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Tolerance Factor and Cooperative Tilting Effects in Vacancy-Ordered Double Perovskite Halides

  • Annalise E. Maughan
    Annalise E. Maughan
    Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872, United States
    More by Annalise E. Maughan
    Orcidhttp://orcid.org/0000-0002-3292-4799
  • , 
  • Alex M. Ganose
    Alex M. Ganose
    Kathleen Lonsdale Materials Chemistry, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom
    Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
    Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
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  • , 
  • Mohammed A. Almaker
    Mohammed A. Almaker
    Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872, United States
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  • , 
  • David O. Scanlon
    David O. Scanlon
    Kathleen Lonsdale Materials Chemistry, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom
    Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
    Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
    More by David O. Scanlon
    Orcidhttp://orcid.org/0000-0001-9174-8601
  • , and 
  • James R. Neilson*
    James R. Neilson
    Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872, United States
    *(J.R.N.) E-mail: [email protected]
    More by James R. Neilson
    Orcidhttp://orcid.org/0000-0001-9282-5752
Cite this: Chem. Mater. 2018, 30, 11, 3909–3919
Publication Date (Web):May 23, 2018
https://doi.org/10.1021/acs.chemmater.8b01549

Copyright © 2022 American Chemical Society. This publication is licensed under CC-BY.

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Abstract

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Lattice dynamics and structural instabilities are strongly implicated in dictating the electronic properties of perovskite halide semiconductors. We present a study of the vacancy-ordered double perovskite Rb2SnI6 and correlate dynamic and cooperative octahedral tilting with changes in electronic behavior compared to those of Cs2SnI6. Though both compounds exhibit native n-type semiconductivity, Rb2SnI6 exhibits carrier mobilities that are reduced by a factor of ∼50 relative to Cs2SnI6. From synchrotron powder X-ray diffraction, we find that Rb2SnI6 adopts the tetragonal vacancy-ordered double perovskite structure at room temperature and undergoes a phase transition to a lower-symmetry monoclinic structure upon cooling, characterized by cooperative octahedral tilting of the [SnI6] octahedra. X-ray and neutron pair distribution function analyses reveal that the local coordination environment of Rb2SnI6 is consistent with the monoclinic structure at all temperatures; we attribute this observation to dynamic octahedral rotations that become frozen in to yield the low-temperature monoclinic structure. In contrast, Cs2SnI6 adopts the cubic vacancy-ordered double perovskite structure at all temperatures. Density functional calculations show that static octahedral tilting in Rb2SnI6 results in marginally increased carrier effective masses, which alone are insufficient to account for the experimental electronic behavior. Rather, the larger number of low-frequency phonons introduced by the lower symmetry of the Rb2SnI6 structure yield stronger electron–phonon coupling interactions that produce larger electron effective masses and reduced carrier mobilities relative to Cs2SnI6. Further, we discuss the results for Rb2SnI6 in the context of other vacancy-ordered double perovskite semiconductors, in order to demonstrate that the electron–phonon coupling characteristics can be predicted using the geometric perovskite tolerance factor. This study represents an important step in designing perovskite halide semiconductors with desired charge transport properties for optoelectronic applications.

Introduction

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Inorganic perovskite halide-based materials have presented a paradigm shift in the search for new high performance semiconductors. The high amplitude lattice dynamics and low elastic moduli, yet excellent carrier dynamics and electronic behavior, suggest that elucidation of structure–dynamics–property relationships is required in order to fully understand these materials. (1,2) The perovskite family is typified by the general formula ABX3, and the structure is formed by a network of corner sharing [BX6] octahedra bridged by A-site cations in the cuboctahedral void. A hallmark of the perovskite family is cooperative octahedral tilting (static and dynamic) of the [BX6] framework, which typically arises from a size mismatch between the A-site cation and the cuboctahedral void formed by 12 neighboring X-site anions, as predicted by the Goldschmidt tolerance factor, . (3−5) Octahedral tilting distortions are well-known to influence the optoelectronic properties of perovskite halides, as the electronic states of the B- and X-site ions dominate the conduction and valence band edges. (6) Static octahedral tilting distortions affect the electronic dispersions of the B and X states through deviation in the X–B–X bond angle from 180°, yielding smaller carrier mobilities and higher resistivities. (7) Changes in these bond angles also affect the relative energies of the conduction and valence band edges, resulting in a widening of the band gap with correspondingly larger octahedral tilting distortions. (8−11) Further, dynamic octahedral tilting of the [BX6] framework has recently been implicated in the formation of polarons via electron–phonon coupling, which are hypothesized to protect carriers and screen crystallographic defects, thereby prolonging carrier excited state lifetimes and reducing carrier mobilities. (12−17)
Vacancy-ordered double perovskites are a subset of the ordered double perovskite family with the general formula A2BX6. The structure is formed by a face-centered lattice of isolated [BX6] octahedral units bridged by A-site cations. Similar to ordered double perovskites, vacancy-ordered double perovskites undergo successive octahedral tilting distortions according to the group-subgroup relationship a0a0a0 (Fm3̅m) → a0a0c+ (P4/mnc) → a+b–b– (P21/n), (18,19) and the proclivity of these materials to undergo octahedral tilting can be predicted by the “radius ratio”, which is defined as the ratio of the A-site cation radius to the radius of the cuboctahedral site formed by the X12 cage. (20,21) As in conventional ABX3 perovskites, the electronic properties of vacancy-ordered double perovskites are directly dictated by the electronic states of the B- and X-site ions at the valence and conduction band edges. Close-packing of the halogen framework provides dispersive electronic states and a framework for mobile carriers, while the interaction of the B-site cations with the coordinating X-site anions dictates the band positions, optical gaps, and tolerance to crystallographic defects. (21−24) Although the electronic states of the A-site cations are typically far from the frontier electronic states in vacancy-ordered double perovskites, (21) varying the identity of the A-site can indirectly influence characteristics such as carrier mobilities and band gap through changes in the dynamics of the surrounding inorganic framework. Our recent study of the vacancy-ordered double perovskites A2SnI6, where A = Cs+, CH3NH3+ (methylammonium), and CH(NH2)2+ (formamidinium) revealed significant lattice anharmonicity in the hybrid compounds, resulting in more tightly bound polarons and subsequently reduced carrier mobilities across the series. (25) Due to the large sizes and hydrogen bonding capabilities of the CH3NH3+ and CH(NH2)2+ cations, the anharmonic effects appear to manifest as random rotations of the isolated [SnI6] octahedral units rather than through typical cooperative tilting motifs.
In this contribution, we present a study of the tin(IV) iodide-based vacancy-ordered double perovskite Rb2SnI6 and compare the observed structural and electronic properties to those of Cs2SnI6 and other A2SnI6 compounds. Both Rb2SnI6 and Cs2SnI6 exhibit native n-type conductivity, though Rb2SnI6 exhibits carrier mobilities that are reduced by ∼50-fold relative to Cs2SnI6. Crystallographic analysis reveals that Rb2SnI6 crystallizes in the tetragonal modification of the vacancy-ordered double perovskite structure at room temperature due to cooperative tilting of the [SnI6] octahedral units. Upon cooling, Rb2SnI6 undergoes further octahedral tilting to a lower-symmetry monoclinic structure. X-ray and neutron pair distribution function analysis reveal that the local coordination environment is best described by the monoclinic structure at all temperatures. This can be rationalized through bond valence sum analysis, which suggests that the Rb+ ion coordination is optimized in the monoclinic structure. Calculation of the phonon dispersions indicate that, although the tetragonal structure is dynamically stable, the lowest-frequency optical phonon polarization corresponds to cooperative octahedral tilting that yields the monoclinic structure. This strongly implicates octahedral rotational dynamics in driving the structural phase transition. Density functional calculations reveal that cooperative octahedral tilting in Rb2SnI6 results in marginally smaller electron effective masses due to subtle changes in the close-packed iodide framework, which alone cannot account for the changes in carrier mobility observed experimentally. Rather, we find that the lower symmetry of Rb2SnI6 relative to Cs2SnI6 yields stronger electron–phonon coupling due to the larger number of nondegenerate low-frequency phonons that contribute to the dielectric response of the lattice and produce more tightly bound polarons that reduce charge carrier mobilities. From these results we show that simple models such as bond valence sum and the perovskite tolerance factor serve as effective predictors for charge transport behavior in vacancy-ordered double perovskite semiconductors.

Methods and Materials

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Materials Synthesis

SnI4 (23,25) and Cs2SnI6 (23,26) were prepared by previously reported methods. Rb2SnI6 was synthesized via solution precipitation. To a 20 mL scintillation vial was added 0.2 g of SnI4, 10.0 mL of isopropanol, and 1.0 mL of hydriodic acid (57% aq., unstabilized). The solution was heated to T = 60 °C while stirring to dissolve the entire mass of SnI4. In a separate scintillation vial was added a stoichiometric amount of Rb2CO3 to 2.0 mL of hydriodic acid and stirred to dissolve the solid. We note that it is important to minimize the length of time over which the Rb2CO3 is allowed to sit in the hydriodic acid, to prevent the formation of RbI3. Once the solids had dissolved in both solutions, the RbI solution was added all at once to the warm SnI4/isopropanol solution. The scintillation vial was capped and stirred gently for 30 min to cool to room temperature. The solution was further cooled in an ice bath for an additional 30 min while stirring to induce precipitation. The black precipitate was collected via centrifugation and washed with a small amount of cold isopropanol. The product was dried in air at T = 60 °C for 24 h. We note that unstabilized hydriodic acid is required for successful precipitation of the final product; use of HI containing the H3PO2 stabilizer typically results in precipitation of bright orange crystals, which are presumably due to precipitation of the SnI4 precursor.

Structural Characterization

High-resolution synchrotron powder X-ray diffraction data for Rb2SnI6 were collected from the diffractometer on beamline 11-BM-B at the Advanced Photon Source, Argonne National Laboratory, at T = 295 K and T = 100 K. (27) The data were analyzed with the Rietveld method implemented in gsas/expgui. (28,29) VESTA was used to visualize and render all crystal structures presented in this publication. (30)
Synchrotron X-ray scattering data suitable for pair distribution function (PDF) analysis were collected at beamline 11-ID-B at the Advanced Photon Source, Argonne National Laboratory, using 86 keV photons and sample–detector distance of 25 cm. Powdered samples of Rb2SnI6 were loaded into polyimide capillaries and measured in transmission geometry at room temperature using a PerkinElmer amorphous silicon image plate detector. (31) The program Fit2D (32) was used to calibrate the sample to detector distance and detector alignment with data from a CeO2 powder standard. Raw scattering data was integrated into Q-space spectra, applying a mask and polarization correction during integration. Experimental PDFs were extracted using PDFgetX2 (33) and analyzed using PDFgui. (34) The normalized total scattering pattern, S(Q), was produced in PDFgetX2 by subtracting polyimide container scattering, utilizing the appropriate sample composition, and applying standard corrections for the area detector setup. (31) The pair distribution function pattern, G(r), was calculated via sine Fourier transformation of the total scattering data utilizing a maximum Q of 23.8 Å–1. Values of Qdamp = 0.034127 Å–1 and Qbroad = 0.021102 Å–1 were extracted from refinement of a TiO2 anatase standard in PDFgui and used for further refinement.
Neutron scattering measurements were performed on the NOMAD instrument at the Spallation Neutron Source, Oak Ridge National Laboratory. A powdered sample of Rb2SnI6 was loaded into a 6 mm vanadium sample can and sealed under a He atmosphere. Total scattering data were collected at T = 2, 10, 150, 250, and 300 K in the cryostat sample environment. Data were normalized against scattering data collected for a vanadium rod, and background scattering from the vanadium can was subtracted. Total scattering data of Cs2SnI6 at T = 10 K were collected on a powdered sample of Cs2SnI6 sealed into a 6 mm vanadium sample can under a He atmosphere utilizing the cryostat sample environment. Data were normalized against scattering data collected for a vanadium rod, and background scattering from the vanadium can was subtracted. Total scattering data of Cs2SnI6 at T = 90 K and T = 300 K were collected on powdered samples of Cs2SnI6 sealed into a quartz capillary (capillary diameter = 3.0 mm) in the multisample changer. Data were normalized against scattering data collected for an empty glass capillary, and background scattering from the empty capillary was subtracted.
For all neutron total scattering experiments, the data were merged to the total scattering structure function using the IDL codes developed for the NOMAD instrument. (35) The pair distribution function was then produced through the sine Fourier transform of the total scattering structure function utilizing Qmax = 31.4 Å–1. For Rb2SnI6 at all temperatures, values of Qdamp = 0.0245 Å–1 and Qbroad = 0.0196 Å–1 were extracted from refinement of a silicon standard in PDFgui. For Cs2SnI6 at T = 90 K and T = 300 K, values of Qdamp = 0.0201 Å–1 and Qbroad = 0.0196 Å–1 were extracted from refinement of a diamond standard in PDFgui. For Cs2SnI6 at T = 10 K, values of Qdamp = 0.01766 Å–1 and Qbroad = 0.01918 Å–1 were extracted from refinement of a silicon standard. Analysis of the nPDFs was performed using PDFgui.

Optical and Electronic Properties

UV–visible diffuse reflectance spectroscopy was performed on powdered samples of Rb2SnI6 diluted to 15 wt % in BaSO4, using BaSO4 as a baseline. Spectra were acquired using a Thermo Nicolet Evolution 300 spectrophotometer with a Praying Mantis mirror setup from λ = 600–1000 nm at a scan rate of 240 nm/min.
Electrical resistance measurements were performed on cold-pressed polycrystalline pellets of Rb2SnI6 using a Physical Properties Measurement System (Quantum Design, Inc.). The measurements were collected in a linear 4-probe configuration using Pt wires and Ag-paste contacts. Similar results were obtained when Au paste was used. Hall measurements were collected on a cold-pressed polycrystalline pellet in the Van der Pauw configuration at T = 300 K with Pt wires contacted to the edge of the cylindrical pellet using Ag paste. 4-probe sample geometry: l = 0.35(1) cm, w = 0.35(1) cm, h = 0.05(1) cm. Hall probe sample thickness: t = 0.08(1) cm.

Density Functional Theory Calculations

Density functional theory calculations were performed using the Vienna Ab initio Simulation Package (VASP). (36−39) A plane-wave basis set was employed, with the interactions between core and valence electrons described using the Projector Augmented Wave (PAW) method. (40) This study employed two functionals: PBEsol, (41) a version of the Perdew, Burke and Erzorhof (PBE) functional (42) revised for solids, and the hybrid functional HSE06 which combines 75% exchange and 100% of the correlation energies from PBE, together with 25% exact Hartree–Fock (HF) exchange at short-range. (43) PBEsol has been shown to reproduce the structural and vibrational properties of many compounds containing weakly dispersive interactions, such as in the vacancy-ordered double perovskites (25) and other layered halide systems. (44,45) The band structures were calculated along a reciprocal space path defined between all high-symmetry points in the Brillouin zone, as detailed by Bradley and Cracknell. (46) For density of states, band structure, and high-frequency dielectric response calculations, special attention was paid to capturing the effects of electron–electron interactions through use of the HSE06 hybrid functional. (47,48) Furthermore, the relativistic effects seen in Sn and I were taken into account through use of scalar relativistic pseudopotentials and explicit treatment of spin–orbit coupling effects. (49) This combination of HSE06+SOC has been shown to provide an accurate description of the electronic structure of many metal–halide containing semiconductors. (50,51) A k-point sampling mesh of Γ centered 3 × 3 × 2 and plane wave energy cutoff of 350 eV were found to converge the total energy to 1 meV/atom for all systems studied. The structures were relaxed using HSE06 until the forces on all atoms totaled less than 0.01 eV Å–1. The HSE06 optimized crystal structures can be found online in a public repository. (52)
The ionic contribution to the dielectric constants were calculated using the PBEsol functional within density functional perturbation theory (DFPT), (53) with a denser 6 × 6 × 4 Γ-centered k-point mesh necessary to reach convergence. The phonon band structure was calculated using the finite-displacement method in a 3 × 3 × 2 (324 atom) supercell using a 1 × 1 × 1 k-point mesh. The DFPT calculations were performed on structures relaxed using PBEsol, due to the high computational expense of performing DFPT calculations using hybrid DFT. The high-frequency real and imaginary dielectric functions were calculated from the optical transition matrix elements within the transversal approximation, (54) obtained at a denser 6 × 6 × 3 Γ-centered k-point mesh. Band alignments were performed using a vacuum–slab model (35 Å slab, 35 Å vacuum) with a (101) surface. The corresponding electrostatic potential was averaged along the c-direction using the MacroDensity package, (55) with the energy of the potential at the plateau used as the external vacuum level. Slab calculations were performed using the HSE06 functional, with a correction for the valence band energy and band gap taken from the HSE06+SOC calculated bulk.
Polaron mobilities were calculated within a temperature-dependent Feynman model, as implemented in the codes produced by Frost. (56) A full definition of the methodology is described elsewhere in the literature. (57) Crucially, the electron–phonon coupling is approximated without empirical parameters, using a highly idealized model. (58−60) The band structure is represented only as the effective mass approximation, with the physical response of the lattice given by the optical and static dielectric constants and an effective phonon-response frequency. This method has recently been applied to calculate the electron mobilities in the hybrid perovskites (57) and other vacancy-ordered double perovskites. (25)

Results

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Structural Characterization

Rubidium tin(IV) iodide crystallizes in the tetragonal modification of the vacancy-ordered double perovskite structure (space group P4/mnc) at room temperature, in contrast to a previous report of a cubic structure. (61) The structure is characterized by in-phase cooperative octahedral rotations about the c-axis by ∼11° (a0a0c+ in Glazer notation) relative to the cubic aristotype, as shown in Figure 1a,b. The tetragonal structural model was refined against high-resolution synchrotron powder X-ray diffraction (SXRD) data shown in Figure 2a,b.

