Directional dependence of surface morphological stability of heteroepitaxial layers

Surface morphological stability in coherent heteroepitaxial layers is analyzed focusing on the directional dependence of surface undulations created by surface diffusion. The critical stability condition is defined in terms of the free energy of the system which is assumed to be the sum of the elastic strain energy and the surface free energy. The displacement and stress fields of the semi-infinite anisotropic solid with the slightly undulating surface are calculated by using the surface admittance tensor and the vector complex potential function. Numerical results for the Si1−xGex/Si systems show that the critical wavelength of the 〈100〉 surface undulations is smaller than that of the 〈110〉 surface undulations, which means that surface undulations are likely to be formed in the 〈100〉 directions. It is also found that the critical wavelength decreases with the increase of Ge fraction. These tendencies are in good agreement with the observations in annealing experiments for the Si1−xGex/Si systems in the literature. If the substrate is assumed to be rigid, the range of layer thickness where the system is absolutely stable against a surface undulation of any wavelength exists. Finally, the growth rate of the amplitude of surface undulations is estimated from an evolution equation for the surface shape. It is shown that even if anisotropy is taken into account, the growth rate of the amplitude takes the maximum value when the wavelength is 4/3 times the critical wavelength, which is the same as in the isotropic approximation.