# Super-roughening: A new phase transition on the surfaces of crystals with quenched bulk disorder

## Abstract

We present and study a model for surface fluctuations and equilibrium crystal shapes in solids with quenched bulk translational disorder but infinitely long-ranged orientational order. Strictly speaking, such surfaces have no sharp surface phase transition. However, for reasonable values of the bulk correlation length B ( B30 A should be sufficient), an experimentally sharp super-roughening transition occurs at a temperature TSR. This transition separates a high-temperature rough phase of the surface from a low-temperature super-rough phase that, counterintuitively, is even rougher. Specifically, the root-mean-square equilibrium vertical fluctuation in the position of the interface h2 1/2 diverge like lnL as the length L of the surface for T>TSR (just as in ordered solids for T greater than the roughening temperature TR), while h2 lnL1/2 for T<TSR. This causes the correlation function C(qz;x)== eziq[h(x)-h(0)] measured in surface-sensitive scattering experiments (e.g., anti-Bragg x-ray scattering) to go from algebraic decay C(qz;x) x - (qz) in the rough phase to short-ranged order C(qz;x) x-h (qz)ln( x ) in the super-rough phase. The functional dependence of (qz) on qz differs from that for fluctuating surfaces of both bulk ordered solids (above TR) and liquids. We identify an experimentally measurable correlation length SR that diverges as T TSR- as exp[ATSR2/(TSR-T)2], where A is a constant of order ln-4 B/a and a is a lattice constant. The equilibrium crystal shapes do not have facets in either the rough or the super-rough phase. At low temperatures in the super-rough phase, however, nearly flat regions appear, with a radius of curvature scaling like (B)-1. © 1990 The American Physical Society.