Annalise E.
Maughan
,
Arnold A.
Paecklar
and
James R.
Neilson
*

Department of Chemistry, Colorado State University, Fort Collins, CO 80523-1872, USA. E-mail: james.neilson@colostate.edu

Received
17th July 2018
, Accepted 17th October 2018

First published on 19th October 2018

Anharmonic lattice dynamics are intimately linked with optical and electronic properties in perovskite halide semiconductors. Vacancy-ordered double perovskites are a subset of the perovskite halide family containing isolated octahedral units. The absence of polyhedral connectivity engenders the vacancy-ordered double perovskites with additional degrees of dynamic freedom, which presents an ideal structural framework to study dynamic–property relationships in perovskite halide semiconductors. In the present study, we examine the structure and bonding origins of anharmonicity in the vacancy-ordered double perovskites Cs_{2}Sn_{1−x}Te_{x}I_{6}. While X-ray diffraction indicates that all members adopt the cubic vacancy-ordered double perovskite structure, the local coordination environment probed by X-ray pair distribution function (XPDF) analysis reveals asymmetry of the Cs–I/I–I pair correlation that smoothly decreases with increasing tellurium content. Temperature-dependent neutron total scattering suggests that this asymmetry in the PDF occurs due to anharmonic lattice dynamics arising from octahedral tilting and Cs^{+} displacements, as supported by Reverse Monte Carlo simulations of the Cs_{2}SnI_{6} and Cs_{2}TeI_{6} end members. We further correlate the trends in asymmetry and anharmonicity with the bond valence sum of the Cs^{+} ion, and find that the anharmonicity vanishes when the bonding preferences of the Cs^{+} are satisfied by the size of the cuboctahedral void. This study presents a simple and effective approach for understanding the origin of anharmonicity in vacancy-ordered double perovskite materials.

Halide-based perovskite materials are redefining the paradigm of semiconductor materials design principles, in that they appear to follow a set of structure–dynamics–property relationships that are distinct from conventional semiconductors. The perovskite structure adopts the general formula ABX_{3}, and the structure is composed of corner sharing BX_{6} octahedra with A-site cations in the 12-coordinate void. Anharmonic lattice dynamics in perovskite halides have been shown to arise from rotational instabilities of the soft, deformable BX_{6} octahedral framework coupled with motions of the A-site cation.^{16–18} These dynamic instabilities yield an anharmonic double potential well, resulting in an instantaneous local structure characterized by cooperative octahedral tilting that averages to a higher-symmetry, untilted structure.^{16–19} Anharmonicity has been further correlated with the remarkable photoconversion efficiencies of halide perovskite materials in photovoltaic devices.^{20,21} Of particular note is the observation of long carrier excited-state lifetimes,^{22–24} which are hypothesized to arise from the formation of polarons that protect photogenerated charge carriers and prevent recombination.^{15,24–30} The intimate link between anharmonic lattice dynamics and functional properties motivates a fundamental understanding of the structural and compositional origins of anharmonicity in perovskite halides.

Vacancy-ordered double perovskites present a materials family to study anharmonicity in a lattice with additional dynamic degrees of freedom. The vacancy-ordered double perovskite structure is formed by doubling the conventional ABX_{3} perovskite unit cell and removing every other B-site cation to form a face-centered lattice of isolated BX_{6} octahedral units bridged by A-site cations in the void. Alternatively, the structure can be thought of as an ordered double perovskite of the formula A_{2}B□X_{6}, with rock salt ordering of BX_{6} and □X_{6} octahedra, where □ denotes a vacancy, as shown in Fig. 1. Anharmonicity in the vacancy-ordered double perovskites Cs_{2}SnI_{6}, (CH_{3}NH_{3})_{2}SnI_{6}, and (CH(NH_{2})_{2})_{2}SnI_{6} has been correlated with reduced carrier mobilities.^{31} Replacing the inorganic Cs^{+} with the molecular methylammonium (CH_{3}NH_{3}^{+}) and formamidinium (CH(NH_{2})_{2}^{+}) cations is accompanied by a significant reduction in electron mobilities due to softer, more anharmonic lattice dynamics that result in stronger electron–phonon coupling interactions. In order to leverage anharmonicity as a design principle for perovskite halide semiconductors, a fundamental understanding of the structural and compositional origins of anharmonic lattice dynamics are required.

