Soft mode superconductivity in SrTiO3− x and CaySr1− yTiO3− x

Abstract

In these nearly ferroelectric titanates, the effective electron-electron interaction has an attractive part caused by the virtual exchange of soft optical phonons which have been observed in the tetragonal phase with Raman scattering and neutron diffraction. These phonons have even symmetry ( A 1 g at q = 0) and their electron-phonon matrix element is finite, independent of the symmetry character of the electron states. The matrix element is proportional to the order parameter for the cubic → tetragonal phase-transition, namely, the angle of rotation ϕ of the oxygen octahedra around the tetragonal axis (ϕ is different for the two titanates). The Coulomb part of the effective interaction is determined by the longitudinal dielectric constant for a degenerate polar semiconductor. In the important low-frequency range where the soft phonons provide an attractive interaction, the Coulomb repulsion is weak because the lattice polarization can follow the motion of a conduction electron. With this effective interaction, the superconducting transition temperature T c of SrTiO 3− x and Ca y Sr 1− y TiO 3− x is calculated as a function of the electron concentration. The deformation potential for the interaction between electrons and soft optical phonons is used as an adjustable parameter to fit T c at one experimental point. The results are in good agreement with the experimental data.