Linear response study of strong electron-phonon coupling in yttrium under pressure

Abstract

Linear response methods are applied to identify the increase in electron-phonon coupling in elemental yttrium that is responsible for its high superconducting critical temperature ${T}_{c}$, which reaches nearly $20\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ at $115\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. While the evolution of the band structure and density of states is smooth and seemingly modest, there is strong increase in the $4d$ content of the occupied conduction states under pressure. We find that the transverse mode near the $L$ point of the fcc Brillouin zone, already soft at ambient pressure, becomes unstable (in harmonic approximation) at a relative volume $V∕{V}_{0}=0.60$ $(P\ensuremath{\approx}42\phantom{\rule{0.3em}{0ex}}\mathrm{GPa})$. The coupling to transverse branches is relatively strong at all high-symmetry zone boundary points $X$, $K$, and $L$. Coupling to the longitudinal branches is not as strong, but extends over more regions of the Brillouin zone and involves higher frequencies. Evaluation of the electron-phonon spectral function ${\ensuremath{\alpha}}^{2}F(\ensuremath{\omega})$ shows a very strong increase with pressure of coupling in the $2--8\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ range, with a steady increase also in the $8--20\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ range. These results demonstrates strong electron-phonon coupling in this system that can account for the observed range of ${T}_{c}$.