In this contribution, we assess the structural origins of anharmonicity in the series of inorganic vacancy-ordered double perovskites Cs_{2}Sn_{1−x}Te_{x}I_{6}. X-ray pair distribution function analysis reveals asymmetry in the local coordination environment of Cs_{2}SnI_{6}, which systematically decreases and vanishes with increasing tellurium content. Neutron total scattering of Cs_{2}SnI_{6} reveals that the peak asymmetry becomes increasingly pronounced at higher temperatures, indicating that this feature is likely due to a dynamic effect rather than a static structural distortion. We attribute the subtle deviations in the local coordination environment of the Cs_{2}Sn_{1−x}Te_{x}I_{6} series to anharmonic lattice dynamics brought about by octahedral rotations and Cs^{+} displacements, consistent with the concave-down shape of the atomic displacement parameter vs. temperature curves for cesium and iodine. This assertion is supported by pseudo-rigid-body Reverse Monte Carlo simulations of Cs_{2}SnI_{6} and Cs_{2}TeI_{6}. From the RMC-optimized structures, we find that the Cs–I and I–I_{inter} partial pair correlations exhibit a broad, asymmetric distribution, indicating that the anharmonicity arises from these atom pairs. We further quantified the trend in anharmonicity in the XPDFs of Cs_{2}Sn_{1−x}Te_{x}I_{6} using a modified anharmonic Toda potential and find a strong correlation of the peak asymmetry with the calculated bond valence of the Cs^{+} cation, suggesting that the anharmonicity arises due to dissatisfied bonding preferences of the Cs^{+} cation within the cuboctahedral void, giving rise to dynamic octahedral rotations coupled to displacements of the Cs^{+} ions. The relationship between bonding and anharmonicity provides a hand-hold for tuning the vibrational properties in vacancy-ordered double perovskite semiconductors.

Neutron total scattering measurements of Cs_{2}SnI_{6} were performed on the nanoscale ordered materials diffractometer (NOMAD) at the Spallation Neutron Source, Oak Ridge National Laboratory. For measurements collected at T = 90, 300, and 500 K, a powdered sample of Cs_{2}SnI_{6} was loaded and sealed in a quartz capillary (capillary diameter = 3.0 mm) in the multisample changer. Data were normalized against scattering data collected for an empty quartz capillary, and background scattering from the quartz capillary was subtracted. For measurements at T = 10 K, a powdered sample of Cs_{2}SnI_{6} was loaded into a 6 mm vanadium canister under He atmosphere. Data were normalized against scattering collected for a vanadium rod, and background scattering from the vanadium can was subtracted.

Temperature-dependent neutron total scattering data were merged to the total scattering structure function using the IDL codes developed for the NOMAD instrument.^{37} The pair distribution function was then produced through the sine Fourier transform of the total scattering structure function using Q_{max} = 31.4 Å^{−1}. For Cs_{2}SnI_{6} at T = 90, 300, and 500 K, values of Q_{damp} = 0.0201 Å^{−1} and Q_{broad} = 0.0196 Å^{−1} were extracted from refinement of a diamond standard in PDFgui. For Cs_{2}SnI_{6} at T = 10 K, values of Q_{damp} = 0.01766 Å^{−1} and Q_{broad} = 0.01918 Å^{−1} were extracted from refinement of a silicon standard. Analysis of the nPDFs was performed using PDFgui.

