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Cs2AgBiX6 (X = Br, Cl) — New visible light absorbing, lead-free halide perovskite semiconductors Eric T. McClure, Molly R. Ball, Wolfgang Windl, and Patrick M Woodward Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.5b04231 • Publication Date (Web): 10 Feb 2016 Downloaded from http://pubs.acs.org on February 11, 2016
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Chemistry of Materials
Cs2AgBiX6 (X = Br, Cl) — New visible light absorbing, lead-free halide perovskite semiconductors Eric T. McClure†, Molly R. Ball‡, Wolfgang Windl‡, and Patrick M. Woodward†* ‡
Department of Materials Science and Engineering, The Ohio State University, 2041 College Road, Columbus, Ohio 43210, United States
†
Department of Chemistry and Biochemistry, The Ohio State University, 100 W. 18th Avenue, Columbus, Ohio 43210, United States
ABSTRACT: The double perovskites Cs2AgBiBr6 and Cs2AgBiCl6 have been synthesized from both solid state and solution routes. X-ray diffraction measurements show that both compounds adopt the cubic double perovskite structure, space group Fm 3 m , with lattice parameters of 11.2711(1) Å (X = Br) and 10.7774(2) Å (X = Cl). Diffuse reflectance measurements reveal band gaps of 2.19 eV (X = Br) and 2.77 eV (X = Cl) that are slightly smaller than the band gaps of the analogous lead halide perovskites, 2.26 eV for CH3NH3PbBr3 and 3.00 eV for CH3NH3PbCl3. Band structure calculations indicate that the interaction between the Ag 4d-orbitals and the 3p/4p-orbitals of the halide ion modifies the valence band leading to an indirect band gap. Both compounds are stable when exposed to air, but Cs2AgBiBr6 degrades over a period of weeks when exposed to both ambient air and light. These results show that halide double perovskite semiconductors are potentially an environmentally friendly alternative to the lead halide perovskite semiconductors.
Introduction In a remarkably short period of time metal-halide perovskites have gone from relative obscurity to an intensely studied class of materials. The efficiencies of solar cells constructed from metal-halide perovskites have risen from 3.8% in a dye-sensitized solar cell configuration [1], to NREL certified 20.1% in a planar heterojunction cells [2]. The most studied materials by far are the AMX3 perovskites where A is an alkyl ammonium cation, such as CH3NH3+, M is Pb2+, and X is a halide ion (I−, Br−, Cl−). The allure of the organohalide perovskites for photovoltaic and LED applications stems from the fact that they contain cheap, earth-abundant elements and are amenable to a variety of processing techniques, including solution-based deposition methods [3]. Metal-halide perovskite solar cells have already reached efficiency levels that are commercially viable. The remaining obstacles to commercialization are threefold: (i) develop cost effective, large scale manufacturing routes, (ii) improve the moisture stability so that modules can operate in an outdoor environment without resorting to expensive encapsulation methods, and (iii) discover metalhalide alternatives that don’t contain environmentally harmful elements like lead [3]. The first obstacle is an industrial challenge that lies outside of the scope of fundamental research, but the latter two challenges fall squarely in the realm of materials chemistry. While there are many reports of CH3NH3PbX3 perovskites being stable
for hundreds of hours when left exposed to the air in the dark [4,5], their stability plummets when simultaneous exposed to moisture and sunlight [6]. Regarding the presence of lead there are varying opinions in the scientific community about the viability of working with lead. Lead is present in everyday items like car batteries and solder, and equally toxic elements like Cd can be found in commercial photovoltaic modules [7,8]. At the same time most of the world is trying to move away from materials containing lead. While there may be a place for applications of the lead halide perovskites in solar energy conversion devices, there can be little doubt that discovery of a lead-free material whose properties matched perovskites like CH3NH3PbI3 or CH3NH3PbBr3 would be a boon. In short that is the goal of this work, to find metal-halide perovskites whose properties approximate those of the extensively studied CH3NH3PbX3 perovskites without incorporating toxic elements like Pb, Cd, Tl, or Hg. The Sn2+ ion is most obvious substitute for Pb2+, and tinhalide perovskites with attractive optical and electrical transport properties can be made [9-11]. Unfortunately, the tin-halide perovskites are extremely unstable in air, degrading over time even when handled in an inert atmosphere glove box. It seems unlikely that the Sn-based perovskites will ever have the stability to be commercially viable alternatives to the Pb-based perovskites. The role of the organic CH3NH3+ cation seems to be less critical. The optical band gaps of CsPbBr3 (2.25 eV) [12] and CsPbCl3 (2.97 eV) [13] are quite similar to their organohalide
