The profile of a decaying crystalline cone

TL;DR

This paper investigates the decay dynamics of crystalline cones below the roughening transition, analyzing surface diffusion mechanisms, instabilities, and scaling behaviors through models and simulations.

Contribution

It introduces a detailed step flow model for cone decay, derives analytical scaling laws, and explores the effects of step permeability on decay behavior.

Findings

In attachment-detachment limited kinetics, step bunching instability occurs with weak repulsive interactions.

The height profile exhibits a flat region scaling as t^{1/4} in radius.

Step permeability does not alter the fundamental scaling behavior, only parameter renormalization.

Abstract

The decay of a crystalline cone below the roughening transition is studied. We consider local mass transport through surface diffusion, focusing on the two cases of diffusion limited and attachment-detachment limited step kinetics. In both cases, we describe the decay kinetics in terms of step flow models. Numerical simulations of the models indicate that in the attachment-detachment limited case the system undergoes a step bunching instability if the repulsive interactions between steps are weak. Such an instability does not occur in the diffusion limited case. In stable cases the height profile, h(r,t), is flat at radii r<R(t)\sim t^{1/4}. Outside this flat region the height profile obeys the scaling scenario \partial h/\partial r = {\cal F}(r t^{-1/4}). A scaling ansatz for the time-dependent profile of the cone yields analytical values for the scaling exponents and a differential equation for the scaling function. In the long time limit this equation provides an exact description of the discrete step dynamics. It admits a family of solutions and the mechanism responsible for the selection of a unique scaling function is discussed in detail. Finally we generalize the model and consider permeable steps by allowing direct adatom hops between neighboring terraces. We argue that step permeability does not change the scaling behavior of the system, and its only effect is a renormalization of some of the parameters.