Traditional free-motion RMC simulations were performed in which all atoms were independently allowed to displace randomly within the Cartesian reference. The Sn–I/Te–I bond lengths were constrained between 2.65–3.05 Å and the I–Sn–I and I–Te–I bond angles were constrained to 80–100° and 170–180°. The simulations were optimized against the experimental data from 0.9–17.45 Å for G(r) and 0.5–25 Å^{−1} for S(Q) − 1.

Pseudo-rigid-body RMC simulations were also implemented, in which the [SnI_{6}] and [TeI_{6}] octahedra were randomly rotated and tilted as rigid units, prior to and after their free relaxation. Similarly to the free-motion RMC simulations, the Sn–I/Te–I bond lengths were constrained between 2.65–3.05 Å and the I–Sn–I and I–Te–I bond angles were constrained to 80–100° and 170–180°. After initiating the refinements with only the intraoctahedron bonds, the refinement range was increased step-wise to include longer-range pair correlations while all atoms were permitted to displace in the Cartesian reference. After the fitting range reached r = 17.45 Å and after a finite number of atomic displacements, the octahedra were constrained as rigid bodies (Sn/Te–I bond lengths and angles were frozen) and allowed to tilt about the center of the octahedra around all three Euler angles up to a maximum tilt angle of 10°. The entire fitting process was reiterated twice from the start to achieve the final configuration.

VESTA was used to visualize and render all crystal structures presented in this publication.^{40}

The local coordination environment probed by XPDF analysis is consistent with solid solution behavior between Cs_{2}SnI_{6} and Cs_{2}TeI_{6}. In Fig. 2a, the X-ray pair distribution function (XPDF) of each member of the Cs_{2}Sn_{1−x}Te_{x}I_{6} series are modeled with the cubic structural model with appropriate fractional occupancies of tin and tellurium. Over medium and long length scales (r > 6 Å), the XPDFs are fairly well described by the cubic structural model, consistent with each member of the series adopting the cubic vacancy-ordered double perovskite structure; however, the fit quality is worse closer to Cs_{2}SnI_{6} (Fig. 2b). As shown in Fig. 3a, tellurium substitution is accommodated by a linear increase in the lattice parameter, consistent with Vegard's law. Further, the larger tellurium ion results in an increase in the average B–I bond length (Fig. 3b) at the expense of the inter-octahedral I–I contact distances along the 〈110〉 directions (Fig. 3c). This also results in a slight increase in the average Cs–I bond lengths across the series (Fig. 3d). The structural parameters extracted from the XPDF fits in Fig. 3 are plotted with the previously reported parameters from analysis of high-resolution synchrotron powder X-ray diffraction data.^{32}

Fig. 3 Structural parameters for the Cs_{2}Sn_{1−x}Te_{x}I_{6} solid solution from refinement of the cubic structural models against high-resolution synchrotron powder X-ray diffraction (SXRD) data (filled symbols) and X-ray pair distribution function analysis (open symbols). In (a), the lattice parameters for each member of the solid solution follow Vegard's law. In (b), the average B–I bond lengths increase linearly with substitution of the larger tellurium ion at the expense of the interoctahedral I–I contact distance along the 〈110〉 direction shown in (c). In (d), the average Cs–I bond length increases. The dashed lines represent linear regressions performed for each data set. Error bars are shown for the parameters extracted from the XPDF fits. The error bars for the SXRD parameters are within the size of the symbol and are therefore omitted for clarity. The structural parameters from the SXRD data are taken from ref. 32. |

Although all members of the series appear to adopt nearly identical crystalline structures by both SXRD and XPDF, the local coordination environment over short length scales reveals subtle differences across the series. The first nearest-neighbor pair correlation at r ∼ 2.85–2.9 Å due to Sn–I/Te–I bonds moves to higher r due to an increase in the average B–I bond length with substitution of the larger tellurium ion. Despite mixed Sn–I and Te–I bond lengths, this peak remains symmetric across the series and is well-described by the cubic structural model, consistent with regular, undistorted BX_{6} octahedral units.