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analogs CH3NH3PbBr3 (2.26 eV) and CH3NH3PbCl3 (3.00 eV), vide infra. Furthermore, Kulbak, et al. have shown that the efficiencies of solar cells made from CsPbBr3 are comparable to equivalent cells made from CH3NH3PbBr3, with the advantage that the all-inorganic CsPbBr3 cells have better thermal stabilities and can operate longer without degrading [14,15]. Unfortunately the perovskite polymorph of CsPbI3, which has the most suitable band gap for solar cell applications, is very difficult to stabilize at room temperature [9]. Hence, the role of the CH3NH3+ ion is largely to stabilize the perovskite form of APbI3 at room temperature. Simple combinatorics of ionic charges in the A+M2+X3 formula does not lend much hope that alternatives to the APbX3 perovskites with comparable band gaps, carrier mobilities, and good moisture stability can be found. There are few if any suitable 2+ cations with a d10s2 or d10s0 configuration that are both non-toxic and stable against oxidation. Fortunately the situation improves if we expand the search from ternary A+M2+X3 perovskites to quaternary A2M+M3+X6 double perovskites. There are numerous combinations of 1+ and 3+ ions with suitable electron configurations that are quite stable when exposed to the air, such as Cu+, Ag+, Bi3+, Sb3+, and In3+. In this paper we document our initial investigations into the structural and optical properties of metal-halide double perovskites by reporting on the synthesis and properties of Cs2AgBiX6 (X = Br, Cl). These compounds have band gaps and moisture stability that are comparable to their CH3NH3PbX3 analogs, demonstrating that double perovskites are a promising alternative to lead halide perovskites.
Experimental Starting materials CsCl and CsBr were prepared by reacting Cs2CO3 (99+%, Strem Chemicals) with HCl (Sigma Aldrich, 37%) and HBr (Fluka, ≥ 48%), respectively. The solutions were evaporated and the resulting solids were filtered and washed with ethanol. AgCl and AgBr were precipitated from as prepared aqueous solutions of AgNO3 (99.9+%, Alfa Aesar) and NaCl (ACS Reagent, GFS Chemicals) or KBr (99+%, Alfa Aesar). BiBr3 was prepared by reacting Bi2O3 (≥99.0%, J.T. Baker) with HBr (Fluka, ≥ 48%). The mixture was heated until fully dissolved, evaporated to dryness, and then filtered and washed with ethanol. Pure BiCl3 could not be prepared from this method, thus a commercial reagent was used (≥98%, Aldrich). Polycrystalline Cs2AgBiX6 samples were prepared by precipitation from a solution of the hydrohalic acid and hypophosphorous acid. A mixture of 8 mL of 12.1 M HCl (or 8.84 M HBr) and 2 mL of a 50% solution of H3PO2 was added to a round bottom flask and heated to 120 °C. Then 1.89 mmol of AgCl and an equal amount of BiCl3 (1.41 mmol AgBr and BiBr3) were dissolved in the hot solution. Next 3.78 mmol CsCl (2.82 mmol CsBr) was added to the flask, immediately triggering the formation of a precipi-
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tate. The precipitate was collected on filter paper, washed with ethanol, and then dried overnight. Polycrystalline Cs2AgBiX6 samples were prepared via a solid state route as well by mixing cesium, silver, and bismuth halide salts in a 2:1:1 molar ratio. Reagents were ground together for 20 minutes and placed in an alumina crucible as a loose powder. The samples were heated in a box furnace in air at 210 °C for 10 hours. It was found that at least two heating cycles with grinding in-between was needed to obtain nearly phase pure samples. X-ray powder diffraction (XRPD) data were collected on a Bruker D8 powder diffractometer (40 kV, 50 mA, sealed Cu X-ray tube) equipped with an incident beam Ge 111 monochromator and Lynx Eye position sensitive detector. Rietveld refinements of laboratory XRPD data were carried out using TOPAS Academic software package to determine the crystal structure [16]. A Rigaku MiniFlex II bench top X-ray powder diffractometer (30 kV, 15 mA, sealed Cu X-ray tube) with a NaI scintillation detector was also used for phase identification, pattern indexing, and for probing air and light stability of the synthesized compounds. UV-Visible diffuse reflectance data were collected over the spectral range of 200-1100 nm with an Ocean Optics USB4000 spectrometer equipped with a Toshiba TCD1304AP (3648-element linear silicon CCD array). The spectrometer was used in conjunction with an Ocean Optics DH-2000-BAL deuterium and halogen UV-Vis-NIR light source and a 400 µm R400-7-ANGLE-VIS reflectance probe. Electronic band structure and density of states (DOS) calculations were obtained with density functional theory implemented in the Vienna Ab-initio Simulation Package [17,18]. These calculations were performed using projector augmented wave (PAW) potentials [19] based on the PBE exchange-correlation functional [20] within the HSE hybrid approach [21-23] with an exact exchange fraction of 0.26. Cutoff energies of 250 and 280 eV were used for Cs2AgBiBr6 and Cs2AgBiCl6, respectively. A 13 × 13 × 13 grid of k-points was used for the calculations. The calculations include spin-orbit coupling.
