Lattice-Misfit Stresses in a Circular Bi-Material Gallium-Nitride Assembly

Abstract

A simple and physically meaningful analytical ( "mathematical" ) predictive model is developed using two-dimensional (planestress) theory-of-elasticity approach (TEA) for the evaluation of the effect of the circular configtiration of the substrate (wafer) on the elastic lattice-misfit (mismatch) stresses (LMS) in a semiconductor and particularly in a gallium nitride (GaN) film grown on such a substrate. The addressed stresses include (I) the inteifacial shearing stress supposedly responsible for the occurrence and growth of dislocations, for possible delaminations, and for the cohesive strength of the intermediate strain buffering material, if any, as well as (2) normal radial arid circumferential (tangential) stresses acting in the film cross-sections and responsible for the short-and long-term strength (fracture toughness) of the film. The TEA results are compared with the formulas obtained using strength-of-materials approach (SMA). This approach considers, instead of the actual circular substrate, an elongated bi-material rectangular strip of unit width and of finite length equal to the wafer diameter. The numerical example is carried out, as an illustration, for a GaN film grown on a silicon carbide (SiC) substrate. It is concluded that the SMA model is acceptable for understanding the physics of the state of stress and for the prediction of the normal stresses in the major midportion of the assembly. The SMA model underestimates, however, the maximum interfacial shearing stress at the assembly periphery and, because of the very nature of the SMA, is unable to address the circumferential stress. The developed TEA model can be used, along with the author's earlier publications and the (traditional and routine) finite-element analyses (FEA), to assess the merits and shortcomings of a particular semiconductor crystal growth (SCG) technology, as far as the level of the expected LMS are concerned, before the actual experimentation and/or fabrication is decided upon and conducted.