Understanding Octahedral Tilting in Lead Halide ... - ACS Publications

Mar 15, 2017 - Matthew R. Linaburg, Eric T. McClure, Jackson D. Majher, and Patrick M. ... Department of Chemistry and Biochemistry, The Ohio State ...
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Cs1−xRbxPbCl3 and Cs1−xRbxPbBr3 Solid Solutions: Understanding Octahedral Tilting in Lead Halide Perovskites Matthew R. Linaburg, Eric T. McClure, Jackson D. Majher, and Patrick M. Woodward* Department of Chemistry and Biochemistry, The Ohio State University, 100 W. 18th Avenue, Columbus, Ohio 43210, United States S Supporting Information *

ABSTRACT: The structures of the lead halide perovskites CsPbCl3 and CsPbBr3 have been determined from X-ray powder diffraction data to be orthorhombic with Pnma space group symmetry. Their structures are distorted from the cubic structure of their hybrid analogs, CH3NH3PbX3 (X = Cl, Br), by tilts of the octahedra (Glazer tilt system a−b+a−). Substitution of the smaller Rb+ for Cs+ increases the octahedral tilting distortion and eventually destabilizes the perovskite structure altogether. To understand this behavior, bond valence parameters appropriate for use in chloride and bromide perovskites have been determined for Cs+, Rb+, and Pb2+. As the tolerance factor decreases, the band gap increases, by 0.15 eV in Cs1−xRbxPbCl3 and 0.20 eV in Cs1−xRbxPbBr3, upon going from x = 0 to x = 0.6. The band gap shows a linear dependence on tolerance factor, particularly for the Cs1−xRbxPbBr3 system. Comparison with the cubic perovskites CH3NH3PbCl3 and CH3NH3PbBr3 shows that the band gaps of the methylammonium perovskites are anomalously large for APbX3 perovskites with a cubic structure. This comparison suggests that the local symmetry of CH3NH3PbCl3 and CH3NH3PbBr3 deviate significantly from the cubic symmetry of the average structure.

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coordinate cavities of the octahedral framework.15 The tolerance factor is given by the expression τ = dAX/(√2dBX), where dAX and dBX are the ideal A−X and B−X bond distances, respectively. Despite its simplicity the tolerance factor has proven to be a powerful tool for predicting the behavior of perovskites. Recent first-principles calculations on lead halide perovskites by Lee et al. showed that quantities such as the stability of the cubic structure, the Pb−I−Pb angles, the band gap, and the electron effective mass all evolve smoothly with changes in the tolerance factor.16 What is rarely discussed, however, are the challenges in comparing tolerance factors calculated for the halide perovskites with those calculated for oxide and fluoride perovskites. For example, in their study Lee et al. computed tolerance factors ranging from 0.89 to 0.65 for the iodide perovskites,16 yet in a comprehensive study of oxide perovskites Lufaso and Woodward did not find any ABO3 perovskites with tolerance factors smaller than 0.87.17 To explore the structural and optical response of halide perovskites to changes in tolerance factor Cs1−xRbxPbBr3 and Cs1−xRbxPbCl3 solid solutions have been studied. As will be shown, both conventional bond valence parameters and ionic radii overestimate the lengths of the Pb−X bonds and underestimate the lengths of the A−X bonds. This combination

INTRODUCTION Lead halide perovskites have been the subject of intense research activity over the past few years. This emerging class of materials holds great promise for applications such as solutionprocessed solar cells, light-emitting diodes (LEDs), and quantum dots.1−4 While the lead iodide perovskites are the most promising candidates for use in single-junction photovoltaic devices, the bromide and chloride analogues are of interest for use in a variety of optoelectronic applications, including LEDs, tandem solar cells, and radiation detectors.5−9 The ABX3 perovskite structure consists of a three-dimensional network of corner-connected octahedra. One of the most attractive features of the perovskite structure is its flexibility. By changing the size of the A-site cation, one can apply so-called chemical pressure to the octahedral framework. In many cases chemical pressure leads to a distortion of the structure, including tilts of the octahedra and out-of-center displacements of the octahedral cations. Through these distortions chemical pressure can be used to tune a wide variety of physical properties.10−13 For example, a computational study by Filip et al. suggested that chemical pressure can in theory be used to tune the band gap of APbI3 perovskites from 1.1 to 1.8 eV.14 In practice, the limits of how much chemical pressure can be applied before the perovskite structure is no longer stable have not been carefully explored. The tolerance factor is a simple geometric quantity that gives a measure of how well the A-site cation fits into the 12© 2017 American Chemical Society

