Step-step Repulsion Research Articles

Burton-Cabrera-Frank theory for surfaces with alternating step types

Burton-Cabrera-Frank (BCF) theory has proven to be a versatile framework to relate surface morphology and dynamics during crystal growth to the underlying mechanisms of adatom diffusion and attachment at steps. For an important class of crystal surfaces, including the basal planes of hexagonal close-packed and related systems, the steps in a sequence on a vicinal surface can exhibit properties that alternate from step to step. Here we develop BCF theory for such surfaces, relating observables such as alternating terrace widths as a function of growth conditions to the kinetic coefficients for adatom attachment at steps. We include the effects of step transparency and step-step repulsion. A general solution is obtained for the dynamics of the terrace widths, assuming quasi-steady-state adatom distributions on the terraces. An explicit simplified analytical solution is obtained under widely applicable approximations. From this we obtain expressions for the full-steady-state terrace fraction as a function of growth rate. Fits of the theoretical predictions to recent experimental determinations of the steady state and dynamics of terrace fractions on GaN (0001) surfaces during organometallic vapor phase epitaxy give values of the kinetic coefficients for this system. In Appendixes, we also connect a model for diffusion between kinks on steps to the model for diffusion between steps on terraces, which quantitatively relates step transparency to the kinetics of atom attachment at kinks, and consider limiting cases of diffusion-limited, attachment-limited, and mixed kinetics.

On the kinetics of the early stage of growth of III-V nanowires

The Au-assisted III-V nanowire growth is often preceded by the formation of pyramid-like pedestals under catalyst droplets during pre-growth annealing. In the present paper, we propose a kinetic step-flow model of the transition from the growth of pedestals to the formation of nanowires with constant diameters. In our model an atomic step at the pedestal sidewall appears when the edge of the 2D island growing at the liquid-solid interface under the droplet attains the triple phase line. The steps move along the sidewall downward to the pedestal base at the expense of adatoms attaching to the step edges from adjacent terraces. The analytical expressions for the step velocities yielding a set of equations for time-dependent positions of steps are obtained. It is shown that the transition to nanowire growth might take place as the result of the step bunching at the pedestal top when the upper terrace becomes shorter than a certain critical value at which the step-step repulsion is significant. The numerical simulations of the pedestal growth demonstrate a rapid transition to nanowire growth at low temperatures and the absence of such a transition at high temperatures, which is in accordance with the experimental observations.

Step bunching and macrostep formation in 1D atomistic scale model of unstable vicinal crystal growth

We devise a new 1D atomistic scale model of vicinal growth based on Cellular Automaton. In it the step motion is realized by executing the automaton rule prescribing how adatoms incorporate into the vicinal crystal. Time increases after each rule execution and then nDS diffusional updates of the adatoms are performed. The increase of nDS switches between the diffusion-limited (DL, nDS=1) and kinetics-limited (KL, nDS>>1) regimes of growth. We study the unstable step motion by employing two alternative sources of instability – biased diffusion and infinite inverse Ehrlich-Schwoebel barrier (iiSE). The resulting step bunches consist of steps but also of macrosteps since there is no step-step repulsion incorporated explicitly into the model. This complex pattern formation is quantified by studying the time evolution of the bunch size N and macrostep size Nm in order to find the proper parameter combinations that rescale the time and thus to obtain the full time-scaling relations including the pre-factors. For the case of biased diffusion the time-scaling exponent β of N is 1/2 while for the case of iiSE it is 1/3. In both cases, the time-scaling exponent βm of Nm is ~3β/4 in the DL regime and 3β/5 in the KL one.

