Step-step Interactions Research Articles

Emergence of local geometric laws of step flow in homoepitaxial growth.

Below the roughening transition, crystal surfaces exhibit nanoscale line defects, steps, that move by exchanging atoms with their environment. In homoepitaxy, we analytically show how the motion of a step train in vacuum under strong desorption can be approximately described by nonlinear laws that depend on local geometric features such as the curvature of each step, as well as suitably defined effective terrace widths. We assume that each step edge, a free boundary, can be represented by a smooth curve in a fixed reference plane for sufficiently long times. Besides surface diffusion and evaporation, the processes under consideration include kinetic step-step interactions in slowly varying geometries, material deposition on the surface from above, attachment and detachment of atoms at steps, step edge diffusion, and step permeability. Our methodology relies on boundary integral equations for the adatom fluxes responsible for step flow. By applying asymptotics, which effectively treat the diffusive term of the free boundary problem as a singular perturbation, we describe an intimate connection of universal character between step kinetics and local geometry.

Step-bunching instability of growing interfaces between ice and supercooled water

SignificanceStep-bunching instability (SBI) is one of the interfacial instabilities driven by self-organization of elementary step flow associated with crystal-growth dynamics, which has been observed in diverse crystalline materials. However, despite theoretical suggestions of its presence, no direct observations of SBI for simple melt growth have been achieved so far. Here, with the aid of a type of optical microscope and its combination with a two-beam interferometer, we realized quantitative in situ observations of the spatiotemporal dynamics of the SBI. This enables us to examine the origin of the SBI at the level of the step-step interaction. We also found that the SBI spontaneously induces a highly stable spiral growth mode, governing the late stage of the growth process.

Revisiting step instabilities on crystal surfaces. Part II: General theory

The quasistatic approximation is a useful but questionable simplification for analyzing step instabilities during the growth/evaporation of vicinal surfaces. Using this approximation, we characterized in Part I of this work the effect on stability of different mechanisms and their interplay: elastic step-step interactions, the Schwoebel barrier, and the chemical coupling of the diffusion fields on adjacent terraces. In this second part, we present a stability analysis of the general problem without recourse to the quasistatic approximation. This analysis reveals the existence of a supplementary mechanism, which we label the “dynamics effect” as it follows from accounting for all the convective and transient terms in the governing equations. This effect can be stabilizing or destabilizing depending on the ratio of step attachment/detachment kinetics to terrace diffusion kinetics. Further, we find that this dynamics effect remains significant in the slow deposition/evaporation regime, thereby invalidating the classical postulate underlying the quasistatic approximation. Finally, revisiting experiments of crystal growth on Si(111)-7 × 7 and GaAs(001), our analysis provides an alternative explanation of the observed step bunching, one that does not require the mechanisms previously invoked in the literature.

Slope selection in unstable multilayer growth in 1+1 dimensions: Step flow models with downward funneling.

We study analytically and numerically aspects of the dynamics of slope selection for one-dimensional models describing the motion of line defects, steps, in homoepitaxial crystal growth. The kinetic processes include diffusion of adsorbed atoms (adatoms) on terraces, attachment and detachment of atoms at steps with large yet finite, positive Ehrlich-Schwoebel step-edge barriers, material deposition on the surface from above, and the mechanism of downward funneling (DF) via a phenomenological parameter. In this context, we account for the influence of boundary conditions at extremal steps on the dynamics of slope selection. Furthermore, we consider the effect of repulsive, nearest-neighbor force-dipole step-step interactions. For geometries with straight steps, we carry out numerical simulations of step flow, which demonstrate that slope selection eventually occurs. We apply perturbation theory to characterize time-periodic solutions of step flow for slope-selected profiles. By this method, we show how a simplified step flow theory with constant probabilities for the motion of deposited atoms can serve as an effective model of slope selection in the presence of DF. Our analytical findings compare favorably to step simulations.

