Analytical strain relaxation model for Si_1-xGe_x /Si epitaxial layers

An approximate but accurate analytical solution is presented for the system of differential equations used by Houghton to model kinetically limited strain relaxation in Si_1-xGe_x alloys layers growing on Si substrates [J. Appl. Phys. 70, 2136 (1991)]. This solution makes it much easier to compare the relaxation model with experimental data. The analytical results are used to refit the Houghton model parameter n0 (representing the initial heterogeneous density of dislocation sources) to published relaxation data, including post-1991 experimental work. The fits, which include experiments in which the growth temperature ranged from 450 to 750 °C, show considerable scattering in n0, but suggests that n0 increases as the growth temperature is lowered. Since this trend was not apparent in the original Houghton work, a detailed analysis is carried out for samples grown and annealed at temperatures below 450 °C. For this purpose, the Houghton model is extended to include the reduction in effective stress as the strain relaxation advances as well as the effect of dislocation pinning. The analysis confirms that n0 increases as the growth temperature is lowered. Possible physical reasons are discussed, and an empirical fit to the temperature dependence of n₀ is used to generate revised predictions of apparent critical thicknesses.