Figure 1

Figure 1. Crystal structures of Rb2SnI6 at T = 295 K and T = 100 K. In (a) and (c), the structures are projected down the c-axis to highlight the octahedral tilting and rotation, while unit cell descriptions are shown in (b) and (d). Rubidium atoms are shown in pink, tin atoms in blue, and iodine atoms in purple.

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Figure 2

Figure 2. Rietveld refinements of the tetragonal and monoclinic structural models of Rb2SnI6 against high resolution synchrotron powder X-ray diffraction patterns collected at T = 295 K (a, b) and T = 100 K (c, d) on the high-resolution 11-BM diffractometer. Data are shown as black circles, the fit is the orange line, and the difference is shown in blue. Plots (b) and (d) highlight the splitting of the most intense reflection (202) at Q ∼ 1.9 Å–1 upon cooling through the phase transition.

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Upon cooling to T = 100 K, Rb2SnI6 undergoes a phase transition to monoclinic symmetry (space group P21/n), characterized by out-of-phase tilting of the octahedra in the ab plane by ∼5° (a–a–c+), in addition to the 11° tilt about the c-axis. The transition is evidenced by splitting of the (202) reflection in the SXRD patterns shown in Figure 2c,d and is consistent with the series of symmetry-lowering phase transitions commonly observed in other members of the vacancy-ordered double perovskite family and with those predicted for ordered double perovskites. (18,62,63) Structural and statistical parameters generated from Rietveld refinements of the tetragonal and monoclinic structural models against the SXRD data can be found in Table 1, and crystallographic information files are included as Supporting Information.
Table 1. Structural Parameters and Refinement Statistics for Rb2SnI6 at T = 295 K and T = 100 K
 295 K100 K
crystal systemtetragonalmonoclinic
space groupP4/mncP21/n
a (Å)8.07056(6)7.9426(1)
b (Å)8.07056(6)7.9758(9)
c (Å)11.7650(1)11.7876(2)
α (deg)9090
β (deg)9090.484(1)
γ (deg)9090
Uiso(Rb) (Å2)0.0564(8)0.0245(7)
Uiso(Sn) (Å2)0.0245(6)0.0194(7)
Uiso(I1) (Å2)0.0587(7)0.0296(5)
Uiso(I2) (Å2)0.0521(4)0.0262(5)
Uiso(I3) (Å2)–0.0219(5)
red. χ25.3739.046
wR8.41%10.92%
Temperature-dependent neutron scattering data collected on the NOMAD beamline at the Spallation Neutron Source, Oak Ridge National Laboratory, provide insight into the phase behavior of Rb2SnI6. Low-fidelity scans were collected continuously upon cooling from T ∼ 298 K to T ∼ 10 K. The reflections shift to higher Q with decreasing temperature consistent with thermal contraction of the lattice upon cooling and show subtle evidence of a phase transition near T ∼ 189 K (Figure S1, as elaborated in the SI).

Electronic and Optical Properties

Electrical resistivity measurements of Rb2SnI6 performed in a 4-probe configuration reveal semiconducting behavior, as evidenced by increasing resistivity upon cooling shown in Figure 3. At T = 300 K, Rb2SnI6 exhibits a bulk resistivity of ρ = ∼5 × 105 Ω·cm, which is approximately 5 orders of magnitude larger than that reported for Cs2SnI6 (ρ = 12 Ω·cm). (23) Hall effect measurements at T = 300 K reveal that Rb2SnI6 exhibits native n-type conductivity, consistent with the formation of iodine vacancy defect states that form as shallow donors to the conduction band, as in other tin(IV)-iodide based vacancy-ordered perovskites. (22,23,25) The higher resistivity of Rb2SnI6 compared with Cs2SnI6 arises from a reduction in both carrier concentration and carrier mobility. From Hall effect measurements, Rb2SnI6 exhibits a carrier concentration on the order of ne ∼ 1012 cm–3 and a carrier mobility of μe ∼ 0.22(3) cm2 V–1 s–1, which are reduced by factors of ∼104 and ∼50 respectively compared with ne ∼ 5(1) × 1016 cm–3 and μe ∼ 8.6(5) cm2 V–1 s–1 reported for Cs2SnI6. (23)

Figure 3

Figure 3. Temperature-dependent resistivity data of rubidium tin(IV) iodide collected using a 4-probe configuration with Pt wires and Ag paste.

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Analysis of UV–visible diffuse reflectance spectroscopy data for Rb2SnI6 yields an optical gap of 1.32(2) eV, consistent with the black color of the compound. The optical gap was extracted by converting the raw reflectance data to pseudoabsorbance, F(R), using the Kubelka–Munk function. The optical gap was determined by extrapolating the linear absorption onset region to zero absorbance, as shown in Figure 4. The optical gap of 1.32(2) eV is slightly larger than the value of 1.23(3) eV determined for Cs2SnI6 by the same method. (23,25)

Figure 4

Figure 4. UV–visible diffuse reflectance spectrum for Rb2SnI6. The data were converted to pseudoabsorbance via the Kubelka–Munk transform, and the optical gap was determined by extrapolating the onset region to zero absorbance, as shown by the black fit. The pink line is the transformed data, the fit is the black line, and zero absorbance is shown as the gray dashed line. The intersection point of the fit and zero absorbance is shown by the black dot.

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Local Coordination Environment

X-ray and neutron pair distribution function (XPDF, nPDF) analyses were employed to correlate changes in the local coordination environment with the observed structural and electronic behavior. The XPDF of rubidium tin(IV) iodide at T = 300 K is shown in Figure 5. The local coordination environment of Rb2SnI6 is moderately well-described by the tetragonal structural model observed in the powder X-ray diffraction data. Modeling the data with this structural model requires the inclusion of large, anisotropic atomic displacement parameters (ADPs) for the iodine and rubidium atoms, as represented by the thermal ellipsoids on the structural models shown adjacent to the fit. Yet, the model still does not capture the shape of the next-nearest neighbor I–I pair correlation at r ∼ 4 Å. The large ADPs required to provide an adequate fit to the data suggest the presence of disorder (static or dynamic) of the iodine and rubidium atoms within the structure. The fit to the XPDF is significantly improved, both statistically and visually, with the use of the low-temperature monoclinic structural model determined from the T = 100 K SXRD data, as shown in Figure 5b. The addition of out-of-phase octahedral tilting into the ab plane permitted by the lower symmetry of the monoclinic structure results in significantly smaller isotropic ADPs for all atoms, as shown by the thermal ellipsoids in the corresponding structural model.

Figure 5

Figure 5. X-ray pair distribution function of Rb2SnI6 fit to the (a) tetragonal (P4/mnc) and (b) monoclinic (P21/n) structural models. In (a), the XPDF is fit to the tetragonal (P4/mnc) model using anisotropic atomic displacement parameters for iodine and rubidium ions and in (b) the XPDF is fit to the monoclinic (P21/n) structural model using isotropic displacement parameters for all atoms. The thermal ellispoids in the corresponding structural models are shown at 95% probability.  

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This analysis indicates that the local coordination environment at T = 300 K is best modeled by the monoclinic structural modification, despite the higher-symmetry tetragonal structure observed by SXRD. This observation is consistent with previous reports of other perovskite halide materials, in which dynamic and cooperative octahedral rotations manifest as a lower-symmetry instantaneous structure (64) in the local coordination environment, yet they average to higher symmetry structures observed by diffraction. (14,16,25,65−68) The large thermal ellipsoids observed for both iodine and rubidium atoms implies that there is strong coupling between octahedral tilting and Rb+ displacements, as has been previously presented. (5)
Temperature-dependent neutron pair distribution function analysis (nPDF) was performed to probe changes in the local coordination environment of Cs2SnI6 and Rb2SnI6 as a function of temperature. At all temperatures, the local coordination environment of Cs2SnI6 is well-described by the cubic structural model, with no evidence of symmetry-lowering phase transitions, consistent with the corresponding neutron diffraction patterns shown in Figure 6b and with our previous report. (23) We note that a slight asymmetry of the next-nearest neighbor I–I pair correlation at r ∼ 4 Å is observed at T = 300 K, which has been previously attributed to anharmonic lattice dynamics. (25) The fits of the cubic structural model to the nPDF are shown in Figure 6a, and Rietveld refinements of the cubic structural model against the corresponding diffraction data are shown in Figure 6b.

Figure 6

Figure 6. (a) Temperature-dependent neutron pair distribution function analysis (nPDF) of Cs2SnI6 extracted from neutron total scattering collected on NOMAD. The x-axis is split to highlight the low-r pair correlations. Rietveld refinements of the corresponding neutron diffraction patterns collected from bank 2 (31° bank) of NOMAD are shown in (b). Both the nPDF and neutron diffraction data are modeled by the cubic (Fm3̅m) structural model at all temperatures. Black circles are the data, the orange lines are the fits, and the blue lines are difference curves. In (b), the gray tick marks indicate the positions of predicted reflections from the Fm3̅m structure.

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Temperature-dependent nPDFs of Rb2SnI6 provide further insight into the structural behavior observed by SXRD and XPDF. For T > 150 K, the nPDF can be modeled with the tetragonal structural model with the inclusion of large, anisotropic ADPs for the iodine atoms or with the monoclinic structural model with isotropic ADPs, consistent with analysis of the XPDF shown in Figure 5. Upon cooling to T = 150 K, a low-r shoulder to the next-nearest neighbor I–I pair correlation at r = 4 Å appears, which is only captured by the monoclinic structural model, as shown in Figure 7a. The appearance of this shoulder results in a poor fit to the nPDF when modeling with the higher-symmetry tetragonal structure. Rietveld refinements of the corresponding neutron diffraction data are shown Figure 7b. Data collected at temperatures above T = 150 K are modeled with the tetragonal P4/mnc structure, while data collected at T = 150 K and below are modeled by the P21/n structure.

Figure 7

Figure 7. (a) Temperature-dependent neutron pair distribution function analysis (nPDF) of Rb2SnI6 extracted from neutron total scattering collected on NOMAD. The nPDFs are best modeled by the monoclinic (P21/n) structure at all temperatures. The x-axis is split to highlight the low-r pair correlations, and the data are offset vertically for clarity. Rietveld refinements of the neutron diffraction data from bank 2 (31° bank) of NOMAD are shown in (b). For T > 150 K, the data are modeled with the tetragonal (P4/mnc) structure. For T ≤ 150 K, the data are modeled by the monoclinic (P21/n) structure. Black circles are the data, the orange lines are the fits, and the blue lines are difference curves. In (b), the gray tick marks indicate the positions of predicted reflections from the P4/mnc and P21/n structures. The x-axis is split to highlight the lower-Q reflections, and the data are offset vertically for clarity.

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Electronic Structure Calculations

Density functional calculations were performed to gain insight into the optical and electronic properties of rubidium tin(IV) iodide. The orbital-projected band structures of Rb2SnI6 in the P4/mnc and P21/n structures, calculated using HSE06+SOC, are shown in Figure 8. Both symmetries yield direct band gaps, with fundamental band gaps of 1.13 and 1.32 eV occurring at the Γ point (Table 2). Spin–orbit coupling was found to play a role on the size of the band gaps, with a band gap renormalization of 0.19 and 0.21 eV seen, respectively, mainly though raising of the valence band maximum. For both compounds, the orbital projections reveal the upper valence band to be dominated by I 5p states, with the conduction band minimum derived from hybridization between Sn 5s and I 5p states, in agreement with other A2SnI6 perovskite materials. (23,25) We note that the calculated fundamental band gap of the P4/mnc phase is significantly smaller than the optical band gap measured in experiment (1.32 eV). Analysis of the transition dipole moments reveals that, consistent with other A2SnI6 materials, (23) the fundamental band gap is dipole disallowed in both symmetries. Instead, the optical band gap results from transitions originating from the bands ∼0.30 eV below the valence band maximum.

Figure 8

Figure 8. Band structures calculated using HSE06+SOC for the (a) P4/mnc and (b) P21/n phases of Rb2SnI6. The color of the band indicates the orbital contribution to that band, with Sn 5s, Sn 5p, and I 5p represented by red, green, and blue, respectively. The resulting color of the bands is obtained by mixing each color in proportion to the orbital contributions. The valence band maximum is set to 0 eV in all cases.

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Table 2. Band Gaps (Eg), Conduction Bandwidths (ΔεCB), and Charge Carrier Effective Masses (m*), Calculated Using HSE06+SOC, for the P4/mnc and P21/n Phases of Rb2SnI6a
phaseEgΔεCBme*mh*
P4/mnc1.130.990.390.98
P21/n1.320.810.441.07
a

Band gaps and widths are provided in eV, and effective masses are given in units of the bare electron mass, m0.

The carrier effective masses, calculated based on parabolic fits of the band edges in three directions around the Γ point, indicate electrons will be relatively mobile, with masses of 0.39 m0 and 0.44 m0 seen for the P4/mnc and P21/n phases, respectively. These are marginally larger than the electron effective masses in Cs2SnI6 (me* = 0.25 m0), (25) likely due to deviations in the close-packed iodine sublattice from octahedral tilting. These changes further manifest as a reduction in the conduction bandwidth, ΔεCB, from 1.39 eV seen in Cs2SnI6, to 0.99 and 0.81 eV, for the P4/mnc and P21/n phases, respectively.
DFT calculations of isolated slab structures surrounded by vacuum reveal the band gap differences between the two phases of Rb2SnI6 are driven by changes in the position of both the valence band maximum and the conduction band minimum. For the P4/mnc phase, the HSE06+SOC calculated ionization potential and electron affinity are 5.82 and 4.70 eV, respectively (Figure 9). Moving to the low temperature P21/n phase, the ionization potential is deeper relative to the vacuum level (5.92 eV), whereas the electron affinity becomes more shallow (4.60 eV). Again, these changes can likely be attributed to subtle distortions of the close-packed iodine sublattice and the reduction in bandwidth of both the upper valence and the lower conduction bands. We note that the band alignments are close to those calculated for Cs2SnI6, which shows an ionization potential and electron affinity of 5.79 and 4.80 eV, respectively; this is not unexpected due to the similar orbital composition of the valence and conduction bands in these materials.

Figure 9

Figure 9. Calculated band alignment (HSE06+SOC) of the P4/mnc and P21/n structured phases of Rb2SnI6 relative to those of Cs2SnI6.

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To further investigate the carrier mobilities in Rb2SnI6, we have calculated the limits of electron mobility within a temperature-dependent Feynman polaron transport model. This method has recently been applied to the cubic hybrid perovskites (57) and other members of the A2SnI6 series. (25) All required parameters were calculated ab initio and are provided in Table 3. The electron effective masses necessary for the calculations, me*, are provided in Table 2. Within this model, the extent of electron–phonon coupling (α) is used to evaluate the polaron mobility (μeH, calculated within the Hellwarth model), (58) along with the associated phonon-drag mass-renormalization (mr*) and relaxation time (τ). The Hellwarth electron mobilities, μeH, at T = 300 K are presented in Table 4. This model is highly idealized, with the physical response of the lattice parametrized by the optical and static dielectric constants and effective-response frequency. Despite this, previous work applying this method to the hybrid perovskites has shown excellent agreement with experiment. We note that this method does not take into account other scattering processes; therefore, the results will form an upper bound for electron mobilities for a perfect crystal.
Table 3. Parameters of the Feynman Polaron Modela
space groupε∞εSf
P4/mnc3.618.384.01
P21/n3.648.484.32
a

High frequency (ε∞) and static (εS) dielectric constants are given in units of the permittivity of free space (ε0). Frequency (f) is in THz.

