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Instabilities in MBE Growth

We address the problem of MBE growth in one horizontal and one vertical direction in the presence of Schwoebel barriers. The time-independent growth equation introduced previously is shown to be identical to that for a classical particle in a potential well. We solve this equation using periodic boundary conditions and find time-independent solutions consisting of a periodic array of mounds. We derive the “dispersion relation”, i.e. the amplitude as a function of wavelength for these mounds. The equation of motion is derivable from a free energy indicating that there is a most stable ground state, which is independent of the initial conditions. The mounds are marginally unstable and there is a minimum wavelength below which no mounds exist. The wavelength of the mounds coarsens slowly in time according to Λ ∼ tα, with α ≈ 1/4.