Formulas for the force dipole interaction of surface line defects in homoepitaxy

Abstract

Using asymptotics, we derive explicit, simplified formulas for integrals representing the force dipole interaction energy per unit length between line defects (steps) of the same sign that form perturbations of circles in homoepitaxy. Our starting point is continuum linear elasticity in accordance with the classic model by Marchenko and Parshin (1981 Sov. Phys.—JETP 52 129). In the case with concentric circular steps, we define a small geometric parameter, δ2, which expresses the smallness of interstep distance relative to the circle radii. We invoke the Mellin transform with respect to δ2 and derive systematically an approximation for the requisite integral. This technique offers an alternative to an exact evaluation in terms of elliptic integrals. We then demonstrate the use of the Mellin transform when calculating the force dipole interaction energy between smoothly, slowly varying steps that form perturbations of circles. We discuss the implications of our results for small-amplitude modulations of circular step profiles.