Role of step stiffness and kinks in the relaxation of vicinal (001) with zigzag [110] steps

We present a kinetic Monte Carlo study of the relaxation dynamics and steady state configurations of 〈110〉 steps on a vicinal (001) simple cubic surface. This system is interesting because 〈110〉 (fully kinked) steps have different elementary excitation energetics and favor step diffusion more than 〈100〉 (nominally straight) steps. In this study we show how this leads to different relaxation dynamics as well as to different steady state configurations, including that 2-bond breaking processes are rate determining for 〈110〉 steps in contrast to 3-bond breaking processes for 〈100〉−steps found in previous work [Surface Sci. 602, 3569 (2008)]. The analysis of the terrace-width distribution (TWD) shows a significant role of kink-generation-annihilation processes during the relaxation of steps: the kinetic of relaxation, toward the steady state, is much faster in the case of 〈110〉−zigzag steps, with a higher standard deviation of the TWD, in agreement with a decrease of step stiffness due to orientation. We conclude that smaller step stiffness leads inexorably to faster step dynamics towards the steady state. The step-edge anisotropy slows the relaxation of steps and increases the strength of step-step effective interactions.

Step-step interactions on GaAs (110) nanopatterns

The step-step interactions on vicinal GaAs (110) surface patterns have been extracted from the quantitative analysis of the terrace width distribution (TWD). We have specifically studied the interactions in near-equilibrium faceting and kinetics-driven step bunching and meandering formed by spontaneous self-organization or through the modification of GaAs growth kinetics by atomic hydrogen. We show that the experimental TWDs determined from atomic force microscopy measurements can be accurately described by a weighed sum of a generalized Wigner distribution and several Gaussians. The results of our calculations indicate that straight facets are formed during high temperature homoepitaxy due to attractive interactions between [11¯0] steps. At low temperatures, steady state attractive interactions in [11¯0] step bunches are preceded by a transition regime dominated by entropic and energetic repulsions between meandering [11¯n]-type steps (n ≥ 2), whose population density exceeds that of the [11¯0] bunched steps. In addition, it has been found that atomic H reduces the attractive interactions between [11¯0] bunched steps and enhances entropic and dipole-induced energetic repulsions between H-terminated [11¯n] steps through the inhibition of As–As bond formation at step edges. Our analysis has evidenced a correlation between the value of the adjustable parameter that accounts in our model for the specific weight of the secondary peaks in the TWD (β) and the extent of transverse meandering on the vicinal surface.

Non-conserved dynamics of steps on vicinal surfaces during electromigration-induced step bunching

We report new results on the non-conserved dynamics of parallel steps on vicinal surfaces in the case of sublimation with electromigration and step-step interactions. The derived equations are valid in the quasistatic approximation and in the limit $f^{-1}\gg l_D\gg l_{\pm} \gg l_i$, where $f$ is the inverse electromigration length, $l_D$ the diffusion length, $l_{\pm}$ the kinetic lengths and $l_i$ the terrace widths. The coupling between crystal sublimation and step-step interactions induces non-linear, non-conservative terms in the equations of motion. Depending on the initial conditions, this leads to interrupted coarsening, anticoarsening of step bunches or periodic switching between step trains of different numbers of bunches.

Anticoarsening and complex dynamics of step bunches on vicinal surfaces during sublimation

A sublimating vicinal crystal surface can undergo a step bunching instability when the attachment-detachment kinetics is asymmetric, in the sense of a normal Ehrlich-Schwoebel effect. Here we investigate this instability in a model that takes into account the subtle interplay between sublimation and step-step interactions, which breaks the volume-conserving character of the dynamics assumed in previous work. On the basis of a systematically derived continuum equation for the surface profile, we argue that the nonconservative terms pose a limitation on the size of emerging step bunches. This conclusion is supported by extensive simulations of the discrete step dynamics, which show breakup of large bunches into smaller ones as well as arrested coarsening and periodic oscillations between states with different numbers of bunches.

Elastic displacements and step interactions on metallic surfaces: Grazing-incidence x-ray diffraction andab initiostudy of Au(332)

We have studied the energetics, relaxation, and interactions of steps on the Au(332) vicinal surface using a combination of grazing incidence x-ray diffraction, anisotropic linear elasticity theory, and ab initio density-functional theory. We find that the initial force distribution on a bulk-truncated surface, as well as the resulting pattern of atomic relaxations, can be reproduced excellently by a buried dipole elastic model. The close agreement obtained between experimental and calculated x-ray diffraction profiles allows us to precisely determine the value of the elastic dipole density at the steps. We also use these results to obtain an experimental estimate of the surface stress on an unreconstructed Au(111) facet, ${\ensuremath{\tau}}_{\text{Au}(111)}=2.3\ifmmode\pm\else\textpm\fi{}0.4\text{ }{\text{Nm}}^{\ensuremath{-}1}$, and the value of the step-step elastic interaction energy $A=950\ifmmode\pm\else\textpm\fi{}150\text{ }\text{meV}\text{ }\text{\AA{}}$.

