ELECTROMIGRATION INDUCED STEP INSTABILITIES ON SILICON SURFACES DISSERTATION

Transcription

1 ELECTROMIGRATION INDUCED STEP INSTABILITIES ON SILICON SURFACES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Brian J. Gibbons, B.S. ***** The Ohio State University 2006 Dissertation Committee: Approved by: Professor Jonathan P. Pelz, Advisor Professor L. Stanley Durkin Professor Nandini Tivedi Professor Fengyuan Yang Advisor Department of Physics

2 ABSTRACT Understanding the processes that govern the motion and arrangement of steps on vicinal semiconductor surfaces been a long-standing problem of great scientific interest. From a technological view point, understanding the behavior of surface steps is very important for semiconductor device processing. Recently there has been great interest in processes that lead to the spontaneous arrangement of steps into large surface structure ranging in size from a few nm up to many µm. One fascinating example of this is step bunching on slightly miscut silicon surfaces heated to high temperatures using a direct current, and although the process of step bunching on Si surfaces is well documented experimentally, it remains poorly understood. The first observations of step bunching on the Si(111) surface were made by Latyshev et al. [Surf. Sci. 213, 157 (1989)]. They found that steps bunch for only one direction of the current used to heat the wafer relative to the vicinal step-up and stepdown direction. Surprisingly, it was also found that the current direction required for bunching changes multiple times with increasing temperature. In particular it was found that surface steps bunch for a step-down current in Temperature Regimes I (~ C) and III (~ C), but for a step-up current in Regime II (~ C). ii

3 It is generally accepted that diffusing atoms may acquire a small effective charge that causes them to drift in a direction either parallel or anti-parallel to the applied direct current. Using this concept Stoyanov [Jpn. J. Appl. Phys. 30, 1 (1991)] was the first to propose a sharp-step model based on surface diffusion (with an added drift term) to explain step bunching on silicon surface. This model was shown to produce step bunching only when atoms drift in the step-down direction relative to the vicinal surface and therefore could not explain the experimentally observed reversals in the bunching current with temperature. Over the years, several mechanisms have been proposed to explain the reversals in the step bunching current. However, each has subsequently been found to be incompatible with experiment. In this thesis I will report on a sequence of experiments designed to address the complex step bunching behavior observed on Si(111). Prior to our studies it was proposed that increased step permeability [S. Stoyanov, Surf. Sci. 416, 200 (1998)] might be responsible for the change from step-down to step-up current bunching under sublimation conditions. Step permeability means that there is a significant direct flow of atoms from one terrace to another. According to this model, if steps are permeable enough then the surface will be unstable (stable) toward bunching for a step-up current under net sublimation (growth) conditions. By studying how Si deposition conditions affect step bunching during step-up direct current heating we find that this model cannot account for step bunching for step-up current in Regime II. We arrived at this conclusion by experimentally showing that there was no transition from step instability to stability when changing from net sublimation to net growth conditions in Regime II. iii

4 Another issue concerning step bunching on the Si(111) is whether the diffusion of atoms across terraces or the attachment/detachment of atoms to and from steps is rate limiting. We address this issue by measuring how the number of steps N and the minimum terrace width l min in a bunch depend on initial surface miscut in all three temperature regimes. This is the first report of how bunching depends on surface miscut, and by comparing experiments to both analytical predictions and numerical simulations of the Stoyanov s sharp-step model we conclude that diffusion across terraces is rate limiting during step-down bunching in Regimes I and III for a wide range of surface miscut. This is contrary to previous beliefs that the attachment/detachment of atoms from step-edges was extremely slow compared to the diffusion on terraces. Because the original sharp-step model cannot explain step-up current bunching we were unable to conclude anything about Regime II, but it is interesting to note that we observe nearly the same dependencies of N and l min on miscut in all three step bunching regimes. This suggests that the fundamental mechanism for step bunching may be similar at all temperatures. This finding is very important since it adds plausibility to the recent suggestions that if the hopping rate of attachment/detachment to step-edges became faster than the terrace hopping rate, then a step bunching instability should occur for a step-up atom drift [Suga et al., Jpn. J. Appl. Phys. 39, 4412 (2000) and T. Zhao et al., Phys. Rev. B 71, (2005)]. Finally, I will also discuss a new continuum formulation of the fast attachment model mentioned above, and show that it is capable of explaining the multiple transitions form step-down to step-up current induced bunching on Si(111) at high temperatures by iv

5 assuming modest temperature-dependent changes to the energy barrier of step attachment/detachment or terrace diffusion. This continuum formulation adds a direct drift-attachemnt term to the boundary conditions describing how atoms attach to and break away from steps. I show that this model can produce bunching for a step-down or step-up atom drift when the step attachment/detachment rate is slower or faster than the rate of diffusion on terraces, respectively. Numerical simulations using physically reasonable parameters show this model can reproduce the experimentally observed step bunching behavior in all three step bunching Regimes reported in this thesis, provided there exist modest (~0.3 ev) temperature-dependent changes to the energy barriers of attachment/detachment and/or terrace diffusion. Step bunching experiments on the Si(001) surface, where step bunching is know to occur for both directions of current at all temperatures are also discussed in this thesis. Studies of step bunching on Si(001) are reported for a variety of Si deposition conditions and a range of initial surface terraces. As with Si(111) we find that the deposition conditions do not greatly effect step bunching on Si(001). These are the first reports of their kind, and should prove useful for test future models of step bunching on the Si(001) surface. Lastly, I will discuss experiments involving the flattening of silicon craters to produced large nearly step free terrace that may have many uses ranging from surface science studies to providing an ultra-low-miscut starting surface for future semiconductor device technology. v

6 Dedicated to my family and friends vi

7 ACKNOWLEDGMENTS I wish to first thank my adviser, Jonathan P. Pelz, for all of the advise, encouragement, and support he offered throughout this research project. His help and patience made this work both possible and enjoyable. Next I wish to thank Rita Rokhlin and Tom Kelch for providing critical technical support. Without access to the Atomic Force Microscope and the Graduate Student Shop, the work here could never have been carried out. I must also thank the Instrument Makers in the Physics Machine Shop John Spaulding, Pete Gosser, Jon Shover, and Bryan Smith. The experimental equipment designed and built here exists because of their work. I would also like thank my lab mates past and present: Jon-Fredrik Nielsen, David Lee, Yi Ding, Wei Cai, Camelia Marginean, and especially Eric Heller, Cristian Tivarus, and Kibog Park. Working with them has truly been an enjoyable experience. I would like to thank a number of other Physics Graduate Students and local friends. This list includes: Wes and Christy Pirkle, Pat and Susan Randerson, Ed and Wendy Smith, Matt and Chastity Whitaker, Meredith Howard, Leslie Schradin, and Andrei Modoran. Finally, I must thank Barry and Jen Van Patten, Tracy and Romely Frazier, and Chris Kennett for an amazing amount of support. vii

8 VITA March 25, Born Cleveland, Ohio B.S. Physics Northern Arizona University Flagstaff, Arizona 1999 present... Graduate Teaching and Research Associate The Ohio State University Columbus, Ohio PUBLICATIONS B.J. Gibbons, J. Noffsinger, J.P. Pelz, Influence of Si deposition on the electromigration induced step bunching instability on Si(111), Surf. Sci. 575, L51 (2005). FIELDS OF STUDY Major Field : Physics viii

9 TABLE OF CONTENTS Page Abstract.. ii Dedication. vi Acknowledgments... vii Vita. viii List of Tables... xiii List of Figures xiiii Chapters: 1. Overview Introduction The Silicon (111) and (001) vicinal surfaces Atomic Surface Structure Vicinal Surfaces Basic surface evolution Macroscopic evolution: The theory of thermal smoothing The Burton-Cabrera-Frank (BCF) step model Direct-current induced step bunching on Si(111) The traditional sharp-step surface electromigration model The observed step bunching transitions on Si(111) Experimental Methodology and Setup Ultra high vacuum (UHV) system The sample transfer section ix

10 3.1.2 The main annealing chamber Sample preparation Dimpling procedure Sample cleaning: Chemical Sample cleaning: UHV thermal Dimpled surface profiles Results: Electromigration Induced Step Instabilities Step instabilities on Si(111): The effects of Si deposition Introduction and motivation Experimental procedures Regime I: Deposition effects Regime II: Deposition effect Crossing step behavior In-phase step wandering The effects of local surface miscut on Si(111) step bunching The rate limiting mechanism on Si(111) during DC heating Introduction Experimental procedures Experimental observations Comparison to theory Discussion of findings Fast vs. slow attachment and the bunching transitions on Si(111) Introduction Drift-attachment continuum equations Numerical simulations Discussion of findings Electromigration induced step bunching on Si(001) Brief review of the Si(001) vicinal surface Step bunching on Si(001) Experimental observations x

11 5. High Temperature Silicon Crater Floor and Wall Evolution Introduction Double oxidation technique for crater patterning Experimental observations and discussion Appendices: A. Numerical simulation code A.1 Sharp-step model A.2 Drift-attachment model. 152 References. 161 xi

12 LIST OF TABLES Table Page 4.1 Parameter values used in sharp-step numerical simulations Parameter values used in drift-attachment numerical simulations 107 xii

13 LIST OF FIGURES Figure Page 2.1 Unit cell of Si crystal lattice and Si (111) and (001) surface Schematic of (111) and (001) vicinal surface and marcroscopic view Schematic of microscopic surface processes and step geometry with hopping potential profile seen by and adatom Schematic representation of the adatom concentration profiles on a surface Diagram showing atomic flow driven by step-edge chemical potential AFM images of step bunching Regimes on Si(111) Diagram of the step pairing instability Sample profile from sharp-step numerical simulations Step bunching behavior predicted by the permeable step model Schematic diagram showing the UHV system used in these studies Optical micrograph of the critical experimental equipment Schematic representation of the sample annealing stage Diagram of basic circuit used to samples with AC Sample AFM profile of a Si(111) surface after flash cleaning Profiles typical of dimples ground into the Si surface. 59 xiii

14 3.7 1 st derivative of profiles in Fig AFM image showing pinning sites created by the e-beam evaporator AFM image showing step-up bunching in Regime I with a Si flux AFM image showing step-up bunching in Regime II with a Si flux Explanation of crossing step density dependence on Si deposition conditions Schematic diagram and AFM of in-phase step wandering AFM image of in-phase step wandering and profile of the morphology Plots show how bunching in Regimes I and II depend on local miscut and Si deposition conditions Plots of bunching behavior dependence on terrace with for Regime I Plots of bunching behavior dependence on terrace with for Regime II Plots of bunching behavior dependence on terrace with for Regime III Plots of how l min N 2/3 depends on terrace width Plots of sharp-step numerical simulations of step-down bunching in Regime I bunching dependence on terrace width Plots of sharp-step numerical simulations of step-down bunching Regime III bunching dependence on terrace width Schematic diagram showing step geometry and adatom hopping energy barrier profile used to derive the drift-attachment continuum equations Numerical simulation results from the drift-attachment model for step-down bunching in Regime I xiv

15 4.16 Numerical simulation results from the drift-attachment model for step-down bunching in Regime III Numerical simulation results from the drift-attachment model for step-up bunching in Regime II Schematic diagram showing the origin of anisotropic diffusion on Si(001) AFM images of step bunching behavior on Si(001) Plot of experimental data showing how step-down current step bunching on Si(001) depends on terrace width Plot of experimental data showing how step-up current step bunching on Si(001) depends on terrace width Schematic representation of the crater flattening process Schematic diagram of the process steps used to fabricate crater in Si surfaces and an optical micrograph of the photo step D AFM image and profile of a typical crater Schematic of ideal and nonideal crater orientation relative to the global surface miscut and experimental data for flattened craters AFM images showing flattened crater on the Si(111) and Si(001) surface D AFM image of a crater on Si(001) annealed above the roughening temperature and corresponding profiles (before, after, and numerical simulation) xv

16 CHAPTER 1 OVERVIEW The semiconductor industry is a multibillion dollar a year entity. Most people associate the transistor and its electronic properties with the semiconductor industry, and although correct, the transistor is only the end product of an extremely complicated fabrication process. This process consists of numerous steps involving the growth and etching of a variety of materials at elevated temperatures. Therefore, a complete understanding of how these materials evolve under various conditions is very important. Trying to understand these processes scientifically has proved challenging, and continues to drive scientific inquiry. As we will see later, at elevated temperatures atomic-height surface steps are responsible for the evolution of crystalline surfaces, and their arrangement/motion is driven by (often competing) thermodynamic and kinetic processes involving the diffusion of adatoms on terraces, the attachment/detachment of adatoms at the step-edges, and in some case the desorption (deposition) of adatoms from (onto) the surface. One particular fascinating observation of surface step motion was made by Latyshev et al. [1]. They found that heating slightly-miscut Si(111) with direct current (DC) in an ultra high vacuum (UHV) environment leads to large scale changes in surface morphology such as 1

17 step bunching. The early reports of Latyshev et al. [1] showed that step bunching on Si(111) only occurs for one direction of the applied current relative to the vicinal step-up and step-down directions, but that the current direction required for bunching reverses multiple times with increasing temperature. This is in contrast to slightly-miscut Si(001) surfaces [2], where step bunching occurs for both directions of the applied current at all temperatures. It is generally accepted that step bunching results from an electric-fieldinduced adatom drift ( electromigration ), but the physical mechanism leading to the multiple temperature-dependent reversals in the current direction required for bunching on Si(111) is still under debate. It should also be mentioned that a satisfactory explanation for step bunching on Si(001) has also not been found. In this thesis, I will discuss several experiments carried out to study the evolution of the silicon (111) and (001) surface heated to high temperatures in an ultra high vacuum (UHV) environment. In particular, the step bunching transitions on Si(111) will be addressed. First, by studying how Si(111) step bunching depends on Si flux conditions we are able to concluded that the transition from step-down to step-up current bunching does not result from increased step transparency, contrary to previous suggestions [3, 4]. We have also measured how the number of steps N and the minimum terrace width l min in a step bunch depend on initial surface miscut. By directly comparing these measurements to both analytic scaling predictions [6 8] and numerical simulations of the sharp-step model [5] we find that the terrace diffusion rate is similar to the adatom step attachment rate during step-down current bunching. If the surface processes are taken to be thermally activated, then this implies that the energy barrier for attachment to a step is only slightly larger than the energy barrier for diffusion on terraces during step-down 2

18 bunching. This last finding adds plausibility to recent suggestions that the transition from step-down to step-up bunching on Si(111) may result from the step attachment rate becoming faster than terrace diffusion [9 12], which according to our findings would only require modest temperature-dependent changes in the surface activation energies. Finally, by treating the step-edge attachment process in the same way as diffusion on the terraces we derive a new set of continuum equations that includes a drift-attachment term analogous to the drift-diffusion term originally added by Stoyanov [5] to the Burton-Cabrera-Frank (BCF) step model [13] to describe surface electromigration. The drift-attachment model is shown to produce step-down (step-up) bunching when the stepedge attachment rate is slower (faster) than the terrace diffusion rate. More attention is given to the Si(111) surface in this thesis since it is isotropic (and exhibits very interesting step behavior), but I will also offer some short discussions of the more technologically important Si(001) surface. The organization of this thesis is as follows. Chapter 2 will cover the background information needed for successful reading of this dissertation. There I will briefly discuss the atomic structure of Si(111) and Si(001) surfaces, and define a vicinal surface. A review of surface evolution will be given from a macroscopic point of view using the continuum theory of Mullins [14, 15]. The motion of surface steps will be discussed using the original (BCF) theory of crystal growth [13], and for the work presented in chapter 4, I will discuss how the BCF model was modified by Stoyanov [4] to account for the drift-diffusion of adatoms, which for a Si wafer results from direct-current (DC) heating. Using the Si(111) surface as an example, the basic physics of how drift-diffusion leads to step bunching will reviewed. Chapter 2 will be 3

19 closed with a review of the original observations of bunching on Si(111) by Latyshev et al. [1], and a brief discussion of the models proposed to explain the observed behavior. The main goal is to introduce the reader to the terminology and ideas used in later chapters. Chapter 3 contains information about the experimental methods used to conduct the research presented in this thesis. Chapter 4 will be devoted mostly to understanding the physics behind step bunching on Si(111). There are three main sections to chapter 4, and in the first part I will discuss studies carried out to determine whether or not step transparency plays a role in step-up bunching on Si(111). These studies were done by adding a variable flux of Si adatoms to the surface using an electron beam evaporator during step bunching experiments at different temperatures. We find that step-up bunching is unaffected by the Si deposition conditions, in direct contradiction with the report of Métois et al. [4] and suggested transparent step model [3]. During these studies we also measured how the height of a bunch H at a fixed temperature and annealing time depends on the initial surface miscut θ. This is the first experimental report of how the step bunching behavior depends on surface miscut, and we find that H increase systematically with miscut for both step-up and step-down bunching, suggesting that the mechanism(s) responsible for both step-up and step-down are similar. In the second part of chapter 4 will expand on how the surface miscut influences step bunching. I will present more detailed studies of how the initial terrace width l 0 on Si(111) surfaces influences the rate of step bunching (the number of steps in a bunch N) and the minimum terrace width l min within a bunch during both step-up and step-down 4

20 bunching. Experimentally we find N l and lmin l 0 in all three step bunching regimes with only slight variations in the exponents with temperature. By comparing these findings with analytic solutions to and numerical simulations of the sharp-step model we find that the kinetic length d characterizing the relative rates of diffusion and step attachment/detachment is relatively small (< ~20 nm) when step-down current is required for step bunching. Physically, this means that surface mass transport is diffusion limited on Si(111) vicinal surfaces with initial terrace widths l 0 > 20 nm. As will be discussed, if one considers the microscopic surface processes responsible for surface diffusion and step attachment, then d << l 0 implies that the energy barrier for step attachment E A is similar in magnitude to the barrier for diffusion on a terrace E D. This is an important finding since for nearly a decade researchers have believed this kinetic length d to be large (i.e., d >> l 0 ), and therefore E A >> E D. Although we find N and l min to experimentally depend on l 0 during step-up bunching in the same way as step-down bunching, we are unable to conclude anything based on the conventional sharp-step model [5] since it can only explain step-down bunching. Therefore, in the third section of chapter 4 we develop a fully consistent model for Si(111) by considering the microscopic processes of adatom diffusion on terraces and adatom attachment/detachment at step-edges. In the model the flow of adatoms onto the step-edges is treated in a symmetric/analogous way as diffusion on the terraces. We do this by adding a drift-attachment term to the boundary conditions to the traditional sharp-step model [5] describing how atoms attach/break away from step. Simulations using physically reasonable parameters show this model can reproduce the experimentally observed step bunching in all three step bunching regimes. By directly 5

21 comparing simulations with experiments we find that this model can explain all of the step bunching behavior on Si(111) provided we assumed the attachment/detachment and/or diffusion energy barriers undergo modest temperature-dependent changes on the order of ~0.2 ev. It should be noted that this model is mathematically equivalent to the that of Suga et al. [9], and numerically similar to that proposed by Zhao et al. [10 12]. Chapter 4 is closed with a discussion of step bunching on Si(001) under various deposition flux conditions. Finally, in chapter 5 I will discuss the fabrication of craters in silicon and the evolution of the crater floor and sidewalls during thermal annealing. The motivation for experiments involving craters is two-fold. First, it is known that extended annealing of crystals with craters patterned into them results in the complete removal (or severe reduction) of the initial wafer miscut from the crater floor [16, 17]. Secondly, the low atomic step density on the crater floor after annealing has proved useful in a variety of surface science experiments [18 23]. Although the mechanism responsible for flattening is well understood, the process has been observed to take a long time. It was thought that trench formation during the crater fabrication process (via plasma-etching) was responsible for the long flattening times. We addressed this issue by fabricating trenchfree-craters using a novel double-thermal-oxidation technique. Interestingly, a small trench and peak is found to form during the flattening process as a result of the rearrangement of atomic steps on the crater floor, or the large scale flow of adatoms away from region of high curvature when above the surface roughening temperature on the Si(001) surface. I will discuss this phenomenon as well as some other interesting observations of the crater s morphological evolution during high temperature annealing. 6

22 CHAPTER 2 INTRODUCTION 2.1 The silicon (111) and (001) vicinal surfaces Atomic surface structure Before discussing the aspects of surface evolution it is important to have a qualitative understanding of the silicon surface. Bulk silicon crystallizes in the diamond structure with 8 atoms per conventional cubic cell and an edge of length of a = 5.43 Å [24], as shown in Fig. 2.1(a). Figure 2.1(b) shows the ideal bulk terminated (1x1) surface structure that occurs when a Si crystal is terminated along the [111] directions. This bulk termination is shown in Fig. 2.1(b) where the larger red solid circles represent the atoms in the top surface layer with one dangling bond pointing up into vacuum, while the smaller blue solid circles are the atoms in the second surface layer. The Si(111)-(1x1) surface unit cell (with 2 atoms per unit cell) is highlighted in turquoise blue in Figs. 2.1(a) and 2.1(b) with each segment of the diamond having a length of a / Å. This bulk termination is not stable, and it is generally accepted (after thermal treatment) that the (111) surface converts to a (7x7) Takayanagi type reconstruction [25], which at elevated temperatures (> ~850 C) becomes a disordered 1x1 phase [26]. It has been reported that a high density of surface adatoms accompanies this 1x1 phase [27]. It 7

23 should also be mentioned that a variety of reconstructions related to the (7x7) have also been observed on the Si(111) surface [27]. Since all experiments involving Si(111) reported in this thesis were carried out above 850 C only the 1x1 phase is of interest. In contrast terminating the crystal along the [001] direction results in completely different surface. From Fig. 2.1(a) it is seen that the top most atoms in the (110) plane have two bonds down toward the bulk atoms, leaving two bonds above the surface in the vacuum. The presence of these dangling bonds cause the surface to under go a reconstruction forming dimers along either the [110] or [ 110] directions, as shown in Fig. 2.1(c). This reconstruction is termed (2x1), and gives the Si(001) surface many of its interesting properties. (001) crystallographic planes occur every a/ Å, however, the arrangement of the surface bonds (hence, the surface reconstruction) rotates by 90 from plane to plane. This is clearly seen by identifying the surface unit cells for two terminated surfaces separated by a/4, as outlined in turquoise in Fig. 2.1(c). The properties of the Si(001)-(2x1) surface have been studied in great detail and a review of this surface and all of its properties can be found in the review of Zandvliet [28] and the text of Dabrowski and Müssig [29]. Although we will not need to directly deal with the atomic nature/arrangement of these surfaces, it is necessary to understand that the Si(111) and (001) surfaces are quite different. This is an important fact, and as one might guess the reconstruction of the Si(001) surface leads to a fair amount of anisotropy while the Si(111) is for the most part isotropic above ~850 C. 8

24 (a) a = 5.43 Å [001] a/4 (b) [010] [100] (c) a = 5.43 Å S B - step S A - step < 2 11 > < 110 > < 01 1 > Surface dimer Fig 2.1. (a) Ball and stick model of the silicon unit cell [Image from: Condensed Matter Sciences Department (ORNL)]. The bulk terminated Si(111) surafe is shown in (b). The light blue diamond outlines the surface unit cell, which can also be identified in (a). The Si(001) dimer reconstructed surface is shown in (c) [Image from The Thesis of Eric Heller, The Ohio State University]. The corresponding 2x1 unit cell is outline in light blue Vicinal surfaces A vicinal surfaces results when a crystal is not terminated or cut perfectly parallel to a high index crystal plane. This always occurs in practice, and results in a vicinal stepped surface. Figures 2.2(a) and 2.2(b) show a one-dimensional (1D) representation of the Si(111) and Si(001) stepped vicinal surfaces. From the Figure it is clear that a vicinal surface is characterized by a sequence of atomically flat terraces separated by atomic height steps. In the case of Si(111) the terrace corresponds to the 9

25 (a) (b) (c) h ~3.14 Å h ~1.36 Å l 0 θ θ S S S A B A [111] Average terrace width [001] l 0 Macroscopic view of surface step-edge terrace Crystal planes Fig One-dimensional schematic diagrams of the terrace and step structure for the (111) and (001) silicon surfaces are shown in (a) and (b), respectively. The Relationship between surface step structure and the macroscopic surface is shown in (c). (111) plane with an atomic step height equal to the distance between (111) planes h 3.14 Å. On Si(001) there are two types of step-edges separating the (001) terraces due to the rotating dimer reconstruction discussed above. The steps are labeled S A and S B, where S A -type (S B -type) steps have dimer rows running parallel (perpendicular) to the step-edge on the step-up side and have a height h 1.36 Å. The alternation of the stepedges is shown in Fig. 2.2(b), and the corresponding dimer orientation for each step-edge 10

26 is clearly labeled in Fig. 2.1(c). Finally, the degree of miscut θ, the atomic step height h, and the initial terrace width l 0 are related by the simple expression tan θ = h/l 0. It is exactly the motion and arrangement of these atomic steps that will be discussed here. Considering a system at elevated temperatures in the absence of sublimation or deposition, and thinking back to the atomic surface structure of the last section, we see that step rearrangement requires the steps to exchange atoms. This is a process that involves an atom breaking away from a step-edge, migrating or diffusing across the terrace, and attaching to a step-edge. As we will see this process can be quite complicated, but luckily when studying how steps move we do not need to keep track of all the migrating atoms. 2.2 Basic surface evolution The evolution of a surface can be viewed on a variety of scales. On the macroscopic scale we view a surface as a sequence of hills and valleys, as depicted by the solid blue line in Fig. 2.2(c). However, we know that the truncation of a crystal results in a sequence of terraces and step-edges, and under clean conditions this terrace and step structure must truly be responsible for the macroscopic surface structure we observe. As shown in Fig. 2.2(c), the crystal planes intersecting the macroscopic surface give rise to the terrace and step-edge structure. On the microscopic level, it is a local variation in slope or miscut θ that gives rise to the macroscopic surface. This concept is important, and we shall briefly look at how to treat surface evolution on these two scales. First, surface evolution will be discussed from a macroscopic continuum point of view then it will be treated on the atomic scale. 11

27 2.2.1 Macroscopic evolution: The theory thermal smoothing We will approach the concept of thermal smoothing from the stand point of macroscopic thermodynamics. The reason for this is that the local surface chemical potential can be related to local surface curvature/morphology. This very concept is powerful since regions on the macroscopic surface of Fig. 2.2(c) where the local surface curvature is negative (hills) will have a high chemical potential compared to regions of positive curvature (valleys) with a low chemical potential. In an effort to lower the total surface free energy, adatoms will flow from regions of high chemical potential (hills) to those of lower chemical potential (valleys), causing the crystal to surface to smooth out. The process described above is basically the Gibbs-Thomson effect, and this concept was used by Mullins [4, 5] to describe the thermal smoothing of surfaces many decades ago. A complete discussion of how to arrive at Mullins [14, 15] original equations for surface evolution can be found in any book or review on crystal growth [30 32]. I will briefly outline the main points here. We start by assuming that the xy-plane corresponds to a low index {hkl} surface plane, and that the crystal surface can be described by a continuous function z(x,y), where z represents the surface height from the low index {hkl} plane. In the lowest energy state the entire crystal surface would be flat, or in other words parallel to {hkl}, leaving z(x,y) independent of x and y. The surface free energy F due to surface morphology (deviations from the {hkl} plane) is given by [6 8]: F = γ ( nˆ ) da, (2.1) where γ (nˆ ) is the surface free energy per unit area (surface tension) and da is a microscopic unit of surface area with a local surface normal: 12

