  Heavily doped semiconductors have emerged as tunable low-loss plasmonic
materials at mid-infrared frequencies. In this article we investigate nonlinear
optical phenomena associated with high concentration of free electrons. We use
a hydrodynamic description to study free electron dynamics in heavily doped
semiconductors up to third-order terms, which are usually negligible for noble
metals. We find that cascaded third-harmonic generation due to second-harmonic
signals can be as strong as direct third-harmonic generation contributions even
when the second-harmonic generation efficiency is zero. Moreover, we show that
when coupled with plasmonic enhancement free electron nonlinearities could be
up to two orders of magnitude larger than conventional semiconductor
nonlinearities. Our study might open a new route for nonlinear optical
integrated devices at mid-infrared frequencies.
