Calculation of critical layer thickness versus lattice mismatch for Ge<i>x</i>Si1−<i>x</i>/Si strained-layer heterostructures

TLDR

Previous theories, such as Matthews et al., attribute the onset of interfacial misfit dislocations to the lack of mechanical equilibrium for threading dislocations, whereas this work adopts a different energy‑balance approach. The study calculates the critical layer thickness hc for GeₓSi₁₋ₓ strained layers on Si substrates across the full composition range 0 ≤ x ≤ 1.0. The calculation assumes misfit dislocation generation is governed solely by energy balance, with dislocations forming when the film’s strain‑energy density exceeds that of a screw dislocation at a distance equal to the film thickness, leading to screw and edge dislocations at the interface for thicker films. The predicted hc versus lattice mismatch values agree excellently with experimental measurements for GeₓSi₁₋ₓ on Si.

Abstract

A calculation of the critical layer thickness hc for growth of GexSi1−x strained layers on Si substrates is presented for 0≤x≤1.0. The present results are obtained assuming misfit dislocation generation is determined solely by energy balance. This approach differs therefore from previous theories (e.g., Matthews et al. ), in which the absence of mechanical equilibrium for grown-in threading dislocations determines the onset of the generation of interfacial misfit dislocations. It is assumed that interfacial misfit dislocations will be generated when the areal strain energy density of the film exceeds the energy density associated with the formation of a screw dislocation at a distance from the free surface equal to the film thickness h. For films thicker than this critical value, screw (and edge) dislocations will be generated at the film/substrate interface. Values obtained for the critical thickness versus lattice mismatch are in excellent agreement with measurements of hc for GexSi1−x strained layers on Si substrates.