Theory of Transitions from Large to Nearly-Small Polarons, with Application to Zr-Doped Superconducting SrTiO3

Expressions for the matrix elements of the Hamiltonian and overlap integrals between large-polaron and nearly-small-polaron states are given in terms of various large- and nearly-small-polaron parameters. The dependence of the mass of the ground state on the bare-electron mass in a region of transition between large- and nearly-small-polaron states is found. Parameters of the theory are evaluated using the continuum-polarization model of electron-phonon interactions. Modifications due to anisotropy and the presence of many longitudinal polar modes of lattice vibration are discussed briefly. Making use of a recent analysis by Reik of free-carrier optical-absorption results in SrTi${\mathrm{O}}_3$ using small-polaron theory, it is shown that the rate of mass rise deducible from superconductivity data on Zr-doped SrTi${\mathrm{O}}_3$ can be explained by the theory if the bare transverse mass $m_{\mathrm{bt}}$ is taken to be $m_{\mathrm{bt}}\simeq 0.6m_e$, where $m_e$ is the free-electron mass, and if small-polaron binding energies are assumed to be larger by a factor of about 1.4 than their values in the continuum-polarization model. The dependence on Zr concentration of the ratio of phonon-induced to Coulomb intervalley electron-electron interactions deducible from the superconductivity data is discussed within the framework of the model.