Giant non-linear susceptibility of hydrogenic donors in silicon and germanium

Implicit summation is a technique for the 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 χ (3) / n 3D = 2.9 to 580 × 10 −38 m 5 /V 2 depending on the frequency, even while avoiding resonance. Using samples of a reasonable density, n 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 χ (3) should exceed that of bulk InSb and χ (3) L should exceed that of graphene and resonantly enhanced quantum wells.