Nonempirical hybrid functionals for band gaps of inorganic metal-halide perovskites

Nonempirical hybrid functionals are investigated for band-gap predictions of inorganic metal-halide perovskites belonging to the class $\mathrm{Cs}B{X}_3$, with $B$ = Ge, Sn, Pb and $X$ = Cl, Br, I. We consider both global and range-separated hybrid functionals and determine the parameters through two different schemes. The first scheme is based on the static screening response of the material and thus yields dielectric-dependent hybrid functionals. The second scheme defines the hybrid functionals through the enforcement of Koopmans' condition for localized defect states. We also carry out quasiparticle self-consistent $GW$ calculations with vertex corrections to establish state-of-the-art references. For the investigated class of materials, dielectric-dependent functionals and those fulfilling Koopmans' condition yield band gaps of comparable accuracy $(\sim 0.2$ eV), but the former only require calculations for the primitive unit cell and are less subject to the specifics of the material.

Figure 1

Workflow for the construction of dielectric-dependent hybrid (DDH) functionals. The procedure for updating the free parameters depends on the considered hybrid-functional form and is described in the text.

Figure 2

Scheme for the construction of a hybrid functional satisfying Koopmans' condition through a defect D. Occupied and unoccupied single-particle energy levels are given as a function of the free parameter of the adopted hybrid functional form. The point of intersection corresponds to the fulfillment of Koopmans' condition. The band edges of the host material are also shown as a function of the hybrid functional parameter. The dashed vertical line represents the resulting band-gap estimate.

Figure 3

Band-gap difference (in eV) for ${\mathrm{CsPbI}}_3$ between calculations with pseudopotential sets PP1 and PP2, which differ by the treatment of semicore electrons. The band gaps are obtained with a CAM-type hybrid functional in which the long-range fraction of Fock exchange is set to ${\alpha }_{\ell }=1/{\varepsilon }_{\infty }$. The figure shows the band-gap difference as a function of the short-range fraction of Fock exchange ${\alpha }_{\text{s}}$ and the inverse screening length $\mu$. Isovalues are shown by solid lines.

Figure 4

Convergence of (a) the dielectric constant ${\varepsilon }_{\infty }$ and (b) the band gap within the self-consistent DD-PBE0 scheme for ${\mathrm{CsPbI}}_3$. The values for the PBE starting point and the converged DD-PBE0 result are highlighted. The present results are obtained with PP1. PP2 pseudopotentials show an analogous behavior.

Figure 5

Inverse dielectric function ${\varepsilon }_{\text{G}}^{-1}(\mathrm{q}\to 0,\omega =0)$ of ${\mathrm{CsPbI}}_3$ as calculated using linear response theory (circles) and fitted according to the model function (solid line). The fitted values for the functional parameters ${\alpha }_{\ell }$ and $\mu$ and for the root-mean-square error (RMSE) are given.

Figure 6

Band edges and defect energies displayed versus the mixing parameter $\alpha$ for the iodine vacancy ${\text{V}}_{\text{I}}$ (green circle) and various local potential probes (red squares) in ${\mathrm{CsPbI}}_3$. ${\mathrm{X}}^{*}$ refers to the potential probe with the smallest degree of hybridization with the band edges. The indicated band gaps are obtained with the functional $\text{PBE}0\left(\alpha \right)$.

Figure 7

Isocontour plots for K-CAM band gaps of ${\mathrm{CsPbI}}_3$ as a function of the short-range fraction of Fock exchange ${\alpha }_{\text{s}}$ and the inverse screening length $\mu$. The parameter ${\alpha }_{\ell }$ is fixed to $1/{\varepsilon }_{\infty }$. The displayed band gaps are obtained with (a) PP1 and (b) PP2 pseudopotentials. The red lines indicate the fulfillment of Koopmans condition, and the regions shaded in red deviations below 0.1 eV. The point in parameter space corresponding to the K-PBE0 functional is highlighted with a yellow square.