Analytical theory of crystal growth

The motion of the crystal–vapor interface is examined by means of the kinetic solid-on-solid model, a restricted version of the kinetic Ising model. We formulate an exact kinetic equation for the model and discuss some of the implications of the mean field and random distribution approximations. These methods require numerical integration of a differential-difference equation and are accurate only in the limit of high temperatures or high deposition rates. We then introduce a new set of approximations which permit the kinetic equations to be solved analytically and discuss their validity. We obtain simple analytic expressions for both the interface width and the growth rate under steady-state conditions. The qualitative dependence of the interface width on both temperature and deposition rate is correctly described by the theory and the predicted growth rates compare favorably with recent Monte Carlo calculations over a very wide range of temperatures and deposition rates.