Effects of band edge positions on defect structure in lead halide perovskites: A case study on the Br vacancy in $\mathrm{Cs}\mathrm{Pb}{\mathrm{Br}}_3$

In practical density-functional-theory-based simulations of defects in lead halide perovskites (LHPs), it is often assumed that spin-orbit coupling (SOC) and self-interaction corrections (SIC) mostly affect the electronic structure, whereas their effects on the structural properties are minor. Therefore, SOC and SIC are usually excluded during structural relaxation or molecular dynamics simulations. With such an assumption, previous studies predicted several unusual features of Br vacancies ($V_{\text{Br}}$) in $\mathrm{Cs}\mathrm{Pb}{\mathrm{Br}}_3$, including the formation of a Pb dimer, negative-$U$ character, and highly dynamic defect level. In this work, it is shown that SOC and SIC can play important roles in determining the equilibrium geometry and potential energy surface of $V_{\text{Br}}$ in $\mathrm{Cs}\mathrm{Pb}{\mathrm{Br}}_3$. Including SOC and SIC for structural relaxation results in a normal shallow level feature of $V_{\text{Br}}$, instead of the negative-$U$ character, and the Pb dimer becomes unstable in this case. Moreover, the highly dynamic defect level of $V_{\text{Br}}$ is absent when SOC is included for molecular dynamic trajectory generation. The contrasting results compared to previous studies are understood by the sensitive dependence of the occupation of the $V_{\text{Br}}$-associated states on the band edge positions of $\mathrm{Cs}\mathrm{Pb}{\mathrm{Br}}_3$, which is greatly affected by the strong SOC and SIC. These results thus highlight the importance of the correct prediction of the band edge positions for accurate modeling of defects in LHPs.

Figure 1

Calculated charge transition levels for $V_{\text{Br}}$ using different schemes for structural relaxation (SR) and single-point calculation (SPC).

Figure 2

The band structure and the charge density of the electronically active states for $V_{\text{Br}}^0, V_{\text{Br}}^{-}$, and $V_{\text{Br}}^{+}$. Cs atoms are not plotted for clarity. The Fermi level is set to zero.

Figure 3

(a) An illustration on the formation mechanism of the Pb-Pb dimer and the defect-localized state when PBE without SOC is used for structural relaxation. (b) The origin of the shallow level when SOC is included.

Figure 4

The total energy as a function of $d_{\text{Pb-Pb}}$ for $V_{\text{Br}}^0, V_{\text{Br}}^{+}$, and $V_{\text{Br}}^{-}$. The structures are relaxed by PBE, PBE-SOC, and HSE-SOC.

Figure 5

(a) The Kohn-Sham eigenvalues around the band gap for $V_{\text{Br}}^0$ along a 6000-fs MD trajectory generated by PBE without SOC. The defect-localized state (DLS) is inside the band gap. (b) The distance between the two Pb atoms around the vacancy site along the same MD trajectory as in (a). (c, d) The same as (a, b) but the MD trajectory is generated using PBE-SOC.