Table 4. Hellwarth Electron Mobilities at T = 300 K (μeH, cm2 V–1 s–1), Electron–Phonon Coupling (α), Effective Mass Renormalization (mr*, units of m*), and Polaron Relaxation Time (τ, ps), Calculated within a Temperature-Dependent Polaron Model, for the P4/mnc and P21/n Phases of Rb2SnI6
space groupμeHαmr*τ
P4/mnc24.42.831.710.07
P21/n19.82.981.640.06
The calculated Hellwarth electron mobilities for the P4/mnc (24 cm2 V–1 s–1) and P21/n (20 cm2 V–1 s–1) phases of Rb2SnI6 are significantly smaller than those calculated for Cs2SnI6 (98 cm V–1 s–1). The increase in electron effective masses seen in the Rb2SnI6 phases will play a crucial role; however, when considered alone, they are insufficient to account for the dramatic difference between the experimental mobilities. Instead, the trend can be attributed to two additional factors: First, the Rb2SnI6 phases show considerably larger static dielectric constants (∼8.4) than Cs2SnI6 (7.2), due to an increase in the ionic component of the dielectric response. Analysis of the zone center phonon eigenvalues and eigenvectors reveals that this is due to an increase in the number of low-frequency polar phonon modes that can contribute to the dielectric response of the lattice. These additional modes and their increased polarity result from the symmetry breaking in the Rb2SnI6 phases, which reduces the degeneracy of several modes in the highly degenerate Cs2SnI6 phonon spectrum. Second, the high-frequency dielectric constants of the Rb2SnI6 phases are slightly reduced, likely due to the changes in the iodine sublattice. Together, these factors result in significantly larger electron–phonon coupling constants (α) of 2.83 and 2.98 for the P4/mnc and P21/n phases of Rb2SnI6, in comparison to just 1.45 for Cs2SnI6. The increased coupling, combined with larger effective masses, results in significant effective mass renormalization and small polaron relaxation times. We note that the effective-response frequency is larger in Rb2SnI6 than in Cs2SnI6, suggesting a stiffening of the lattice. Further investigation into the IR response of the modes indicates that the symmetry breaking also allows for increased polarity of the higher-frequency phonon modes, thereby increasing the effective-response frequency, with the stiffness of the lattice remaining largely unchanged.
Further examination of the phonon band structure of tetragonal Rb2SnI6 reveals that the tetragonal structure is dynamically stable within the harmonic approximation of forces calculated by DFPT, as evidenced by the lack of imaginary phonon modes (Figure 10). The lowest-energy optical phonon occurs at 0.25 THz (1.03 meV, 8.27 cm–1) at the Γ point, as denoted by the orange circle in Figure 10. Analysis of the polarization eigenvectors indicates that this mode corresponds to displacements of the iodine and rubidium ions within the structure, which are consistent with cooperative octahedral tilting out of the ab plane coupled to Rb+ displacements and map onto the lower-symmetry (a–a–c+) monoclinic structure observed at low temperatures. This observation is consistent with the structural behavior of Rb2SnI6 observed experimentally. At higher temperatures, dynamic octahedral tilting averages to the tetragonal structure observed by diffraction, yet it gives rise to the lower symmetry suggested from the local coordination environment. Upon cooling, the octahedral tilts freeze in to yield the lower-symmetry monoclinic structure observed in the crystallography. This notion is supported by previous studies of the vacancy-ordered double perovskite family, which have shown that the symmetry-lowering phase transitions in these materials are driven by condensation of the octahedral rotary phonon mode. (65,69,70) The lack of imaginary modes in these calculations suggests that anharmonicity may play a role in driving the structure to the monoclinic ground state.

Figure 10

Figure 10. Phonon band structure of rubidium tin(IV) iodide in the tetragonal P4/mnc structure (a0a0c+). The lowest-frequency optical mode at 0.25 THz (1.03 meV, 8.27 cm–1) (denoted by the orange circle) corresponds to displacements of the rubidium and iodine atoms, as shown by the displacement vectors in the structural representations. Together, these displacements map to the octahedral tilting out of the ab plane coupled with Rb+ displacements observed in the lower-symmetry monoclinic structure, (a–a–c+).

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Discussion

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Rb2SnI6 adopts the tetragonal variant of the vacancy-ordered double perovskite structure at room temperature characterized by cooperative octahedral tilting and undergoes a symmetry-lowering phase transition from tetragonal (P4/mnc) to monoclinic (P21/n) upon cooling. Temperature-dependent phase transitions in perovskites are typically driven by a size mismatch between the A-, B-, and X-site ions, and cooperative rotation and tilting of the octahedral units occur to improve coordination to the A-site cation by the neighboring X-site anions by collapsing the A-site void. (4,5,18,71) Bond valence sum (BVS) analysis of Cs2SnI6 reveals that the coordination provided to the Cs+ ion by the I12 cage results in a Cs valence of 1.156, indicating that the cesium is potentially overbonded by the surrounding iodine network in the cubic structure (using tabulated parameters). This analysis provides an explanation for the observation that Cs2SnI6 does not appear to be susceptible to temperature-dependent phase transitions or distortions of the local coordination environment. BVS analysis of the previously reported cubic structure of Rb2SnI6 yields an under-bonded BVS for the Rb+ ions of 0.891, indicating that the Rb–I bond distances are too long to provide adequate coordination to the rubidium ions in that structural model. (61) The bond valence of rubidium is improved in the tetragonal (0.939) and monoclinic (1.069) structural models, consistent with the observation that Rb2SnI6 adopts both the tetragonal and monoclinic variants of the vacancy-ordered double perovskite family, as well as with DFT-based calculations that show the dominating role of electrostatics in determining octahedral tilting patterns in perovskite halides. (4) Further, this lends support to our hypothesis that dynamic octahedral tilting occurs readily at room temperature to yield the tetragonal structure by diffraction yet reduced symmetry in the local coordination environment. This notion is further supported by analysis of the phonon spectrum of Rb2SnI6, which indicates that the lowest-energy optical phonon corresponds to octahedral tilting and rubidium ion displacements that together map onto the lower-symmetry monoclinic structure. Parameters used in the BVS analysis are shown in Table 5.
Table 5. Bond Valence Sum Analysis for Cs–I and Rb–I Bonds in Cs2SnI6 and Rb2SnI6
 B (72)R0 (Å) (72)BVS
Cs–I (Fm3̅m)0.6092.69261.156
Rb–I (Fm3̅m)0.6382.45090.891
Rb–I (P4/mnc)0.6382.45090.939
Rb–I (P21/n)0.6382.45091.069
Analysis of the band effective masses suggests that static cooperative octahedral tilting does not play a strongly significant role in dictating the charge transport behavior in Rb2SnI6. While octahedral tilting in Rb2SnI6 induces subtle changes in the close-packed iodine framework and yields slightly larger carrier effective masses than in the cubic Cs2SnI6 (by a factor of 1.5–1.8), these structural changes alone are not sufficient to account for the trend in carrier mobilities observed experimentally. Rather, calculation of the electron–phonon coupling characteristics of Rb2SnI6 indicates that the lower symmetry due to octahedral tilting yields a larger number of low-frequency polar phonons that contribute to a large static dielectric constant and subsequently stronger electron–phonon coupling that significantly reduces carrier mobilities. Therefore, interpreting the electronic properties from the perspective of static cooperative octahedral tilting is insufficient to fully understand the charge transport behavior in vacancy-ordered double perovskites, and a dynamic understanding of these structural deviations is necessary to account for the observed behavior.
While lattice dynamics play a critical role in dictating the electronic properties of vacancy-ordered double perovskites, the interplay between octahedral tilting and charge transport in Rb2SnI6 can be generalized to a set of guiding principles for predicting charge transport behavior in vacancy-ordered double perovskites based upon the geometric model of the perovskite tolerance factor. A highly simplified use of the Goldschmidt tolerance factor, , where B = Sn(IV) and X = I (rSn = 0.69 Å and rI = 2.2 Å) (73) capture the observed trends in carrier mobilities for these vacancy-ordered double perovskites. Of the A2SnI6 family (A = Rb+, Cs+, CH3NH3+, and CH(NH2)2)+), the tolerance factor for Cs2SnI6 is 0.998 and Cs2SnI6 exhibits the largest carrier mobility of ∼9 cm2 V–1 s–1 (Figure 11). Incorporation of the larger CH3NH3+ and CH(NH2)2+ ions yields tolerance factors of 1.07 and 1.16 and carrier mobilities of ∼2.5 cm2 V–1 s–1 and ∼0.36 cm2 V–1 s–1, respectively. (25) Replacement of Cs+ with Rb+ yields a tolerance factor of 0.959 and a carrier mobility of μe ∼ 0.22(3) cm2 V–1 s–1. Further, the trends in carrier mobilities observed experimentally are captured by the computationally derived Hellwarth electron mobilities (μeH), as also shown in Figure 11. Therefore, we might expect that vacancy-ordered double perovskite materials with tolerance factors closest to unity will exhibit weaker electron–phonon coupling interactions and higher carrier mobilities. Cations that are too large for the A-site can introduce soft, anharmonic lattice dynamics that produce tightly bound polarons that localize charge carriers and impede charge transport, as in (CH3NH3)2SnI6 and (CH(NH2)2)2SnI6. (25) In contrast, cations that are too small yield structures of reduced symmetry, as characterized by cooperative octahedral tilting and rotation. This slightly distorts the close-packed halogen sublattice and increases the number of low-lying phonon modes that contribute to stronger electron–phonon coupling interactions and reduce carrier mobilities.

Figure 11

Figure 11. Experimentally (μe,expt.) and computationally derived Hellwarth (μeH) electron mobilities of the A2SnI6 vacancy-ordered double perovskites plotted as a function of perovskite tolerance factor. Experimental electron mobilities are shown as filled purple circles on the left axis, while the calculated Hellwarth electron mobilities are denoted by open orange squares on the right axis. Values for the carrier mobilities of Cs2SnI6, (CH3NH3)2SnI6, and (CH(NH2)2)2SnI6 are taken from a previous study. (25) For Rb2SnI6, the μeH value is calculated from the tetragonal structure.

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Conclusions

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Replacing Cs+ with the smaller Rb+ ion in the vacancy-ordered double perovskite Rb2SnI6 is accompanied by significant changes in structural and electronic behavior. Electrical conductivity measurements of Rb2SnI6 indicate that, similarly to Cs2SnI6, Rb2SnI6 is a native n-type semiconductor. However, the carrier mobility is reduced by a factor of ∼50 for Rb2SnI6 compared to Cs2SnI6. Structural analysis through high-resolution synchrotron powder X-ray diffraction indicates that Rb2SnI6 crystallizes in a lower-symmetry structural modification of the vacancy-ordered double perovskite structure, characterized by cooperative tilting of the [SnI6] octahedra (a0a0c+). Upon cooling, Rb2SnI6 undergoes further octahedral tilting to a monoclinic structure (a–a–c+). X-ray and neutron pair distribution function analysis reveals that the bonding environment is best described by the monoclinic structure at all temperatures, which becomes thermally averaged to higher symmetry at ambient temperature. No structural distortions are observed for Cs2SnI6 to T = 10 K. Density functional calculations reveal that the reduced carrier mobilities observed experimentally in Rb2SnI6 relative to Cs2SnI6 arise from stronger electron–phonon interactions and subsequently smaller Hellwarth electron mobilities in Rb2SnI6 due to an increase in the number of low-frequency phonons that contribute to the dielectric response of the lattice, rather than a trivial increase in band effective mass. This observation suggests that the polaron characteristics and charge transport behavior in the vacancy-ordered double perovskite family can be tuned by introducing cooperative octahedral tilting distortions through judicious choice of A-site cation and further predicted bond valence sum and the perovskite tolerance factor.

Supporting Information

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.8b01549.

  • Temperature-dependent neutron diffraction data (PDF)

  • Crystallographic information file for tetragonal Rb2SnI6 at T = 295 K (CIF)

  • Crystallographic information file for monoclinic Rb2SnI6 at T = 100 K (CIF)

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Author Information

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  • Corresponding Author
  • Authors
    • Annalise E. Maughan - Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872, United States;  Orcidhttp://orcid.org/0000-0002-3292-4799
    • Alex M. Ganose - Kathleen Lonsdale Materials Chemistry, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom;  Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom;  Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom
    • Mohammed A. Almaker - Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872, United States
    • David O. Scanlon - Kathleen Lonsdale Materials Chemistry, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, United Kingdom;  Diamond Light Source Ltd., Diamond House, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom;  Thomas Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom;  Orcidhttp://orcid.org/0000-0001-9174-8601
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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This work was supported by Grant DE-SC0016083 funded by the U.S. Department of Energy, Office of Science. J.R.N. and A.E.M. acknowledge support from Research Corporation for Science Advancement through a Cottrell Scholar Award, and J.R.N. thanks the A.P. Sloan Foundation for assistance provided from a Sloan Research Fellowship. 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. A portion of this research at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. This work made use of the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), via membership of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202). D.O.S. acknowledges support from the EPSRC (EP/N01572X/1). D.O.S. acknowledges membership of the Materials Design Network. A.M.G. acknowledges Diamond Light Source for the cosponsorship of a studentship on the EPSRC Centre for Doctoral Training in Molecular Modelling and Materials Science (EP/L015862/1). The authors thank Dr. K. Page, Dr. D. Olds, and Dr. T.-M. Usher from the NOMAD instrument at the Spallation Neutron Source, Oak Ridge National Laboratory, and Dr. T.-M. Usher and Dr. M. McDonnell for collection of X-ray total scattering data. The authors also thank Prof. M. M. Reynolds for the use of her optical spectrometer and the expert beamline scientists at 11-BM at the Advanced Photon Source.