28 nˆ { z ˆ ˆ xi + z y j + kˆ} = 1 / 2. (2.2) {1 + z + z } 2 x 2 y Here z α = z/ α is the local surface slope relative to the {hkl} plane, and î, ĵ, and kˆ correspond to the usual right-handed coordinate axes. It is not easy to integrate da over the surface z ( x, y), so the surface area element da is often projected onto the xy-plane (the low index {hkl} plane) so that da dxdy [30 32]. This is easily done by recognizing that dxdy ˆ ˆ) / 2 = ( k n da = da/{1 + z x z y}, which upon substitution gives the following expression for the surface free energy: 2 ( nˆ){1 + zx + 2 1/ 2 F = γ z } dxdy. (2.3) y The integrand of Eq. (2.3) represents the surface energy per unit area projected on the low index {hkl} plane, and is referred to as the projected surface free energy per unit area. Assuming an infinitesimally small surface element and small slopes with z α = z/ α << 1, the surface tension can be approximated by a Taylor series expansion γ ( n ˆ) = γ + ( γ / 2)( z x + z ) and {1 + z + z } 1/ 1+ (1/ 2)( z + z ), where γ 0 is the 0 1 y x y surface tension of the {hkl} plane and γ 1 is the second order variation in the surface tension due to local surface slope [30 32]. It should be mentioned that the first order variation in the surface tension with local surface slope vanishes since {hkl} was assumed to be a low index plane with a local minimum in the surface free energy. With these approximations, Eq. (2.3) becomes: σ 2 2 F( x, y) = γ 0 + ( zx + z y ) dxdy, (2.4) 2 13 x y

29 where σ = γ 0 + γ 1 is defined to the be the surface stiffness or rigidity [30 32], which characterizes how hard it is for the surface to be deformed from (or fluctuate about) the low energy {hkl} plane. Equation (2.4) gives an expression for the surface free energy in terms of local slope z α = z/ α, and as pointed out by Jeong et al. [30], the chemical potential µ ( x, y) represents the average free energy change due to the removal of an atom from the position (x, y), and is given by: δf( x, y) µ ( x, y) = Ωv, (2.5) δz where Ω = a 2 a is the atomic volume, where a the in-plane atomic length (a x = a y = a ) v z and a z is the atomic length perpendicular to the surface. Since the surface free energy F depends on surface slope and the chemical potential in Eq. (2.5) is proportional to the variation of F with slope, we see that the chemical potential will be proportional to the local curvature of the surface. It should be pointed out the reference chemical potential has been taken to be zero, and with the use of variational methods, it has been shown that Eqs. (2.4) and (2.5) lead to [30 32]: 2 µ ( x, y) Ωvσ z( x, y). (2.6) Using Eq. (2.6), it can be verified that regions with negative (positive) curvature with have a positive (negative) chemical potential, and as we will see next this is what drives the flow of surface adatoms, which leads to a change in the local surface height z ( x, y). Assuming that surface diffusion is the dominant mechanism for thermal r r 2 smoothing and taking the route of Mullins [14, 15] by assuming dz / dt = a a ( j), z 14

30 r where j = ( c D / k T ) ( x, y) eq B µ is the chemical potential driven surface adatom flux, the change in the surface height z as a function of time can be written as [30]: 2 4 dz( x, y) az a ceqdσ 2 2 = dt k T B ( z( x, y) ), (2.7) where c eq is the equilibrium adatom density and D is the surface diffusion constant, and k B T is the usual product of the Boltzmann constant and temperature. It should be pointed out that the effects of sublimation have been neglected. Lastly, it should be stated that Eq. (2.7) is a strictly continuum model, which does not specifically allow for surface steps. This is due the fact that steps create a discontinuity in the surface free energy. It costs energy to create a step on a surface, and this must be reflected in the total surface free energy F by the additions of a step free energy term. This fact will be briefly discussed later. There is another form of transport called evaporation-condensation kinetics, which one can think of as a hot sample in equilibrium with its vapor [4, 5]. Rather than diffusing along the surface adatoms evaporate from regions of higher chemical potential, travel through the vapor phase, and finally condense at regions of lower chemical potential. I will not treat this case in detail here since all of the experiments presented in this thesis were done in ultra high vacuum where an evaporating adatom does not return to the surface (cold walls), but it should be pointed out that the surface evolution in Eq. (2.7) for this mechanism becomes proportional to 2 z(x, y) The Burton-Cabrera-Frank (BCF) step model One very weak point of the thermal smoothing treatment of Mullins [14, 15] is that it is only valid for a surface in thermodynamic equilibrium above the surface 15

31 roughening temperature where the step free energy goes to zero. Usually the system under consideration is not in perfect thermodynamic equilibrium and is almost never rough so step-edges are present, as in the case of step-flow growth or sublimation. In these cases it is the microscopic kinetics that determines the motion of atomic steps, and thus the evolution of surface morphology. One standard treatment of step motions is called the Burton, Cabrera, and Frank (BCF) theory of crystal growth [13], and we shall examine a slightly modified version of this step model here. In the original work [13] great attention is paid to the statistical mechanics of the kink sites at the step-edge (see Fig. 2.3(a)). Here the main focus will be on the motion of the step-edge without detailed treatment of the kinks. Formulation of the original BCF model Figure 2.3(a) shows the kinetic processes that operate under typical experimental conditions, and all of these basic surface processes involve the breaking of bonds. If we represent atoms by cubes with one bond per face, then the lowest energy state for an atom is when all of its bonds are saturated, i.e., all faces of the cube are in contact with faces of other cubes. Thus, lowest energy state is that of an atom in a solid (built from atomic cubes ) with six nearest neighbors. If the solid is terminated, then a flat surface would be created that is composed of a plane of cubes with one face or bond exposed to vacuum (called a dangling bond). In the atomic cube model considered here, 5 bonds must be broken to remove an atom from the surface layer, and the hole left behind in the surface is called a vacancy. If the removed atom is placed on top of the surface layer, it is referred to as an adatom. As discussed above, if the surface termination is not perfect (or miscut) then steps will be present. The step-edges bound regions of the crystal 16

32 (a) step edge kink κ D terrace R sub R dep (b) Adatom κ attachment/detachment rate D terrace diffusion R sub desorption R dep deposition l n E a E d E=0 x x n+1 xn n E = E a - E d a E f x Fig (a) General view of the atomic processes on vicinal surface. (b) Step geometry and labeling (upper) used in the BCF step model, and corresponding energy landscape for a diffusing adatom (lower). 17

33 surface called terraces, and from simple bond breaking arguments it is easier to remove (fewer bonds to break) an atom from a step-edge and place on it the terrace. The process of breaking bonds can be accomplished with thermal energy, and with an ample amount, adatoms will be released from step-edges on to the surface. Once on the surface the adatoms can migrate/diffuse around until they reach a step-edge where they can lower their energy by attaching. A kink sight in the step-edge, as shown in Fig. 2.3(a), can saturate the most bonds, and therefore, tends to be the place where adatom attach to and detach from. If the thermal energy is great enough then adatoms may desorb from the surface into vacuum if they do not find a step-edge to attach to in a sufficient time. It is less likely for an atom to evaporate directly from the surface layer or from a step-edge since the number of bonds that must be broken is larger for these sites. Lastly, it is possible to deposit atoms onto the surface using and external source. The BCF model is one which attempts to capture all of the essential atomic level physics above but on a slightly courser scale (often called coarse graining [30]). To formulate the model we shall assume the 1D step geometry shown in Fig. 2.3(b), which is a good starting point for describing the systems presented in this thesis. I first review the standard BCF model, which was developed to model step motions when sublimation or growth are allowed, but does not yet consider step motions in the presences of an electric field. This will be treated later. The kinetic processes are taken to be thermally activated where an adatom hopping a distance a from one site to another must overcome an energy barrier as depicted in the surface energy landscape schematic of Fig. 2.3(b). Therefore, diffusion on the terraces can be described by an attempt frequency ν t0 and an energy 2 barrier E d separating each atomic site on the surface such that D a ν exp( E / k T ) 18 t0 d B

34 with units nm 2 /s. Adatom attachment/detachment at the step-edge is characterized by an attachment rate κ with units nm/s, and is described by an attempt frequency ν A0 and an energy barrier E A such that κ aν A exp( E / k ). This form for κ assumes that steps 0 A BT are impermeable [12], which mean that if an adatom hops over the attachment barrier then it must incorporate into the step before it can hop off. Physically, this is the expected case if the kink density along a step is extremely high so an adatom on a step can always quickly find a kink. If the kink density is low, then it is expected that steps become partially permeable, meaning that an adatom that hops onto a step can hop back off before incorporating into the step, either back to the original terrace to the other terrace on the opposite side of the step [12]. In this case, there will effectively be direct adatom flow from one terrace to the other that is proportional to the adatom concentration difference on either side of the step. This will not be considered in detail here. The formation of adatoms at a step site is characterized by an energy E F with an attempt frequency ν F 0, and from detailed balance the equilibrium adatom concentration (atoms/nm 2 ) at the n th 2 step is given by c a ν / ν )exp( E / k T ). It should be eq ( F 0 a0 pointed out that the concentration at the step site was taken to be 1, which physically means that there is a high kink density along a step so there is always a step site (atom) available for attachment (detachment). The Adatom lifetime τ before desorption/sublimation in UHV is described by an energy barrier E sub and attempt 1 frequency ν sub such that τ ν exp( E / k T ), where it is assumed that the chamber sub sub walls are cold so that an adatom does not return to the surface. Thus, the rate of B F B 19

35 sublimation is given by R sub c(x)/τ in units of atoms/nm 2 -s. Finally, we allow for the deposition of atoms onto the surface at a rate of R dep from an external source. At elevated temperatures (assuming no deposition, i.e. R dep = 0) the step-edges will be in steady-state with the sea of adatoms on the adjacent terraces, so as adatoms evaporate from the terraces, the step-edges must try and reestablish local steady-state by releasing more atoms. So as time progresses, there is a net loss of atoms at the step-edges that causes them move or flow in the step-up direction [the +x-direction according to Fig. 2.3(b)]. By depositing atoms on the surface at a rate R dep, it is possible to create near balanced (R dep = R sub ) or net growth (R dep > R sub ) conditions where the steps either do not move on average (zero net loss/gain of material) or move in the direction opposite that during sublimation (net gain of material), respectively. From this simple picture it is clear that if we want to characterize the motion of steps on the surface, we must determine the net rate at which atoms are exchanged between the step-edges and adjacent terraces. To do this we must determine the concentration of adatoms on the terraces subjected to the appropriate boundary conditions describing the exchange of atoms at the step-edges. Considering a closed region on the terrace and Applying Fick s second law, the change in the adatom concentration with time can be expressed as [13]: dc ( x) djs cn ( x) = dt dx τ n + R, (2.8) where j s is the surface current of diffusing adatoms, τ is the desorption life time, and R is the rate of deposition. Equation (2.8) amounts to saying that the time rate of change in the concentration in the closed region on the terrace is equal to net flux into or out of the 20

36 region. Taking j s to be given by j D[ dc ( x) dx] s n / situation ( dc n ( x) / dt 0 ) Eq. (2.10) can be written as: 2 d c ( x) cn( x) = D dx τ n 0 + 2, and assuming a quasi-static R. (2.9) One assumption here is that the step edges are moving with an average velocity v 0 that is much smaller than the typical surface diffusion velocity D/λ of an adatom, where λ Dτ is the adatom diffusion length [33]. If this were not the case we would have to treat the terrace concentration due to fast moving steps by adding the term v [ dc ( x) / dx] to Eq. (2.9), where v s is step velocity. This specific case really only applies to extreme growth conditions, and was treated by Ghez and Iyer [33]. It will not be addressed here since under the most extreme conditions encounter in the work here ν 0 ~ nm/s, D ~ nm 2 /s, and λ ~ 10 µm, so that ν 0 /( D / λ) << 1, hence step motion can be safely neglected in Eq. (2.9). As shown in Fig. 2.3(b) the n th terrace is bounded by the x n (x n+1 ) step on the left (right), so to calculate c n (x) in Eq. (2.9) we need to formulate the boundary conditions describing how the steps exchange mass/atoms with the terrace. In the case of a finite step attachment/detachment rate κ and forbidding direct jumps of adatoms over the stepedge from one terrace to another ( impermeable ), Chernov [34] proposed that the boundary conditions be written as: [ c c ( x = x )] dcn ( x) D = κ eq n n (2.10a) dx xn s n dcn ( x) D dx xn + 1 [ c c ( x = x )] = κ, (2.10b) eq n n+ 1 21

37 where κ is assumed to be equal for both the step-up and step-down side of a step and c eq was define above. Physically, the quantity κ ceq represents the flux of atoms from the step to the adjacent terraces while the quantities κ c x = x ) and κ c x = x ) n ( n n( n+ 1 represent the flux from the n th terrace onto the n th and n th +1 step, respectively, and Eq. (2.10) arises by balancing these fluxes with the diffusion flux on the terrace at the stepedges. Using Eqs. (2.9) and (2.10), the concentration on the n th terrace can be fully determined, and looks like those depicted in Fig. 2.4 for the case of net sublimation (green line) and net growth (orange line). It should be pointed out that the addition of κ by Chernov [11] is often considered the first modification to the traditional BCF [13] where κ was assumed large enough so that the terrace concentration is held at c eq at the step-edges. Thus, in the original BCF case [13], the concentration profiles in Fig. 2.4 would touch the c eq line at the step edges signifying that the step-edges maintain local equilibrium with the terraces. R sub < R dep c eq c n-1 (x) c n (x) x x n+1 n x n-1 x n-1 v n R sub > R dep Fig Example of concentration profiles from the BCF step model showing how step move under net sublimation (green) and net growth (orange) conditions. v n 22

38 Examining Fig. 2.3(b), we see that the n th step has the n th and n th -1 terrace adjacent to it. Therefore, the net flux to/from the n th step will be determined by the differences in concentration on the left side of the step c c ( x = x )] and the right [ eq n 1 n side of the step c c ( x = x )], and so the velocity of n th step is given by: [ eq n n {[ c c x = x )] + [ c c ( x x )]} dxn 2 v n = = a κ eq n ( n eq n 1 = n, (2.11) dt where the positive direction it taken to the right. Note that this expression assumes the step orientation shown in Fig. 2.4, i.e., a positive surface slope with step-up surface orientation as one moves to the right. If we had assumed a negative surface slop or stepdown orientation, then the expression on the right side of the equation would be multiplied by a negative sign. Consider next how the concentration profiles shown in Fig. 2.4 would cause steps to move. In the case of net sublimation R sub > R dep (net growth R sub < R dep ) the adjacent terrace concentrations on both sides of the step-edge are slightly smaller (larger) than c eq as shown in green (orange) in Fig Thus, there is a net loss (gain) of material at the n th when R sub > R dep (R sub < R dep ) causing the step to move in the step-up (step-down) direction as indicated by the green (orange) with a velocity given by Eq. (2.11). Using Eqs. (2.9) (2.11), one can, in principle, predict the evolution of a known step distribution provided the parameters D, κ, τ, and c eq are known. The treatment here was strictly 1D, so it should be pointed out that modifications to this model have been made to account for the 2D nature of the step-edge, e.g., step stiffness and the possible finite density of kink sites along the step. These modifications are important when studying step-edge fluctuations, and although not dealt with here a detailed discussion can be found in [30] and references therein. Lastly, it was assumed 23

39 that there was no external electromigration force acting on the adatoms causing a directional drift. This is only true when an indirect method such as electron bombardment or an alternating current is used to heat the sample. As presented so far, the BCF model is independent of step density, and therefore independent of surface miscut. One consequence of this is that surface evolution in the model is shape preserving, i.e., the steps simply flow keeping their original distribution. As a result, there is no tendency for the steps to rearrange into the lowest surface energy state, in direct contradiction with experiments on most surfaces. Furthermore, this independence on miscut is in contrast to the continuum description of Mullins [14, 15] discussed above where the surface energy is assumed to depend on the slope/curvature of the surface relative to a low index {hkl} crystal plane. In a BCF-type model, one way to add the effects of a slope- and curvature-dependent surface free energy is to assume that it costs energy to add a steps to an initially flat surface, and that the steps repel each other. The first to introduce step interactions to the BCF model was Natori [35], and this addition came in the form of a modified equilibrium adatom concentration c eq at the stepedge. To start we need to briefly return to the concept of surface tension γ, or surface energy per unit area. In the classical analysis of Mullins [14, 15] step-edges were ignored. To extend the theory to cases where the surface has a well defined terrace and step structure, Gruber and Mullins [36] derived an expression for surface tension that includes a step creation energy and a step-step repulsion energy. As discussed by Gruber and Mullins [36], one physical origin of repulsive step-step interactions arises from the fact that a real 2D step-edge on a surface is confined to a region by its neighboring steps. The steps can fluctuate along their length, but cannot cross one another since the 24

40 configuration is energetically unfavorable. As shown by Gruber and Mullins [36], this gives rise to an effective repulsive step-step interaction that depends of l -2 where l is the distance between two steps. Another physical origin of a step-step interaction with a similar l -2 dependence is from stress-dipole-mediated interactions [90]. Due to surface tension on the upper- and lower- terraces around a step, a stress dipole will exist at a step, resulting in strain in the surround material [90]. Neighboring strain dipoles produce an interaction energy that also scales as l -2. Over the years the so-called Gruber-Mullins form of γ has morphed into a reduced free energy density (per unit length of terrace) of the form [37]: β 3 f ( θ ) = f0 + tanθ + g tanθ, (2.12) h where f 0 is the surface free energy density of a perfectly flat low index {hkl} surface plane (the terraces), β is the energy required to create an single isolated step on the surface, h is the height of an atomic step, g is the step interaction parameter between neighboring steps with units ev/nm 2, and θ is the surface miscut as depicted in Figs. 2.2(a) and 2.2(b). Recalling the definition of miscut above, it is seen that the factor tanθ / h is just the number of steps per unit length (or step density). Assuming the step orientation shown in Fig. 2.3(b), taking the local slope on the n th terrace to be tan θ = h / l n, and interpreting the step-edge chemical potential as the free energy change when a step-edge moves due to the attachment of a single adatom, Liu and co-workers showed that the chemical potential of an adatom attaching to the n th step is given by [37]: µ n = 2ga h, (2.13) 3 3 ln 1 ln 25

41 where a 2 is the area of a square surface unit cell and l n (l n-1 ) is the terrace width in front of (behind) the n th step as depicted in Fig. 2.3(b). I again note that this assumes the positive surface slop step orientation depicted in Fig If the surface slope were negative, then the right side of Eq. (2.13) should be multiplied by -1. Before discussing how this chemical potential is implemented in BCF-type model it is worth pointing out how Eq. (2.13) was used by Liu et al. [37] to develop numerous atomic-layer models of step motion. The simplest physical picture of how the chemical potential drives step rearrangement in the atomic-layer models is shown in Fig. 2.5, and can be thought of as follows. Suppose that on average the steps do not move (R dep = R sub ), so if we start out with a sequence of equally spaced parallel steps all step-edge chemical potentials will be equal to zero, as depicted in Fig. 2.5(a). This situation corresponds to the lowest energy configuration or equilibrium state, where there is no need for the step-edges to exchange mass. Now suppose the n th step is displaced to right as shown in Fig. 2.5(b). Then according to Eq. (2.13) the n th -1, n th, and n th +1 steps will have chemical potentials that are above, below, and above the reference value of zero, respectively. This situation is depicted in Fig. 2.5(b). More importantly, the chemical potential at the n th -1 and n th +1 steps will be greater than that of the n th step, causing a net adatom flow on the terraces from the both the n th -1 and n th +1 steps to the n th step. As a result the n th step will move back toward its equilibrium position. Note that it appears as though the n th step is repelled by the n th +1 step, and should the step be perturbed to left toward the n th -1 it would be repelled by the n th -1 step. These step-step interactions or repulsions drive the surface into the lowest energy/equilibrium state where all of the steps are equally spaced. From Fig. 2.2(c) it is seen that this relaxation amounts to 26

42 removing the curvature from the surface through the transfer of adatoms between the step-edges. This is exactly the same behavior observed in the continuum model of Mullins [14, 15] above except on a more microscopic scale. (a) l n-2 = l 0 x n-2 Uniformly distributed steps x n-1 µ n-1 = 0 l l n-1 = l n = l l n+1 = l x x n+1 n µ µ n = 0 n+1 = 0 perturbed step distribution (b) Adatom flow µ n-1 > 0 µ = 0 µ n+1 > 0 µ n < 0 l l l n-2 = l n-1 > l l n < l n+1 = l x x n+1 n x n-2 x n-1 x n+2 x n+2 Fig (a) A uniformly spaced train of steps where all step-edge chemical potentials are zero. (b) Changes to the step-edge chemical potential when one step is perturbed from its original position. In connection to the BCF [13] picture of step motion, Natori [35] suggested that step interactions could modify the local equilibrium adatom concentration c eq around a step site, and proposed that the equilibrium adatom concentration at the n th step should be writtted as: c n eq µ / kbt = c e n c (1 + µ / k T ), (2.14) eq eq n B 27

43 where c eq is still the (true) equilibrium concentration of atoms on a surface with straight, isolated step-edges. It is typical to say that n c eq is the equilibrium concentration at or of the n th step, rather than to say that it is the atomic concentration around the step that would be required to have in zero net flow onto the step. One can think of this 2 modification in the following way. Recalling that c = a exp( E / k T ), where E F is the adatom formation energy [as shown in Fig. 2.3(b)], Eq. (2.14) may be written as c 2 = a exp{ ( EF n ) / kbt}. Taking E F µ n as the effective formation energy, we n eq µ see that if µ n < 0 (µ n > 0) the equilibrium adatom concentration at the step-edge will decrease (increase). This, of course, leads to a corresponding change in the flux of n adatoms κ c eq from the n th step onto the adjacent terraces. Equations (2.9) (2.11), (2.13), and (2.14) form a complete set of continuum equations that can be used to model/simulate step flow growth and sublimation, as well as predict surface evolution. Much of the work here will concentrate on direct-current (DC) induced step bunching eq F B which requires another modification of the model above. The phenomenon of DC induced step bunching will be introduced next, as well as the drift-diffusion sharp-step model used to explain it. 2.3 Direct-current induced step bunching on Si(111) Step bunching in 1D is characterized by an initially uniform train of steps with spacing l 0 that is transformed into a distribution consisting of regions of closely spaced steps with spacing less than l 0 (step bunches) separated by large nearly step free terraces. This phenomenon has been experimentally observed on a variety of surfaces under a number of different conditions, and one of the most fascinating systems is silicon. For 28

44 example, it has long been known that when Si(111) is heated to high temperature in UHV using a direct current (DC) results in step bunching under sublimation conditions [1]. Interestingly, high temperature annealing is not the only path to step bunching. Recently it has been found that etching Si(111) in potassium-hydroxide (KOH) can also lead to step bunching [38]. The first observation of step bunching on Si(111) was made by Latyshev et al. [1]. They found that the stability of a vicinal train of surface steps toward step bunching depends on both temperature and direction of heating current/e-field. The observation can be summed up in a few short statements, but before that we need to clarify the language to be used. When discussing the DC effects we shall speak in terms of the applied electric field (E-field) or the current that it gives rise to. Furthermore, when an E- field or current is directed up (down) the vicinal stair case in Fig. 2.6, we shall refer to it as being step-up (step-down). More generally, when any quantity is described as being step-up or step-down we always mean relative to the vicinal orientation of steps. With this we may describe the observations of Latyshev et al. [1] in the following way. Heating a vicinal Si(111) sample with a step-down E-field leads to step bunching when the temperature falls in Regime I (~ C) or Regime III (~ C), but in Regime II (~ C) the steps retain their evenly spaced character, and if fact appears to have enhanced stability as compared to annealing without DC conditions [37, 57]. In the case of a step-up E-field a vicinal train of steps is only observed to bunch in Regime II, and remain evenly spaced (stable) in Regimes I and III. It should be mentioned that another switch to step-up bunching was also reported above 1300 C [1]. 29

45 E r (a) vicinal (T < 950 C) Regime I bunched 1 µm 1.25 µm (b) bunched vicinal (1040 C 1140 C ) Regime II 5 µm 1 µm (c) vicinal (T > 1150 C ) Regime III bunched 1 µm 10 µm Fig The direction of the DC relative to the vicinal step-up and step-down direction is indicated above the figure, and corresponds to the AFM images directly below. Images (a) (c) show the observed step bunching behavior on Si(111). The bunched images are derivative-mode AFM images, and should be viewed as if illuminated from left-to-right. 30

46 Figures 2.6 (a) (c) show AFM images of this interesting observation. The schematic diagram above the images shows the vicinal step-up (left) and step-down (right) current directions corresponding the images below. Figure 2.6(a) shows the result of heating a Si(111) surface in Regime I with a step-down (step-up) current right panel (left panel) where an initially uniform distribution of steps became bunched (remained vicinal). In the right image of Fig. 2.6(a), the step bunches are identified as the vertical bright feature, and the darker regions separating them are step free terraces. Figure 2.6(b) shows AFM images of Si(111) heated in Regime II where a step-down (step-up) current has left the surface vicinal (bunched), opposite that of Regime I. Finally, Fig. 2.6(c) shows the effects of heating a Si(111) in Regime III, where step bunching has changed back to the step-down current direction as in Regime I. These observations are fascinating, and to date no sufficient explanation for this behavior has been found. One thing to notice is that in Regimes II and III there are crossing steps populating the wide terraces between step bunches due to the sublimation conditions [39]. Crossing steps are also seen at lower temperatures in Regime I, but they are rare since the sublimation rate is low The traditional sharp-step surface electromigration model Stoyanov [5] was the first to propose a model to explain step bunching on Si(111). The model is inherently 1D, which is a good approximation for Si(111) since the step bunches are generally observed to be straight and parallel when the heating current is passed perpendicular to the step-edges, as seen from Fig It was suggested that a diffusing surface adatoms may acquire a small effective charge q eff causing them to drift with a velocity v = DF k T, in the direction either parallel (q eff > 0) or anti- drift / B 31

