . . . 29 3.7 Evolution of a pit instability during step flow growth. . . . . . 30 4.1 Step density S and (b) adatom density n vs. T . . . . . . . . 34 4.2 Surface morphology at different temperatures and slopes. . . . 34 4.3 Adatom density with two different step-edge configurations. . 36 4.4 Nucleation length ln and a/θmin vs. temperature. . . . . . . . 37 4.5 A typical plot of the interface width versus line length. . . . . 39 v List of Figures 4.6 Test of Eq. (4.7), with simulation data and a line of best fit. . 41 4.7 Schematic representation of facet formation on an initially mounded surface. . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.8 Experimental atomic force microscope cross sections of 1D gratings on GaAs (001). . . . . . . . . . . . . . . . . . . . . . 43 4.9 Grating profiles for kMC simulations on a patterned surfaces. 44 4.10 Temperature dependence of the facet slope. . . . . . . . . . . . 46 A.1 Γe vs. temperature. . . . . . . . . . . . . . . . . . . . . . . . . 57 A.2 Fractal dimension on a surface with nucleation suppressed, and the surface slope constrained to a/ln, as shown in Fig. 2.6. 59 B.1 Step density vs. surface slope where step edges have been straightened. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B.2 Step density vs. surface slope with a lateral binding energy of El = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi Acknowledgements I would like to thank my two research supervisors, Jörg Rottler and Tom Tiedje, for their scientific insight and general help in navigating the rocky waters of academia. No matter what was going on, they always managed to make time to discuss things with me. And thank you to Gren Patey for agreeing to review my thesis on such short notice. Thank you to my friends and family for their encouragement and under- standing through the last couple of years. I swear I wasn’t screening my calls, Mom – I just wasn’t home! Above all, I’d like to thank Amanda. None of this would have been possible without her support. vii Chapter 1 Introduction A major challenge in the field of epitaxial growth is to understand and control the relationship between macroscopic surface shapes and underlying atomic processes that create these shapes[1, 2]. As the science of surface growth adavances, the focus has shifted from macroscopic morphology, such as sur- face smoothness [3], to the fabrication of repeatable nanoscale structures [4]. To successfully continue this transition, a greater theoretical understanding of the governing parameters of self assembly is needed. Two good examples of the type of nanostructure driving this shift in focus are quantum dots and nanowires. These are structures whose applications have the potential to revolutionize many areas of technology – as soon as robust methods of self assembly are discovered. In this thesis, we present results from computational studies of surface growth. We examine two specific macroscopic growth phenomena that arise out of simple rules for atomistic self-assembly: slope selection on patterned substrates, and the “step bunching” instability thought to be present on vicinal surfaces [5, 1]. Surface growth is a discrete and stochastic process, and systems are often held in far-from-equilibrium during growth. Thus, kinetic effects dominate the surface morphology, and the growth can be difficult to treat analytically. Here we use kinetic Monte Carlo (KMC) simulations of 1 Chapter 1. Introduction a solid-on-solid model to describe and predict the evolution of the crystal surface – from atomic level detail to macroscopic morphology. Molecular beam epitaxy (MBE) is a process of surface growth which has applications in both research and industry. As simply described as possible, MBE is the directional deposition of atoms onto a substrate, where they diffuse and assemble into a crystal. It is typically performed under ultra- high-vacuum (UHV) conditions. In industry MBE is used in the manufacture of semiconductor devices; for example, silicon and gallium arsenide (GaAs) crystals used in the construction of microprocessors can be grown using MBE. Additionally, the precise control over surface structure in MBE is invaluable in a research context. The source and substrate can vary widely; in the case of homoepitaxy, the substrate material is the same as the source, and in heteroepitaxy it is not. In this study, we examine homoepitaxy with system parameters chosen to match those of GaAs. This allows comparison of our results with existing experiments. However, the results are general, and are not tied to any particular substance. Other important parameters in MBE include the shape or orientation of the substrate. A substrate miscut just a few degrees from the plane of a facet can have completely different growth properties than a flat surface. Also, the evolution of preexisting surface shapes can lead to a wide variety of surface phenomena which are not seen on flat stable surfaces, such as faceting and relaxation of large amplitude features. In this work, we examine growth on GaAs substrates with preexisting grooves. These distinctions are not only of theoretical interest. For instance, a slightly miscut surface can lead to what is called “step flow growth”. This 2 Chapter 1. Introduction growth mode often results in surfaces that are smoother, and therefore more suitable to most practical applications [6]. Furthermore, in semiconductor device manufacturing, regrowth on patterned substrates is commonly used as a means to control the lateral distribution of deposited material in order to synthesize optical waveguides or electron confinement structures [7, 8, 9]. This thesis is organized as follows. In chapter 2, we discuss some theoret- ical background which is relevant to the following chapters, as well as some details about the simulation methods used. In chapter 3, the phenomenon of step flow in crystal growth is examined, with particular reference to the “step bunching” instability in one dimension. Chapter 4 describes how the dependence of surface morphology on vicinal angle leads to a novel mecha- nism of kinetic facet formation in epitaxal growth on patterned surfaces. In the Appendices, we revisit some peripheral topics raised in chapter 4 in more detail. Finally, we conclude with chapter 5. 3 Chapter 2 Theory of Epitaxial Crystal Growth 2.1 Continuum theory in the sub-monolayer regime As described in the introduction, epitaxy is a conceptually simple process. There is a competition between the equilibrating process of surface diffusion and the adatom flux, which pushes the system away from equilibrium. This competition generates a wide variety of surface morphologies which are ob- served during epitaxial growth. In the latter part of this thesis, we focus on discrete models for surface growth. However, continuum theory plays a role as well: theoretical insight gained from the continuum theory of sub-monolayer growth is invaluable when interpreting the results from simulations of dis- crete models. The continuum theory in the sub-monolayer regime is well developed, and describes how we may parametrize the surface in terms of a basic length scale, and the diffusion and deposition rates. The general approach to the theory of surface evolution has been to start with the absolute minimum detail necessary to describe a given growth pro- cess or state, and add complexity if needed. Along this line of thought, the 4 Chapter 2. Theory of Epitaxial Crystal Growth basic atomic processes required to actually develop any ordered structure at all are deposition, diffusion, and nucleation. 2.1.1 Nucleation length In the presence of an adatom flux, the surface adatom density n begins to rise. The adatom density is defined as the number of free atoms on the surface per unit area. In this theory, it is assumed that surface diffusion leads to a uniform adatom density over the entire surface. As the adatoms diffuse on the surface, and their density rises due to the flux F , they eventually meet each other and nucleate immobile “islands”, which begin to act as adatom sinks. As islands form on the surface, they collectively gather more and more adatoms, eventually suppressing the adatom density n below the level at which any new islands will form. At this point, the island density nI has reached its peak. There is now an identifiable length scale on the surface: the average distance between island centers. We call this length the nucleation length ln, and it can be determined through the island density nI [1]. In particular, ln ∝ 1/√nI . This provides an easy way to obtain ln on a simulated surface. 