Optical properties of Cu-chalcogenide photovoltaic absorbers from self-consistent $GW$ and the Bethe-Salpeter equation

Self-consistent $GW$ calculations and the solution of the Bethe-Salpeter equation are to date the best available approaches to simulate electronic excitations in a vast class of materials, ranging from molecules to solids. However, up to now numerical instabilities made it impossible to use these techniques to calculate optical absorption spectra of the best-known thin-film absorbers for solar cells: ${\mathrm{Cu}(\mathrm{In},\mathrm{Ga})(\mathrm{S},\mathrm{Se})}_2$ chalcopyrites and ${\mathrm{Cu}}_2{\mathrm{ZnSn}(\mathrm{S},\mathrm{Se})}_4$ kesterites/stannites. We show here how to solve this problem using a finite-difference method in $k$ space to evaluate the otherwise diverging dipole matrix elements, obtaining an excellent agreement with experiments. Having established the validity of this approach, we use it then to calculate the optical response of the less studied, but promising, ${\mathrm{Cu}}_2{\mathrm{ZnGe}(\mathrm{S},\mathrm{Se})}_4$ compounds, opening the way to predictive calculations of still unknown materials.

Figure 1

Crystal structures of ${\mathrm{Cu}}_2{\mathrm{ZnSnSe}}_4$ kesterite (left) and stannite (right), after Ref. [11].

Figure 2

LDA RPA spectra of silicon using LDA wave functions and energies for $k$ meshes from $10\times 10\times 10$ to $48\times 48\times 48$ points, calculated using the commutator expression $[\mathbf{r},\mathcal{H}]$, Eq. (15), and the $k$ derivative $i{\nabla }_{\mathbf{k}}$, Eq. (16).

Figure 3

Imaginary part of ${\epsilon }_M$ of ${\mathrm{CuGaSe}}_2$ from $\mathrm{sc}GW$ RPA for light polarization $\left|\right|$ to the $c$ axis, using (a) only the commutator expression; (b) only the $k$ derivative; and (c), the combination of both (mixed, $\delta =10$ millihartrees). The numbers 8 and 16 denote $k$-point meshes of $8\times 8\times 8$ and $16\times 16\times 16 k$ points, respectively.

Figure 4

Overlaps of the normalized, mixed $\mathrm{sc}GW$ RPA spectrum of ${\mathrm{CuGaSe}}_2$ with the normalized $i{\nabla }_{\mathbf{k}}$ spectrum for $16\times 16\times 16 k$ points and with the experimental one, as a function of $\delta$ for a $k$-point mesh of $8\times 8\times 8 k$ points and for light polarization $\left|\right|$ to the $c$ axis.

Figure 5

Imaginary part of ${\epsilon }_M$ of ${\mathrm{CuGaSe}}_2$ from $\mathrm{sc}GW$ RPA and $\mathrm{sc}GW$ BSE calculations. A $k$ mesh of $8\times 8\times 8$ points is used. Experimental data are from Alonso et al. [4]. The vertical lines indicate the calculated optical gaps using $\mathrm{sc}GW$ RPA and $\mathrm{sc}GW$ BSE (see Sec. 3c).

Figure 6

The same as Fig. 5 for ${\mathrm{CuInSe}}_2$. A $k$ mesh of $10\times 10\times 10$ points is used. Experimental data are from Alonso et al. [4].

Figure 7

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnSnS}}_4$ kesterite.

Figure 8

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnSnSe}}_4$ kesterite. Experimental data are from Choi et al. for an unknown polarization direction [13].

Figure 9

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnSnS}}_4$ stannite.

Figure 10

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnSnSe}}_4$ stannite. A $k$ mesh of $10\times 10\times 10$ points is used.

Figure 11

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnGeS}}_4$ kesterite. Experimental data are from León et al. [14] for the stannite structure for an unknown polarization direction.

Figure 12

The same as Fig. 5 for ${\mathrm{Cu}}_2{\mathrm{ZnGeSe}}_4$ kesterite.

Figure 13

Overlap between experimental and $\mathrm{sc}GW$ BSE spectra of ${\mathrm{CuGaSe}}_2$ (“CGSe”) and ${\mathrm{CuInSe}}_2$ (“CISe”).

Figure 14

Optical band gaps from experiment, compared with the noninterpolated photoemission gaps calculated using the $\mathrm{sc}GW$ method, and the optical gaps calculated using $\mathrm{sc}GW$-RPA and $\mathrm{sc}GW$-BSE, for ${\mathrm{CuGaSe}}_2$ (CGSe), ${\mathrm{CuInSe}}_2$ (CISe), ${\mathrm{Cu}}_2{\mathrm{ZnSnS}}_4$(S,Se) [CZTS, CZTSe], and ${\mathrm{Cu}}_2{\mathrm{ZnGeS}}_4$(S,Se) [CZGS, CZGSe] (see also Table 2). KES and ST stand for kesterite and stannite structures, respectively.