Critical Thickness of Nano-scale Lattice Mismatched Heterostructures

Nano-epitaxy has attracted great attention in the last several years. Theoretical study can date back to more than twenty years ago when Luryi and Suhir (LS) studied critical thickness of a strained epi layer grown on a ridged strip (Luryi, 1986). They used an analytic expression for strain and predicted that the critical thickness increased to infinity when the strip width was small enough (in the nm range). Following the same approach, Zubia and Hersee (ZH) considered a strained layer grown on a compliant patterned substrate and got the same conclusion (Zubia, 1999). In both studies, a critical thickness is defined when the strain energy exceeds the dislocation energy. However, numerous experiments reveal that the lattice mismatch can not be completely accommodated by misfit dislocations and the residual strain does not go to zero at the critical thickness. In addition, Van Mieghem et al. investigated strain and stress in a parallelepiped strained layer using finite element method (FEM) and found that the analytic formula for strain was deficient (Van Mieghem, 1994). In this work, we investigate strain distribution and critical thickness of a strained layer grown on a nano-scale patterned compliant substrate using FEM. We take into account the residual strain distribution and show that the predicted critical thickness is much larger than those predicted values using the LS model.