47 parallel (q eff < 0) to the applied E-field according to the adatom force F = q E. This eff directional-drift of surface adatoms is analogous to the well-established phenomenon of bulk electromigration, and it is thought that the adatom force F is composed of a direct force term F direct resulting from the E-field induced drift and a wind force term F wind resulting from a transfer of momentum from the charge carriers [40, 41]. To simplify things the adatom effective charge is written using q eff, which can be either positive or negative depending the magnitude and direction of F wind relative to F direct. To account for this directional-drift, Stoyanov proposed that the diffusion equation in Eq. (2.11) should be modified to become [5]: 2 d c ( x) DF dcn( x) cn( x) = D dx k T dx τ n B subject to the corresponding boundary conditions: 32 R, (2.15) n [ c c ( x = x )] dcn( x) F D + cn( x = xn) = κ eq n n, (2.16a) dx x kbt n dcn( x) D dx xn+ 1 F + c k T B n n+ 1 [ c cn( x xn+ 1) ] ( x = x + 1) n = κ =. (2.16b) eq Here all parameters are the same as outlined above in the original BCF case with incorporating step-step interactions as defined by Eq. (2.14). These boundary conditions follow from the same arguments above, except here there is an additional term on the left-hand-side describing the drift-diffusion on the terraces. The modified diffusion equation shown in Eq. (2.15) will be referred to as the drift-diffusion equation, and it should be pointed out that the applied E-field was assumed to have no effect on the attachment or detachment processes at the step-edge, so there is no analogous drift- n c eq

48 attachment term on the right-hand-side of Eq. (2.16). Neglecting the drift-attachment term is only valid when d D/κ >> l 0, or when E a >> E d according to Fig. 2.3(b), and as will be discussed in chapter 4 this often referred to as the attachment/detachment limit. In this limit, the velocity of the n th step remains the same as that given by Eq. (2.11) above, and now Eqs. (2.15), (2.16), and (2.11) can be used to evolve a 1D sequence of surface steps on the Si(111) when heated with DC. At this point, two questions remain: (1) Why does a directional drift of adatoms cause step bunching? and (2) Why does step bunching have such a complicated dependence on the temperature and current direction? In the next few paragraphs I will directly address question (1), and along the way it will be discovered that question (2) can not be answered using what has been discussed so far. Question (2) has remained unanswered since the original discovery of step bunching on Si(111) by Latyshev et al. [1] nearly 15 years ago, and has motivated most of the work in chapter 4 of this thesis. Step pairing in the sharp-step model Numerous authors have discussed the criteria for step-edge and/or terrace stability in BCF-type models. The first to analyze the stability of an infinite train of parallel steps using the original BCF model [13] was Bennema and Gilmer [42]. This work did not include the effects of drift-diffusion, but did account for the Ehrlich-Schwoebel effect [68, 69]. Stoyanov [5] was the first to analyze terrace stability in the presence of driftdiffusion. Since this early analysis, step bunching in the presence of an electromigration induced drift has been studied by Pierre-Louis [43] assuming very general step kinetics. Similarly, Liu and Weeks have also reported on the stability of steps [44], and more recently Zhao et al. [10 12] have examined step instabilities on both Si(111) and Si(001) 33

49 when diffusion near a step-edge is allowed to differ than that on a terrace. Rather than reproduce these works here, I will briefly discuss the stability of steps on a surface only to reveal the conditions necessary to have step bunching with a directional drift of adatoms. The reason for this simple approach stems from the fact that a full analysis using Eqs. (2.15) and (2.16) is extraordinarily complex, and so it is easy to lose sight of the basic physics behind step bunching. The best way to start is to considering the simplest possible case where there is no net sublimation or deposition so that we may write Eq. (2.15) as: 2 d cn ( x) DF dcn( x) = D. (2.17) dx k T dx 0 2 B which has a solutions of the form c ( x) = A Bexp( Fx / k T ). Typically F << 1, so to n + make things easier we can expand the solution to linear order in F c ( x) = A B( Fx / k T ) without the loss of any important physics. Using the boundary n + B n n conditions in Eq. (2.16), neglecting step-step interactions ( c = c +1 = c ), assuming impermeable steps so that the boundary conditions do not couple adjacent terraces, and taking x = 0 at the center of the n th terrace the concentration to linear order in F on the n th terrace is given by: F ceq ln + 2d 1 + x kbt cn( x) =, (2.18) l + 2d n where l 2 x l / 2 and d D / κ is defined as the kinetic length. Note that if this n / n approximate solution is used then the right side of Eq. (2.17) differs very slightly from zero, but no basic physics has been lost since the deviation is very small. From Eq. 34 B eq eq eq

50 (2.18) it is seen that the direction of the eletromigration force F = q E will determine the sign of the slope of the steady-state concentration. This is an important concept and for the step geometry given in Fig. 2.3(b) a step-up (step-down) adatom force will cause an adatom pile-up on the right (left) side of the n th terrace, as illustrated in Fig Using Eq. (2.18) with the geometry in Fig. 2.3(b) the velocity of n th step-edge is given by: eff v n = a 2 κ 2c eq c eq [ l + 2d(1 Fl / 2k T )] c [ l + 2d(1 + Fl / 2k T )] n l n + 2d n B + eq n 1 l n 1 + 2d n 1 B, (2.19) where l n and l n-1 are the widths of the terraces in front and behind the n th step, respectively. Here, we are interested in determining under what conditions the n th step will be stable or unstable toward step pairing. One simple way to test this is to choose a direction (+/- step-up/step-down) for the adatom force, then perturb the n th step from its equilibrium position and see whether or not it returns to its original position. More clearly, we perturb the n th step to the right such that l n (l n-1 ) decreases (increases), then if we find v n positive (negative) we know the step will be unstable (stable) toward step pairing for that particular force direction. This is not easily done using Eq. (2.19), but if we consider two strong limits we can readily determine the qualitative requirements for step pairing. In the complete diffusion limit (d 0) and the complete attachment/detachment limit (d ) Eq. (2.19) reduces to: v 0, (when d 0) (2.20a) n 2 F v { } n a κ ln ln 1, 2kBT (when d ). (2.20b) 35

51 (a) Step-up adatom force (F > 0) c n-1 (x) c n (x) l l l n-2 = l n-1 > l l n < l = l n x x n-1 n-2 v n x x n Stability n+1 (b) Step-up adatom force (F < 0) c n-1 (x) c n (x) x n+2 l l l n-2 = l n-1 > l l n < l = l n x x n-1 n-2 v n x x n+1 xn n Step pairing instability x n+2 Fig Electromigration induced adatom pile-ups in the sharp-step model using the conventional boundary conditions of Eqs for a step-up adatom force (a) and stepdown adatom force (b). As discussed in the text, in this sharp-step model only a stepdown adatom force leads to a step pairing instability. First, in the complete diffusion limit of Eq. (2.20a), where d 0, the step velocity is found to go zero, so no instability is expected. This limit is characterized by κ, which according to the boundary conditions of Eq. (2.16) means that the steps maintain local equilibrium with the adatoms on the terrace near the step-edge. Therefore, there is no net attachment/detachment of adatoms to the step-edges. Hence, the step remains stationary. If the kinetics in this system are interpreted using the energy barriers in Fig. 2.3(b), then one might question the validity of this limit since it would require E a -. This issue will be addressed in chapter 4. In the opposite attachment/detachment limit (d 36

52 ) of Eq. (2.20b), we see that v F l l }. Now, if the adatom force is in the n { n n 1 step-up direction (F > 0), and we displace the n th step to the right (l n < l n-1 ), then we find v n < 0. Thus, the step returns to its equilibrium position, i.e., the step train is stable. If we heat the sample such that there is a step-down adatom force (F < 0), and consider a small fluctuation that perturbs the n th step again to the right (l n < l n-1 ), then this time we find v n > 0. Thus, the n th step will now run into or pair with the n th + 1 step in front of it, and it this step pairing that is thought to lead to large scale step bunching. Considering the complete attachment/detachment limit (d ) where c ( x) = c [1 ( F / k T ) x], it n eq + easy to see that it is the adatom concentrations at the right and left sides of the step at position x n will be controlled by the size of the upper (l n ) and lower (l n-1 ) terraces, respectively. Using this fact, and examining Fig. 2.7, it is clear that the necessary requirement for step pairing is to have a pile-up and dearth of adatoms on the upper and lower side of the step-edge, respectively. In the sharp-step model of Stoyanov [5], it happens that this requirement is met only for a step-down adatom force. It should be pointed out that these strong limits are unrealistic and do not occur in real experimental systems. Therefore, it is necessary to consider a slightly softer version of them where the kinetic length d is compared to the initial terrace width l 0. Following the statements of Liu et al. [44], we will consider the surface kinetics to be diffusion limited when d << l 0 and attachment/detachment limited when d >> l 0. These limits will be discussed in more detail in chapter 4. Finally, before discussing the nature of the step bunching transitions observed on Si(111) it is worth saying a few words about my simulations of step bunching using the sharp-step model. B 37

53 Typical surface step profiles from sharp-step model simulations Figure 2.8 shows a typical numerical simulation of step bunching using Eqs. (2.15) and (2.16), with R = c eq /τ (near balanced flux conditions) and a step-down adatom force F. The main plot shows a small portion of a 1000 step simulation. The black points mark the step positions of the initial step train before the simulation, and the line connecting them is to guide the eye. In order for the instability to occur in the simulations a small perturbation was added to each starting step position, which can be seen by the fact that the black points do not form a perfectly straight line. If the steps were exactly equally spaced, then it would be in unstable equilibrium, and would not change with time. This is not realistic, since there will always be some perturbations on a real surface at finite temperature. Reasonable estimates for the initial perturbations are easily obtained by examining initial step distributions on real samples. This will be discussed more in chapter 4 when numerical simulations are compared to experiments. The average initial step spacing (miscut) can also be adjusted, which allows step bunching behavior to be simulated at a variety of miscuts for direct comparison to experiments. The red data points (and red guiding line) show the evolved surface. The parameters used to generate step-bunched profiles will be discussed in Chapter 4. Clearly there are regions of high step density (step bunches) and large step free regions (wide terraces). The number of steps occupying a bunch can be easily determined from the numerical data, making it possible to determine the average number of steps in a bunch for the full simulation of 1000 steps for a given time of simulation. The inset shows a close-up of the center step bunch in the main plot. One thing to notice is that the step bunch has a sigmoid shape with a minimum terrace width at the center of the bunch 38

54 due to step-step interactions, as discussed above. We will also be interested in extracting the average value for this minimum terrace width after a fixed simulation time for comparison to experiments. The code used to evolve step profiles from Eqs. (2.15) and (2.16) is provided in Appendix A along with the parameter file read by the program. Simulation details will be discussed in more detail in later chapters. y (nm) x (nm) Fig Typical surface profile from numerical simulations of the sharp-step model. The black circles represent the initial step distribution, and the red circles represent the bunched surface The observed bunching step bunching transitions One long standing puzzle of step bunching on Si(111) has been the transitions from step-down to step-up current bunching between regimes I and II and from step-up to step-down current bunching between regime II and III as shown in Fig There have been a few mechanism proposed to explain these transitions, and they will be highlighted in this section. 39

55 Within the framework of the traditional sharp-step model of Stoyanov [5], it was shown that step bunching can only result when there is an adatom pile-up (dearth) at the upper (lower) edges of steps. Furthermore, in the analysis above it was assumed that adatoms drift in the direction of the applied heating current/e-field (q eff > 0). In this case one can readily describe the step-down DC induced bunching in Regimes I and III but not the step-up DC bunching in Regime II. The first (an perhaps the most obvious) possible explanation for the transitions between step-up and step-down bunching is that adatoms drift parallel to the applied E-field in Regimes I and III, and anti-parallel to the applied E-field in Regime II. Recalling F = q eff E, this means that the sign of q eff must change between Regimes I and II and Regimes II and III. An explanation for how this might happen, using first principles was, given by Kandel and Kaxiras [45]. They assumed that q eff was comprised of a direct component q d and a wind component q w (q eff = q d + q w ), where q d is the component due to electrostatic interactions and q w is the component resulting from the scattering of charge carriers from the DC off of the adatom, just as in bulk electromigration [40, 41]. Note that these two components act to oppose one another. Assuming an enhanced density of free surface electrons and that Si(111) experiences a surface melting phase a few hundred degrees below the bulk melting temperature, Kandel and Kaxiras [45] argued that it is possible to have a situation where the sign of q eff = q d + q w changes from positive to negative between Regimes I and II and from negative to positive between Regimes II and III. They argued that the first transition was purely due to a dominating wind force while the second was due to the incomplete surface melting. Although this scenario is appealing, it should be pointed out that more recent experiments indicate that adatoms on Si(111) drift parallel to the applied 40

56 electric field (q eff > 0) at all temperatures [46]. In these experiments, Degawa et al. [46], used large scale morphological changes to the shape of vertical-side-wall trenches in Si(111) during DC annealing at various temperatures. By considering the shape after annealing and the possible directions for macroscopic surface mass flow (adatom drift), they were able to conclude that q eff must be positive at all temperatures. Therefore, it now seems unlikely that reversals in q eff are responsible for the observed step bunching transitions. To explain step bunching with a step-up adatom force Stoyanov [3] later proposed a continuum model with a kinetic process not included in his original model of electromigration induced step bunching. In the original sharp-step model it was assumed that steps were impermeable, i.e. that adatoms hopping onto a step must solidify into the crystal phase, i.e., they must attach to the step edge. It was argued that if the kink or attachment site density was low enough, then many adatoms may simply hop over the step onto the next terrace. This mechanism (if fast enough) means there is a direct exchange of adatoms between neighboring terraces, and is often referred to as step transparency, step permeability, or sometimes kinetics with out local conservation of adtaoms. In his transparent step continuum model Stoyanov [3] argued that a step bunch can be considered a continuum source of adatoms during sublimation with a generating power proportional to the step density and the local undersaturation. Note that this model is not of the BCF class discussed above since step motion is not determined only by the two immediately neighboring terraces. Although no direct mechanism leading to bunching was identified, it was argued that the electromigration induced adatom concentration asymmetry for a step-up adatom force could lead to stable 41

57 step bunches under sublimation conditions provided the electromigration effect compensates the Gibbs Thomson effect (curvature related mass transport, see above). Under net growth conditions one should only find stable step bunches for a step-down force. This mechanism is shown schematically in Figs. 2.9 (a) (d). Since step bunching in Regime I has been observed to occur on Si(111) only for a step-down adatom force under both sublimation or growth conditions [4, 47], this model therefore assumes that transparent steps do not exist in Regime I. (a) c eq Net sublimation c(x) mass flow E r Step velocities (c) c eq c(x) Net growth mass flow E r relaxation (b) c eq mass flow E r c(x) relaxation (d) c eq c(x) E r mass flow Fig. 2.9 Predicted step bunching behavior from the transparent step model under net sublimation (growth) conditions (a) and (b) [(c) and (d)]. 42

58 Experimental observations of step bunching were made in Regimes II and III under both sublimation and net growth conditions by Métois and Stoyanov [4]. They reported that bunching in Regime III behaves just as in Regime I, i.e., step bunching only for a step-down adatom force under both sublimation and growth conditions. However, they reported that in Regime II, step bunching occurs for a step-up (step-down) adatom force under sublimation (growth) conditions. These findings match the predictions of the transparent continuum step model, so it was then proposed [4] that the step bunching transitions result from a change from non-transparent to transparent steps between Regimes I and II and change from transparent back to non-transparent steps between Regimes II and III. Some of the experiments described in Chapter 4 were designed to directly test this proposed model. Finally, it has been recently proposed [9 12] that if hopping toward a step-edge (or diffusion in a small region around a step) were to become faster than on the terrace, the pile-ups and dearths in Fig. 2.7(a) would reverse, leading to step-bunching for a stepup force. This suggestion is appealing because it can provide an explanation for all observed bunching behavior on Si(111). The majority of this thesis, from here forward, will be devoted to sorting out which of these mechanism(s), if any, is truly responsible for the observed step bunching transitions on Si(111). 43

59 CHAPTER 3 EXPERIMENTAL METHODS Studying the motion and arrangement of surface atomic steps requires a clean, controlled environment in which experiments can be carried out. The samples themselves must be clean and have a well defined surface structure. A clean surface and vacuum environment is needed because surface contaminants/impurities can pin steps, resulting in unwanted step arrangements and step bunching [48]. To achieve this, an ultra high vacuum (UHV) system was designed and assembled with the capability of processing many samples with limited downtime. The silicon samples used to study the process of step bunching were prepared using a dimpling process, giving them the desired surface geometry with a low number of surface contaminants and virtually no defects. In this chapter I will outline the equipment and procedures used to carry out the experiments presented in chapters 4 and 5 of this thesis. 3.1 Ultra high vacuum system The basic setup and major UHV components to be discussed in this section are show schematically in Fig The UHV system consists of two separate sections, and each serves a specific function. The main chamber is based on a Perkin-Elmer TBN-X style vacuum chamber with a ~500 l/s ion pump, and is also equipped with a titanium 44

60 sublimation pump to help achieve and maintain UHV conditions. To eliminate the need for constant venting, pump downs, and baking the main chamber is connected to a sample transfer section by a gate valve. This allows the main chamber, where annealing experiments are carried out, to always be kept under clean UHV conditions with a base pressure 1x10-10 torr. Figure 3.2 shows a photograph looking into the annealing chamber showing the sample tray, pincer, annealing stage, quartz crystal deposition sensing head, and electron beam (e-beam) evaporator. Sample transfer section Ion pump (20 l/s) Main annealing chamber Gate valve Ion gauge transfer arm Load lock Roughing port 6 veiw port w/ shutter e-beam evap. Fig Schematic diagram (top view) showing the layout of the UHV system The sample transfer section The sample transfer section consists of a magnetically coupled transporter arm, sample tray, load lock, and a 20 l/s ion pump. Typical base pressures on this side of the system range between ~5x10-7 and ~1x10-8 torr depending on how recently the transfer section was vented, and whether or not it was baked upon pump down. To increase the 45

61 throughput of the experimental process, the sample tray was designed to accommodate up to 5 samples. The samples to be annealed should be placed in the thin slots in the front half of the tray, and the wider slots in the rear are for annealed sample transferred out of the main chamber. The sample tray can by seen in Fig Sample tray pincer E-beam evap. Quartz x-tal monitor Annealing stage Fig Photograph looking through the 6 view port into the main annealing chamber where sample heating is carrier out. The major components discussed in this chapter are labeled. When preparing to load samples one should check to ensure that the gate valve between transfer section and main chamber is closed, as well as the gate valve to the 20 l/s ion pump. The ion pump should be left running when closed off. Venting the transfer section is done with compressed nitrogen gas using the roughing port attached to the bottom of the load lock. Once vented, samples are easily loaded through the load lock 46

62 access door into the sample tray. With the access door closed securely, the transfer section can be pumped down using the turbo-molecular pump. Before opening the gate valve to the 20 l/s ion pump, the pressure measured at the turbo pump should be at or below ~1x10-6 torr. This is usually achieved after about 3.5 hours of pumping, and at this point the gate valve to the 20 l/s ion pump can be opened and the roughing port valve to the turbo pump closed The annealing chamber Sample annealing is carried out in the main chamber, so it is important that the pressure on this side of the system be as low as possible. Furthermore, the constituent gases giving rise to the base pressure should ideally be hydrogen (mass 2), nitrogen (mass 28), and oxygen (mass 16). The main chamber is outfitted with an Uthe Technology International residual gas analyzer (RGA) that can be used to check the gas components, and leak detection using compressed helium gas. Note that an extremely large nitrogen peak might signify a leak, while an extremely large water peak (mass 18) may signify a poor bake out or a leaking water cooling line. Given the age of the RGA (ca. 1973), it is not possible to provide a background mass spectrum for the main chamber here. Typically, a base pressure below 2x10-10 torr is easily achieved as measured by an ion gauge, but this may fluctuate depending on the recent sample transfer and annealing history. Should the main chamber be vented, standard bake out procedures should be used to reestablish the base pressure. Wobble stick and pincer The top 6 port on the chamber accepts a 6 flange with four 1/4 diameter copper feedthroughs and the wobble stick with pincer. These copper feedthroughs 47

63 provide electrical contact to the annealing stage (see below), while the wobble stick and pincer provide a means of transferring samples from the sample tray to the annealing stage. The wobble stick was purchased from Thermionics Vacuum Supply and customized with 3 of additional stroke. The pincer shown in Fig. 3.2 is made from a stainless steel and molybdenum. The majority of the pincer is stainless steel, but the jaws of the pincer that come into contact with the silicon sample are made of molybdenum. Stainless steel should never be brought into contact with silicon since it has a large concentration of nickel, which can contaminate the surface and pin steps. Annealing stage Copper clamps provide the base for the annealing stage and allow the entire assembly to be attached to the copper feedthroughs in the annealing chamber. Figure 3.3 shows a simple schematic of the annealing stage from which one can figure out the order of assembly. The entire assembly is held together with 2-56 molybdenum threaded rod and 2-56 molybdenum hex nuts. The key components to the annealing stage are the Copper clamp Alumina hat washer Tantalum sample post Moly support/ spacer Fig Schematic diagram showing the basic assembly of the sample annealing stage. 48

64 alumina hat washers. These provide electrical isolation between the copper clamps and the two molybdenum sample posts, on top of which the sample sits. This isolation is important since it ensures the applied voltage drops across silicon sample to be annealed. The molybdenum spacer plate provides structural support and ensures that the sample sits flat and level so that there is minimal sample strain, which can influence step motion [49, 50]. Lastly, the molybdenum sample posts serve as a holder for the sample, and the silicon sample is held in place by tungsten foil clips. Tungsten was chosen over tantalum and molybdenum because it can be worked into the desired shape, was stiff enough to withstand the sample transfer process, and can survive repeated thermal cycling. It should be noted that only the refractory metals, i.e., tantalum, tungsten, molybdenum, and rhenium are suitable for direct contact with Si at high temperatures since they will not contaminate the surface. Rod fed electron beam evaporator To study how step motion depends on net sublimation or growth conditions, a Tectra e-flux electron beam evaporator ( modified to accept 6 mm diameter rods of Si, was added to the main chamber. The evaporator is water cooled, which helps keep the main chamber pressure at a reasonable level ( 2x10-9 torr) during long depositions at high flux rates. The copper cooling shroud of this instrument can be seen in Fig This instrument also has an electrode that can used to monitor the deposition flux, but has been found to be unreliable when depositing Si. The control unit has analog meters for reading/controlling the rod (accelerating) voltage and the rod (emission) current. The product of these two gives the rod power which can be used as a rough guide for monitoring the deposition rate. A digital readout for the emission current 49

65 has been added to the control unit to allow for a more accurate determination of the rod power, however, the quartz microbalance equipment (discussed below) should be used to determine the deposition rate, not the rod power. To date the highest possible Si flux rate achieved is 0.25 Å/s for sample-to-source distance of ~5 cm. The user manual should be read before using this instrument since high voltages are need when evaporating Si. Quartz microbalance/deposition monitor In order to monitor the deposition rate/thickness the main chamber was outfitted with a quartz crystal microbalance sensor, from Sycon instruments ( The sensing head is controlled using the STM-100 deposition thickness/rate monitor and oscillator, with an accuracy of 1 Å (thickness) with a deposition rate resolution of 0.1 Å/s, as quoted by the Sycon. The sensing head is completely blocked ( shadowed ) from the hot sample source, so it sees Si flux only from the evaporator. This can be seen in Fig. 3.2 from the deposition pattern on the rear wall of the chamber behind the quartz crystal monitor. As stated above, the highest possible deposition rate at the sample is 0.25 Å/s. At this rate one can rely on the realtime reporting of the monitor, but below this value time averaging needs to be done. For example, if a deposition rate of 0.1 Å/s is desired then every minute the thickness reading should increase by ~6 Å, and for rates below 0.1 Å/s a larger time interval should be used. Small corrections to the rate can be made by adjusting the accelerating voltage, filament current, or rod position. Caution should be exercised when adjusting the rod position since too large of a movement forward (backwards) will cause a drastic increase (decrease) in the deposition rate. Care should also be taken when adjusting the filament current, as a small change corresponds to an exponential change in the number of 50

66 electrons emitted from the tungsten filament. Lastly, anytime the main chamber is vented the deposition monitoring system should be recalibrated and a new tooling factor determined. This can be done by completing a deposition for a fixed time onto a room temperature sample with a shadow mask, and then measuring the step height using the KLA-Tencore Alpha-step surface profilometer in Professor Epstein s lab at The Ohio State University. 3.2 Sample Preparation In order to study the effects of both a step-up and step-down current on step bunching, as well as miscut, a spherical dimple was ground into the silicon samples. This was done using a VCR D300 dimpler instrument ( usually used for preparing samples for study with transmission electron microscopy. Typical surface profile will be discussed shortly. The polishing wheel used was ~6 in diameter, and the samples were nominally 25x6x0.4 mm 3 cut from an n-type ( Ω-cm) Si wafer. It is important to note that this dimpler was designed to us a much small polishing wheel, and that getting proper dimples with a 6 diameter wheel is tricky (see below). After dimpling the sample is chemically cleaned (see below) before being loaded into the UHV system, with further thermal cleaning (see below) done in UHV using standard techniques before beginning the experimental anneal Dimpling Procedure The procedure for producing symmetric dimples is as much art as science. If the procedure given here is followed, then one should be able to consistently produce highly symmetric dimples with a depth control of a few microns. It should be noted that this 51

67 procedure was perfected by Sylvia Schaepe, an Ohio State undergraduate physics major, during a summer internship. Mount the Si sample to the brass sample chuck by doing the following: 1) Turn the hot plate to 600 C and place the brass sample puck on it. 2) Once hot, apply crystal bond adhesive to the sample puck. 3) Place the sample on the puck polished side up, and carefully center the sample using the wooded end of a cotton swab. There are lines scribed on the chuck that can be used as a guide. 4) Remove the sample chuck from hot plate, and cool on the copper block. Check to see if the sample is centered on the chuck by doing the following: 1) Mount the sample chuck on the dimpler. 2) Check the centering using the microscope and reticle. -First focus the sample and then turn the eye piece so that the reticle marks run horizontal. -Turn the sample so its long edges are approximately horizontal. -Chose a major reticle division mark and line it up with the top of the Si sample by moving the microscope. If the sample is centered, then after a 180 rotation what was previously the bottom of the sample will be the top and should line up with the original major division. If it does not line up, then the chuck should be removed and placed back on hot plate. Once the crystal bond has melted sufficiently the sample position can be adjusted using the wooden end of the cotton swab by applying pressure to the edges of the sample. The chuck can then be 52

68 cooled, and the centering of sample checked again. This procedure should be repeated until the sample is centered, and once centered do the following. Secure the sample chuck to the dimpler. Cut a piece of the Buehler adhesivebacked polishing felt into a ~1/8 thick strip and the cover the brass wheel with it. Using dionized (DI) water moisten the polishing felt. Using an eye dropper generously coat the felt with the Buehler Masterment polishing fluid, and place a few drops of the polishing fluid on the Si sample. Start the dimpler while the wheel is not in contact with the Si sample. It is best to start with the arm speed on a low setting like 20 revolutions per minute. Once the wheel is in contact with the sample, the arm speed should be increased to 40 revolutions per minute. During the process of polishing one should do the following until the desired polishing time is reached: Every 1 minute add two drops of DI water to the felt while spinning. Every 5 minutes add 5 drops of polishing fluid to the felt while spinning and 2 drops of the extender. Once the desired time/depth is reached, lift the wheel up from the Si sample, reduce the arm speed to zero, and turn the dimpler off. Rinsed the sample with DI water and blow dry with compressed nitrogen gas. Remove the sample from the chuck. Technical note: The depth of the dimple can be controlled by adjusting the polishing time. There is a vertical stop micrometer on the instrument that could also be used, but 53

