Robustness of electronic screening effects in electron spectroscopies: Example of ${\mathrm{V}}_2{\mathrm{O}}_5$

In bulk and low-dimensional extended systems, the screening of excitations by the electron cloud is a key feature governing spectroscopic properties. Widely used computational approaches, especially in the framework of many-body perturbation theory, such as the $GW$ approximation and the resulting approximate Bethe-Salpeter equation, are explicitly formulated in terms of the screened Coulomb interaction. In the present work, we explore the effect of screening in absorption and electron energy loss spectroscopy, concentrating on the effect of local distortions on the screening and elucidating the resulting changes in the various spectra. Using the layered bulk oxide ${\mathrm{V}}_2{\mathrm{O}}_5$ as prototype material, we show in which way local distortions affect the screening, and in which way changes in the screening impact electron energy loss and absorption spectra including excitons. We highlight cancellations that make many-body effects in the spectra very robust with respect to structural modifications, while the band structure undergoes significant changes and the nature of the excitations may also be affected. This yields insight concerning the structure-properties relations that are crucial for the use of ${\mathrm{V}}_2{\mathrm{O}}_5$ as energy storage material, and more generally, that may be used to optimize the analysis and the calculation of electronic spectra in complex materials.

Figure 1

Crystal structure and electronic properties of ${\mathrm{V}}_2{\mathrm{O}}_5$. (a) ${\mathrm{V}}_2{\mathrm{O}}_5$ layers are stacked along the $z$ axis. V atoms are grey and O atoms are red. Highlighted in yellow is one $\mathrm{V}\text{−}{\mathrm{O}}_b$-V rung. The orthorhombic unit cell (space group $Pmmn$) is represented by the black lines. (b) The elementary units of the ${\mathrm{V}}_2{\mathrm{O}}_5$ layers are ${\mathrm{VO}}_5$ pyramids, formed by one V atom, two chain oxygen atoms (${\mathrm{O}}_{c1}$ and ${\mathrm{O}}_{c2}$), one bridge oxygen (${\mathrm{O}}_b$), and one vanadyl oxygen (${\mathrm{O}}_{v1}$). The corresponding bond lengths are reported in Table 1. (c) LDA band structure and corresponding projected density of states (PDOS). The bold line indicates the top valence band. (d) The first Brillouin zone with the labels of high-symmetry $\mathbf{k}$ points.

Figure 2

Loss function $-\text{Im}{\epsilon }_M^{-1}(\mathbf{q}\to 0,\omega )$ (top) and corresponding imaginary part ${\epsilon }_2(\mathbf{q}\to 0,\omega )$ (middle panels), and real part ${\epsilon }_1(\mathbf{q}\to 0,\omega )$ (bottom) of the macroscopic dielectric function of ${\mathrm{V}}_2{\mathrm{O}}_5$ in the three Cartesian directions $x$ (left), $y$ (center), and $z$ (right), computed within the TDDFT-RPA (green), the $GW$-IPA (orange) and the BSE (blue).

Figure 3

Dispersion of the loss function $-\text{Im}{\epsilon }_M^{-1}(\mathbf{q},\omega )$ as a function of momentum transfer $\mathbf{q}$ in the three Cartesian directions, computed within TDDFT-RPA with (solid lines) and without (dashed lines) crystal local field effects, and comparison to EELS experimental data [62] (black dots). A vertical offset has been added to the spectra for improved clarity. The size of the theoretical or experimental momentum transfer is reported next to each spectrum.

Figure 4

LDA band structures of the ideal and distorted structures. The bold dashed lines indicate the top-valence band. The considered displacement corresponds to the largest one in Table 1. The band structure of the crystal distorted along the $B_{1g}^1$ mode (left-most panel) almost entirely overlaps that of the ideal structure. The size of the orange circles is proportional to the contribution $|A_{\lambda }^{vc\mathbf{k}}{\tilde{\rho }}_{vc\mathbf{k}}|$ to the lowest energy exciton in the $y$ polarization direction (see Sec. 3b).

Figure 5

The absorption spectra, in the three Cartesian polarization directions, (left) $x$, (middle) $y$, and (right) $z$, of the crystal structure distorted along the $B_{2g}^6$ mode, for the case of the largest displacement, using the BSE with $W_{\mathrm{ideal}}$ (orange lines) or the screened interaction $W_{\mathrm{dist}}$ of the distorted structure (blue lines).

