Phonons and superconductingp-state pairing in Pd

Abstract

We present an analytical calculation of the electron-phonon parameters for $s$- and $p$-state pairing in Pd, ${\lambda }_0^{\mathrm{ph}}$ and ${\lambda }_1^{\mathrm{ph}}$. The calculation is based on Doniach's six half-sphere Fermi-surface model for the heavy $d$ electrons, a Debye and an Einstein model for intra- and intersphere phonon-scattering processes, respectively, and the LCMTO (linear combination of muffin-tin orbitals) representation to obtain the electron-phonon matrix elements. The group-theoretical method of Allen is employed in solving the anisotropic gap equation at the multiply sheeted Fermi surface for $s$- and $p$-state pairing interactions. The phonon-mediated interaction, $I^{\mathrm{ph}}(\overrightarrow{\mathrm{k}},{\overrightarrow{\mathrm{k}}}^{\prime })$ is attractive for all momentum transfers, $\overrightarrow{\mathrm{k}}-{\overrightarrow{\mathrm{k}}}^{\prime }$, and yields the BCS parameter, ${\lambda }_0^{\mathrm{ph}}=0.153\mathrm{}$, in agreement with Papaconstantopoulos et al. Our results for $p$-state pairing is ${\lambda }_1^{\mathrm{ph}}=0\mathrm{}$, when Einstein phonon model is used for both intra- and intersheet scattering processes and ${\lambda }_1^{\mathrm{ph}}=0.089\mathrm{}{\lambda }_0^{\mathrm{ph}}$ when the Debye model is applied to the small-momentum intrasheet transitions of the Cooper pairs. The weakness of the $p$-state interaction is due to the cancellation between the contributions from intrasheet scattering and from large-momentum intersheet scattering processes. A detailed discussion of the results is presented in terms of the parameters of our electron-phonon model.