Received: December 19, 2016 Revised: March 12, 2017 Published: March 15, 2017 3507

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ously determined. Presumably problems with twinning of single-crystal samples when cooled through the phase transitions that occur over a narrow temperature range between 47 and 37 °C are partly to blame for the lack of clarity. Therefore, we collected high-resolution synchrotron X-ray powder diffraction data at the Advanced Photon Source and analyzed it to determine the structure of CsPbCl3. Using the approach first described by Glazer and Ishida, the pattern of octahedral tilting was ascertained by inspection of the reflection conditions.24 When indexed on a doubled cubic unit cell (ignoring any peak splitting), the presence of (odd, odd, odd) peaks indicates out-of-phase tilts, while (odd, odd, even) peaks indicate in-phase tilts. In the case of CsPbCl3 the presence of the (113) reflection, located at roughly 7.1°, signals out-of-phase tilts of the octahedra, while the (103) and (213) reflections arise from in-phase tilts of the octahedra (see Figure 1). The remaining superstructure peaks visible in Figure 1 are

leads to unrealistically small tolerance factors. To rectify the situation, bond valence parameters are derived that can be used to make meaningful estimates of the bond distances in the inorganic APbX3 halide perovskites. This allows for predictions to be made about the lower tolerance factor limit where the perovskite structure will be stable, as well as the range over which the band gap can be tuned. Finally, it is shown that the observed band gaps of the cubic hybrid perovskites CH3NH3PbX3 (X = Cl, Br) are considerably larger than what would be expected from their crystal structures.

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EXPERIMENTAL SECTION

Reagents used were PbBr2 (98+%, Alfa Aesar, Ward Hill, MA), PbCl2 (99+%, J.T. Baker Chemical Co., Phillipsburg, NJ), RbCl (99%, Alfa Aesar), and RbBr (99.8%, Alfa Aesar). Precursors CsCl and CsBr were synthesized in bulk by neutralizing concentrated hydrochloric acid (Fisher Scientific, Waltham, MA) or hydrobromic acid (Purum p.a. ≥48%, Fluka) with Cs2CO3 (99+%, Strem Chemicals, Newburyport, MA). The neutralization results in the precipitation of either CsCl or CsBr, which were collected using vacuum filtration and washed thoroughly with cold ethanol. X-ray powder diffraction (XRPD) patterns were collected to confirm phase purity of precipitated precursors. Polycrystalline samples of RbxCs1−xPbX3 (X = Cl, Br) were synthesized by solid state reactions. Reagents were combined in stoichiometric amounts and then ground using an agate mortar and pestle. After the reagents were ground, the mixture was transferred to a 25 mL alumina crucible and placed in a box furnace to react for 20 h at 425 °C. For some samples an additional cycle of grinding and heating was employed to improve phase purity. Synchrotron X-ray powder diffraction (calibrated wavelength = 0.413369 Å) collected on the 11-BM beamline at the Advanced Photon Source (APS) was used to characterize the CsPbCl3 sample. XRPD data for CsPbBr3 was collected with a Bruker D8 advance diffractometer (40 kV, 40 mA; sealed Cu X-ray tube) equipped with a Lynx Eye XE-T position-sensitive detector and a SiO2 Johansson monochromator. A scan range of 10−130° 2θ and a step size of 0.015° was used for the measurement. XRPD patterns for all other samples were collected with a Bruker D8 advance diffractometer (40 kV, 50 mA; sealed Cu X-ray tube) equipped with a Lynx Eye positionsensitive detector and a Ge 111 incident beam monochromator, which selects only Kα1 radiation. A scan range of 8−75° 2θ and a step size of 0.015° were used for the measurements. Structure refinements were performed with the Rietveld method18 as implemented in the TOPAS software package.19 UV−vis 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.