2.1.2 Critical nucleus size As we have seen, the length scale of the surface is set by the density of islands nI . In the above section, we have assumed that these clusters of two or more adatoms are permanent and immobile. A more nuanced approach allows for adatom detachment as well as attachment. In this case, we take into account the binding energies Eb, and must identify the largest cluster size that is not 5 Chapter 2. Theory of Epitaxial Crystal Growth considered stable: the critical nucleus size n∗. At low temperatures or high fluxes, this critical size is expected to be n∗ = 1. At higher temperatures the smallest stable nucleus is expected to not only be larger, but to exhibit some symmetry which enhances binding energy. For example, on a cubic lattice, a cluster with 4 members is much more stable than a three member cluster, as each adatom is bound to two nearest neighbours. In the range of experimentally relevant temperatures and flux rates, the critical nucleus size is therefore either one or three. As an illustration of this, Fig. 2.1 shows the ultimate survival rate of two-adatom clusters on a surface simulated using kinetic Monte Carlo methods. The temperature range is typical for experiments at a flux F of one monolayer per second (ML/s). At the lower end of the range, we can see that almost all the dimers survive, indicating a critical nucleus size of n∗ = 1. At higher temperature the dimers do not usually survive, indicating a transition to n∗ = 3 [1]. 2.1.3 Scaling with temperature and flux Having defined a surface length scale, we can consider the factors which affect this length. In general, these factors include the substrate binding energy Es, the flux rate, the surface temperature, and the critical nucleus size. We can include Es and the temperature in one parameter: the diffusion rate for a free adatom D. This rate follows an Arrhenius law, with the substrate binding energy Es as the activation energy. Here we have ignored the relative strength of the substrate and lateral binding energies – for a more detailed parametrization which includes this, see appendix A.1. By considering the number of sites that an adatom will typically visit, and 6 Chapter 2. Theory of Epitaxial Crystal Growth D im er Su rv iv al Pr o ba bi lit y T (K) 600 650 700 750 800 850 0 0.2 0.4 0.6 0.8 1 Figure 2.1: Eventual survival rate for 2-adatom clusters, as a function of temperature. In this simulation the flux F = 1. Simulation details are given in section 2.3.2 the density of sub-critical clusters, we can determine the rate of stable island formation. The total number of islands formed is proportional to this rate multiplied by the layer filling time 1/(Fa2), where a is the lattice spacing. This allows us to write the nucleation length as a function of the diffusion to flux ratio Γ = D/F , and the critical nucleus size [1]: ln ∝ aΓχ, (2.1) where χ = n∗ 2 + 4n∗ , (2.2) Again using kinetic Monte Carlo simulations of a flat surface, we can apply 7 Chapter 2. Theory of Epitaxial Crystal Growth these equations to obtain an estimate for the critical nucleus size. Using ln ∝ 1√ nI (2.3) we obtain nI ∝ Γ−2χ. (2.4) With equations (2.1) and (2.4), we can use the cluster density data from simulated surfaces to examine their scaling properties. l n (a) Γ n I (1/a2 ) 10−2 100 102 104 106 108 10−4 10−2 100 100 101 102 Figure 2.2: Cluster density and nucleation length as a function of Γ = D/F . The dashed line is a power law fit to the primary region of interest ( 650−820 K). The result of these simulations can be seen in Fig. 2.2, where we have plotted nI and ln against Γ. As we can see from the varying slope of the curve, the power law exponent χ is not constant. Through the instantaneous 8 Chapter 2. Theory of Epitaxial Crystal Growth slope, we can determine this exponent, and using Eq. (2.2), an effective critical nucleus size. By fixing the flux F = 1, we plot these two quantities as a function of temperature in Fig. 2.3. The transition from n∗ = 1 to n∗ = 3 occurs over approximately the same temperature range as indicated in Fig. 2.1, where the effective critical nucleus size n∗ was measured directly. This indicates that nucleation theory is applicable in this temperature range. T (K) n ∗ χ 500 600 700 800 900 0.2 0.4 0.6 1 2 3 4 Figure 2.3: Exponent χ and critical nucleus size n∗, obtained from the in- stantaneous slope of Fig. 2.2. 2.2 Evolution of macroscopic shape The previous continuum theory is only applicable in the sub-monolayer growth regime. In the limit of long growth times and long length scales, the be- haviour of the surface is better described by non-linear continuum differential 9 Chapter 2. Theory of Epitaxial Crystal Growth equations [10]. Although a full discussion of these equations is beyond the scope of this work, there are some aspects of long term growth which are important to the present study. 2.2.1 Kinetic step edge barriers The ultimate behaviour of a growing surface is determined by the direction of its adatom current. So far we have assumed that diffusion is isotropic, but there is ample evidence that adatoms have direction preferences based on the kinetic energy landscape of the surface. This “diffusion bias” can manifest itself in an uphill or downhill adatom current. The case of anisotropy without height preference is also interesting [11], but is not relevant to the phenomena discussed here. The diffusion bias determines the stability of a surface: an uphill diffusion bias causes the roughness of a surface to grow without bound, while a downhill bias causes it to smooth out. A well known example of a kinetic barrier which results in a diffusion bias is the Ehrlich Schwoebel (ES) barrier [12, 13]. The rationale behind this barrier is that adatoms traveling to a lower terrace experience an energetically unfavourable state in mid-jump, due to lack of neighbours. This results in a kinetic barrier which tends to push adatoms back from the downhill step edge, leading to a net uphill current and an unstable surface. Some substances, such as GaAs, exhibit stable epitaxial growth, remaining smooth over time. This suggests a downhill diffusion bias, which could be caused by a number of atomic mechanisms [14, 15]. Two examples of plausible barriers that produce a downhill diffusion bias are the negative ES barrier EES, and the incorporation barrier Eib, pictured in Fig. 2.4. 10 Chapter 2. Theory of Epitaxial Crystal Growth Figure 2.4: Kinetic energy landscape of a growing crystal surface, depicting a negative ES (EES) and positive incorporation (Eib) barrier. 2.2.2 Faceting in and out of equilibrium A facet is generally defined as an area where the crystal surface is flat. Both in and out of equilibrium, facets are an example of local behaviour leading to non-local effects. Namely, the local atomic interactions lead to long-distance surface correlations. In equilibrium, facets result from the minimization of the surface free energy. This surface energy is locally represented by the surface tension. The minimum in the free energy implies that the surface tension must be the same at all points on the surface – otherwise a surface atom could migrate and reduce the total free energy. In three dimensions with continuous symmetry, this results in a spherical crystal surface. However, because crystals have a symmetry which is dependent on lattice structure, the surface tension is anisotropic. In this case, the equilibrium crystal shape 11 Chapter 2. Theory of Epitaxial Crystal Growth can be determined by the Wulff construction [2]. In a system that is held out of equilibrium, like a growing crystal, the situation is slightly different. In this case, the facets are determined by the surfaces that have a constant growth rate, with largest facets being the ones that grow slowest. Two dimensional surface growth is somewhat of a special case: there is often only one direction that new material arrives from, meaning that the surface shape is unchanged in the absence of some unbalanced adatom current. Kinetic faceting on surfaces is determined by surface currents. An excel- lent example of this is the mechanism which produces pyramidal mounds of a given slope in epitaxially grown metal surfaces [16, 17]. In materials with a positive ES barrier, adatom current is uphill, meaning that any mounds on the surface will increase in slope without bound. However, practically this uphill current is often balanced out by a slope dependent “downhill funnelling” current. This funnelling has been attributed to a number of dif- ferent mechanisms, and when it is equal to the uphill current the slope of the mound stabilizes and generates a facet [17]. In chapter 4, we detail another mechanism for kinetic facet formation. 