Experimental quantitative study into the effects of electromigration field moderation on step bunching instability development on Si(111)

We experimentally studied the effects of a moderated electromigration field on the dynamics of the step bunching process on the Si(111) surface at 1130 \ifmmode^\circ\else\textdegree\fi{}C (regime II) and 1270 \ifmmode^\circ\else\textdegree\fi{}C (regime III). The surfaces with step bunch morphologies were created by annealing vicinal Si(111) at fixed temperatures while the applied electric field $E$ was adjusted for every experiment. Scaling relations, ${y}_{m}~{h}^{\ensuremath{\alpha}}{E}^{q}$, between the slope of a step bunch ${y}_{m}$, step bunch height $h$, and electromigration field $E$ were experimentally probed. Scaling exponents $\ensuremath{\alpha}$ \ensuremath{\approx} 2/3 and $q$ \ensuremath{\approx} 1/3 were extracted from the step bunch morphologies created by annealing Si(111) in the regime III (1270 \ifmmode^\circ\else\textdegree\fi{}C), which are in good agreement with the predictions of the generalized BCF theory. Scaling exponents $\ensuremath{\alpha}$ \ensuremath{\approx} 3/5 and $q$ \ensuremath{\approx} 1/3 were extracted from the morphologies created by annealing in regime II (1130 \ifmmode^\circ\else\textdegree\fi{}C). This result was compared to the scaling relations derived within the frame of the transparent step model, which correctly predicts the formation of the step bunching instability by step-up adatom electromigration. The scaling relation obtained by experiment was found to differ from the model predictions. We measured values of critical electric field (${E}_{\mathrm{cr}}$), i.e., minimum electric field required for the step bunching to take place. A relatively weak field of $E$ > 0.5 V/cm was found to be sufficient to initiate the step bunching process in regime II. This contrasts with regime III, where ${E}_{\mathrm{cr}}=1.0$ and 2.0 V/cm were measured for Si miscut from the (111) plane by 1.1\ifmmode^\circ\else\textdegree\fi{} and 2.5\ifmmode^\circ\else\textdegree\fi{}, respectively. The increased values of ${E}_{\mathrm{cr}}$ were attributed to the enhanced step-step repulsion in regime III. The theoretically predicted formation of compressed step density waves was observed upon annealing in both regimes with $E<{E}_{\mathrm{cr}}$.

Layer-by-layer nucleation mechanism for quantum dot formation in strained heteroepitaxy

We study the spontaneous formation of quantum dots in the form of three-dimensional (3D) islands on facetted surfaces in heteroepitaxy. Island development from fast kinetic Monte Carlo (KMC) simulations at low deposition rates is found to follow a layer-by-layer nucleation pathway characterized by energetics driven continuous lateral expansion interrupted by a sequence of independent two-dimensional (2D) upper-layer nucleation events. The process involves only unstable 2D upper-layer nuclei but no unstable 3D nucleus. We have calculated analytically the elastic strain energy of an island in the form of an axisymmetric stepped mound using a small-slope approximation. The total free energy of a system with a 3D island and an adatom bath is obtained. Our theory explains island formation via a free energy driven layer-by-layer nucleation mechanism. Upper-layer nucleation energy barrier, nucleation time, critical radius, and island step spacings are estimated. The relevance of entropic step-step repulsion is discussed. Our theory satisfactorily explains the 3D KMC simulations and may describe the initial evolution of islands in the form of stepped mounds observed in experiments.

Scaling and universality in models of step bunching: the “C+–C-” model

We recently introduced a novel model of step flow crystal growth – the so-called “C+–C-” model [B. Ranguelov et al., C.R. Acad. Bulgare Sci. 60, 389 (2007)]. In this paper we aim to develop a complete picture of the model’s behaviour in the framework of the notion of universality classes. The basic assumption of the model is that the reference (“equilibrium”) densities used to compute the supersaturation might be different on either side of a step, so CL/CR ≠ 1 (L/R stands for left/right in a step train descending from left to right), and that this will eventually cause destabilization of the regular step train. Linear stability analysis considering perturbation of the whole step train shows that the vicinal is always unstable when the condition CL /CR >1 is fulfilled. Numerical integration of the equations of step motion combined with an original monitoring scheme(s) results in obtaining the exact size- and time- scaling of the step bunches in the limit of long times (including the numerical prefactors). Over a broad range of parameters the surface morphology is characterized by the appearance of the minimal interstep distance at the beginning of the bunches (at the trailing edge of the bunch) and may be described by a single universality class, different from those already generated by continuum theories [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002), J. Krug et al., Phys. Rev. B 71, 045412 (2005)]. In particular, the scaling of the minimal interstep distance lmin in the new universality class is shown to be lmin = (Sn /N)1/(n+1), where N is the number of steps in the bunch, n is the exponent in the step-step repulsion law U ~ 1/d0 n for two steps placed a distance d0 apart and Sn is a combination of the model parameters. It is also shown that N scales with time with universal exponent 1/2 independent of n. For the regime of slow diffusion it is obtained for the first time that the time scaling depends only on the destabilization parameter CL/CR. The bunching outside the parameter region where the above scaling exists cannot be assigned to a specific universality class and thus should be considered non-universal.