Figure 1. Refined fit to the low-angle region of the synchrotron XRPD pattern of CsPbCl3. Labeled hkl reflections are indexed on a 2ap × 2ap × 2ap cell, not the orthorhombic Pnma cell. The two peaks marked with a green arrow belong to a Cs4PbCl6 impurity phase.

of the (even, even, odd) type. They arise from A-site cation displacements that are normally present only when both types of tilts are present. The presence of both in- and out-of-phase tilts is a clear signature of a−b+a− tilting and orthorhombic Pnma space group symmetry, despite the nearly tetragonal dimensions of the unit cell. For the sake of completeness, the structure of CsPbBr3 was refined from monochromatic laboratory XRPD data. The obtained structure, which also has Pnma symmetry, agrees well with the earlier report by Stoumpos et al.7 Structural details for both CsPbCl3 and CsPbBr3 are given in Table 1, while bond distances and Pb−X−Pb bond angles are given in Table 2. Fits to the diffraction patterns are given in the Supporting Information. The structure of CsPbCl3 is shown in Figure 2. The Shannon radii for the ions in question are r(Cs+) = 1.88 Å, r(Pb2+) = 1.19 Å, r(Cl−) = 1.81 Å, and r(Br−) = 1.96 Å (all chosen with the appropriate coordination numbers).25 Using these radii, one would predict Pb−Cl and Pb−Br distances of 3.00 and 3.15 Å, respectively. Comparison with the experimental distances given in Table 2 shows that both estimates are far too large. Travis et al. have discussed the limitations of using ionic radii for compounds other than oxides and fluorides.26 Bond distances for atom pairs involving the heavier halides tend to be shorter than predicted by ionic radii due to the effects of covalency. They derive a radius for sixcoordinate Pb2+ of 0.98 Å in bromides and 0.99 Å in chlorides. With use of those values, bond distances of 2.80 and 2.94 Å are predicted for the Pb−Cl and Pb−Br bonds, respectively. While

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RESULTS There are conflicting reports in the literature about the roomtemperature crystal structure of CsPbCl3. The structure was initially reported by Møller in 1958 as tetragonal distortion of the perovskite structure.20 In 1971, Hirotsu reported an orthorhombic unit cell where all three lattice parameters were doubled from the simple cubic perovskite cell, but did not attempt to determine the structure.21 Structure determination was attempted by Nitsch et al. in 1995 who reported an orthorhombic perovskite structure with Pmmm symmetry.22 There are also reports of a monoclinic structure with P21/m symmetry in the literature.23 Given the sheer volume of papers on CsPbCl3 it is surprising that the room-temperature structure has not been unambigu3508

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Chemistry of Materials Table 1. Details of the Structural Refinements of CsPbCl3 and CsPbBr3 at Room Temperature CsPbCl3 space group Pnma a (Å) 7.90193(1) b (Å) 11.24778(1) c (Å) 7.89928(1) volume (Å3) 702.081(2) χ2 1.441 Atomic Coordinatesa Cs x 0.5124(1) Cs z 0.0026(2) Cl/Br(1) x 0.2114(6) Cl/Br(1) y −0.0200(2) Cl/Br(1) z 0.7129(6) Cl/Br(2) x −0.0036(5) Cl/Br(2) z −0.0335(5) Atomic Displacement Parameters Cs Biso 5.36(1) Pb Biso 1.603(5) Cl/Br(1) Biso 4.71(5) Cl/Br(2) Biso 5.65(8)

CsPbBr3 Pnma 8.2517(1) 11.7534(1) 8.2032(1) 795.59(2) 1.232 0.5287(4) 0.0060(6) 0.2063(5) −0.0243(3) 0.7061(5) −0.0027(8) −0.0428(7) 5.0(1) 1.13(7) 3.3(1) 4.0(2)

a

Wyckoff positions are as follows: Cs on 4c (x, 1/4, z), Pb on 4a (0, 0, 0), Cl/Br(1) on 8d (x, y, z), Cl/Br(2) on 4c (x, 1/4, z).