2.3 Stochastic modeling and simulation 2.3.1 Solid-on-solid model By nature, epitaxy is a discrete process. The sub-monolayer and long term evolution of the surface is adequately described by continuum equations; however, the scale in between is home to a rich array of surface morphologies 12 Chapter 2. Theory of Epitaxial Crystal Growth which are not yet well described by any continuum approach. In order to bridge the gap between experiment and continuum theory, we simulate the surface via discrete models of epitaxy. Given currently available computing power, the typical size of these simulations is on the order of one to ten microns, and the time scales examined are up to a few hours [18]. We have chosen to use the solid-on-solid model, a lattice model named for its dissallowance of overhangs or covered vacancies. For simplicity, and for specific applicability to GaAs, we model epitaxial growth on a cubic lattice [19] with lattice constant a. In this particular solid-on-solid model, every adatom on the surface is bound by a substrate binding energy Esub, and a lateral binding energy NnElat which depends on the number of nearest neighbours Nn. The allowed processes, as shown in Fig. 2.5, are deposition, attachment, nucleation, detachment, and step edge crossing. There are also Figure 2.5: The solid-on-solid model 13 Chapter 2. Theory of Epitaxial Crystal Growth specific implementations of the solid-on-solid model which limit the slope of the surface. For instance, it is sometimes useful to restrict the height of steps to one adatom [20]. In the simulations considered here, it is not entirely clear what the energy landscape looks like around a double step; however, the situation occurs so infrequently that this restriction is unnecessary. The model is usually implemented with periodic boundary conditions. For simulation of vicinal surfaces, we can alter these boundaries slightly. As shown in Fig. 2.6, we can enforce a given surface slope θ = dh/dx by adding or subtracting a given height when an adatom crosses a surface boundary. Figure 2.6: An illustration of periodic boundary conditions on a vicinal sur- face. 2.3.2 Kinetic Monte Carlo method The model is advanced through time by the kinetic Monte Carlo (kMC) method, which is used to simulate a system which has a known group of 14 Chapter 2. Theory of Epitaxial Crystal Growth events, each occurring at a given rate. KMC is also used to model the evo- lution of chemical reactions [21, 22]. This method, when combined with the solid-on-solid model, allows access to time scales that are completely inacces- sible to more detailed computational methods, such as molecular dynamics. Unfortunately, it has the disadvantage that the list of possible events most be known in advance. For this reason, kMC is less able to predict emergent behaviour than more fine grained methods [23]. When applied to the solid-on-solid model, the allowed events are adatom deposition and hopping. Deposition occurs at the externally controlled flux rate F , and hopping occurs at rates determined by Arrhenius’ rate law: ν = ν0e −βE, (2.5) where β = 1/kBT . The rate constant v0 is normally determined empirically. In this kMC implementation, the activation energy E is given by E = Esub +NnElat +NESEES, (2.6) where Esub is the substrate binding energy that every adatom shares, and NnElat is the lateral binding energy, which is dependent on the number of nearest neighbours m. The Ehrlich Schwoebel barrier EES allows us to con- trol the diffusion bias on the surface, and is dependent on the number of adjacent inter-terrace hop locations NES. We typically use binding energies of Esub = 1.25 eV, Elat = 0.35 eV, and EES = −0.05 eV. These energies are chosen to match those of GaAs [24]. The general algorithm for kMC is as follows: 1) Enumerate all the possible events, and their rates Ri 15 Chapter 2. Theory of Epitaxial Crystal Growth 2) Add the rates together, obtaining the total rate Rt = ∑ i Ri (2.7) 3) Obtain the probability that any event will occur next, based on its rate, and normalized by Rt: Pi = Ri Rt (2.8) 4) Randomly choose an event to execute, weighted by it’s probability 5) As kMC simulates Poisson processes, in which the time between events is given by an exponential distribution, we increment the system time by drawing a random time from this distribution using a random num- ber rn ∈ (0, 1]: Ti+1 = Ti + ln rn Rt (2.9) 6) Start the next iteration The most time consuming portions of this algorithm are steps 3 and 4. In the past, these steps have been accomplished by obtaining a cumulative probability and stepping through the list, a method which yields an event execution time which is linear in the number of possible events ne. Later, a more advanced approach using balanced binary trees was used, leading to an execution time of order O(log(ne)) for each event [25]. By taking advantage of the finite (and relatively small) number of types of events, and by using a slightly different memory structure we can further shorten the execution and update time to be logarithmic in the number of possible event types. In the systems we simulate ne is on the order of 10 6 to 107, but the number of 16 Chapter 2. Theory of Epitaxial Crystal Growth event types is only 25, so the improved algorithm leads to a large speedup, especially in larger systems [26]. The kMC algorithm can be incredibly efficient, as it is a rejection free algorithm: every timestep results in some evolution of the surface. This efficiency only applies when all the fastest rates are relevant to the evolution of the system. In surface growth modelling at higher temperatures, this is not always the case: much of the computing time is spent calculating the trajectory of adatoms across flat, relatively vacant surfaces. This amounts to a random walk, which is already a well understood process. In order to “freeze out” this fast motion, a larger step size can be adapted for free adatoms [18]. This adaptive step size improves execution times considerably above system temperatures of 800K, sometimes by as much as ten times. 17 Chapter 3 Diffusion Bias and Step Flow Growth We have seen in Chapter 2 that a positive Ehrlich-Schwoebel (ES) barrier ultimately leads to unstable surface evolution. By creating a downhill diffu- sion bias, a negative ES barrier stabilizes the surface with respect to surface roughness. However, according to Villain and Pimpinelli [1], the presence of this bias leads to another type of instability: the step bunching instabil- ity. Step bunching refers to the terrace edges on a vicinal surface merging together and producing macrosteps. In the step flow growth regime, the majority of the adatoms arriving at the surface eventually attach to already existing terrace edges, causing the terraces to grow outward. The step bunching instability pertains to the evolution of the spacing of these terrace edges. Island nucleation has no effect on these phenomena; in this chapter we will neglect its effects. In the ideal continuum model, the size of a terrace is subject to either negative or positive feedback, depending on the type of diffusion barrier present. For example, an ES barrier directs deposited adatoms towards uphill step edges. Each adatom that attaches onto this uphill step reduces the size of it’s terrace of origin, a negative feedback effect. In contrast, an incorporation 18 Chapter 3. Diffusion Bias and Step Flow Growth or inverse ES barrier produce the opposite effect: adatoms are directed to the downward step, and large terraces grow. The step bunching instability is thought to be evidence of a negative diffusion bias, and in the aforementioned theory, the magnitude of this bias has no effect on the ultimate stability of a step train [27]. However, GaAs is thought to have an inverse ES barrier [24, 28], and yet does not consistently exhibit step bunching during growth [29]. The contradiction is solved by closer examination of the required conditions for unrestrained step bunching. We find that the existence of a diffusion bias is a necessary, but not sufficient condition for instability. Due to specific effects related to the non-continuous nature of the crystal surface, and the fluctuations introduced by the stochastic nature of growth, the bias must be of sufficient strength. Of the two types of barriers that lead to a negative diffusion bias, we find that only the incorporation barrier can lead to step bunching, while an inverse ES barrier does not. We examine this problem at three varying levels of detail: 1) a one- dimensional difference equation, generated via the continuum theory of Bur- ton, Cabrera and Frank (BCF) [34], 2) a one-dimensional stochastic model, with no detachment or nucleation, and 3) a hybrid model, in which we add deposition noise to the flux in the analytical model. In addition, our general findings are supported by a full kMC simulation of vicinal surfaces in two dimensions, and including diffusion and detachment. 