69 since the diameter of the wheel is many times larger than the intended wheel, modifications would be needed to implement it Chemical sample cleaning procedure After the dimpling and removing the sample the following procedure should be used to clean the sample. Using Acetone (ACE) and a cotton swab scrub the sample (gently) on each side. This should be done twice for both sides of the sample. To ultra-sonic clean with solvents do the following: Three beakers are needed for each of the three solvents to be used: trichloroethalyne (TCE), ACE, and methanol (METH).When using the ultra-sonic cleaner a plastic wafer tray lid should be placed between the beaker and the bottom of the cleaner. To avoid metal contamination, be sure that the beakers have not been used to clean metal parts. 1) Place samples in beaker 1 and fill with ~150 ml of TCE. Ultra-sonic clean for 5 minutes. Drain TCE from beaker. 2) Place samples in beaker 2 and fill with ~150 ml of ACE. Ultra-sonic clean for 5 minutes. Drain ACE from beaker. 3) Place sample in beaker 3 and fill with ~150 ml of METH. Ultra-sonic clean for 5 minutes. Drain METH from beaker. All used solvents should be disposed in the proper solvents waste can. Not left in the fume hood to evaporate! Rinse samples with DI water and blow dry with compressed nitrogen gas. The samples are ready, at this point, for loading into the UHV system. But, it is a good idea to take the time to inspect them using the KLA-Tencore surface profilometer in 54

70 Professor Epstein s lab. This can be done after the first ACE scrubbing step above to save chemicals. Typical surface profiles of dimples samples will be discussed below in section UHV thermal cleaning procedure Prior to heating a sample under the desired experimental conditions, it needs to be thermally cleaned to ensure it is free of surface contaminants and oxide layers. The method used here for cleaning is a standard procedure used to prepare silicon for surface science studies [51]. It consists of a sample degas and then a sequence of short high temperature flashes. Heating must be done using an indirect method or alternating current, to ensure that the steps do no rearrange themselves during the cleaning procedure. The sample temperature is monitored with an optical pyrometer. It should be noted that the pyrometer reading is not the true sample temperature. When measuring the temperature through a 6 glass view port the pyrometer under estimates the real temperature by ~40 C. More information on the pyrometer reading can be found in Jeff Seiple s Thesis (The Ohio State University). The temperatures reported in this chapter are direct readings from the pyrometer and not the real sample temperature. The temperatures reported in chapters 4 and 5 are real sample temperatures. Alternating current (AC) resistive heating was used to degas and flash the sample. When the sample is at room temperature it is extremely resistive, so initially a large voltage needs to be dropped across the sample to get an ample amount of current flowing. This can be accomplished using a variac. For this a heat gun (used for wire shrink tubing) was placed in series with the Si sample. This device is capable of drawing ~0 12 A when controlled using the variac, and has the added benefit of protecting the sample 55

71 from melting. The basic circuit for this is shown in Fig For the sample size used here (15 x 0.6 x 0.4 mm 3 ), there is only a small chance of melting the sample with this set up. Even at a full sample power of ~70 W (~5.5 V and 12.5 A), the sample will only reach ~1250 C (pyrometer). If melting occurs, it typically happens at the clips where there is a contact resistance. This is rare, and it is usually a crack in the sample, introduced during the sample transfer, that causes the sample to break. V Si Heat gun A M Variac autotransformer Fig Basic circuit used for alternate-current sample heating. Initial sample degas Once the Si sample is in the annealing stage it needs to be degassed at ~740 C (as read on the pyrometer) for several hours. This degas temperature is low enough to leave the protective native oxide intact while the sample and annealing stage degas. Degassing times may vary, and a good rule to follow is to make note of the base pressure in the main chamber (say for example, 1.5x10-10 torr) before the sample transfer, and then degas for as long as it takes to get back to or very close to the base pressure. Depending on when the last degas occurred this can take anywhere from hours. 56

72 Care must be taken when starting the sample degas to keep the temperature from rising above 740 C (pyrometer). It takes roughly 35 V (on multimeter in Fig. 3.4) to get current flowing in the sample, and once the current rises there is a sharp increase in sample temperature and a decrease in the voltage across the sample. The voltage decreased is caused by the fact that the sample resistance deceases strongly with increasing temperature. Therefore, the voltage should be increased slowly while watching the pyrometer temperature, and once the temperature and current start the sharp rise the Variac should be scaled backed drastically. At this point the degas temperature of 740 C (pyrometer) can be uniformly approached using the Variac. Sample flash cleaning After the sample degas is complete, the sample should be taken through a sequence of short, high temperature flashes. First, the Variac voltage, sample current, and sample voltage need to be established. By turning up the Variac until the pyrometer reads ~1220 C, one can figure out the dial voltage the Variac needs to be turned durning the flashing sequence. The sample should be left at this temperature for 5 seconds while the sample current and voltage are recorded. Typical sample currents and voltages during a flash are 12.5 A and 5.5 V (~70 W), with a variac dial setting of V. Ultimately, these numbers will depend on the sample size and contact resistance between the sample and the clips. Note that Variac dials are prone to failure, so care must be taken when establishing the dial setting. Dial failure is due to slipping of the set screw that holds the dial in place. This first high temperature exposure will desorb the native oxide, exposing a bare Si surface. 57

73 After determining these values the sample should be put through a series of 30 flashes under the conditions above. Starting from a pyrometer reading below 300 C, the Variac dial should be quickly turned to the value determine above. The pyrometer should read at least 1220 C, and if not the Variac should be adjusted so it does. The sample should be left at this temperature for ~5 seconds, after which the dial of variac should be quickly turned to zero to quench the sample. Typical cooling rates are of the order ~250 C/s. The sequence of quickly ramping up the sample temperature and quenching it constitutes one flash, and this sequence should be repeated 30 times. It is a good idea to wait ~3 s after the quench to let the pressure in the chamber recover. During each flash the Si(111) surface loses approximately ~41 Å or ~13 surface bilayers of Si, leaving a clean surface composed of a contaminant free layer. The sequence of 30 flashes ensures that all surface contaminants are gone. Figure 3.5 shows a typical 3x3 µm 2 AFM scan of the Si(111) surface after this cleaning procedure. The miscut here is ~0.14 or equivalently l 0 ~ 129 nm. 600nm Fig AFM image of a typical Si(111) vicinal surface after flash clean using the procedure described in the text. Image size is 3x3 µm 2. 58

74 3.3 Dimpled Surface Profiles Figure 3.6 shows typical surface profiles of dimples polished for 2, 5, 10, and 20 minutes. Longer polishing times result in deeper dimples when the procedure above is followed. One thing to notice is that the dimples are not perfectly symmetric about the deepest point at ~5000 µm. There is always some asymmetry present, and a number of things were tried to remove this imperfection. Although puzzling, it does not affect the experimental results. Following the dimpling procedure above, a 20 minute polishing time should consistently give dimples that are ~40 45 µm deep. Note that the weight of the dimpler arm will also affect the final dimple depth. Depth (µm) minutes 5 minutes 10 minutes 20 minutes Distance (µm) Fig Typical dimpled surface profiles. The polishing times are indicated on the plot. (The profiles were taken using the Alpha-Step Profiler in Professor Epstein s laboratory.). Aside from providing both step-up and step-down direct current annealing data, the dimples also provide a range of local surface miscuts. Figure 3.7 shows the first derivative, corresponding to the macroscopic variation in the surface miscut, of the 59

75 profiles in Fig Dimples polished for minutes provide a substantial linear region over which the rate of change in the surface slope is gradual. For example, the dashed lines indicate an approximately linear region for both the 10 and 20 minute dimples. The range of miscut in this region is ~±0.5 [l 0 57 nm for Si(111)] over a distance of ~1000 µm. This corresponds to ~0.001 degrees/µm, and for a large area AFM scan of ~25 µm this translates into a variation in local miscut of ~0.025 across the scan, which is tolerable for the work to be presented here. All data shown in this thesis was taken from dimples polished for 20 minutes, where miscuts ranging from ~0.1 to ~1 were accessible with on a small variation in miscut across the scan. Finally, notice that the asymmetry in the depth profiles of Fig. 3.6 only has noticeable affects beyond the range of miscut we are interested in here. Miscut (degrees) minutes 5 minutes 10 minutes 20 minutes Distance (µm) Fig st derivatives of the dimple profiles shown in Fig

76 CHAPTER 4 RESULTS: ELECTROMIGRATION INDUCED INSTABILITIES 4.1 Step instabilities on Si(111): The effects of Si deposition In this section I will discuss the effects of Si deposition on electromigration induced step instabilities on the Si(111) (1x1) surface, with a focus on step bunching in Temperature Regimes I (~ C) and II (~ C) on dimpled samples that have a range of initial surface miscut angles (±0.5 ). It should be noted that the temperatures reported through out this chapter are actual estimated sample temperatures, i.e., pyrometer values correct plus a 40 C to account for a glass view port. Three primary findings will be discussed. First I will discuss electromigration-induced step bunching on Si(111) for Regimes I and II. We find that a step-down electric current is required to induce bunching under both net sublimation and depositions conditions in temperature Regime I, in agreement with previous reports. However, for temperature Regime II a step-up current is required to induce step bunching for both net deposition and net sublimation conditions, in contradiction with the report of Métois et al. [4] and suggested step permeability model of Stoyanov [3]. Secondly, the observation of a strong reduction in the number of crossing steps on the wide terraces for near equilibrium Si flux conditions will be discussed. Lastly, I will present data showing that the rate of 61

77 step bunching has a nearly linear dependence on the initial sample miscut angle in both Regimes I and II, and is found to be independent of net deposition/sublimation conditions Introduction and motivation As discussed previously in this thesis, understanding the processes that govern the motion of vicinal surface steps has been a longstanding problem of great fundamental interest in surface science. For semiconductors, understanding the behavior of surface steps is technologically critical for the growth of epitaxial overlayers and for device processing. Recently there has been great interest in processes that lead to spontaneous rearrangement of steps into surface structures of size ranging from a few nm up to many µm [52 56]. Since 1989 [1] it has been known that heating a slightly-miscut Si(111) with a DC in an UHV environment leads to large scale changes in surface morphology such as the formation of step bunches. The earliest reports of these surface electromigration phenomena by Latyshev and coworkers [1], showed that step bunching on Si(111) only results for one direction of the current relative to the vicinal step-up or step-down direction, but also that this required current direction reverses multiple times with increasing temperature. In temperature Regime I (~ C) and Regime III (~ C) bunching occurs only for step-down heating current, while in Regime II (~ C) and Regime IV (>1320 C) bunching occurs only for step-up heating current. In all temperature Regimes the opposite current direction maintains an initially-vicinal surface and accelerates the relaxation of an initially step-bunched surface [57]. Figure 2.6, in chapter 2 of this thesis, clearly shows AFM images of the step distributions for the first three step bunching Regimes. 62

78 Still controversial is the physical origin of these temperature-dependent reversals of the current direction required for step bunching. It is generally accepted that the diffusing surface species (thought to be individual Si adatoms for Si(111)) each have an effective charge q eff causing them to drift (or flow) either parallel (for q eff > 0) or antiparallel (for q eff < 0) to the applied electric field. It is also generally agreed that a step-down adatom flow will cause a bunching instability provided steps have a sufficiently small attachment probability and are sufficiently impermeable (i.e., a diffusing surface atom must incorporate into a step before it can cross onto the adjacent terrace [6]). Recalling from chapter 2, one general model proposed that q eff changes sign as the temperature is increased so that adatom flow is parallel to the applied electric current in Regime I and Regime III, and anti-parallel in Regime II and Regime IV [45]. In this case there would be step-down adatom flow (and hence bunching) only for stepdown (step-up) electric current in Regimes I and III (Regimes II and IV). However, recent experiments indicate that q eff is positive at all temperatures on Si(111). Another proposed model is that adatom flow is always parallel to the applied electric field (i.e., q eff is always positive) but that temperature-dependent changes to the step permeability [3, 4] or to the relative adatom diffusivity close to a step edge [9 12] lead to bunching even for step-up adatom flow. In this section, I will focus on the permeable step model proposed by Stoyanov [3] and supporting experimental evidence [4] which hold that significantly increased step permeability in Regime II causes step bunching for a step-up adatom flow. This mechanism was briefly discussed in chapter 2, and is shown schematically in Fig. 2.9 there. One key prediction of this model is that permeable steps will bunch for a stepup adatom flow only under net sublimation conditions (when the Si sublimation rate 63

79 R sub is larger than an applied Si deposition rate R dep ), while a step-down flow is required to produce bunching under net deposition conditions (R sub < R dep ). Métois et al. [4] reported experimental observations that showed bunching in Regime II, indeed, follows this predicted pattern, with step-up current required for bunching under net sublimation, and step-down current required under net deposition conditions. This would appear to rule out competing models [9 12] which predict that the required current direction for step bunching should be the same under either net sublimation or net deposition conditions. Given the importance of this finding, the complexity of carrying out DC annealing experiments at high temperature in the presence of a flux, and the fact that the result has not been independently verified we have set out to test this experimental finding for a range of deposition conditions and local surface miscut angles Experimental procedures The Si samples used in this study were cut from an n-type Si(111) wafer ( Ω-cm) into 25 x 6 x 0.4 mm 3 rectangular strips with the long edge parallel to 112. Shallow spherical dimples of radius of ~76 mm were ground into the sample surface providing a ±0.5 range of surface miscut angles (see Figs. 3.5 and 3.6) and cleaned as discussed in chapter 3. To minimize contamination during annealing, unmounted wafer samples where introduced through a load lock into the UHV annealing chamber (base pressure <1 x Torr) and then inserted in-situ into a well-degassed sample holder consisting of two ~2.5 cm long Mo heating supports held far from anything else in the system (see chapter 3). Samples were degassed in UHV for ~2 hours at 780 C by passing 60 Hz alternating-current through the sample, and then cleaned by flashing 64

80 repeatedly at ~1220 C (each flash lasting ~5 10s) for a total time ~2.5 5 minutes, as discussed in chapter 3 section The Samples were then annealed for 3.0 h at 940 C or 1090 C under various deposition conditions. Si deposition (up to ~0.25 Å/s) was done using the commercial rod-fed electron beam evaporator discussed in chapter 3 held ~5 cm from the Si wafer. The Si deposition rate was monitored using a quartz crystal that was shielded from the hot sample to ensure it only received Si flux from the deposition source, and the sample temperature was monitored using an optical pyrometer. The sublimation rate at a given temperature was determined by measuring the deposition flux that produced near zero net sublimation/deposition rate R net (R dep R sub ). How zero net sublimation/deposition conditions were determined will be discussed below. The pressure remained below ~4x10-10 Torr during long anneals in the absence of Si flux, and below ~1x10-9 Torr for the cases of highest Si deposition flux due to outgassing of the hot electron beam source. After annealing under a given sublimation/deposition condition, the samples were transferred out of UHV and characterized ex-situ with AFM. On samples annealed with the electron beam evaporator operating, widely-spaced pinning sites were observed, but these only affected step bunching within 5 10 µm of the pinning site, as shown in Fig. 4.1(a). As done previously [47], the shape of pinned steps close to pinning sites was used to accurately determine whether a given annealing condition produced net sublimation or net deposition conditions. For example, in Fig.4.1(b) the surface steps move under stepflow sublimation from left-to-right. When these moving steps encounter an impurity (marked by the light blue arrow) they become pinned, causing them to bend into the observed patterned. The single step seen on the wide region behind the impurity in Fig. 65

81 4.1(b) is one that has broken away from the impurity. This same idea can be applied to Fig. 4.1(c), with a left-to-right vicinal step-up direction, where the shape of the pinned steps indicates net growth conditions. a) b) 25µm c) 2µm 5µm Fig (a) Typical large area scan of bunched surface (annealed at T=1090 C) with a high Si deposition flux (R dep 0.17 Å/S R net 0.01 Å/S). Close up views of pinning site under slight (b) net sublimation and (c) net deposition conditions. In net sublimation (net deposition) the steps retract from (advance past) the pinning site Regime I: Deposition effects Figure 4.2(a) shows a schematic sample cross-section near the dimple bottom, with the applied electric field directed from right-to-left during the high temperature DC anneal. Figures 4.2(b) 4.12(e) correspond to this geometry, so the left (right) side of the dimple corresponds to step-up (step-down) current. Figures 4.2(b) and 4.2(c) show stepup and step-down areas of a sample annealed for 3 hours in Regime I at 940 C in the absence of the Si deposition flux (free sublimation with R sub Å/s), while Figs. 66

82 (a) Step-up current Regime I 940 C 3 hours Dimpled sample surface during anneal E r Step-down current Free sublimation ( Å/s) (b) 1.25 µm (c) 1.25 µm Net growth (+0.01 Å/s) (d) 1.25 µm (e) 1.25 µm Fig (a) Schematic view of vicinal stepped surface close to the bottom of a spherical dimple. For right-to-left applied electric field (and conventional current) direction, the left side (right side) has step-up (step-down) current. (b) and (c): AFM images of the surface annealed for 3.0 h at 940 C under free sublimation conditions (R net Å/s) for (b) step-up and (c) step-down current. (d) and (e): surfaces annealed at 940 C under net deposition of 0.01 Å/s for (d) step-up and (e) step-down current conditions. Bunching is only seen for step-down current for both net-sublimation and net-deposition conditions. (c) and (e) are derivative-mode AFM images, which appear as if illuminated from the left. 67

83 4.2(d) and 4.2(e) show step-up and step-down regions of a different sample annealed at the same temperature but under a Si flux R dep Å/s, and so R net Å/s. We find that step bunching occurs only for step-down current under both net sublimation and net deposition conditions, consistent with past reports for Regime I [4, 47]. Under free sublimation [Fig. 4.2(c)] the lowest step in each bunch has separated slightly from the bunch. This was also observed by Homma et al [58], who suggest this is due to step-flow growth when free adatoms on the wide terraces attach to the bounding step-edges during the sample quench. Notice that the top step in the bunch has not separated from the bunch. This is also explained by the same idea of free adatoms on the terrace attaching the step-edge during the quench. As the sample cools the adatoms on the terrace very close to upper edge of the bunch still have enough thermal energy to diffuse and attach to the top step. The addition of atoms to the step-edge pushes it forward into the bunch. Fig. 4.2(e) shows slightly different behavior under net deposition conditions, where now a number of crossing steps [39] can be observed on the wide terraces between bunches. During annealing, the net adatom deposition flux on the terraces causes significant step-flow growth of the lowest step of each bunch, which continuously break away from the bottom of one bunch and cross over to the top edge of the adjacent bunch. This is in contrast to the free sublimation sample in Fig. 4.2(c), where no crossing steps are seen due to the extremely low sublimation rate R sub Å/s. As discussed later, crossing steps should exist under either net sublimation or net deposition conditions [39], but not close to balanced deposition/sublimation conditions. Determining the deposition rate that produce terraces that are free of crossing steps 68

84 provides a way to determine the sublimation rate at a particular temperature, since R net = R dep R sub R dep = R sub under near balanced flux conditions Regime II: Deposition effects Figure 4.3 shows the bunching behavior at 1090 C (well into Regime II) where under free sublimation conditions step bunching is known to occur only for step-up current [4, 1, 47, 58, 59, 60]. Again, Fig. 4.3(a) shows a schematic of the surface near the bottom of the dimple with Figs. 4.3(b) 4.3(g) below corresponding to this geometry, so that the left (right) side of the dimple corresponds to a step-up (step-down) current. Figures 4.3(b) and 4.3(c) show that bunching is only observed for step up current under free sublimation (R sub 0.16 Å/s), consistent with these previous reports. Figures 4.3(d) and 4.3(e) show bunching behavior at 1090 C under near balanced sublimation/deposition conditions (R dep R sub 0.16 Å/s), where again bunching is only observed for step-up current. In this near-balanced case a dramatic reduction in the number of crossing steps on the bunched surface is observed, consistent with the report of Stoyanov et al. [61]. The nature of this reduction will be discussed in more detail below. Figures 4.3(f) and 4.3(g) show bunching behavior at 1090 C under net-deposition conditions, with R dep 0.25 Å/s and hence R net 0.09 Å/s. Bunching is clearly observed only for step-up current, in direct disagreement with the report [4] that net deposition conditions in Regime II cause bunching to occur only for step-down current. This measurement has been repeated more than 5 times under a variety of net deposition/sublimation conditions, and still only step-up current bunching has been observed in Regime II. These observations are inconsistent with the step-permeability model [3] which predicts a reversal of the required current direction for step bunching in 69

85 (a) Step-up current Regime II 1090 C 3 hours Dimpled sample surface during anneal E r Step-down current Free sublimation (-0.16 Å/s) (b) 5 µm (c) 1.25 µm ~Equilibrium (~0 Å/s) (d) 5 µm (e) 1.25 µm Net growth (+0.09 Å/s) (f) 5 µm (g) 1.25 µm Fig AFM images for Si(111) samples DC annealed for 3.0 h at 1090 C. The DC is oriented from right to left in all images as in Fig (a) and (b): annealed under free sublimation conditions (with R net Å/S), (c) and (d): annealed under near-balanced deposition conditions (R net 0 Å/s), and (e)) and (f): annealed under net deposition conditions (R net Å/S). In all cases the step train remains uniform for step-down current (right side), but bunch under step-up current (left side). (b), (d), and (f) are derivative-mode AFM images, which should be viewed as if illuminated from the left. 70

86 Regime II. However, these observations are consistent with the model which assumes sign reversal of the effective charge in Regime II [45], and the recent models which assume diffusion near step-edges becomes very fast in Regime II [9 12]. Both of these models predict that net deposition/sublimation have no effect on the required current direction for bunching. It should be mentioned that we have since discussed our measurements with Stoyanov, who also made measurements of step bunching under near balanced flux conditions [61] and saw step-up bunching. He agrees that his permeable step model [3] cannot be correct. But it still appears that some of his co-workers believe it may be correct [62] Crossing step behavior Although there were no qualitative changes to the bunching behavior, we do observe a marked reduction in the density of crossing steps under near-balanced flux conditions (R dep R sub ). This reduction is clearly seen in Fig. 4.3 where many crossing steps are found on the terraces between the step bunches under free sublimation in Fig. 4.3(b), but under near-balanced deposition conditions, Fig. 4.3(d), virtually no crossing steps are observed. Crossing steps reappear under net deposition conditions, as shown in Fig. 4.3(f). A similar observation of this kind of behavior was made by Stoyanov and coworkers [61]. The behavior of the crossing step density with Si flux can be understood within the simultaneous bunching/debunching instability model of Kandel and Weeks [39] in the following way. The applied electric field causes the adatom concentration to vary across a wide terrace between bunches as shown schematically in Fig. 4.4, assuming here 71

87 (a) Direction of adatom pile up x (b) c(x) Net sublimation (-0.16 Å/s) c eq Stable (c) Stable (d) Unstable Unstable equilibrium flux conditions (~0 Å/s) c(x) c(x) c eq Stable Net growth (+0.09 Å/s) c eq Stable Fig (a) Schematic of a bunched surface, assuming that the applied field causes a higher adatom concentration on the down-hill edge of a wide terrace between bunches. (b)-(d) the corresponding adatom concentration across the terrace (solid line) relative to the equilibrium concentration (dashed line) for (b) net sublimation, (c) balanced, and (d) net deposition conditions. For balanced conditions, the bottom (top) step of the bunch on the left (right) side of the terrace will both be stable against breaking away from the bunch. The surface topographies corresponding to these situations are shown to the right. 72

88 that surface electromigration causes the adatoms to pile-up on the down-hill side of the wide terrace [Fig. 4.4(a)]. Under strong net sublimation conditions [Fig. 4.4(b)], the adatom concentration is below the equilibrium value c eq on both sides of the large terrace. This will result in a net loss of adatoms from the top step in the right bunch, causing it to break away and cross over to the left bunch. In the case of strong net deposition [Fig. 4.4(d)], the adatom concentration is above c eq on both sides of the large terrace, so now the bottom step of the left bunch is unstable against crossing over. However, for a certain range of near-balanced conditions the concentration will be above c eq on the right side and below c eq on the left side, causing stability for the bounding steps on both sides of the wide terrace, as shown in Fig. 4.4(c) In-phase step wandering Before discussing how the physical properties of step bunching depend on local surface miscut, it should be pointed out that so-called in-phase step wandering (IPSW) has been reported in Regime II for a step-down directed heating current after long heating times [63]. IPSW can be thought of as sequence of vicinal surface steps (not step bunches) with in-phase sinusoidal undulations of the same period. This is depicted schematically in Fig. 4.5(a), and the resulting surface topography, as seen using AFM, is shown in Fig. 4.5(b). The image was taken from a sample that was annealed at 1140 C for 3 hours under free sublimation conditions, and is shown here to emphasize what this instability looks like on a macroscopic level. 73

89 (a) E r (b) Step-down E r Fig (a) Diagram showing a vicinal surface with IPSW, which is observed to occur on the Si(111) surfaces heated with a step-down DC in Regime II. (b) Actual topography seen using AFM (sample annealed at T ~ 1140 C under free sublimation conditions). IPSW was clearly observed on two samples out of the eight that we studied in Regime II. The IPSW occurred on those two samples for step-down current at fairly high local miscut ( 0.3 ) under near balanced conditions (R net and Å/s). Figures 4.6(a) and 4.6(b) show the AFM data of regions with IPSW for the net sublimation sample with R net Å/s and the slight net growth sample with R net Å/s, respectively. The white arrows indicate the step-down current direction. Accompanying Fig. 4.6(b) is a line profile along the red, dashed line showing the large scale sinusoidal morphology caused by the IPSW. From the line profile the period/wavelength of the IPSW is seen to be ~8 µm with a trough-to-peak amplitude 74

90 (a) z (nm) (b) y (µm) Fig (a) 35x50 µm 2 AFM image of IPSW annealed at T = 1090 C with R net = Å/s for 3 hours. The white arrow corresponds to the to the step-down current direction, and the red, dashed line corresponds to the profile shown in (b). The yellow circle outlines a place where two regions of IPSW (with two different phases) meet. ranging between nm. The dashed, yellow circle indicates what is thought to be a place where two (separately nucleated) regions of different phase meet [63]. It should be pointed out that very weak IPSW was also observed on the net growth sample with R net 0.09 Å/s (data not shown). The ~3 h annealing time used here is short compared to the ~4 48 h times use in the previous reports of IPSW [63], making it hard to determine whether the net deposition/sublimation conditions affect the formation of IPSW. A number of theoretical treatments of IPSW have been discussed in the literature [9 12, 64, 65] The effects of local surface miscut on Si(111) step bunching As discussed above, and shown in Fig. 3.6 of chapter 3, the dimple geometry used here provides a range of local suface miscuts or initial terrace widths according to Figs. 2.2(a) and 2.2(b). By inspecting different regions within the dimple it is possible to 75

