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

We analyze 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 Drude formulla 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 Drude theory fails and the conductivity collapses with decreasing $n$. We argue that this is the metal-insulator transition (MIT) in spite of the fact that due to the very large Bohr radius of hydrogen-like donor state $a_B \simeq 700$ nm the Mott criterion of MIT $na_B^3 \simeq 0.02$ predicts $10^{5}$ times smaller $n_c$. We 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$^{-3}$. We extend this MIT theory to doped STO films with thickness $d \leq 100$ nm and calculate critical MIT electron concentration $n_c(d)$. We find that doped STO films on insulating STO bulk crystals or in a low dielecric constant environment like silicon oxide have the same $n_c(d)$ which grows with decreasing $d$. 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 $N= 10^{19}$ cm$^{-3}$. The theory of the percolation MIT developed here for STO films is valid for all other compensated semiconductor films.