Table 2. Bond Distances, Selected Bond Angles, and Bond Valence Sums for CsPbCl3 and CsPbBr3 CsPbCl3 Bond Distances (Å) Cs−X

3.547(4) ×2 3.708(5) ×1 3.820(4) ×2 3.835(5) ×1 4.064(5) ×2 4.087(5) ×1 4.195(5) ×1 4.485(4) ×2 Pb−X 2.8245(4) ×2 2.826(5) ×2 2.842(5) ×2 Selected Bond Angles (deg) Pb−X(1)−Pb 169.2(2) Pb−X(2)−Pb 160.5(2) Bond Valence Sums (Literature Parameters)a Cs 0.66 Pb 2.66 X(1) 1.11 X(2) 1.10 Bond Valence Sums (Proposed Parameters)b Cs 0.87 Pb 1.98 X(1) 0.96 X(2) 0.93

Figure 2. Crystal structure of CsPbCl3, looking down [010] (top) and [101] (bottom) directions, showing the in-phase and out-of-phase tilts, respectively. The cesium ions are shown as blue spheres, the chloride ions as green spheres, and the lead-centered octahedra as gray polyhedra.

CsPbBr3 3.673(5) ×2 3.809(8) ×1 3.887(8) ×1 3.946(5) ×2 4.171(6) ×2 4.403(8) ×1 4.411(8) ×1 4.850(6) ×2 2.9593(7) ×2 2.965(5) ×2 2.969(5) ×2

coordination cage around Cs+, but in each structure most of the bonds are longer than the ionic radii predictions of 3.69 Å (Cs−Cl) and 3.84 Å (Cs−Br). This suggests that ionic radii underestimate the Cs−X bond lengths. Distorted environments like those seen here for cesium are much more amenable to analysis using the bond valence approach. With use of bond valence parameters from the literature, bond valence sums of 0.66 and 0.72 are observed for Cs+ in CsPbCl3 and CsPbBr3, respectively. While this would seem to suggest that Cs+ is substantially underbonded in these two structures, an unusually low bond valence sum is a common feature of structures that contain 12-coordinate Cs+ ions surrounded by a network of corner-connected octahedra, like the perovskite structure (see Tables S1 and S3 in the Supporting Information). This is true to a lesser extent with 12-coordinate Rb+ ions. To address the inability of ionic radii and valence parameters to accurately predict bond lengths in halide perovskites, the Inorganic Crystal Structure Database (ICSD) was searched for chlorides and bromides containing 12-coordinate Cs+ and Rb+ ions. The resulting compounds were used to derive new Cs−X and Rb−X (X = Cl, Br) bond valence parameters for use in perovskites and perovskite-like structures. New Pb−X bond valence parameters are also proposed based on the bond distances seen in CsPbCl3 and CsPbBr3. Full details are given in the Supporting Information. The resulting bond valence parameters are given in Table 3. These bond valence parameters are used throughout the remainder of the paper to estimate tolerance factors. Cs1−xRbxPbCl3 and Cs1−xRbxPbBr3 solid solutions were prepared to see how the structure responds to changes in

166.4(2) 157.3(1) 0.72 2.78 1.16 1.17 0.97 2.01 0.97 1.00

a

Calculated using parameters from ref 27. bCalculated using parameters described in the text and given in Table 3.

both slightly underestimate the bond distances seen here, they are in considerably better agreement with the experimentally observed values than estimates made from Shannon radii. Comparing the Cs−X distances with the predictions of ionic radii is a more difficult task because of the distorted 3509

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Chemistry of Materials Table 3. Existing Bond Valence R0 Parameters from Ref 27 and Those Proposed for Use in Halide Perovskites as Discussed in the Texta Existing Parameters