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    Stoumpos, C. C.; Malliakas, C. D.; Kanatzidis, M. G. Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and Near-Infrared Photoluminescent Properties. Inorg. Chem. 2013, 52, 9019– 9038,  DOI: 10.1021/ic401215x
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    7
    Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and Near-Infrared Photoluminescent Properties
    Stoumpos, Constantinos C.; Malliakas, Christos D.; Kanatzidis, Mercouri G.
    Inorganic Chemistry (2013), 52 (15), 9019-9038CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    A broad org.-inorg. series of hybrid metal iodide perovskites AMI3, where A is the methylammonium (MeNH3+) or formamidinium (HC(NH2)2+) cation and M is Sn (1 and 2) or Pb (3 and 4) are reported. The compds. were prepd. through a variety of synthetic approaches, and the nature of the resulting materials is discussed in terms of their thermal stability and optical and electronic properties. The chem. and phys. properties of these materials strongly depend on the prepn. method. Single crystal x-ray diffraction anal. of 1-4 classifies the compds. in the perovskite structural family. Structural phase transitions were obsd. and studied by temp.-dependent single crystal x-ray diffraction in the 100-400 K range. The charge transport properties of the materials are discussed in conjunction with diffuse reflectance studies in the mid-IR region that display characteristic absorption features. Temp.-dependent studies show a strong dependence of the resistivity as a function of the crystal structure. Optical absorption measurements indicate that 1-4 behave as direct-gap semiconductors with energy band gaps distributed at 1.25-1.75 eV. The compds. exhibit an intense near-IR luminescence (PL) emission in the 700-1000 nm range (1.1-1.7 eV) at room temp. Solid solns. between the Sn and Pb compds. are readily accessible throughout the compn. range. The optical properties such as energy band gap, emission intensity, and wavelength can be readily controlled for the isostructural series of solid solns. MeNH3Sn1-xPbxI3 (5). The charge transport type in these materials was characterized by Seebeck coeff. and Hall-effect measurements. The compds. behave as p- or n-type semiconductors depending on the prepn. method. The samples with the lowest carrier concn. are prepd. from soln. and are n-type; p-type samples can be obtained through solid state reactions exposed in air in a controllable manner. In the case of Sn compds., there is a facile tendency toward oxidn. which causes the materials to be doped with Sn4+ and thus behave as p-type semiconductors displaying metal-like cond. The compds. appear to possess very high estd. electron and hole mobilities that exceed 2000 cm2/(V s) and 300 cm2/(V s), resp., as shown in the case of MeNH3SnI3 (1). The authors also compare the properties of the title hybrid materials with those of the all-inorg. CsSnI3 and CsPbI3 prepd. using identical synthetic methods.
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  8. 8
    Amat, A.; Mosconi, E.; Ronca, E.; Quarti, C.; Umari, P.; Nazeeruddin, M. K.; Grätzel, M.; De Angelis, F. Cation-induced band-gap tuning in organohalide perovskites: Interplay of spin–orbit coupling and octahedra tilting. Nano Lett. 2014, 14, 3608– 3616,  DOI: 10.1021/nl5012992
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    8
    Cation-Induced Band-Gap Tuning in Organohalide Perovskites: interplay of Spin-Orbit Coupling and Octahedra Tilting
    Amat, Anna; Mosconi, Edoardo; Ronca, Enrico; Quarti, Claudio; Umari, Paolo; Nazeeruddin, Md. K.; Gratzel, Michael; De Angelis, Filippo
    Nano Letters (2014), 14 (6), 3608-3616CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
    Organohalide lead perovskites have revolutionized the scenario of emerging photovoltaic technologies. The prototype MAPbI3 perovskite (MA = MeNH3+) has dominated the field, despite only harvesting photons >750 nm (∼1.6 eV). Intensive research efforts are being devoted to find new perovskites with red shifted absorption onset, along with good charge transport properties. Recently, a new perovskite based on the formamidinium cation ((NH2)2CH+ = FA) showed potentially superior properties in terms of band gap and charge transport compared to MAPbI3. The results were interpreted in terms of the cation size, with the larger FA cation expectedly delivering reduced band-gaps in Pb-based perovskites. To provide a full understanding of the interplay among size, structure, and org./inorg. interactions in detg. the properties of APbI3 perovskites, in view of designing new materials and fully exploiting them for solar cells applications, the authors report a fully 1st-principles study on APbI3 perovskites with A = Cs+, MA, and FA. The results evidence that the tetragonal-to-quasi cubic structural evolution obsd. when moving from MA to FA is due to the interplay of size effects and enhanced H bonding between the FA cations and the inorg. matrix altering the covalent/ionic character of Pb-I bonds. Most notably, the obsd. cation-induced structural variability promotes markedly different electronic and optical properties in the MAPbI3 and FAPbI3 perovskites, mediated by the different spin-orbit coupling, leading to improved charge transport and red shifted absorption in FAPbI3 and in general in pseudocubic structures. The theor. model constitutes the basis for the rationale design of new and more efficient organohalide perovskites for solar cells applications.
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  9. 9
    Yang, Y.; Ostrowski, D. P.; France, R. M.; Zhu, K.; Van De Lagemaat, J.; Luther, J. M.; Beard, M. C. Observation of a hot-phonon bottleneck in lead-iodide perovskites. Nat. Photonics 2016, 10, 53– 59,  DOI: 10.1038/nphoton.2015.213
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    9
    Observation of a hot-phonon bottleneck in lead-iodide perovskites
    Yang, Ye; Ostrowski, David P.; France, Ryan M.; Zhu, Kai; van de Lagemaat, Jao; Luther, Joseph M.; Beard, Matthew C.
    Nature Photonics (2016), 10 (1), 53-59CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)
    We study the carrier dynamics in planar Me ammonium lead iodide perovskite films using broadband transient absorption spectroscopy. We show that the sharp optical absorption onset is due to an exciton transition that is inhomogeneously broadened with a binding energy of 9 meV. We fully characterize the transient absorption spectrum by free-carrier-induced bleaching of the exciton transition, quasi-Fermi energy, carrier temp. and bandgap renormalization const. The photo-induced carrier temp. is extd. from the transient absorption spectra and monitored as a function of delay time for different excitation wavelengths and photon fluences. We find an efficient hot-phonon bottleneck that slows down cooling of hot carriers by three to four orders of magnitude in time above a crit. injection carrier d. of ∼5 × 1017 cm-3. Compared with mol. beam epitaxially grown GaAs, the crit. d. is an order of magnitude lower and the relaxation time is approx. three orders of magnitude longer.
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  10. 10
    Prasanna, R.; Gold-Parker, A.; Leijtens, T.; Conings, B.; Babayigit, A.; Boyen, H.-G.; Toney, M. F.; McGehee, M. D. Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics. J. Am. Chem. Soc. 2017, 139, 11117– 11124,  DOI: 10.1021/jacs.7b04981
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    10
    Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics
    Prasanna, Rohit; Gold-Parker, Aryeh; Leijtens, Tomas; Conings, Bert; Babayigit, Aslihan; Boyen, Hans-Gerd; Toney, Michael F.; McGehee, Michael D.
    Journal of the American Chemical Society (2017), 139 (32), 11117-11124CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Tin and lead iodide perovskite semiconductors of the compn. AMX3, where M is a metal and X is a halide, are leading candidates for high efficiency low cost tandem photovoltaics, in part because they have band gaps that can be tuned over a wide range by compositional substitution. We exptl. identify two competing mechanisms through which the A-site cation influences the band gap of 3D metal halide perovskites. Using a smaller A-site cation can distort the perovskite lattice in two distinct ways: by tilting the MX6 octahedra or by simply contracting the lattice isotropically. The former effect tends to raise the band gap, while the latter tends to decrease it. Lead iodide perovskites show an increase in band gap upon partial substitution of the larger formamidinium with the smaller cesium, due to octahedral tilting. Perovskites based on tin, which is slightly smaller than lead, show the opposite trend: they show no octahedral tilting upon Cs-substitution but only a contraction of the lattice, leading to progressive redn. of the band gap. We outline a strategy to systematically tune the band gap and valence and conduction band positions of metal halide perovskites through control of the cation compn. Using this strategy, we demonstrate solar cells that harvest light in the IR up to 1040 nm, reaching a stabilized power conversion efficiency of 17.8%, showing promise for improvements of the bottom cell of all-perovskite tandem solar cells. The mechanisms of cation-based band gap tuning we describe are broadly applicable to 3D metal halide perovskites and will be useful in further development of perovskite semiconductors for optoelectronic applications.
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  11. 11
    Linaburg, M. R.; McClure, E. T.; Majher, J. D.; Woodward, P. M. Cs1–xRbxPbCl3 and Cs1–xRbxPbBr3 solid solutions: Understanding octahedral tilting in lead halide perovskites. Chem. Mater. 2017, 29, 3507– 3514,  DOI: 10.1021/acs.chemmater.6b05372
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    11
    Cs1-xRbxPbCl3 and Cs1-xRbxPbBr3 Solid Solutions: Understanding Octahedral Tilting in Lead Halide Perovskites
    Linaburg, Matthew R.; McClure, Eric T.; Majher, Jackson D.; Woodward, Patrick M.
    Chemistry of Materials (2017), 29 (8), 3507-3514CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    The structures of the lead halide perovskites CsPbCl3 and CsPbBr3 were detd. 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 were detd. 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 av. structure.
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  12. 12
    Zhu, X.-Y.; Podzorov, V. Charge carriers in hybrid organic–inorganic lead halide perovskites might be protected as large polarons. J. Phys. Chem. Lett. 2015, 6, 4758– 4761,  DOI: 10.1021/acs.jpclett.5b02462
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    12
    Charge Carriers in Hybrid Organic-Inorganic Lead Halide Perovskites Might Be Protected as Large Polarons
    Zhu, X.-Y.; Podzorov, V.
    Journal of Physical Chemistry Letters (2015), 6 (23), 4758-4761CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    Large polaron that provides the protection of charge carrier is explained in terms of carrier diffusion length,electron hole recombination rates, charge carrier and Hall mobility and low carrier scattering rate.
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  13. 13
    Zhu, H.; Miyata, K.; Fu, Y.; Wang, J.; Joshi, P. P.; Niesner, D.; Williams, K. W.; Jin, S.; Zhu, X.-Y. Screening in crystalline liquids protects energetic carriers in hybrid perovskites. Science 2016, 353, 1409– 1413,  DOI: 10.1126/science.aaf9570
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    13
    Screening in crystalline liquids protects energetic carriers in hybrid perovskites
    Zhu, Haiming; Miyata, Kiyoshi; Fu, Yongping; Wang, Jue; Joshi, Prakriti P.; Niesner, Daniel; Williams, Kristopher W.; Jin, Song; Zhu, X.-Y.
    Science (Washington, DC, United States) (2016), 353 (6306), 1409-1413CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
    Hybrid lead halide perovskites exhibit carrier properties that resemble those of pristine nonpolar semiconductors despite static and dynamic disorder, but how carriers are protected from efficient scattering with charged defects and optical phonons is unknown. Here, we reveal the carrier protection mechanism by comparing three single-crystal lead bromide perovskites: CH3NH3PbBr3, CH(NH2)2PbBr3, and CsPbBr3. We obsd. hot fluorescence emission from energetic carriers with ∼102-picosecond lifetimes in CH3NH3PbBr3 or CH(NH2)2PbBr3, but not in CsPbBr3. The hot fluorescence is correlated with liq.-like mol. reorientational motions, suggesting that dynamic screening protects energetic carriers via solvation or large polaron formation on time scales competitive with that of ultrafast cooling. Similar protections likely exist for band-edge carriers. The long-lived energetic carriers may enable hot-carrier solar cells with efficiencies exceeding the Shockley-Queisser limit.
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  14. 14
    Neukirch, A. J.; Nie, W.; Blancon, J.-C.; Appavoo, K.; Tsai, H.; Sfeir, M. Y.; Katan, C.; Pedesseau, L.; Even, J.; Crochet, J. J.; Gupta, G.; Mohite, A. D.; Tretiak, S. Polaron stabilization by cooperative lattice distortion and cation rotations in hybrid perovskite materials. Nano Lett. 2016, 16, 3809– 3816,  DOI: 10.1021/acs.nanolett.6b01218
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    14
    Polaron Stabilization by Cooperative Lattice Distortion and Cation Rotations in Hybrid Perovskite Materials
    Neukirch, Amanda J.; Nie, Wanyi; Blancon, Jean-Christophe; Appavoo, Kannatassen; Tsai, Hsinhan; Sfeir, Matthew Y.; Katan, Claudine; Pedesseau, Laurent; Even, Jacky; Crochet, Jared J.; Gupta, Gautam; Mohite, Aditya D.; Tretiak, Sergei
    Nano Letters (2016), 16 (6), 3809-3816CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
    Soln.-processed organometallic perovskites have rapidly developed into a top candidate for the active layer of photovoltaic devices. Despite the remarkable progress assocd. with perovskite materials, many questions about the fundamental photophys. processes taking place in these devices, remain open. High on the list of unexplained phenomena are very modest mobilities despite low charge carrier effective masses. Also, expts. elucidate unique degrdn. of photocurrent affecting stable operation of perovskite solar cells. These puzzles suggest that, while ionic hybrid perovskite devices may have efficiencies on par with conventional Si and GaAs devices, they exhibit more complicated charge transport phenomena. Here the authors report the results from an in-depth computational study of small polaron formation, electronic structure, charge d., and reorganization energies using both periodic boundary conditions and isolated structures. Using the hybrid d. functional theory, volumetric strain in a CsPbI3 cluster creates a polaron with binding energy of ∼300 and 900 meV for holes and electrons, resp. In the MAPbI3 (MA = CH3NH3) cluster, both volumetric strain and MA reorientation effects lead to larger binding energies at ∼600 and 1300 meV for holes and electrons, resp. Such large reorganization energies suggest appearance of small polarons in organometallic perovskite materials. The fact that both volumetric lattice strain and MA mol. rotational degrees of freedom can cooperate to create and stabilize polarons indicates that to mitigate this problem, formamidinium (FA = HC(NH2)2) and cesium (Cs) based crystals and alloys, are potentially better materials for solar cell and other optoelectronic applications.
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  15. 15
    Sendner, M.; Nayak, P. K.; Egger, D. A.; Beck, S.; Müller, C.; Epding, B.; Kowalsky, W.; Kronik, L.; Snaith, H. J.; Pucci, A.; Lovrincic, R. Optical phonons in methylammonium lead halide perovskites and implications for charge transport. Mater. Horiz. 2016, 3, 613– 620,  DOI: 10.1039/C6MH00275G
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    15
    Optical phonons in methylammonium lead halide perovskites and implications for charge transport
    Sendner, Michael; Nayak, Pabitra K.; Egger, David A.; Beck, Sebastian; Mueller, Christian; Epding, Bernd; Kowalsky, Wolfgang; Kronik, Leeor; Snaith, Henry J.; Pucci, Annemarie; Lovrincic, Robert
    Materials Horizons (2016), 3 (6), 613-620CODEN: MHAOBM; ISSN:2051-6355. (Royal Society of Chemistry)
    Lead-halide perovskites are promising materials for opto-electronic applications. Recent reports indicated that their mech. and electronic properties are strongly affected by the lattice vibrations. Herein we report far-IR spectroscopy measurements of CH3NH3Pb(I/Br/Cl)3 thin films and single crystals at room temp. and a detailed quant. anal. of the spectra. We find strong broadening and anharmonicity of the lattice vibrations for all three halide perovskites, which indicates dynamic disorder of the lead-halide cage at room temp. We det. the frequencies of the transversal and longitudinal optical phonons, and use them to calc., via appropriate models, the static dielec. consts., polaron masses, electron-phonon coupling consts., and upper limits for the phonon-scattering limited charge carrier mobilities. Within the limitations of the model used, we can place an upper limit of 200 cm2 V-1 s-1 for the room temp. charge carrier mobility in MAPbI3 single crystals. Our findings are important for the basic understanding of charge transport processes and mech. properties in metal halide perovskites.
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  16. 16
    Yaffe, O.; Guo, Y.; Tan, L. Z.; Egger, D. A.; Hull, T.; Stoumpos, C. C.; Zheng, F.; Heinz, T. F.; Kronik, L.; Kanatzidis, M. G.; Owen, J. S.; Rappe, A. M.; Pimenta, M. A.; Brus, L. E. Local polar fluctuations in lead halide perovskite crystals. Phys. Rev. Lett. 2017, 118, 136001,  DOI: 10.1103/PhysRevLett.118.136001
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    16
    Local polar fluctuations in lead halide perovskite crystals
    Yaffe, Omer; Guo, Yinsheng; Tan, Liang Z.; Egger, David A.; Hull, Trevor; Stoumpos, Constantinos C.; Zheng, Fan; Heinz, Tony F.; Kronik, Leeor; Kanatzidis, Mercouri G.; Owen, Jonathan S.; Rappe, Andrew M.; Pimenta, Marcos A.; Brus, Louis E.
    Physical Review Letters (2017), 118 (13), 136001/1-136001/6CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
    Hybrid lead-halide perovskites have emerged as an excellent class of photovoltaic materials. Recent reports suggest that the org. mol. cation is responsible for local polar fluctuations that inhibit carrier recombination. We combine low-frequency Raman scattering with first-principles mol. dynamics (MD) to study the fundamental nature of these local polar fluctuations. Our observations of a strong central peak in the cubic phase of both hybrid (CH3NH3PbBr3) and all-inorg. (CsPbBr3) leadhalide perovskites show that anharmonic, local polar fluctuations are intrinsic to the general lead-halide perovskite structure, and not unique to the dipolar org. cation. MD simulations indicate that head-tohead Cs motion coupled to Br face expansion, occurring on a few hundred femtosecond time scale, drives the local polar fluctuations in CsPbBr3.
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  17. 17
    Miyata, K.; Meggiolaro, D.; Trinh, M. T.; Joshi, P. P.; Mosconi, E.; Jones, S. C.; De Angelis, F.; Zhu, X.-Y. Large polarons in lead halide perovskites. Sci. Adv. 2017, 3, e1701217,  DOI: 10.1126/sciadv.1701217
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    17
    Large polarons in lead halide perovskites
    Miyata, Kiyoshi; Meggiolaro, Daniele; Trinh, M. Tuan; Joshi, Prakriti P.; Mosconi, Edoardo; Jones, Skyler C.; De Angelis, Filippo; Zhu, X.-Y.
    Science Advances (2017), 3 (8), e1701217/1-e1701217/9CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
    Lead halide perovskites show marked defect tolerance responsible for their excellent optoelectronic properties. These properties might be explained by the formation of large polarons, but how they are formed and whether org. cations are essential remain open questions. We provide a direct time domain view of large polaron formation in single-crystal lead bromide perovskites CH3NH3PbBr3 and CsPbBr3. We found that large polaron forms predominantly from the deformation of the PbBr3-_frameworks, irresp. of the cation type. The difference lies in the polaron formation time, which, in CH3NH3PbBr3 (0.3 ps), is less than half of that in CsPbBr3 (0.7 ps). First-principles calcns. confirm large polaron formation, identify the Pb-Br-Pb deformation modes as responsible, and explain quant. the rate difference between CH3NH3PbBr3 and CsPbBr3. The findings reveal the general advantage of the soft [PbX3]- sublattice in charge carrier protection and suggest that there is likely no mechanistic limitations in using all-inorg. or mixed-cation lead halide perovskites to overcome instability problems and to tune the balance between charge carrier protection and mobility.
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    Woodward, P. M. Octahedral tilting in perovskites. I. Geometrical considerations. Acta Crystallogr., Sect. B: Struct. Sci. 1997, 53, 32– 43,  DOI: 10.1107/S0108768196010713
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    18
    Octahedral tilting in perovskites. I. Geometrical considerations
    Woodward, Patrick M.
    Acta Crystallographica, Section B: Structural Science (1997), B53 (1), 32-43CODEN: ASBSDK; ISSN:0108-7681. (Munksgaard)
