Integral $ab$ $initio$/DFT and experimental TDPAC approach enlightening the $aftereffects$ phenomenon: probing electronic properties in $α$-Al$_2$O$_3$:$^{111}$In($\rightarrow$ $^{111}$Cd) at the atomic scale

Abstract: By means of an integral experimental and $ab$ $initio$/DFT approach we contribute here to enlighten and quantify the origin of dynamic hyperfine interactions (HFIs) assigned to the electron-capture (EC) decay aftereffects (ECAE) phenomenon observed in time-differential perturbed $\gamma$-$\gamma$ angular correlation (TDPAC) experiments in oxides doped with ($^{111}$In (EC)$\rightarrow$)$^{111}$Cd as probe-atom. In previous works [Darriba et al., Phys. Rev. B 105, 195201 (2022)] we proposed an $ab$ $initio$ scenario in which the fluctuating electric-field gradients (EFG) producing the dynamic HFI were related with fluctuating electronic environments close to the $^{111}$Cd nucleus, succeeding to identify the environment which produce the final static EFG when the dynamic ($on-off$) process have stopped. In this work we show that in addition it is possible to obtain, for each temperature and HFI observed, the set of initial electronic configurations close to the probe nucleus as well as their related EFGs among which the system fluctuates to generate these dynamic HFIs. For this, we demonstrate analytically and checked experimentally the conditions to stablish the equivalence between the two approaches most used to analyze this type of dynamic HFIs, proposed by B\"averstam et al. and by Lupascu et al.. To unravel the unexpected TDPAC results in $^{111}$In($\rightarrow$ $^{111}$Cd)-implanted $\alpha$-Al$_2$O$_3$ single crystals reported in the literature, we perform a complete $ab$ $initio$/DFT study of Cd-doped $\alpha$-Al$_2$O$_3$ semiconductor and a detailed defect formation energy analysis as a function of the charge state of the Cd impurity. The presence of an unexpected second interaction was a key factor to provide experimental support to identify and quantify the different charge states the $^{111}$Cd atom goes through during its electronic recovery process.