Results Synthesis When synthesizing the bromine compound in the solid state, evidence for the perovskite phase was not seen in the XRPD pattern until the sample was heated to 150 °C. At a heating temperature of 185 °C the target compound was the dominant phase in the sample mixture. After heating to 210 °C the yellow-orange sample was nearly phase pure. For the solid state synthesis of the chlorine compound, evidence of the perovskite phase was seen upon room temperature grinding of the reactants. Since bismuth (III) chloride is hygroscopic, the sample mixture took on moisture from the air while being ground, causing the
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sample to clump together in a gray mass that had the appearance of wet clay. The sample was left to sit on the lab bench for 10-15 minutes, and during that time the gray mass dried out. Additional grinding led to the formation of a fine powder with a pale yellow color, similar to the final product. Evidently the moisture taken on by the reagents promotes reactivity and is then displaced once the double perovskite phase forms. Two cycles of heating to 210 °C are still needed to obtain phase pure samples. While samples made via solid state and solution routes gave very similar products, small levels of secondary phases were visible in the XRPD patterns of the solid state samples whereas samples made via solution synthesis method exhibited excellent phase purity. Consequently the solution processed samples were used for subsequent characterization.
Crystal Structure The refined fits to the XRPD patterns of Cs2AgBiBr6 and Cs2AgBiCl6 are shown in Figure 1. Both compounds adopt the cubic double perovskite structure with Fm 3 m space group symmetry. Reflections with odd-odd-odd Miller Indices signaling a rock salt ordering of Ag+ and Bi3+ ions are readily apparent. Close inspection of the XRPD patterns reveals no signs of secondary phases.
Figure 1. Rietveld refinements of the XRPD patterns for Cs2AgBiCl6 (upper) and Cs2AgBiBr6 (lower). The black dots, red, and gray lines are the experimental pattern, calculated fit, and difference curve, respectively. Red tick marks at the bottom give the expected peak positions. The inset shows the fit to the (111) and (200) reflections.