    The 23 Glazer tilt systems describing octahedral tilting in perovskites were studied. In tilt systems a+a+a-, a+b+b-, a+a+c-, a+b+c-, a0b+b- and a-b+c- it is not possible to link together a three-dimensional network of perfectly rigid octahedra. In these tilt systems small distortions of the octahedra must occur. The magnitude of the distortions in the a+a+a- and a0b+b- tilt systems are estd. A table of predicted space groups for ordered perovskites, A2MM'O6, for all 23 tilt systems is also given.
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    Howard, C. J.; Stokes, H. Group-theoretical analysis of octahedral tilting in perovskites. Acta Crystallogr., Sect. B: Struct. Sci. 1998, 54, 782– 789,  DOI: 10.1107/S0108768198004200
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    19
    Group theoretical analysis of octahedral tilting in perovskites
    Howard, Christopher J.; Stokes, Harold T.
    Acta Crystallographica, Section B: Structural Science (1998), B54 (6), 782-789CODEN: ASBSDK; ISSN:0108-7681. (Munksgaard International Publishers Ltd.)
    A group-theor. anal. is made of the structures derived from the aristotype cubic perovskite (Pm‾3m) by the simple tilting of rigid octahedral units. The tilting is mediated by the irreducible representations R4+ and M3+ or the two in combination. These result in 15 possible structures, compared with the 23 possibilities suggested previously by Glazer. The anal. makes the group-subgroup relations apparent.
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    Brown, I. D. The crystal structure of K2TeBr6. Can. J. Chem. 1964, 42, 2758– 2767,  DOI: 10.1139/v64-409
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    20
    The crystal structure of K2TeBr6
    Brown, I. D.
    Canadian Journal of Chemistry (1964), 42 (12), 2758-67CODEN: CJCHAG; ISSN:0008-4042.
    Crystals of K2TeBr6 are monoclinic, space group P21/n-C52h, with a 7.521, b 7.574, and c 10.730 A., β 89°40'. At. positions were found by 3-dimensional x-ray diffraction anal. (least sqs. R = 0.12). The crystals possess a K2PtCl6 structure which is distorted to allow a more efficient packing of the comparatively large anions than is possible with the undistorted cubic form. The stereochem. of the octahedral TeBr6- ion (Te-Br = 2.71 A.) is discussed.
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  21. 21
    Cai, Y.; Xie, W.; Ding, H.; Chen, Y.; Thirumal, K.; Wong, L. H.; Mathews, N.; Mhaisalkar, S. G.; Sherburne, M.; Asta, M. Computational Study of Halide Perovskite-Derived A2BX6 Inorganic Compounds: Chemical Trends in Electronic Structure and Structural Stability. Chem. Mater. 2017, 29, 7740– 7749,  DOI: 10.1021/acs.chemmater.7b02013
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    21
    Computational Study of Halide Perovskite-Derived A2BX6 Inorganic Compounds: Chemical Trends in Electronic Structure and Structural Stability
    Cai, Yao; Xie, Wei; Ding, Hong; Chen, Yan; Thirumal, Krishnamoorthy; Wong, Lydia H.; Mathews, Nripan; Mhaisalkar, Subodh G.; Sherburne, Matthew; Asta, Mark
    Chemistry of Materials (2017), 29 (18), 7740-7749CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    The electronic structure and energetic stability of A2BX6 halide compds. with the cubic and tetragonal variants of the perovskite-derived K2PtCl6 prototype structure are investigated computationally within the frameworks of d.-functional-theory (DFT) and hybrid (HSE06) functionals. The HSE06 calcns. are undertaken for seven known A2BX6 compds. with A = K, Rb, and Cs; and B = Sn, Pd, Pt, Te, and X = I. Trends in band gaps and energetic stability are identified, which are explored further employing DFT calcns. over a larger range of chemistries, characterized by A = K, Rb, Cs, B = Si, Ge, Sn, Pb, Ni, Pd, Pt, Se, and Te; and X = Cl, Br, I. For the systems investigated in this work, the band gap increases from iodide to bromide to chloride. Further, variations in the A site cation influences the band gap as well as the preferred degree of tetragonal distortion. Smaller A site cations such as K and Rb favor tetragonal structural distortions, resulting in a slightly larger band gap. For variations in the B site in the (Ni, Pd, Pt) group and the (Se, Te) group, the band gap increases with increasing cation size. However, no obsd. chem. trend with respect to cation size for band gap was found for the (Si, Sn, Ge, Pb) group. The findings in this work provide guidelines for the design of halide A2BX6 compds. for potential photovoltaic applications.
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  22. 22
    Xiao, Z.; Zhou, Y.; Hosono, H.; Kamiya, T. Intrinsic Defects in a Photovoltaic Perovskite Variant Cs2SnI6. Phys. Chem. Chem. Phys. 2015, 17, 18900– 18903,  DOI: 10.1039/C5CP03102H
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    Intrinsic defects in a photovoltaic perovskite variant Cs2SnI6
    Xiao, Zewen; Zhou, Yuanyuan; Hosono, Hideo; Kamiya, Toshio
    Physical Chemistry Chemical Physics (2015), 17 (29), 18900-18903CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    Cs2SnI6, a rarely studied perovskite variant material, is recently gaining a lot of interest in the field of photovoltaics owing to its nontoxicity, air-stability and promising photovoltaic properties. In this work, we report intrinsic defects in Cs2SnI6 using first-principles d. functional theory calcns. It is revealed that iodine vacancy and tin interstitial are the dominant defects that are responsible for the intrinsic n-type conduction in Cs2SnI6. Tin vacancy has a very high formation energy (>3.6 eV) due to the strong covalency in the Sn-I bonds and is hardly generated for p-type doping. All the dominant defects in Cs2SnI6 have deep transition levels in the band gap. It is suggested that the formation of deep defects can be suppressed significantly by employing an I-rich synthesis condition, which is inevitable for photovoltaic and other semiconductor applications.
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  23. 23
    Maughan, A. E.; Ganose, A. M.; Bordelon, M. M.; Miller, E. M.; Scanlon, D. O.; Neilson, J. R. Defect tolerance to intolerance in the vacancy-ordered double perovskite semiconductors Cs2SnI6 and Cs2TeI6. J. Am. Chem. Soc. 2016, 138, 8453– 8464,  DOI: 10.1021/jacs.6b03207
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    23
    Defect Tolerance to Intolerance in the Vacancy-Ordered Double Perovskite Semiconductors Cs2SnI6 and Cs2TeI6
    Maughan, Annalise E.; Ganose, Alex M.; Bordelon, Mitchell M.; Miller, Elisa M.; Scanlon, David O.; Neilson, James R.
    Journal of the American Chemical Society (2016), 138 (27), 8453-8464CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    Vacancy-ordered double perovskites of the general formula A2BX6 are a family of perovskite derivs. composed of a face-centered lattice of nearly isolated [BX6] units with A-site cations occupying the cuboctahedral voids. Despite the presence of isolated octahedral units, the close-packed iodide lattice provides significant electronic dispersion, such that Cs2SnI6 has recently been explored for applications in photovoltaic devices. To elucidate the structure-property relationships of these materials, we have synthesized solid-soln. Cs2Sn1-xTexI6. However, even though tellurium substitution increases electronic dispersion via closer I-I contact distances, the substitution exptl. yields insulating behavior from a significant decrease in carrier concn. and mobility. D. functional calcns. of native defects in Cs2SnI6 reveal that iodine vacancies exhibit a low enthalpy of formation, and that the defect energy level is a shallow donor to the conduction band rendering the material tolerant to these defect states. The increased covalency of Te-I bonding renders the formation of iodine vacancy states unfavorable and is responsible for the redn. in cond. upon Te substitution. Addnl., Cs2TeI6 is intolerant to the formation of these defects, because the defect level occurs deep within the band gap and thus localizes potential mobile charge carriers. In these vacancy-ordered double perovskites, the close-packed lattice of iodine provides significant electronic dispersion, while the interaction of the B- and X-site ions dictates the properties as they pertain to electronic structure and defect tolerance. This simplified perspective based on extensive exptl. and theor. anal. provides a platform from which to understand structure-property relationships in functional perovskite halides.
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  24. 24
    Saparov, B.; Sun, J.-P.; Meng, W.; Xiao, Z.; Duan, H.-S.; Gunawan, O.; Shin, D.; Hill, I. G.; Yan, Y.; Mitzi, D. B. Thin-film deposition and characterization of a Sn-deficient perovskite derivative Cs2SnI6. Chem. Mater. 2016, 28, 2315– 2322,  DOI: 10.1021/acs.chemmater.6b00433
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    24
    Thin-Film Deposition and Characterization of a Sn-Deficient Perovskite Derivative Cs2SnI6
    Saparov, Bayrammurad; Sun, Jon-Paul; Meng, Weiwei; Xiao, Zewen; Duan, Hsin-Sheng; Gunawan, Oki; Shin, Donghyeop; Hill, Ian G.; Yan, Yanfa; Mitzi, David B.
    Chemistry of Materials (2016), 28 (7), 2315-2322CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    In this work, we describe details of a two-step deposition approach that enables the prepn. of continuous and well-structured thin films of Cs2SnI6, which is a one-half Sn-deficient 0-D perovskite deriv. (i.e., the compd. can also be written as CsSn0.5I3, with a structure consisting of isolated SnI64- octahedra). The films were characterized using powder X-ray diffraction (PXRD), SEM (SEM), thermogravimetric anal. (TGA), UV-vis spectroscopy, photoluminescence (PL), photoelectron spectroscopy (UPS, IPES, XPS), and Hall effect measurements. UV-vis and PL measurements indicate that the obtained Cs2SnI6 film is a semiconductor with a band gap of 1.6 eV. This band gap was further confirmed by the UPS and IPES spectra, which were well reproduced by the calcd. d. of states with the HSE hybrid functional. The Cs2SnI6 films exhibited n-type conduction with a carrier d. of 6(1) × 1016 cm-3 and mobility of 2.9(3) cm2/V·s. While the computationally derived band structure for Cs2SnI6 shows significant dispersion along several directions in the Brillouin zone near the band edges, the valence band is relatively flat along the Γ-X direction, indicative of a more limited hole minority carrier mobility compared to analogous values for the electrons. The ionization potential (IP) and electron affinity (EA) were detd. to be 6.4 and 4.8 eV, resp. The Cs2SnI6 films show some enhanced stability under ambient air, compared to methylammonium lead(II) iodide perovskite films stored under similar conditions; however, the films do decomp. slowly, yielding a CsI impurity. These findings are discussed in the context of suitability of Cs2SnI6 for photovoltaic and related optoelectronic applications.
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    Maughan, A. E.; Ganose, A. M.; Candia, A. M.; Granger, J. T.; Scanlon, D. O.; Neilson, J. R. Anharmonicity and Octahedral Tilting in Hybrid Vacancy-Ordered Double Perovskites. Chem. Mater. 2018, 30, 472– 483,  DOI: 10.1021/acs.chemmater.7b04516
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    25
    Anharmonicity and Octahedral Tilting in Hybrid Vacancy-Ordered Double Perovskites
    Maughan, Annalise E.; Ganose, Alex M.; Candia, Andrew M.; Granger, Juliette T.; Scanlon, David O.; Neilson, James R.
    Chemistry of Materials (2018), 30 (2), 472-483CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    The advantageous performance of hybrid org.-inorg. perovskite halide semiconductors in optoelectronic applications motivates studies of their fundamental crystal chem. In particular, recent studies have sought to understand how dipolar, dynamic, and org. cations such as methylammonium (CH3NH3+) and formamidinium (CH(NH2)2+) affect phys. properties such as light absorption and charge transport. To probe the influence of org.-inorg. coupling on charge transport, we prepd. the series of vacancy-ordered double perovskite derivs. A2SnI6, where A = Cs+, CH3NH3+, and CH(NH2)2+. Despite nearly identical cubic structures by powder X-ray diffraction, replacement of Cs+ with CH3NH3+ or CH(NH2)2+ reduces cond. through a redn. in both carrier concn. and carrier mobility. We attribute the trends in electronic behavior to anharmonic lattice dynamics from the formation of hydrogen bonds that yield coupled org.-inorg. dynamics. This anharmonicity manifests as asymmetry of the interoctahedral I-I pair correlations in the X-ray pair distribution function of the hybrid compds., which can be modeled by large atomistic ensembles with random rotations of rigid [SnI6] octahedral units. The presence of soft, anharmonic lattice dynamics holds implications for electron-phonon interactions, as supported by calcn. of electron-phonon coupling strength that indicates the formation of more tightly bound polarons and reduced electron mobilities with increasing cation size. By exploiting the relatively decoupled nature of the octahedral units in these defect-ordered perovskite variants, we interrogated the impact of org.-inorg. coupling and lattice anharmonicity on the charge transport behavior of hybrid perovskite halide semiconductors.
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    Lee, B.; Stoumpos, C. C.; Zhou, N.; Hao, F.; Malliakas, C.; Yeh, C.-Y.; Marks, T. J.; Kanatzidis, M. G.; Chang, R. P. Air-stable molecular semiconducting iodosalts for solar cell applications: Cs2SnI6 as a hole conductor. J. Am. Chem. Soc. 2014, 136, 15379– 15385,  DOI: 10.1021/ja508464w
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    Air-Stable Molecular Semiconducting Iodosalts for Solar Cell Applications: Cs2SnI6 as a Hole Conductor
    Lee, Byunghong; Stoumpos, Constantinos C.; Zhou, Nanjia; Hao, Feng; Malliakas, Christos; Yeh, Chen-Yu; Marks, Tobin J.; Kanatzidis, Mercouri G.; Chang, Robert P. H.
    Journal of the American Chemical Society (2014), 136 (43), 15379-15385CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    The authors introduce a new class of mol. iodosalt compds. for application in next-generation solar cells. Unlike tin-based perovskite compds. CsSnI3 and MeNH3SnI3, which have Sn in the 2 + oxidn. state and must be handled in an inert atm. when fabricating solar cells, the Sn in the mol. iodosalt compds. is in the 4+ oxidn. state, making them stable in air and moisture. As an example, using Cs2SnI6 as a hole transporter, the authors can successfully fabricate in air a solid-state dye-sensitized solar cell (DSSC) with a mesoporous TiO2 film. Doping Cs2SnI6 with additives helps to reduce the internal device resistance, improving cell efficiency. In this way, a Z 907 DSSC delivers 4.7% of energy conversion efficiency. By using a more efficient mixt. of porphyrin dyes, an efficiency near 8% with photon confinement was achieved. This represents a significant step toward the realization of low-cost, stable, lead-free, and environmentally benign next-generation solid-state solar cells.
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    Wang, J.; Toby, B. H.; Lee, P. L.; Ribaud, L.; Antao, S. M.; Kurtz, C.; Ramanathan, M.; Von Dreele, R. B.; Beno, M. A. A dedicated powder diffraction beamline at the Advanced Photon Source: Commissioning and early operational results. Rev. Sci. Instrum. 2008, 79, 085105,  DOI: 10.1063/1.2969260
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    A dedicated powder diffraction beam-line at the Advanced Photon Source: Commissioning and early operational results
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    Review of Scientific Instruments (2008), 79 (8), 085105/1-085105/7CODEN: RSINAK; ISSN:0034-6748. (American Institute of Physics)
    A new dedicated high-resoln. high-throughput powder diffraction beam-line was built, fully commissioned, and opened to general users at the Advanced Photon Source. The optical design and commissioning results are presented. Beam-line performance was examd. using a mixt. of the NIST Si and Al2O3 std. ref. materials, as well as the LaB6 line-shape std. Instrumental resoln. ≤1.7 × 10-4 (ΔQ/Q) was obsd. (c) 2008 American Institute of Physics.
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    Toby, B. H. EXPGUI, a graphical user interface for GSAS. J. Appl. Crystallogr. 2001, 34, 210– 213,  DOI: 10.1107/S0021889801002242
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    Journal of Applied Crystallography (2001), 34 (2), 210-213CODEN: JACGAR; ISSN:0021-8898. (Munksgaard International Publishers Ltd.)
    A description and justification of the EXPGUI program is presented. This program implements a graphical user interface and shell for the GSAS (Generalized Structure and Anal. Software) single-crystal and Rietveld package. Use of the Tcl/Tk scripting language allows EXPGUI to be platform independent. Also included is a synopsis of how the program is implemented.
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    Momma, Koichi; Izumi, Fujio
    Journal of Applied Crystallography (2011), 44 (6), 1272-1276CODEN: JACGAR; ISSN:0021-8898. (International Union of Crystallography)
    VESTA is a 3D visualization system for crystallog. studies and electronic state calcns. It was upgraded to the latest version, VESTA 3, implementing new features including drawing the external morphpol. of crysals; superimposing multiple structural models, volumetric data and crystal faces; calcn. of electron and nuclear densities from structure parameters; calcn. of Patterson functions from the structure parameters or volumetric data; integration of electron and nuclear densities by Voronoi tessellation; visualization of isosurfaces with multiple levels, detn. of the best plane for selected atoms; an extended bond-search algorithm to enable more sophisticated searches in complex mols. and cage-like structures; undo and redo is graphical user interface operations; and significant performance improvements in rendering isosurfaces and calcg. slices.
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    Journal of Applied Crystallography (2003), 36 (6), 1342-1347CODEN: JACGAR; ISSN:0021-8898. (Blackwell Publishing Ltd.)
    An image-plate (IP) detector coupled with high-energy synchrotron radiation was used for at. pair distribution function (PDF) anal., with high probed momentum transfer Qmax ≤ 28.5 Å-1, from cryst. materials. Materials with different structural complexities were measured to test the validity of the quant. data anal. Exptl. results are presented for cryst. Ni, cryst. α-AlF3, and the layered Aurivillius type oxides α-Bi4V2O11 and γ-Bi4V1.7Ti0.3O10.85. Overall, the diffraction patterns show good counting statistics, with measuring time from one to tens of seconds. The PDFs obtained are of high quality. Structures may be refined from these PDFs, and the structural models are consistent with the published literature. Data sets from similar samples are highly reproducible.
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    Farrow, C.; Juhas, P.; Liu, J.; Bryndin, D.; Božin, E.; Bloch, J.; Proffen, T.; Billinge, S. PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals. J. Phys.: Condens. Matter 2007, 19, 335219,  DOI: 10.1088/0953-8984/19/33/335219
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    Farrow, C. L.; Juhas, P.; Liu, J. W.; Bryndin, D.; Bozin, E. S.; Bloch, J.; Proffen, Th; Billinge, S. J. L.
    Journal of Physics: Condensed Matter (2007), 19 (33), 335219/1-335219/7CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
    PDFfit2 is a program as well as a library for real-space refinement of crystal structures. It is capable of fitting a theor. three-dimensional (3D) structure to at. pair distribution function data and is ideal for nanoscale investigations. The fit system accounts for lattice consts., at. positions and anisotropic at. displacement parameters, correlated at. motion, and exptl. factors that may affect the data. The at. positions and thermal coeffs. can be constrained to follow the symmetry requirements of an arbitrary space group. The PDFfit2 engine is written in C++ and is accessible via Python, allowing it to inter-operate with other Python programs. PDFgui is a graphical interface built on the PDFfit2 engine. PDFgui organizes fits and simplifies many data anal. tasks, such as configuring and plotting multiple fits. PDFfit2 and PDFgui are freely available via the Internet.
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    Nuclear Instruments & Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms (2012), 287 (), 68-75CODEN: NIMBEU; ISSN:0168-583X. (Elsevier B.V.)
    The Nanoscale Ordered MAterials Diffractometer (NOMAD) is neutron time-of-flight diffractometer designed to det. pair distribution functions of a wide range of materials ranging from short range ordered liqs. to long range ordered crystals. Due to a large neutron flux provided by the Spallation Neutron Source SNS and a large detector coverage neutron count-rates exceed comparable instruments by one to two orders of magnitude. This is achieved while maintaining a relatively high momentum transfer resoln. of a δQ/Q∼0.8% FWHM (typical), and a possible δQ/Qof0.24% FWHM (best). The real space resoln. is related to the max. momentum transfer; a max. momentum transfer of 50 Å-1 can be obtained routinely and the max. momentum transfer given by the detector configuration and the incident neutron spectrum is 125 Å-1. High stability of the source and the detector allow small contrast isotope expts. to be performed. A detailed description of the instrument is given and the results of expts. with std. samples are discussed.
