Abstract
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 X(3)⁄n3D = 2:9 to 580x10-38m5V2 depending on the frequency even while avoiding resonance. Using samples of reasonable density n3D 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 X(3) should exceed that of bulk InSb and X(3)L should exceed that of graphene and resonantly enhanced quantum wells.