Optical response and band structure of ${\mathrm{LiCoO}}_2$ including electron-hole interaction effects

The optical response functions and band structures of ${\mathrm{LiCoO}}_2$ are studied at different levels of approximation, from density functional theory (DFT) in the generalized gradient approximation (GGA) to quasiparticle self-consistent $\mathrm{QS}GW$ (with $G$ for Green's function and $W$ for screened Coulomb interaction) without and with ladder diagrams $\left(\mathrm{QS}G\widehat{W}\right)$ and the Bethe Salpeter Equation (BSE) approach. The $\mathrm{QS}GW$ method is found to strongly overestimate the band gap and electron-hole or excitonic effects are found to be important. They lower the quasiparticle gap by only about 11% but the lowest energy peaks in absorption are found to be excitonic in nature. The contributions from different band to band transitions and the relation of excitons to band-to-band transitions are analyzed. The excitons are found to be strongly localized. A comparison to experimental data is presented.

Figure 1

Band structure of $R\overline{3}m{\mathrm{LiCoO}}_2$ in GGA (left), and $\mathrm{QS}GW$, $\mathrm{QS}G\widehat{W}$ (middle). The Brillouin zone high symmetry point labeling follows Refs. [10, 11]. The rightmost figure shows the exciton spectrum of eigenvalues of the two-particle BSE compared with the band gaps at different levels of theory. The $\widehat{W}$ is used here for the exciton levels.

Figure 2

Interband optical response function ${\varepsilon }_2\left(\omega \right)$ calculated in long-wavelength approximation (no local field or excitonic effects). The partial contribution from a given band pair to the spectrum are shown as a color scale for each band pair: (a) band numbering ($\mathrm{QS}G\widehat{W}$), (b) $\mathbf{E}\parallel \mathbf{c}$, and (c) $\mathbf{E}\perp \mathbf{c}$.

Figure 3

Macroscopic dielectric function ${\varepsilon }_2\left(\omega \right)$ for both polarizations comparing RPA [Eq. (4)] with the long-wavelength limit [Eq. (3)] without local field effects.

Figure 4

Macroscopic optical dielectric function ${\varepsilon }_2\left(\omega \right)$ in RPA and BSE approximation both based on the $\mathrm{QS}G\widehat{W}$ band structure. Top to bottom, $z$ polarization, in-plane polarization, and exciton density of states without optical matrix elements and exciton spectrum.

Figure 5

Real part of $\varepsilon \left(\omega \right)$ in RPA and BSE for $\mathbf{E}\perp \mathbf{c}$.

Figure 6

Contributions to the BSE optical absorption from different bands. The gray filled spectrum is the RPA reference spectrum. ${\mathcal{I}}_{1,\infty }$ shows the difference between BSE, while including all valence bands but only the first conduction band, with RPA, ${\mathcal{I}}_{2,\infty }$ shows the difference between BSE including conduction bands 1,2 with only conduction band 1. ${\mathcal{I}}_{\infty ,1}$ means all conduction bands included but only the top valence band in BSE relative to RPA, ${\mathcal{I}}_{\infty ,2}$ means the difference between including valence bands 1,2 vs only 1 while keeping all conduction bands.

Figure 7

Band contributions to the excitons in the range 1.6–1.9 eV.

Figure 8

Convergence of the gap as function of iteration number in $\mathrm{QS}GW$ and $\mathrm{QS}G\widehat{W}$.

Figure 9

Comparison of imaginary part of the dielectric function ${\varepsilon }_2\left(\omega \right)$ (for $\mathbf{E}\parallel \mathbf{c}$) with $W$ calculated in RPA and $\widehat{W}$ calculated including ladder diagrams both in the underlying band structure and BSE.

Figure 10

Exciton projected band structure for LiF.