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 (${\mathrm{\epsilon }}_{\mathrm{\infty }}$) and the nonlinear optical susceptibilities for second-harmonic generation (${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$) 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 ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$, 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.