91 extract information about how step bunching depends on miscut in the presence of a Si flux. Figures 4.7 and 4.8 show a plot of the average bunch height as a function of local miscut angle for several samples annealed for 3.0 h at 940 C and 1090 C under various deposition conditions, respectively. Each data point is the calculated average of numerous bunches (~10) at a given miscut, and the error bars represent the standard deviation of the mean. In Fig. 4.7, the data points for the 940 C anneal in Regime I correspond to the following net deposition conditions ( ) R net Å/s and ( ) 0.01 Å/s. Likewise, the 1090 C Regime II data points in Fig. 4.8 correspond to the net deposition conditions: ( ) R net Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, and ( ) Å/s. For reference, the dashed lines are power law fits, with best-fit exponents 0.93 and 0.92 for Figs. 4.7 and 4.8, respectively. From these two data sets, it is clear that the average bunch height increases systematically (approximately linearly) with sample miscut in both Regimes I and II. Interestingly, the bunching rate does not depend strongly on the deposition conditions, consistent with the images shown in Figs. 4.2 and 4.3. The fact that the bunching rate appears to have a nearly identical dependence on miscut in both Regimes I and II may indicate that the mechanism driving the bunching instability may not be all that different between the two Regimes. This idea will be explored further in the next section by examining a full range of bunching behavior with miscut for all three temperature Regimes to determine the rate limiting process during step bunching. 76

92 Height (nm) Heigth (nm) (a) T = 940 C H B ~ θ Step-down current Initial miscut (degrees) (b) T = 1090 C H B ~ θ Step-down current Initial miscut θ 0 (degrees) Fig Plot of the average bunch height after a 3.0 anneal as a function of sample miscut for Regimes I and II. (a) 940 C anneal for net conditions ( )R net Å/s and ( ) 0.01 Å/s. (b) 1090 C anneal for net conditions: ( ) R net Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, ( ) Å/s, and ( ) Å/s. For reference, the solid lines are power law fits, with best-fit exponents 0.92 and 0.93 for (a) and (b) respectively. There exists a clear near-linear increase in bunch height with initial surface miscut angle, but with little dependence on net deposition/sublimation conditions. 77

93 4.2 The rate limiting mechanism on Si(111) during DC heating Introduction As discussed in Chapter 2 step motion results from the terrace diffusion of surface adatoms towards (or away from) steps, and the attachment/detachment (attachment for short) of adatoms to steps. The relative rate of terrace diffusion and step attachment is a critical factor in a number of surface phenomena; including, island nucleation, island growth/decay, step-edge fluctuations [66], and denuded zone formation [22]. However, it is difficult to directly measure adatom diffusion and attachment/detachment at elevated temperatures because atomic motion is too fast. But, the phenomenon of step bunching on vicinal Si(111) surfaces can be used to investigate the relative rates of diffusion and attachment at high temperatures (940 C 1290 C). Here I will expand upon how the rate of step bunching (actually the number of steps in a bunch after a given time) and the minimum terrace width within a step bunch depend on the initial terrace width of the vicinal surface. These measured dependencies are compared with scaling predictions from analytic solutions [6 8] and numerical simulations of the conventional sharp-step model for electromigration proposed by Stoyanov [5] to determine whether adatom transport onto the steps is diffusion limited or attachment limited at a particular temperature and initial miscut angle. The experimental data and numerical modeling indicate that surface mass transport is diffusion limited for a wide range of temperatures (940 C 1290 C) and initial terrace widths (20 nm 150 nm). More specifically, these measurements indicate that the kinetic length parameter d < ~20 nm, where d D t /κ, D t is the terrace diffusion constant, and κ is the step attachment rate [67] when compared to analytic predictions 78

94 and numerical simulations using the 1D BCF based sharp-step model discussed in chapter 2. As will be discussed, this relatively small value of d indicates that the energy barrier for attachment to a step is comparable in magnitude to the energy barrier for diffusion on a terrace in this 1D sharp-step model Experimental procedures The experimental procedures used here are exactly the same as those discussed above except that a 62 mm diameter polishing wheel was also used to produce dimples with wider range of initial terrace widths (20 nm 150 nm). The dependence of the number of steps in a bunch N and the minimum terrace spacing within a bunch l min on the initial starting terrace width l 0 can be determined by analyzing AFM images measured over different areas of a dimple. N is easily determined using the relation N = H B /h, where H B is the average height of a set of step bunches from a sample region measured using AFM that had a particular initial terrace width l 0, and h = nm is the atomic step height on the Si(111) surface. l min was determined using l min = h/m max, where m max is the average maximum slope of this set of bunches, relative to the Si(111) crystal surface. Typically ~7 10 step bunches from a single AFM image were analyzed and averaged together for each value of l 0. Finally, l 0 for a particular AFM image ~10 35 µm-wide was determined using l 0 = h/m 0, where m 0 is the average slope over the entire image Experimental observations The data taken at 940 C (Regime I shown above) will be reanalyzed and new data under free sublimation conditions for 3 hours at 1090 C (Regime II step-up current bunching) and for 3 minutes at 1290 C (Regime III 1290 C) will be presented. The reason for taking new data for Regime II is that the original data shown in Fig. 4.7(b) 79

95 was taken from 5 different samples, and as result has significant scatter. But it should be pointed out that the new data is consistent with the data in Fig. 4.7(b), but does have less scatter, as intended. Typical AFM images corresponding to these studies are exactly those shown in Fig. 2.6 of chapter 2 for the three temperature Regimes. Figures show the dependence of N and l min on l 0 for each of these three temperature Regimes. Each data set was fit to a power law to aid in the characterization of the scaling of N and l min with l 0, and for direct comparison to the numerical simulation results below. Figure 4.8 shows the step bunching data in Regime I (with step-down current) obtained from two different samples after a 3 hour DC anneal at 940 C under both net sublimation ( ) and net growth ( ) conditions. It is clearly seen that N decreases with l 0 roughly as N l , and l min increases roughly as l min l Figure 4.9 shows step bunching data in Regime II (with step-up current) from a sample after a 3 hour DC anneal at 1090 C under free sublimation conditions. A similar scaling dependence of N and l min with l 0 is observed to that found in Regime I, but with a larger step bunching rate (~2.5 times faster than at 940 C). Figure 4.10 shows step bunching data for a sample in Regime III (with step-down current) after a much shorter 3 minute DC anneal at 1290 C under free sublimation conditions. The scaling of N and l min with l 0 is consistent with that found in Regimes I and II, but with a much larger rate of bunching (~60 times faster than at 1090 C, and ~150 times faster than at 940 C). One thing to note is that the dependence of l min on l 0 appears to become slightly stronger with increasing temperature. However, there is a fair amount of scatter in the data [particularly for 940 o C - Fig. 4.8(b)] making it hard conclude that this apparent temperature dependence is real. 80

96 N lmin (nm) (a) (b) T=940 C step-down current N ~ l l min ~ l T=940 C step-down current l o (nm) Fig (a) Plot of same data shown in Fig. 4.7 above (shown in a different form). (b) Dependence of l min on the initial surface terrace l 0 at 940 C (Regime I). 81

97 N lmin (nm) (a) T=1090 C step-up current N ~ l (b) l min ~ l T=1090 C step-up current l o (nm) Fig Plot of data showing how (a) the number of steps N and the (b) minimum terrace width l min in a bunch depend on the initial surface miscut l 0 at 1090 C (Regime II). 82

98 N lmin (nm) (a) T=1290 C step-down current N ~ l (b) l min ~ l T=1290 C step-down current l o (nm) Fig Plot showing how (a) the number of steps N and the (b) minimum terrace width l min in a bunch depend on the initial surface miscut l 0 at 1290 C (Regime III). 83

99 4.2.4 Comparison to theory In order to determine the rate limiting surface process (terrace diffusion or stepedge attachment) I will compare the data in Fig to analytic scaling predictions [6 8] from the original electromigration-induced step bunching model proposed by Stoyanov [5] for step-down adatom flow under extreme diffusion limited (d << l 0 ) and extreme attachment limited (d >> l 0 ) conditions [44]. This sharp-step model is a modified form of the Burton, Cabrera, and Frank theory of crystal growth [13], and as discussed in chapter 2 includes effects of an electromigration force acting on surface adatoms, and an attachment barrier at step edges. To explore the conditions between the extremes of diffusion limited and attachment/detachment limited kinetics the data here will also be compared to 1D numerical simulations of this model. Formalism: review of the sharp-step model The top portion of Fig. 2.3(b) shows the geometry and notation used in this model, while the lower portion of Fig. 2.3(b) shows the corresponding energy-barrier landscape for adatom diffusion and step-edge attachment. It should be pointed out that the attachment barrier E a is symmetric in this model, and hence does not include an asymmetric Ehrlich-Schwoebel barrier [68, 69]. The adatom concentration c n (x) on the n th terrace and the velocity v n of the n th step can be determined by the following set of equations [5]: D t 2 d cn ( x) FDt 2 dx k T B dcn ( x) cn ( x) + R= 0 dx τ (4.1) dcn x D ( ) t dx x= x n F n cn( x= xn ) = κ ( cn( x= xn ) ceq) (4.2a) kbt 84

100 dcn( x) Dt dx x= x n+ 1 F k B cn( x= x T n+ 1 = κ ( c ( x= x c ) eq (4.2b) n+ 1 ) n n+ 1) v n n {( c x= x ) c ) + ( c ( x= x c n )} dxn = a 2 n 1 n eq n n ) eq dt κ (. (4.3) Here D t is the terrace diffusion constant, F = q E is the electromigration force due to eff the applied field E and effective adatom charge q eff, τ is the lifetime for an adatom to desorb from the surface, R is the adatom deposition rate from an external source, k B is the Boltzmann constant, T is temperature, κ is the step attachment/detachment rate, and a = nm is the surface lattice constant. Also, as discussed in chapter 2, step-step n repulsions are included in the usual way [35] by letting c = c ( 1+ / k T ), where 3 3 ( 1/ l 1/ l ) 2 3 n = 2ga h n 1 n eq eq µ n µ is the step-edge chemical potential [37], c eq is the equilibrium B adatom concentration, g is a step interaction parameter, h is the atomic step height, and l n (l n-1 ) is the width of the n th (n th -1) terrace as shown in Fig. 2.3(b). The boundary conditions [Eqs. (4.2a) and (4.2b)] ensure mass conservation at the step edges. To characterize the relative rate of terrace diffusion to step attachment/detachment the kinetic length scale d = / κ will be used, which is easily identified in the boundary D t conditions. The nature of the step bunching instability for step-down adatom flow (with F < 0 according to the geometry in Fig. 2.3(b)) and terrace stability for step-up adatom flow (F > 0) was discussed in terms of adatom pile-ups in chapter 2. Briefly, it is the adatom pile-up and dearth on the upper and lower side of a step, respectively, that leads to the step bunching instability in the original sharp-step model of Stoyanov [5]. 85

101 Comparison: analytic scaling predictions of the sharp-step model Recently, Sato and Uwaha [7] showed that for diffusion limited kinetics (d << l 0 neglecting evaporation or deposition) the number of steps N and the minimum terrace width l min within a bunch should be related by: 2 3 1/ 3 2 / 3 8a gh l minn =, (for d << l 0 ) (4.4) F d while in the opposite attachment limit (d >> l 0 ) Krug et al. [8] arrived at a similar relation: 1/ / 3 16a gh l minn =. (for d >> l 0 ) (4.5) F l0 It should be pointed out that Eq. (4.4) was also independently derived by Stoyanov et al. [6] in the special case of d = a, and Eq. (4.5) is implicitly derived in the work of Liu et al. [44]. Examining Eqs. (4.4) and (4.5) one immediately sees that the key difference between the two is that the quantity (l min N 2/3 ) scales as l -1/3 0 in the attachment limited case, while it is independent of l 0 in the diffusion limited case. Therefore, the rate limiting kinetics within such a model can be determined by simply making plots of (l min N 2/3 ) vs. l 0 using the data in Figs , and comparing them with these two predictions. Before presenting the data an estimate of what should be found can be made. Using the best fit power laws as a guide one finds the product (l min N 2/3 ) l , l , and l for Figs. 4.8 (Regime I), 4.9 (Regime II), and 4.10 (Regime III), respectively. This indicates that the kinetics should be mostly diffusion limited in temperature Regimes I and III where step bunching occurs for a step-down current. 86

102 lminn 2/3 (nm) lminn 2/3 (nm) lminn 2/3 (nm) (a) T=940 C step-down current (b) T=1090 C step-up current (c) l (nm) T=1290 C step-down current l o (nm) Fig. 11: (a) (c): Measured dependence of the quantity l min N 2/3 on l 0 using the data in Figs. 2 4, respectively. Horizontal and l 0-1/3 power law dashed lines are shown for reference. 87

103 Note, Regime I appears to have a slight inverse dependence on l 0, which will be discussed below. Figures 4.11(a), 4.11(b), and 4.11(c) show the experimentally measured scaling of l min N 2/3 with l 0 for the data shown in Figs. 4.8, 4.9, and 4.10, respectively. A power law ~l -1/3 0 and a horizontal line are included on each plot for comparison. It is very clear that the product l min N 2/3-1/3 shows absolutely no sign of an l 0 scaling for temperature Regimes II and III (Figs. 4.11(b) and 4.11(c), respectively), down to the lowest measured values of l 0 20 nm. This indicates that diffusion limited kinetics hold down to at least l 0 20 nm, and hence the kinetic length d < 20 nm. As noted, the quantity l min N 2/3 for Regime I [Fig. 4.11(a)] appears to have a slight inverse dependence on l 0. This slight inverse dependence may indicate a weak diffusion limit in Regime I. Furthermore, it should be mentioned that there may be a slight difference between the cases of net sublimation ( ) and net growth ( ). This apparent difference can be seen in Fig. 4.11(a), although it is not so apparent in Figs. 4.8(a) and 4.8(b), where the data is displayed in a different way. Since the data for sublimation and growth came from different samples annealed at different times, it is possible that the apparent dependence could be due to small differences in the experimental conditions, such as sample temperature or impurities from the Si deposition source. Overall, we still conclude that the data for all three temperature Regimes are much better described by the diffusion limited prediction in Eq. (4), with l min N 2/3 roughly independent of l 0. Finally, since the above sharp-step model only produces step bunching only for a step-down adatom flow, and since it is currently believed that adatom flow on Si(111) is in the same direction as the direct electric current [46], Eqs. (4.4) and (4.5) can only be directly related to the measured data in Regimes I and III, where step bunching occurs for 88

104 step-down current. It is nevertheless interesting to note that the observed dependences of N and l min on l 0 appear to be similar in Regime II, where bunching occurs for step-up flow. This suggests that the fundamental mechanism for step bunching may be similar at all temperatures. I will return to this point in section 4.3 below. Comparison: numerical simulation using the sharp-step model To help gauge how diffusion limited the system is, and to clear up the possible discrepancy in Regime I, numerical simulations were performed of a real-sized system with realistic parameters using the sharp-step model outlined by Eqs. (4.1) (4.3). These numerical simulations also make it possible to directly model how N and l min each independently depend on l 0, allowing for a direct comparison to the experimental data in Figs. 4.8 and One important issue with this model is that it has a large number of free parameters, not all of which are well known. In fact, previous studies using this model have shown that, for a particular value of l 0, different combinations of parameters with a wide range of values for d = D t / κ could produce experimentally-observed stepbunching and de-bunching behavior [57, 44]. Therefore, it was not possible to use that measured data to conclusively determine the relevance of terrace diffusion to step-edge attachment by evaluating d. However, we have found that if a range of l 0 values are considered, then only certain values of d are consistent with the experimentally observed bunching behavior shown in Figs. 4.8 and Obviously this 1D model cannot explain two dimensional (2D) step structures such as step wandering [63] (section above) or step bending [70], but should be adequate to describe the essentially straight and parallel step bunches seen on Si(111) under step-down DC heating. The program code used for these simulations can be found in Appendix A. It should be mentioned that 89

105 near-balanced deposition conditions (R ~ c eq /τ) were assumed, but test simulation indicate that the deposition has little effect on the bunching behavior. The following procedure was used to conduct the numerical simulations. First, a core set of fundamental parameter values was estimated from the literature for T 940 o C and T 1290 o C, as shown in the top part of Table 4.1. We also used the experimental estimates of the applied electric field of E -7x10-7 V/nm for T 940 o C and E -5x10-7 V/nm for T 1290 o C, leaving three adjustable parameters: κ, g, and q eff. Then a test value for d (which sets κ via the relationκ = D t / d ) was chosen, and simulations were run using that value for a particular initial terrace width of l 0 = 130 nm, while making adjustments to the effective charge q eff and the step repulsion parameter g until rough agreement for both N and l min was found between the simulations and experiments. This was straightforward to do because N is strongly dependent on q eff, but weakly dependent on g, while l min is weakly dependent on q eff and strongly dependent on g. Once rough agreement was found for l 0 = 130 nm, the parameters were held fixed and simulations were run for various values of l 0. This process was then repeated for different values of d. A quick note about the initial surface profiles should be made. Recall from chapter 2 section that to have a step bunching instability there must be a slight perturbation to the uniform step train, i.e., the step-edges cannot be equally spaced. This always occurs in experiments due to sample (wafer) preparation and/or thermal fluctuations of the step-edges, as reflected in Fig Thus, for the simulations we must add a slight perturbation to the uniformly spaced step train. This was done by measuring the mean terrace width l 0 and the standard deviation σ from AFM images of a surface 90

106 prior to DC annealing, such as that shown Fig. 3.4, over different regions of the dimple. Then the initial surface profile was simply generated assuming a normal distribution with measured mean l 0 and standard deviation σ. This was done for miscut angles of ~0.08 (l 0 ~ nm, σ ~ nm), ~0.14 (l 0 ~ 129 nm, σ ~ nm), ~0.25 (l 0 ~ nm, σ ~ nm), and ~0.3 (l 0 ~ nm, σ ~ nm), while the values for other miscut regions were determined using l 0 = h/tan θ and choosing a reasonable value for σ. E des (ev) [2, 71] τ 0 (s) E d (ev) [2, 71] D 0 (nm 2 /s) a 2 c eq (BL) [27] X X d (nm) q eff ( e ) g (ev/nm 2 ) E (ev) Table 4.1: Top: Fixed activation energy parameters from the literature. The prefactor D 0 was found using E d = 1.1 ev [2, 71] with the relation λ = Dτ, and taking the diffusion length at 1090 C to be ~25 µm. Bottom: Adjustable parameters used in the sharp step model for Regimes I and III shown in Figs. 7 and 8, respectively. Values for q eff, g, and E are listed in the format (940 C) (1290 C). The last column gives the corresponding values of E=E a E d (see Fig. 2.3(b)). Figure 4.12 shows simulation results from the sharp step model for T = 940 C (Regime I) with a step-down adatom flow, corresponding to F < 0 for the geometry shown in Fig. 2.3(b). Table 4.1 shows the adjustable parameter values of for the three values of d (500 nm, 50 nm, and 2 nm) shown here. It is clear from the figure that both N and l min show a marked dependence on the initial terrace width l 0, which depends strongly on the value of d. For example, for d = 2 nm N is found to decrease strongly with l 0 roughly as l 0-1, while for d ~ 500 nm this scaling dependence becomes much weaker 91

107 N (a) N~l (d=2 nm) N~l (d=50 nm) N~l (d=500 nm) lmin (nm) lminn 2/3 (nm) (b) l o (nm) l min ~l (d=500 nm) l min ~l (d=50 nm) l min ~l (d=2 nm) (c) l o (nm) l min N 2/3 ~l (d=500 nm) l min N 2/3 ~l (d=50 nm) l min N 2/3 ~l (d=2 nm) l o (nm) Fig 4.12: Sharp-step model numerical simulations for a 3-hour step-down 940 C anneal (Regime I) for different values of d, showing the dependence on l 0 of (a) number of steps in a bunch N and (b) minimum terrace width in a bunch l min. Power law fits are included for reference and comparison to Fig

108 (e.g., N ~ l ), as seen in Fig. 4.12(a). Figure 4.12(b) shows that the predicted dependence of the minimum terrace width in a bunch l min depends even more strongly on l 0, and actually switches from a decreasing dependence for d > ~100 nm to an increasing dependence for d < ~100 nm, in sharp disagreement with the experimental data shown in Fig Comparing the simulation results (Fig. 4.12) to the experiments (Fig. 4.8), it is clearly seen that the best agreement between the two occurs for small values of d < ~20 nm. In Fig. 4.12(c) we show how the simulated value of l min N 2/3 depends on l 0 for different values of d at 940 C. Here we see that even for d as small as 2 nm we observe a weak decrease of l min N 2/3 with increasing l 0, consistent with the data shown in Fig. 4.12(a). All of the above comparisons of simulated and measured behavior indicate diffusion limited kinetics at 940 C for all miscuts studied here. Similarly, Fig shows the simulation results for T = 1290 C (Regime III) with a step-down adatom flow, and Table 4.1 shows the parameter values of for each value of d shown here at this temperature. A Similar dependence of both N and l min on l 0 and d is observed as that in Regime I, again with a change in the scaling dependence of l min on l 0 for d > ~100 nm. As in Regime I, the best match between experiment (Fig. 4.10) and simulations (Fig. 4.14) is for values of d < ~20 nm. It should be noted that an unphysically large value for q eff is required in Regime III, because experimentally step bunching is was so fast. We have found that if a diffusion barrier smaller than 1.1 ev is assumed, then the required value of q eff becomes smaller. There are indications that some transition in the surface occurs around ~ C, which has been postulated to be incomplete surface melting [61, 83, 85]. I will come back to this point later in the chapter. 93

109 N lmin (nm) (a) N~l (d=2 nm) N~l (d=50 nm) N~l (d=500 nm) l (nm) (b) l min ~l (d=2 nm) l min ~l l 0 min ~l (d=50 nm) (d=500 nm) l o (nm) Fig 4.13: Sharp-step model numerical simulations for a 3-minute step-down 1290 C anneal (Regime III) for different values of d, showing the dependence on l 0 of (a) number of steps in a bunch N and (b) minimum terrace width in a bunch l min. Power law fits are included for reference and comparison to Fig Before moving on to a discussion of these findings it should be pointed out that simulations (not shown here) for T = 1090 C (in Regime II) were also conducted assuming a step-up current and q eff < 0. Given that this still corresponds to a step-down adatom force/flow, the same scaling dependencies should be expected. Indeed, comparison of these simulations with Regime II data leads to the same conclusion as in Regimes I and III, namely that d 20 nm in order to reproduce the experimentally 94

110 observed scaling behavior. However, no attempt is made to draw a conclusion from these Regime II simulations since experiments [46] indicate that q eff > 0 for all temperatures on Si(111). We will return to the Regime II data in section Discussion of findings Previous experimental observations of step bunch decay (debunching) in temperature Regime I have suggested that the high-temperature kinetics on Si(111) were attachment limited with d > ~500 nm [57]. But, as stated above, these experiments were performed at a particular miscut (fixed l 0 ), and could be adequately described by a range of values for d. However, by varying l 0, the measurements discussed here clearly indicate that the kinetics are in fact diffusion limited with d < ~20 nm. With this established, I will next discuss a simple, but very appealing, interpretation of this finding. Assuming that adatom diffusion can be described by a simple hopping process, where an adatom must hop with an attempt frequency ν t0 over diffusion barrier of height E d between two adjacent terrace sites separated by a distance a, we may write the thermally activated diffusion constant as D t a 2 ν t0 exp(-e d /k B T). Similarly, if hopping toward and attaching to a step-edge is characterized in a same way with an attempt frequency ν a0, barrier E a, and hop length a, the step attachment rate can be written as κ aν a0 exp(-e a /k B T). If ν t0 ν a0 is assumed for simplicity, then the kinetic length can be written as d = D / κ a exp( E / k T ), where E = E a E d as shown in Fig. 2.3(b), which t B can also be expressed as E E k T ln( d / a). With these assumptions, the a d + B conclusion that d is relatively small implies that E is correspondingly small. For example, d 20 nm implies E 0.4 ev while d 2 nm gives E 0.2 ev. Since it has 95

111 been reported that E d 1.1 ev [2, 71], it is possible to conclude that the attachment barrier E a is of similar magnitude as the terrace diffusion barrier E d, at least in Regimes I and III. The value of E for each value of d simulated here for both 940 and 1290 C is shown in Table 4.1. Note that here d has been interpreted in terms of possible activation energy differences between terrace diffusion hopping and step attachment hopping, by making the ad-hoc assumption ν t0 ν a0, and also assuming the steps were impermeable. It is, of course, possible that ν t0 ν a0 and that steps could be partially permeable. In this case, the simple relation d a exp( E / k T ) may not be accurate. However the observed B scaling dependence on l 0 of N and l min still indicate diffusion-limited kinetics down to at least l 0 20 nm. With regard to the step-up-bunching observed in Regime II, several authors have recently pointed out that if the attachment/detachment hopping rate in Regime II somehow became faster than the terrace hopping rate, then a step bunching instability should occur for step-up adatom flow [9 12]. Although the physical mechanism for such a change in the relative hopping rates has not been so far identified, this general idea nevertheless provides an appealing scenario of how the required direction for stepbunching could reverse at different temperatures. However, if the attachment hopping rate were much larger than the diffusion hopping rate in Regimes I (as previously suggested [9]) and III, drastic changes in terrace and/or step processes would be required to have attachment become faster than diffusion in Regime II. This seems unlikely. On the other hand, if the relative attachment and diffusion hopping rates were of similar magnitude in Regimes I and III, then only modest 96

112 temperature-dependent changes in terrace or step-edge processes would be required to cause the reversals between step-down and step-up bunching. Hence, the finding that the attachment and diffusion energy barriers have comparable magnitudes ads plausibility to fast-attachment models [9 12] of step-up bunching in Regime II. The idea of fastattachment will be addressed in the next section, where the boundary conditions used in the original sharp-step model [5] will be reexamined. 4.3 Fast vs. slow attachment and the bunching transitions on Si(111) Introduction As shown so far, it is generally agreed that step bunching results from a directional adatom drift on the terraces between steps, resulting from a surface electromigration adatom force F = q eff E, where E is the applied DC electric field, and q eff is the adatom effective charge [5]. As was shown in chapter 2, if an attachment barrier exists at step edges, then a step-down (step-up) adatom force will cause adatoms to pileup on the upper (lower) edge of each step. By using continuum equations to describe surface adatom flow in such a sharp-step model, it is then straightforward to show that adatom pile-ups at upper or lower step edges result in step bunching or enhanced terrace stability, respectively [5, 44]. Since q eff appears to be positive on Si(111) [46], the stepdown bunching in Regimes I and III is readily understood, but the step-up bunching in Regime II is not. How to explain the observed step bunching transitions remains a puzzle. Recently, several authors [9 12] have pointed out that if in Regime II the stepedge attachment hopping rate ν a somehow becomes faster than the terrace hopping rate ν t, then the adatom pile-ups would be reversed, and step bunching should now occur only for a step-up adatom force. These authors also argued that the continuum equations in 97

