Surface Science 244 (1991) 149-159 North-Holland 149 Diffraction from stepped surfaces in thermal equilibrium N.C. Bartelt, T.L. Einstein and Ellen D. Williams Department of Physics, University of Maryland, College Park, MD 20742-41 II, USA Received 29 June 1990; accepted for publication 14 September 1990 We have performed Monte Carlo simulations of the diffraction from simple two-dimensional models of vicinal surfaces in order to aid interpretation of measured diffraction profiles. At low temperature, we find the sharp diffraction features predicted from the analogy of stepped surfaces with two-dimensional incommensurate phases. These sharp features vanish only near the roughening temperature of the low-index surface corresponding to the terraces between steps. If one fits experimental data having sharp diffraction features to models of step disorder which do not include the ordering influence of step wandering, one can severely overestimate the amount of disorder. We emphasize that long-range correlations in step positions are more important than the local order in step edge structure or step separations for interpreting sharp diffraction features from steps. After much theoretical effort, it has become well-established that asymptotically the height-height correlations for rough surfaces diverge logarithmically (with a prefactor having a universal component at low temperature). We show explicitly how to use diffraction data to access this behavior for stepped surfaces. In the process, we evaluate the accuracy of a popular approximate expression for the diffracted intensity. 1. Introduction Steps on surfaces play an important role in many surface processes. To understand these processes the characterization of the configuration of steps is necessary. In particular, one would like to determine from diffraction experiments such quantities as. the disorder in step edges and the fluctuations in the distances between steps. Unfor- tunately, such analysis of distribution functions from measured diffraction profiles is fraught with problems of non-uniqueness. Thus to provide a basis for interpreting experimental results, we dis- cuss the expected diffraction signatures for vari- ous physical models which might govern step be- havior under experimental conditions. For stepped surfaces below the roughening temperature of the nearby low-index surface, the fundamental source of disorder in thermal equilibrium is step wander- ing. As we shall illustrate throughout this paper, the equilibrium step behavior cannot be predicted correctly in a one-dimensional description, since step wandering is omitted by definition in one dimension. In this paper we address the problem of characterization of equilibrium step behavior by computing the diffraction from simple model surfaces using the Monte Carlo method. Diffraction experiments are most sensitive to step disorder at conditions where neighboring ter- races scatter out-of-phase [l]. At out-of-phase con- ditions, straight steps which are arranged in a perfectly ordered staircase will give rise to “split beams”, the spacing between the split beams being proportional to the inverse of the step spacing. The introduction of step wandering into the step configuration will broaden these beams. The amount of step disorder consistent with a diffrac- tion pattern with well-defined splittings, i.e., with a well-defined step periodicity, has been the sub- ject of a number of discussions based on purely statistical one-dimensional models [2], without ad- dressing the issue of the physical mechanisms re- sponsible for the disorder. In devising models of step disorder, it is important to respect the behav- ior one expects for correlations in the step posi- tions. Step distributions which arise from a one-di- mensional statistical mechanical model with a given set of interactions will have much smaller height-height correlations than one expects of a two-dimensional surface described by the same 0039-6028/91/!§03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)