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

Abstract

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.