First-principles study of excitons in optical spectra of silver chloride

Silver chloride is a material that has been investigated and used for many decades. Of particular interest are its optical properties, but only few fundamental theoretical studies exist. We present first-principles results for the optical properties of AgCl, obtained using time-dependent density functional theory and many-body perturbation theory. We show that optical properties exhibit strong excitonic effects, which are correctly captured only by solving the Bethe-Salpeter equation starting from quasiparticle self-consistent $GW$ results. Numerical simulations are made feasible by using a model screening for the electron-hole interaction in a way that avoids the calculation of the static dielectric constant. Finally, the localization of bright and dark excitons is discussed.

Figure 1

Calculated band structure of silver chloride. The top valence energy has been aligned to zero in all cases. Red lines are the LDA calculation, the dots the $G^0{W}^0$ results, and the crosses the QSGW results. The dashed and dotted-dashed lines represent the conduction bands shifted from the LDA by 1.9 and 2.98 eV, respectively.

Figure 2

Projected density of states of silver chloride calculated in KS-LDA compared to photoemission spectra from Mason [80] and Tejeda et al. [78] at a photon energy $h\nu =1486.6$ eV. In each curve the top valence has been aligned to zero and the intensity scaled to the maximum of the most prominent peak. In the inset: zoom on the calculated unoccupied PDOS.

Figure 3

QSGW QP energy corrections as a function of the corresponding KS-LDA energies. The corrections have been set to zero for the top valence state, whose KS-LDA energy is also set to zero. The horizontal blue line is the value of the scissor correction used in the calculation of absorption spectra in Sec. 5a.

Figure 4

Extinction coefficient as a function of energy. The RPA calculation based on the KS-LDA band structure (black solid curve) is compared to experimental data from Ref. [25] measured at 4 K (red curve), at 90 K (blue curve), and at room temperature (orange curve).

Figure 5

Comparison of the extinction coefficient measured at 4 K with several TDDFT approximations. Since the energy range is limited to 3–7 eV, these spectra have been calculated with only four conduction bands.

Figure 6

Comparison between the experimental extinction coefficient from Ref. [25] at different temperatures and MBPT calculations (with 2048 shifted $\mathbf{k}$ points and 19 bands): $G_0{W}_0$-RPA obtained with 1.9-eV scissor correction (pink dashed line) $G_0{W}_0+\mathrm{BSE}$ obtained with the full screening matrix $W_{\mathbf{G}{\mathbf{G}}^{\prime }}\left(\mathbf{q}\right)$ (black solid line) or only its diagonal contribution (black dashed line).

Figure 7

Effects of the model electron-hole screening on BSE results in AgCl. Left panel: ab initio LDA-RPA (crosses) and model (solid lines) dielectric constants $\epsilon (\mathbf{q},\omega =0)$ as a function of $\left|\mathbf{q}\right|$. The ab initio results are obtained using four-times shifted $4\times 4\times 4$ (red) or $6\times 6\times 6$ (black) Monkhorst-Pack grids of $\mathbf{k}$ points, corresponding to 256 or 864 $\mathbf{k}$ points in the full Brillouin zone, respectively. The black cross at $q\to 0$ also contains the correction for the nonlocal pseudopotential, while the red cross does not. Model results are fitted to the ab initio results at $q=0$ (respective color code) or at $q=0.15$ a.u. (blue), where results on the smaller grid are already converged. Right panel: BSE spectra obtained with 864 shifted $\mathbf{k}$ points, 8 valence and 6 conduction bands. The screening of the electron-hole interaction is taken from the results shown in the left panel: either the diagonal of the ab initio screened Coulomb interaction (dashed black), or the model screening fitted to the 864 $\mathbf{k}$ points result at $q=0$ (solid black), or to the 256 $\mathbf{k}$ points result at $q=0$ (red) or at $q=0.15$ a.u. (blue).

Figure 8

Bulk silicon: Static dielectric function as function of wave vector, for different $\mathbf{k}$-point grid sizes. Plus (circle) symbols are results from calculations including (excluding) the commutator with the nonlocal pseudopotential. In the inset: zoom on $\mathbf{q}=0$.

Figure 9

Silver chloride: static dielectric constant as function of momentum transfer calculated using different levels of theory: Kohn-Sham LDA, $G^0{W}^0$, evQSGW, and QSGW. All calculations are done using a four-times shifted $4\times 4\times 4$ Monkhorst-Pack $\mathbf{k}$-point grid and 340 bands. The commutator with the nonlocal potentials (pseudopotential and self-energy, when applicable) is neglected. The fit of the model results at ${\mathbf{q}}^0=(-0.125,-0.125,0)$ in units of the reciprocal lattice ($q=0.15$ a.u.) yields the dielectric constants.

Figure 10

Absorption spectra calculated in the RPA (dashed lines) and from the BSE (solid lines), using scissor corrections and model screenings obtained from $G_0{W}_0$ (black lines) or QSGW (blue lines).

Figure 11

Extinction coefficient spectra: comparison between experimental results [25, 31], and $G_0{W}_0+\mathrm{BSE}$ and $\mathrm{QSGW}+\mathrm{BSE}$ calculations obtained with a grid of shifted 2048 $\mathbf{k}$ points and 25 bands. The horizontal black arrow indicates the estimated shift of the 6.6-eV peak that would be obtained by using QSGW corrections instead of the scissor operator.

Figure 12

(Left) Projection of $|A_{\lambda }^t{\tilde{\rho }}_t|$ on the band structure for one of the two dark excitons. The size of the circles is proportional to $|A_{\lambda }^t{\tilde{\rho }}_t|$. (Center) Same analysis for one of the three bright excitons. The scale is the same for both panels. (Right) Corresponding cumulative functions $S_{\lambda }\left(\omega \right)$ for the bright and dark excitons (black and blue lines, respectively).

Figure 13

(Upper panel) Projection of $|A_{\lambda }^t{\tilde{\rho }}_t|$ on the band structure for one of the three excitons forming the peak at 6.6 eV in the $\mathrm{QSGW}+\mathrm{BSE}$ spectrum (see Fig. 11). Full black circles correspond to transition with energy smaller than the energy ${\omega }_0$ given by the black vertical line in the bottom panel, full orange circles correspond to transitions at higher energy. The scale is the same as in Fig. 12. (Bottom panel) The corresponding real part of the cumulative function $I_{\lambda }\left(\omega \right)$ (the imaginary part, not shown, has a similar behavior). Contributions for $\omega >{\omega }_0$ (orange part of the curve) add constructively to form the exciton oscillator strengths, while the contributions for $\omega <{\omega }_0$ (in black) cancel.

Figure 14

Electron density distribution for the two degenerate lowest-energy dark excitons. Shown are cuts along the [101] plane for the exciton calculated in $W$-RPA (left panels) and $W$-QSGW (right panels), where the hole has been placed close to the Ag atom (top panels) and close to the Cl atom (bottom panels). Cl (Ag) atoms are represented by green (gray) balls, while the hole position is black.

Figure 15

Same as the previous figure for the three degenerate bright excitons.

Figure 16

Convergence of the BSE absorption spectrum with respect to the number of plane waves using a four-times shifted $8\times 8\times 8$ grid of $\mathbf{k}$ points.

Figure 17

Convergence of the $G^0{W}^0+\mathrm{BSE}$ absorption spectrum as a function of the number of $\mathbf{k}$ points in the full Brillouin zone with a 0.1-eV Lorentzian broadening.