19 Chapter 3. Diffusion Bias and Step Flow Growth 3.1 Analytical step flow model To obtain a difference equation describing the evolution of a given step train, we begin with the BCF equation for step density in one dimension, with no detachments and no evaporation. Assuming that diffusive motion is much faster than step motion, the result is a steady-state diffusion equation relating the adatom density n and flux F : D ∂2n ∂x2 = −F. (3.1) The solution to this equation is n(x) = −Fx 2 2D + Ax+B. (3.2) We can solve this equation for A and B using the adatom current at x = 0 and x = l. In order to do this, we make use of Fick’s law, relating surface current and adatom density at the terrace boundaries: j(x = 0) = −D∇n(0), (3.3) j(x = l) = D∇n(l), (3.4) where j is the adatom current at a given location and downward current is defined as positive. As we are considering a system with no dettachment, the current at the edge of a terrace is entirely outward-facing and is determined solely by a hopping frequency and the fraction of hopping adatoms that will exit the terrace. The sum of this exit current is determined by the total terrace area and the adatom deposition rate. In all cases, the total magnitude of exit current must be Flt, where lt is the length of the terrace. When there is a 20 Chapter 3. Diffusion Bias and Step Flow Growth step edge barrier, however, this current becomes unbalanced. For example, an adatom at the edge of a terrace, in the presence of a barrier, has an off- terrace hopping rate [1] νd = ν0e −β(Esub+Eb), and a rate for hopping back into the terrace νu = ν0e −β(Esub), where β = 1/kBT . Taking the ratio of these values, we obtain the ES and incorporation lengths: les = ae βEes , and lib = ae βEib , (3.5) which can be interpreted as the effective distance to a step edge when the adatom is actually a distance a away [19]. Current at a terrace boundary is proportional to the adatom density ρ, and the current is scaled according to the aforementioned lengths. Using Fick’s law, we obtain our boundary conditions: D∇n(0) = −n(0)D lib , (3.6) D∇n(l) = n(l)D les . (3.7) Solving for A and B, we obtain A = F 2D l(l + 2les) 1 + lib + les , and (3.8) B = Alib. (3.9) Thus we obtain a final expression for the adatom density on a terrace. First defining λn = ln + 2les ln + lib + les , (3.10) we obtain n(x) = F 2D (−x2 + (lx+ libl)λn) . (3.11) 21 Chapter 3. Diffusion Bias and Step Flow Growth No Barrier n (1/ a) Positive ES Negative ES x (a) n (1/ a) Positive IB x (a) 0 10 20 300 10 20 30 0 0.1 0.2 0 0.1 0.2 Figure 3.1: Comparison of the adatom density profiles for different step-edge barriers. Note that negative ES barriers do not have a large effect on adatom density. We plot this distribution for various barrier values in Fig. 3.1. Note that negative barrier values have far less effect on the density profile than positive barriers. Given the solution for the adatom density, we can solve for the current at the terrace edges. The velocity of a given step is then given by the downward current off of the upper terrace, plus the upward current from the lower terrace: vn = −jn(0) + jn−1(l). (3.12) Here positive current is defined as downhill, and the labels are given by figure 3.2. 22 Chapter 3. Diffusion Bias and Step Flow Growth Figure 3.2: Schematic representation of a step train. Of primary interest is the evolution of the terraces themselves. The rate of change with respect to time of a given terrace length is given by ∂tl = vn+1 − vn. (3.13) Using Ficks law, and our solution to the diffusion equation, we can solve for ∂tl: ∂tl = F [ ln(1− λn) + ln−1 ( λn−1 2 − 1 ) + ln+1 ( λn+1 2 )] (3.14) Typically we are not interested in individual terraces, but in larger sys- tems consisting of a very large number of them. In simulations this is usu- ally represented by periodic boundary conditions and some smaller number of terraces( around 10-100) [30]. We created a system of 80 terraces, with an average length of 64 lattice spacings, which was then evolved numerically using Eq. (3.14). Starting terrace lengths were perturbed by a small random amount. An important consideration when solving a system of terrace lengths is the behaviour when the lengths go to 0. Lengths below 0 are not physical, and there is nothing in Eq. (3.14) preventing l from going below 0, so this 23 Chapter 3. Diffusion Bias and Step Flow Growth condition must be built into the integrator. Ordinarily, this is done via a “step repulsion” force [31]. However, a continuous repulsion often introduces some longer range interactions between step edges which are not explicitly prescribed by the adatom dynamics. We have developed a more direct method implemented within the inte- grator itself. At each time step, each terrace length ln is adjusted based on contributions from ln, ln−1, and ln+1. When ln ≈ 0, the adatom flux is small, and the remaining adatom contribution comes from the neighbouring terraces. If this contribution causes the terrace to have a length of less than 0, the sign of the effect is reversed, and the terrace is forced to grow instead. This rule is meant to emulate the discrete step flow case; when an adatom falls from the terrace above, it will lengthen the upper terrace, and shorten the lower one. When the lower terrace has a length of 0, this cannot happen, and a reasonable assumption is that the adatom will instead fall past the l = 0 terrace, thereby lengthening it by one unit, and shortening the lower terrace by one. This rule has a large effect on the macroscopic behaviour of the system. With an ES barrier such that les < 1, the system without this rule is unstable against unrestrained bunching, but with this rule, the bunching is limited to a large extent. In order to see why this is the case, imagine that a large terrace has appeared in a system (perhaps by random fluctuations). As this terrace is larger than most others, it will be larger than terrace n+2, meaning that terrace n + 1 will shrink quite quickly to size 0. With terrace n + 1 at around 0 length, the leading edge of the large terrace will move forward at a velocity of half its normal speed, due to approximately every second adatom 24 Chapter 3. Diffusion Bias and Step Flow Growth σ /σ ma x Monolayers No barrier Strong negative ES barrier Incorporation barrier 0.1 1 10 100 0.1 0.02 0.05 0.1 Figure 3.3: Evolution of step bunching from an analytical model. Note the weak bunching behaviour for negative ES barriers. RMS deviation σ is scaled by the maximum bunching possible σmax. falling past the next terrace, and failing to lengthen terrace n. This causes an abnormally fast shrinking of large terraces, and ensures that steps will not bunch to a large extent. In Fig. 3.3, we see the RMS deviation from the average terrace length σ (a useful measure of bunching) plotted as a function of time. The deviation is scaled by the maximum possible for this step train σmax: when all the step edges have merged into one. We can see that the extent of bunching for a strongly negative ES barrier (−0.2eV) is almost negligible. Even the bunching produced by a 0.2eV incorporation barrier is not large compared to the maximum value possible. We conclude that, in the case of this semi- analytical model, a negative ES barrier does not lead to significant bunching, 25 Chapter 3. Diffusion Bias and Step Flow Growth and while the step bunching produced by an incorporation barrier is signifi- cant, it is restrained in extent. 