In contrast, deviations in the next-nearest-neighbor (nnn) pair correlation at r ∼ 4.1 Å, due to I–I and Cs–I pairs, are observed across the series, manifesting as a slight asymmetry of the high-r side of the peak present in the difference curves in Fig. 2. The apparent asymmetry is most pronounced in Cs_{2}SnI_{6} and gradually decreases with increasing tellurium content, consistent with the nearly monotonic decrease in R_{wp} with x shown in Fig. 2b. Our previous work found that the XPDFs of the intermediate members could be obtained by a linear combination of the XPDFs of the Cs_{2}SnI_{6} and Cs_{2}TeI_{6} end members,^{32} indicating that this apparent asymmetry evolves smoothly as a function of tellurium content. As the BI_{6} octahedra remain relatively undistorted and all members of the solid solution adopt nearly identical crystal structures by X-ray diffraction, we propose that this asymmetry may be due to anharmonic lattice dynamics rather than a static structural distortion.

Neutron total scattering experiments of Cs_{2}SnI_{6} reveal a temperature-dependence of the asymmetry observed in the local coordination environment. Neutron diffraction data of Cs_{2}SnI_{6} collected from the 31° bank (bank 2) of NOMAD at T = 10, 90, 300, and 500 K reveal that Cs_{2}SnI_{6} adopts the cubic vacancy-ordered double perovskite structure at all measured temperatures, as shown in the Rietveld refinements in Fig. 4. The neutron pair distribution functions (nPDF) extracted from total scattering data are shown in Fig. 5. At all temperatures the nPDFs were modeled with the cubic vacancy-ordered double perovskite structure of Cs_{2}SnI_{6}, consistent with previous reports^{32,41} and with the corresponding diffraction data that indicate Cs_{2}SnI_{6} adopts the cubic structure at all temperatures (Fig. 4). At T = 10 K, the pair correlations are sharp, narrow, and symmetric, consistent with low-amplitude harmonic thermal vibrations at this temperature. Increasing temperature to T = 90 K and T = 300 K is accompanied by broadening of all pair correlations. The nnn pair correlation becomes visibly asymmetric with increasing temperature, with a slight tailing on the high-r side of the peak revealed in the difference curves. At T = 500 K, the peaks in the nPDF are significantly broadened and dampened, and we observe significant asymmetry of the nnn pair correlation that is not captured by the cubic structural model. The nPDFs for Cs_{2}SnI_{6} at T = 10, 90, and T = 300 K are taken from our previous study and re-fit here for comparison with the T = 500 K data.^{41}

Fig. 4 Rietveld refinements of temperature-dependent neutron diffraction of Cs_{2}SnI_{6} at T = 10, 90, 300, and T = 500 K from the 31° bank (bank 2) of the NOMAD instrument at the Spallation Neutron Source, Oak Ridge National Laboratory. The data are modeled with the cubic vacancy-ordered double perovskite structure at all temperatures. Black circles are the data, the orange line is the fit, the blue line is the difference, and the grey tick marks represent positions of anticipated reflections for the Fmm vacancy-ordered double perovskite structure. The data at T = 10, 90, and T = 300 K have been previously reported.^{41} |

Fig. 5 Temperature-dependent neutron pair distribution function analysis of Cs_{2}SnI_{6} at T = 10, 90, 300, and T = 500 K. The data are modeled with the cubic vacancy-ordered double perovskite structure at all temperatures. Black circles are the data, orange lines are the fits, and grey lines are the difference curves. The x-axis is split to highlight the low-r pair correlations and the increasing asymmetry of the next-nearest-neighbor pair correlation at r ∼ 4 Å with increasing temperature. The nPDFs at T = 10, 90, and T = 300 K have been previously reported and are re-fit here for comparison with the T = 500 K data.^{41} |