Figure 6

Absorption spectra of the distorted crystal structures (orange and blue lines) compared to the absorption spectra in the ${\mathrm{V}}_2{\mathrm{O}}_5$ ideal structure (gray lines). The spectra are obtained from the solution of the BSE (solid lines) and from the $GW$-IPA (dashed lines), where e-h interactions are neglected. The $d_{\max }$ values are the largest atomic displacement in the unit cell with respect to the ideal crystal (see Table 1). For the $B_{1u}^5$ and $B_{2g}^6$ phonon modes both a small (blue lines) and large (orange lines) displacement are considered. The arrows for the structures distorted along the $B_{2g}^6$ and $B_{1u}^5$ modes indicate the lowest exciton energy, which is perfectly dark in the ideal ${\mathrm{V}}_2{\mathrm{O}}_5$ structure (black arrows), but not in the $y$ direction in the distorted structures (orange arrows).

Figure 7

Contributions to the oscillator strength of the lowest energy exciton in the $y$ polarization direction for the crystal structures distorted along $B_{2g}^6$ (first histograms from the left), $B_{1u}^5$ (second histograms), $B_{1g}^1$ (third histograms) modes in comparison to the ideal ${\mathrm{V}}_2{\mathrm{O}}_5$ case (fourth histograms). Blue (orange) columns represent the real (imaginary) parts of ${\sum }_{vc\mathbf{k}}{A}_{\lambda }^{vc\mathbf{k}}{\tilde{\rho }}_{vc\mathbf{k}}$, where the sum over $\mathbf{k}$ is restricted to one half of the first Brillouin zone. The $\mathbf{k}$ points belonging to each subset are related to those of the other subset by inversion symmetry $\mathbf{k}\leftrightarrow -\mathbf{k}$. In the $B_{1g}^1$ and ideal structures, the columns have same heights but opposite signs, so the exciton is dark. In the other distorted structures, the columns are higher and their signs do not cancel, so the exciton becomes bright.

Figure 8

TDDFT-RPA loss function spectra $-\text{Im}{\epsilon }_M^{-1}(\mathbf{q},\omega )$ as a function of the momentum transfer $\mathbf{q}$ along the $x$ (left), $y$ (middle), and $z$ (right) directions, calculated for the structure distorted along the $B_{2g}^6$ mode (solid lines) and the undistorted structure (dashed lines). A vertical offset has been added to the spectra for improved clarity. The size of the momentum transfer is reported next to each spectrum.

Figure 9

Static macroscopic dielectric function ${\epsilon }_M(\mathbf{q},\omega =0)=1/{\epsilon }_{\mathbf{G}=0,{\mathbf{G}}^{\prime }=0}^{-1}(\mathbf{q},\omega =0)$ for ideal and distorted structures (blue and orange dots, respectively) calculated within the RPA. The values at $\left|\mathbf{q}\right|\to 0$ have been calculated along the direction (1,2,3) in reciprocal lattice units and are meant to represent an average of the different spatial directions.

Figure 10

Spatial distribution of the wave function $|{\mathrm{\Psi }}_{\lambda }({\mathbf{r}}_h,{\mathbf{r}}_e){|}^2$ of the lowest energy dark (top) and bright excitons in $y$ polarization (bottom) for fixed position of the hole ${\mathbf{r}}_h^0$ close to a bridge oxygen ${\mathrm{O}}_b$ atom (light blue dot): Distorted crystal structure along the $B_{2g}^6$ mode (left) and ideal crystal structure (right). The isosurface value corresponds to 2% of the maximum.

Figure 11

Convergence with respect to the Brillouin zone sampling of spectral features of ${\mathrm{V}}_2{\mathrm{O}}_5$ in the ideal crystal structure. (Top) BSE absorption spectra along $x$ polarization direction. (Middle and bottom) JDOS of BSE and $GW$-IPA for different grids. Middle panel, 7 occupied bands and 4 unoccupied bands. (Bottom) 13 occupied and 10 unoccupied bands.

Figure 12

$GW$-IPA absorption spectra, in the three Cartesian polarization directions, (left) $x$, (middle) $y$ and (right) $z$, of the crystal structure distorted along the $B_{2g}^6$ mode with the largest displacements (solid lines) compared to those of ideal crystal structure (dashed lines): Full $GW$-IPA (blue lines) and scissor-corrected (sc) $GW$-IPA (green lines). For the distorted structure, the full $GW$ spectra (orange lines) and scissor-corrected (sc) $GW$ spectra (purple lines) are also obtained with the screened Coulomb interaction $W_{\mathrm{ideal}}$ calculated in the ideal ${\mathrm{V}}_2{\mathrm{O}}_5$.