Efficient and accurate calculation of band gaps of halide perovskites with the Tran-Blaha modified Becke-Johnson potential

We report on a reoptimization of the Tran-Blaha modified Becke-Johnson (TB-mBJ) potential dedicated to the prediction of the band gaps of three-dimensional (3D) and layered hybrid organic-inorganic perovskites (HOPs) within pseudopotential-based density functional theory methods. These materials hold promise for future photovoltaic and optoelectronic applications. We begin by determining a set of parameters for 3D HOPs optimized over a large range of materials. Then we consider the case of layered HOPs. We design an empirical relationship that facilitates the prediction of band gaps of layered HOPs with arbitrary interlayer molecular spacers with a computational cost considerably lower than that of more advanced methods like hybrid functionals or $GW$. Our study also shows that substituting interlayer molecular chains of layered HOPs with Cs atoms is an appealing and cost-effective route to band gap calculations. Finally, we discuss the pitfalls and limitations of TB-mBJ for HOPs, notably its tendency to overestimate the effective masses due to the narrowing of the band dispersions. We expect our results to extend the use of TB-mBJ for other low-dimensional materials.

Figure 1

Electronic structure of ${\mathrm{MAPbI}}_3$ and ${\mathrm{CsSnI}}_3$ in their orthorhombic phases. Left: comparison of LDA and ppTB-mBJ band structures of (a) ${\mathrm{MAPbI}}_3$ and (b) ${\mathrm{CsSnI}}_3$. Right: wave functions (one of the complex spinor components) at the VBM and CBM calculated using ppTB-mBJ for (a) ${\mathrm{MAPbI}}_3$ and (b) ${\mathrm{CsSnI}}_3$.

Figure 2

(a) The 2D HOP layered structures considered in this work. The space group is given in parentheses. The distance $L$ is defined as the separation between the different layers of inorganic octahedra sandwiched between the molecular spacers. (b) Band structure of the complete GAMA structure as an example of 2D confinement including the molecules. One notices a flat dispersion along the $W\text{−}T$ direction which corresponds to the stacking axis in real space. ${\mathrm{Z}}_r$ corresponds to $[1/4,1/4,-1/4]$.

Figure 3

Variation of $c$ and the average ${\langle \mathbf{\nabla }\rho /\rho \rangle }^{1/2}$ [Eq. (2)] with respect to the interlayer distance $L$ in 2D HOP structures. The values at $L=0$ correspond to 3D HOPs and were calculated as the average of the different 3D structures in Table 1. The values of $c$ are given in Tables S2 and S3 of the SM.

Figure 4

Variation of the band gap and effective masses with respect to $c$ for (a) ${\mathrm{MAPbI}}_3$ ($Pnma$) and (b) C4 ($Pbca$) structures. The conflicting behavior of $c$ to optimize the band and the effective masses is highlighted.

Figure 5

Band structure of the 2D HOP C4, with the full $\mathrm{BA}$ molecular chain included in the calculations, obtained with (a) LDA and (b) ppTB-mBJ. (c) The wave function at $\mathrm{\Gamma }$ of the localized states within the band gap obtained with ppTB-mBJ. These misplaced states are due to the $\mathrm{BA}$ molecules of the organic interlayer spacer.