Temperature-dependent peak asymmetries have been observed in the XPDF of the related perovskite CsSnBr_{3}.^{42,43} At T = 300 K, the nearest-neighbor pair correlation due to Sn–Br bonds is symmetric, but becomes noticeably asymmetric on the high-r side of the peak at higher temperatures. The emergence of this asymmetry with temperature, termed “emphanisis”, has been attributed to dynamic off-centering of the Sn^{2+} ion within the SnBr_{6} octahedra, which arises from stereochemically-active 5s^{2} electrons. While the formal [Kr]4d^{10}5s^{0} electron configuration of Sn^{4+} in Cs_{2}SnI_{6} precludes the presence of stereochemically-driven structural distortions, the temperature dependence of the peak asymmetry suggests that this effect arises from high-amplitude anharmonic lattice vibrations rather than a static structural distortion.

To gain insight into the atomistic contributions to the anharmonicity and peak asymmetry in Cs_{2}SnI_{6}, we extracted values for the atomic displacement parameters (ADPs) from the Rietveld refinements of the neutron diffraction data shown in Fig. 4. The iodine ADPs were refined anisotropically; U_{11} corresponds to displacements along the Sn–I bond, while U_{22} = U_{33} corresponds to displacements perpendicular to the Sn–I bond. As shown in Fig. 6, the ADPs for Cs and I (U_{22} = U_{33}) increase monotonically (though not linearly) with increasing temperature, while the ADPs for Sn and I (U_{11}) increase only slightly from T = 10 K to T = 500 K. In systems with harmonic interactions, the relationship between the atomic displacement parameter (U_{iso}) and temperature is well-described by a Debye–Waller model.^{44,45} In Cs_{2}SnI_{6}, however, the ADPs for Cs and I U_{22} = U_{33} follow a concave-down shape with increasing temperature, a trend which has previously been attributed to anharmonic dynamics in the VAl_{10+δ} system due to rattling of the Al atoms within the structural voids.^{45} Similarly, neutron diffraction studies of CsPbX_{3} halide perovskites reveal anomalously large atomic displacement parameters of the Cs and X ions that diverge from the harmonic Debye–Waller model, indicating the presence of anharmonic effects due to coupled displacements of the Cs^{+} and X^{−} ions.^{46} The similarities observed between the VAl_{10+δ} and CsPbX_{3} systems and Cs_{2}SnI_{6} suggests that the trends in ADP vs. temperature arises from anharmonic dynamics of the Cs and I atoms in Cs_{2}SnI_{6}. Furthermore, the observation that iodine displacements perpendicular to the Sn–I bond (U_{22} = U_{33}) follow the same trend as the atomic displacement parameter for cesium and are significantly larger than iodine displacements along the Sn–I bond (U_{11}) may suggest the presence of [SnI_{6}] octahedral rotations coupled to Cs^{+} displacements as the source of anharmonicity. Prior nuclear quadrupole resonance studies of the vacancy-ordered double perovskite family support this assertion, as dynamics in these materials originate predominantly from rotations of the rigid octahedral units.^{47–50}

Reverse Monte Carlo (RMC) simulations of Cs_{2}SnI_{6} and Cs_{2}TeI_{6} were performed to provide atomistic insights into the asymmetry observed in the XPDFs. As the dominant lattice dynamics in vacancy-ordered double perovskites arise from octahedral rotations rather than deformations of the octahedra,^{47–50} we elected to use a pseudo-rigid-body RMC approach, in which the isolated [SnI_{6}] and [TeI_{6}] octahedra were allowed to tilt as rigid bodies, to encourage chemically reasonable descriptions of the anharmonicity. The constraint of rigid-bodies has been shown to improve RMC simulation results, especially in cases with dynamics such as rigid-unit modes.^{51} From this approach, we find that the XPDFs are best described by structures with random rotations of the SnI_{6} and TeI_{6} octahedra and displacements of the Cs^{+} ions away from their crystallographic positions, as shown in the optimized supercells in Fig. 7 and in the fits to the XPDF shown in Fig. 8a and b. We have also performed traditional free-motion RMC simulations to ensure that the outcome is consistent with our pseudo-rigid-body RMC approach. These calculations are independently consistent with those of our pseudo-rigid-body approach, though the quality of fit is not as good for the same number of moves. These results are shown in the ESI.†

