Pushing the boundary of the stability and band gap Pareto front by going towards high-entropy perovskites

Abstract

Lead-free Cs₂BX₆ (B = Zr⁴⁺, Sn⁴⁺, Te⁴⁺, Hf⁴⁺, Re⁴⁺, Os⁴⁺, Ir⁴⁺, and Pt⁴⁺ and X = Cl⁻, Br⁻, and I⁻) vacancy-ordered double perovskites have gained significant attention due to their high performance in solar cell devices. Besides mitigating toxicity concerns associated with the use of lead, the presence of a formally tetravalent B-site in Cs₂BX₆ has been demonstrated to improve the stability against air and moisture. Recently, experimental studies have shown that high-entropy forms of vacancy-ordered double perovskites can be synthesized and stabilized at room temperature, which opens new opportunities for designing better solar cell absorbers. In this work, we employed high throughput density functional theory (DFT) calculations using the HSE06 hybrid functional to study 546 medium-to-high-entropy vacancy-ordered double perovskites. Our results show that Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ perovskites can break the existing linear scaling relationships between the bandgap and formation energy observed in the pure Cs₂BX₆ and Cs₂B{XX′}₆ perovskites, which enables materials that simultaneously exhibit an optimal band gap of ∼1.3 eV for single-junction solar cells along with a low formation energy. Electronic structure analysis reveals that this can be attributed to the weak coupling between the BX₆ octahedra in Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆. Based on these findings, we identified the analytical equations that can be used to efficiently predict the band gap and formation energy of high-entropy perovskites from their constituent pure perovskites. Our study offers simple and practical guidelines for the design and synthesis of novel high-entropy perovskites with improved photovoltaic performance.

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Guoxiang Hu

Guoxiang (Emma) Hu is an assistant professor in the School of Materials Science and Engineering at the Georgia Institute of Technology, Atlanta, USA. She obtained her Ph.D. in physical chemistry in 2018 from the University of California, Riverside, and was a postdoctoral scholar in the Center for Nanophase Materials Sciences at the Oak Ridge National Laboratory, USA. Her research focuses on using quantum mechanical (QM) modeling and machine learning (ML) to study functional materials for energy-related applications including catalysis and optoelectronics. Emma's group also develops and applies advanced computational methods integrating ML with QM modeling at different levels of theory for accurate and efficient modeling of complex interfaces and quantum materials.

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Introduction

At present, lead halide perovskites are the best performing solution-processed photovoltaic materials with a power conversion efficiency record of 26.7% in single-junction perovskite solar cells and 34.6% in perovskite/Si 2T tandem solar cells in 2024.^1–5 However, the ultimate commercialization of these materials still faces serious practical challenges due to their toxicity from lead and their instability against moisture, air, and temperature.^6–10 Alternative compositional and structural derivatives of the perovskite family have consequently been explored to assess their potential in photovoltaic applications.^11–16 Among these novel perovskites developed over the past decade, Cs₂BX₆ (B = Zr⁴⁺, Sn⁴⁺, Te⁴⁺, Hf⁴⁺, Re⁴⁺, Os⁴⁺, Ir⁴⁺, and Pt⁴⁺ and X = Cl⁻, Br⁻, and I⁻) vacancy-ordered double perovskites have attracted particular attention following the recent demonstrations of their high performance in solar cell devices.^17–21 Besides mitigating toxicity concerns associated with the use of lead, the presence of a formally tetravalent B-site in Cs₂BX₆ can also improve air and moisture stability relative to the divalent B-site in conventional CsBX₃ perovskites.^22–24

High-entropy halide perovskites are an emerging class of materials that combine the intriguing properties of halide perovskites with the concept of high entropy.^25,26 Common advantages of high-entropy materials, which contain five or more different elements distributed in a single crystal structure, include entropic stabilization (at high temperature) and a tunable continuum of materials properties.^25,27–29 This innovative design strategy has been widely explored in metals and oxides,^30–33 but its application to halide perovskites is a relatively new concept and holds great untapped potential. A recent experimental study has demonstrated a viable route to synthesize and stabilize high-entropy vacancy-ordered double perovskites at room temperature such as the five-element Cs₂{SnTeReIrPt}₁Cl₆ and six-element Cs₂{SnTeReOsIrPt}₁Cl₆ compositions.²⁵ This opens up new opportunities for designing novel materials based on high-entropy vacancy-ordered double perovskites for photovoltaic devices beyond the state-of-the-art lead halide perovskites.

