Non-universal equilibrium crystal shape results from sticky steps

Abstract

The anisotropic surface free energy, Andreev surface free energy and equilibrium crystal shape (ECS) z = z(x,y) are calculated numerically using a transfer matrix approach with the density matrix renormalization group (DMRG) method. The adopted surface model is a restricted solid-on-solid (RSOS) model with ‘sticky’ steps, i.e. steps with a point-contact-type attraction between them (p-RSOS model). By analyzing the results, we obtain a first-order shape transition on the ECS profile around the (111) facet; and on the curved surface near the (001) facet edge, we obtain shape exponents having values different from those of the universal Gruber–Mullins–Pokrovsky–Talapov (GMPT) class. In order to elucidate the origin of the non-universal shape exponents, we calculate the slope dependence of the mean step height of ‘step droplets’ (bound states of steps) ⟨n(p)⟩ using the Monte Carlo method, where p = (∂z/∂x,∂z/∂y) and ⟨⋅⟩ represents the thermal average. Using the result of the |p| dependence of ⟨n(p)⟩, we derive a |p|-expanded expression for the non-universal surface free energy feff(p), which contains quadratic terms with respect to |p|. The first-order shape transition and the non-universal shape exponents obtained by the DMRG calculations are reproduced thermodynamically from the non-universal surface free energy feff(p).