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    Physical Review B: Condensed Matter and Materials Physics (1994), 49 (20), 14251-69CODEN: PRBMDO; ISSN:0163-1829.
    The authors present ab initio quantum-mech. mol.-dynamics simulations of the liq.-metal-amorphous-semiconductor transition in Ge. The simulations are based on (a) finite-temp. d.-functional theory of the 1-electron states, (b) exact energy minimization and hence calcn. of the exact Hellmann-Feynman forces after each mol.-dynamics step using preconditioned conjugate-gradient techniques, (c) accurate nonlocal pseudopotentials, and (d) Nose' dynamics for generating a canonical ensemble. This method gives perfect control of the adiabaticity of the electron-ion ensemble and allows the authors to perform simulations over >30 ps. The computer-generated ensemble describes the structural, dynamic, and electronic properties of liq. and amorphous Ge in very good agreement with expt.. The simulation allows the authors to study in detail the changes in the structure-property relation through the metal-semiconductor transition. The authors report a detailed anal. of the local structural properties and their changes induced by an annealing process. The geometrical, bounding, and spectral properties of defects in the disordered tetrahedral network are studied and compared with expt.
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    Kresse, G.; Furthmüller, J. Efficiency of Ab Initio Total Energy Calculations for Metals and Semiconductors Using a Plane Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15– 50,  DOI: 10.1016/0927-0256(96)00008-0
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    Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
    Kresse, G.; Furthmuller, J.
    Computational Materials Science (1996), 6 (1), 15-50CODEN: CMMSEM; ISSN:0927-0256. (Elsevier)
    The authors present a detailed description and comparison of algorithms for performing ab-initio quantum-mech. calcns. using pseudopotentials and a plane-wave basis set. The authors will discuss: (a) partial occupancies within the framework of the linear tetrahedron method and the finite temp. d.-functional theory, (b) iterative methods for the diagonalization of the Kohn-Sham Hamiltonian and a discussion of an efficient iterative method based on the ideas of Pulay's residual minimization, which is close to an order N2atoms scaling even for relatively large systems, (c) efficient Broyden-like and Pulay-like mixing methods for the charge d. including a new special preconditioning optimized for a plane-wave basis set, (d) conjugate gradient methods for minimizing the electronic free energy with respect to all degrees of freedom simultaneously. The authors have implemented these algorithms within a powerful package called VAMP (Vienna ab-initio mol.-dynamics package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semi-conducting surfaces, phonons in simple metals, transition metals and semiconductors) and turned out to be very reliable.
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    Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169– 11186,  DOI: 10.1103/PhysRevB.54.11169
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    Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
    Kresse, G.; Furthmueller, J.
    Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    The authors present an efficient scheme for calcg. the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set. In the first part the application of Pulay's DIIS method (direct inversion in the iterative subspace) to the iterative diagonalization of large matrixes will be discussed. This approach is stable, reliable, and minimizes the no. of order Natoms3 operations. In the second part, we will discuss an efficient mixing scheme also based on Pulay's scheme. A special "metric" and a special "preconditioning" optimized for a plane-wave basis set will be introduced. Scaling of the method will be discussed in detail for non-self-consistent and self-consistent calcns. It will be shown that the no. of iterations required to obtain a specific precision is almost independent of the system size. Altogether an order Natoms2 scaling is found for systems contg. up to 1000 electrons. If we take into account that the no. of k points can be decreased linearly with the system size, the overall scaling can approach Natoms. They have implemented these algorithms within a powerful package called VASP (Vienna ab initio simulation package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semiconducting surfaces, phonons in simple metals, transition metals, and semiconductors) and turned out to be very reliable.
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    Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758– 1775,  DOI: 10.1103/PhysRevB.59.1758
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    From ultrasoft pseudopotentials to the projector augmented-wave method
    Kresse, G.; Joubert, D.
    Physical Review B: Condensed Matter and Materials Physics (1999), 59 (3), 1758-1775CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    The formal relationship between ultrasoft (US) Vanderbilt-type pseudopotentials and Blochl's projector augmented wave (PAW) method is derived. The total energy functional for US pseudopotentials can be obtained by linearization of two terms in a slightly modified PAW total energy functional. The Hamilton operator, the forces, and the stress tensor are derived for this modified PAW functional. A simple way to implement the PAW method in existing plane-wave codes supporting US pseudopotentials is pointed out. In addn., crit. tests are presented to compare the accuracy and efficiency of the PAW and the US pseudopotential method with relaxed-core all-electron methods. These tests include small mols. (H2, H2O, Li2, N2, F2, BF3, SiF4) and several bulk systems (diamond, Si, V, Li, Ca, CaF2, Fe, Co, Ni). Particular attention is paid to the bulk properties and magnetic energies of Fe, Co, and Ni.
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    Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406,  DOI: 10.1103/PhysRevLett.100.136406
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    Physical Review Letters (2008), 100 (13), 136406/1-136406/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Popular modern generalized gradient approxns. are biased toward the description of free-atom energies. Restoration of the first-principles gradient expansion for exchange over a wide range of d. gradients eliminates this bias. We introduce a revised Perdew-Burke-Ernzerhof generalized gradient approxn. that improves equil. properties of densely packed solids and their surfaces.
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    Generalized gradient approximation made simple
    Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
    Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106,  DOI: 10.1063/1.2404663
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    Influence of the exchange screening parameter on the performance of screened hybrid functionals
    Krukau, Aliaksandr V.; Vydrov, Oleg A.; Izmaylov, Artur F.; Scuseria, Gustavo E.
    Journal of Chemical Physics (2006), 125 (22), 224106/1-224106/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    This work reexamines the effect of the exchange screening parameter ω on the performance of the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional. We show that variation of the screening parameter influences solid band gaps the most. Other properties such as mol. thermochem. or lattice consts. of solids change little with ω. We recommend a new version of HSE with the screening parameter ω = 0.11 bohr-1 for further use. Compared to the original implementation, the new parametrization yields better thermochem. results and preserves the good accuracy for band gaps and lattice consts. in solids.
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    Ganose, A. M.; Savory, C. N.; Scanlon, D. O. Electronic and defect properties of (CH3NH3)2Pb(SCN)2I2 analogues for photovoltaic applications. J. Mater. Chem. A 2017, 5, 7845– 7853,  DOI: 10.1039/C7TA01688C
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    Electronic and defect properties of (CH3NH3)2Pb(SCN)2I2 analogues for photovoltaic applications
    Ganose, Alex M.; Savory, Christopher N.; Scanlon, David O.
    Journal of Materials Chemistry A: Materials for Energy and Sustainability (2017), 5 (17), 7845-7853CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
    In the past 5 yr, hybrid halide perovskites have emerged as a class of highly efficient photovoltaic (PV) absorbers, with excellent electronic properties and low cost synthesis routes. Unfortunately, despite much research effort, their long-term stability is poor and presents a major obstacle toward commercialisation. The layered perovskite (CH3NH3)2Pb(SCN)2I2 (MAPSI) has recently been identified as a promising PV candidate material due to its enhanced stability and favorable electronic properties. We demonstrate, using relativistic hybrid d. functional theory, that the MAPSI structural motif can be extended to include a range of other metals, halides and even pseudohalides. In this way, the electronic structure of MAPSI can be tuned without affecting its stability with respect towards decompn. These results indicate the possibility of Pb-free MAPSI analogs, with suitable properties for photovoltaic top cells in tandem devices.
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    Biswas, D.; Ganose, A. M.; Yano, R.; Riley, J.; Bawden, L.; Clark, O.; Feng, J.; Collins-Mcintyre, L.; Sajjad, M.; Meevasana, W.; Hoesch, M.; Rault, J.; Sasagawa, T.; Scanlon, D.; King, P.; Kim, T. K. Narrow-band anisotropic electronic structure of ReS2. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 085205,  DOI: 10.1103/PhysRevB.96.085205
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    Narrow-band anisotropic electronic structure of ReS2
    Biswas, D.; Ganose, Alex M.; Yano, R.; Riley, J. M.; Bawden, L.; Clark, O. J.; Feng, J.; Collins-Mcintyre, L.; Sajjad, M. T.; Meevasana, W.; Kim, T. K.; Hoesch, M.; Rault, J. E.; Sasagawa, T.; Scanlon, David O.; King, P. D. C.
    Physical Review B (2017), 96 (8), 085205/1-085205/7CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    We have used angle-resolved photoemission spectroscopy to investigate the band structure of ReS2, a transitionmetal dichalcogenide semiconductor with a distorted 1T crystal structure. We find a large no. of narrow valence bands, which we attribute to the combined influence of structural distortion and spin-orbit coupling. We further show how this leads to a strong in-plane anisotropy of the electronic structure, with quasi-one-dimensional bands reflecting predominant hopping along zigzag Re chains. We find that this does not persist up to the top of the valence band, where a more three-dimensional character is recovered with the fundamental band gap located away from the Brillouin zone center along kz. These expts. are in good agreement with our d.-functional theory calcns., shedding light on the bulk electronic structure of ReS2, and how it can be expected to evolve when thinned to a single layer.
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    Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 2003, 118, 8207– 8215,  DOI: 10.1063/1.1564060
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    Hybrid functionals based on a screened Coulomb potential
    Heyd, Jochen; Scuseria, Gustavo E.; Ernzerhof, Matthias
    Journal of Chemical Physics (2003), 118 (18), 8207-8215CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Hybrid d. functionals are very successful in describing a wide range of mol. properties accurately. In large mols. and solids, however, calcg. the exact (Hartree-Fock) exchange is computationally expensive, esp. for systems with metallic characteristics. In the present work, we develop a new hybrid d. functional based on a screened Coulomb potential for the exchange interaction which circumvents this bottleneck. The results obtained for structural and thermodn. properties of mols. are comparable in quality to the most widely used hybrid functionals. In addn., we present results of periodic boundary condition calcns. for both semiconducting and metallic single wall carbon nanotubes. Using a screened Coulomb potential for Hartree-Fock exchange enables fast and accurate hybrid calcns., even of usually difficult metallic systems. The high accuracy of the new screened Coulomb potential hybrid, combined with its computational advantages, makes it widely applicable to large mols. and periodic systems.
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    Hybrid functionals based on a screened Coulomb potential. [Erratum to document cited in CA139:042043]
    Heyd, Jochen; Scuseria, Gustavo E.; Ernzerhof, Matthias
    Journal of Chemical Physics (2006), 124 (21), 219906/1CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The screening parameter quoted in this paper (ω=0.15) is not the one that was actually used in the code. In order to reproduce our results, two ω values are needed: ω =0.15 2≈0.106 for Hartree-Fock and ω=0.15x21/3 ≈0.189 for the PBE part. This erratum applies to all of our ωPBEh and HSE calcns. published to date. The authors emphasize that all the results obtained with this functional are reproducible using the two ω values quoted. The validity of all the published HSE values is not affected. Optimization of a single value of ω for a functional of the type reported in this paper will be presented elsewhere.
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    Hobbs, D.; Kresse, G.; Hafner, J. Fully unconstrained noncollinear magnetism within the projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 2000, 62, 11556– 11570,  DOI: 10.1103/PhysRevB.62.11556
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    Fully unconstrained noncollinear magnetism within the projector augmented-wave method
    Hobbs, D.; Kresse, G.; Hafner, J.
    Physical Review B: Condensed Matter and Materials Physics (2000), 62 (17), 11556-11570CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    Spin-polarized calcns. in solids were generally confined to a global quantization axis to simplify both the theor. model and its implementation in self-consistent codes. This approxn. is justified as many materials exhibit a collinear magnetic order. However, in recent years much interest was directed towards noncollinear magnetism in which the magnetization d. is a continuous vector variable of position. We develop the all-electron projector augmented-wave (PAW) method for noncollinear magnetic structures, based on a generalized local-spin-d. theory. The method allows both the at. and magnetic structures to relax simultaneously and self-consistently. The algorithms were implemented within a powerful package called VASP (Vienna ab initio simulation package), which was used successfully for a large variety of different systems such as cryst. and amorphous semiconductors, simple liqs., and transition metals. The approach was used to study small clusters of Fe and Cr; some of these clusters show noncollinear magnetic arrangements.
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    Ganose, A. M.; Butler, K. T.; Walsh, A.; Scanlon, D. O. Relativistic electronic structure and band alignment of BiSI and BiSeI: candidate photovoltaic materials. J. Mater. Chem. A 2016, 4, 2060– 2068,  DOI: 10.1039/C5TA09612J
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    Relativistic electronic structure and band alignment of BiSI and BiSeI: candidate photovoltaic materials
    Ganose, Alex M.; Butler, Keith T.; Walsh, Aron; Scanlon, David O.
    Journal of Materials Chemistry A: Materials for Energy and Sustainability (2016), 4 (6), 2060-2068CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
    Bi-based solar absorbers are of interest due to similarities in the chem. properties of Bi halides and the exceptionally efficient Pb halide hybrid perovskites. While they both experience the same beneficial relativistic effects acting to increase the width of the conduction band, Bi is non-toxic and non-bioaccumulating, meaning the impact of environmental contamination is greatly reduced. We use hybrid d. functional theory, with the addn. of spin orbit coupling, to examine 2 candidate bismuth contg. photovoltaic absorbers, BiSI and BiSeI, and show that they possess electronic structures suitable for photovoltaic applications. We calc. band alignments against commonly used hole transporting and buffer layers, which indicate band misalignments are likely to be the source of the poor efficiencies reported for devices contg. these materials. We suggest alternative device architectures expected to result in improved power conversion efficiencies.
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    Savory, C. N.; Ganose, A. M.; Scanlon, D. O. Exploring the PbS-Bi2S3 series for next generation energy conversion materials. Chem. Mater. 2017, 29, 5156– 5167,  DOI: 10.1021/acs.chemmater.7b00628
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    Exploring the PbS-Bi2S3 Series for Next Generation Energy Conversion Materials
    Savory, Christopher N.; Ganose, Alex M.; Scanlon, David O.
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    As photovoltaics become an ever more important part of the global energy economy, the search for inexpensive, earth-abundant solar absorbers has grown rapidly. The binary compds. PbS and Bi2S3 have both seen success in previous photovoltaic studies; however, bulk PbS has a small band gap, restricting its efficiency, and Bi2S3, while strongly absorbing, can be limited by its layered structure. The mixed PbS-Bi2S3 series has previously been the focus of mostly structural studies, so in this article, we examine the electronic structure of the known members of this series using hybrid d. functional theory. We find that the lead bismuth sulfides are able to retain optimal properties, such as low carrier effective masses and strong absorption, from both parent phases, with band gaps between 0.25 and 1.32 eV. PbBi2S4 emerges from our computational screening as a possible earth-abundant solar absorber, with a predicted max. efficiency of 26% at a film thickness of 0.2 μm and with the retention of the three-dimensional connectivity of lead and bismuth polyhedra.
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    Baroni, Stefano; De Gironcoli, Stefano; Dal Corso, Andrea; Giannozzi, Paolo
    Reviews of Modern Physics (2001), 73 (2), 515-562CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
    This article reviews with many refs. the current status of lattice-dynamical calcns. in crystals, using d.-functional perturbation theory, with emphasis on the plane-wave pseudopotential method. Several specialized topics are treated, including the implementation for metals, the calcn. of the response to macroscopic elec. fields and their relevance to long-wavelength vibrations in polar materials, the response to strain deformations, and higher-order responses. The success of this methodol. is demonstrated with a no. of applications existing in the literature.
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    Gajdoš, M.; Hummer, K.; Kresse, G.; Furthmüller, J.; Bechstedt, F. Linear optical properties in the projector-augmented wave methodology. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 045112,  DOI: 10.1103/PhysRevB.73.045112
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    Linear optical properties in the projector-augmented wave methodology
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    In this work we derive closed expressions for the head of the frequency-dependent microscopic polarizability matrix in the projector-augmented wave (PAW) methodol. Contrary to previous applications, the longitudinal expression is utilized, resulting in dielec. properties that are largely independent of the applied potentials. The improved accuracy of the present approach is demonstrated by comparing the longitudinal and transversal expressions of the polarizability matrix for a no. of cubic semiconductors and one insulator, i.e., Si, SiC, AlP, GaAs, and diamond (C), resp. The methodol. is readily extendable to more complicated nonlocal Hamiltonians or to the calcn. of the macroscopic dielec. matrix including local field effects in the random phase or d. functional approxn., which is demonstrated for the previously mentioned model systems. Furthermore, d. functional perturbation theory is extended to the PAW method, and the resp. results are compared to those obtained by summation over the conduction band states.
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    Frost, J. M. Calculating polaron mobility in halide perovskites. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 195202,  DOI: 10.1103/PhysRevB.96.195202
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    Calculating polaron mobility in halide perovskites
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  • Abstract

