Importance of valence-band anharmonicity and carrier distribution for third- and fifth-harmonic generation in Si:B pumped with intense terahertz pulses

We observe third-harmonic generation (THG) and fifth-harmonic generation (FHG) of free holes in the valence band of bulk Si:B at cryogenic temperature upon irradiation with intense terahertz pulses from a free-electron laser, polarized along the -X direction. The intensities of both signals increase as a function of pulse energy following power laws with respective exponents of 4.2 and 6.2, larger than the exponents of 3 and 5 expected for x (3) and x (5) processes with constant susceptibilities and a fixed number of holes. The larger values are attributed to the increase in the density of mobile holes, which results from impact ionization of boron dopants by thermally activated holes in the electric field of the terahertz pulses as already observed in our studies of THG in Si:B reported in Phys. Rev. B 102, 075205 (2020). We apply Monte Carlo simulations of the nonlinear heavy-and light-hole field response, coupled with a finite-difference time-domain treatment of the pump-pulse propagation in the sample, which reproduce the experimental THG:FHG intensity ratio reasonably well. An analysis of the local response demonstrates that, in our pump regime, the harmonic generation is dominated by the band anharmonicity as opposed to the energy-dependent momentum scattering rates or interband scattering. Comparison with the results of a simple one-dimensional model and scrutiny of the three-dimensional band structure shows that one must account for the extent of the carrier distribution transverse to the P -X axis as the anharmonicity grows rapidly away from this axis.