Scaling properties of step bunches induced by sublimation and related mechanisms

Abstract

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 d d = Ds/K d is larger than the average distance l between the steps on the vicinal surface; here Ds is the surface diffusion coefficient and K d is the step kinetic coefficient. In the opposite limit d d ≪ l 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 form L ∼ H α , lmin ∼ H −γ between the width L, the height H, and the minimum interstep distance lmin 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 case d d ≫ l. The analysis of the continuum equation reveals the existence of two types of stationary bunch profiles with different scaling properties. Through comparison with numerical simulations of the discrete step equations, we establish the value γ = 2/(n + 1) for the scaling exponent of lmin in terms of the exponent n of the repulsive step-step interaction, and provide an exact expression for the prefactor in terms of the energetic and kinetic parameters of the system. For the bunch width L we observe significant deviations from the expected scaling with exponent γ = 1 − 1/α, which are attributed to the pronounced asymmetry between the leading and the trailing edges of the bunch, and the fact that bunches move. 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.

Antiband instability on vicinal Si(111) under the condition of diffusion-limited sublimation

“…20 It should be noted that l j +1 and l j −1 behave differently in the case of non-negligible desorption, as was revealed by numerical integration of step motion equations within the framework of one-dimensional model. 3 These numerical calculations of l j +1 reproduce the result obtained for a conserved system 20 but give considerably larger values of l j −1 . As an approximation, we will use the expression derived for a conserved system 20 but written with somewhat different notation:…”

“…[1][2][3][4][5][6][7][8][9][10] When Si(111) is annealed by direct current (dc) driven along the miscut direction, adatom surface electromigration causes the atomic steps to gather together and creates step bunches separated by wide terraces. These step bunches grow with time and reach tens to hundreds of nanometers in height, while the terraces between them can grow to several micrometers in width.…”

Nonlinear Dynamics of Surface Steps

“…Related quantities of interest are the minimal terrace size l min in the bunch and the size l 1 of the first terrace in the bunch. On the basis of the continuum equation (1.17) one finds that, asymptotically for large N [44] 21) in good agreement with numerical simulations of the discrete model [45]. Note that W ∼ Nl min , as one would expect, but l 1 ≫ l min .…”

“…Together Eqs. (1.14,1.15,1.16) define a two-parameter family of nonlinear many-body problems which have been investigated in detail in [44][45][46]. In the following two sections some pertinent results of this study will be summarized.…”

“…Coarse graining the discrete equations of motion (1.14), one arrives first at a "Lagrangian" continuum description for the step positions x i or the terrace sizes l i = x i+1 − x i by converting the layer index i into a continuous surface height h = ih 0 (here h 0 denotes the height of an elementary step) [43,48]. In a second step this is transformed into an "Eulerian" evolution equation for the surface height profile h(x, t) or, equivalently, the step density m = ∂h/∂x , which reads, for the model (1.14) [44,45] …”