Impacts of cation ordering on bandgap dispersion of double perovskites

Since the methylammonium lead tri-iodide is first adopted as a sensitizer of dye-sensitized solar cells,¹ the photon conversion efficiency (PCE) enhancement speed of metal-halide perovskite (MHP) solar cells has skyrocketed. The biggest concerns regarding MHPs are poor stability and the toxicity of Pb-based materials.^2,3 There have been many experimental and theoretical studies to develop robust Pb-free MHPs.^4–7 While Sn-based MHP solar cells exhibit the highest PCE,^8,9 the stability of Sn-based perovskites is worse than that of Pb-based solar cells due to the multivalent feature.^10,11

Recently, there have been many efforts to develop Pb-free perovskites by expanding the concept of the common perovskite lattice family. Double perovskite is one promising candidate for Pb-free MHPs, which can be made by replacing two divalent Pb²⁺ ions with one monovalent B⁺ ion and one trivalent B³⁺ ion (A₂B⁺B³⁺X₆). Since Volonakis et al. first suggested the double perovskite concept and synthesized Cs₂AgBiCl₆,¹² there have been many improvements and discoveries in developing new-type double perovskites. The same research group made Cs₂AgBiBr₆ and provided a detailed analysis of its electronic structures.¹³ Research on molecular cation double perovskite ((CH₃NH₃)₂AgBiBr₆) was also published by Wei et al.¹⁴ Ag-Bi-based double perovskites usually exhibit an indirect bandgap so that high PCE cannot be expected. As the bandgaps of synthesized double perovskites are too wide to be used for solar cells, many theoretical studies have focused on modulating bandgaps by alloying these materials with metal ions,¹⁵ or using defect engineering,¹⁶ and order-disorder transition.¹⁷ 

One noticeable feature in double halide perovskites is that the variation of the bandgaps is relatively large in experimental values, i.e., there is large bandgap dispersion. For example, the experimentally reported bandgap of Cs₂AgBiCl₆ is in the range of 2.2 eV–2.77 eV and that of Cs₂AgBiBr₆ is between 2.06 eV and 2.3 eV.^12,18 Savory et al. conjectured that the cation disorder can significantly alter the bandgap: they showed that the bandgap of a disordered supercell can be smaller by 1.2 eV than that of an ordered supercell.⁶ These prior studies have motivated us to investigate the role of cation ordering in the electronic structures of double perovskites. The cation ordering can be visualized with octahedron ordering, as shown in Fig. 1.

In this study, we report the impacts of cation ordering on electronic structure changes in double perovskites; the bandgap dispersion of synthesized double perovskites would be originated from this cation ordering variation. While there is only one atom configuration in the primitive cell of Cs₂AgBiCl₆, there can be three different cation orderings in an orthorhombic unit cell, as shown in Fig. 1. Common rock-salt type double perovskites correspond to C-type [Fig. 1(c)].

Density functional theory (DFT) calculations, performed using the VASP package, are used to present the electronic structures and to provide a theoretical description of halide perovskites.^19,20 DFT has been widely used to reveal the fundamentals and unique properties of MHPs.^21–26 The calculated lattice structures, energies/atoms, and bandgaps are summarized in Table I. C-type octahedron alignment has minimum energy and shows the largest bandgap. Changes of cation ordering significantly influence the bandgap despite the relatively small lattice energy difference.

The calculated E-k diagrams according to lattice symmetry and density of states (DOS) are presented in Figs. 2(a)–2(c) for the structures in Figs. 1(a)–1(c). As spin orbit coupling (SOC) is essential to reflect the band structure changes,^27–29 the presented data were obtained with PBEsol + SOC (PBESOC). Even considering the hybrid density functional method (HSE06), the qualitative trends are not changed (see Fig. S1 of the supplementary material). Although C-type has the minimum energy configuration, it exhibits an indirect bandgap. On the other hand, the band structure of B-type changes to the direct bandgap and, additionally, the bandgap can be decreased. A photon can be successfully absorbed at this direct bandgap point; this can be verified with the calculated absorption curve for B-type (Fig. S2 of the supplementary material). A-type perovskites even show a metallic band structure; the bandgap is not open, even when we consider HSE06 [Fig. S1(a) of the supplementary material].

