Zone-center phonons in $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$: A comprehensive ab initio study including Raman spectra and electron-phonon interaction

The eigenfrequencies and eigenvectors of the $\Gamma$ point phonons of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$ in its high-temperature phase are calculated ab initio in the frozen phonon approximation, and the Raman spectra for the $B_{1g}$, $B_{2g}$, and $B_{3g}$ modes are computed. All theoretical results are compared to measured data from literature and are used to improve both the assignment of experimental peaks as well as the detailed description of the eigenvectors. Furthermore, hopping parameters and exchange matrix elements are extracted from the ab initio band structure. Analyzing the change of these model parameters for atomic displacements along the phonon eigenvectors, electron-phonon, and spin-phonon coupling parameters are obtained.

Figure 1

(Color online) The crystal structure (Ref. 9) of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$: (a) the view along the ladders, (b) the view from the top. The bold gray dashed lines correspond to relatively weak bonds between the ladders. The unit cell is indicated by a thin black dashed line.

Figure 2

(Color online) $B_{1g}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 3

(Color online) $B_{2g}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 4

(Color online) $B_{3g}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$.The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 5

(Color online) $B_{1u}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 6

(Color online) $B_{2u}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 7

(Color online) $B_{3u}$ eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The arrows show the direction and magnitude of the eigenvectors for each eigenmode, i.e., they are proportional to the displacement times the square root of the mass of the corresponding atom.

Figure 8

Theoretical Raman spectra for the $B_{1g}$ (upper panel), the $B_{2g}$ (central panel), and the $B_{3g}$ modes (lower panel) of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$. The inset shows details of the $B_{3g}$ spectra in the frequency range between 100 and $250{\mathrm{cm}}^{-1}$. All spectra are calculated for an incoming photon energy of $2.54\mathrm{eV}$ $\left(514\mathrm{nm}\right)$.

Figure 9

Change of the dielectric function with respect to displacements along the eigenvectors, ${\mid d{\epsilon }_{ij}∕dQ\mid }^2$, as a function of the photon energy $\omega$, for the $B_{1g}$ (upper left panel), the $B_{2g}$ (right panel), and the $B_{3g}$ (lower left panel) eigenmodes of $\mathrm{Na}{\mathrm{V}}_2{\mathrm{O}}_5$.