Proposed Parameters

2.791 2.95 2.652 2.78 2.53 2.68

2.89 3.06 2.67 2.84 2.42 2.56

Cs−Cl R0 (Å) Cs−Br R0 (Å) Rb−Cl R0 (Å) Rb−Br R0 (Å) Pb−Cl R0 (Å) Pb−Br R0 (Å) a

transforming to the nonperovskite polymorph, as signaled by a color change, in anywhere from a few minutes to a few seconds when exposed to air. However, if synthesized and kept in a moisture-free environment such as an evacuated sealed glass ampule, the perovskite phase will persist for an extended period of time. As can be seen in the XRPD patterns of Figure 3, the peak splitting that arises from the orthorhombic distortion steadily increases as the Rb-content increases. To quantitatively assess the structural response to rubidium substitution, the structure of each sample was refined using the Rietveld method. The results are given in the Supporting Information, while the lattice parameters and unit cell volume are plotted as a function of the tolerance factor in Figures 4 and 5. Because Rb+ is smaller than

For all values b = 0.37 Å.

tolerance factor. XRPD patterns for both series of solid solutions are shown in Figure 3. In both cases Rb+ was incrementally substituted for Cs+ at 10% intervals until the perovskite structure could no longer be stabilized at room temperature. For the Cs1−xRbxPbCl3 system, rubidium could be incorporated all the way up to x = 0.8, whereas in the Cs1−xRbxPbBr3 system a stable perovskite structure could be maintained only long enough for characterization of samples with x ≤ 0.6. In both solid solutions, as the Rb-content increases, the room-temperature stability of the perovskite phase decreases. For the chloride series, the Cs0.2Rb0.8PbCl3 sample begins to degrade within an hour if left exposed to the ambient laboratory atmosphere. For the bromide series the moisture sensitivity is heightened, with the most Rb-rich samples (0.7 ≤ x ≤ 0.9)

Figure 4. Unit cell parameters and volume for Cs1−xRbxPbCl3 samples (0 ≤ x ≤ 0.8). The tolerance factor decreases as the Rb content increases.

Cs+, the perovskite tolerance factor decreases as the Rb content increases. The unit cell volume follows a linear Vegard’s Law trend, indicating that the Rb content of the perovskite phase is evolving as expected from the nominal sample compositions. As the tolerance factor increases, the a lattice parameter contracts while the b and c lattice parameters expand. If the octahedra were perfectly rigid, all three lattice parameters should expand with increasing tolerance factor. A small contraction of the a lattice parameter has been seen in other perovskite systems.28 It is caused by small distortions in the intraoctahedral X−Pb−X bond angles, and the fact that the a lattice parameter is the least sensitive of the three to changes in octahedral tilting. Diffuse reflectance measurements were taken from polycrystalline specimens. The absorption coefficient α was

Figure 3. X-ray powder diffraction patterns for selected samples in the Cs1−xRbxPbCl3 (upper) and Cs1−xRbxPbBr3 (lower) solid solution series. The gray hash marks at the bottom of each graph indicate the reflection positions for the Cs0.2Rb0.8PbCl3 and Cs0.4Rb0.6PbBr3 samples, respectively. 3510

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Figure 5. Unit cell parameters and volume for Cs1−xRbxPbBr3 samples (0 ≤ x ≤ 0.6). The tolerance factor decreases as the Rb content increases. Figure 6. Diffuse reflectance (upper) and pseudoabsorbance, as obtained by the Kubelka−Munk transformation (lower), for Cs1−xRbxPbCl3 (0 ≤ x ≤ 0.6) and CH3NH3PbCl3.

