Excitonic wave function, correlation energy, exchange energy, and oscillator strength in a cubic quantum dot

Abstract

We present a variational calculation of the envelope wave function of an exciton inside a cube. We show that all sixfold integrals can be reduced to threefold integrals with a change of variables which shortens tremendously the computer calculations. Then (i) we calculate numerically the Bohr radius, the correlation energy (the difference between the energy of the exciton and of the uncorrelated electron-hole pair), the exchange energy, and the oscillator strength of the exciton for any value of the cube side and (ii) we obtain asymptotic expression of these four quantities for a large cube. This allows one to discern the difference between a large but finite cubic semiconductor and an infinite semiconductor. We show that the correlation energy tends to its bulk limit value in infinite volume as the inverse of the cube side while the effective Bohr radius and the exchange energy tend to their limiting value as the inverse of the square of the side. Finally our results are applied to porous silicon: the experimental exchange energy 10 meV corresponds to a cube of side 28 Å which is quite reasonable in this compound.