3.2 Stochastic model In order to confirm that the results from the semi-analytical model are valid, we compare them to step bunching results from a kMC simulation of a 1D system. In this model, adatoms are deposited with a given rate F and diffuse with a rate D. All diffusion events happen with equal probability, and an ES barrier leads to a preferred hopping direction at the step leading to a lower terrace. In this case, we have also chosen to neglect all nucleation and step detachment. As in the analytical model, the stochastic treatment fails to reproduce step bunching behaviour to any large extent. In Fig. 3.4, we show the extent of bunching as a function of system time (again measured by RMS deviation from mean terrace length σ). The relative extent of bunching is not very high for any size barrier, indicating that the system is not unstable towards step bunching. The behaviour for a system with a negative ES barrier is very similar to a system with no barrier at all, and in contrast to the analytical solution, both these systems do exhibit significant bunching. 3.3 Hybrid model The results from the stochastic system are appreciably different from the results of the analytical model. As the major difference between the two 26 Chapter 3. Diffusion Bias and Step Flow Growth σ /σ ma x Monolayers No barrier Negative ES barrier Positive ES barrier 0.2 0.5 1 2 5 10 20 50 0 0.02 0.04 0.06 0.08 Figure 3.4: Evolution of step bunching via a stochastic model. Note the similarity in bunching behaviour for zero and negative ES barriers. RMS deviation σ is scaled by the maximum bunching possible σmax. models is the random fluctuations present in the stochastic model, we can seek to bring the two sets of results in line through the addition of “noise” to the solution of the analytical model. The noise in the system is largely in the form of variations in the instantaneous adatom flux. As the deposition of adatoms is a Poisson process, we replace the original uniform flux that any terrace receives by a flux value based on a random draw from an appropriate Poisson distribution. As we can see in Fig. 3.5 , this added noise produces results that are very similar to the results from the purely stochastic model. We can draw two conclusions from this data: 1) the step bunching behaviour, in the absence of a very strong diffusion bias, is primarily dependent on the “noise” present 27 Chapter 3. Diffusion Bias and Step Flow Growth σ /σ ma x Monolayers Stochastic, negative ES barrier Hybrid, negative ES barrier Hybrid, incorporation barrier 0.1 1 10 20 0 0.02 0.04 0.06 0.08 0.1 Figure 3.5: Comparison of the evolution of surface roughness between a fully stochastic model, and an analytical model with noise added. The curves are averaged over 150 runs. Note the agreement between the stochastic and hybrid models. in the system, and 2) an inverse ES barrier is insufficient to produce unstable step bunching during growth. 3.4 Step bunching in 2D The one dimensional results from the above sections are confirmed by kMC simulations in two dimensions. We performed simulations of a much more realistic (and still quite simple) solid-on-solid model which included step detachment and nucleation. We found no evidence of step bunching in the presence of an inverse ES barrier. This model leads to very similar results: 28 Chapter 3. Diffusion Bias and Step Flow Growth evenly spaced step trains remain relatively evenly spaced as time evolves. Furthermore, on surfaces where steps are initially bunched, the bunching decreases over time, regardless of the magnitude of the negative ES barrier. This can be seen in Fig. 3.6: an initially bunched surface, and its unbunched configuration after a long growth period. Figure 3.6: Step unbunching with a negative ES barrier. This study has also produced an interesting observation concerning the long term stability of a surface experiencing step flow growth. It has long been thought that, even with a positive ES barrier, a surface undergoing step flow growth would be stable if the rate at which a terrace filled in was less than the time required to nucleate a new island [32]. The idea behind this theory is that if terraces fill in faster than islands can grow, the growing terraces will “absorb” any islands that might form, stabilizing the surface against fluctuation. We find that this is not actually the case: given sufficient growth time, any system with a positive ES barrier is always unstable, regardless of the vicinal angle. This instability can be explained by reference to the “effective ES bar- rier” of a mound on a terrace. If a monolayer island forms on a terrace, 29 Chapter 3. Diffusion Bias and Step Flow Growth Figure 3.7: Evolution of a pit instability during step flow growth. an adatom has a chance to escape that mound which is based on the ES barrier; if that chance is high enough, the island will probably be absorbed before another island can nucleate on its surface. If, by random fluctuation, a mound appears on the surface, an adatom has a lesser chance of escaping to the terrace; it must cross many individual ES barriers on it’s way down. Effectively, the mound has an ES barrier which is larger than that of a mono- layer island. Eventually, due to random fluctuations, a mound possessing an effective ES barrier large enough to destabilize the system will appear. Given enough time, this can happen for any positive ES barrier, and any vicinal slope. The principle of an effective ES barrier can also be applied to “pits” which appear on the surface. Here we define pits as surface vacancies more than a monolayer in depth. If a pit reaches a critical depth, the time for 30 Chapter 3. Diffusion Bias and Step Flow Growth an adatom to reach the bottom of the pit will be greater than the terrace filling time. In this case the pit is unstable, and will continue to grow. The development of such a pit instability can be seen in Fig. 3.7. 31 Chapter 4 Kinetic Faceting During Patterned Regrowth 4.1 Step density as a function of vicinal angle The step density S is defined as the total length of step edges on a surface, divided by the total surface area. This is a useful parameter because it contains information about the degree to which the surface is ordered, and is also experimentally measurable. In this section, we examine the step density in different growth regimes through kMC simulation. In Fig. 4.1(a) we calculate S as a function of vicinal angle θ for several different temperatures under steady state growth conditions. As one would expect, S rises linearly with θ for large values of θ; in this regime, the surface morphology is dominated by terrace edges, and higher slopes imply more of these edges. However, this behaviour does not hold in regions of lower vici- nal angle. For T > 450◦C the step density is non-monotonic as a function of surface slope θ [33]. In particular, the step density has a minimum at a non- zero slope θmin. The position and character of the minimum are temperature dependent: at higher temperatures the position shifts to smaller angles and 32 Chapter 4. Kinetic Faceting During Patterned Regrowth becomes more sharply defined. At lower temperatures, the minimum disap- pears entirely (see Appendix A for more information on the low temperature regime). The adatom density n is closely related to the step density. Figure 4.1(b) shows n as a function of surface slope. We note that it has a maximum at slopes close to, but slightly smaller than θmin in the temperature range 400‰ to 600‰ studied here. As mentioned in chapters 2 and 3 a vicinal surface grows either in a step flow mode, in which all deposited adatoms attach to existing step edges, or in a nucleation dominated regime in which new steps are continuously being created by nucleation of islands [1]. The middle part of Fig. 4.2 shows a top view of a surface at a slope corresponding to the step density minimum at 600‰. This picture suggests that the growth process makes a transition from island nucleation to step flow near the slope at which the step density is minimized. It follows that at higher slopes the surface consists of a staircase of parallel steps, while at lower slopes the step pattern includes closed loops surrounding monolayer islands. This observation leads to an explanation for the existence of the step density minimum. Step edges in the form of parallel lines collect adatoms more efficiently than step edges in the form of closed loops. The essence of the difference can be seen by comparing the adatom collection rates for idealized patterns of step edges: a cross-hatch step pattern represents edges in the form of closed loops, which we compare with a parallel line pattern of step edges. To facilitate analysis, we assume a continuum picture similar to that of Burton, Cabrera, and Frank [34], where the step edges are perfectly absorbing, and 33 Chapter 4. Kinetic Faceting During Patterned Regrowth n (10 −3 /a2 ) θ ⋆ ● ◾S (1/a) θ ● ⋆ ◾ 0 0.05 0.1 0.15 0.20 0.05 0.1 0.15 0.2 0 0.25 0.5 0.75 1 0 0.1 0.2 0.3 0.4 Figure 4.1: (a) Step density S and (b) adatom density n on vicinal surfaces as a function of slope θ for 400, 450, 550 and 600‰ (◦,O,♦,). Figure 4.2 shows the step edge patterns at 400‰ and 600‰ for the slopes indicated in the main figure. Figure 4.2: Surface morphology at different temperatures and slopes, as in- dicated by the symbols in Fig. 4.1. 