A simple method for calculating strain distributions in quantum dot structures

Abstract

A simple method is presented for calculating the stress and strain distributions arising from an initially uniformly strained quantum dot of arbitrary shape buried in an infinite isotropic medium. The method involves the evaluation of a surface integral over the boundary of the quantum dot and is therefore considerably more straightforward to implement than alternative stress evaluation techniques. The technique is ideally suited to calculating strain distributions within disordered arrays of pyramidal quantum dots prepared by Stranski–Krastanow growth. The strain distribution for a cuboidal quantum dot is presented and compared to that of a rectangular quantum wire.

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

“…Following Downes et a/., 13 the Lame potential u during relaxation can be described by a scalar potential …”

“…Another variant of these methods is the boundary element approach; this, however, can be mathematically complex. 12 A simple and elegant method for calculating strain fields around a single, isotropic, cubic dot has been presented by Downes et a/.. 13 This method is based on a simplification of Eshelby's classic inclusion theory. 14 The method first identifies a set of vectors such that the divergence of each gives the Green's function for the stress components o#.…”

A general treatment of deformation effects in Hamiltonians for inhomogeneous crystalline materials

“…There are basically two main approaches, an atomic approach 8 and a continuum approach. [9][10][11] If the deformation has been found using an atomic approach, it is necessary to use the information about how the individual atoms are shifted to construct a C 2 map, but this is outside the scope of this article, i.e., it will just be assumed that the deformation has been found using a continuum approach and that this results in a C 2 map ͑the C 2 demand will be weakened slightly in Sec. II C͒.…”

“…͑7͔͒ and a positive Jacobian determinant, where ⌫ is a finite set of piecewise smooth surfaces ͑two-dimensional manifolds͒. This set of functions includes the deformations considered in quantum-well, as well as quantum-wire and quantum-dot structures ͑ac-cording to continuum mechanical models [9][10][11] ͒. The problem with the functions belonging to A is that they are not necessarily once differentiable on B because the first derivative can have discon-tinuities at interfaces.…”

Real-space Green’s tensors for stress and strain in crystals with cubic anisotropy

“…Large systems involving many nanostructures with graded composition pose no difficulty and analytic expressions are sometimes tractable to provide deeper insight into the physics, revealing trends and simple relations. [12][13][14][15][16][17][18][19][20] For example, within the isotropic approximation, full analytical expressions for the strain distribution have been derived for ellipsoidal, cuboidal, and truncated-pyramidal dots 13,16 and for quantum wires of arbitrary shape 12 and near to free surfaces. 15 The Green's technique is a powerful method for the calculation of QD-or QWI-induced strain distributions and its ease of application makes it attractive to practitioners in the field.…”