Fig. 7 Supercell structures of Cs_{2}SnI_{6} and Cs_{2}TeI_{6} optimized from pseudo-rigid-body RMC simulations. |

To determine the atom pair contributions to the asymmetry observed in the next-nearest-neighbor pair correlation at r ∼ 4 Å, the partial radial distribution functions (RDF) for the intraoctahedral I–I (I–I_{intra}), interoctahedral I–I (I–I_{inter}), and Cs–I pairs were extracted from the RMC-optimized supercells. As shown in Fig. 9, the partial RDFs are relatively consistent between both Cs_{2}SnI_{6} and Cs_{2}TeI_{6} and show only subtle variations between the two compounds. In both compounds, the I–I_{intra} RDFs for Cs_{2}SnI_{6} and Cs_{2}TeI_{6} are fairly well-described by a Gaussian function. In contrast, both the I–I_{inter} and Cs–I partials exhibit an asymmetric peak shape evidenced by the deviations from a Gaussian function shown in Fig. 9, indicating that the overall peak asymmetry arises due to these atom pairs. It is important to note that the I–I_{inter} and Cs–I pairs in both Cs_{2}SnI_{6} and Cs_{2}TeI_{6} exhibit slightly asymmetric peak shapes, and therefore we cannot unambiguously assign one compound as being more anharmonic than the other from these simulations. Instead, these simulations are consistent with the notion that anharmonicity in these compounds arises from octahedral rotations coupled with displacements of the Cs^{+} cations.

In order to quantify the trends in anharmonicity across the intermediate members of the Cs_{2}Sn_{1−x}Te_{x}I_{6} series, the asymmetry of the nnn pair correlation was modeled with a modified Toda potential, which has been previously used to describe anharmonic interactions between nearest-neighbors in a linear atomic chain.^{52,53} This modified Toda potential, U(r), takes the form

(1) |

Fig. 10 (a) Toda potential fits to the next-nearest-neighbor pair correlation in the X-ray pair distribution function analysis for Cs_{2}Sn_{1−x}Te_{x}I_{6}. The data are shown as black circles and the fit is the orange line. The PDF data are fit with one Toda potential peak, and are offset vertically for comparison and clarity. In (b and c), the interatomic distance (b) and degree of anharmonicity (β) are plotted as a function of x in Cs_{2}Sn_{1−x}Te_{x}I_{6}, respectively. The colored tick marks in (a) represent the contact distances for Cs–I (teal), intraoctahedral I–I (pink), and interoctahedral I–I (purple) atom pairs taken from the refinements of the cubic model against the XPDF data from Fig. 2. Dashed lines in (b and c) represent linear regressions. |

We propose that the subtle deviations in the Cs–I/I–I pair correlations of the Cs_{2}Sn_{1−x}Te_{x}I_{6} series arise from anharmonic lattice dynamics originating from [BI_{6}] octahedral rotations coupled with displacements of the Cs^{+} ions. Octahedral rotations in vacancy-ordered double perovskites have been studied at length by nuclear quadrupole resonance, which reveals that these modes are the dominant source of dynamics in these materials.^{47–50} Furthermore, prior studies of inorganic perovskite halides CsPbX_{3} and CsSnX_{3} have shown that anharmonic lattice dynamics originate from cooperative tilting of the BX_{6} octahedral units coupled with small displacements of the A-site cations within the cuboctahedral void.^{17,18,54} In the present study, the presence of octahedral rotations coupled with Cs^{+} displacements is supported by analysis of the neutron total scattering experiments of Cs_{2}SnI_{6}; the concave-down shape of the curves for the Cs atoms and the I U_{22} = U_{33} atomic displacement parameters with increasing temperature follow a similar trend observed for localized vibrations associated with an ion rattling in a cage.^{45} Notably, the iodine atomic displacement parameter parallel to the Sn–I bond (U_{11}) remains relatively constant while the perpendicular displacements (U_{22} = U_{33}) increase significantly with temperature. This observation indicates that iodine displacements perpendicular to the Sn–I bond dominate over those parallel to the Sn–I bond, lending further support to the notion of octahedral tilting as the primary source of dynamics and anharmonicity in Cs_{2}SnI_{6} (Table 1).

