Dmitry E. Kiselov
We develop a theory of superconducting pairing in low-density Strontium titanate based upon the idea of quadratic coupling of electron density to soft transverse optical phonons [1]. This coupling leads to static attractive potential between electrons which decays in real space exponentially with the length $l_{eff}$ that scales inversely with soft optical gap $\omega_T$ . For low electron densities $n \leq 10^{18} cm^{-3}$ attraction between electrons can be considered local and superconducting transition temperature $T_c$ is calculated using old results [2]. We use independently obtained magnitude g of the quadratic coupling strength and find $T_c(n)$ dependence in agreement with experimental data [3] for low doping. Next, we show that suppression of $T_c$ by hydrostatic pressure [4] and strong increase of $T_c$ due to isotop substitution $^{16}O \to\,^{18}O$ observed in [5] are qualitatively explained within our theory.
[1] K. L. Ngai, Phys. Rev. Lett. 32, 215 (1974)
[2] L. P. Gor’kov and T. K. Melik-Barkhudarov, Sov. Phys. – JETP 40, 1452 (1961)
[3] X. Lin, et al, Phys. Rev. Lett. 112, 207002 (2014).
[4] C. Enderlein, et al, Nature Comm. (2020)
[5] A. Stucky, et al, Scientific Reports, 6:37582 (2016)