Discontinuity Mass of Finite Difference Calculation in InAs-GaAs Quantum Dots

Abstract

Recently, theoretical analysis of the electronic properties of quantum dot has attracted a great attention when modern nanotechnology has made it possible to fabricate a realistic quantum dots in laboratory [. Quantum dot structures which provide electron confinement in three dimensions can be grown by the so called self-assembly effect or Stranski-Krastanov growth mode. Particular interest attracts ordering effects in StranskiKrastanow growth which proceeds on a lattice-mismatched substrate via formation of essentially three-dimensional islands. This is especially true for the InAs-GaAs system where the lattice mismatch is high and the nucleation process is rapid. Although, quantum dots have being studied experimentally but large amount of numerical studies of electron confined states also have been developed to simulate electronic and optical properties in quantum dots. The single band effective mass is one of the formalism of envelope function which has been widely used to solve quantum dot systems. However, the effective massm*is usually position dependent in semiconductor heterostrutures. Consequently, the concerning about the form of the boundary conditions to impose on different material interface arisen [3]. According to the present works [2, , the position dependent Hamiltonian is given by: . wherem=m(r) is the position dependent effective mass of an electron in conduction band. The constant α, β, and γ is arbitrary set to satisfy α + β + γ = -1. Various approximations regarding the actual constant of α, β, and γ in position dependent effective mass have been observed, example Gora & William (by putting α = -1 and β = γ = 0), Zhu & Kroemer (α = γ = -1/2 and β = 0), and BenDaniel-Duke (α = γ = 0 and β = -1). Among them, β = 1 (known as the Ben DanielDuke Hamiltonian [) is most popular method for solving mass continuity problem on the classic Hamiltonian [. Extensively, these interface condition was been used to solved most of the heterostructure problem such as quantum dots [. However, there is a qualitative argument based upon the Ben DanielDuke choice violates the Heisenberg uncertainty principle and the issue of the correct effective-mass equation was further questioned by Pistol, M. E. which he claims that all the possible equations lead to the same interfacial conditions on the envelope function [. In this paper, we will investigate the effect of discontinuity mass within interface of two semiconductor materials inside InAs-GaAs quantum dot by using the classic constant mass Hamiltonian (CH), position dependent effective mass Hamiltonian (PDH) and Ben Daniel and Duke Hamiltonian (BDH). The most common analytic methods are solving the transcendental equation obtained by matching the interface boundary condition on the envelope function. But this kind of method will suffer from complexity of model quantum dots that contain multiple layer or geometry that unable to derive into analytic formulation. Thus, this study will focus on comparison between difference finite difference formalism to illustrate the mass discontinuity effect on the numerical solution.