x | XPDF | SXRD | ||
---|---|---|---|---|

Cs–I length (Å) | BVS | Cs–I length (Å) | BVS | |

0 | 4.115 | 1.161 | 4.118 | 1.156 |

0.1 | 4.117 | 1.158 | 4.119 | 1.154 |

0.25 | 4.119 | 1.154 | 4.122 | 1.149 |

0.5 | 4.124 | 1.145 | 4.126 | 1.140 |

0.75 | 4.130 | 1.134 | 4.132 | 1.130 |

0.9 | 4.133 | 1.127 | 4.134 | 1.125 |

1 | 4.137 | 1.119 | 4.137 | 1.119 |

Anharmonicity in the vacancy-ordered double perovskites Cs_{2}Sn_{1−x}Te_{x}I_{6} can be correlated with the bonding preferences of the cesium cation within the cuboctahedral void. In Fig. 11, the degree of anharmonicity extracted from the Toda potential fits are plotted as a function of the Cs^{+} bond valence sum. For x = 1 (Cs_{2}TeI_{6}), the degree of anharmonicity is the lowest and corresponds with a bond valence sum of ∼1.12, suggesting that the Cs^{+} is most optimally bonded in Cs_{2}TeI_{6}. As tellurium is replaced with tin, the bond valence sum of Cs^{+} increases concomitantly with an increase in the degree of anharmonicity, reaching a maximum bond valence of ∼1.16 for Cs_{2}SnI_{6}. This analysis suggests that the anharmonicity is minimized when the size of the cuboctahedral void satisfies the bonding preferences of the Cs^{+} cation. Conversely, increasingly anharmonic lattice dynamics are therefore expected as the bond valence of the Cs^{+} ion diverges from ideal coordination.

Fig. 11 The degree of anharmonicity from the Toda potential fits plotted as a function of the Cs^{+} bond valence sum derived from XPDF analysis. |

The bond valence sum has previously been applied to other perovskite halide systems to predict the presence of dynamic and cooperative octahedral tilting distortions.^{41,54,56,57} Bond valence sum calculations of the vacancy-ordered double perovskite Rb_{2}SnI_{6} indicate that the coordination to the smaller Rb^{+} ion is optimized by symmetry-lowering cooperative octahedral tilting distortions,^{41} as is also observed in the Cs_{1−x}Rb_{x}PbX_{3} (X = Cl^{−}, Br^{−}) series.^{57} In the Cs_{2}Sn_{1−x}Te_{x}I_{6} series, the Cs^{+} coordination is nearly optimal in the cubic structural models, consistent with the observation that neither Cs_{2}SnI_{6} nor Cs_{2}TeI_{6} undergo structural phase transitions down to T = 10 K.^{32} Rather, the slight deviations in bond valence sum in this system manifest as a small degree of anharmonicity. Therefore, anharmonic effects in vacancy-ordered double perovskites may be expected when the bond valence sum of the A-site cation deviates slightly from ideal, while more significant structural changes due to cooperative octahedral tilting may be expected if the A-site is significantly underbonded. As the properties of halide perovskites are intimately linked to (anharmonic) lattice dynamics, the bond valence sum provides a simple tool for predicting the presence and extent of anharmonic behavior and may further be leveraged as a design principle for materials with desired structure–dynamic–property relationships.

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## Footnote |

† Electronic supplementary information (ESI) available: Data from additional Reverse Monte Carlo simulations. See DOI: 10.1039/c8tc03527j |

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