In this work, we performed high throughput density functional theory (DFT) calculations using the hybrid functional HSE06 to study over five hundred medium-to-high-entropy vacancy-ordered double perovskites. We calculated the band gap and formation energy of these perovskites and found that high-entropy perovskites exhibit emerging properties that are not present in their pure counterparts. We further analyzed their density of states (DOS) and revealed that this can be attributed to the unique electronic structure of vacancy-ordered double perovskites with weakly coupled octahedra. Finally, we identified the analytical equations that can be used to efficiently predict the band gap and formation energy of high-entropy perovskites using the values of their constituent pure perovskites. Our work demonstrated new opportunities for applications of high-entropy halide perovskites in solar cell devices.

Results and discussion

Vacancy-ordered double perovskites are a specialized class of perovskite materials in which specific atomic sites within the crystal structure are deliberately left vacant, leading to a unique arrangement of atoms. In conventional Cs₂BB′X₆ (B = Cu⁺, Ag⁺, Au⁺, K⁺, In⁺, and Ti⁺ and B′ = Bi³⁺, Sb³⁺, and Tl³⁺) double perovskites, two different cations B and B′ occupy alternating positions within the lattice. However, in Cs₂BX₆ vacancy-ordered double perovskites, as shown in Fig. 1, one of the cation sites is left vacant, creating a pattern of missing atoms within the structure. In this work, we considered 8 elements (Zr, Hf, Re, Os, Ir, Pt, Sn and Te) at the B-site, and 3 elements (Cl, Br, and I) at the X-site. Besides the pure Cs₂BX₆ perovskites, we also investigated three types of mixed perovskites: (1) Cs₂B{XX′}₆ with the X-site mixed, (2) Cs₂{B₁B₂B₃B₄}₁X₆ with the B-site mixed, and (3) Cs₂{B₁B₂B₃B₄}₁{XX′}₆ with both B- and X-sites mixed. For X-site mixing, we examined the combinations of Cl₄Br₂, Cl₂Br₄, Br₄I₂, and Br₂I₄, considering their atomic radii and structural symmetry.^25,34 Previous studies have shown that the large difference in atomic radii between the Cl⁻ and I⁻ anions leads to low ionic miscibility, preventing Cl⁻ from substituting into the I⁻ lattice, particularly for relatively high concentration levels.^35–37 This in total yields 546 perovskite structures: (1) 24 pure Cs₂BX₆ perovskites (8 elements at the B-site × 3 elements at the X-site), (2) 32 Cs₂B{XX′}₆ perovskites (8 elements at the B-site × 4 combinations at the X-site), (3) 210 Cs₂{B₁B₂B₃B₄}₁X₆ perovskites (70 combinations at the B-site × 3 elements at the X-site), and (4) 280 Cs₂{B₁B₂B₃B₄}₁{XX′}₆ perovskites (70 combinations at the B-site × 4 combinations at the X-site).

Fig. 1 The chemical space and atomic structures of the pure and mixed vacancy-ordered double perovskites investigated in this work.

Fig. 2 presents the DFT-calculated formation energy and band gap for the 546 vacancy-ordered double perovskites. The computed values for the lattice constant, band gap, formation enthalpy, entropy, and Gibbs free energy at 298 K are provided in Table S1.† A linear scaling relationship between them can be observed for Cs₂BX₆ and Cs₂B{XX′}₆ perovskites: the band gap increases as the magnitude of the formation energy increases. This has been widely reported for vacancy-ordered double perovskites and can be roughly explained by molecular orbital theory.^38–40 Typically, perovskites with stronger chemical (covalent) bonds have more negative formation energies (i.e. larger magnitude) along with a wider splitting between the bonding and anti-bonding molecular orbitals (i.e. larger band gap). As a result, perovskites with desirable band gaps (1.1–1.5 eV)⁴¹ for light absorption normally exhibit relatively poor stability. As highlighted by a grey box in Fig. 2, only very few Cs₂BX₆ and Cs₂B{XX′}₆ perovskites are in this band gap range and their formation energy are relatively high, corresponding to low stability. Table 1 lists the chemical composition, band gap, and formation energy of these perovskites. Our computational predictions of the band gaps agree very well with previous computational and experimental studies.^{22,40,42–44} It is also worth noting that all these perovskites are made of expensive metals (i.e., Pt, Os, or Ir). However, very excitingly, we found that the B-site-mixed Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ medium-to-high-entropy perovskites can break the linear scaling relationship between the formation energy and band gap, allowing for significantly more candidates exhibiting an optimal band gap and high stability for solar absorbers.

