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

Abstract

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)/n3D = 2.9 to 580 × 10−38 m5/V2 depending on the frequency, even while avoiding resonance. Using samples of a 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 χ(3) should exceed that of bulk InSb and χ(3)L should exceed that of graphene and resonantly enhanced quantum wells.

Highly efficient THz four-wave mixing in doped silicon

“…Doped silicon at low temperature has already been shown to produce giant values for the imaginary part of the non-linear refractive index (via multi-photon absorption) 12 , and there have been theoretical predictions that the real part of the non-linear refractive index is also very large 13 and of large experimental non-linearities 14 but there have been no experimental reports of χ (3) till now, largely because of the challenge of quantitative non-linear THz metrology.…”

The multi-photon induced Fano effect

“…( 5 )) appearing here is over an infinite number of intermediate states. In a similar way to the problem of two-photon absorption between bound states, implicit summation 20 , 21 , 35 can reduce this difficulty substantially. In this technique becomes the solution of partial differential equation (PDE) that does not require construction of .…”

“…These values have some significant uncertainty: the experimentally fitted value of Γ has an error of about 30%, and there is also potentially large systematic uncertainty in the experimental field amplitude due to the problem of pulse temporal profile estimation. Furthermore, the hydrogen scaling theory does not take account of the valley multiplicity, mass anisotropy or central cell correction 21,26 . The effect of these theoretical corrections is illustrated by looking at the 2p 0 transition, as the scaled hydrogen model would predict that all components of the 2p would have the same q and Γ.…”

“…Using multi-photon transitions, both the Fano shape index and width are controllable, and here we show that by making use of a THz-coupling beam the ICS can be obtained with a relatively weak beam. We used hydrogenic donor impurities in silicon, which have very large multi-photon cross-sections 20,21 producing a shape index close to unity for an easily accessible field amplitude of only a few 10s of kV/cm. Many-electron atoms possess discrete states in which each electron remains localized, but the total energy of all the electrons is enough to ionize one of them.…”

“…Just as silicon double donors are, in many respects, semiconductor analogues of free helium atoms and exhibit the classic Fano auto-ionization structures 5 , so silicon shallow donors are analogues of free hydrogen atoms. They exhibit a reduced Hartree energy E H and an increased Bohr radius a B 26,27 , and this scaling tends to enhance nonlinear optical effects and reduce the intensities needed 20,21 ; hence, their interest for ICS here.…”