Density functional theory of superconductivity in doped tungsten oxides

We apply density functional theory for superconductors (SCDFT) to doped tungsten oxide in three forms: electrostatically doped ${\mathrm{WO}}_{3}$, perovskite ${\mathrm{WO}}_{3\ensuremath{-}x}{\mathrm{F}}_{x}$, and hexagonal ${\mathrm{Cs}}_{x}{\mathrm{WO}}_{3}$. We achieve a consistent picture in which the experimental superconducting transition temperature ${T}_{c}$ is reproduced, and superconductivity is understood as a weak-coupling state sustained by soft vibrational modes of the ${\mathrm{WO}}_{6}$ octahedra. SCDFT simulations of ${\mathrm{Cs}}_{x}{\mathrm{WO}}_{3}$ allow us to explain the anomalous ${T}_{c}$ behavior observed in most tungsten bronzes, where ${T}_{c}$ decreases with increasing carrier density. Here, the opening of structural channels to host Cs atoms induces a softening of strongly coupled W-O modes. By increasing the Cs content, these modes are screened and ${T}_{c}$ is strongly reduced.