We derive the nonlinear-optical coefficients of insulators using a fully quantum-mechanical theory of the electron-photon interaction. Using the minimal-couping interaction, we find an alternative interpretation for the absence of the vector-potential-squared (${\mathit{A}}^2$) term in the second-order response. Specializing to the case of up-conversion, we use the resulting expression, together with empirical tight-binding band-structure calculations, to compute the second-order susceptibility ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ of bulk semiconductors in the static limit and of semiconductor heterostructures at resonance. These nonlinear-optics calculations are based on the empirical tight-binding model without additional parameter fit beyond the band-structure model.

The calculated bulk values of ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$(0) are much smaller than available experimental values. This provides an independent and sensitive test on the accuracy of the higher tight-binding conduction states and also reveals the need for more conduction states. Calculations for the heterostructures were done for GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As and HgTe/${\mathrm{Hg}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Cd}}_{\mathit{x}}$Te quantum wells. The ‘‘macroscopic’’ asymmetry of the quantum structures is reflected in the optical properties. Our calculations of intersubband ${\mathrm{\chi }}^{\mathrm{}\left(2\right)\mathrm{}}$ for the ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As system confirm similar results obtained using one-band calculations and agree with recent experimental results. The results predict interesting differences with respect to the ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As system due to the inverted nature of the HgTe band structure. We obtain a nonzero interconduction subband c1-c3 optical coupling in a symmetric HgTe quantum well, even at the zone center of the Brillouin zone. Our analysis of ‘‘symmetric’’ quantum wells predicts the possibility of second-order nonlinearity due to the tetragonal ${\mathit{D}}_{2\mathit{d}}$ symmetry. This possibility had been overlooked in previous discussions of nonlinearity based on the quantum-well picture. The contribution due to interconduction subband transitions in the symmetric wells is negligible. On the other hand, for the p-doped wells of both material systems, we predict a susceptibility that is two orders of magnitude larger than typical bulk values and, hence, should be amenable to experimental verification.

• Received 11 August 1994

DOI:https://doi.org/10.1103/PhysRevB.50.14421