Step-step interaction on vicinal Si(001) surfaces studied by scanning tunneling microscopy

We report on measurements of step-step interaction on a flat Si(111)−(7×7) surface and on vicinal Si(001) surfaces with miscut angles ranging between 0.2° and 8°. Starting from scanning tunneling microscopy images of these surfaces and describing steps profile and interactions by the continuum step model, we measured the self-correlation function of single steps and the distribution of terrace widths. Empirical parameters, such as step stiffness and step-step interaction strength, were evaluated from the images. The present experiment allows to assess the dependence of the step-step repulsion on miscut angle, showing how parameters drawn from tunneling images can be used to interpolate between continuum mesoscopic models and atomistic calculations of vicinal surfaces.

Evaporation and growth of crystals: Propagation of step-density compression waves at vicinal surfaces

We studied the step dynamics during crystal sublimation and growth in the limit of fast surface diffusion and slow kinetics of atom attachment-detachment at the steps. For this limit we formulate a model free of the quasi-static approximation in the calculation of the adatom concentration on the terraces at the crystal surface. Such a model provides a relatively simple way to study the linear stability of a step train in a presence of step-step repulsion and an absence of destabilizing factors (as Schwoebel effect, surface electromigration etc.). The central result is that a critical velocity of the steps in the train exists which separates the stability and instability regimes. When the step velocity exceeds its critical value the plot of these trajectories manifests clear space and time periodicity (step density compression waves propagate on the vicinal surface). This ordered motion of the steps is preceded by a relatively short transition period of disordered step dynamics.

Step-position distributions and the pairwise Einstein model for steps on crystal surfaces

The Pairwise Einstein Model (PEM) of steps not only justifies the use of the Generalized Wigner Distribution (GWD) for Terrace Width Distributions (TWDs), it also predicts a specific form for the Step Position Distribution (SPD), i.e., the probability density function for the fluctuations of a step about its average position. The predicted form of the SPD is well approximated by a Gaussian with a finite variance. However, the variance of the SPD measured from either real surfaces or Monte Carlo simulations depends on $\Delta y$, the length of step over which it is calculated, with the measured variance diverging in the limit $\Delta y \to \infty$. As a result, a length scale $L_{\rm W}$ can be defined as the value of $\Delta y$ at which the measured and theoretical SPDs agree. Monte Carlo simulations of the terrace-step-kink model indicate that $L_{\rm W} \approx 14.2 \xi_Q$, where $\xi_Q$ is the correlation length in the direction parallel to the steps, independent of the strength of the step-step repulsion. $L_{\rm W}$ can also be understood as the length over which a {\em single} terrace must be sampled for the TWD to bear a "reasonable" resemblence to the GWD.

Scaling properties of step bunches induced by sublimation and related mechanisms

This work provides a ground for a quantitative interpretation of experiments on step bunching during sublimation of crystals with a pronounced Ehrlich-Schwoebel (ES) barrier in the regime of weak desorption. A strong step bunching instability takes place when the kinetic length is larger than the average distance between the steps on the vicinal surface. In the opposite limit the instability is weak and step bunching can occur only when the magnitude of step-step repulsion is small. The central result are power law relations of the between the width, the height, and the minimum interstep distance of a bunch. These relations are obtained from a continuum evolution equation for the surface profile, which is derived from the discrete step dynamical equations for. The analysis of the continuum equation reveals the existence of two types of stationary bunch profiles with different scaling properties. Through a mathematical equivalence on the level of the discrete step equations as well as on the continuum level, our results carry over to the problems of step bunching induced by growth with a strong inverse ES effect, and by electromigration in the attachment/detachment limited regime. Thus our work provides support for the existence of universality classes of step bunching instabilities [A. Pimpinelli et al., Phys. Rev. Lett. 88, 206103 (2002)], but some aspects of the universality scenario need to be revised.

