Optical properties of periodic systems within the current-current response framework: Pitfalls and remedies

We compare the optical absorption of extended systems using the density-density and current-current linear response functions calculated within many-body perturbation theory. The two approaches are formally equivalent for a finite momentum $\mathbf{q}$ of the external perturbation. At $\mathbf{q}=\mathbf{0}$, however, the equivalence is maintained only if a small $q$ expansion of the density-density response function is used. Moreover, in practical calculations, this equivalence can be lost if one naively extends the strategies usually employed in the density-based approach to the current-based approach. Specifically, we discuss the use of a smearing parameter or of the quasiparticle lifetimes to describe the finite width of the spectral peaks and the inclusion of electron-hole interaction. In those instances, we show that the incorrect definition of the velocity operator and the violation of the conductivity sum rule introduce unphysical features in the optical absorption spectra of three paradigmatic systems: silicon (semiconductor), copper (metal) and lithium fluoride (insulator). We then demonstrate how to correctly introduce lifetime effects and electron-hole interactions within the current-based approach.

Figure 1

Optical absorption in bulk LiF (a) and Si (b) at the IP level. The spectra obtained by replacing ${\omega }^{+}$ by $\omega +i\eta$ in Eq. (9), for the density-based approach, Eq. (7), (grey shadow) and the current-based approach, Eq. (6) with either $1/{\omega }^2$ (red dashed line) or $1/z^2$ (blue continuous line), are compared. Results with the numerical recipe $\overline{z}=\sqrt{\omega +2i\omega \eta }$ are also shown. Here, $\eta =0.1$ eV for Si and $\eta =0.2$ eV for LiF. The conductivity sum rule is always enforced as explained in the main text.

Figure 2

Optical absorption in bulk Cu (a) and Si (b) at the IP level. Red dashed lines: spectra obtained in the current-based approach using Eq. (34) without enforcing the conductivity sum rule [Eq. (15)]; the various lines correspond to different number of bands. Grey shadow: spectra obtained in the density-based approach [Eq. (7)] and in the current-based approach by enforcing the conductivity sum rule [Eq (15)]. Brown shadow: Drude term added via a Drude model (for Copper). Blue dots, red dots, and black continuous bold line: experimental data from Refs. [35, 36], and [37], respectively.

Figure 3

Optical absorption in bulk Cu (a) and Si (b) at the IP level. Red dashed line: spectra obtained in the current-based approach using the complex one-particle energies (54) in Eq. (9). Grey shadow: spectra obtained in the density-based approach [Eq. (7)] and in the current-based approach by using the complex one-particle energies (54) and the velocity operator (55) [with (56)] in Eq. (9). Brown shadow: as in Fig. 2 Blue dots, red dots, and bold continuous black line: as in Fig. 2.

Figure 4

Optical absorption in bulk LiF (a) and Si (b) at the BSE level. Black continuous line: spectra obtained in the current-based approach [Eq. (6)] neglecting the corrections (58) and (62). Red dashed line: spectra obtained in the current-based approach [Eq. (6)] considering only the correction (58). Blue dot-dashed line: spectra obtained in the current-based approach [Eq. (6)] considering only the correction (62). Grey shadow: spectra obtained in the density-based approach [Eq. (7)] and in the current-based approach considering the corrections (58) and (62). Black dots: experimental data are from Refs. [44, 45].