# A critical thickness condition for triangular strained quantum wires grown in V-grooves on a patterned substrate

- Authors
- Journal
- Acta Materialia 1359-6454
- Publisher
- Elsevier
- Volume
- 44
- Issue
- 1
- Identifiers
- DOI: 10.1016/1359-6454(95)00173-x
- Disciplines

## Abstract

Abstract The physical system under study is a quantum wire of triangular cross-section grown epitaxially in a V-shaped groove on a patterned (100) surface of a cubic substrate. It is assumed that the walls of the groove are {111} planes of the substrate material, so that the wire extends along a 〈110〉 direction. The critical thickness, or depth, of the wire is defined as the smallest thickness for which formation of a strain-relaxing misfit dislocation within the wire is possible. An analysis is presented which leads to an estimate of this critical thickness for a given lattice mismatch between the wire and its substrate. Because of the non-uniformity of the elastic mismatch strain field in the wire, a computational approach is adopted to determine the critical thickness as a function of lattice mismatch. First, a simulation of the system by means of the numerical finite element method leads to a result for the driving force on a threading dislocation that leaves behind a misfit dislocation as it advances along the wire. Both the free surface effect and the mismatch strain effect are taken into account in the simulation, and the vanishing of this driving force leads to the critical thickness condition. Using the results of this detailed analysis as guidance, it is then shown that independent upper and lower bounds on the critical thickness may be obtained, and these bounds are provided. The mathematical expressions are both simple and quite general, being valid for all values of the elastic constants and wire thicknesses. Moreover, the bounds are sufficiently restrictive so that the value of critical thickness is actually established to an accuracy adequate for most practical purposes. It is found that during growth a wire is, as expected, more stable than a uniform quantum well of the same composition and thickness, but less stable than a buried wire in the completed structure. Comparison is made with the available experimental observations.

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