Tail state formation in solar cell materials: First principles analyses of zincblende, chalcopyrite, kesterite, and hybrid perovskite crystals

Tail state formation in solar cell absorbers leads to a detrimental effect on solar cell performance. Nevertheless, the characterization of the band tailing in experimental semiconductor crystals is generally difficult. In this paper, to determine the tail state generation in various solar cell materials, we have developed a quite general theoretical scheme in which the experimental Urbach energy is compared with the absorption edge energy derived from density-functional theory (DFT) calculation. For this purpose, the absorption spectra of solar cell materials, including CdTe, ${\mathrm{CuInSe}}_2$ (CISe), ${\mathrm{CuGaSe}}_2$ (CGSe), ${\mathrm{Cu}}_2{\mathrm{ZnSnSe}}_4$ (CZTSe), ${\mathrm{Cu}}_2{\mathrm{ZnSnS}}_4$ (CZTS), and hybrid perovskites, have been calculated by DFT particularly using very-high-density $k$ meshes. As a result, we find that the tail state formation is negligible in CdTe, CISe, CGSe, and hybrid perovskite polycrystals. However, coevaporated CZTSe and CZTS layers exhibit very large Urbach energies, which are far larger than the theoretical counterparts. Based on DFT analysis results, we conclude that the quite large tail state formation observed in the CZTSe and CZTS originates from extensive cation disordering. In particular, even a slight cation substitution is found to generate unusual band fluctuation in CZTS(Se). In contrast, ${\mathrm{CH}}_3{\mathrm{NH}}_3{\mathrm{PbI}}_3$ hybrid perovskite shows the sharpest absorption edge theoretically, which agrees with experiment.

Figure 1

Schematic representation of absorption-coefficient (α) spectra: experimental α $\left({\alpha }_{\mathrm{Ex}}\right)$ and DFT α $\left({\alpha }_{\mathrm{DFT}}\right)$ spectra. The $E_{\mathrm{U}}$ and $E_{\mathrm{DFT}}$ indicate the Urbach energy and DFT-derived absorption edge energy, respectively.

Figure 2

Three different crystal structures of CZTS(Se) employed in the tail state analyses: (a) kesterite, (b) stannite, and (c) primitive-mixed CuAu (PMCA) structures. The arrows indicate the $a$ and $c$ axes of the crystals and $z$ indicates the position of the cationic plane.

Figure 3

Experimental α spectra of various solar cell absorbers (open circles) and the result of the Urbach analyses performed assuming $ln\alpha \propto E/E_{\mathrm{U}}$ (solid lines). For the experimental results, those reported for CdTe [20], CISe [21], CGSe [21], CZTSe [13], CZTS [22], ${\mathrm{MAPbI}}_3$ [14], and ${\mathrm{FAPbI}}_3$ [15] are shown.

Figure 4

(a) Band structures of CdTe calculated by HSE06 and PBE, (b) ${\varepsilon }_2$ spectra of CdTe calculated by HSE06 and PBE, (c) normalized oscillator strength $\left(f\right)$ of CdTe in the zincblende Brillouin zone, (d) variation of ${\alpha }_{\mathrm{DFT}}$ with $k$-point mesh used in the DFT calculation, and (e) variation of $E_{\mathrm{DFT}}$ with the number of total $k$ points used in the DFT calculation. In (a), all the conduction bands of the PBE result are shifted upward by $\mathrm{\Delta }{E}_{\mathrm{g}}=0.7$ eV. In (c), the result corresponds to $f$ for the optical transition from the first valence band to the first conduction band. The black lines indicate the cross section of the contour for the energy separation between the first valence and conduction bands. In (d), the solid lines indicate the result of $E_{\mathrm{DFT}}$ analysis.

Figure 5

${\alpha }_{\mathrm{Ex}}$ (open circles) and ${\alpha }_{\mathrm{DFT}}$ (solid lines) of (a) CdTe, (b) CISe, (c) CZTSe, and (d) ${\mathrm{MAPbI}}_3$, together with (e) enlarged α spectra near the $E_{\mathrm{g}}$ regions. The ${\alpha }_{\mathrm{DFT}}$ has been shifted toward higher energy by the energy indicated as $\mathrm{\Delta }{E}_{\mathrm{g}}$ to improve matching with the experimental results. For the DFT calculation of CZTSe, the kesterite structure was assumed. For ${\mathrm{MAPbI}}_3$, the PBE spectrum obtained by incorporating the SOC interaction is also shown.

Figure 6

Urbach energy $\left(E_{\mathrm{U}}\right)$ as a function of the absorption edge energy derived from DFT $\left(E_{\mathrm{DFT}}\right)$. The $E_{\mathrm{U}}$ values were estimated from the analyses of Fig. 3, whereas $E_{\mathrm{DFT}}$ was calculated from ${\alpha }_{\mathrm{DFT}}$ of Fig. 5. For CZTSe and CZTS, the kesterite phases are assumed.

Figure 7

DOS distribution of the valence and conduction states in ${\mathrm{MAPbI}}_3$, CdTe, and CISe.

Figure 8

(a) ${\alpha }_{\mathrm{DFT}}$ spectra of CZTS obtained assuming the kesterite, stannite, PMCA, and three-phase-mixed structures (solid lines), together with ${\alpha }_{\mathrm{Ex}}$ (open circles) and (b) variation of $E_{\mathrm{DFT}}$ with stannite volume fraction in CZTSe and CZTS. In (a), the experimental data are consistent with Fig. 3. In (b), the $E_{\mathrm{U}}$ positions are indicated by the dotted lines.

Figure 9

Cation-disordered structures of CZTS for a 16-atom cell. In a 32-atom supercell shown in (e), one pair of Cu and Zn atoms indicated by the red circles is replaced assuming a kesterite structure.

Figure 10

α spectra calculated for different CZTS crystal structures shown in Figs. 2 and 9 (solid lines) and α spectrum obtained from experiment (open circles).

Figure 11

Band alignments for different CZTS crystal structures shown in Figs. 2 and 9. In the figure, the conduction-band positions have been shifted upward by $\mathrm{\Delta }{E}_{\mathrm{g}}=0.40$ eV to compensate the underestimated $E_{\mathrm{g}}$ in the PBE calculations. The numerical values shown within each band represent the relative energy positions of each band.