113 this case should maintain the same form as originally proposed by Stoyanov [5] in Eqs. (4.1) and (4.2), but that the value of the equilibrium adatom concentration c eq and/or the step-edge attachment rate κ should be changed. In this section I will address two issues. First, I will address the form and physical meaning of the continuum boundary conditions in the presence of an applied adatom force. It is proposed that an adatom drift-attachment/detachment term should be added to the boundary conditions, which is directly analogous to the drift-diffusion term that Stoyanov added to the terrace diffusion equation [5]. These proposed boundary conditions are shown to be mathematically equivalent to those proposed by Suga et al. [9], and in an important special case produce numerically equivalent results as those proposed by Zhao and co-workers [10 12]. However, we believe that the boundary conditions derived here give a better physical picture of step-edge attachment/detachment in the presence of an applied adatom force, and maintain the definitions and physical meanings of both c eq and κ to be identical to those in the original continuum model [5]. Secondly, these continuum equations are used to simulate step-up bunching in Regime II (assuming fast step-edge attachment) and are directly compared to the measurements presented earlier in this chapter. For completeness, simulations of step-down bunching in Regime I and III were also carried out using the new boundary conditions and are compared to the experiments and simulations presented in section 4.2 above. Using physically reasonable terrace and step parameter values, these simulations can reproduce the measured step bunching behavior in all three temperature Regimes, provided we assume that there are modest temperature-dependent changes (~ ev) in the activation energy barrier for attachment and/or terrace diffusion. 98

114 4.3.2 Drift-attachment continuum equations In this section we derive a set of continuum equations describing the 1D step bunching process on Si(111). We start by considering the microscopic adatom-hopping potential energy landscape depicted in Figs. 4.14(b) and 4.14(c), and restrict our discussion to a 1D hopping model with impermeable steps, where the attachment energy barriers are symmetric with respect to each step site when F = 0. As usually done, we assume that diffusing adatoms must hop between any two terrace sites separated by the surface lattice constant a, with a certain terrace attempt rate ν t0 over diffusion energy barriers, each with peak energy E D located midway between the terrace sites. We assume that all terrace sites have the same attempt rate ν t0 and a zero rest energy, as shown in Figs. 4.14(b) and 4.14(c). Adatoms must hop over an attachment/detachment barrier with peak energy E A to reach a site on a step edge, which is also separated by a distance a from the adjacent terrace site. Adatom formation (i.e., an adatom breaking away from a step edge) is characterized by an attempt rate ν F0 and rest energy E F0 µ n, where µ n is the step-edge chemical potential due to step-step interactions [35] discussed in chapter 2. For convenience, the formation energy of step n is defined as E F,n E F0 µ n. The adatom concentration C i is the fractional adatom occupancy at terrace site i, and C step = 1 is taken as the conventional assumption for impermeable steps meaning that there is always an adatom available for detachment and a step site available for attachment. To include the effects of electromigration, the applied electric field is assumed to raise and lower the energy barriers to the left and right side of a terrace site by an amount +Fa/2k B T and Fa/2k B T), respectively, as illustrated in Fig. 4.14(c). 99

115 (a) (b) l n x n +x x n+1 a a E D E A (c) E=0 E F0 -µ n E r x n,l i-1 i x i+1 n,s x n,s Fig (a) Schematic diagram showing the step orientation and labeling used in deriving the continuum equations. The energy barrier landscape seen by a diffusing adatom is shown in (b) with no applied electric field and (c) with an applied electric field. 100

116 101 The key additional assumption (also made explicitly by Suga et al. [9] and implicitly by Zhao et al. [12]) is an analogous change to the attachment/detachment barrier to the left and right of each step, by an amount +Fa/2k B T and Fa/2k B T, respectively. F > 0 is assumed, and we first consider adatom hopping with between a step (located at x n,s in Fig. 4.15) and the adjacent terrace site x n,l at the far left side of terrace n. The hopping rate from the step to site x n,l is } 2) / / ( exp{, 0 T k Fa E E J B A n F F + = ν, and the hopping rate from the adjacent terrace site x n,l back to the step site is given by } 2) / / ( exp{ 0, T k Fa E C J B A t L n + = ν. Expanding to linear order in F, the net adatom hopping rate across the attachment barrier just to the left of the site at x n,l is: ( ) ( ) + + = T k Fa v C T k E v v C T k E v T k E J B t L n B n F F t L n B n F F B A L n 2 ) / exp( ) / exp( ) / exp( 0,, 0 0,, 0,. (4.6a) With the definition ) / )exp( / (, 0 0, T k E C B n F t F eq n ν ν, Eq. (4.6a) can be written as: ( ) ( ) ( ) ( ) + + = + + = T k Fa C C a C C a T k Fa C C C C T k E v J B L n eq n L n eq n B L n eq n L n eq n B A t L n 2 2 ) / exp(,,,,,,,, 0, κ κ, (4.6b) where the step attachment rate is defined as ) / exp( 0 T k E v a B A t κ. Considering the site i in Fig. 4.14(c), and using similar reasoning the net flux across any barrier between terrace sites can be shown to be: ( ) ( ) + + = + + T k Fa C C C C T k E v J B i i i i B D t i 2 ) / exp( 1 1 0, (4.7a)

117 102 Changing to the continuum adatom density c(x i ) = C i / a 2 and adatom surface flux j(x i )= J i /a, Eq. (4.7a) can be written: [ ] [ ] ) ( ) ( ) / exp( 2 ) ( ) ( ) ( ) ( ) / exp( ) ( i B i B B D t B i i i i B D t i x c T k F D i x dx dc D x c T k F i x dx dc T k E v a T k Fa x c x c x c x c T k E av x j + = , (4.7b) where the diffusion constant is defined to be ) / exp( 0 2 T k E v a D B D t. From Eq. (4.7b), it is straightforward to derive (in the usual way) the terrace drift-diffusion equation, by considering a small strip of width x centered at a position x on a terrace, and requiring that in steady state: R x c x dx dc T k DF x dx c d D R x c x x x j x x j dt x dc B = = τ τ ) ( ) ( 2)) / ( 2) / ( ( 0 ) ( 2 2. This is the same drift-diffusion equation proposed by Stoyanov [25] shown in Eq. (4.1) above. Using the same change to continuum variables, the flow across the terrace boundary in Eq. (4.6b) can be written: ( ) ( ) + + = T k Fa x c c x c c x j B L n n eq L n n eq L n 2 ) ( ) ( ) (,,, κ κ. (4.8) By equating the adatom flux over an attachment barrier with the flux over the adjacent terrace barrier [given by Eq. (4.7b)], we arrive at our proposed boundary conditions: ( ) T k Fa x c c x c c x c T k DF x dx dc D B L n n eq L n n eq L n B L n 2 ) ( )] ( [ ) (,,,, κ κ, (4.9a)

118 dc D dx x n, R DF + k T B c( x n, R ) ( ) + Fa n+ 1 c( xn, R ) ceq 2kBT n+ 1 κ [ c( xn, R ) ceq ] + κ, (4.9b) Where Eq. (4.9b) is evaluated at the right terrace boundary, and is derived in the same way as the left terrace boundary. These are equivalent to the conventional boundary conditions of Eqs. (4.2a) and (4.2b), except for the addition of a driftattachment/detachment term (or drift-attachment for short) on the far right of Eqs. (6a) and (6b). This term represents directional adatom flow to or from a step, which is directly produced by the applied adatom force F. It has the same physical origin as the drift-diffusion term for terrace sites, namely small, asymmetric changes to the energy barriers surrounding a particular site. Before conducting numerical simulations using these newly formulated boundary conditions it is worth discussing the drift-attachment term in more detail. Consequences of the drift-attachment term For physically reasonable parameter values, we expect that the concentrations c ( x,r ) n, c ( x n,l ), and n c eq should deviate only by a very small fractional amount (typically less than a part in 10 4 ) from the equilibrium concentration c eq of an isolated n step. Since ( Fa / kb T ) << 1, the product [ ceq+ c( xn, L )]( Fa / 2kBT ) can with negligible error also be written as κ [ c( x, )]( Fa / k T ) or κ [ c( x, )]( Fa / k T ). n L B It is insightful to write these new terms using the kinetic length d D/κ, in which case Eqs. (4.9a) and (4.9b) become: dc a DF n D + 1 c( xn, L ) κ [ ceq c( xn, L )], (4.10a) dx x d kbt n, L n R B

119 dc a DF n D c( xn, R ) κ [ c( xn, R ) ceq ], (4.10b) dx x d kbt n, R where in the last step we have also used the approximations mentioned above. These appear similar to the conventional sharp-step boundary conditions of Eq. (4.2) except that now there is an effective drift-diffusion term on the left side, which can be positive, zero, or negative depending on the ratio (a/d). If a/d << 1, the boundary conditions in Eq. (4.10) approach the conventional form in Eq. (4.2) above. Physically, the driftattachment adatom flow in this slow attachment limit is much less than the driftdiffusion flow, and hence can be safely neglected. According to step geometry in Fig. 4.14(a), an applied force directed to the right produces the conventional adatom pile-up situation [5, 44], with c x ) < c( x ) for a step-up force (resulting in terrace ( n, L n, R stability) and c x ) > c( x ) for a step-down force (resulting in step bunching). ( n, L n, R However, the drift-attachment term cannot be neglected when d is comparable to or less than a. If a/d = 1, then the drift-diffusion and drift-attachment flows are equal, and adatom drift disappears from the boundary conditions altogether. In this case there is a driven directional adatom flow across the surface, but with no adatom pile-ups at step edges, and hence no step bunching instability. When d < a, ( fast attachment ) the drift-attachment/detachment flow is larger than the drift-diffusion flow, causing the effective drift-diffusion term to become negative, hence reversing the adatom pile-ups. As noted previously [9 12], such a reversal results in step bunching for step-up flow, and terrace stability for step-down flow. 104

120 Comparison to models of Suga et al. and Zhao et al. It is worth comparing the formulation of the model here with that of other authors. Suga et al. [9] started with an essentially identical hopping model, and proposed that the resulting continuum boundary conditions should maintain the same form as originally proposed in Eq. (4.2), but with the parameters κ and c eq re-defined to be asymmetric functions of the applied adatom force: L / R 0 κ = κ exp( mfa / 2k ) and c = c exp( Fa / k T ), (4.11) L / R BT eq eq ± where L (R) means the left (right) side of the terrace. Substituting these re-defined parameters into the conventional boundary conditions in Eq. (4.2) and expanding to first order in Fa/k B T, one directly recovers Eqs. (4.9a) and (4.9b). Hence the underlying mathematics and physics of the boundary conditions proposed here are the same as those of Suga et al. [9]. Zhao et al. [12] also started with a similar (but more generalized) hopping model, and then mapped the problem onto the conventional boundary conditions [Eq. (4.2)] with a re-defined attachment parameter that is positive for slowattachment conditions, negative for fast-attachment conditions, and infinite when the attachment and diffusion hopping rates are equal. The boundary conditions here are not mathematically equivalent to those of Zhao et al. [12], as can be seen by considering the case F = 0. However, we have compared solutions to the drift-diffusion equation using Zhao s boundary conditions to solutions with the drift-attachment term [Eq. (4.9)], and find that for many conditions the solutions are extremely close to each other. Although the boundary conditions here are either identical or produce numerically very similar solutions, it is believe that the formulation here is useful because (1) it more directly reveals the effect of F on adatom flow at steps, (2) it treats terrace drift and step-edge 105 B

121 drift in an analogous way, and (3) it leaves the definitions and physical meanings of both κ and c eq the same as in the conventional boundary conditions Numerical simulations In this section I will present a comparison of numerical simulations of step bunching behavior on a vicinal surface using Eqs. (4.1) with the new boundary conditions given in Eq. (4.9), to the experimental observations (discussed in section 4.2 above) on Si(111) of how the average number of steps N and minimum terrace width l min in a bunch depends on the initial average terrace width l 0. From this comparison we hope to (1) determine whether a 1D model with fast attachment conditions can reasonably account for the observed step-up bunching behavior on Si(111) in temperature Regime II, and (2) if so, limit the range of possible parameter values. A similar comparison for stepdown bunching in temperature Regimes I and III was presented and discussed in section 4.2 above. In those simulations the conventional boundary conditions of Eq. (4.2) were used, and only slow attachment conditions were considered (with 10-3 < a/d < 0.2). To check for consistency we have run new simulations with parameters identical to those shown in Table 4.1 using the new boundary conditions in Eq. (4.9). We find similar bunching behavior in Regimes I and III as that shown in Figs and 4.13, and it is conclude that a/d was sufficiently small in those simulations that drift-attachment term in boundary conditions of Eq. (4.9) was negligible, i.e., the boundary conditions are nearly the same as those in Eq. (4.2). The procedure used to select other parameter values is identical to that discussed above in section 4.4.4, and values used in the simulations can be found in Table 4.2. Each simulation consisted of 1000 steps for which the concentration of adatoms on the n th 106

122 terrace was found by solving diffusion equation in Eq. (4.1) assuming quasi-static conditions and using the newly formulated boundary conditions in Eq. (4.9). In these new simulations adatom sublimation and deposition were assumed to be zero. Test simulations indicate that modest adatom sublimation or deposition has little effect on the bunching behavior, consistent with results presented here in section 4.1 in Figs. 4.1, 4.2, 4.7, and 4.8. After finding the solutions for c n (x) on each terrace, the n th step was propagated using its calculated velocity: v n n n Fa [ c c x) ] + [ c c ( x) ] + [ c ( x) c ( x ] dxn 2 = = a ( 1 1 ) dt κ eq n eq n n n 2kT. (4.12) Because a drift-attachment term is now present in the boundary conditions, the notation in the code used to run these simulations is slightly different than the code used for the simulations presented above, so a copy of it can be found in Appendix A of this thesis. E D (ev) [2, 71] 1.1 D 0 (nm 2 /s) 3.80 X E F (ev) [27] 0.16 T ( C) E A : E D (ev) d (nm) q eff ( e ) g (ev/nm 2 ) : : : : : : Table 4.2: Top: Activation energy parameters from the literature. Bottom: Adjustable parameters used in the drift-attachment model for Regimes I, III, and II shown in Figs. 4.15, 4.16, and 4.17, respectively. The first column gives the corresponding values for E A and E D [see Fig. 4.14(b)] (note that E D was changed in the last row at 1290 C). 107

123 Before presenting the simulation result for step-up bunching in Regime II it is worth briefly discussing simulations of step-down bunching in Regimes I and III. Figure 4.15 shows the new simulation results for Regime I with a step-down adatom force at T 940 C after 3 hours assuming d = 2 nm (and hence a/d 0.2) with other parameter values listed in Table 4.2. We find that N l and l min l , which is very similar to the previous simulations shown in Fig for these parameters using the traditional boundary conditions of Eq. (4.2). The simulated bunching behavior for d = 2 nm is consistent with the experimental observations in Fig. 4.9, where we found N l and l min l for step-down current bunching at 940 C. N N ~ l l min ~ l l 0 (nm) lmin (nm) Fig Numerical simulation of step-down bunching in Regime I (940 C for 3 hours) using the drift-attachment model (with slow attachment conditions) of how the number of step N ( ) and minimum terrace width l min (o) depend on the initial terrace width l 0. The results from the drift-attachment model (d = 2 nm slow attachment) are nearly identical to those of the sharp-step model (with d = 2 nm) shown in Fig

124 Simulations assuming d = 1 nm at 940 C were also performed, and essentially the same scaling was found as those with d = 2 nm, provided the values of q eff and g were adjusted as shown in Table 4.2. However, the correct scaling behavior could not be obtained for values of d larger than ~20 nm, in agreement with the finding of section 4.2 above where the conventional boundary conditions were used. Simulations at 1290 C for 3 minutes were also conducted in temperature Regime III (shown in Fig. 4.16) where step-down bunching also occurs, and we observed similar scaling of N and l min with l 0 as in Regime I, using the values of q eff and g shown in Table 4.2. Again, experiment could only be matched for values of d < ~20 nm, as found in Regime I. According to this model, the requirement that d < ~20 nm physically means that the attachment energy barriers E A must be of similar size as the diffusion barrier E D in Regimes I and III. N N ~ l l min ~ l l 0 (nm) lmin (nm) Fig Numerical simulation of step-down bunching in Regime III (1290 C for 3 minutes) using the drift-attachment model (with slow attachment conditions) of how the number of step N ( ) and minimum terrace width l min (o) depend on the initial terrace width l 0. The results from the drift-attachment model (d = 2 nm slow attachment) are nearly identical to those of the sharp-step model (with d = 2 nm) shown in Fig

125 Next, step bunching behavior in Regime II is considered, where a step-up adatom force is required for bunching. In this case, the BCF-type model considered here will only lead to step bunching if fast attachment conditions (E A < E D ) are assumed and the drift-attachment boundary conditions such as those in Eq. (4.9), or those suggested by Suga et al. [9] or Zhao et al. [12] are used. Figure 4.17 shows the simulation results for a step-up adatom force (F > 0) at 1090 C after 3 hours, assuming E A = 0.9 ev and E D = 1.1 ev (resulting in d = 0.07 nm) with other parameter values shown in Table 4.2. The simulations indicate that N l and l min l , consistent with the experimentally measured behavior shown in Fig These observed dependences on l 0 are also very similar to those observed for step-down bunching in Regimes I and III, and to the simulated behavior in those Regimes when d < ~20 nm N N ~ l l min ~ l lmin (nm) l 0 (nm) 0 Fig Numerical simulation of step bunching in Regime II (1090 C for 3 hour) using the drift-attachment model (with fast attachment conditions) of how the number of step N ( ) and minimum terrace width l min (o) depend on the initial terrace width l

126 One thing to note from the parameter values in Table 4.2 is that an increasing larger effective charge q eff is needed as the temperature is increased from Regime I to Regime III. The values reported here for Regimes II and III in Table 4.2 are considerably larger than previously reported values for Si(111) [57, 59, 72]. However, we have noticed that if D is made larger while holding d = D/κ fixed, we can only match experiments if there is a corresponding decrease in both q eff and g. In other words, if both D and κ were to become a factor of 10 larger, then we would need to decrease both q eff and g by a factor of 10 to match experiment. Similar behavior was also found and discussed by Fu et al. [57], and is reflected in Table 4.2 by the set of simulations conducted at 1290 C using E A = 0.9 ev and E D = 0.7 ev (d = 2 nm) where an effective charge similar to that in Regime I is required to match experiments. It should be noted that a surface transition has been reported to occur around ~ C [61, 83, 85], and is speculated to be incomplete surface melting. If this is the case, then one would expect a sharp increase in the rate of diffusion. This is appealing since it could explain the transition from step-up to step-down bunching between Regimes II and III Discussion of findings The addition of a drift-attachment term to the conventional boundary conditions has been shown to produce step-up (step-down) bunching provided fast attachment (slow attachment) conditions are assumed. This is appealing since this model can describe all of the experimentally observed step bunching behavior on Si(111) provided one assumes that the step attachment E A and/or terrace diffusion energy E D barriers under go modest temperature-dependent changes (~ ev). These findings allows us to proposed the following scenario for step bunching on Si(111). In temperature Regime I (~

127 C) step-down bunching results from slow attachment at step with E A slightly larger than E D, while in Regime II (~ C) step-up bunching results from fast attachment at the step characterized by E A being slightly smaller than E D. The transition from slow to fast attachment may be the result of a modest reduction in the step attachment barrier E A relative to the terrace diffusion energy barrier E D. Finally, step-down bunching in Regime III (~ C) may be the result from a return to slow attachment conditions with E A again slightly larger than E D. This last transition from fast to slow attachment may likely be the result of a change in the high temperature diffusion process. For example, should diffusion become anomalously fast around ~1200 C, then one might expect E D to become slightly smaller than E A (Hence, slow attachment conditions) in Regime III. Although there has been no attempt to directly verify a sudden jump in the diffusion rate on Si(111) experimentally, it should be mentioned that there appears to be a surface transition around C [61, 83, 85] that is thought be incomplete or partial surface melting. Partial surface melting has been observed on the Ge(111) just a few hundred degrees below the bulk melting temperature [73, 74]. It is reasonable to think that such a phase may lead to faster adatom diffusion, and would add plausibility to the explanation of the second transition proposed here. 4.4 Electromigration induced step bunching on Si(001) Up to now I have only discussed step bunching phenomenon on vicinal Si(111) surfaces. Before closing this chapter, it is worth while mentioning step bunching on the more technologically important Si(001) vicinal surfaces. Here I will briefly outline the various mechanisms currently believed to be responsible of the observed step bunching 112

128 behavior. Then I will present and discuss a few experiments carried out on the Si(001) surface Brief review of the Si(001) vicinal surface As with Si(111), the exact nature of the step bunching instability on vicinal Si(001) surfaces is still under debate. Currently, there are three basic views/models concerning Si(001) electromigration induced step bunching, and all three are based on the (2x1) surface reconstruction of vicinal Si(001) shown in Fig. 2.1(c). Therefore, it is necessary to highlight some important aspects of the reconstructed Si(001) surface in order to understand the models. The energetic details of this surface will not be discussed here, but a comprehensive review can found in Ref. [28] and the references therein. First, as discussed in chapter 2 section 2.1.1, at low surface miscut towards a <110> crystal direction the (2x1) dimer reconstruction leads to two kinds of atomic height surface step edges with height h = 1.36 Å. (1) S A steps, which have dimer rows running parallel to the upper edge of the step and (2) S B steps, which have dimer rows running perpendicular to the upper edge of the step. It costs more energy to create a kink in a purely S A -type step than a purely S B -type step, and as a result S A steps are much straighter than S B steps. Secondly, there have been many interesting observations regarding the nature of surface diffusion on Si(001). The diffusing species on Si(001) is thought to be an addimer (a pair of adatoms), and these diffusing addimers have been observed using atom-tracking scanning tunneling microscope (STM) at temperatures up to ~128 C [75]. In these experiments it was found that the addimers diffuse mostly on top of the dimer rows shown in Fig. 2.1(c), and as a result, diffusion is highly anisotropic at temperatures below ~128 C [75] with fast (slow) diffusion occurring parallel (perpendicular) to the dimer 113

129 rows. This complicated anisotropy is shown schematically in Fig. 4.18, and can also be understood by examining the atomic surface structure in Fig. 2.1(c) and noting that the step-up vicinal surface direction is from left-to-right. The terrace on the upper side of the S B step has a fast diffusion direction that is perpendicular to the step-edge, and a slow diffusion direction that is parallel to the step-edge. Similarly, the terrace on the upper D f D s D f D s Addimer S A Surface dimer Surface dimer row S B Fig Schematic view of the diffusion process on the Si(001)-(2x1) surface. Anisotropic diffusion is characterized by slow (fast) diffusion perpendicular (parallel) to the dimer rows on the terrace. side of the S A step has slow diffusion direction perpendicular to the step-edge, and a fast diffusion direction that is parallel to the step. I shall refer to terraces on which fast (slow) diffusion occurs in the direction perpendicular to the step-edges as fast (slow) diffusion terraces. Finally, it is thought that diffusing addimer species remains stable up to ~650 C [20, 21], and possibly beyond, but there is still debate about whether or not diffusion 114

130 is anisotropic above ~650 C. Tromp et al. [20, 21] made low energy electron microscopy (LEEM) studies of denuded zones that formed on large step free terraces after low-temperature growth followed by annealing, and argued that diffusion on Si(001) may become more isotropic above 650 C. However, a more recent analysis of LEEM data of denuded zones following high-temperature growth indicated that has shown significant anisotropy in diffusion remains for temperatures up to at least 700 C [22] Step bunching on Si(001) As with Si(111), step bunching on Si(001) under direct current heating was first observed by Latyshev et al. [2]. At temperatures above ~900 C, they found that step bunching occurs for both directions (step-up and step-down) of the direct current, and normal step flow was observed without step bunching when Si(001) is heated with an alternating current. One very interesting feature of this bunching instability is that during the initial stages of step bunching the steps are observed to pair or couple in a way that depends on the direction of the heating current [2]. Taking the step-up direction to be from left-to-right, as shown in Fig. 2.2(b), it was found that for a step-up (step-down) current the steps would pair in the left-to-right order S B -S A (S A -S B ), so that the large terrace covering the majority of the surface would have dimer rows running parallel (perpendicular) to the step pair [2]. Recalling that the diffusion of addimers is fast along the dimer row direction, continued pairing of the step pairs (step bunching) leaves the surface covered by a majority of large step free regions on which diffusion is fast or slow in the direction perpendicular to the step bunches when heated with step-down or step-up current, respectively. 115

131 The generally-accepted explanation for step pairing was given by Stoyanov [5]. The qualitative nature of the pairing mechanism can be understood in the following way. First, consider the S A step in Fig and take the adatom drift to be in the step-up direction (left-to-right). It is seen that diffusion on the upper (lower) terrace is slow (fast) in the direction perpendicular the step, so there will be fewer addimers detaching from the step to the upper terrace than attaching to it from the lower terrace. This means that the S A step gains atoms, and as results moves to the left. Now consider the S B, where diffusion on the upper (lower) terrace is fast (slow) in the direction perpendicular the step. This case is opposite that of the S A step, so the S B has a net loss of mass and moves to the right. The two steps move until they the pair in the left-to-right order of S B -S A. A similar argument can be applied for the case of a step-down addimer drift. Using the same 1D BCF-type sharp-step model with an electromigration driftdiffusion term, Stoyanov [5] argued that in the limit of zero desorption or deposition (and neglecting step-step interactions) the growth rate of fast diffusion D f terrace of width l f is 2 expressed by dl / dt= 2( F / k T ) a c [ D D ], while the growth rate of the slow f B eq s f diffusion D s terrace of width l s is given by dl / dt= dl dt. Here the electromigration s f / force on an addimer F is taken to be positive in the vicinal step-up direction according to Fig. 2.2(b). It is clearly seen that step pairing is a consequence of the electromigration induced drift of the addimers coupled with anisotropic diffusion. In the case of a stepdown direct current (F < 0) the slow diffusion terraces shrink while the fast diffusion terraces grow. This corresponds to a left-to-right pairing of S A -S B, and as a result the surface will be covered primarily by fast diffusion terraces. In the opposite case, a stepup direct current (F > 0) will cause slow diffusion terraces to grow and fast diffusion 116

