  In this paper we investigate the effect of Cd doping at ultra-low
concentrations in SnO$_2$ both experimentally, by measuring the temperature
dependence of the electric quadrupole hyperfine interactions with
time-differential perturbed angular correlation (TDPAC) spectroscopy using
$^{111}$Cd as probe nuclei, and theoretically, by performing first-principles
calculations based on the density functional theory. TDPAC spectra were
successfully analyzed with a time-dependent on-off model for the perturbation
factor. These results show combined dynamic plus static interactions whose
electric-field gradients were associated in this model to different stable
electronic configurations close to the Cd atoms. The dynamic regime is then
originated in fast fluctuations between these different electronic
configurations. First-principles calculations results show that the Cd impurity
introduces a double acceptor level in the top of the valence band of the doped
semiconductor and produces isotropic outward relaxations of the nearest oxygen
neighbors. The variation of the calculated electric-field gradient tensor as a
function of the charge state of the Cd impurity level shows an interesting
behavior that explains the experimental results, giving strong support from
first-principles to the electron-capture after-effects proposed scenario. The
electron-capture decay of the parent $^{111}$In to $^{111}$Cd as well as the
double acceptor character of the $^{111}$Cd impurity and the electric nature of
the host are shown to contribute to the existence of these type of
time-dependent hyperfine interactions.
