Diffraction from stepped surfaces. I. Reversible surfaces

In electron or atom diffraction experiments on surfaces, the angular shapes of the diffracted beams depend upon the distribution of steps over the surface. In this paper we analyze diffracted beam profiles from stepped surfaces that are reversible. A reversible surface is one in which the pair correlation function over the surface is symmetric with respect to positive and negative directions. We show that the intensity profile across a diffracted beam can be separated into a sharp central spike due to the limit of the correlation function at large separation plus wings or shoulders due to the finite extent of the step disorder. Simple functional expressions for these angular profiles are obtained by a Markov method of treating a one-dimensional geometric distribution of steps. The result explicitly displays the deep structure found for the general case. The method reduces the calculation to a simple eigenvalue problem so that even the continuously changing step distributions that occur in epitaxial growth can be treated easily. As in the general case, the resulting intensity profile is a sharp central spike plus a step-broadened term which now is a sum of Lorentzians. The widths of the Lorentzians are the logarithms of the eigenvalues of the matrix of probabilities which describe the step distribution over the surface. This matrix method, which treats the surface as a Markov chain, also points the correct way to account for correlations between surface atoms for two-dimensional distributions of steps. For a two-dimensional surface one must consider a Markov Random Field as opposed to a simple multiplication of two one-dimensional results. We compare the results of the general calculation to the Si epitaxy experiments of Gronwald and Henzler. The coverage and momentum transfer dependencies of the shapes of the calculated profiles agree with their measurements. The calculation is also applied to the RHEED measurements of Van Hove et al. during GaAs MBE. The measured intensity oscillations can be accounted for by a cyclically changing one-dimensional geometric distribution of steps among three layers in which the third-layer scattering increases with time.