There are so far two main models for epitaxial multilayers which relate the wafer curvature to the strain state and the material properties of each layer. Ohlsen et al. (J. App. Phys. 1977) derived an analytical expression for the stress in each layer and the final wafer curvature for a given layer stack. This model however does require one to define strain states in each layer which is not straightforward to predict prior to the actual hetero-epitaxy of complex layers with large misfit dislocation densities such as in the GaN-on-Si system. Usher et al. (Phys. Rev. B 2003) proposed an atomistic approach for analyzing pseudo-morphic multi-layer structures, and were able to provide a closed form equation for radius of curvature. However, none of the reports have explicitly considered the thermal strain in the calculation. In this presentation we would address these limitations and provide a complete description of the wafer curvature including the effects of both lattice mismatch and coefficient of thermal expansion mismatch.
We extend the atomistic approach to calculate the radius of curvature for non-pseudomorphic systems like III/V-on-Si systems. The density of misfit dislocations at each interface is used as a parameter to define the effective lattice constant of partially relaxed epilayers at the interface. Subsequently the lattice constant at any point in each layer is defined with respect to the interface lattice constant and is used to calculate the strain. Then the total strain energy of the resulting system is calculated as a function of system curvature, with the minimum energy solution corresponding to the predicted final curvature of the wafer. For better accuracy each epitaxial layer may be divided to separate sublayers with individual misfit dislocation density values. For illustration GaN-3xAlGaN-AlN-Si system was considered and radius of curvature was predicted. The accuracy of the model depends on the accuracy with which the misfit dislocation density is known. Experimental techniques like TEM and electron beam induced current (EBIC) can be used to estimate or quantify the number of dislocated atoms. Finally, we will include the thermal effect within the atomistic approach to be able to predict in-situ curvature of III/V-on-Si systems which will be helpful in designing the growth experiments for various devices.