Band-structure calculation of dispersion and anisotropy in χ→(3)for third-harmonic generation in Si, Ge, and GaAs

Abstract

We perform a full band-structure calculation of the dispersion, anisotropy, and magnitude of the nonlinear response \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$ for optical third-harmonic generation in Si, Ge, and GaAs, using both an empirical tight-binding (ETB) and a semi--ab initio band-structure technique. The calculation is performed with use of standard perturbation theory within the random-phase approximation and neglect of local-field corrections, with the minimal-coupling interaction Hamiltonian. The sign of \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$(0) is positive, independent of the tight-binding approximation, and in agreement with experimental results from the literature. We find that intraband contributions to \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$ are small, in contrast with earlier results obtained by using the dipole interaction Hamiltonian. While both expressions for \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$ yield a positive \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$(0), the minimal-coupling expression is dominated by the interband response, whereas the dipole expression is dominated by the intraband response. The resulting anisotropy in \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$ obtained from the ETB bands for all materials agrees with the experiment better than other calculations to date at frequencies far below the optical band gap. For silicon, the anisotropy calculated from the ETB band structure also agrees well with recent measurements on the dispersion of the anisotropy at frequencies comparable to the optical band gap. The values of \ensuremath{\Vert}\ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$(0)\ensuremath{\Vert} obtained from the ETB bands overestimate the experimental values by factors of 2 (for Si and GaAs) to 5 (for Ge), while the results obtained from the semi--ab initio bands underestimate the experimental values by a factor of approximately 2 (for Si) and 7 (for Ge and GaAs). While the (ETB) results for Si and GaAs are approximately within experimental error, the difference between the results from the two band-structure calculations reflects the high sensitivity of \ensuremath{\chi}\ensuremath{\rightarrow} $^{(3)}$ to the details of the energy bands and wave functions.