Fig. 2 DFT-calculated formation energy and band gap for 546 vacancy-ordered double perovskites including Cs₂BX₆, Cs₂B{XX′}₆, Cs₂{B₁B₂B₃B₄}₁X₆, and Cs₂{B₁B₂B₃B₄}₁{XX′}₆.

Table 1 The band gap E_g and formation energy E_f of the five Cs₂BX₆ and Cs₂B{XX′}₆ perovskites with 1.1–1.5 eV band gaps

	Cs2PtI6	Cs2PtBr4I2	Cs2OsCl2Br4	Cs2OsBr4I2	Cs2IrCl4Br2
E g (eV)	1.13	1.49	1.12	1.22	1.33
E f (eV)	−7.72	−9.15	−9.55	−7.91	−10.55

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Table S2† presents 99 promising vacancy-ordered double perovskites with band gaps ranging from 1.1 to 1.5 eV. It is important to highlight that, as solar-absorbing materials, these perovskites may exhibit dipole-allowed transition gaps (referred to as “allowed gaps”) that are larger than their direct band gaps.⁴⁵ For example, the direct band gap of Cs₂SnI₆ was calculated to be 0.7 eV, while its dipole-allowed transition gap was found to be 1.1 eV. Additionally, these materials may exhibit significant exciton binding energies, which can impede charge separation and reduce open-circuit voltages in solar cells.⁴⁶ Therefore, in addition to the fundamental band gaps (VBM–CBM) shown in Fig. 2, it is crucial to further investigate the optical band gap and exciton binding energy of these promising candidates in future studies. Moreover, while formation energy provides an initial estimate of the stability of predicted perovskites, phonon calculations and finite-temperature molecular dynamics simulations can be further employed for a more comprehensive evaluation of their dynamic and thermodynamic stability.

We then sought to understand the origin of the breaking of the linear scaling relationship between the formation energy and band gap of the B-site-mixed Cs₂{B₁B₂B₃B₄}₁X₆ perovskites. As shown in Fig. 3a, we found that the formation energy of Cs₂{B₁B₂B₃B₄}₁X₆ can be predicted very well by the average of the formation energy of the constituent pure perovskites using the following equation:

where E_form (B₁, B₂, B₃, B₄) is the formation energy of Cs₂{B₁B₂B₃B₄}₁X₆ and E_form(B_i) is the formation energy of the constituent pure perovskite, whereas for the band gap, the corresponding equation does not work:where E_g (B₁, B₂, B₃, B₄) is the band gap of Cs₂{B₁B₂B₃B₄}₁X₆ and E_g(B_i) is the band gap of the constituent pure perovskite. As illustrated in Fig. 3b, eqn (2) generally overestimates the band gap of Cs₂{B₁B₂B₃B₄}₁X₆ with the predicted values consistently exceeding those calculated using DFT.

Fig. 3 The comparison between (a) the predicted formation energy from eqn (1) and the DFT-calculated formation energy and (b) the predicted band gap from eqn (2) and the DFT-calculated band gap.

The dramatically different correlations with the constituent pure perovskites for the formation energy and band gap shown in Fig. 3 explain the breaking of scaling observed in Fig. 2. To gain a better understanding of the correlation of the band gap of Cs₂{B₁B₂B₃B₄}₁X₆ with their constituent pure perovskites, we performed a detailed analysis of their electronic structures. We found that for the diamagnetic metal ions (i.e., Hf⁴⁺, Pt⁴⁺, Sn⁴⁺, Te⁴⁺, and Zr⁴⁺), the projected density of states is very similar in the mixed Cs₂{B₁B₂B₃B₄}₁X₆ and pure Cs₂BX₆ perovskites. This also applies to the paramagnetic metal ion Re⁴⁺ with each t_2g orbital being singly occupied (Fig. 4a). As for the paramagnetic metal ions Ir⁴⁺ and Os⁴⁺, we found that the pure perovskites exhibit degenerate t_2g orbitals (i.e., d_xz, d_yz, and d_xy orbitals), which agrees with previous reports.^47–49 However, as shown in Fig. 4b and c, in the mixed perovskites, the breaking of symmetry leads to splitting of the t_2g orbitals. It is noted that while the t_2g orbitals are different for the mixed and pure perovskites, the energy range of the projected density of states is not significantly changed.

