Soft-Mode Superconductivity inSrTiO3−x

The transition temperature $T_c$ of $\mathrm{SrTi}{\mathrm{O}}_{3-x}$ is calculated as a function of the electron concentration. The effective electron-electron interaction has an attractive part caused by the intravalley exchange of soft optical phonons that have been observed in the tetragonal phase with Raman scattering and neutron diffraction. The Coulomb part is determined by the longitudinal dielectric constant for a degenerate polar semiconductor. The deformation potential for the interaction between electrons and soft phonons of symmetry $A_{1g}$ is used as an adjustable parameter to fit $T_c$ at one experimental point.