Spontaneous Formation of Ridges on Patterned Mesas and Their Role in the Evolution of

ABSTRACTMesa structures fabricated on Si(111) surfaces have been found experimentally to develop step arrays with large spacing of the order of a micron or more after annealing at temperatures where sublimation becomes important. Ridges around the edges initially develop during annealing and form barriers to step motion before eventually breaking down. This produces an array of steps of the same sign with a few wide terraces. Computer simulations using one dimensional Burton, Cabrera and Frank (BCF) theory including attachment-detachment rates and step-step repulsion for this configuration show that the terraces evolve under different dynamics depending on the terrace widths. For large terrace widths, sublimation dominates the step dynamics and the Ehrlich-Schwoebel effect is negligible. Sinusoidal terrace width distributions result in this case. The experimentally measured step distribution has such a sinusoidal shape suggesting that the step dynamics is sublimation dominated on the mesas after ridge breakdown.

Spontaneous Formation of Ridges on Patterned Mesas and Their Role in the Evolution of Step Arrays

AbstractMesa structures fabricated on Si(111) surfaces have been found experimentally to develop step arrays with large spacing of the order of a micron or more after annealing at temperatures where sublimation becomes important. Ridges around the edges initially develop during annealing and form barriers to step motion before eventually breaking down. This produces an array of steps of the same sign with a few wide terraces. Computer simulations using one dimensional Burton, Cabrera and Frank (BCF) theory including attachment-detachment rates and step-step repulsion for this configuration show that the terraces evolve under different dynamics depending on the terrace widths. For large terrace widths, sublimation dominates the step dynamics and the Ehrlich-Schwoebel effect is negligible. Sinusoidal terrace width distributions result in this case. The experimentally measured step distribution has such a sinusoidal shape suggesting that the step dynamics is sublimation dominated on the mesas after ridge breakdown.

Si(111) step fluctuations at high temperature: Anomalous step-step repulsion

CRMC2-CNRS, Campus Luminy, Case 913, F-13288 Marseille Cedex 9, France~Received 18 March 2002; revised manuscript received 28 June 2002; published 12 September 2002!Using reflection electron microscopy we examine the step fluctuations of Si ~111! at 1100 °C. Evaporation iscompensated by a replenishing flux. The step fluctuation behavior is qualitatively similar to that at 900 °C~where sublimation is negligible!, with unexplained quantitative differences. We focus on the three parametersof the step continuum model of vicinals. The step stiffness scales with an increase in T from 900 °C aspredicted by an appropriate lattice model. The kinetic coefficient is larger than scaling of the parameters from900 °C would predict. The step-step correlations are assessed in traditional and novel ways; step repulsions areat least 6 times as strong as predicted from lower temperatures, suggesting nonequilibrium effects probably dueto electromigration.DOI: 10.1103/PhysRevB.66.115310 PACS number~s!: 05.40.2a, 05.70.Np, 68.35.Md, 68.37.2dI. INTRODUCTION

Electric-current-induced step bunching on Si(111)