Fig. 4 (a) Comparison of the projected density of states on Re 5d orbitals in Cs₂ReCl₆ (blue) and Cs₂{ZrSnRePt}₁Cl₆ (red). (b) Comparison of the projected density of states of Os 5d orbitals in Cs₂OsCl₆ (blue) and Cs₂{ZrSnOsPt}₁Cl₆ (red). (c) Comparison of the projected density of states on Ir 5d orbitals in Cs₂IrCl₆ (blue) and Cs₂{ZrSnIrPt}₁Cl₆ (red).

The highly overlapped energy range of the electronic states in the pure and B-site-mixed perovskites shown in Fig. 4 can be attributed to the relatively weak coupling between the neighboring BX₆ octahedra in the vacancy-ordered double perovskites. Consequently, the conduction band minimum (CBM) and valence band maximum (VBM) of the pure perovskites can be used to estimate the band gap of the B-site-mixed perovskites. Fig. 5 shows the alignment of the density of states between the B-site-mixed and the constituent pure perovskites using Cs₂{ZrSnTeHf}₁Cl₆ as an example. As one can see, the band gap of a B-site-mixed perovskite is essentially determined by the highest VBM and the lowest CBM of the constituent pure perovskites, as expressed by the following equation:

where E_gap(B₁, B₂, B₃, B₄) is the band gap of the B-site-mixed perovskite. CBM_Bi refers to the CBM and VBM_Bi refers to the VBM of the constituent pure perovskites, respectively (i = 1, 2, 3, and 4). We further analyzed the 99 promising vacancy-ordered double perovskites listed in Table S2.† Notably, 94 of them contain either Ir or Os, indicating that these two elements are particularly beneficial for achieving the desired band gap. This can be attributed to the unique band edges of Cs₂IrCl₆ and Cs₂OsCl₆. As shown in Fig. S1,† Cs₂IrCl₆ exhibits the lowest CBM, while Cs₂OsCl₆ has the highest VBM among the eight constituent pure perovskites. The desired band gap of 1.1–1.5 eV can only be achieved when either Ir or Os is present.

Fig. 5 Alignment of the density of states for the B-site-mixed Cs₂{ZrSnTeHf}₁Cl₆ perovskite and its constituent pure perovskites: Cs₂ZrCl₆, Cs₂SnCl₆, Cs₂TeCl₆, and Cs₂HfCl₆.

We then applied eqn (3) to all B-site-mixed Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ perovskites, and Fig. 6a shows the comparison between the predicted band gap from eqn (3) and the DFT-calculated band gap. We found that eqn (3) can qualitatively predict the band gap of Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆, especially when there are one or fewer paramagnetic metal ions (e.g., Re⁴⁺, Os⁴⁺, and Ir⁴⁺). The red circles in Fig. 6a correspond to Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ with multiple paramagnetic metal ions. Excluding these data points, the root-mean-square-error (RMSE) of prediction is reduced from 0.79 eV to 0.67 eV (Fig. 6b). The poor performance of eqn (3) for Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ with multiple paramagnetic metal ions can be attributed to the complex interactions between the spins of the paramagnetic metal ions, which gives rise to different band edges from the pure perovskites.

Fig. 6 Comparison of the predicted and DFT-calculated band gaps for Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ perovskites. (a) Predictions using pure perovskite band edges. Red circles represent perovskites with multiple paramagnetic metal ions (e.g., Re⁴⁺, Os⁴⁺, and Ir⁴⁺). (b) Predictions using pure perovskite band edges with the red circles removed. (c) Predictions using band edges obtained from randomly chosen B-site-mixed perovskites.