estimated from the reflectance data using the Kubelka−Munk transformation.29 The band gap can then be estimated by plotting α vs the incident photon energy and extrapolating back to the x-axis. This approach is valid for direct band gap semiconductors, which is assumed to be the case for all samples studied here based on previously reported transmission measurements of single crystals of CsPbBr3 and CsPbCl3.30 The diffuse reflectance plots for Cs1−xRbxPbCl3 and Cs1−xRbxPbBr3 are given in Figures 6 and 7. As the photon energy increases, the absorption coefficient rises sharply, peaks, and drops off before rising again. Sebastian et al. have studied optical absorption and photoluminescence as a function of temperature in single-crystal and polycrystalline samples of CsPbCl3 and CsPbBr3. They conclude that the lower energy transition corresponds to excitation across the band gap, while the higher energy features are due to excitonic transitions from trap states that lie above the conduction band edge.30 Following that precedent, we have used the lower energy onset of absorbance to determine the band gap. We obtain values of 2.91 and 2.27 eV for CsPbCl3 and CsPbBr3, respectively. These are in good agreement with the values of 2.88 and 2.25 eV reported by Sebastian et al.30 It was not possible to determine the band gaps of samples with x > 0.6 because the sample degraded too quickly to characterize when exposed to incident light. The evolution of the band gap as a function of the tolerance factor is plotted for both solid solution series in Figure 8. As the tolerance factor decreases, the magnitude of the octahedral tilting distortion increases as does the band gap. It has been shown that the bands narrow, leading to an increase in band gap as the Pb−X−Pb bonds become increasingly bent from the linear geometry of the undistorted cubic structure.14,16 Over the accessible compositional range of 0 ≤ x ≤ 0.6 the band gap of the Cs1−xRbxPbCl3 system increases by 0.15 eV, and the

Cs1−xRbxPbBr3 system by 0.2 eV (see Supporting Information for table of values). Interestingly, the band gap evolves linearly with tolerance factor, particularly for samples in the Cs1−xRbxPbBr3 series (R2 = 0.992).

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DISCUSSION Table 4 contains the perovskite tolerance factors for CsPbCl3 and CsPbBr3 calculated using Shannon ionic radii, bond valence parameters from ref 27, and the revised bond valence parameters given in Table 3. When ionic radii are used, the tolerance factor for CsPbCl3 is 0.87, which for an oxide or fluoride perovskite is typically at the lower range of the perovskite stability field. In their study of the structures of 50 different oxide perovskites with Pnma symmetry and a−b+a− tilting, Lufaso and Woodward found examples with tolerance factors ranging from 0.99 to 0.87, with those at the lower tolerance factor end possessing highly distorted structures.17 CsPbCl3 does not behave like a perovskite that is so distorted it is on the verge of adopting a different structure. It becomes cubic on heating above 47 °C (320 K)21,23 and the tolerance factor can be lowered significantly by replacing Cs+ with the smaller Rb+ before the perovskite structure finally becomes unstable. A similar tolerance factor, 0.893, is obtained using bond valence parameters from the literature. If on the other hand the bond valence parameters given in Table 3 are adopted, a more reasonable tolerance factor of 0.953 is obtained for CsPbCl3. It is instructive to compare with the oxide perovskite SrSnO3 which has a comparable tolerance factor of 0.961. Its Sn−O−Sn bond angles, 160.5° and 159.7°,32 are smaller but not wildly different from the 160.5° and 169.2° bond angles seen in CsPbCl3. SrSnO3 also undergoes phase 3511

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behavior of these two compounds is far from identical, consider the comparison between CsPbCl3 and CaSnO3. The latter compound has a tolerance factor of 0.927, which is still larger than the ionic radii estimate of the tolerance factor in CsPbCl3 (see Table 4), yet the Sn−O−Sn bond angles of 146.9° and 148.0° are much more bent than those seen in CsPbCl3.32 Furthermore, there is no reported phase transition to higher symmetry on heating CaSnO3. Simply put, the tolerance factors obtained for lead halide perovskites using either ionic radii or existing bond valence parameters cannot be compared with those obtained for oxide or fluoride perovskites. On the other hand, the tolerance factors calculated with the perovskite-specific bond valence parameters proposed in this study allow useful comparisons to be made with the vast literature of oxide and fluoride perovskites, where the bond angles and stability of the perovskite structure can be qualitatively estimated from the tolerance factor. Having established meaningful tolerance factors, we are able to investigate how the structure and properties evolve as the tolerance factor changes. As demonstrated in Figures 3 and 4, the unit cell volume evolves linearly with tolerance factor. A linear fit gives the following relationships between unit cell volume (in Å3) and tolerance factor, τ:

Figure 8. Band gaps of Cs1−xRbxPbCl3 and Cs1−xRbxPbBr3 as a function of tolerance factor.

