In this work, the effects of quantum confinement on the ground state energy of a correlated electron–hole pair in a spherical and in a disc-like quantum dot have been investigated as a function of quantum dot size. Under parabolic confinement potential and within effective mass approximation Ritz's variational method is applied to Hylleraas-like trial wavefunction. An efficient method for reducing the main effort of the calculation of terms like rehk exp(−λreh) is introduced. The main contribution of the present work is the introduction of integral transforms which provide the calculation of expectation value of energy and the related matrix elements to be done analytically over single-particle coordinates instead of Hylleraas coordinates.

In this study the electronic eigenstructure of an exciton in a parabolic quantum dot (QD) has been calculated with a high accuracy by using Finite element method (FEM). We have converted the coordinates of electron-light-hole system to relative and center of mass coordinate, then placed the Spherical Harmonics into Schrödinger equation analytically and obtained the Schrödinger equation which depends only on the radial variable. Finally we used FEM with only radial variable in order to get the accurate numerical results. We also showed first 21 energy level spectra of exciton depending on confinement and Coulomb interaction parameters.

In this work, we propose an efficient method of reducing the computational effort of variational calculation with a Hylleraas-like trial wavefunction. The method consists of introducing integral transforms for the terms as r k 12 exp (−λr 12 ) which provide the calculation of the expectation value of energy and the relevant matrix elements to be done analytically over single-electron coordinates instead of Hylleraas coordinates. We have used this method to calculate the ground state energy of a two-electron system in a spherical dot and a disk-like quantum dot separately. Under parabolic confinement potential and within effective mass approximation size and shape effects of quantum dots on the ground state energy of two electrons have been investigated. The calculation shows that our results even with a small number of basis states are in good agreement with previous theoretical results.

A study of variational wave functions for calculation of the ground-state energies of excitons confined in a two-dimensional (2D) disc-like and three-dimensional (3D) spherical parabolic GaAs quantum dots (QDs) is presented. We have used four variational trial wave functions constructed as the harmonic-oscillator basis multiplied by different correlation functions. The proposed correlation function formed by including linear expansion in terms of Hylleraas-like coordinates to the Jastrow factor is able to capture nearly exactly the ground-state energies of 3D excitons, and it properly account for the results of 2D excitons. Quantum Monte Carlo techniques combined with the proposed wave function are a powerful tool for studying excitons in parabolic QDs.