Structural parameters extracted from the refinements of Cs2AgBiCl6 and Cs2AgBiBr6 are given in Table 1, while atomic coordinates and displacement parameters are given in Table 2. A visualization of the double perovskite structure is shown in Figure 2. Like the basic perovskite structure there is a three-dimensional framework of corner connected octahedra, with Cs+ ions occupying the cuboctahedral cavities in the framework. The double perovskite structure is then obtained by alternating Ag+ and Bi3+ centered octahedra in all three directions building up a superstructure that is typically referred to as rock salt ordering. Bond distances and bond valence sums are tabulated in Table 3. In both compounds the Ag+ and Bi3+ ions show complete ordering into a rock-salt supercell of the simple perovskite structure. The halide ion undergoes small displacements toward the bismuth site, leading to Bi−X distances that are slightly shorter than the Ag−X distances. The possible presence of Ag/Bi antisite disorder was investigated in the Rietveld refinements, but the occupancies did not refine to values that indicated anything other than complete ordering. Given the similarity of the Bi−X and Ag−X distances it is somewhat surprising that antisite mixing between the two ions is not observed. The relatively low bond valence sum for the Cs+ ion and rather large values of the displacement parameters (Beq) suggest that fairly large dynamic motions of the octahedra, most likely octahedral rotations, are taking place at room temperature. It would not be surprising to see a phase transition involving cooperative octahedral tilting take place upon cooling below room temperature. Table 1. Details of the X-ray powder diffraction experiments and Rietveld refinements. Cs2AgBiCl6 Crystal system
Cs2AgBiBr6 Cubic
Space group
Fm-3m (#225)
a (Å) 3
V (Å ) −3
Density (g cm )
10.7774(2)
11.2711(1)
1251.82(6)
1431.86(5)
4.220(2)
4.927(9)
Radiation Cu Kα (Å)
1.5418
Collection Limits (° 2θ)
10.00 – 120.00
Step Size (° 2θ) Rwp 2
χ
0.01460 29.289
28.929
1.126
1.159
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Table 2. Refined atomic positions and atomic displacement parameters for Cs2AgBiCl6 (upper) and Cs2AgBiBr6 (lower, in italics). Site
Wyckoff site
x
y
z
Ag
4a
0
0
0
Bi
4b
0.5
0.5
0.5
Cs
8c
0.25
0.25
0.25
0
0
Cl Br
24e
0.2513(9) 0.2504(4)
Beq 2.1(2) 2.5(2) 0.95(7) 1.63(7) 2.77(9) 4.0(1)
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Optical properties In order to determine the optical band gaps UV-Vis diffuse reflectance spectra of the samples were measured (Figure 3). Spectra for the methylammonium lead halide samples are also shown (in red) for comparison. Aside from some minor differences seen at photon energies above the absorption onset there are striking similarities between the Cs2AgBiX6 double perovskites and their CH3NH3PbX3 analogs. However, the onset of absorption is not quite as sharp in the double perovskites. We believe this change can be attributed to the indirect band gap of the double perovskites. This hypothesis is supported by band structure calculations discussed below.
3.1(2) 3.78(9)
+
Figure 2. Refined crystal structure of Cs2AgBiCl6. The Cs ions are shown as gray spheres, the chloride ions as small green spheres, while the Ag and Bi centered octahedra are shown as blue and green polyhedra, respectively. Table 3. Bond lengths and bond-valence sums. X = Cl
X = Br
Cs – X (× 12)
3.810(7)
3.985(3)
Ag – X (× 6)
2.708(9)
2.822(5)
Bi – X (× 6)
2.680(9)
2.813(5)
Cs
0.76
0.73
Ag
1.07
1.18
Bi
3.49
3.55
X
1.01
1.03
Bond lengths (Å)
Bond valence sums Figure 3. Diffuse reflectance spectra for Cs2AgBiCl6 and CH3NH3PbCl3 (top), and Cs2AgBiBr6 and CH3NH3PbBr3 (bottom).
In order to extract an optical band gap the reflectance data were transformed to pseudo-absorption data using the Kubelka-Munk equation, which expresses the absorbance as a function of reflectance: F(R) = α = (1-R)2/(2R), where R is the reflectance and α is the optical absorption coefficient. Kubleka-Munk plots for the silver-bismuth
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and lead compounds are shown in the Supplemental Information. The Tauc Method is then applied to the Kubelka Munk transformed data. It utilizes the equation [F(R)hν]1/ϒ vs. hν, where ϒ is equal to ½ for direct transitions (applicable to CH3NH3PbX3 compositions) and 2 for indirect transitions (applicable to Cs2AgBiX6 compositions). Using this approach band gaps of 2.19 eV and 2.77 eV are extracted for Cs2AgBiBr6 and Cs2AgBiCl6, respectively. Meanwhile, those for CH3NH3PbBr3 and CH3NH3PbCl3 are found to be 2.26 eV and 3.00 eV.
Band Structure Calculations Band structure calculations were performed to further investigate the electronic structure of the two compounds. Calculations were also performed on lead halide systems for comparison using Cs+ as the A-site cation for simplicity. This approach is justified by the established fact that neither the CH3NH3+ nor the Cs+ ions make significant contributions to the band structure near the Fermi level [12,24-26]. The space group symmetry for the lead compounds was taken to be cubic with Pm 3 m symmetry, both for ease of comparison with the cubic double perovskites and because the average structures of CH3NH3PbBr3 and CH3NH3PbCl3 are both cubic. The band structure diagrams for the chlorides are shown in Figure 4, and the bromides in Figure 5.
