PHYSICAL REVIEW B VOLUME 52, NUMBER 7 15 AUGUST 1995-1 Smoothing of rough surfaces Angel Sanchez Theoretical Division and Center lor Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Departamento de Matematicas, Escuela Politicnica Superior, Universidad Carlos Ill, E-28911 Leganes, Madrid, Spain A. R. Bishop, David Cai, and Niels Gr!Zlnbech-Jensen Theoretical Division and Center lor Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 19 April 1995) Simulations of surface smoothing (healing) by Langevin dynamics in large systems are reported. The surface model is described by a two-dimensional discrete sine-Gordon (solid-on-solid) equation. We study how initially circular terraces decay in time for both zero and finite temperatures and we compare the results of our simulations with various analytical predictions. We then apply this knowledge to the smoothing of a rough surface obtained by heating an initially flat surface above the roughening temperature and then quenching it. We identify three regimes in terms of their time evolution, which we are able to associate with the resulting terrace morphology. The regimes consists of a short initial stage, during which small scale fluctuations disappear; an intermediate, longer time interval, when evolution can be understood in terms of terraces and their interaction; and a final situation in which almost all terraces have been suppressed. We discuss the implications of our results for modeling rough surfaces. I. INTRODUCTION The study of the morphology of growing surfaces is very important from both fundamental and applied viewpoints. 1 ,2 Since the pioneering work of Burton, Cabr- era, and Frank,3 it has been clearly established that the surface of a crystal at equilibrium is macroscopically Hat at low temperature and rough above some temper- ature T R, the "roughening temperature.,,4 For the case of a growing surface, i.e., a nonequilibrium situation, the roughening temperature is only slightly reduced with re- spect to the equilibrium case (see, e.g., Ref. 5 and ref- erences therein). In view of this, the question arises as to whether an initially rough surface (Le., in equi- librium above TR, or growing at temperatures close to it) smooths when quenched below TR, and if so what are the relevant time scales, and what rough features are preserved in the final structure. Posed as the study of the relaxation of a grooved surface, this problem was al- ready discussed more than thirty years ago by Mullins, 6 under the assumption that all surface properties were independent of the orientation. However, Mullins's the- ory fails below TR, because the anisotropic surface en- ergy displays cusps at particular angles. This difficulty was addressed by Martin and Perraillon 7 and by and Villain, 8 who showed how step structure could be taken into account. Additionally, Monte Carlo simula- tions showed that the atomistic structure of the surface leads to a number of new subtleties in the smoothing pro- cess (see Ref. 9 and references therein). These studies led to a good understanding of the smoothing process of a si- nusoidal profile above the roughening temperature (TR), although the situation is less clear as regards smoothing below TR.9 0163-1829/95/52(7)/5433( 12)/$06.00 52 Our purpose here is to move a step further and ad- dress the problem of the smoothing of a rough surface instead of that of an artificially grooved profile. The case of such a rough surface has not received much attention. Villain 10 phrased the question as follows: For tempera- tures T > T R, the surface of a crystal is rough, and the Huctuations of the height h(r) diverge at long distances, C(r) == ([h(r' + r) - h(r')]2) '" C(T) In r, (1) where C(T) is a temperature-dependent constant, rand r' are two-dimensional (2D) vectors giving the position at which height is measured, and r == Irl. If now the crystal is cooled down, from T > TR to T « TR, it is still rough. How does the roughness decrease with time t? From the theoretical analysis in Ref. 10, there were two main conclusions: On the one hand, smaller terraces disappear first, and after a time t the minimum terrace size '" t exp( -EoJT)JT (disregarding a few small terraces due to shrinkage of large ones); on the other hand, for an infinite surface C(r) is still given by the same expression as above, and only at short distances is the height difference much smaller. The prediction about Rmin was in contradiction with the results in Ref. 6 (R!'in '" t) and in Ref. 7 (R!.in '" t), and therefore, as noted by Villain in Ref. 10, there are subtleties in the theoretical treatment of the problem. Motivated by this unclear situation of sinusoidal pro- file smoothing below TR,9 and by the lack of any exper- imental or numerical result on smoothing of fully rough surfaces, in this work we undertook the study of this problem by means of Langevin dynamics simulations of a 2D discrete sine-Gordon ('" solid-on-solid) model. We report on our results as follows. In Sec. II we present our 5433 @1995 The American Physical Society