Modeling Quantum Structures with the Boundary Element Method

A detailed analysis of the boundary element formulation of the electronic states of quantum structures is presented. Techniques for minimizing computation time by reducing the number of boundary integrals, utilizing the repetitive nature of embedded multiple quantum structures, and eliminating boundary elements for modeling the effects of quantum wells that contain quantum structures are discussed. Boundary element solutions for isolated and coupled quantum wires and pyramidal quantum dots are presented. The results for the coupling of such quantum structures clearly show the splitting of ground and excited states into “bonding” and “antibonding” states for both wires and dots. The numerical algorithm is shown to accurately capture these symmetry properties of a system of quantum structures if the boundary element and quadrature points are properly organized. For an asymmetric system of coupled dots, depending on the energy level, the center of charge is shown to be either above or below that for a pair of uncoupled dots with the same dot separation and orientation. Results are also presented showing the increase in energy levels resulting from the additional confinement arising from placing quantum dots into a quantum well.