Metal-Insulator Transition in $n$-type bulk crystals and films of strongly compensated SrTiO$_3$.

We start by analyzing experimental data of Spinelli \textit{et al}~\cite{spinelli2010} for conductivity of $n$-type bulk crystals of SrTiO$_3$ (STO) with broad electron concentration $n$ range of $4\times 10^{15}$ - $4 \times10^{20} $ cm$^{-3}$, at low temperatures. We obtain good fit of the conductivity data, $\sigma(n)$, by the Drude formula for $n \geq n_c \simeq 3 \times 10^{16} $ cm$^{-3}$ assuming that used for doping insulating STO bulk crystals are strongly compensated and the total concentration of background charged impurities is $N = 10^{19}$ cm$^{-3}$. At $n< n_c$ 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 hydrogen-like donor state $a_B \simeq 700$ nm with which the Mott criterion of MIT $na_B^3 \simeq 0.02$ 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 $N=10^{19}$ cm$^{-3}$. 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 $d \leq 130$ 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. We also study the random potential in a graphene sample on the top of an insulating STO bulk crystal, and find reasonable agreement with experiments using the same $N= 10^{19}$ cm$^{-3}$.