Terrace-width distributions of touching steps: Modification of the fermion analogy with implications for measuring step-step interactions

Using Monte Carlo simulations, we compute the terrace-width distributions (TWDs) of surfaces in which steps can touch each other, forming multiple-atomic height steps, but cannot cross (no overhangs), and so inconsistent with the standard mapping to spinless fermions. Our results show that the generalized Wigner distribution with minor modifications at small step separations, gives a very good fit for TWDs of touching steps. The interaction strength derived from the fit parameter $(\ensuremath{\varrho})$ indicates an effective attraction between steps. The strength of this effective attraction decreases for larger mean-step separations and decreasing step-touching energies; describable via finite-size scaling. Hence, accurate extraction of the true repulsion strength requires multiple vicinalities.

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.

Effect of length scales in directing step bunch self-organization during annealing of patterned vicinal Si(111) surfaces: Comparison with a simple near-equilibrium model

We compare the experimental observations of the effect of changing the length scale of an artificially imposed pattern on self-organization during annealing of stepped Si(111) surfaces with the predictions of the simple quasi-one-dimensional near-equilibrium model proposed by Weeks et al. [Dynamics of Crystal Surfaces and Interfaces, edited by P. Duxbury and T. Spence (Plenum, New York, 1997), pp. 199--216] in which step stiffness competes with the effect of repulsive step-step interactions. We find that this model reproduces the observed convergence of in-plane and out-of-plane relaxation at large length scales, but faster relaxation in plane at small periods. Simulations based on this model require step energetic parameters which dramatically depart from their equilibrium values to obtain agreement with the observed pattern period at which relaxation becomes isotropic. We find that this model does not reproduce the observed asymptotic quartic dependence of the relation time on spatial period.

Contact Angle Singularity in 4He Crystals

Equilibrium shape of the interface between superfluid and crystalline 4He near the (0001) orientation is analyzed. We observe a singular dependence of the contact angle on the wall inclination with respect to the gravity. The energy of a step on the basal plane is measured. From the analysis of the meniscus profile the step-step interaction constant is estimated.

Vicinal silicon surfaces: From step density wave to faceting

We investigate faceting mechanisms induced by electromigration in the regime where atomic steps are transparent. For this purpose we study several vicinal orientations by means of in situ (optical diffraction, electronic microscopy) as well as ex situ (atomic force microscopy, microprofilometry) visualization techniques. The data show that faceting proceeds in two stages. The first stage is short and leads to the appearance of a step density wave, with a wavelength roughly independent of the surface orientation. The second stage is much slower and leads to the formation of a hill-and-valley structure, the period of which depends on the initial surface orientation. A simple continuum model enables us to point out why the wavelength of the step density wave does not depend on the microscale details of the surface. The final wavelength is controlled by the competition between elastic step-step interaction and facet edge energy cost. Finally, the surface stress angular dependence is shown to emerge as a coarsed-grained picture from the step model.

Growth Mode Diagram for the Epitaxial Growth on the Vicinal Surface of Strained Si (001)

We present the linear stability analysis for the epitaxial thin film growth on the vicinal surface of strained Si and the growth mode diagrams of the epitaxial growth under various operation conditions. Competition between step-step elastic interactions and the asymmetry of incorporation of adatoms from the terraces to step edge is considered. Force monopoles at steps and their interaction lead to it on the vicinal surface while kinetic asymmetry of the adatom incorporation at steps due to Ehrlich-Schwoebel barrier prevents the step bunching instability. Growth mode on the vicinal surface is determined by the competition between elastic step-step interactions and Ehrlich-Schwoebel barrier.

