Possible Pairing without Superconductivity at Low Carrier Concentrations in Bulk and Thin-Film Superconducting Semiconductors

Simultaneous equations for the BCS parameter $\Delta$ and for the Fermi energy in the BCS state are studied at low carrier concentrations in bulk and thin-film superconducting semiconductors at $T=0\mathrm{}$ for large cutoff energies, and solutions are plotted in the form of universal curves. For electron-electron attraction greater than some critical value in bulk material or for any attractive interaction in very thin films, the pair-binding energy tends to a constant limit for low $n$, and pairing without superconductivity is expected in some temperature range. Solutions of simultaneous equations for $T_c$ as obtained by formal application of the BCS theory, and for the Fermi energy at $T_c$, are also obtained in bulk material; but it is shown that at low carrier concentrations the temperature $T_p$ at which pairing takes place is given by $T_p=2\mathrm{}{T}_c$, while it is thought that superconductivity will not set in until a lower temperature, of the order of the Bose-Einstein condensation temperature of the pairs, is reached. By combining the above theory with a previously published model for superconductivity in Zr-doped SrTi${\mathrm{O}}_3$, predictions for this material in the region of low carrier concentrations are made.