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

1990; American Physical Society; Volume: 41; Issue: 1 Linguagem: Inglês

10.1103/physrevb.41.632

1095-3795

John Toner, David P. DiVincenzo,

Stochastic processes and statistical mechanics

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 ${\ensuremath{\xi}}_{B}$ (${\ensuremath{\xi}}_{B}$\ensuremath{\gtrsim}30 A\r{} 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 〈${h}^{2}$〉\ifmmode\bar\else\textasciimacron\fi{} $^{1/2}$ diverge like \ensuremath{\surd}lnL as the length L of the surface \ensuremath{\rightarrow}\ensuremath{\infty} for T>${T}_{\mathrm{SR}}$ (just as in ordered solids for T greater than the roughening temperature ${T}_{R}$), while 〈${h}^{2}$〉\ifmmode\bar\else\textasciimacron\fi{} $^{1/2}\mathrm{ln}\mathit{L}$ for T${T}_{\mathrm{SR}}$.This causes the correlation function C(${\mathit{q}}_{\mathit{z}}$;x)==〈${\mathrm{e}}_{\mathit{z}}^{\mathit{iq}}$[h(x)-h(0)]〉 measured in surface-sensitive scattering experiments (e.g., anti-Bragg x-ray scattering) to go from algebraic decay C(${q}_{z}$;x)\ensuremath{\propto}\ensuremath{\Vert}x${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\eta}({q}_{z})}$ in the rough phase to short-ranged order C(${\mathit{q}}_{\mathit{z}}$;x)\ensuremath{\propto}\ensuremath{\Vert}${\mathrm{x}}^{\mathrm{\ensuremath{-}}\mathit{h}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$(${\mathit{q}}_{\mathit{z}}$)ln(\ensuremath{\Vert}x\ensuremath{\Vert}) in the super-rough phase. The functional dependence of \ensuremath{\eta}(${q}_{z}$) on ${q}_{z}$ differs from that for fluctuating surfaces of both bulk ordered solids (above ${T}_{R}$) and liquids. We identify an experimentally measurable correlation length ${\ensuremath{\xi}}_{\mathrm{SR}}$ that diverges as T\ensuremath{\rightarrow}${T}_{\mathrm{SR}{}^{\mathrm{\ensuremath{-}}}}$ as exp[${\mathrm{AT}}_{\mathrm{SR}{}^{2}/({T}_{\mathrm{SR}\mathrm{\ensuremath{-}}\mathrm{T}{)}^{2}}}$], where A is a constant of order ${\mathrm{ln}}^{\mathrm{\ensuremath{-}}4}$\ensuremath{\Vert}${\ensuremath{\xi}}_{\mathrm{B}}$/a\ensuremath{\Vert} 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 (${\ensuremath{\xi}}_{B}$${)}^{\mathrm{\ensuremath{-}}1}$.