Metal-insulator transition in n -type bulk crystals and films of strongly compensated SrTiO 3

We start by analyzing experimental data of Spinelli et al. [Phys. Rev. B 81, 155110 (2010)] for the conductivity of $n$-type bulk crystals of ${\mathrm{SrTiO}}_{3}$ (STO) with broad electron concentration $n$ range of $4\ifmmode\times\else\texttimes\fi{}{10}^{15}$--$4\ifmmode\times\else\texttimes\fi{}{10}^{20}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$, at low temperatures. We obtain a good fit of the conductivity data, $\ensuremath{\sigma}(n)$, by the Drude formula for $n\ensuremath{\ge}{n}_{c}\ensuremath{\simeq}3\ifmmode\times\else\texttimes\fi{}{10}^{16}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ assuming that used for doping insulating STO bulk crystals are strongly compensated and the total concentration of background charged impurities is $N={10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$. At $nl{n}_{c}$, the conductivity collapses with decreasing $n$ and the Drude theory fit fails. We argue that this is the metal-insulator transition (MIT) in spite of the very large Bohr radius of hydrogenlike donor state ${a}_{B}\ensuremath{\simeq}700$ nm with which the Mott criterion of MIT for a weakly compensated semiconductor, $n{a}_{B}^{3}\ensuremath{\simeq}0.02$, predicts ${10}^{5}$ times smaller ${n}_{c}$. We try to explain this discrepancy in the framework of the theory of the percolation MIT in a strongly compensated semiconductor with the same $N={10}^{19}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$. In the second part of this paper, we develop the percolation MIT theory for films of strongly compensated semiconductors. We apply this theory to doped STO films with thickness $d\ensuremath{\le}130$ nm and calculate the critical MIT concentration ${n}_{c}(d)$. We find that, for doped STO films on insulating STO bulk crystals, ${n}_{c}(d)$ grows with decreasing $d$. Remarkably, STO films in a low dielectric constant environment have the same ${n}_{c}(d)$. This happens due to the Rytova-Keldysh modification of a charge impurity potential which allows a larger number of the film charged impurities to contribute to the random potential.