Quasiparticle band structure and excitonic optical response in $V_2{O}_5$ bulk and monolayer

The electronic band structure of $V_2{O}_5$ is calculated using an all-electron quasiparticle self-consistent (QS) $GW$ method, including electron-hole ladder diagrams in the screening of $W$, named $\mathrm{QS}G\hat{W}$ and using a full-potential linearized muffin-tin-orbital basis set. The optical dielectric function calculated with the Bethe-Salpeter equation (BSE) exhibits excitons with large binding energy, consistent with spectroscopic ellipsometry data and other recent calculations using a pseudopotential plane-wave-based implementation of the many-body-perturbation theory approaches. Convergence issues are discussed. Sharp peaks in the direction perpendicular to the layers at high energy are found to be an artifact of the truncation of the numbers of bands included in the BSE calculation of the macroscopic dielectric function. The static (electronic screening only) dielectric constant ${\varepsilon }_1(\omega =0)$ gives indices of refraction in good agreement with experiment. The exciton wave functions are analyzed in various ways. They correspond to charge transfer excitons with the hole primarily on oxygen and electrons on vanadium, but depending on which exciton, the distribution over different oxygens changes. The dark exciton at 2.6 eV is the most localized and has the highest weight on the bridge oxygen, while the lowest bright excitons for in-plane polarizations at 3.1 eV for $E\parallel a$ and 3.2 eV for $E\parallel b$ have their higher weight on the chain and vanadyl oxygens. The exciton wave functions have a spread of about 5–15Å, with asymmetric character for the electron distribution around the hole depending on which oxygen the hole is fixed at. The same method applied first to bulk layered $V_2{O}_5$ is here applied to monolayer $V_2{O}_5$. The monolayer quasiparticle gap increases inversely proportional to interlayer distance once the initial interlayer covalent couplings are removed which is thanks to the long-range nature of the self-energy and the reduced screening in a two-dimensional system. The optical gap on the other hand is relatively independent of interlayer spacing because of the compensation between the self-energy gap shift and the exciton binding energy, both of which are proportional to the screened Coulomb interaction $\hat{W}$. Recent experimental results on very thin layer $V_2{O}_5$ obtained by chemical exfoliation provide experimental support for an increase in gap.