34 Chapter 4. Kinetic Faceting During Patterned Regrowth the adatom density n is determined by a 2D diffusion equation: D∇2n = −F. (4.1) We also assume that island nucleation occurs immediately if the adatom density becomes larger than some critical value nc. Based on this picture, a stable configuration of step edges requires n < nc everywhere. Numerical solutions show that the minimum step density required for stability is ∼ 1.5 times higher for steps in the cross-hatch pattern than for the parallel line step density. This explains why the step density is lowest at the onset of step flow growth: the terrace edges are configured as parallel lines and the lines are just far enough apart to suppress nucleation. This can be seen most clearly in Fig. 4.3, which represents two steady-state surface configurations. We see that, while the line spacing is slightly higher in Fig. 4.3(a), the total length of step edges is lower in Fig. 4.3(b). In general agreement with Fig. 4.1(b), which shows a peak in the adatom density around θmin, the solution to Eq. (4.1) shows that the mean value of n is ∼ 1.4 times higher for step edges in the form of parallel lines. As we can see in Fig. 4.3, this is because n is uniform in the direction parallel to the step edges; for the case of a cross-hatch pattern, n exhibits dips in this direction which lower the average. 4.1.1 Estimating θmin via the nucleation length In order to quantitatively describe the location of the step density minimum, We must quantify the transition between the step flow and nucleation dom- inated regimes. It is natural to compare the typical spacing between vicinal 35 Chapter 4. Kinetic Faceting During Patterned Regrowth (a) Cross hatch pattern (b) Parallel lines Figure 4.3: Adatom density obtained by solving Eq. (4.1) with two different step-edge configurations, and absorbing boundary conditions. steps ls to the typical distance between nucleated islands; the latter quantity is the nucleation length ln introduced in chapter 2. For ls  ln one expects step flow growth and for ls  ln, island nucleation will take place. A simple estimate for the crossover slope θmin is therefore a θ ∼ ln. (4.2) We plot these two quantities in Fig. 4.4. At low temperatures, we see that a/θmin diverges, but at higher temperatures (427 ◦C-600◦C), the relationship is quite good: the two quantities have the same form. However a/θmin is offset from ln by some nearly constant value; clearly there are variables not being taken into account. In our previous analysis, we have assumed that nucleation can occur when terrace edges are more than ln apart, and the surface is no longer in a pure step-flow growth mode. The problem with this assumption is that it implies that the terrace edges are completely straight and smooth. In reality this is not the case: the step edges have an effective area, which increases at lower 36 Chapter 4. Kinetic Faceting During Patterned Regrowth a/θmin ln T (K) Le n gt h (a ) 650 700 750 800 850 5 10 15 20 25 30 35 Figure 4.4: Nucleation length ln, obtained on a flat surface using cluster den- sity data (see section 2.1.1), and average terrace spacing at the step density minimum; plotted vs. temperature. temperatures. This edge roughening is examined in more detail in the next section. In addition, we have assumed that the nucleation length, which is only defined on a flat surface, is applicable to the reduced dimensions of a terrace on a vicinal surface. The flat-surface nucleation length develops in the presence of an initially uniform step density, whereas the step density on a terrace is altered by the presence of step edges, as discussed in section 3.1. 37 Chapter 4. Kinetic Faceting During Patterned Regrowth 4.1.2 Improving the estimate of θmin A more refined estimate of the step density minimum is possible. In this approach, we include the roughness of the step edges explicitly and calculate the nucleation length on vicinal terraces directly. We introduce the nucle- ation length on a vicinal surface lnv, which is calculated with separate kMC simulations in a geometry with two perfectly absorbing parallel boundaries that represent the step edges. lnv(T ) is then taken as the separation between the boundaries where stable islands first begin to appear. Step edges resulting from a miscut crystal go straight across the surface (at least over a distance scale much larger than the terrace spacing), which allows us to determine their mean position. This is essentially a line of best- fit, and we can define the rms distance of the real edge from the best-fit as its width wrms. We obtain the interface width of each individual terrace edge by identi- fying and following line edges, and computing their RMS deviation over the extent of the edge. Over the course of surface growth, we average the RMS value for each length scale. This process typically results in a wrms which depends on the length over which it was measured, as well as temperature. This measurement-length dependence can be seen in Fig. 4.5, which is a plot of wrms over different length scales at a temperature of 400 ◦C. As we must choose one value for wrms, we take the width measured over a length scale equal to the nucleation length ln; this the characteristic length of most surface structure [2]. If we assume absolutely straight step edges, then the slope θmin should be at the transition from layer-by-layer to step flow growth [1]. Quantitatively, 38 Chapter 4. Kinetic Faceting During Patterned Regrowth 100 101 102 103 100.2 100.3 100.4 100.5 100.6 Length (a) Li ne  W id th  (a ) l n  ~ 8.1 Figure 4.5: A typical plot of the interface width versus line length. Taken from a simulated vicinal surface at T = 400◦C and slope θ = 0.125. this happens when the average terrace width is lt = c1lnv, (4.3) where c1 is a constant of order one, which is chosen to match simulation data. At widths smaller than this, the step edges are too close together to allow nucleation. In the case of non-straight edges, the width available for nucleation is then some effective terrace length leff , as the edges have some intrinsic width, and take up some area themselves. The step density minimum actually occurs when leff = c1lnv. (4.4) 39 Chapter 4. Kinetic Faceting During Patterned Regrowth In order to capture the effect of the deviation of the step edges from straight lines, we calculate the rms displacement wrms of a step edge around its mean position; c2wrms is then an effective thickness of the step edges, where c2 is a constant. The effective terrace length is then given by leff = lt − c2wrms, (4.5) The step edges are therefore effectively closer together than estimated by the average terrace spacing lt = a/θ. Starting with the condition for the minimum (Eq. (4.3) ), we can solve for an improved estimate of lt at the minimum: lt = c1lnv + c2wrms. (4.6) In terms of the minimum slope, θmin, this becomes a θminlnv = c1 + c2 wrms lnv . (4.7) Using c1 and c2 as adjustable parameters, we plot this relationship in Fig. 4.6 over a large range of θmin, obtaining excellent agreement with the simulations. 4.2 Slope selection We now make a connection between the minimum step density and a novel mechanism of slope selection applicable to epitaxial regrowth on patterned surfaces. The physical origin of this slope selection can be understood with reference to growth on a convex surface, as shown schematically in Fig. 4.7. 