Energy-Band Theory of the Second-Order Nonlinear Optical Susceptibility of Crystals of Zinc-Blende Symmetry

The zero-frequency limit of the second-order nonlinear optical susceptibility ${\ensuremath{\chi}}_{\mathrm{ijk}}^{(2)}$ is calculated within the framework of energy-band theory. Both vector and scalar potential representations of the electric field are investigated. The apparent divergences of standard finite-frequency expressions in the limit of zero frequency are shown to vanish. The two- and three-band contributions in the scalar potential representation, describing the electric field effect on the coherent (phase factor) and cell-periodic parts of the Bloch function, respectively, are shown to combine as required by gauge invariance to yield the result obtained in the vector potential representation, where only three-band terms contribute. In crystals of zinc-blende symmetry, virtual electronic processes involving one valence and two conduction bands, originating from the ${\ensuremath{\Gamma}}_{15}$ valence and ${\ensuremath{\Gamma}}_{1}$ and ${\ensuremath{\Gamma}}_{15}$ conduction states at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=0$, dominate virtual-hole processes involving one conduction and two valence bands. The sign of ${\ensuremath{\chi}}_{123}^{(2)}$ is related to the Brillouin-zone average of the ordering with respect to energy of the lowest conduction bands. The magnitude of ${\ensuremath{\chi}}_{123}^{(2)}$ is related to the average over the Brillouin zone of the inverse fifth power of the local energy separation ${E}_{\mathrm{cv}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ between the valence and lowest conduction bands, in contrast to the inverse third power average of the same quantity known for the linear susceptibility ${\ensuremath{\chi}}^{(1)}$. This enhance the contribution to ${\ensuremath{\chi}}_{123}^{(2)}$ from smallgap regions of the band structure in a manner similar to that also found for ${\ensuremath{\chi}}^{(3)}$. The value of ${\ensuremath{\chi}}_{123}^{(2)}$ calculated for zinc-blende crystals in both the constant energy gap and the parabolic criticalpoint models is consistently lower than that observed experimentally. Extended calculations using more realistic analytic approximations to the local energy-band structure suggest that this discrepancy is due in part to band nonparabolicity effects with additional contributions from enhancement of oscillator strength from the electron-hole Coulomb interaction.