We develop a theory of superconducting pairing in low-density strontium titanate due to the quadratic coupling of electron density to soft transverse optical phonons first proposed by Ngai [Phys. Rev. Lett. 32, 215 (1974)]. This leads to a static attractive potential between electrons with decay length $l_{\mathrm{eff}}$ that scales inversely with soft optical gap ${\omega }_T$. For low electron densities $n\le {10}^{18}$ ${\mathrm{cm}}^{-3}$, attraction between electrons is static and local in space; thus the transition temperature $T_c$ was found using known results for low-density electron gas. The $T_c\left(n\right)$ dependence for low doping was calculated and found to be in agreement with experimental data. In addition, we show that suppression of $T_c$ by hydrostatic pressure and strong increase in $T_c$ due to isotope substitution ${}^{16}\mathrm{O}{\to }^{18}\mathrm{O}$ observed experimentally also can be explained within our theory.

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• Electron-phonon coupling
• Isotope effect
• Optical phonons
• Pairing mechanisms
• Superconductivity