Tight-binding bonding orbital model for third-order nonlinear optical susceptibilities in group-IV crystals

The nonresonant part of the third-order nonlinear optical susceptibility ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ has been calculated for diamond, silicon, and germanium by using the tight-binding Bloch states constructed from $s{p}^3$ hybridized orbitals. Only the nearest-neighbor bond interactions have been retained in the calculations. The three independent energy parameters, which enter in the model, are fitted by comparing the calculated and known experimental values of the valence-band width at $\overrightarrow{\mathrm{k}}=0\mathrm{}$, the direct energy gap, and the linear dielectric constant. An explicit exponential form of the $s$- and $p$-type atomic orbitals is used in order to obtain the required dipole matrix elements in each case. Although the sign of the calculated ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ agrees with the experimental values in each case, the agreement in magnitude of ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ is best attained for diamond where the tight-binding model is expected to be very good.