Optical properties, electron-phonon coupling, and Raman scattering of vanadium ladder compounds

The electronic structure of two V-based ladder compounds, the quarter-filled $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ in the symmetric phase and the isostructural half-filled $\mathrm{Ca}{\mathrm{V}}_2{\mathrm{O}}_5$, is investigated by ab initio calculations. Based on the band structure we determine the dielectric tensor $\epsilon \left(\omega \right)$ of these systems in a wide energy range. The frequencies and eigenvectors of the fully symmetric $A_g$ phonon modes and the corresponding electron-phonon and spin-phonon coupling parameters are also calculated from first principles. We determine the Raman scattering intensities of the $A_g$ phonon modes as a function of polarization and frequency of the exciting light. All results—i.e., shape and magnitude of the dielectric function, phonon frequencies, and Raman intensities—show very good agreement with available experimental data.

Figure 1

The crystal structure of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ and $\mathrm{Ca}{\mathrm{V}}_2{\mathrm{O}}_5$. (a) The view along the ladders, (b) the view from the top. The dashed lines correspond to relatively weak bonds between the ladders. X indicates either Na or Ca, respectively.

Figure 2

Band structure and density of states (in states per unit cell and eV) of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$.

Figure 3

Band structure and density of states (in states per unit cell and eV) of $\mathrm{Ca}{\mathrm{V}}_2{\mathrm{O}}_5$.

Figure 4

Valence electron density of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ in the $(x,z)$ plane (upper panel) and the $(x,y)$ plane (lower panel), which both contain the V positions. The logarithmic contour lines range from $0.07e∕{\mathrm{\AA }}^3\text{to}2.3e∕{\mathrm{\AA }}^3$.

Figure 5

Diagonal components of the dielectric function ${\epsilon }_{ii}\left(\omega \right)$ of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$.

Figure 6

Diagonal components of the dielectric function ${\epsilon }_{ii}\left(\omega \right)$ of $\mathrm{Ca}{\mathrm{V}}_2{\mathrm{O}}_5$.

Figure 7

Eigenvectors of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$.

Figure 8

Raman intensity $I_R$ for $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ in three different geometries at $\omega =2.41\mathrm{eV}$ $(\lambda =514.5\mathrm{nm})$ and $T=300\mathrm{K}$.

Figure 9

Raman intensity $I_R$ for $\mathrm{Ca}{\mathrm{V}}_2{\mathrm{O}}_5$ in three different geometries at $\omega =2.41\mathrm{eV}$ $(\lambda =514.5\mathrm{nm})$ and $T=300\mathrm{K}$.

Figure 10

Derivatives of the dielectric functions ${\mid \partial \epsilon ∕\partial Q\mid }^2$ with respect to displacements along the eigenvectors of the phonon modes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ as indicated by their phonon frequencies. The $zz$ component is considerable only for the highest frequency and is therefore omitted in the other panels.