40 Chapter 4. Kinetic Faceting During Patterned Regrowth c1 + c2x c1 = 0.89 ± 0.02 c2 = 0.29 ± 0.01 Simulation data Linear fit wrms/lnv a /θ min l n v 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Figure 4.6: Test of Eq. (4.7), with simulation data and a line of best fit. The plot includes temperatures from 450◦C to 600◦C at 5◦ intervals. The high adatom density in the region with slope corresponding to θmin will lead to lateral diffusion of adatoms to nearby areas where the adatom density is lower and the step density is higher. The lateral diffusion of adatoms away from the slope with minimum step density means that neighbouring regions will grow faster, tending to expand the area with slope matching the minimum in the step density. This process will create an extended region with slope equal to θmin, or in other words, a facet [35, 36]. The initial surface must be patterned because we are restraining our analysis to the case where growth is stable. In this case, macroscopic structure will not spontaneously appear, and the conditions for facet formation will not be met. Much effort has gone into the theory behind epitaxial regrowth on non- singular surfaces. However, the models only address the formation of high- 41 Chapter 4. Kinetic Faceting During Patterned Regrowth Figure 4.7: Schematic representation of facet formation on an initially mounded surface. The shaded arrows represent adatom diffusion out of areas with low step density, and the dashed arrows represent the areas of lowest growth. symmetry crystal facets based on their equilibrium interfacial properties [37, 35, 38]. The facets at θf we refer to in this paper are of a different physical origin, and are better understood as vicinal regions of a particular slope. It is important to note that this facet is kinetically defined: in equilibrium, the crystal surface would form facets as prescribed by the Wulff construction (section 2.2.2). This faceting behaviour has been observed in a number of experiments dealing with epitaxial regrowth on patterned surfaces [20, 39, 40]. Figure 4.8 shows atomic force microscope traces of GaAs growth on a surface with an existing grating pattern. As growth progresses, a clear facet appears on either side of the grating peak, as predicted by our model. We are unable to determine the expected facet slope of this experimental surface, as we 42 Chapter 4. Kinetic Faceting During Patterned Regrowth are missing sufficient experimental step density data to accurately determine θmin. x (µm) H ei gh t- ar b. o ffs et (nm ) Starting surface 580°C 2000nm 600nm 1.7○ -2 0 2 0 50 100 150 200 250 300 350 Figure 4.8: Experimental atomic force microscope cross sections of 1D grat- ings on GaAs (001) with the [110] direction in the plane of the page. The bot- tom line shows the initial microfabricated grating profile before any growth has taken place. The next lines are after deposition of 600 nm and 2000 nm of GaAs at 580‰. The straight line indicates the facet slope θf [20]. In order to verify that these facets are occurring for the reasons detailed above, we construct similar surfaces in a simulated environment. The sim- ulation model is the simple model described in chapter 2, which is detailed enough to exhibit kinetic slope selection via the step density minimum. Fig- ure 4.9(a) shows the effect of regrowth on a prepatterned surface in this kMC simulation. After deposition of 2000 monolayers, an initially sinusoidal profile 43 Chapter 4. Kinetic Faceting During Patterned Regrowth H ei gh t- ar b. o ffs et (a) x (a) a) 600°C 555°C Starting surface 1.8○ 1.5○ x (a) Starting surface 120 ML 300 ML 1600 ML b) 600°C 0 200 400 600 8000 800 1600 2400 3200 0 20 40 60 80 100 120 0 50 100 150 200 250 Figure 4.9: a) Grating profiles for kMC simulations on a sinusoidally pat- terned surface at two different temperatures after deposition of 2000 mono- layers on 3200×600 sites. b) Smoothing progression of a smaller surface with a different starting geometry. Each point in the 2D profiles is an average over the 600 surface sites in the third dimension. 44 Chapter 4. Kinetic Faceting During Patterned Regrowth has developed facets with a well defined slope ±θf near the grating peak. The facets grow in size and become better defined as the simulation goes on, and the general shape of the evolving surface is in good agreement with the exper- imental results. The slope θf is weakly temperature dependent and decreases with increasing temperature. In Fig. 4.9(b) we see the surface evolution of a different initial grating, whose profile has a trapezoidal shape. Although it is more difficult to pick out due to the smaller surface dimensions, the same faceting behaviour is observed for this geometry. The long-term smoothing behaviour of this surface is in good agreement with experiment [20]. Another feature of the surface shape also becomes evident at higher temperatures: the buildup of material on the “shoulders” of the sinusoidal ridge in figure 4.9(a) causes a steepening of the slope away from the facet. These regions tend to collect adatoms from neighbouring lower slope regions, which leads to the oscillatory surface topography observed in the simulations. In Fig. 4.10, we plot the facet slope θf in the regrowth simulation, along with the value of the step density minimum θmin obtained from the simulations on vicinal surfaces as function of inverse temperature. Remarkably, θf closely tracks θmin in the temperature range 550‰ to 600‰. The experimentally determined slope is also in good agreement with both theoretical and simulation results. The ac- cessible temperature range of the simulations is constrained by physics and by practical considerations: at lower temperature the facet slope becomes indistinct and at higher temperature we are limited by processor time. The temperature dependence of the facet slope also contains information about island nucleation through its relationship to the step density. Accord- ing to nucleation theory [1], the step density is controlled by the formation 45 Chapter 4. Kinetic Faceting During Patterned Regrowth 1000/T (1/K) Sl o pe 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 0.02 0.05 0.1 0.2 Figure 4.10: Arrhenius plot of the temperature dependence of: () the facet slope θf from the regrowth simulations (Fig. 4.9); (◦) the slope at the mini- mum in the step density θmin ; (- -) the reciprocal of the nucleation length at zero slope represented by aS0/2, where S0 is the step density at zero slope, as discussed in the text; and (—) the calculated threshold slope for step flow growth from Eq. (4.7) (see text). of stable nuclei on the surface and depends on growth rate F and adatom diffusion constant D according to S ∼ Γ(−n∗/2n∗+4) where n∗ is the size of a critical nucleus, and Γ = D/F , as introduced in chapter 2. We can use this relation and the data in Fig. 4.10 to estimate the size of the critical nucleus. The temperature dependence of the step density on the high temperature end of Fig. 4.10 corresponds to an activation energy of 0.42 eV. From this value, and the known activation energy of D (1.25 eV), we conclude that the 46 Chapter 4. Kinetic Faceting During Patterned Regrowth critical nucleus size is n∗ = 3. 47 Chapter 5 Conclusions Using simple geometric arguments and discrete simulations, we have exam- ined two specific aspects of crystal growth. In particular this study pertains to epitaxial growth of crystal surfaces on both patterned and flat substrates. In chapter 3, we studied the step flow regime of crystal growth, and found that an existing continuum theory insufficiently described the relevant behaviour with regard to the “step bunching” instability. The continuum ap- proach predicted that step flow was unstable in the presence of any downhill diffusion bias. We provide evidence that this is not the case and identify two reasons: bunches next to large terraces are unstable, and deposition noise drives the surface away from ideal bunching behaviour. 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Physical Review B, 50(9):6057, 1994. 