Predicted Transition Temperatures of Very Thin Films and Whiskers of Superconducting Semiconductors—Application to SrTiO3

Approximate solutions of the BCS energy-gap equation with constant kernel are found for superconducting semiconductors at carrier concentrations such that the distance of the Fermi level above the bottom of the band is less than or of the order of the relevant phonon energy. Results of Tavger and Demikhovskii for the energy gap in effectively two-dimensional films are generalized to all values of the Fermi energy, and approximate gaps for effectively one-dimensional whiskers are calculated. Screening by free carriers in quasi-one- and two-dimensional systems is studied by the Lindhard method; the results lend some support to the hypothesis that in many-valleyed semiconductors the interaction strength between electrons will not be greatly altered by production in suitably oriented thin-film or whisker form. With the usual BCS relation between transition temperature and energy gap, and using a simple model with an interaction which is equal to a constant for electrons within a phonon energy 0.099 eV of the Fermi level and zero otherwise, an effective interaction strength $V$ as a function of carrier concentration in bulk SrTi${\mathrm{O}}_3$ is deduced from published experimental results on concentration dependence of transition temperature in this material. Hence, assuming a localized interaction of the same strength occurs in effectively two-dimensional films or one-dimensional whiskers, which assumption implies, respectively, effective interactions $V_2$ and $V_1$ given by $V_2=\frac{3}{2}V$ and $V_1={\left(\frac{3}{2}\right)}^{2}V$, transition temperatures for films and whiskers are predicted as a function of carrier concentration for several different thicknesses. Transition temperatures higher than those of any known superconductor are predicted on this model for (111) films of 7 and 14 Å thicknesses, and for [111] whiskers of ${\left(7\mathrm{\AA }\right)}^2$, ${\left(14\mathrm{\AA }\right)}^2$, and ${21\mathrm{\AA })}^2$ cross sections, but at such small thicknesses the model can be expected to give only indications of trends.