We experimentally investigated step bunching induced by direct current on vicinal $\mathrm{Si}(111)``1\ifmmode\times\else\texttimes\fi{}1''$ surfaces using scanning electron microscopy and atomic force microscopy. The scaling relation between the average step spacing ${l}_{b}$ and the number of steps N in a bunch, ${l}_{b}\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\alpha}},$ was determined for four step-bunching temperature regimes above the $7\ifmmode\times\else\texttimes\fi{}7\ensuremath{-}``1\ifmmode\times\else\texttimes\fi{}1''$ transition temperature. The step-bunching rate and scaling exponent differ between neighboring step-bunching regimes. The exponent \ensuremath{\alpha} is 0.7 for the two regimes where the step-down current induces step bunching (860\char21{}960 and 1210\char21{}1300 \ifmmode^\circ\else\textdegree\fi{}C), and 0.6 for the two regimes where the step-up current induces step bunching (1060\char21{}1190 and g1320 \ifmmode^\circ\else\textdegree\fi{}C). The number of single steps on terraces also differs in each of the four temperature regimes. For temperatures higher than 1280 \ifmmode^\circ\else\textdegree\fi{}C, the prefactor of the scaling relation increases, indicating an increase in step-step repulsion. The scaling exponents obtained agree reasonably well with those predicted by theoretical models. However, they give unrealistic values for the effective charges of adatoms for step-up-current-induced step bunching when the ``transparent'' step model is used.

Can one have preroughening of vicinal surfaces?

We discuss the possibility that, besides roughening, a vicinal surface could display preroughening (PR), and consider the possible mechanisms for its promotion. Within the framework of a terrace-step-kink model, it turns out that a PR transition is possible, and could be induced by a short-range repulsion between parallel kinks along the same step or on adjacent steps, or even by some kind of extended range step-step repulsion. We discuss the possible relevance of this phenomenon to the anomalous roughening behaviour recently reported for Ag(115).

Properties and dynamic interaction of step density waves at a crystal surface during electromigration affected sublimation

Electromigration affected sublimation is a complicated phenomenon, involving surface transport coupled to a process of atom exchange between the two-dimensional gas of adatoms and the crystal phase. The case of intensive exchange is theoretically treated and equations of step motion are derived for the case of ``nontransparent'' steps (kinetics with local conservation of adatoms). The numerical integration of these equations manifests step bunching (a formation of step density waves) at step-down direction of the electromigration of adatoms. We studied some properties of the step density waves: the amplitude (the maximum slope of the bunch) and its dependence on the number of steps in the bunch, the kinematic wave velocity and the dynamic interaction of waves of different amplitudes. The central result of this work is the dependence of the minimum interstep distance (in the steady state shape of the bunch) on the model parameters. This dependence, extracted from numerical study, is presented in terms of scaling laws ${l}_{\mathrm{min}}\ensuremath{\sim}{N}^{\ensuremath{-}r}{(A/F)}^{q}$, where $N$ is the number of steps in the bunch, $A$ is the magnitude of step-step repulsion, and $F$ is the force, inducing electromigration of the adatoms. Both scaling exponents $r$ and $q$ depend on the power $n$ in the step-step repulsion dependence on the interstep distance ${(U=A/l}^{n})$ and, therefore, they are a key to the problem of experimental evaluation of $n$. A striking result of this model is the constant value of ${l}_{\mathrm{min}}$ in a wide range of values of the average diffusion distance ${\ensuremath{\lambda}}_{s}.$ Thus one cannot relate the temperature dependence of ${l}_{\mathrm{min}}$ to the temperature dependence of ${\ensuremath{\lambda}}_{s}.$ Numerical analysis of the dynamics of steps at a crystal surface of small misorientation angle reveals two types of dynamic interaction of bunches of steps: ``bunch size exchange'' and ``effective coalescence.'' The former type of interaction is rather interesting --- a smaller (and faster) bunch approaches a larger one and they travel together until the initially larger bunch achieves (by losing steps) a size, smaller than the size of its partner, and runs away of it.

Effects of Dynamic Step-Step Repulsion and Autocatalysis on the Morphology of Etched Si(111) Surfaces

Steady-state morphologies of ${\mathrm{NH}}_{4}\mathrm{F}$-etched Si(111) surfaces were investigated with scanning tunneling microscopy and kinetic Monte Carlo simulations. In stirred solutions, dynamic step-step repulsion, which is mediated by small etch pits, leads to the production of surfaces that are significantly smoother than the equilibrium limit as evidenced by terrace width distributions. In unstirred solutions, surfaces with a low density of steps show profound step bunching, while those with a high density of steps do not. This effect is attributed to step-step attractions mediated by spatially localized autocatalytic reactions.

