Instabilities at vicinal crystal surfaces: Competition between electromigration of adatoms and kinetic memory effect

Abstract

We studied the step dynamics during sublimation and growth when the adatoms on the crystal surface have a drift velocity ${D}_{s}F/kT$, where ${D}_{s}$ is the surface diffusion coefficient and $F$ is a force acting on the adatoms ($F$ is related to the electric current heating the crystal). In the limit of fast surface diffusion and slow kinetics of atom attachment-detachment at the steps, we formulate a model that is free of the quasistatic approximation in the calculation of the adatom concentration on the terraces. The linear stability analysis of a step train results in an instability condition in the form $\ensuremath{-}(f{\ensuremath{\tau}}_{s}^{\ensuremath{'}}/3\ensuremath{\epsilon})+(V/{V}_{\text{cr}})\ensuremath{-}1>0$, where ${\ensuremath{\tau}}_{s}^{\ensuremath{'}}$ is the dimensionless lifetime of an adatom before desorption, $f$ and $\ensuremath{\epsilon}$ are the dimensionless electromigration force and the force of step repulsion, respectively, and $V$ and ${V}_{\text{cr}}$ are the velocity of steps in the train and the critical velocity, respectively. As seen, instability is expected when either the velocity $V$ is larger than ${V}_{\text{cr}}$ (this instability is related to the ``kinetic memory effect'') or $\ensuremath{-}f{\ensuremath{\tau}}_{s}^{\ensuremath{'}}/3\ensuremath{\epsilon}>1$, i.e., when the electromigration force $f$ is negative (but strong enough to dominate over the term $V/{V}_{\text{cr}}$), which means step-down direction of the drift. In the latter case, the initial stage of the step bunching process dramatically depends on the value of the ratio $\ensuremath{-}f/\ensuremath{\epsilon}$. When $\ensuremath{-}f/\ensuremath{\epsilon}\ensuremath{\ge}1$ the instability starts with a formation of small bunches (or pairs) of steps, whereas at $\ensuremath{-}f/\ensuremath{\epsilon}\ensuremath{\ll}1$, the instability starts with a formation of relatively large bunches containing ${N}_{\text{initial}}=2\ensuremath{\pi}/{q}_{\text{max}}=2\ensuremath{\pi}/{(\ensuremath{-}f/3\ensuremath{\epsilon})}^{1/2}$ steps. A numerical integration of the equations for the time evolution of the adatom concentrations and the equations of step motion reveals two different step bunching instabilities: (1) step density waves (small bunches that do not manifest any coarsening) induced by the kinetic memory effect and (2) step bunching with coarsening (eventually leading to a formation of very large bunches) when the dynamics is dominated by the electromigration. For the latter case, we obtained very instructive illustrations of the dynamical phase transition [V. Popkov and J. Krug, Phys. Rev. B 73, 235430 (2006)] during sublimation and growth of a vicinal crystal surface.