132 terraces to shrink, leaving the surface covered by mostly slow diffusion terrace with a left-to-right step pairing sequence of S B -S A. These predictions match the experimental observations of Latyshev and coworkers [2], and this model has been successfully used to interpret experimental observations of the drift/motion of ad-islands nucleated Si(001) terraces [76]. One long standing issue with this model is that it only predicts step pairing and not large scale step bunching on Si(001). Next I will briefly describe three proposed mechanisms that could be responsible for the continued pairing of paired steps, hence step bunching on Si(001). The first mechanism used to explain a large range of bunching behavior on Si(001) was offered by Nielsen et al. [77]. The under lying assumption is that the effective charge q eff might be a tensor rather than scalar quantity that could cause both the magnitude and direction to depend on the orientation of the electric field relative to the local surface reconstruction (dimer row direction). They argued that such changes to q eff can lead to a variety of interesting surface addimer flows. Nielsen and coworkers [51] used this model to explain a wide range of observed step bunching behavior on dimpled Si(001) surfaces. The second model to describe step bunching on Si(001) surfaces was proposed by Sato and co-workers [78]. They investigated step bunching behavior using numerical simulations of the sharp-step model [5] assuming the addimer terrace density is in equilibrium with the steps at the step positions. This assumption amounts to a very large step attachment rate κ. From simulations, Sato et al. [78] found that bunching is stronger in the case of a step-down addimer drift than the case with a step-up drift, and it was found that the addimer concentration varied across the step pairs/bunches in way that 117

133 favored further bunching. Analyzing the steady-state shape of a step bunch at a fixed miscut, Sato et al. [78] found that the average terrace width in a bunch l b varied with the number of steps in a bunch N as 1/ 2 l b ~ N in the case with step-step interactions of the form described in chapter 2. The third model to describe step-bunching on Si(001) to be discussed here is relatively new idea proposed by Zhao et al. [10, 12]. The application of this model to Si(001) is quite complicated, so I will only make a few brief comments about it. First, it was not only assumed that diffusion is anisotropic on a terrace, but also that each S A and S B step is surrounded by a region (possibly just a few lattice spacings) where diffusion is different than that on the terrace [10, 12]. Furthermore, it was assumed that diffusion in the step region is different on the step-up and step-down side of the step just like diffusion on the terraces [10, 12]. This means that the terrace on the step-up side of an S A step will have two small step regions with slow diffusion D s slow and a terrace portion with slow diffusion D slow. A similar thing is found for terrace on the step-up side of an S B step. Zhao and coworkers [10, 12] argued that this arrangement leads to step pairing and large scale step bunching for both step-up and step-down direct current. It was also argued that this model [15] could reproduce the bunching patterns observed by Nielsen et al. [77] on dimpled Si(001) samples, and provides an interesting explanation for the bunching process on Si(001) Experimental observations In this section I will discussed step bunching experiments carried out on Si(001) under a variety of Si deposition conditions at a fixed temperature. The same dimpling and cleaning procedure discussed in chapter 3 (and above) was used to produce Si(001) 118

134 samples with varying local miscut. Annealing was done in the same UHV system described in Chapter 3 at a temperature of 1060 C for 6 hours. Three Si flux conditions were explored in this study: (1) free sublimation, (2) near balanced flux, and (3) net growth. Si flux was provided by the rod fed e-beam evaporator (see Ch. 3), and at the highest deposition rate of ~0.13 Å/s the pressure in the chamber remained below ~5x10-10 torr. As before, the sublimation rate at this temperature was determined by find the deposition rate R dep that produced near balanced flux conditions R net = R dep R sub 0, which was determined from the shape of pinned steps in ex-situ AFM images and the existence crossing step free terraces. At the temperature used here (1060 C) the free sublimation rate was found to be R sub ~ Å/s. After annealing the samples were transferred out of UHV and examined ex-situ using AFM. Figure 4.19 shows the experimentally observed step bunching behavior on Si(001). Figure 4.19(a) shows the step-up and step-down surface orientation corresponding to Figs. 4.19(b) 4.19(g) directly below. All images are first derivatives (viewed as if illuminated from left-to-right) from regions with a local miscut of ~0.15 l 0 ~ 52 nm except for panel (f) which has a local miscut of ~0.1 l 0 ~ 78 nm. Panels (b) and (c) show the behavior for a step-up and step-down DC annealing under free sublimation conditions (R net ~ Å/s), respectively. Under sublimation conditions step bunching is observed for both a step-up and step-down DC, in agreement with the observations of Latyshev et al. [2]. Step bunching is also found to occur for both DC directions under near balanced and net growth Si deposition conditions as shown in Figs (d) (e) and Figs (f) (g), respectively. The absence of crossing steps in Figs. 119

135 (a) Step-up 1060 C 6 hrs. E r Step-down (b) (c) 5 µm (d) Free sublimation ( Å/s) (e) 4 µm 5 µm (f) Near balanced (0 Å/s) (g) 5 µm 5 µm Net growth ( Å/s) 5 µm Fig Derivative-mode AFM images (viewed as if illuminated from the left) of the observed step bunching behavior for the Si(001) surface. (a) Schematic view of the surface during DC annealing showing the direction of the applied current. All images (b) (g) are for samples annealing at 1060 C for 6 hours under the following deposition conditions: (b) and (c) free sublimation R net ~ Å/s, (d) and (e) near balanced R net ~ 0 Å/s, and (f) and (g) net growth R net ~ Å/s. Step bunching occurs for both directions of the heating current, and the black, double headed arrows indicated the dimer row directions on the majority terraces (determined using the shape of the vacancy pits marked by the blue arrows). 120

136 4.19 (d) and (e) indicates near balanced flux conditions. However, under such near balanced conditions we do observe large monolayer deep h = 1.36 Å vacancy pits (marked by blue arrows) on the terraces between step bunches. By considering the shape of these elliptical vacancy pits, one can determine the direction of the dimer rows on the large, step free terraces separating the bunches. According to Bartelt et al. [79] adislands (or vacancy pits) on Si(001) are elongated in one direction due to the anisotropy between the S A and S B step free energies. Physically, the islands elongate in such a way to minimize the Gibbs-Thomson chemical potential of the island, and since the energy of the S A is lower than that of the S B step, the equilibrium island shape is one that maximizes the length of S A step [79]. Therefore, the long axis of the vacancies in Fig (d) and (e) is parallel to the lower energy S A step-edge, and so the dimer rows on the terrace surrounding the vacancy run parallel to the long axis of the vacancy (see Figs. 2.1(c) and 4.18.). As indicated by the black double-headed arrows in Figs (d) and (e), the dimer rows on the large terraces run perpendicular (parallel) to the step-edges when the DC is passed in the step-down (step-up) directions. This is consistent with the observations of Latyshev et al. [2] under free sublimation, and means that the surface on the step-down (step-up) current side of dimple is mostly covered by terraces with fast (slow) diffusion in the direction perpendicular to the step-edge. The dimer rows were observed to have this orientation for all of the samples studied here independent of the Si flux conditions, i.e., deposition does not change the order of step pairing. It should be pointed out that vacancy pits, which probably formed during the quench from high temperatures, were seen on all of the samples studies, and this observation differs from the ad-island formation reported by others following a quench from high temperature [21]. 121

137 We do not understand the different behavior we observe, but speculate that it could be connected to the cooling rate during a quench. Finally, one last qualitative observation should be pointed out. On all of the samples studied here it was found that step bunching persisted to a higher miscut when the DC was passed in the step-up direction. For example, under free sublimation step bunching stopped at ~0.2 l 0 ~ 39 nm (~0.47 l 0 ~ 17 nm) when DC was passed in the step-down (step-up) direction. The reason for this behavior is not understood at the time. The effects of local surface miscut on the step bunching process were explored in the same way as discussed above for Si(111). The average number of steps N and the average terrace width l ave in a bunch after a 6 hour DC annealing time was measured for a fixed initial terrace width l 0, and a simple power law was fit to the data to aid in the characterization of these dependencies. The values for N and l min were extracted from the data using the same method discussed above except here on Si(001) the atomic step height is h = 1.36 Å, and instead of measuring l min the average terrace width in a bunch l ave was determined by the relation l ave = L B /N where L B is the width of a bunch. Figures 4.20 (a) and (b) show the measured dependencies of N and l ave on the initial starting terrace width l 0 for step-down DC bunching. Data was taken from three different samples under free sublimation ( ) and ( ) with R net ~ Å/s and net growth conditions ( ) with R net ~ Å/s. The bunching rate (defined as the number of steps in a bunch after a given annealing time) and average terrace width in a bunch depend on the initial surface terrace width as N l and l ave l

138 N lave (nm) (a) T = 1060 C Step-down current (b) N ~ l l ave ~ l T = 1060 C Step-down current l 0 (nm) Fig Plot showing how (a) the number of steps N and the (b) minimum terrace width l min in a bunch depend on the initial surface miscut l 0 on Si(001) DC annealed (1060 C 6 hours) with a current in the step-down direction. Step-up DC bunching behavior was also extracted from these same three samples and is shown in Figs (a) and (b) under free sublimation ( and ) and net growth conditions ( ). Similar to step-down DC bunching, N and l ave are found to depend on l 0 as approximate power-laws. More specifically, N l and l ave l At first glance, it is tempting to conclude that the differing dependence on l 0 between step-up and stepdown bunching indicates that one side is more diffusion- or attachment/detachment limited than the other. However, there is sufficient scatter in the data that may account 123

139 for this. Furthermore, if the step-up bunching data is restricted to the same range of l 0 as the step-down bunching data the scaling dependencies become nearly identical (a) T = 1060 C Step-up current N N ~ l lave (nm) (b) l ave ~ l T = 1060 C Step-up current l 0 (nm) Fig Same plot as Fig. 4.21, but with a current in the step-up direction. Finally, there are no scaling predictions for step bunching on Si(001) like those for Si(111). The three basic models [5, 10, 12, 78] discussed above have only been shown to lead to step pairing, except recent numerical simulations by Sato et al. [78] using the first model discussed above [5] where step bunching was observed for both current directions assuming the steps are in local equilibrium with the terrace. Using the measured scaling behavior shown in Figs and 4.21 it is interesting to note that for Si(001) the product of l ave N 2/3 is practically independent of l 0 just like Si(111). 124

140 CHAPTER 5 HIGH TEMPERATURE SILICON CRATER FLOOR AND WALL EVOLUTION In this chapter I will discuss experiments conducted to study the high temperature evolution of craters formed on the Si surface. After outlining the procedure used to create the craters on the wafer surface, I will address two things in this chapter. First, I will discuss how simple step flow can be used to create large surfaces with an ultra-low step density on crater floors. Then I will finish with a short discussion of the evolution of the crater morphology during high temperature annealing from the macroscopic point of view. 5.1 Introduction Achieving the desired electrical behavior in silicon based devices with active regions on the submicron scale requires atomically smooth surfaces. Understanding exactly how to control the atomic scale morphology of surfaces has been a challenge, and continues to be an active area of scientific inquiry. Recently, a way to produce large, atomically flat, nearly step free regions on Si(111) was discovered by Homma et al. [16], and independently demonstrated on Si(001) by Tanaka et al. [17]. This technique makes use of step flow sublimation on wafer regions bounded by walls ( craters ) where steps can flow until they collide with the crater wall. The result is a large, nearly step free 125

141 terrace on the crater floor that may some day be used as an atomically smooth starting surface in industry, and is already widely used in science to study things such as step motion [18], etching [19], nucleation [20, 21, 22], and Si/Ge alloying [23]. The process of flattening craters developed in Refs. [16, 17] is depicted schematically in Fig. 5.1(a), and will be described briefly here. The left side of Fig. 5.1(a) shows the initial starting surface just as heating begins with the red arrows indicating the direction of step flow, while the right side shows the surface during high temperature annealing. As a result of step flow sublimation, the steps on the bottom of the crater floor move toward the crater wall leaving behind a large step free (001) or (111) silicon plane/terrace. As flattening progresses the step free region becomes unstable against the nucleation of macrovacancies [16 18] due to the sublimation of adparticles [either adatoms on Si(111) or addimers on Si(001)], setting an upper limit for how large the step free area can be, as shown by the left image of Fig. 5.1(b). It should be mentioned that the addition of a small deposition flux has been shown to greatly enhance the stability/size of step free terraces on Si(001) [18]. Finally, as discussed by Tanaka et al. [17], if the annealing time is too long the sidewalls no longer serve as effective barriers, as indicated by the side wall breach in the right image of Fig. 5.1(b). This also occurs if the craters are not deep enough. One issue with the flattening process is that long annealing times are needed to remove the initial surface miscut from the crater floor [17]. As pointed out by Tanaka et al. [17] this is due to two things: (1) The initial wafer miscut and (2) depressions or trenches that form during the crater fabrication process, both of which greatly increase the volume of material that must be removed during flattening. The problem with initial 126

142 (a) Initial During anneal <001> <111> (b) Later stage of heating Vacancy formation Side wall Breach Fig Schematic diagram showing the crater flattening process. (a) Shows the initial stages stages of heating where significant flattening occurs. The later stage of heating are shown in (b) where both macrovacancy formation and a possible sidewall breach occur. The red arrows indicate the direction of step flow. miscut is easily dealt with since well oriented low miscut wafers are available, but avoiding trench formation during plasma etching is not so easy. Therefore, I will outline a novel process based on the thermal oxidation of silicon and simple photolithography for making craters without these trench features. This process has the advantage of being clean, very easy to carry out, and does not require an expensive plasma etching system. Surprisingly, it is found that small trenches form around the perimeter of the crater floor due to step rearrangement during the flattening process. This feature is observed on Si(111), and also on Si(001) below the roughening temperature T R. For Si(001) above T R, the peaks become amplified and a significant mound is formed around the outer 127

143 perimeter of the crater. These morphological changes above T R can be explained using the continuum theory of Mullins [14, 15], as discussed in chapter 2, when diffusion is the dominant pathway for mass flow on the surface. The organization of the rest of this chapter is as follows. First, I will discuss the method used to fabricate craters of various shapes and sizes on Si(001) and Si(111) wafers. Then I will discuss the observations made concerning flattening and the evolution of the crater walls and surrounding areas. 5.2 Double oxidation technique for crater patterning The process of thermally oxidizing silicon is a standard technique that is widely available (see Ref. [80] for a comprehensive review). According to the Deal-Grove model of thermal oxidation [81] the oxide thickness d ox as a function of time is given approximately by [80]: 2 dox + Adox = B( t+τ ), (5.1) where A, B, and τ are constants that depend on the process and initial surface conditions. It should be pointed out that the constants A and B can be looked up [80] or measured for a specific process. τ can be calculated for cases where the wafer starts with an initial oxide film of thickness d 0 at time t = 0 using Eq. (5.1), which physically means that the oxidation is treated as if started at time -τ [80]. If oxidation is done in a dry environment (O 2 gas only), then even if d 0 = 0 τ still has a well defined value to account for the rapid growth of thin oxides [80]. For thick oxides the linear term in Eq. (5.1) can be ignored, and the resulting oxide thickness as function of time is given by 2 d ox B( t+τ ). This indicates that the rate of oxidation slows as the oxide thickens. 128

144 Making use of the last fact, the process of forming craters on the Si surface is shown schematically in Fig. 5.2, and can be described as follows. Starting with an oxide free Si wafer, an initial oxide with a thickness 1 µm is grown under wet or steam oxidation conditions since the oxidation rate is faster than the dry oxidation technique. After growth, photoresist is spun onto the wafer and photolithography is used to pattern craters of the desired shapes and sizes as shown in the photo step of Fig The photoresist is then developed, and the underlying, exposed oxide is etched down to the Si substrate. The photoresist is then stripped and the wafer cleaned. The result is a Si wafer with a thick oxide with holes that extend down to the Si substrate in the shape of the desired features. Next, the wafers are oxidized a second time. The oxidation rate of the regions where the Si substrate is exposed is much faster than the regions covered by the thick oxide according the Deal-Grove model [81]. This means that during the second oxidation the newly grown oxide will be thicker over the exposed regions. Thus, more silicon will be consumed in the exposed regions during the second oxidation, as shown in Fig The final step is to strip all of the oxide leaving behind craters in the silicon surface with desired shapes and dimensions. It should be mentioned that the depth of these features can be controlled by adjusting the time of the second oxidation according to Eq. (5.1), and a more detailed account of this process will be given next. All steps below were carried out in the Microelectronics Cleanroom in the Department of Electrical and Computer Engineer at The Ohio State University. Step 1: Initial clean and 1 st oxidation Before going into the oxidation furnace the wafers were cleaned using the standard RCA clean [80]. First the wafers were put through degrease sequence 129

145 consisting of a 10 minute ultra-sonic bath in actone (ACE) with, a 5 minute deionized (DI) water rinse, a 10 minute ultra-sonic bath in methanol (METH) with sonication, and a final 5 minute (DI) rinse. Then the wafers were put through Standard Clean One (SC1) using a mixture (by volume 1:5:1) of NH 4 OH:(DI)H 2 O:H 2 O 2 for 10 minutes with the beaker submersed in a ~70 C water bath, which removes organics and many metals that may be present on the surface [80]. The wafers were removed and rinsed in DI for 5 minutes. After rinsing the wafers were put through Standard Clean Two (SC2) using a mixture (by volume 1:5:1) of HCL:(DI)H 2 O: H 2 O 2, followed by a 5 minute DI rinse. In SC2 the Cl halogen removes heavy alkali ions and cations [80]. The final step in the clean was to strip the native (or any chemically grown) oxide on the wafer using a 1:1 HF:(DI)H 2 O solution for ~10 seconds and final DI rinse. This last step leaves an oxide free surface. After cleaning the wafers were oxidized using the wet oxidation furnace in the OSU microelectronics clean room at a temperature of ~1100 C following the standard procedure outlined the EE 637 Microelectronics Laboratory Manual. Photoligthography and etch step After cooling, the wafers were put through a standard photolithograph process. The spinner was set to 5000 rpm with an acceleration of 10,000 rpm/s to spin on a layer of the adhesion promoter [80] hexamethyldisilazane (HDMS) for 10 seconds, followed by a layer of Shipley 1818 photoresist spun for 30 seconds. After spin coating, the wafers were soft-baked for 20 minutes at ~95 C, to evaporate the solvent base of the photoresist [80]. The wafers were then patterned by contact photolithography using a designed photomask and the MJB-3 aligner with a lamp setting of 405 nm and a 20 second exposure time. Once exposure was complete the wafers were developed for 1 130

146 minute using the MF-320 developer with agitation. The final step before etching was a hard bake for 20 minutes at 120 C. The optical image in Fig. 5.2 shows an image of the Initial thick oxide growth SiO 2 Si Photo step holes PR Etch step & PR strip 100x Optical image 5µm 10µm 20µm 40µm 50µm 2 nd oxidation SiO 2 Final oxide removal craters Fig Schematic diagram showing the steps for creating craters in the silicon surface a sequence of thermal oxidations. The optical image was taken after exposing and developing the photoresist (just before the etch step). The feature size is labeled on the image. There is considerable rounding of the 5 µm features. 131

147 sample at the end of the photo step. The mask features were preserved down to the 10 µm mask level, but at the 5 µm level the features were transferred as circles. Note that there is also some rounding of the 10 µm features. Rounding of the features may be the result of any imperfections in the steps above. Etching of the SiO 2 layer was done using buffered HF (BHF), and a test etch was performed to determine the time required to remove the oxide and expose the bare Si surface. The patterned wafers were immersed in the BHF for the determined time +10%. After optically inspecting the wafers the photoresist was stripped using a 7 minute ACE bath with ultra-sonic agitation, followed by a 7 minute METH bath with ultra-sonic agitation and a final 5 minute DI rinse. Final clean and 2 nd oxidation Before completing the second oxidation, the wafers were put through SC1 and SC2 described above. The HF dip was omitted since the initially grown oxide is crucial to the process. After cleaning the wafers were oxidized a second time using the same procedure above. Figure 5.3 show a three-dimensional AFM scan of a portion of a 20 µm wide crater and the corresponding surface profile for a Si(001) sample oxidized for 8 hours twice. The measured depth of the crater is ~375 nm, which is ~125 nm less than the expected depth for this process as determined using Eq. (5.1) and the fact that for each 1 µm of oxide grown 0.46 µm of silicon is consumed. This discrepancy may be due to the extremely long oxidation times, during which the bubbler providing steam for the wet oxidation conditions had to be refilled with water numerous times. As a result, the furnace environment was not uniform over the entire time of oxidation. Note that the width of the crater floor is ~5 µm larger than the mask dimension of 20 µm and that the 132

148 sidewalls are not vertical. This is due to the fact that as the oxide grew vertically in the exposed region during the second oxidation it also grew horizontally by consuming silicon from the sidewalls. An important thing to note about these craters is that there are no trenches around the lower edge of the sidewalls. This may be important for forming large flat craters using high-temperature heating, as discussed above. z (nm) x (µm) Fig Typical 20 µm level crater produced using the procedure above. The top image is an AFM image (three-dimensional rendering), and the lower plot is a line profile through crater indicating a depth of ~375 nm. 133

149 5.3 Experimental observations and discussion There have been numerous reports [16, 17, 82 86] on crater flattening, here I will only briefly outline a few things that have not been explicitly discussed in the literature already. As pointed out by Tanaka et al. [17] the ability to flatten the bottom of a crater depends greatly on miscut. This is obvious from Fig. 5.1(a), where if one considers a crater of fixed width, then a larger miscut means more material needs to be removed from the crater floor. There is also one more simple issue involving the crater orientation relative to the vicinal step-up direction that has not been mentioned in the literature. This is shown schematically in Fig. 5.4(a), where the square crater with the vicinal step-up direction oriented along the diagonal (right figure) has 2 times more material by volume to remove than a square crater with the vicinal step-up direction running perpendicular to one of the crater edges (left figure). Therefore, one can consider the orientation on the left side of Fig. 5.4(a) ideal since under equal annealing times and conditions it should flatten faster than the nonideal orientation on the right side of Fig. 5.4(a). The advantage of both low miscut and crater orientation is shown in Fig. 5.4(b) for Si(001) samples annealed at 1080 C for various times made using the procedure above. The filled black symbols represent the area of the flattened region for craters of widths 50, 40, and 20 µm in the ideal orientation on 0.3 Si(001). The corresponding open symbols are data from the same sized craters with the nonideal orientation also on wafers with a miscut of 0.3. Craters with the orientation shown in the left schematic of Fig. 5.4(a) flatten faster than those with the orientation shown on the right. It should be noted that all measurements taken at a fixed time (for example 20 hours) are from the same sample since craters with both orientations exist on a single sample (see Fig. 5.2). 134

150 (a) (b) Flat Area (µm 2 ) Ideal Vicinal step-up direction w Crater floor θ w Lowest crystal plane Top view of crater Material to remove 50 µm ideal (0.3 ) 40 µm ideal (0.3 ) 20 µm ideal (0.3 ) 20 µm ideal (0.1 ) Crater floor Lowest crystal plane average height of surface Fig (a) Schematic diagram showing the ideal (left) and nonideal (right) orientation of craters relative to the vicinal step-up direction. (b) Plot showing the area of the flattened region for both crater orientations. w Vicinal step-up direction w Anneal time (hours) 135

151 For completeness, the advantage of lower miscut is shown by the red diamonds ( ) in Fig. 5.4(b), which shows flattening data for an ideally oriented 20 µm crater etched in Si(001) with a miscut of 0.1. A similar comparison was also made for the 50, 40, and 10 µm, but the data is not shown here. Larger flat regions for a given annealing temperature and fixed time can be obtained with lower miscut silicon, as mentioned (but not observed) by Tanaka et al. [17]. The new observation made here, although simple, is that the orientation of the crater relative to the vicinal surface miscut is also important. Figure 5.5 shows AFM images typical of flattened craters. Figure 5.5(a) shows a 20 µm mask level (~30 µm actual) crater in 0.1 Si(111) annealed using AC at a temperature of 1080 C for 1 hour. The image should be viewed as if illuminated from the upper right corner, and the vicinal step-up direction runs from the lower left corner to the upper right. The initial step distribution (with a terrace spacing of l 0 ~ 189 nm) has been removed by the step flow sublimation process shown in Fig. 5.1(a), and extended annealing beyond the flattening time has resulted in macrovacancy nucleation near the center of the crater floor marked by the x. The nucleation of macrovacanies results in closed, circular step loops that flow out away from the center (The diameter of the inner most step loop is ~10.5 µm.) [16 18, 82 86]. The process that leads to breaching of the crater wall, where significant step annihilation occurs at the rim of the crater, is marked by the green arrow in the lower left corner of Fig. 5.5(a). The two annihilating steps pinch-off creating two steps that recede along the crater rim, as indicated by the dashed arrows. Similar features were observed by Umbach et al. [87] at the tops of onedimensional sinusoidal profiles, and these features have since been modeled by Erlebacher et al. [88]. 136

152 (a) Si(111) (b) Si(001) x ~10.5 µm 6 µm (c) 3.0µm Fig (a) A completely flat 20 µm (mask level) crater in Si(111) annealed using AC at 1180 C for 1 hour. As a result of macrovacancy formation, circular step loops exist on the crater floor. The green and black dashed arrows show the breaching process (see text). (b) A completely flat 10 µm (mask level) crater in Si(001) and (c) the corresponding image of the crater floor where only small vacancy pits are seen. Images (a) and (b) are derivative-mode AFM images that should be viewed as if illuminated from the upper right. 137

153 Although circular step loops are present on the crater floor, a completely step free terrace can be achieved by annealing the crater at ~950 C for a short period of time. This method appears to be limited to craters with dimensions ~20 µm for Si(111), and according to the measurements of Nielsen et al. [18] ~15 µm for Si(001) in the case of zero Si deposition flux. This was demonstrated for a 10 µm mask level (~13 µm actual) crater on Si(001), as shown in Figs. 5.5(b) and 5.5(c). The sample was first annealed at 1100 C for 22 hours then at 970 C for 3 hours, and the AFM image of the crater floor in Fig. 5.5(c) shows that the flat region is completely step free. Next I will address the formation of mounds and trenches on in and around Si(001) annealed above the roughening temperature. Step flow on Si(111) persists to much higher temperatures than on Si(001). The reason for this is that the steps on Si(001) are thought to roughen at T R ~ 1200 C [79]. Here it has been found that craters on Si(001) do not flatten when annealed above T R, consistent with previous reports [82]. Figure 5.7(a) shows a ~10 µm crater on Si(001) annealed for 5 hours above T R at ~1200 C, where no flattening has occurred. The solid, red line in Fig. 5.7(b) shows a profile through the center of the crater before annealing, and the solid, blue circles shows the corresponding post-anneal profile. Both a trench and mound has formed around the perimeter of the crater floor and rim, respectively. These feature can be qualitatively described by the continuum theory of Mullins [14, 15], which was discussed in chapter 2. According to this theory, when surface diffusion is the primary mode of mass transport on the surface, then 1D macroscopic surface evolution can be described by [14, 15]: 138