Chemical bonding characteristics of C-type are the same as those in prior results.¹³ Considering the DOS in Figs. 2(d)–2(f), valence band maxima (VBMs) of B- and C-types consist of Ag-d and Cl-p hybridization. The charge density profile shown in Fig. 3(a) indicates that VBM is formed by anti-bonding between Ag-d and Cl-p. The conduction band minimum (CBM) of C-type is made by hybridization among Bi-p, Ag-s, and Cl-p [Fig. 2(f)]. To compare the indirect and direct gap CBM orbitals, charge density profiles for “2” and “3” states in Fig. 2(c) are represented in Figs. 3(b) and 3(c), respectively. The hybridization between Bi-p and Cl-p is predominant for the direct gap CBM [Fig. 3(b)], while there is a contribution of Ag-s for the indirect gap CBM [Fig. 3(c)]. The direct gap CBM of B-type, i.e., “4” state in Fig. 2(b), has chemical bonding characteristics similar to those of type-C [Fig. 3(d)].

The direct-indirect band transition between B and C-type double perovskites can be explained with the orbital configuration changes. Hybridization among Ag-s, Bi-p, and Cl-p can be found at CBM of C-type [Figs. 2(f), 3(b), and 3(c)], while there is only hybridization between Bi-p and Cl-p at CBM of B-type [Figs. 2(e) and 3(d)]. CBM of halide perovskites can show the direct bandgap by negating the contribution of the metal-s orbital, which has been shown by Tran et al.’s report.³⁰ C-type is a 3-dimensional corner-shared network with BiCl₆ and AgCl₆ octahedron so that there is 3-dimensional deployment of antibonding between Ag-s and Cl-p [Fig. S3(a) of the supplementary material]. For B-type, a Bi-Cl octahedron is aligned with 1-D so that CBM can be made with only Bi-p and Cl-p hybridization without Ag-s [Fig. S3(b) of the supplementary material]. As the energy level of Ag-5s is higher than that of Bi-6p_1/2, the bandgap can be decreased for B-type. Thus, the energy level of CBM of B-type at the Γ-point can be lowered and the direct bandgap can be achieved.

Considering larger lattice structures, the bandgap dispersion can also be detected. Using the site occupancy disorder method,³¹ we construct 10 different 2 × 1 × 1 supercells containing 40 atoms. C-type cation alignment (configurational degeneracy: 2) shows the lowest energy and the largest bandgap, but the energy of the highly degenerate lattice structure (degeneracy: 16, ID#6 in Table S.I of the supplementary material) is larger only by 24 meV/atom and the bandgap is lower by 0.96 eV. Therefore, different cation ordering can be generated within bulk Cs₂BiAgCl₆ and the effects on the bandgap cannot be disregarded.

Monte Carlo (MC) sampling coupled with simulated annealing using cluster expansion (CE)³² of the configuration energy is used to find the minimum energy configuration at ambient temperature. The energy difference between DFT calculations and CE estimated values is below 5 meV/atom (Fig. S4 of the supplementary material). At 300 K, for a 2 × 2 × 2 supercell, 10 lattice configurations close to minimum energy configuration were obtained; these are used as atomic position data for DFT calculations. Bandgap dispersion is represented in Fig. 4(a). While two lowest minimum energy configurations, i.e., CN#1 vs. CN#2, exhibit almost the same lattice energies, the bandgaps are different by 0.55 eV. Considering 10 different low-lying energy configurations, the bandgap can be modulated by 1 eV.

