Strain distributions in quantum dots of arbitrary shape

Abstract

A method based on the Green’s function technique for calculating strain in quantum dot (QD) structures has been developed. An analytical formula in the form of a Fourier series has been obtained for the strain tensor for arrays of QDs of arbitrary shape taking into account the anisotropy of elastic properties. Strain distributions using the anisotropic model for semiconductor QDs are compared to results of a simplified model in which the elastic properties are assumed to be isotropic. It is demonstrated that, in contrast to quantum wells, both anisotropic and isotropic models give similar results if the symmetry of the QD shape is less than or equal to the cubic symmetry of the crystal. The strain distribution for QDs in the shape of a sphere, cube, pyramid, hemisphere, truncated pyramid, and flat cylinder are calculated and analyzed. It is shown that the strain distributions in the major part of the QD structure are very similar for different shapes and that the characteristic value of the hydrostatic strain component depends only weakly on the QD shape. Application of the method can considerably simplify electronic structure calculations based on the envelope function method and plane wave expansion techniques.

Symmetry ofk∙pHamiltonian in pyramidalInAs∕GaAsquantum dots: Application to the

“…The analytical formulae from which e ij (q) can be derived in a crystal with cubic symmetry are given in Ref. 33. After integration, the characteristic functions can all be expressed as a linear combination of integrals of the type…”

Self-assembled (In,Ga)As/GaAs quantum-dot nanostructures: strain distribution and electronic structure

“…For most cases, however, the anisotropy only modifies the strain distributions slightly. 16 ' 19 Classical and atomistic elasticity methods have been shown to give similar results for small strains. 20 Differences become apparent for strains larger than 5%, particularly in the case of semiconductor quantum dots.…”

“…In any event, it has been shown in a recent paper that as long as the symmetry of the shape of the structure is less than or equal to the cubic symmetry of the crystal, both anisotropic and isotropic models give similar results. 16 We caution however that even though this is true for quantum-dot nanostructures, it is not so for quantum-well heterostructures.…”

Quantum box energies as a route to the ground state levels of self-assembled InAs pyramidal dots

“…In order to justify the choice of the parameters used in our calculations we, nevertheless, performed a set of calculations of the strain distribution in pyramidal structures with aspect ratio Q ranging from 1 to 4.5 using a method based on the Green's function technique, 5 taking into account the anisotropy of the elastic properties as well. 13 The carriers' strained confining potentials were then calculated as a function of position along the growth direction, in the framework of the eight-band k"p theory ͑see, for example Ref.…”

“…The problem, here, does not lay therefore in the pyramidal approximation ͑in a previous article 4 we have shown that the transition energies of lens shaped structures can be successfully reproduced by using pyramids with the same dimensions. In fact, as illustrated by Andreev et al, 5 the strain distribution is similar for both shapes throughout most of the dot, differing only in the upper part, due to the sharp edges of the pyramid and smooth boundary of the hemisphere͒, the problem lies in the use of theoretical data relative to structures with a given QϭQ th to fit the spectra of structures with different aspect ratios (Q exp Q th ). A small difference in only one of the dimensions results in dramatic differences in the electronic energy levels, ͑as it is shown in Table I for square-based pyramids with the same base length bϭ200 Å but different heights hϭ100 Å and hϭ70 Å. Qϭ1 and 1.428, respectively͒, so that the conclusions drawn from the comparison are at least inaccurate, if not misleading.…”