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

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 ${\xi }_B$ (${\xi }_B$≳30 Å should be sufficient), an experimentally sharp ‘‘super-roughening’’ transition occurs at a temperature $T_{\mathrm{SR}}$. 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 〈$h^2$〉¯ ${}_{}{}^{1/2}{}_{}{}^{}$ diverge like √lnL as the length L of the surface →∞ for T>$T_{\mathrm{SR}}$ (just as in ordered solids for T greater than the roughening temperature $T_R$), while 〈$h^2$〉¯ ${}_{}{}^{1/2}{}_{}{}^{}\ln \mathit{L}$ for T$T_{\mathrm{SR}}$.