Multi-valley theory for giant non-linear susceptibility of hydrogenic donors in silicon and germanium

Implicit summation is a technique for conversion of sums over intermediate states in multiphoton absorption and the high-order susceptibility in hydrogen into simple integrals. Here we derive the equivalent technique for hydrogenic impurities in multi-valley semiconductors. While the absorption has useful applications it is primarily a loss process, conversely the non-linear susceptibility is a crucial parameter for active photonic devices. For Si:P we predict the hyperpolarizability ranges from $\chi^{(3)}/n_{\text{3D}}=2.9$ to $580 \times 10^{-38} \text{m}^5/\text{V}^2$ depending on the frequency even while avoiding resonance. Using samples of reasonable density $n_{\text{3D}}$ and thickness $L$ to produce third harmonic generation at 9 THz, a frequency that is difficult to produce with existing solid state sources, we predict that $\chi^{(3)}$ should exceed that of bulk InSb and $\chi^{(3)}L$ should exceed that of graphene and resonantly enhanced quantum wells.