Crystal surfaces with correlated disorder: Phase transitions between roughening and super-roughening.

A theory for surface transitions in the presence of a disordered pinning potential is presented. Arbitrary disorder correlations are treated in the framework of a dynamical functional renormalization group. The roughening transition, where surface roughness and mobility behave discontinuously, is shown to turn smoothly into the continuous superroughening transition, when the range of disorder correlations is decreased. Implications for random-field XY -models and vortex glasses are discussed. PACS: 64.40, 64.70.Pf, 68.35.Rh, 75.60.Ge Typeset using REVTEX 1 The shape of a fluctuating crystal surface is affected by the pinning potential provided by the crystal planes. The interface exhibits a roughening transition, if these planes are perfect, and it exhibits a superroughening transition, when these planes are strongly disordered. The roughening transition is well understood theoretically and verified in different experimental situations (see e.g. [1,2] and references therein). Above the roughening temperature TR the interface is thermally rough, whereas below TR the interface becomes smooth, since it locks into the pinning potential. However, for quenched bulk disorder [3] and substrate disorder [4,5] it was shown, that the transition has a different nature. This so-called superroughening transition occurs at TSR = TR/2. Above TSR the interface is thermally rough again, but below TSR the disordered pinning potential increases the roughness of the interface. In this Letter we address the question: how are the roughening transition and the superroughening transition related? What happens “in-between”, if we switch from a pinning potential with long-range correlations to short-range correlations? As shown below, new physics emerges for logarithmic substrate roughness. This can be realized, if the crystal is grown on a logarithmically rough substrate. Such a substrate could be generated by quenching it from a temperature above its roughening transition. In order to have a stable substrate surface, the roughening temperature of the substrate should be higher than that of the crystal. Our analysis is also relevant for a XY -model in the presence of a random field, as long as vortices can be neglected [6]. Different types of sources of the random field may have different multipolar character, which lead to a power-law decay of correlations. Finally, the consideration applies to two-dimensional vortex glasses [7,8]. Their pinning can be caused by local suppressions of the condensate density, which also relaxes with power laws and leads in turn to a power-law decay of correlations. This power varies with the dimensionality of the defects. At present there is a controversial discussion about the actual roughness of the superrough phase. Renormalization group (RG) calculations [4,5], variational calculations [9] and simulations [10] give an inconsistent picture. In view of this controversy we focus on topics 2 consistently described by these different approaches: the phase boundary and the nature of the phase transition. The present study is based on a functional dynamical renormalization group applying to the limit of very weak pinning. The interface profile is described by φ as a function of time t and the two-dimensional lateral coordinate r. Its kinetics is captured by the overdamped equation of motion m∂tφ(t, r) = K∇ φ(t, r)− χ sin [φ(t, r)− d(r)] + +F + ξ(t, r). (1) Here m is a mobility, K is the surface stiffness, χ is the amplitude of the periodic pinning potential, which has a quenched disorder phase d representing deformed crystal planes, F is a driving force per unit area, and ξ is thermal noise at temperature T . We implicitly suppose a cutoff Λ for the wave-vector of shape fluctuations, which corresponds to an area 4π/Λ per degree of freedom in the lateral plane. Since we assume regularization in momentum space, we consider φ and d to be defined continuously on the whole lateral plane (see Fig. 1) and not just only on a lattice of points r [11]. The disorder field is characterized by zero mean and a difference correlation [d(r)− d(r′)]2 = 2∆(r − r), which for simplicity is taken to be isotropic. For the following it is convenient to introduce the correlation (g0 := 1 2 mχ) γ0(r) := g0 ei[d(r)−d(0)] = g0 e . (2) In terms of this function, a perfect crystal exhibiting the roughening transition has γ0(r) = g. In the studies of the superroughening transition [3–5], γ0(r) was supposed to decay rapidly for large r. Eq. (2) shows, that this case corresponds to a very rough substrate with a rapidly increasing function ∆(r). Here we allow for a general, possibly slowly decaying correlation γ0(r). In constructing the RG, we choose an approach different from previous treatments. Since we do not want to specialize to a special correlation, we have to perform a functional RG. This can be achieved in the formalism of Martin, Siggia, and Rose [12], which requires the 3 introduction of an additional field φ̃. In this formulation the disorder average can easily be performed. The generating functional then reads