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    Figure 1

    Figure 1. Crystal structures of Rb2SnI6 at T = 295 K and T = 100 K. In (a) and (c), the structures are projected down the c-axis to highlight the octahedral tilting and rotation, while unit cell descriptions are shown in (b) and (d). Rubidium atoms are shown in pink, tin atoms in blue, and iodine atoms in purple.

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    Figure 2

    Figure 2. Rietveld refinements of the tetragonal and monoclinic structural models of Rb2SnI6 against high resolution synchrotron powder X-ray diffraction patterns collected at T = 295 K (a, b) and T = 100 K (c, d) on the high-resolution 11-BM diffractometer. Data are shown as black circles, the fit is the orange line, and the difference is shown in blue. Plots (b) and (d) highlight the splitting of the most intense reflection (202) at Q ∼ 1.9 Å–1 upon cooling through the phase transition.

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    Figure 3

    Figure 3. Temperature-dependent resistivity data of rubidium tin(IV) iodide collected using a 4-probe configuration with Pt wires and Ag paste.

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    Figure 4

    Figure 4. UV–visible diffuse reflectance spectrum for Rb2SnI6. The data were converted to pseudoabsorbance via the Kubelka–Munk transform, and the optical gap was determined by extrapolating the onset region to zero absorbance, as shown by the black fit. The pink line is the transformed data, the fit is the black line, and zero absorbance is shown as the gray dashed line. The intersection point of the fit and zero absorbance is shown by the black dot.

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    Figure 5

    Figure 5. X-ray pair distribution function of Rb2SnI6 fit to the (a) tetragonal (P4/mnc) and (b) monoclinic (P21/n) structural models. In (a), the XPDF is fit to the tetragonal (P4/mnc) model using anisotropic atomic displacement parameters for iodine and rubidium ions and in (b) the XPDF is fit to the monoclinic (P21/n) structural model using isotropic displacement parameters for all atoms. The thermal ellispoids in the corresponding structural models are shown at 95% probability.  

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    Figure 6

    Figure 6. (a) Temperature-dependent neutron pair distribution function analysis (nPDF) of Cs2SnI6 extracted from neutron total scattering collected on NOMAD. The x-axis is split to highlight the low-r pair correlations. Rietveld refinements of the corresponding neutron diffraction patterns collected from bank 2 (31° bank) of NOMAD are shown in (b). Both the nPDF and neutron diffraction data are modeled by the cubic (Fm3̅m) structural model at all temperatures. Black circles are the data, the orange lines are the fits, and the blue lines are difference curves. In (b), the gray tick marks indicate the positions of predicted reflections from the Fm3̅m structure.

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    Figure 7

    Figure 7. (a) Temperature-dependent neutron pair distribution function analysis (nPDF) of Rb2SnI6 extracted from neutron total scattering collected on NOMAD. The nPDFs are best modeled by the monoclinic (P21/n) structure at all temperatures. The x-axis is split to highlight the low-r pair correlations, and the data are offset vertically for clarity. Rietveld refinements of the neutron diffraction data from bank 2 (31° bank) of NOMAD are shown in (b). For T > 150 K, the data are modeled with the tetragonal (P4/mnc) structure. For T ≤ 150 K, the data are modeled by the monoclinic (P21/n) structure. Black circles are the data, the orange lines are the fits, and the blue lines are difference curves. In (b), the gray tick marks indicate the positions of predicted reflections from the P4/mnc and P21/n structures. The x-axis is split to highlight the lower-Q reflections, and the data are offset vertically for clarity.

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    Figure 8

    Figure 8. Band structures calculated using HSE06+SOC for the (a) P4/mnc and (b) P21/n phases of Rb2SnI6. The color of the band indicates the orbital contribution to that band, with Sn 5s, Sn 5p, and I 5p represented by red, green, and blue, respectively. The resulting color of the bands is obtained by mixing each color in proportion to the orbital contributions. The valence band maximum is set to 0 eV in all cases.

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    Figure 9

    Figure 9. Calculated band alignment (HSE06+SOC) of the P4/mnc and P21/n structured phases of Rb2SnI6 relative to those of Cs2SnI6.

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    Figure 10

    Figure 10. Phonon band structure of rubidium tin(IV) iodide in the tetragonal P4/mnc structure (a0a0c+). The lowest-frequency optical mode at 0.25 THz (1.03 meV, 8.27 cm–1) (denoted by the orange circle) corresponds to displacements of the rubidium and iodine atoms, as shown by the displacement vectors in the structural representations. Together, these displacements map to the octahedral tilting out of the ab plane coupled with Rb+ displacements observed in the lower-symmetry monoclinic structure, (a–a–c+).

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    Figure 11

    Figure 11. Experimentally (μe,expt.) and computationally derived Hellwarth (μeH) electron mobilities of the A2SnI6 vacancy-ordered double perovskites plotted as a function of perovskite tolerance factor. Experimental electron mobilities are shown as filled purple circles on the left axis, while the calculated Hellwarth electron mobilities are denoted by open orange squares on the right axis. Values for the carrier mobilities of Cs2SnI6, (CH3NH3)2SnI6, and (CH(NH2)2)2SnI6 are taken from a previous study. (25) For Rb2SnI6, the μeH value is calculated from the tetragonal structure.

