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 d(+)=D-s/K+ is larger than the average distance l between the steps on the vicinal surface; here D-s is the surface diffusion coefficient and K+ is the step kinetic coefficient. In the opposite limit 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 Lsimilar toH(alpha), l(min)similar toH(-gamma) between the width L, the height H, and the minimum interstep distance l(min) 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(+)>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 gamma=2/(n+1) for the scaling exponent of l(min) 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 gamma=1-1/alpha, 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 , Phys. Rev. Lett. 88, 206103 (2002)], but some aspects of the universality scenario need to be revised.