54 Appendix A Surface Characterization In chapter 4.2, we examined a particular type of slope selection, and saw that the mechanism responsible was only valid over a particular temperature range. In particular, the slope of minimum step density is strongly affected by temperature. Starting at 625 ◦C, the slope first increases as the temperature decreases, then drops rapidly to zero at a critical temperature. The initial increase in slope is driven by the temperature dependence of the nucleation length which becomes progressively smaller at low temperatures and can be expected to follow an Arrhenius law according to nucleation theory [1]. However, there is a limit to this mechanism: once ln is smaller than the roughness of the step edges, the distinction between the parallel steps of the step flow regime and the closed loop steps of the nucleation dominated regime is lost, and the crossover vanishes. In this chapter, we study the reasons for this by developing metrics for surface roughness, and examining the effect of different growth parameters on surface morphology. A.1 Edge D/F ratio As we have seen in section 2.1.3, it is useful to parametrize the system via the quantity Γ = D/F , the ratio of diffusion rate and flux. However, this quantity does not take into account the relative sizes of the lateral and substrate 55 Appendix A. Surface Characterization binding energies. We can add a second related parameter to more completely describe the surface. It is natural to assume that diffusive adatom arrival roughens the step edge in a manner similar to one dimensional ballistic deposition, and that edge diffusion smooths out the edge at a rate which depends on the hopping frequency for adatoms with one lateral nearest neighbour. This rate D1 depends on both the substrate and lateral binding energies according to D1 = v0e −β(Elat+Esub), (A.1) where β = 1/kBT , v0 is a rate constant, and Elat and Esub are the lateral and substrate binding energies. This is a significant simplification, as there is an additional smoothing effect caused by adatom attachment from the upper terrace. However, the assumption does hold for small islands, which have little flux on the upper terrace, and also for “finger-like” projections on the front of any edge, which also are not easily “filled in” from behind. Conveniently, in the low temper- ature regime where this analysis is applied, the morphology is dominated by these two features. Given this newly defined diffusion rate, it is useful to look at ratio Γe = D1/Fe, (A.2) where Fe the flux rate on the step edge, which we determine by considering an array of parallel step edges. Each edge receives a total flux which is proportional to the length of the terrace in front of it. This leads to a flux per unit length which is proportional to the terrace spacing lt. We can generalize this to any geometry by noting that the natural distance between steps on any 56 Appendix A. Surface Characterization (flat or weakly vicinal) surface is the nucleation length ln (See section 2.1.1 for more information on ln); therefore, Fe = ln/aF . From simulation cluster density data, we can determine ln according to ln ≈ 1/√ni, where ni is the island density. This gives Γe ≈ De −βElat√ni Fa (A.3) This expression is plotted in Fig. A.1, and becomes approximately one for the temperatures at which the step density minimum vanishes. When Γe ≈ 1, we would expect that adatoms are deposited onto each step edge faster than the edge can smooth out. At this point the characteristic length of the surface approaches one lattice spacing, and the surface is effectively disordered. 400 500 600 700 800 900 1000 10−6 10−4 10−2 100 102 104 T (K) Γ e Figure A.1: Γe vs. temperature, determined according to Eq. (A.3), with ln obtained from simulation. At the point where Γe = 1, T = 640K. 57 Appendix A. Surface Characterization A.2 Fractal dimension Another interesting way to characterize the disorder of the surface is through its fractal dimension [41]. The formal definition of fractal dimension is df = lim LN→∞ logN logLN , (A.4) where LN is a length scale, andN is the number of discrete elements which are present within that length scale. A limit of infinity is not feasible in any real system, so we use a practical definition in which we measure logN/ logLN over a continuum of length scales. In order to determine the fractal dimension of a terrace edge in simulation, we choose a site on a step edge, i, and a number of elements N . For each site i, we measure the straight-line length from edge i to edge i+N . Performing this measurement over many values of N , and many step edges, we obtain a set of LN for each N , and are able to apply the ratio from Eq. (A.4). In Fig. A.2, we plot the fractal dimension of lines on a surface with slope θ = 1/ln for different temperature values and different values of LN . Most of the surfaces are not fractal in nature, as their fractal dimensions Df decay immediately with increasing length scale. However, the surface at T= 673K is fractal on a length scale up to approximately ln. This is the temperature at which the non-zero step density minimum as a function of θ disappears. This is in broad agreement with the crossover temperature of 640K obtained in Fig. A.1. 58 Appendix A. Surface Characterization Length scale (a) D f 798K 673K 0 5 10 15 20 25 0.9 1 1.1 1.2 1.3 1.4 1.5 Figure A.2: Fractal dimension on a surface with nucleation suppressed, and the surface slope constrained to a/ln, as shown in Fig. 2.6. 59 Appendix B Altered Growth Conditions B.1 Straightened step edges In order to confirm the effect of line width wrms on the step density vs. slope curve, we attempt to straighten the step edges, and observe the behavior of the surface. In order to this, we decrease the barrier for lateral adatom motion on a step edge. The edge then equilibrates at a higher rate, as the less tightly bound adatoms are more likely to settle into kinks in the step edge. The effect of this on the step density as a function of surface slope can be seen in Fig. B.1: we see that the curve has a strong minimum at all temperatures. This is in contrast to unstraightened step edges, where the minimum disappears below T= 425◦C. We can also make a prediction about the relationship between the step edge density on a flat surface S0, and the step density at the minimum Smin. The minimum step density Smin occurs at θ/a = 1/ln and is equal to 1/ln, when the step edges are straight (see section 4.1). For a flat surface, finding S0 is slightly more complicated. First we assume that nucleation of islands occurs uniformly, with each island separated by a distance ln, after which the area of each island grows uniformly in time. This continues until the islands are ln across, at which point they merge into each 60 Appendix B. Altered Growth Conditions other and vanish, completing the layer filling. Taking the circumference of an island as CI, we integrate over the island area l2n ∝ t to obtain the expectation value of S0 during layer filling: 〈S0〉 = 1 l4n ∫ l2n 0 CIdA, = 1 l4n ∫ l2n 0 4 √ AdA, = ( 4 l4n )( 2 3 ) (l2n) 3/2, = 8 3 1 ln . Given the step density at both the minimum and at θ = 0, we can deter- mine the slope of the S vs. θ curve in the island nucleation regime: dS dθ = Smin − S0 ln − 0 , (B.1) = −5 3 . (B.2) (B.3) As we can see in Fig. B.1, this result is in good agreement with simulation results. B.2 Change of binding energies Normally the simulations are run with a substrate binding energy Es = 1.25eV, and a lateral binding energy El = 0.35eV. Changing these constants alters the simulation in a number of ways. The D/F ratio is directly de- pendent on Es, and the critical nucleus size and edge smoothing rate are 61 Appendix B. Altered Growth Conditions Figure B.1: Step density S vs. surface slope θ where step edges have been straightened. The dashed line represents the theoretically calculated slope from Eq. (B.2). The temperatures of the solid curves range from 698K to 773K. dependent on El [2]. For example, if we lowered the lateral binding energy to El = 0.25eV, we would expect the surface structures to be smoother at lower temperatures, leading to a disappearance of the step density minimum at lower T. This is exactly what happens, as seen in Fig. B.2. By compari- son with Fig. 4.1, we see that there is a now weak minimum at T = 400◦C where there was none at Elat = 0.35 eV, and that S is lower in general for Elat = 0.25 eV. 62 Appendix B. Altered Growth Conditions 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Surface Slope St ep  E dg e De ns ity   673 K 723 K 773 K 823 K Figure B.2: Step density vs. surface slope with a lateral binding energy of El = 0.25. In the altered binding energies examined above, we find that the general behavior of the step density S at different slopes is very similar to that of the original system, but that the magnitude of S is different. We have tailored our study of slope selection (section 4.1) to GaAs systems, but any system which exhibits distinguishable step edges in the form of both lines and closed loops should be sufficient, even with widely different quantitative properties. This is not a very restrictive condition; much of the research in the field of surface growth is devoted to ordered surfaces, as they are most applicable to device construction. 63


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