One key approximation for eqn (3) is that the positions of projected density of states are identical in the B-site-mixed and pure perovskites. However, realistically, they are not exactly the same, as shown in Fig. 4 and 5. In particular, we found that the VBM of the pure perovskite is ∼0.2 eV higher than that of the B-site-mixed perovskite, while the CBM of the pure perovskite is ∼0.2 eV lower than that of the B-site-mixed perovskite (Fig. 5). Thus, the band gap predicted from eqn (3) using the pure perovskite band edges are on average ∼0.4 eV less than those from DFT calculations. Fig. 6c shows that the RMSE is reduced from 0.67 eV to 0.21 eV, if the VBM and CBM values are obtained from randomly selected B-site-mixed perovskites instead of the pure perovskites. We have further analyzed the standard deviations of the VBM and CBM of the VODPs in various B-site-mixed perovskites and found that they are quite small (less than 0.1 eV). This suggests minimal fluctuations in the band edges of VODPs across various B-site-mixed perovskites and using the average VBM and CBM of the VODPs for eqn (3) will not change the RMSE significantly, as shown in Fig. 6c.

Conclusions

To summarize, we calculated the band gap and formation energy for 546 vacancy-ordered double perovskites and found that the linear scaling relationship observed in Cs₂BX₆ and Cs₂B{XX′}₆ perovskites can be broken in medium-to-high-entropy perovskites. As a result, more than 90 Cs₂{B₁B₂B₃B₄}₁X₆ and Cs₂{B₁B₂B₃B₄}₁{XX′}₆ perovskites were predicted to be promising for solar absorbers with an optimal band gap and high stability. We performed a detailed electronic structure analysis to understand the breaking of the scaling relationship and developed simple analytical equations to predict the band gap of B-site-mixed perovskites using the band gap of their constituent pure perovskites. This will significantly accelerate the design and discovery of novel medium/high-entropy perovskites with desired band gaps for both experimental and computational studies. In addition, the generated data from high throughput DFT calculations using the hybrid functional in this work will be beneficial for the community to further develop machine learning models for materials screening.

Computational methods

Spin-polarized DFT calculations were performed as implemented in the Vienna ab initio simulation package (VASP 6.2.1).^50,51 The electron exchange–correlation was represented by the Perdew–Burke–Ernzerhof revised functional for solids (PBEsol) of generalized gradient approximation (GGA).⁵² While the geometry optimizations were performed using the PBEsol functional, the band gap and density of states calculations were performed using the hybrid HSE06 functional.²² The ion–electron interaction was described with the projector augmented wave (PAW) method.⁵³ A cutoff energy of 500 eV was used for the plane-wave basis set. The energies were converged with a 1 × 10⁻⁵ eV tolerance and the forces were optimized to within 0.01 eV Å⁻¹. The cubic phase of vacancy-ordered double perovskites was used, with a unit cell consisting of 8 Cs atoms, 4 B-site metals, and 24 halides. A 6 × 6 × 6 Monkhorst–Pack k-point grid was used for the geometry optimizations with the PBEsol functional, while a 3 × 3 × 3 Monkhorst–Pack k-point grid was used for the band gap calculations with the HSE06 hybrid functional. It is important to note that while the HSE06 functional is generally effective in predicting band gaps for many semiconductors, it has been shown to underestimate band gaps, particularly in halide perovskites.⁵⁴ Table S3† compares our calculated fundamental band gaps (VBM–CBM) from HSE06 with previous computational studies and experimentally measured optical band gaps. Our calculations are in excellent agreement with prior computational work. However, when compared to experimentally measured optical gaps, our results are consistently lower, which can be attributed to the dipole-allowed transition gaps discussed previously. Thus, determining the optical band gap and employing more accurate functionals, such as PBE0 or dielectric-dependent hybrid functionals, to better align with the experimental results are important avenues for future research.

The formation energy of the pure perovskite E_form(Cs₂BX₆) was calculated using the following equation:⁵⁵

E_form(Cs₂BX₆) = E_total(Cs₂BX₆) − 2E(Cs) − E(B) − 6E(X) (4)

where E_total(Cs₂BX₆) is the total energy of the Cs₂BX₆ unit cell. E(Cs), E(B), and E(X) are the energy per atom for the corresponding elementary substances. For elements that have multiple allotropes, the reference state was chosen to be the form in which the element is most stable under 1 bar. The formation energy of the mixed perovskites was calculated in a similar way to that with eqn (4). The initial atomic structures were obtained from materials projects.⁵⁶

Data availability

All data from this work will be shared, including the data needed to reproduce the results and the raw data from VASP calculations. The data will be shared on the open-source repository Figshare. They will be made available immediately following the publication of the work.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This article is based on work supported as part of the Atomic-C2E project by the U.S. Department of Energy, Office of Science under award number DE-SC-0024716. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-05CH11231 using NERSC award BES-ERCAP0032102.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4nr05013d

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