Metal organic vapor phase epitaxy of BiSbTe3 films on (001) GaAs vicinal substrates

We have investigated the growth of BiSbTe3 films by metal organic vapor phase epitaxy on (001) GaAs vicinal substrates. A detailed experimental study of the surface morphologies of BiSbTe3 films grown on GaAs (001) vicinal surfaces was carried out and is discussed in this paper with a view to understanding the step-step interaction kinetics that result in step bunching. BiSbTe3 layers grown on nominal (001) GaAs substrates exhibit triangular facet structures consisting of atomically flat plateau areas separated by steps. In contrast, the growth of films on vicinal substrates was found to result in regular arrays of terrace step structures. The formation of step bunches and rectangular terraces on such film surface is attributed to the preferential incorporation of adatoms at step or kink sites and lateral growth rate anisotropy.

Orientation-dependent surface and step energies of Pb from first principles

The orientation-dependent surface energies of 35 low-index and vicinal Pb surface orientations, located in the [001], $[\overline{1}10]$, and $[01\overline{1}]$ zones, have been calculated by density-functional theory within the local-density approximation. The highest surface energy anisotropies in these zones are at the (210), (110), and (311) directions. Surface relaxation decreases the surface energy anisotropy significantly. For misorientations smaller than 12\ifmmode^\circ\else\textdegree\fi{} the (projected) surface energy in a given zone increases linearly with step density, while curvature is found at higher misorientations, indicative of repulsive step-step interactions. These results are fully consistent with the orientation-dependent surface energy predicted by the statistical mechanics of the terrace-step-kink model of vicinal surfaces. The step formation energies and surface and step relaxation energies are derived and analyzed. There is good agreement with available experimental data. The calculated surface energies in eV/atom correlate linearly with the number of broken surface bonds. Deviations from perfect linearity are found to be essential for a proper description of the equilibrium crystal shape of Pb.

All-electron first-principles investigations of the energetics of vicinal Cu surfaces

Using first-principles calculations we studied the energetics (surface energy, step energy, stability with respect to faceting) of the low- and high-Miller-index (vicinal) Cu surfaces, namely, the (111), (100), (110), (311), (331), (210), (211), (511), (221), (711), (320), (553), (410), (911), and (332) surfaces. Our calculations are based on density-functional theory employing the all-electron full-potential linearized augmented plane-wave (FLAPW) method. We found that the unrelaxed vicinal Cu surfaces between (100) and (111) are unstable relative to faceting at $0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, while fully relaxed vicinal surfaces between (100) and (111) are stable relative to faceting, which is in agreement with the observed stability of vicinal Cu surfaces at room temperature. Thus atomic relaxations play an important role in the stability of the vicinal Cu surfaces. Using the surface energies of Cu(111), Cu(100), and Cu(110) and employing the effective pair-potential model, which takes into account only the changes in the coordination of the surface atoms, the surface energies of the vicinal Cu surfaces can be calculated with errors smaller than 1.0% compared with the calculated FLAPW surface energies. This result is due to the almost perfect linear scaling of the surface energies of the $\mathrm{Cu}(hkl)$ surfaces as a function of the total number of broken nearest-neighbor bonds. Furthermore, we calculate step-step interactions as a function of terrace widths and step energies of isolated steps.

Length-scale dependence of the step bunch self-organization on patterned vicinal Si(111) surfaces

We use lithographic patterning and high-temperature annealing to investigate the length-scale dependence of self-organization of step bunches on vicinal Si(111) surfaces due to the motion and interference of misorientation-derived straight steps and the circular steps which bound cylindrical pits. Annealing of patterns with small pit diameters produces a rapid relaxation to straight step bunches, while larger structures maintain significant in-plane corrugations for as long as the out-of-plane corrugations persist. This indicates the existence of a characteristic length scale, set by the competition between the Gibbs–Thomson effect and sublimation. Significantly, the eventual relaxation is to nearly uniform step trains indicating that the late stage evolution is dominated by repulsive step-step interactions.

Elastic relaxations and interactions for vicinal and self-organized surfaces: Role of crystalline anisotropy

We present a calculation of the elastic displacements due to periodic lines of force at a surface or buried under the surface. This applies, for example, to steps on vicinal surfaces or to self-organized one-dimensional systems. The choice of a calculation in the reciprocal space allows a detailed description of the topography of the displacement field. Far from the surface, the elastic crystal behaves like a low-$q$ filter and the displacement topography does not depend on the details of the force distribution but mainly on the surface orientation. Near the surface, vortices always appear; their characteristics depend mainly on the force orientation rather than the surface orientation. For step-step interactions, the vortices are responsible for the important role of the lever arm orientation of the dipoles associated with steps.