Eigen-Energies and Eigen-Functions of Symmetroidal Quantum Dots

A general method for evaluating the energy levels and wavefunctions of symmetroidal quantum dots is proposed, via solving the three-dimensional time-independent Schrodinger equation for the envelope function in the effective mass approximation. Energy levels and corresponding wavefunctions were numerically calculated for the cases of a cylinder-shaped dot, a cone-shaped dot and a lens-shaped dot. By examining the convergence of the ground state energy to the asymptotic value as a function of the matrix dimension N for each shape of dots, we found that the errors in the energies with N=400 were less than 1% when compared with N=1600. By examining the wavefunctions of dots with a height (h0) of 10 nm and a diameter (d) of 50 nm (i.e. α= h0/d=0.2), we found that, for all shapes examined, the probability densities showed one hump in the z-direction but two humps in the r-direction in the well regions, and the probability densities diminished rapidly in the barrier regions. To estimate the ground state energies of pyramidal QDs, a method is proposed which closely sandwiches a pyramid with two cones. We found that a 400×400 matrix was sufficient to yield the average energies of two cones and to approximate the ground state energies of pyramidal QDs with errors of less than±1%. When the energy shifts and the overlap integrals were calculated as a function of the volume of the dots for optical transitions, the influence of the shapes of the dots on the magnitudes of the energy levels and overlap integrals were not significant, and the errors were within 4%.