  We report on a strong nonlinear behavior of the photogalvanics and
photoconductivity under excitation of HgTe quantum wells (QWs) by intense
terahertz (THz) radiation. The increasing radiation intensity causes an
inversion of the sign of the photocurrent and transition to its superlinear
dependence on the intensity. The photoconductivity also shows a superlinear
raise with the intensity. We show that the observed photoresponse
nonlinearities are caused by the band-to-band \emph{light} impact ionization
under conditions of a photon energy less than the forbidden gap. The signature
of this kind of impact ionization is that the angular radiation frequency
$\omega=2\pi f$ is much higher than the reciprocal momentum relaxation time.
Thus, the impact ionization takes place solely because of collisions in the
presence of a high-frequency electric field. The effect has been measured on
narrow HgTe/CdTe QWs of 5.7\,nm width; the nonlinearity is detected for
linearly and circularly polarized THz radiation with different frequencies
ranging from $f=0.6$ to 1.07\,THz and intensities up to hundreds of kW/cm$^2$.
We demonstrate that the probability of the impact ionization is proportional to
the exponential function, $\exp(-E_0^2/E^2)$, of the radiation electric field
amplitude $E$ and the characteristic field parameter $E_0$. The effect is
observable in a wide temperature range from 4.2 to 90\,K, with the
characteristic field increasing with rising temperature.
