ELSEVIER Surface Science 366 (1996) 377 393 surface science Statistical multiple diffuse scattering from rough surfaces in RHEED - beyond the distorted-wave Born approximation Z.L. Wang * School of Material Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USA Received 25 January 1996; accepted for publication 8 May 1996 Abstract In reflection high-energy electron diffraction (RHEED) of growing surfaces in molecular beam epitaxy (MBE), diffuse scattering is generated by atom vibrations, point vacancies and growth islands (or surface roughness). Most of the existing RHEED theories have been developed under the first-order diffuse scattering approximation, and thus they are restricted for surfaces whose roughness is relatively low. In fact, crystal surfaces grown by MBE are usually rough; the change of surface coverage from 0 to 1 monolayer accounts for the observed RHEED oscillation. In this paper, a formal dynamical theory of RHEED has been developed to calculate the diffuse scattering produced by both atom vibrations and point vacancies at surfaces. The theory is aimed at recovering the multiple diffuse scattering that has been dropped by the distorted-wave Born approximation (DWBA). With the inclusion of a complex potential in the dynamical calculation, a rigorous proof is given to show that the high-order diffuse scattering terms are recovered in the calculation using the equation originally derived under the DWBA. This conclusion establishes the basis for expanding the RHEED theories developed under the first-order diffuse scattering to cases where the degree of surface roughness is high, allowing dynamical calculation of RHEED rocking curves for any growing surface. The statistical time and structure averages over the distorted crystal potential are evaluated analytically before numerical calculation. The dynamic form factor is calculated with consideration of anisotropic surface atom vibration and point vacancies at a growing surface. Keywords: Distorted wave-born approximation; Point defect; Reflection high-energy electron diffraction (RHEED); Shirt range order; Surface roughness; Thermal diffuse scattering I. Introduction Reflection high-energy electron diffraction (RHEED) is a powerful technique for in-situ obser- vation of surface structural evolution during molec- ular beam epitaxial (MBE) growth. RHEED oscillation is a sensitive technique for monitoring layer-by-layer growth on crystalline surfaces. RHEED has been routinely used to monitor the growth of surface layers, and it has been demon- * Corresponding author. Fax: + 1 404 8949140; e-mail: zhong.wang@mse.gatech.edu Published by Elsevier Science B.V. PII S0039-6028 (96)00800-X strated to exhibit high surface sensitivity because the electron penetration depth into the surface is no more than 1-2 nm. Surface structures can now be determined quantitatively using RHEED, in which dynamical calculation plays an essential role [ 1-3 ]. There are three types of elastic scattering theories which have been proposed for RHEED calcula- tions. The Bloch wave theory developed by Bethe [4] for transmission electron diffraction was first applied in RHEED by Colella [5] and Moon [6]. In this theory, the crystal is considered as a periodic repetition of a unit cell, and the surface is a sharp