Figure 4. Band Structure diagrams for Cs2AgBiCl6 (top) and cubic CsPbCl3 (bottom). The Fermi energy is set to E = 0 and denoted with a dashed line.
The calculated band gaps are in good agreement with the experimental values (Table 4). While the band structures for the Cs2AgBiX6 and lead-halide systems contain many similarities, there is one significant difference—the double perovskites possess an indirect band gap as opposed to the direct band gap seen for the lead halide perovskites. This comes from the fact that the valence band maximum has moved away from the (111) Brillouin zone boundary (the R point in a primitive cubic cell and the L point in a face centered cubic cell, respectively) to the Xpoint. Calculations performed with and without spin orbit coupling show that the band degeneracy present at the Γpoint of the conduction band is lifted when spin-orbit coupling is included. This leads to the emergence of a narrow heavy electron band that is separated from the two next higher energy bands by 1 to 1.5 eV. In order to estimate the carrier mobilities, we determined effective charge carrier masses from the curvature of the band extrema (assuming parabolic bands). Along the R to X direction for CsPbCl3 (CsPbBr3) the electron effective mass is 0.41me (0.34me) and the hole effective mass along the same direction is 0.35me (0.37me). By comparison Cs2AgBiCl6 (Cs2AgBiBr6) has an approximate electron effective mass of 0.53me (0.37me) along the L to W direction, and a hole effective mass of 0.15me (0.14me) along the X to Γ direction. These estimates offer only a qualitative comparison since they neglect possible differences in carrier scattering rates. Although these double perovskites have a less disperse conduction band, it is encouraging that estimates of the hole effective masses for the double perovskites are lighter than their lead analogs.
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ing the band gap and is partially responsible for the indirect band gap of these phases.
Figure 6. Atomic partial density of states plots for Cs2AgBiCl6 (top) and Cs2AgBiBr6 (bottom).
Table 4. Observed and calculated optical band gaps. An HSC hybrid functional with spin orbit coupling was used for the computations. Figure 5. Band Structure diagrams for Cs2AgBiBr6 (top) and cubic CsPbBr3 (bottom). The Fermi energy is set to E = 0 and denoted with a dashed line.
The valence to conduction band transition is primary from filled halogen 3p/4p states to antibonding Ag 5s and Bi 6p states. This can be seen by looking at the partial density of state diagrams (PDOS) in Figure 6. The participation of both Ag and Bi orbitals to the lower energy conduction bands is key to maintaining some dispersion of the conduction band. Although the valence band is largely halogen 3p/4p in character there is extensive admixture of Ag 4d states, which results in the presence of several relatively flat bands at energies between −1.5 and −4 eV. The presence of Ag 4d states plays a role in reduc-
Compound
Experimental
Calculated
CH3NH3PbCl3
3.00 eV
2.97 eV
Cs2AgBiCl6
2.77 eV
2.62 eV
CH3NH3PbBr3
2.26 eV
2.25 eV
Cs2AgBiBr6
2.19 eV
2.06 eV
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Chemical and Light stability To probe the stability of the compounds, samples were exposed to an ambient atmosphere in both light and dark conditions. Each compound was loaded into a flat sample holder used for XRPD measurements and then left exposed to the ambient air. Visual inspection of those samples that were kept in the dark for two weeks, showed no apparent change. Subsequent XRPD measurements, confirmed that no change had occurred. When the samples were left sitting next to a large glass window, thereby exposing them to both the ambient atmosphere and visible light, darkening of the exposed surface was observed, particularly for Cs2AgBiBr6. It should be noted that during the month of exposure, the samples only received about six hours of direct exposure each day. The samples were analyzed by XRPD and UV-Vis diffuse reflectance spectroscopy after two and four weeks exposure to sunlight. As can be seen in Figure 7, the overall features of the reflectance spectrum for Cs2AgBiCl6 remain intact, but the total reflectance decreases ever so slightly, in agreement with the visual darkening of the surface of the sample that was observed. No apparent change was detected in the XRPD patterns of Cs2AgBiCl6 that were exposed to both light and air.
