EQUILIBRIUM SURFACE-ROUGHNESS OF A STRAINED EPITAXIAL FILM DUE TO SURFACE-DIFFUSION INDUCED BY INTERFACE MISFIT DISLOCATIONS

In recent years, developments in the microelectronics industry have led to extensive studies of the growth and characterization of thin solid films and their implementation in electronic and opto-electronic devices. A goal is to produce thin films with minimal bulk and surface defects, For those systems produced by epitaxial growth of a film on a substrate that has a slightly different lattice parameter, the stress associated with the elastic mismatch strain needed to satisfy the constraint of epitaxy provides a driving force for nucleation and growth of undesirable defects in the film material or on its surface. Among the most common defects are interface misfit dislocations, arranged more or less periodically on the film-substrate interface, which partially relax the elastic mismatch strain in the film. It has been observed that, for some material systems, surface roughness or waviness arises which correlates spatially with the positions of interface misfit dislocations. It is suggested here that the waviness along the surface may be a result of surface diffusion which is driven by a gradient in the chemical potential of the material along the surface. The chemical potential gradient arises from the nonuniform strain field of the interface misfit dislocations, as well as from the unrelaxed elastic mismatch strain. The focus here is on the development of a relatively simple model of this system which leads to an estimate of the magnitude and profile of surface waviness under conditions of thermodynamic equilibrium, i.e., after the material responds to the chemical potential gradient by seeking out a new configuration for which stresses are redistributed and the chemical potential is again uniform. The condition of uniform chemical potential for the final shape leads to an integro-differential equation for the equilibrium surface shape which is solved numerically. For representative values of system parameters, estimates of equilibrium surface roughness are obtained which can vary from less than one percent of film thickness to a significant fraction of film thickness. Although transient aspects of the process are not studied here, the characteristic time for achieving an equilibrium configuration is estimated.