Table 4. Tolerance Factors, τ, for CsPbX3 and RbPbX3 Calculated Using Different Methods To Estimate Bond Distances

CsPbCl3 CsPbBr3 RbPbCl3 RbPbBr3

τ (literature bond valence parameters)

τ (revised bond valence parameters)

0.870 0.862 0.832 0.826

0.893 0.886 0.860 0.848

0.953 0.948 0.898 0.896

(1)

unit cell volume(Cs1 − xRbx PbBr3) = (816)τ + 22

(2)

It is instructive to compare these trends with the hybrid lead halide perovskites CH3NH3PbCl3 and CH3NH3PbBr3, both of which have cubic Pm3̅m symmetry at room temperature. The lattice parameters for these two phases are 5.6928(1) and 5.9339(1) Å, respectively (see Supporting Information for XRPD patterns and analysis). This gives Pb−Cl and Pb−Br distances of 2.8464(1) and 2.9670(1) Å, which are similar to the average bond distances of 2.83 and 2.97 Å observed in CsPbCl3 and CsPbBr3, respectively. If the unit cell volumes of CH3NH3PbCl3 and CH3NH3PbBr3 are multiplied by 4, they can be compared directly to the orthorhombic Pnma cell volume. Those values, 738 and 836 Å3 for the chloride and bromide, respectively, can be plugged back into eqs 1 and 2 to estimate a tolerance factor of 1.000 for CH3NH3PbCl3 and 0.998 for CH3NH3PbBr3. It is reassuring to get tolerance factors that are in the range where cubic symmetry is expected. This is not the case if tolerance factors calculated from ionic radii are used to derive eqs 1 and 2. Perhaps the most intriguing comparison between the inorganic Cs 1−x Rb x PbX 3 perovskites and the hybrid CH3NH3PbX3 perovskites is the evolution of the band gap. One can perform linear fits to the plots of band gap as a function of tolerance factor, τ, given in Figure 8:

Figure 7. Diffuse reflectance (upper) and pseudoabsorbance, as obtained by the Kubelka−Munk transformation (lower), for Cs1−xPbBr3 (0 ≤ x ≤ 0.6) and CH3NH3PbBr3.

τ (ionic radii)

unit cell volume(Cs1 − xRbx PbCl3) = (750)τ − 12

Eg (Cs1 − xRbx PbCl3) = 6.70 − (3.96)τ eV

(3)

Eg (Cs1 − xRbx PbBr3) = 8.33 − (6.39)τ eV

(4)

As has been discussed at length in the literature, octahedral tilting changes the orbital overlap at the band extrema, narrowing both conduction and valence bands, which leads to an increase in the band gap.14,16,31,33 So it is not surprising to see the band gap decrease as the tolerance factor increases. Using first-principles calculations on APbI3 perovskites (A = K, Rb, Cs, Fr), Lee et al. found a linear relationship between the band gap and the tolerance factor, in agreement with the experimental results obtained in this study.16 However,

transitions that involve the loss of octahedral tilting at elevated temperatures, similar to CsPbCl3. The first transition being from Pnma (a−b+a−) to Imma (a−b0a−) at 636 °C.32 While the 3512

