A general perturbation method for inhomogeneities in anisotropic and piezoelectric solids with applications to quantum-dot nanostructures

Abstract

a b s t r a c tBy introducing a homogeneous piezoelectric material and its Green's function, we present a new semianalytical three-dimensional perturbation method for general inhomogeneity problems in anisotropic and piezoelectric solids. This method removes the limitations associated with previous analytical methods, which often ignore the anisotropic properties or the difference between the material properties of the inhomogeneity and its surrounding matrix. As an important application, the proposed theory is employed to calculate the elastic and electric fields in a truncated pyramidal InAs/GaAs quantum-dot (QD) nanostructure. Numerical results demonstrate that the anisotropy of the materials and the difference between the material constants of the QD and the matrix have a significant influence on the strain and electric fields. The relative differences of the strain and electric field inside the QD between the simplified isotropic and homogeneous model and the real anisotropic and heterogeneous one may reach 22% and 53%, respectively. The accuracy of the calculated elastic strain and electric fields is improved greatly by a second order approximate solution (OAS). Since the third OAS nearly coincides with the second one, good convergence of the iteration procedure is demonstrated. Moreover, contours of the hydrostatic strain and electric potential within and around the QD are also presented and analyzed.

Elastic strain field due to an inclusion of a polyhedral shape with a non-uniform lattice misfit

“…16,36 In the case of different elastic constants of the matrix and the inclusion (an inhomogeneous inclusion), one can apply the perturbation method based on the concept of effective inclusion. 71,72 ACKNOWLEDGMENTS This work is funded by RFBR (grants #16-29-14031 and #16-02-00397, development of analytical expressions) and by State Programme (Grant No. 0306-2016-0015, creating the computer program for calculation of the strain).…”

“…16,36 In the case of different elastic constants of the matrix and the inclusion (an inhomogeneous inclusion), one can apply the perturbation method based on the concept of effective inclusion. 71,72 is equal to the sum of two integrals, the first of which being the quantity ϕ (mnp),α defined by Eq. (31).…”

Analysis of quantum-dot systems under thermal loads based on gradient elasticity

“…Green's function method [14][15][16] proved to be more efficient, particularly suited for polygonal shapes. The perturbation method [17,18] also allows considering material anisotropy and arbitrary shape of inclusion. A misfitted inclusion embedded in a finite spherical matrix was analyzed in the framework of linear piezoelectricity by Seo et al [19].…”

Аналитическое Выражение Для Распределения Упругой Деформации, Создаваемой Включением В Форме Многогранника С Произвольной Собственной Деформацией

“…1, b, d демонстрирует пример негидростатической собственной деформации. В этом иллюстративном примере различие между параметрами решетки двух сред имеет место только в направлении оси X. Случай негидростатической собственной деформации, помимо самостоятельного значения, возникает в задаче об упругой неоднородности (т. е. включения с модулями упругости, отличными от соответствующих модулей матрицы) как поправка по теории возмущений в рамках метода эффективного включения [14,15].…”