154 2 4 4 dz( x) az a ceqdσ z( x) =, (5.1) 4 dt kbt x where a z = nm is the height of one surface layer, 2 a = 0.29 nm 2 lateral size of the diffusing species (taken to be an addimer [20, 21]), c eq is the equilibrium surface addimer concentration, D is the surface diffusion constant, σ is the surface stiffness (see chapter 2), and k B T is the product of the Boltzmann constant and temperature. To use Eq. (5.1) to predict high temperature crater evolution, we need estimates of the diffusion constant and equilibrium adparticle concentration. For the modeling I will discuss next, we assumed the following literature values of the diffusion constant and equilibrium adparticle concentration, which were determined using the thermally activated relations 2 D= D exp( E / k ) and c = a exp( E / k ), where D 0 = 1.4x10 13 nm 2 /s [18], E d 0 d BT eq f BT = 1.45 ev [18], E f = 0.35 ev [21], and T = 1200 C. The value for the surface stiffness at this temperature is not known, so it was taken to be σ ~ 1.5 ev/nm 2 (the accepted value for Si(111) at ~900 C [37]). The initial surface [red line in Fig. 5.7(b)] was broken up into its Fourier components given z(x, t = 0), and then Eq. (5.1) was solved using the Fourier transform method. The evolved surface profile is shown as the light blue in Fig. 5.7(b). The time of evolution required to match the experimental post-anneal profile is ~500 times larger than the actual 5 hour anneal time. Although large, the discrepancy may be due to the fact that the parameter values were extrapolated from below T R (It is likely that diffusion in the roughened phase is much faster than that predicted.). Nevertheless, it is interesting to note that the profiles are qualitatively identical, and using 139

155 (a) z (nm) mound trench (b) Surf 0 atom flow x (µm) Fig (a) Three-dimensional AFM image of a ~10 µm crater on Si(001) annealed above the roughening temperature at ~1200 C for 5 hours. (b) Plot of the starting crater morphology (red line), actual anneal morphology (blue circles), and the simulated profile (light blue line). The inset in (b) shows the surface and the atom flow drive by crater call morphology. 140

156 Mullins theory the formations trenches and mounds can be described by curvature driven surface adparticle flow. With the aid of the inset in the lower right corner of Fig. 5.6(b) the formation of these features, as well as the evolution of the crater in time, is easily described. The red curve shows the right sidewall of the crater, and the blue curve shows the adparticle flow determined by j= ( a a c Dσ / k T )[ z( x) / ] (see chapter z eq B x 2) over the surface. First consider the lower portion of the surface on the sidewall near the crater floor where the atom flow is zero (marked by the vertical dashed line in the inset). Just to the left of this point the atom flow is positive, i.e., to the right toward the point of zero flow. This means material will accumulate near this point on the sidewall causing it to grow inward, and a trench is formed because there is no flow material from the crater floor to replace the material that is leaving. Mound formation can be accounted for using similar arguments. Features of this type were originally pointed out and described by Mullins [14,15], and more recently they have been used to the control the nucleation of Ge quantum dots on Si(001) [89]. 141

157 APPENDIX A NUMERICAL SIMULATION CODE A.1 Sharp-step model The following program was written using C++ to simulate step bunching on the Si(111) surface using the sharp-step model. It is followed by both the parameter file and a screen shot of how to run the program. //Si(111) 1D surface evolution code. Uses the sharp-step form of BCF theory. This version finds the //optimum time step as before and then randomizes it //using a Gaussian with a standard deviation of 0.5 and chopped at the ends. //drive directives #include <iostream.h> #include <fstream.h> #include <iomanip.h> #include <stdlib.h> #include <time.h> #include <math.h> #include <string.h> #include <stdio.h> // //******globally defined parameters******* char i_file[512]; double parameters(int parameter){ char store[512]; char s_dumb[512]; double r_p; ifstream infile(i_file, ios::in); 142

158 for(int i = 0; i<parameter; i++)infile.getline(s_dumb,512); infile.getline(store,512); r_p = atof(store); if(parameter == 34){cout << endl << "Enter the number of steps to evolve(<10000): "; cin >> r_p;} if(parameter == 35){cout << endl << "Enter the total time of evolution (s): "; cin >> r_p;} infile.close(); return r_p; }//end parse file for parameters double Temp_c; double a; double h; double q_eff; double g; double E_v; double E_f; double tau_0; double E_t; double D_0; double E_d; double R1; double Temp; double kt; double c_eq; double t_e; double D_d; double k_p; //kappa double R; double F; double c1; double c2; double x_pos; int num_steps; double loops; double t_w; double sim_time; //define the doubles in the function concentration so they are no re-defined each time double mu[10000]; // //loop variables int i; int z; // //******************************************** double concentrations(double [],double [], double []); //define the function concentration to caluculate the concentrations int main(){ //assingment of global variables 143

159 cout << endl << "Enter parameter file to read: "; cin >> i_file; Temp_c = parameters(3); a = parameters(5); h = parameters(7); q_eff = parameters(9); g = parameters(11); E_v = parameters(13); E_f = parameters(15); tau_0 = parameters(17); E_t = parameters(19); D_0 = parameters(21); E_d = parameters(23); R1 = parameters(25); k_p = parameters(27); num_steps = int(parameters(34)); loops = parameters(35); Temp = Temp_c+273; kt = 8.617E-5*Temp; c_eq = 0.2*exp(-E_f/kT)/(a*a); cout << endl << "Equilibrium Concentration (Atoms/nm^2) --> (ML): " << c_eq << "-->" << c_eq*a*a; t_e = tau_0*exp(e_t/kt); cout << endl << "Desorption Lifetime(s): " << t_e << endl << "Sub. Rate(ML/s): " << c_eq/t_e*a*a; D_d =D_0*exp(-E_d/kT); cout << endl << "Diff. Constant (nm^2/s): " << D_d << endl << "Diff. length (nm): " << sqrt(d_d*t_e); R = R1*c_eq/t_e; cout << endl << "Depostion Rate (A/s) --> (BL/s): " << R*a*a*3.14 << "-->" << R*a*a; F = q_eff*e_v; cout << "Electromigration force: " << F << endl; c1 = (1/(2*t_e*D_d*kT))*(t_e*D_d*F+sqrt(t_e*t_e*D_d*D_d*F*F+4*kT*kT*t_e*D_d)); c2 = (1/(2*t_e*D_d*kT))*(t_e*D_d*F-sqrt(t_e*t_e*D_d*D_d*F*F+4*kT*kT*t_e*D_d)); //initial starting surface generation double ter_width_pert; double ter_width_start; char filename[256]; int l_count = 0; double t_max, t_div; int start_time = time(0); int print_count = 0; double print_time; char a_file[256]; cout << endl << "Enter the starting terrace width: "; cin >> ter_width_start; cout << endl << "Enter the standard deviation of the terrace distribution (integer, units: nm): "; cin >> ter_width_pert; cout << endl << "Enter the maximum time step (s): "; cin >> t_max; cout << endl << "Enter the time step division factor (>1): "; cin >> t_div; cout << endl << "Enter the output filename: "; cin >> filename; cout << endl << "Enter the printing/progress report time interval (s, 0 = no printing): "; 144

160 cin >> print_time; srand(1); //by putting time(0), a system clock read, the seed will change from run to run double x_step_initial[10000]; double x[10000]; double tshift; for(i = 0; i<num_steps; i++){ //the step array has num_steps elements running from 0 to num_steps-1 tshift = ter_width_pert*((rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand ()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0 )+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_ma X+1.0)+rand()/(RAND_MAX+1.0))-6.0); while( fabs(tshift) > ter_width_start) tshift = ter_width_pert*((rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand ()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0 )+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_ma X+1.0)+rand()/(RAND_MAX+1.0))-6.0); if(i == 0){ x_step_initial[i] = 0; x[i] = x_step_initial[i]; } else{ x_step_initial[i] = x_step_initial[i-1] + ter_width_start + tshift; x[i] = x_step_initial[i]; } } //end initial surface generation double c_ter_left[10000], c_ter_right[10000], v_step[10000]; double t_advance = t_max; sim_time = 0; t_w = ter_width_start; int s_lim = 0; double wlim = 0; double frac = 0.5; //best value for time step randomization double t_temp = t_max, t_store; x[num_steps] = x[num_steps-1] + t_w; //carry value of "terrace" between last and first step for period b.c.'s char out_file[512]; strcpy( out_file, "progress_"); strcat( out_file, filename ); strcat( out_file, ".txt" ); ofstream prog_file; //damping variables double Damp=0; double count_damp=0; // 145

161 //*******step advance loop***************** while(sim_time < loops){ //****animated/progress file printing***** if(print_time!= 0){ if(floor(sim_time/print_time) == print_count l_count%50000 == 0){ prog_file.open(out_file,ofstream::app); prog_file << setiosflags(ios::fixed ios::showpoint) << filename << endl << "Run time:" << time(0)- start_time << '\t' << "Simulation time:" << sim_time << '\t' << "loops turned:" << l_count << '\t' << "t_advance:" << t_advance << '\t' << "Time division factor:" << t_div << '\t' << "t_step:" << t_advance/t_div << '\t' << s_lim << ":" << wlim << endl; prog_file.close(); if(l_count%50000!= 0){ sprintf(a_file,"%i",print_count); strcat(a_file,filename); strcat(a_file,".txt"); ofstream an_data(a_file, ios::out); for(i=0; i<num_steps; i++){ an_data << setiosflags(ios::fixed ios::showpoint) << i << '\t' << x[i] << '\t' << i*0.314 << '\t' << x[i+1]-x[i] << '\t' << v_step[i] << endl; } an_data.close(); print_count = print_count +1; } } } s_lim = 0; //set limiting step marker to 0 wlim = 0; //***end progress/animated file printing concentrations(x,c_ter_left,c_ter_right); //call function to fill concentrations //fill step velocity array for steps 0...num_steps-1 v_step[0] = a*a*k_p*( -( c_ter_right[num_steps-1] - c_eq*(1+mu[0]/kt) ) - ( c_ter_left[0] - c_eq*(1+mu[0]/kt) ) ); if(v_step[0] >=20) Damp=1; for(z = 1; z<num_steps; z++){ v_step[z] = a*a*k_p*( -( c_ter_right[z-1] - c_eq*(1+mu[z]/kt) ) - ( c_ter_left[z] - c_eq*(1+mu[z]/kt) ) ); if(v_step[z] >= 20) Damp=1; } //end fill step velocity array //optimum time step determination for step advance using relative velocities t_advance = t_max; // set t_advance back to max value if(v_step[num_steps-1]-v_step[0] > 0) t_temp = (t_w)/(v_step[num_steps-1]-v_step[0]+0.001); if(t_temp < t_advance) {t_advance = t_temp; s_lim = num_steps-1;} for(z = 0; z<=num_steps-2; z++){ if(v_step[z]-v_step[z+1] > 0 ) t_temp = (x[z+1]-x[z])/(v_step[z]-v_step[z+1]+0.001); if(t_temp < t_advance) {t_advance = t_temp; s_lim = z; wlim = x[z+1] - x[z];} t_temp = t_max; 146

162 } //randomization of time step if(damp=0){ t_store = t_advance; t_advance = t_store*((((rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(r AND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+ra nd()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+ 1.0)+rand()/(RAND_MAX+1.0))-6.0)*frac)+1); while(t_advance <= 0.2*t_store t_advance >=1.8*t_store) t_advance = t_store*((((rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(r AND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+rand()/(RAND_MAX+1.0)+ra nd()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+1.0)+rand()/(rand_max+ 1.0)+rand()/(RAND_MAX+1.0))-6.0)*frac)+1); } //end randomization of time step if(damp=1 && count_damp<5){ t_advance = t_advance/2; count_damp = count_damp + 1; } if(count_damp=5){ Damp=0; count_damp=0;} if(sim_time >= (loops-20) ) t_div = 100; //end optimum time determination //step reassignment for(z = 0; z<num_steps; z++){ x[z] = x[z] + v_step[z]*(t_advance/t_div); } t_w = t_w + (v_step[0]-v_step[num_steps-1])*(t_advance/t_div); x[num_steps] = x[num_steps-1] + t_w; sim_time = sim_time + t_advance/t_div; l_count = l_count + 1; //end step reassignment } //**********end step advance loop************** //file output ofstream f_data(strcat(filename,".txt"), ios::out); f_data << "data file: " << filename << '\t' << "parameter file: " << i_file << endl << endl; f_data<<"temperature(c):"<<temp_c<<'\t'<<"sub. Rate(BL/s):"<<(c_eq/t_e)*(a*a)<<'\t'<<"Dep. Rate(BL/s):"<<R*(a*a)<<'\t'<<"Net Flux(BL/s):"<< (R-(c_eq/t_e))*(a*a)<<endl; 147

163 f_data << "Step interaction parameter(ev/nm^2):"<<g<<'\t'<<"effective charge(e):"<<q_eff<<"desorption lefetime(s):"<<t_e<<endl; f_data << "diff_length(nm), D_d(nm^2/s), D_d*c_eq(1/s), Mobiltiy(nm^3/s), Kappa(nm/s), d:"<<'\t'<<sqrt(d_d*t_e)<<", "<<D_d<<", "<<D_d*c_eq<<", " <<2*c_eq*a*a*a*a*k_p<<", "<<k_p<<", "<<D_d/k_p<<endl; f_data << "Electromigration Force:"<<F<<'\t'<<"Time of evolution (s):"<<sim_time<<", Run time:"<<time(0)-start_time<<", Num loops:"<<l_count<<endl; f_data << "<----Initial surface---->" << '\t' << "<-Final evolved surface->" << '\t' << "<Terrace width>" <<'\t'<< "<--concentrations-->"<<endl; for(i = 0; i<num_steps; i++){ f_data <<setiosflags(ios::fixed ios::showpoint)<<x_step_initial[i]<<'\t'<<i*0.314<<'\t'<<x[i]<<'\t'<<i*0.314<<'\t'<<x[i+1]-x[i]<<'\t' <<c_ter_left[i]<< ", " <<c_ter_right[i]<<endl; } f_data << endl << endl << "parameter file: " << i_file << endl; ifstream infile(i_file, ios::in); char f_p[512]; while(!infile.eof()){ infile.getline(f_p,512); f_data << f_p << endl; } f_data << "t_min:"<<t_max<<", t_div:"<<t_div<<endl; infile.close(); f_data.close(); //end file output return 0; } //close of int main //function to generate the set of concentrations double concentrations(double x[],double c_ter_left[], double c_ter_right[]){ //chemical potential assignment for concentratoin evaluation mu[0] = 2*g*a*a*h*h*h*( pow(t_w,-3)-pow(x[1]-x[0],-3) ); mu[num_steps-1] = 2*g*a*a*h*h*h*( pow(x[num_steps-1]-x[num_steps-2],-3)-pow(t_w,-3) ); mu[num_steps] = mu[0]; //defined to ensure the concentration for terrace between last and first step can be found for(z = 1; z<=num_steps-2;z++){ mu[z] = 2*g*a*a*h*h*h*( pow(x[z]-x[z-1],-3)-pow(x[z+1]-x[z],-3) ); } //end chemical potential assignment for(z = 0; z<num_steps;z++){ x_pos = x[z]; c_ter_left[z] = (D_d*D_d*kT*exp(c2*x[z])*c2*F*t_e*R- D_d*D_d*kT*F*t_e*R*c2*exp(c2*x[z+1])+D_d*D_d*F*F*t_e*R*exp(c2*x[z+1])- D_d*D_d*exp(c2*x[z])*F*F*t_e*R- D_d*kT*kT*k_p*t_e*R*c2*exp(c2*x[z+1])+D_d*kT*kT*exp(c2*x[z])*c2*k_p*c_eq+D_d*kT*kT*k_p* c_eq*c2*exp(c2*x[z+1])-d_d*kt*kt*exp(c2*x[z])*c2*k_p*t_e*r- D_d*kT*exp(c2*x[z])*F*k_p*c_eq+D_d*kT*exp(c2*x[z])*c2*k_p*c_eq*mu[z+1]- D_d*kT*k_p*c_eq*F*exp(c2*x[z+1])+D_d*kT*k_p*c_eq*mu[z]*c2*exp(c2*x[z+1])- D_d*exp(c2*x[z])*F*k_p*c_eq*mu[z+1]-D_d*k_p*c_eq*mu[z]*F*exp(c2*x[z+1])- kt*kt*k_p*k_p*t_e*r*exp(c2*x[z+1])+kt*kt*exp(c2*x[z])*k_p*k_p*t_e*rkt*kt*exp(c2*x[z])*k_p*k_p*c_eq+kt*kt*k_p*k_p*c_eq*exp(c2*x[z+1])- 148

164 149 kt*exp(c2*x[z])*k_p*k_p*c_eq*mu[z+1]+kt*k_p*k_p*c_eq*mu[z]*exp(c2*x[z+1]))/(d_d*kt*kt*k_p* c2*exp(c2*x[z]+c1*x[z+1])- k_p*k_p*kt*kt*exp(c2*x[z]+c1*x[z+1])+d_d*d_d*kt*kt*c1*c2*exp(c2*x[z]+c1*x[z+1])- D_d*kT*kT*c1*k_p*exp(c2*x[z]+c1*x[z+1])-D_d*D_d*kT*c1*F*exp(c2*x[z]+c1*x[z+1])- D_d*D_d*kT*kT*c1*c2*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*c1*F*exp(c1*x[z]+c2*x[z+1])- D_d*kT*kT*c1*k_p*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*F*c2*exp(c1*x[z]+c2*x[z+1])+D_d*kT*k T*k_p*c2*exp(c1*x[z]+c2*x[z+1])+k_p*k_p*kT*kT*exp(c1*x[z]+c2*x[z+1])- D_d*D_d*kT*F*c2*exp(c2*x[z]+c1*x[z+1])+F*F*D_d*D_d*exp(c2*x[z]+c1*x[z+1])- F*F*D_d*D_d*exp(c1*x[z]+c2*x[z+1]))*exp(c1*x_pos)-(- k_p*k_p*exp(c1*x[z])*kt*kt*c_eq+k_p*k_p*exp(c1*x[z+1])*kt*kt*c_eq- F*D_d*exp(c1*x[z+1])*k_p*c_eq*kT- F*D_d*exp(c1*x[z+1])*k_p*c_eq*mu[z]+D_d*D_d*kT*c1*exp(c1*x[z])*F*t_e*R+D_d*kT*kT*c1*exp( c1*x[z])*k_p*c_eq+k_p*k_p*exp(c1*x[z+1])*kt*c_eq*mu[z]-k_p*k_p*exp(c1*x[z+1])*kt*kt*t_e*r- D_d*D_d*kT*c1*exp(c1*x[z+1])*F*t_e*R-F*F*D_d*D_d*exp(c1*x[z])*t_e*R- F*D_d*exp(c1*x[z])*k_p*c_eq*mu[z+1]+F*F*D_d*D_d*exp(c1*x[z+1])*t_e*R+D_d*kT*kT*c1*exp(c1 *x[z+1])*k_p*c_eq+d_d*kt*c1*exp(c1*x[z+1])*k_p*c_eq*mu[z]- D_d*kT*kT*c1*exp(c1*x[z+1])*k_p*t_e*R- D_d*kT*kT*c1*exp(c1*x[z])*k_p*t_e*R+D_d*kT*c1*exp(c1*x[z])*k_p*c_eq*mu[z+1]- F*D_d*exp(c1*x[z])*k_p*c_eq*kT+k_p*k_p*exp(c1*x[z])*kT*kT*t_e*Rk_p*k_p*exp(c1*x[z])*kT*c_eq*mu[z+1])/(D_d*kT*kT*k_p*c2*exp(c2*x[z]+c1*x[z+1])-k_p*k_p* kt*kt*exp(c2*x[z]+c1*x[z+1])+d_d*d_d*kt*kt*c1*c2*exp(c2*x[z]+c1*x[z+1])- D_d*kT*kT*c1*k_p*exp(c2*x[z]+c1*x[z+1])-D_d*D_d*kT*c1*F*exp(c2*x[z]+c1*x[z+1])- D_d*D_d*kT*kT*c1*c2*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*c1*F*exp(c1*x[z]+c2*x[z+1])- D_d*kT*kT*c1*k_p*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*F*c2*exp(c1*x[z]+c2*x[z+1])+D_d*kT*k T*k_p*c2*exp(c1*x[z]+c2*x[z+1])+k_p*k_p*kT*kT*exp(c1*x[z]+c2*x[z+1])- D_d*D_d*kT*F*c2*exp(c2*x[z]+c1*x[z+1])+F*F*D_d*D_d*exp(c2*x[z]+c1*x[z+1])- F*F*D_d*D_d*exp(c1*x[z]+c2*x[z+1]))*exp(c2*x_pos)+t_e*R; x_pos = x[z+1]; c_ter_right[z] = (D_d*D_d*kT*exp(c2*x[z])*c2*F*t_e*R- D_d*D_d*kT*F*t_e*R*c2*exp(c2*x[z+1])+D_d*D_d*F*F*t_e*R*exp(c2*x[z+1])- D_d*D_d*exp(c2*x[z])*F*F*t_e*R- D_d*kT*kT*k_p*t_e*R*c2*exp(c2*x[z+1])+D_d*kT*kT*exp(c2*x[z])*c2*k_p*c_eq+D_d*kT*kT*k_p* c_eq*c2*exp(c2*x[z+1])-d_d*kt*kt*exp(c2*x[z])*c2*k_p*t_e*r- D_d*kT*exp(c2*x[z])*F*k_p*c_eq+D_d*kT*exp(c2*x[z])*c2*k_p*c_eq*mu[z+1]- D_d*kT*k_p*c_eq*F*exp(c2*x[z+1])+D_d*kT*k_p*c_eq*mu[z]*c2*exp(c2*x[z+1])- D_d*exp(c2*x[z])*F*k_p*c_eq*mu[z+1]-D_d*k_p*c_eq*mu[z]*F*exp(c2*x[z+1])- kt*kt*k_p*k_p*t_e*r*exp(c2*x[z+1])+kt*kt*exp(c2*x[z])*k_p*k_p*t_e*r- kt*kt*exp(c2*x[z])*k_p*k_p*c_eq+kt*kt*k_p*k_p*c_eq*exp(c2*x[z+1])- kt*exp(c2*x[z])*k_p*k_p*c_eq*mu[z+1]+kt*k_p*k_p*c_eq*mu[z]*exp(c2*x[z+1]))/(d_d*kt*kt*k_p* c2*exp(c2*x[z]+c1*x[z+1])- k_p*k_p*kt*kt*exp(c2*x[z]+c1*x[z+1])+d_d*d_d*kt*kt*c1*c2*exp(c2*x[z]+c1*x[z+1])- D_d*kT*kT*c1*k_p*exp(c2*x[z]+c1*x[z+1])-D_d*D_d*kT*c1*F*exp(c2*x[z]+c1*x[z+1])- D_d*D_d*kT*kT*c1*c2*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*c1*F*exp(c1*x[z]+c2*x[z+1])- D_d*kT*kT*c1*k_p*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*F*c2*exp(c1*x[z]+c2*x[z+1])+D_d*kT*k T*k_p*c2*exp(c1*x[z]+c2*x[z+1])+k_p*k_p*kT*kT*exp(c1*x[z]+c2*x[z+1])- D_d*D_d*kT*F*c2*exp(c2*x[z]+c1*x[z+1])+F*F*D_d*D_d*exp(c2*x[z]+c1*x[z+1])- F*F*D_d*D_d*exp(c1*x[z]+c2*x[z+1]))*exp(c1*x_pos)-(- k_p*k_p*exp(c1*x[z])*kt*kt*c_eq+k_p*k_p*exp(c1*x[z+1])*kt*kt*c_eq- F*D_d*exp(c1*x[z+1])*k_p*c_eq*kT- F*D_d*exp(c1*x[z+1])*k_p*c_eq*mu[z]+D_d*D_d*kT*c1*exp(c1*x[z])*F*t_e*R+D_d*kT*kT*c1*exp( c1*x[z])*k_p*c_eq+k_p*k_p*exp(c1*x[z+1])*kt*c_eq*mu[z]-k_p*k_p*exp(c1*x[z+1])*kt*kt*t_e*r- D_d*D_d*kT*c1*exp(c1*x[z+1])*F*t_e*R-F*F*D_d*D_d*exp(c1*x[z])*t_e*R- F*D_d*exp(c1*x[z])*k_p*c_eq*mu[z+1]+F*F*D_d*D_d*exp(c1*x[z+1])*t_e*R+D_d*kT*kT*c1*exp(c1 *x[z+1])*k_p*c_eq+d_d*kt*c1*exp(c1*x[z+1])*k_p*c_eq*mu[z]-

165 D_d*kT*kT*c1*exp(c1*x[z+1])*k_p*t_e*R- D_d*kT*kT*c1*exp(c1*x[z])*k_p*t_e*R+D_d*kT*c1*exp(c1*x[z])*k_p*c_eq*mu[z+1]- F*D_d*exp(c1*x[z])*k_p*c_eq*kT+k_p*k_p*exp(c1*x[z])*kT*kT*t_e*Rk_p*k_p*exp(c1*x[z])*kT*c_eq*mu[z+1])/(D_d*kT*kT*k_p*c2*exp(c2*x[z]+c1*x[z+1])-k_p*k_p *kt*kt*exp(c2*x[z]+c1*x[z+1])+d_d*d_d*kt*kt*c1*c2*exp(c2*x[z]+c1*x[z+1])- D_d*kT*kT*c1*k_p*exp(c2*x[z]+c1*x[z+1])-D_d*D_d*kT*c1*F*exp(c2*x[z]+c1*x[z+1])- D_d*D_d*kT*kT*c1*c2*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*c1*F*exp(c1*x[z]+c2*x[z+1])- D_d*kT*kT*c1*k_p*exp(c1*x[z]+c2*x[z+1])+D_d*D_d*kT*F*c2*exp(c1*x[z]+c2*x[z+1])+D_d*kT*k T*k_p*c2*exp(c1*x[z]+c2*x[z+1])+k_p*k_p*kT*kT*exp(c1*x[z]+c2*x[z+1])- D_d*D_d*kT*F*c2*exp(c2*x[z]+c1*x[z+1])+F*F*D_d*D_d*exp(c2*x[z]+c1*x[z+1])- F*F*D_d*D_d*exp(c1*x[z]+c2*x[z+1]))*exp(c2*x_pos)+t_e*R; } return 0; } //close concentration function The following is the parameter file read by the program above. It is important that the lines are preserved. //parameter table for Si(111) //change only the numerical values! Mathematical operations are not supported 1090 //temperature in degrees C //length of surface unit cell (nm) //step-height (distance between (111) planes) (nm) 0.15 //effective charge of adatom (in unit charge) 15 //step-step interaction parameter (ev/nm^2) -7E-7 //applied electric feild (V/nm) + = up-hill : - = down-hill 0 //Calculated formation energy of adatom (0 sets to 0.2ML)(eV) 5.66e-16 //prefactor for desorption lifetime (s) 4.2 //desorption barrier (ev) 3.26e8 //prefactor for diffusion (nm^2/s) 0 //activation energy for diffusion (ev) 1 //depostion rate (A/s) 1.63e8 //kappa 150

166 The following is a screen shot showing how the program is run in the windows environment. Care must be taken when entering the name of the parameter file to read. 151