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Chemistry of Materials comparison with the band gaps of the methylammonium halide perovskites raises some interesting questions. Our estimates of the band gaps of CH3NH3PbCl3 (Eg = 2.97 eV) and CH3NH3PbBr3 (Eg = 2.23 eV) are in good agreement with the work of Leguy et al., who obtained values of 2.97(3) and 2.24(2) eV.34 Quite unexpectedly, we see that the band gap of cubic CH3NH3PbCl3 is larger than that of orthorhombic CsPbCl3 (Eg = 2.91 eV). On the surface the bromides appear to behave more in line with expectations, at least in the sense that the band gap of CH3NH3PbBr3 is smaller than CsPbBr3 (Eg = 2.27 eV). However, if we extrapolate eq 4 to a tolerance factor of unity, the band gap for a cubic APbBr3 perovskite should be 1.94 eV, a value much smaller than the observed band gap of 2.23 eV for CH3NH3PbBr3. There is no way to rationalize the abnormally large band gaps of the hybrid CH3NH3PbX3 perovskites from their average crystallographic structures. One possible way to reconcile this puzzling trend would be if the local structures in the hybrid perovskites like CH3NH3PbCl3 and CH3NH3PbBr3 were distorted from the long-range cubic structure. This conclusion is consistent with a pair distribution function study of the local structure of these two compounds by Worhatch et al.35 That study found large transverse displacements of the halide ions, suggestive of dynamic octahedral tilts, in both compounds. Furthermore, in CH3NH3PbCl3 the pair distribution function could be modeled only by allowing off-center displacements of the lead ions. Given the differences between the optical properties of CH3NH3PbCl3 and Cs1−xRbxPbCl3 perovskites, it is tempting to conclude that the off-centering distortion seen by the PDF method in CH3NH3PbCl3 is either enabled or enhanced by the presence of the nonspherical CH3NH3+ cation. More recently, a study of CH3NH3PbI3 also concluded that the local structure could be modeled only if large, dynamic rotations of the octahedral were invoked.36

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Patrick M. Woodward: 0000-0002-3441-2148 Author Contributions

All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding for this research was provided by the National Science Foundation under award number DMR-1610631. We acknowledge partial support from the Center for Emergent Materials: an NSF MRSEC under award number DMR-1420451. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. We would like to acknowledge the Modern Methods in Rietveld Refinement and Structural Analysis (MMRRSA-2015) Workshop, held in Tallahassee, Florida. The free three-day workshop provided hands on instruction and experience on using modern methods and advanced techniques on refining high-resolution powder diffraction data from structural user facility beamlines. In accordance with the workshop, participants were provided the opportunity to have data collected on a user-provided sample from said user facilities, which provided the diffraction data for the CsPbCl3 sample presented herein.

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CONCLUSIONS Careful analysis of the structures of CsPbCl3 and CsPbBr3 reveal that existing measures for estimating bond distances and tolerance factors of lead halide perovskites are inadequate. New Cs−X, Rb−X, and Pb−X (X = Cl, Br) bond valence parameters are derived for use in halide perovskites. When these parameters are used to calculate tolerance factors, the structural behavior of Cs1−xRbxPbCl3 and Cs1−xRbxPbBr3 solid solutions can be compared with oxide and fluoride perovskites. The band gaps of compounds in these two solid solutions are shown to increase linearly as the tolerance factor increases. Extrapolation of these trends to a cubic perovskite structure shows that the band gaps of the hybrid perovskites, CH3NH3PbCl3 and CH3NH3PbBr3, are much larger than expected. This finding suggests the presence of local deviations from the average cubic symmetry in these two compounds, a conclusion that is consistent with an earlier pair distribution function study.

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fits to the XRPD patterns of CH3NH3PbX3 (X = Cl, Br); diffuse reflection data and band gaps for Cs1−xRbxPbX3 and CH3NH3PbX3 (X = Cl, Br) compositions (PDF) (CIF) (CIF)

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.6b05372. Details of the analysis used to derive new Cs−X, Rb−X, and Pb−X (X = Cl, Br) bond valence parameters; Rietveld refinements and refined structures for Cs1−xRbxPbX3 (X = Cl, Br) compositions; whole pattern 3513

DOI: 10.1021/acs.chemmater.6b05372 Chem. Mater. 2017, 29, 3507−3514

Article

Chemistry of Materials

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