Our experience with the EW and the KPZ equations has already taught us an important lesson: nonlinear terms may be present for certain growth models and, if present, may control the scaling behavior. For the problem of MBE dominated by surface diffusion, there is no reason to ignore the possibility of nonlinear terms, so we must carefully examine when nonlinear terms are relevant and explore the resulting scaling behavior.

In order to address the relevance of certain nonlinear terms that can affect the growth equation (13.9), it will be necessary to use the dynamic RG method, as we did with the KPZ equation. We shall see that the dynamic RG method leads to exact exponents in any dimension, in contrast to the case of the KPZ equation where results are found only for d = 1.

Surface diffusion: Nonlinear effects

If the relevant relaxation process is surface diffusion, then the growth equation must satisfy the continuity equation (13.2). If the surface current is driven by the gradient of the local chemical potential (13.4), we argued that the diffusive growth process in the linear approximation is described by the growth equation (13.9).

For d = 1, 2, Eq. (13.9) predicts α = 3/2, 1 respectively. However, continuum growth equations are valid in the ‘small slope’ approximation, |∇h| « 1, which means that the local slopes are small at every stage of the growth process.