Figure 7. UV-Vis diffuse reflectance spectra showing the light stability of Cs2AgBiCl6 after two and four weeks of light exposure.
The reflectance spectrum for Cs2AgBiBr6, shown in Figure 8, changes drastically after two weeks exposure to light and ambient air, accompanied by the appearance of unidentified phases in the XRPD (see Supporting Information). These observations clearly show degradation of the double perovskite phase over time when simultaneously exposed to light and moisture.
Figure 8. UV-Vis Diffuse spectra showing the light instability of Cs2AgBiBr6 after two and four weeks of light exposure.
Conclusions Given the challenges of finding air-stable, non-toxic alternatives to the lead halide perovskites, it’s highly encouraging to find double perovskites with band gaps that are comparable to their CH3NH3PbX3 analogs. Given the intense level of interest and research activity being devoted to the lead halide perovskites the emergence of double perovskites that possess comparable optical properties is significant. While the structure and properties of Cs2AgBiX6 double perovskites are exciting as a proof of concept there are still challenges to be overcome. The mixing of silver 4d orbitals with the halogen 3p/4p orbitals changes the valence bands sufficiently that an indirect band gap is observed. While not ideal it should be noted that the most widely used material in photovoltaic cells, silicon, has an indirect gap, so this result does not completely rule out their use in solar cells. The air and moisture stability are promising, but the instability of Cs2AgBiBr6 to simultaneous exposure would necessitate encapsulation for long term use. Nonetheless, its stability is similar to that of CH3NH3PbBr3 and CH3NH3PbI3. Finally, while metalhalide perovskites with band gaps of ~2.2 eV, such as CH3NH3PbBr3 and Cs2AgBiBr6, are of interest for use with silicon in tandem solar cells, smaller band gaps are needed for single junction cells [3,14]. It’s worthwhile to compare the properties of Cs2AgBiBr6 with those of the defect ordered perovskites A3Bi2I9 and A3Sb2I9 (A = Cs, Rb, K), which have also been investigated as lead-free alternatives to CH3NH3PbI3 and CH3NH3PbBr3. The band gaps of Cs3Bi2I9 (1.9 eV), Rb3Bi2I9 (2.1 eV), K3Bi2I9 (2.1 eV) and Cs3Sb2I9 (2.05 eV) are slightly smaller but comparable to Cs2AgBiBr6 [27-29]. While the A3Bi2I9 and A3Sb2I9 compounds have direct or near direct band gaps, they also possess fairly flat bands which will translate to heavier carrier effective masses than estimated here for Cs2AgBiBr6 [28,29]. Saparov et al. estimated
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electron and hole effective masses for Cs3Sb2I9 in the [100] direction, where they are smallest, to be 0.44me and 0.60me, respectively [29]. Both values are heavier than those estimated here for Cs2AgBiBr6, particularly the hole effective mass. Although they have not been estimated from calculations the effective carrier masses for A3Bi2I9 phases appear to be even heavier. Finally it’s important to remember that Cs2AgBiCl6 and Cs2AgBiBr6 are just two members of what is likely to be a large family of halide double perovskites. Replacing Bi3+ with ions like Sb3+ and In3+, or replacing Ag+ with ions like Cu+ and Au+, are obvious strategies for tuning the electronic structure to narrow the band gap and move back toward a direct band gap. Double perovskites such as Cs2NaSbCl6 and Cs2NaInCl6 have been reported, but little is known about their optical properties [30], and chemical substitutions such as Ag+ for Na+ and Br− for Cl− have not been explored. The results reported here provide a compelling motivation to prepare and characterize additional halide double perovskites.
ASSOCIATED CONTENT Supporting Information. X-ray diffraction patterns of the samples that have been exposed to light and air, partial density of states plots for CH3NH3PbX3 (X = Br, Cl), KubelkaMunk transformed optical absorption data, and *.cif files of the crystal structures are given in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
AUTHOR INFORMATION Corresponding Author * E-mail:
[email protected] Author Contributions The manuscript was written through contributions of all authors.
ACKNOWLEDGMENT Thanks to Jason Khoury for preliminary synthesis of the methylammonium lead halide perovskites. Funding for this research was provided by the Center for Emergent Materials: an NSF MRSEC under award number DMR-1420451. We also thank the Ohio Supercomputer Center for support under Project No. PAA0010.
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