In Fig. 4(b), DOS and the charge density profiles of CBM and VBM of configuration number CN#1 shown in Fig. 4(a) are represented. Although the bandgap of CN#1 is smaller than that of bulk C-type and CN#2, the band states are well developed throughout the supercell. The extended state-like feature shown in Fig. 4(b) is different from the anti-site (Ag_Bi-Bi_Ag) defect formation. When there is only one-pair of Ag_Bi-Bi_Ag within a 2 × 2 × 2 C-type supercell [Fig. S4(a) of the supplementary material], CBM shows localized profiles which can also be confirmed by considering the mid-gap states within the bandgap in DOS [Fig. S4(b) of the supplementary material]. Therefore, the cation ordering in double perovskites can be regarded as evidence of a narrower bandgap, rather than as evidence of trapping centers of free carriers.

Our results imply that the bandgap dispersion induced by various cation ordering configurations can be reduced by increasing the energy difference between C-type and other configurations. The energy differences between C-type and B-type were calculated for various double perovskites and are presented in Fig. 4(c). We chose Ag, In, and Tl for the B⁺ cation, Bi and Sb for the B³⁺ cation, and Cl and Br for the X⁻ anion for the candidates. A combination of B⁺ = In, Tl and B³⁺ = Bi, Sb was suggested by Meng et al.³³ because the parity forbidden gap problem can be avoided. Among B⁺ cations, Tl shows promising behavior and Bi is commonly better than Sb for the B³⁺ cation. Br exhibits worse selectivity than Cl except when Tl is used for the B⁺ cation. The ionic radius of Tl is larger than that of the other ions so that 1-D or 2-D alignment of TlX₆ may not be favored over 0-D alignment (C-type).

Successful synthesis of Ag-Bi based double perovskites can be partly ascribed to their similar ionic radius which is helpful to form a stable solid solution. However, this good miscibility can induce disorder of cations, which significantly affects the electronic structures of double perovskites. Ordered cation alignment can be achieved by alloying larger ions such as Tl, but this may be harmful to the formation of stable perovskite structures.⁶ Alloying metal ions,¹⁵ or triple metal cations, may be beneficial to enhance the selectivity of the ordered cation structures, which have not been studied yet.

The bandgap variation by ordering variation was intensively studied in the field of semiconductor alloys.³⁴ Wei and Zunger has already shown that the bandgaps of II-VI and III-V semiconductor alloys can be significantly affected by their polytypisms and order-disorder transition.³⁵ The effects of order-disorder in halide perovskites have not been thoroughly examined yet as much as conventional compound semiconductors. We believe that order-disorder issues become more important in the field of developing more robust and less toxic perovskites.

In summary, we have reported that the electronic structures of double perovskites can be severely affected by cation ordering, i.e., octahedron ordering. When BiCl₆ is surrounded by 6 AgCl₆ octahedrons in Cs₂BiAgCl₆ (0-D BiCl₆ alignment), the total energy is lower and the bandgap is larger, with an indirect bandgap feature. As the dimensionality of BiCl₆ increases to 1-D or 2-D, the bandgap decreases greatly and even the 2-D BiCl₆ alignment shows metallic electronic structures. This microscopic change of octahedron configuration can be generated even with small energies and can induce significant bandgap dispersion, even at ambient temperature. Our results show that the selectivity of target cation alignment should be considered in the design and synthesis of multiple metal cation Pb-free perovskites.

See supplementary material for the computational details, E-k diagrams obtained by the HSE06 + SOC calculation, calculated absorption coefficient of the B-type octahedron alignment, charge density profiles of the CBM at the Γ-point for C-type and B-type, comparison of DFT calculation results with CE estimated energies, electronic structures of anti-site (Ag_Bi-Bi_Ag) within the 2 × 2 × 2 C-type supercell, and dispersion of total energy and bandgaps of the 2 × 1 × 1 supercell.

This research was supported by the National R&D Program through the National Research Foundation of Korea (NRF) (Nos. NRF-2015R1A1A1A05001241, NRF-2015M1A2A2055836, and NRF-2017M3A7B4041698). Supercomputing resources including technical support were supported by the Supercomputing Center/Korea Institute of Science and Technology Information (No. KSC-2017-C2-0006).