Optical second-harmonic generation in III-V semiconductors: Detailed formulation and computational results

In an earlier paper [Phys. Rev. Lett. 66, 41 (1991)], we calculated both the dielectric constant (${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\infty}}}$) and the nonlinear optical susceptibilities for second-harmonic generation (${\mathrm{\ensuremath{\chi}}}^{(2)}$) in the static limit for AlP, AlAs, GaP, and GaAs in the local-density approximation with and without a self-energy correction in the form of a ``scissors operator,'' including local-field effects. In this paper, we expand our presentation of this calculation. Agreement with experiment to within 15% for the nonlinear susceptibility is demonstrated where experiments are available (GaP and GaAs); the dielectric constants are in no worse than 4% agreement with experiment. The ``virtual hole'' contributions are reformulated to avoid large numerical cancellations in the case of near degeneracies. The ``virtual electron'' terms dominate over the ``virtual hole'' terms by about one order of magnitude. Local-field corrections are smaller than the main terms by about one order of magnitude. The formulas needed to apply a self-energy correction in the form of a ``scissors operator'' to this problem are presented. The addition of a self-energy correction requires a renormalization of the velocity operator; a failure to include the velocity-operator renormalization leads to a factor-of-2 correction to ${\mathrm{\ensuremath{\chi}}}^{(2)}$, destroying the good agreement with experiment. The neglect of the short-wave charge induced at the second-harmonic frequency is justified. The f-sum rule and another, related sum rule for second-harmonic generation is well satisfied numerically. For well-converged results, a plane-wave-basis-set energy cutoff of 9--12 hartrees is required for GaAs, but only eigenfunctions with eigenvalues less than about 1--2 hartrees need be included. Using a special-points integration scheme, 10 points are not sufficient, 28 points are typically adequate, and for the material considered with the smallest band gap, GaAs, 60 special points are marginally desirable.