Computation of the Exciton Ground State in Arbitrarily Shaped Quantum Dots Using Discrete Variable Representation Method

The sinc-function discrete variable representation method is used to calculate the ground state energy of an electron and its corresponding hole bound inside quantum dots of varying shapes in the strong confinement regime. Under the effective mass approximation, a time-independent Schrödinger equation is numerically solved for the two-particle system with the infinite potential well used to model the dot boundary and the direct electrostatic Coulomb force used to describe the interaction between the particles. Using dot shapes of known energy values, in particular the spherical and ellipsoidal quantum dots, the method was verified and tested for convergence along with an analysis of the level of sparsity of the Hamiltonian matrix which could be helpful to maximize memory efficiency for future problems. The calculations of CdSe rod-shaped quantum dots were compared to actual photoluminescence spectra, and good agreement (3% or less percent difference) was found when the rod width was between 4.8 and 6.4 nm, hovering around the CdSe Bohr exciton radius of 5.6 nm.