EXCITONIC WAVE-FUNCTION, CORRELATION-ENERGY, EXCHANGE ENERGY, AND OSCILLATOR STRENGTH IN A CUBIC QUANTUM-DOT

We present a variational calculation of the envelope wave function of an exciton inside a cube. We show that all sixfold integrals can be re duced to threefold integrals with a change of variables which shortens tremendously the computer calculations. Then(i) we calculate numerica lly the Bohr radius, the correlation energy (the difference between th e energy of the exciton and of the uncorrelated electron-hole pair), t he exchange energy, and the oscillator strength of the exciton for any value of the cube side and (ii) we obtain asymptotic expression of th ese four quantities for a large cube. This allows one to discern the d ifference between a large but finite cubic semiconductor and an infini te semiconductor. We show that the correlation energy tends to its bul k limit value in infinite volume as the inverse of the cube side while the effective Bohr radius and the exchange energy tend to their limit ing value as the inverse of the square of the side. Finally our result s are applied to porous silicon: the experimental exchange energy 10 m eV corresponds to a cube of side 28 Angstrom which is quite reasonable in this compound.