Atomistic Approach for Modeling the Wafer Curvature for Pseudo-Morphic and Non-Pseudo-Morphic Epitaxial Multilayer Structures
Authors: Kadir Abdul , Somasundaram Sivanand , Lee Kenneth E. , Chua Soo Jin , Fitzgerald Eugene A 
Published: February 2020

During last few decades, there have been tremendous progress in the development of III-V based and III-Nitride based devices, namely High Electron Mobility Transistors (HEMT) and Light Emitting Diodes (LEDs). Research interest has recently shifted towards III-V compound semiconductor epitaxy on large diameter (200 mm) Si substrates in order to realize III-V-on-Si and CMOS integration. Due to large lattice and thermal mismatch between III-V and Si materials, e g. GaN-on-Si wafers experience large internal stresses and significant wafer bow. The wafer bow is severe for large wafers (for a given radius of curvature, the wafer bow increases by 125% for 150 mm and 300% for 200 mm diameter wafers compared to 100 mm wafers). This results in wafer fragility issues, as well as difficulties in subsequent handling and processing of such wafers. These issues are typically addressed using thicker Si substrates and careful design of buffer layers. While there are many experimental efforts to overcome these challenges, there are only limited reports that try to properly explain and understand the mechanism of wafer curvature. An appropriate theoretical model which predicts the final wafer curvature for given epilayer designs would greatly improve our ability to design optimal buffer and device layers.
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.