Step structures on Br-chemisorbed vicinal Si(111)

To investigate the effects of chemisorption on step energetics, Si surfaces miscut by 1.2° to 12° from the (111), as well as Si(335) and Si(112), have been exposed to bromine, and studied using scanning tunneling microscopy. As the atomic structure on (111) terraces changes from the (7 × 7) or (5 × 5) reconstruction to a (1 × 1) after Br exposure and annealing at 800-900 K, the density of triple-layer steps on samples miscut towards [211] decreases dramatically. The structures of the steps themselves have been measured with high resolution. The edges of a triple step on the Br-covered surfaces resembles a (112) facet, while it is (113)-like on a clean surface. Br-chemisorption extends the miscut angle range over which the sample can be described as a vicinal Si(111) surface from ≤14° for clean Si to ∼19°. Analysis of kink densities shows that step diffusivity on Br-covered surfaces is governed by the intrinsic step misorientation as on the clean surfaces, however the kink size is registered to the 1 × 1 lattice on the Br-covered surface. The measured terrace width distributions are the same on the clean and Br-covered surfaces, indicating the same ratio of step diffusivity to direct step-interaction energy. The coexistence of single and triple steps on both the clean and Br-chemisorbed surfaces can be described semi-quantitatively with a model based on elasticity theory. The analysis shows that the reduced triple-step density is due to both the seven-fold decrease in step-step repulsion energy and the seven-fold increase in triple-step formation energy after Br-chemisorption. On the equilibrium step-bunched structures that form on surfaces miscut towards [211], removal of the 7 × 7 reconstruction during halogen exposure and annealing below 900 K does not break the step-bunching.

Dynamics of surface profile evolution through surface diffusion.

We have studied the evolution, mediated by surface diffusion, of periodic corrugations on a surface using Monte Carlo simulations on a solid-on-solid (SOS) model. Above the roughening temperature ${\mathit{T}}_{\mathit{R}}$, for both unidirectional and bidirectional sinusoidal corrugations of wavelength L, the amplitude h decays exponentially with time h/${\mathit{h}}_{0}$\ensuremath{\approxeq}exp(-\ensuremath{\alpha}t), with t/${\mathit{L}}^{4}$ scaling in agreement with Herring-Mullins theory. Below the roughening temperature, there is a gradual transition to a power-law decay of the amplitude as the temperature is lowered. The wavelength scaling varies with the substrate temperature and the periodicity of the corrugation in the two orthogonal transverse directions. Well below ${\mathit{T}}_{\mathit{R}}$, the amplitude in unidirectional sinusoidal corrugations evolves with time according to h/${\mathit{h}}_{0}$\ensuremath{\approxeq}(1+\ensuremath{\lambda}t${)}^{\mathrm{\ensuremath{-}}1}$, with t/${\mathit{L}}^{5}$ scaling for diffusion-limited kinetics, in agreement with the theory of Ozdemir and Zangwill [Phys. Rev. B 42, 5013 (1990)]. In bidirectional sinusoidal corrugations, profile decay is driven by a combination of line-tension and step-step entropic repulsion, in agreement with the theory of Rettori and Villain [J. Phys. (Paris) 49, 257 (1988)]. \textcopyright{} 1996 The American Physical Society.

Phase separation and step roughening of vicinal Si(111): An x-ray-scattering study.

We present a high-resolution x-ray-scattering study of the temperature-dependent structure of Si(111) vicinally miscut by 4\ifmmode^\circ\else\textdegree\fi{} along the 〈11\ifmmode\bar\else\textasciimacron\fi{}0〉 direction. At temperatures below 1159 K, the surface phase separates into flat 7\ifmmode\times\else\texttimes\fi{}7 terraces and densely stepped regions, in agreement with previous reports. The angle between the phase-separated regions shows a temperature dependence consistent with both entropic and strain-induced step-step repulsive interactions. At temperatures above 1159 K, the surface is shown to exist as a single, logarithmically rough phase; the observed scattering line shapes demonstrate the presence of a long-range step-step repulsion of a magnitude comparable to the entropic repulsion.