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      Naturwissenschaften (1926), 14 (), 477-85CODEN: NATWAY; ISSN:0028-1042.
      Several laws are cited which govern the relationships between chem. compn. and cryst. form. The crystal structure of a substance is detd. by the size (Bragg's radius) and the polarizability of its components (atoms or atom groups). The at. radius depends on at. no. and "condition" (particularly degree of ionization) of the atom. The polarizability, a function of radius and charge of the atom or ion, increases with increasing radius, decreases with increasing positive charge (cf. Born and Heisenberg). High symmetry of the neighboring atoms tends to diminish it; the polarizing effect of the neighboring atoms depends again on their charge and radius. The distance of 2 polarizable structural units is generally less than the normal distance. The term contrapolarization is used for the dilating influence of strongly polarizing atoms (like Li+ or Zr++++) on such closed groups as ClO4-, CO3--, etc. This influence may even cause these groups to break up entirely. A rule is given for the occurrence of isomorphic crystals: the relative sizes and polarizabilities of the 2 substances must correspond, in order for the chem. formula to be analogous. Isomorphic mixts. will be possible if the radii of the corresponding atoms differ by less than about 15%. For antisomorphic crystals (positive and negative charges reversed) no mixts. are possible. Polymer-isomorphic mixts. occur when a multiple of the crystal unit of one substance is analogous to the unit of the other. If, as a result of thermodynamic influences, a substance ceases to be its own isomorph at different temps. (the polarizability is also a temp. function) polymorphism will appear; the substance undergoes a morphotropic change. Contrapolarization particularly is easily influenced by temp.; it causes such polymorphism as NH4NO3 in its different forms exhibits. At high temps. that cryst. form will be the most stable, which is obtained on substitution of the contrapolarizing cation by a lower homolog. Numerous examples are given.
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    4. 4
      Young, J.; Rondinelli, J. M. Octahedral Rotation Preferences in Perovskite Iodides and Bromides. J. Phys. Chem. Lett. 2016, 7, 918– 922,  DOI: 10.1021/acs.jpclett.6b00094
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      4
      Octahedral Rotation Preferences in Perovskite Iodides and Bromides
      Young, Joshua; Rondinelli, James M.
      Journal of Physical Chemistry Letters (2016), 7 (5), 918-922CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Phase transitions in ABX3 perovskites are often accompanied by rigid rotations of the corner-connected BX6 octahedral network. Although the mechanisms for the preferred rotation patterns of perovskite oxides are fairly well recognized, the same cannot be said of halide variants (i.e., X = Cl, Br, or I), several of which undergo an unusual displacive transition to a tetragonal phase exhibiting in-phase rotations about one axis (a0a0c+ in Glazer notation). To discern the chem. factors stabilizing this unique phase, we investigated a series of 12 perovskite bromides and iodides using d. functional theory calcns. and compared them with similar oxides. We find that in-phase tilting provides a better arrangement of the larger bromide and iodide anions, which minimizes the electrostatic interactions, improves the bond valence of the A-site cations, and enhances the covalency between the A-site metal and Br- or I- ions. The opposite effect is present in the oxides, with out-of-phase tilting maximizing these factors.
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    5. 5
      Bechtel, J. S.; Van der Ven, A. Octahedral tilting instabilities in inorganic halide perovskites. Phys. Rev. Mater. 2018, 2, 025401,  DOI: 10.1103/PhysRevMaterials.2.025401
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      5
      Octahedral tilting instabilities in inorganic halide perovskites
      Bechtel, Jonathon S.; Van der Ven, Anton
      Physical Review Materials (2018), 2 (2), 025401CODEN: PRMHBS; ISSN:2475-9953. (American Physical Society)
      Dynamic instabilities, stabilized by anharmonic interactions in cubic and tetragonal halide perovskites at high temp., play a role in the electronic structure and optoelectronic properties of halide perovskites. In particular, inorg. and hybrid perovskite materials undergo structural phase transitions assocd. with octahedral tilts of the metal-halide octahedra. We investigate the structural instabilities present in inorg. CsMX3 perovskites with Pb or Sn on the metal site and Br or I on the X site. Defining primary order parameters in terms of symmetry-adapted collective displacement modes and secondary order parameters in terms of symmetrized Hencky strain components, we unravel the coupling between octahedral tilt modes and macroscopic strains as well as the role of A-site displacements in perovskite phase stability. Symmetry-allowed secondary strain order parameters are enumerated for the 14 unique perovskite tilt systems. Using first-principles calcns. to explore the Born-Oppenheimer energy surface in terms of symmetrized order parameters, we find coupling between octahedral tilting and A-site displacements is necessary to stabilize Pnma ground states. Addnl., we show that the relative stability of an inorg. halide perovskite tilt system correlates with the vol. decrease from the high-symmetry cubic phase to the low-symmetry distorted phase.
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    6. 6
      Walsh, A. Principles of Chemical Bonding and Band Gap Engineering in Hybrid Organic–Inorganic Halide Perovskites. J. Phys. Chem. C 2015, 119, 5755– 5760,  DOI: 10.1021/jp512420b
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      6
      Principles of Chemical Bonding and Band Gap Engineering in Hybrid Organic-Inorganic Halide Perovskites
      Walsh, Aron
      Journal of Physical Chemistry C (2015), 119 (11), 5755-5760CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
      A review. The performance of solar cells based on hybrid halide perovskites has seen an unparalleled rate of progress, while our understanding of the underlying phys. chem. of these materials trails behind. Superficially, CH3NH3PbI3 is similar to other thin-film photovoltaic materials: a semiconductor with an optical band gap in the optimal region of the electromagnetic spectrum. Microscopically, the material is more unconventional. Progress in our understanding of the local and long-range chem. bonding of hybrid perovskites is discussed here, drawing from a series of computational studies involving electronic structure, mol. dynamics, and Monte Carlo simulation techniques. The orientational freedom of the dipolar methylammonium ion gives rise to temp.-dependent dielec. screening and the possibility for the formation of polar (ferroelec.) domains. The ability to independently substitute on the A, B, and X lattice sites provides the means to tune the optoelectronic properties. Finally, ten crit. challenges and opportunities for phys. chemists are highlighted.
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    7. 7
      Stoumpos, C. C.; Malliakas, C. D.; Kanatzidis, M. G. Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and Near-Infrared Photoluminescent Properties. Inorg. Chem. 2013, 52, 9019– 9038,  DOI: 10.1021/ic401215x
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      7
      Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and Near-Infrared Photoluminescent Properties
      Stoumpos, Constantinos C.; Malliakas, Christos D.; Kanatzidis, Mercouri G.
      Inorganic Chemistry (2013), 52 (15), 9019-9038CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      A broad org.-inorg. series of hybrid metal iodide perovskites AMI3, where A is the methylammonium (MeNH3+) or formamidinium (HC(NH2)2+) cation and M is Sn (1 and 2) or Pb (3 and 4) are reported. The compds. were prepd. through a variety of synthetic approaches, and the nature of the resulting materials is discussed in terms of their thermal stability and optical and electronic properties. The chem. and phys. properties of these materials strongly depend on the prepn. method. Single crystal x-ray diffraction anal. of 1-4 classifies the compds. in the perovskite structural family. Structural phase transitions were obsd. and studied by temp.-dependent single crystal x-ray diffraction in the 100-400 K range. The charge transport properties of the materials are discussed in conjunction with diffuse reflectance studies in the mid-IR region that display characteristic absorption features. Temp.-dependent studies show a strong dependence of the resistivity as a function of the crystal structure. Optical absorption measurements indicate that 1-4 behave as direct-gap semiconductors with energy band gaps distributed at 1.25-1.75 eV. The compds. exhibit an intense near-IR luminescence (PL) emission in the 700-1000 nm range (1.1-1.7 eV) at room temp. Solid solns. between the Sn and Pb compds. are readily accessible throughout the compn. range. The optical properties such as energy band gap, emission intensity, and wavelength can be readily controlled for the isostructural series of solid solns. MeNH3Sn1-xPbxI3 (5). The charge transport type in these materials was characterized by Seebeck coeff. and Hall-effect measurements. The compds. behave as p- or n-type semiconductors depending on the prepn. method. The samples with the lowest carrier concn. are prepd. from soln. and are n-type; p-type samples can be obtained through solid state reactions exposed in air in a controllable manner. In the case of Sn compds., there is a facile tendency toward oxidn. which causes the materials to be doped with Sn4+ and thus behave as p-type semiconductors displaying metal-like cond. The compds. appear to possess very high estd. electron and hole mobilities that exceed 2000 cm2/(V s) and 300 cm2/(V s), resp., as shown in the case of MeNH3SnI3 (1). The authors also compare the properties of the title hybrid materials with those of the all-inorg. CsSnI3 and CsPbI3 prepd. using identical synthetic methods.
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    8. 8
      Amat, A.; Mosconi, E.; Ronca, E.; Quarti, C.; Umari, P.; Nazeeruddin, M. K.; Grätzel, M.; De Angelis, F. Cation-induced band-gap tuning in organohalide perovskites: Interplay of spin–orbit coupling and octahedra tilting. Nano Lett. 2014, 14, 3608– 3616,  DOI: 10.1021/nl5012992
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      8
      Cation-Induced Band-Gap Tuning in Organohalide Perovskites: interplay of Spin-Orbit Coupling and Octahedra Tilting
      Amat, Anna; Mosconi, Edoardo; Ronca, Enrico; Quarti, Claudio; Umari, Paolo; Nazeeruddin, Md. K.; Gratzel, Michael; De Angelis, Filippo
      Nano Letters (2014), 14 (6), 3608-3616CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      Organohalide lead perovskites have revolutionized the scenario of emerging photovoltaic technologies. The prototype MAPbI3 perovskite (MA = MeNH3+) has dominated the field, despite only harvesting photons >750 nm (∼1.6 eV). Intensive research efforts are being devoted to find new perovskites with red shifted absorption onset, along with good charge transport properties. Recently, a new perovskite based on the formamidinium cation ((NH2)2CH+ = FA) showed potentially superior properties in terms of band gap and charge transport compared to MAPbI3. The results were interpreted in terms of the cation size, with the larger FA cation expectedly delivering reduced band-gaps in Pb-based perovskites. To provide a full understanding of the interplay among size, structure, and org./inorg. interactions in detg. the properties of APbI3 perovskites, in view of designing new materials and fully exploiting them for solar cells applications, the authors report a fully 1st-principles study on APbI3 perovskites with A = Cs+, MA, and FA. The results evidence that the tetragonal-to-quasi cubic structural evolution obsd. when moving from MA to FA is due to the interplay of size effects and enhanced H bonding between the FA cations and the inorg. matrix altering the covalent/ionic character of Pb-I bonds. Most notably, the obsd. cation-induced structural variability promotes markedly different electronic and optical properties in the MAPbI3 and FAPbI3 perovskites, mediated by the different spin-orbit coupling, leading to improved charge transport and red shifted absorption in FAPbI3 and in general in pseudocubic structures. The theor. model constitutes the basis for the rationale design of new and more efficient organohalide perovskites for solar cells applications.
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    9. 9
      Yang, Y.; Ostrowski, D. P.; France, R. M.; Zhu, K.; Van De Lagemaat, J.; Luther, J. M.; Beard, M. C. Observation of a hot-phonon bottleneck in lead-iodide perovskites. Nat. Photonics 2016, 10, 53– 59,  DOI: 10.1038/nphoton.2015.213
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      9
      Observation of a hot-phonon bottleneck in lead-iodide perovskites
      Yang, Ye; Ostrowski, David P.; France, Ryan M.; Zhu, Kai; van de Lagemaat, Jao; Luther, Joseph M.; Beard, Matthew C.
      Nature Photonics (2016), 10 (1), 53-59CODEN: NPAHBY; ISSN:1749-4885. (Nature Publishing Group)
      We study the carrier dynamics in planar Me ammonium lead iodide perovskite films using broadband transient absorption spectroscopy. We show that the sharp optical absorption onset is due to an exciton transition that is inhomogeneously broadened with a binding energy of 9 meV. We fully characterize the transient absorption spectrum by free-carrier-induced bleaching of the exciton transition, quasi-Fermi energy, carrier temp. and bandgap renormalization const. The photo-induced carrier temp. is extd. from the transient absorption spectra and monitored as a function of delay time for different excitation wavelengths and photon fluences. We find an efficient hot-phonon bottleneck that slows down cooling of hot carriers by three to four orders of magnitude in time above a crit. injection carrier d. of ∼5 × 1017 cm-3. Compared with mol. beam epitaxially grown GaAs, the crit. d. is an order of magnitude lower and the relaxation time is approx. three orders of magnitude longer.
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    10. 10
      Prasanna, R.; Gold-Parker, A.; Leijtens, T.; Conings, B.; Babayigit, A.; Boyen, H.-G.; Toney, M. F.; McGehee, M. D. Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics. J. Am. Chem. Soc. 2017, 139, 11117– 11124,  DOI: 10.1021/jacs.7b04981
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      10
      Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics
      Prasanna, Rohit; Gold-Parker, Aryeh; Leijtens, Tomas; Conings, Bert; Babayigit, Aslihan; Boyen, Hans-Gerd; Toney, Michael F.; McGehee, Michael D.
      Journal of the American Chemical Society (2017), 139 (32), 11117-11124CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Tin and lead iodide perovskite semiconductors of the compn. AMX3, where M is a metal and X is a halide, are leading candidates for high efficiency low cost tandem photovoltaics, in part because they have band gaps that can be tuned over a wide range by compositional substitution. We exptl. identify two competing mechanisms through which the A-site cation influences the band gap of 3D metal halide perovskites. Using a smaller A-site cation can distort the perovskite lattice in two distinct ways: by tilting the MX6 octahedra or by simply contracting the lattice isotropically. The former effect tends to raise the band gap, while the latter tends to decrease it. Lead iodide perovskites show an increase in band gap upon partial substitution of the larger formamidinium with the smaller cesium, due to octahedral tilting. Perovskites based on tin, which is slightly smaller than lead, show the opposite trend: they show no octahedral tilting upon Cs-substitution but only a contraction of the lattice, leading to progressive redn. of the band gap. We outline a strategy to systematically tune the band gap and valence and conduction band positions of metal halide perovskites through control of the cation compn. Using this strategy, we demonstrate solar cells that harvest light in the IR up to 1040 nm, reaching a stabilized power conversion efficiency of 17.8%, showing promise for improvements of the bottom cell of all-perovskite tandem solar cells. The mechanisms of cation-based band gap tuning we describe are broadly applicable to 3D metal halide perovskites and will be useful in further development of perovskite semiconductors for optoelectronic applications.
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    11. 11
      Linaburg, M. R.; McClure, E. T.; Majher, J. D.; Woodward, P. M. Cs1–xRbxPbCl3 and Cs1–xRbxPbBr3 solid solutions: Understanding octahedral tilting in lead halide perovskites. Chem. Mater. 2017, 29, 3507– 3514,  DOI: 10.1021/acs.chemmater.6b05372
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      11
      Cs1-xRbxPbCl3 and Cs1-xRbxPbBr3 Solid Solutions: Understanding Octahedral Tilting in Lead Halide Perovskites
      Linaburg, Matthew R.; McClure, Eric T.; Majher, Jackson D.; Woodward, Patrick M.
      Chemistry of Materials (2017), 29 (8), 3507-3514CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      The structures of the lead halide perovskites CsPbCl3 and CsPbBr3 were detd. 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 were detd. 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 av. structure.
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    12. 12
      Zhu, X.-Y.; Podzorov, V. Charge carriers in hybrid organic–inorganic lead halide perovskites might be protected as large polarons. J. Phys. Chem. Lett. 2015, 6, 4758– 4761,  DOI: 10.1021/acs.jpclett.5b02462
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      12
      Charge Carriers in Hybrid Organic-Inorganic Lead Halide Perovskites Might Be Protected as Large Polarons
      Zhu, X.-Y.; Podzorov, V.
      Journal of Physical Chemistry Letters (2015), 6 (23), 4758-4761CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Large polaron that provides the protection of charge carrier is explained in terms of carrier diffusion length,electron hole recombination rates, charge carrier and Hall mobility and low carrier scattering rate.
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    13. 13
      Zhu, H.; Miyata, K.; Fu, Y.; Wang, J.; Joshi, P. P.; Niesner, D.; Williams, K. W.; Jin, S.; Zhu, X.-Y. Screening in crystalline liquids protects energetic carriers in hybrid perovskites. Science 2016, 353, 1409– 1413,  DOI: 10.1126/science.aaf9570
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      13
      Screening in crystalline liquids protects energetic carriers in hybrid perovskites
      Zhu, Haiming; Miyata, Kiyoshi; Fu, Yongping; Wang, Jue; Joshi, Prakriti P.; Niesner, Daniel; Williams, Kristopher W.; Jin, Song; Zhu, X.-Y.
      Science (Washington, DC, United States) (2016), 353 (6306), 1409-1413CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
      Hybrid lead halide perovskites exhibit carrier properties that resemble those of pristine nonpolar semiconductors despite static and dynamic disorder, but how carriers are protected from efficient scattering with charged defects and optical phonons is unknown. Here, we reveal the carrier protection mechanism by comparing three single-crystal lead bromide perovskites: CH3NH3PbBr3, CH(NH2)2PbBr3, and CsPbBr3. We obsd. hot fluorescence emission from energetic carriers with ∼102-picosecond lifetimes in CH3NH3PbBr3 or CH(NH2)2PbBr3, but not in CsPbBr3. The hot fluorescence is correlated with liq.-like mol. reorientational motions, suggesting that dynamic screening protects energetic carriers via solvation or large polaron formation on time scales competitive with that of ultrafast cooling. Similar protections likely exist for band-edge carriers. The long-lived energetic carriers may enable hot-carrier solar cells with efficiencies exceeding the Shockley-Queisser limit.
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    14. 14
      Neukirch, A. J.; Nie, W.; Blancon, J.-C.; Appavoo, K.; Tsai, H.; Sfeir, M. Y.; Katan, C.; Pedesseau, L.; Even, J.; Crochet, J. J.; Gupta, G.; Mohite, A. D.; Tretiak, S. Polaron stabilization by cooperative lattice distortion and cation rotations in hybrid perovskite materials. Nano Lett. 2016, 16, 3809– 3816,  DOI: 10.1021/acs.nanolett.6b01218
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      14
      Polaron Stabilization by Cooperative Lattice Distortion and Cation Rotations in Hybrid Perovskite Materials
      Neukirch, Amanda J.; Nie, Wanyi; Blancon, Jean-Christophe; Appavoo, Kannatassen; Tsai, Hsinhan; Sfeir, Matthew Y.; Katan, Claudine; Pedesseau, Laurent; Even, Jacky; Crochet, Jared J.; Gupta, Gautam; Mohite, Aditya D.; Tretiak, Sergei
      Nano Letters (2016), 16 (6), 3809-3816CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)
      Soln.-processed organometallic perovskites have rapidly developed into a top candidate for the active layer of photovoltaic devices. Despite the remarkable progress assocd. with perovskite materials, many questions about the fundamental photophys. processes taking place in these devices, remain open. High on the list of unexplained phenomena are very modest mobilities despite low charge carrier effective masses. Also, expts. elucidate unique degrdn. of photocurrent affecting stable operation of perovskite solar cells. These puzzles suggest that, while ionic hybrid perovskite devices may have efficiencies on par with conventional Si and GaAs devices, they exhibit more complicated charge transport phenomena. Here the authors report the results from an in-depth computational study of small polaron formation, electronic structure, charge d., and reorganization energies using both periodic boundary conditions and isolated structures. Using the hybrid d. functional theory, volumetric strain in a CsPbI3 cluster creates a polaron with binding energy of ∼300 and 900 meV for holes and electrons, resp. In the MAPbI3 (MA = CH3NH3) cluster, both volumetric strain and MA reorientation effects lead to larger binding energies at ∼600 and 1300 meV for holes and electrons, resp. Such large reorganization energies suggest appearance of small polarons in organometallic perovskite materials. The fact that both volumetric lattice strain and MA mol. rotational degrees of freedom can cooperate to create and stabilize polarons indicates that to mitigate this problem, formamidinium (FA = HC(NH2)2) and cesium (Cs) based crystals and alloys, are potentially better materials for solar cell and other optoelectronic applications.
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    15. 15
      Sendner, M.; Nayak, P. K.; Egger, D. A.; Beck, S.; Müller, C.; Epding, B.; Kowalsky, W.; Kronik, L.; Snaith, H. J.; Pucci, A.; Lovrincic, R. Optical phonons in methylammonium lead halide perovskites and implications for charge transport. Mater. Horiz. 2016, 3, 613– 620,  DOI: 10.1039/C6MH00275G
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      15
      Optical phonons in methylammonium lead halide perovskites and implications for charge transport
      Sendner, Michael; Nayak, Pabitra K.; Egger, David A.; Beck, Sebastian; Mueller, Christian; Epding, Bernd; Kowalsky, Wolfgang; Kronik, Leeor; Snaith, Henry J.; Pucci, Annemarie; Lovrincic, Robert
      Materials Horizons (2016), 3 (6), 613-620CODEN: MHAOBM; ISSN:2051-6355. (Royal Society of Chemistry)
      Lead-halide perovskites are promising materials for opto-electronic applications. Recent reports indicated that their mech. and electronic properties are strongly affected by the lattice vibrations. Herein we report far-IR spectroscopy measurements of CH3NH3Pb(I/Br/Cl)3 thin films and single crystals at room temp. and a detailed quant. anal. of the spectra. We find strong broadening and anharmonicity of the lattice vibrations for all three halide perovskites, which indicates dynamic disorder of the lead-halide cage at room temp. We det. the frequencies of the transversal and longitudinal optical phonons, and use them to calc., via appropriate models, the static dielec. consts., polaron masses, electron-phonon coupling consts., and upper limits for the phonon-scattering limited charge carrier mobilities. Within the limitations of the model used, we can place an upper limit of 200 cm2 V-1 s-1 for the room temp. charge carrier mobility in MAPbI3 single crystals. Our findings are important for the basic understanding of charge transport processes and mech. properties in metal halide perovskites.
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    16. 16
      Yaffe, O.; Guo, Y.; Tan, L. Z.; Egger, D. A.; Hull, T.; Stoumpos, C. C.; Zheng, F.; Heinz, T. F.; Kronik, L.; Kanatzidis, M. G.; Owen, J. S.; Rappe, A. M.; Pimenta, M. A.; Brus, L. E. Local polar fluctuations in lead halide perovskite crystals. Phys. Rev. Lett. 2017, 118, 136001,  DOI: 10.1103/PhysRevLett.118.136001
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      16
      Local polar fluctuations in lead halide perovskite crystals
      Yaffe, Omer; Guo, Yinsheng; Tan, Liang Z.; Egger, David A.; Hull, Trevor; Stoumpos, Constantinos C.; Zheng, Fan; Heinz, Tony F.; Kronik, Leeor; Kanatzidis, Mercouri G.; Owen, Jonathan S.; Rappe, Andrew M.; Pimenta, Marcos A.; Brus, Louis E.
      Physical Review Letters (2017), 118 (13), 136001/1-136001/6CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      Hybrid lead-halide perovskites have emerged as an excellent class of photovoltaic materials. Recent reports suggest that the org. mol. cation is responsible for local polar fluctuations that inhibit carrier recombination. We combine low-frequency Raman scattering with first-principles mol. dynamics (MD) to study the fundamental nature of these local polar fluctuations. Our observations of a strong central peak in the cubic phase of both hybrid (CH3NH3PbBr3) and all-inorg. (CsPbBr3) leadhalide perovskites show that anharmonic, local polar fluctuations are intrinsic to the general lead-halide perovskite structure, and not unique to the dipolar org. cation. MD simulations indicate that head-tohead Cs motion coupled to Br face expansion, occurring on a few hundred femtosecond time scale, drives the local polar fluctuations in CsPbBr3.
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    17. 17
      Miyata, K.; Meggiolaro, D.; Trinh, M. T.; Joshi, P. P.; Mosconi, E.; Jones, S. C.; De Angelis, F.; Zhu, X.-Y. Large polarons in lead halide perovskites. Sci. Adv. 2017, 3, e1701217,  DOI: 10.1126/sciadv.1701217
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      Large polarons in lead halide perovskites
      Miyata, Kiyoshi; Meggiolaro, Daniele; Trinh, M. Tuan; Joshi, Prakriti P.; Mosconi, Edoardo; Jones, Skyler C.; De Angelis, Filippo; Zhu, X.-Y.
      Science Advances (2017), 3 (8), e1701217/1-e1701217/9CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
      Lead halide perovskites show marked defect tolerance responsible for their excellent optoelectronic properties. These properties might be explained by the formation of large polarons, but how they are formed and whether org. cations are essential remain open questions. We provide a direct time domain view of large polaron formation in single-crystal lead bromide perovskites CH3NH3PbBr3 and CsPbBr3. We found that large polaron forms predominantly from the deformation of the PbBr3-_frameworks, irresp. of the cation type. The difference lies in the polaron formation time, which, in CH3NH3PbBr3 (0.3 ps), is less than half of that in CsPbBr3 (0.7 ps). First-principles calcns. confirm large polaron formation, identify the Pb-Br-Pb deformation modes as responsible, and explain quant. the rate difference between CH3NH3PbBr3 and CsPbBr3. The findings reveal the general advantage of the soft [PbX3]- sublattice in charge carrier protection and suggest that there is likely no mechanistic limitations in using all-inorg. or mixed-cation lead halide perovskites to overcome instability problems and to tune the balance between charge carrier protection and mobility.
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      Woodward, P. M. Octahedral tilting in perovskites. I. Geometrical considerations. Acta Crystallogr., Sect. B: Struct. Sci. 1997, 53, 32– 43,  DOI: 10.1107/S0108768196010713
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      Octahedral tilting in perovskites. I. Geometrical considerations
      Woodward, Patrick M.
      Acta Crystallographica, Section B: Structural Science (1997), B53 (1), 32-43CODEN: ASBSDK; ISSN:0108-7681. (Munksgaard)
      The 23 Glazer tilt systems describing octahedral tilting in perovskites were studied. In tilt systems a+a+a-, a+b+b-, a+a+c-, a+b+c-, a0b+b- and a-b+c- it is not possible to link together a three-dimensional network of perfectly rigid octahedra. In these tilt systems small distortions of the octahedra must occur. The magnitude of the distortions in the a+a+a- and a0b+b- tilt systems are estd. A table of predicted space groups for ordered perovskites, A2MM'O6, for all 23 tilt systems is also given.
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      Howard, C. J.; Stokes, H. Group-theoretical analysis of octahedral tilting in perovskites. Acta Crystallogr., Sect. B: Struct. Sci. 1998, 54, 782– 789,  DOI: 10.1107/S0108768198004200
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      Group theoretical analysis of octahedral tilting in perovskites
      Howard, Christopher J.; Stokes, Harold T.
      Acta Crystallographica, Section B: Structural Science (1998), B54 (6), 782-789CODEN: ASBSDK; ISSN:0108-7681. (Munksgaard International Publishers Ltd.)
      A group-theor. anal. is made of the structures derived from the aristotype cubic perovskite (Pm‾3m) by the simple tilting of rigid octahedral units. The tilting is mediated by the irreducible representations R4+ and M3+ or the two in combination. These result in 15 possible structures, compared with the 23 possibilities suggested previously by Glazer. The anal. makes the group-subgroup relations apparent.
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      Brown, I. D. The crystal structure of K2TeBr6. Can. J. Chem. 1964, 42, 2758– 2767,  DOI: 10.1139/v64-409
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      The crystal structure of K2TeBr6
      Brown, I. D.
      Canadian Journal of Chemistry (1964), 42 (12), 2758-67CODEN: CJCHAG; ISSN:0008-4042.
      Crystals of K2TeBr6 are monoclinic, space group P21/n-C52h, with a 7.521, b 7.574, and c 10.730 A., β 89°40'. At. positions were found by 3-dimensional x-ray diffraction anal. (least sqs. R = 0.12). The crystals possess a K2PtCl6 structure which is distorted to allow a more efficient packing of the comparatively large anions than is possible with the undistorted cubic form. The stereochem. of the octahedral TeBr6- ion (Te-Br = 2.71 A.) is discussed.
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      Cai, Y.; Xie, W.; Ding, H.; Chen, Y.; Thirumal, K.; Wong, L. H.; Mathews, N.; Mhaisalkar, S. G.; Sherburne, M.; Asta, M. Computational Study of Halide Perovskite-Derived A2BX6 Inorganic Compounds: Chemical Trends in Electronic Structure and Structural Stability. Chem. Mater. 2017, 29, 7740– 7749,  DOI: 10.1021/acs.chemmater.7b02013
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      Computational Study of Halide Perovskite-Derived A2BX6 Inorganic Compounds: Chemical Trends in Electronic Structure and Structural Stability
      Cai, Yao; Xie, Wei; Ding, Hong; Chen, Yan; Thirumal, Krishnamoorthy; Wong, Lydia H.; Mathews, Nripan; Mhaisalkar, Subodh G.; Sherburne, Matthew; Asta, Mark
      Chemistry of Materials (2017), 29 (18), 7740-7749CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      The electronic structure and energetic stability of A2BX6 halide compds. with the cubic and tetragonal variants of the perovskite-derived K2PtCl6 prototype structure are investigated computationally within the frameworks of d.-functional-theory (DFT) and hybrid (HSE06) functionals. The HSE06 calcns. are undertaken for seven known A2BX6 compds. with A = K, Rb, and Cs; and B = Sn, Pd, Pt, Te, and X = I. Trends in band gaps and energetic stability are identified, which are explored further employing DFT calcns. over a larger range of chemistries, characterized by A = K, Rb, Cs, B = Si, Ge, Sn, Pb, Ni, Pd, Pt, Se, and Te; and X = Cl, Br, I. For the systems investigated in this work, the band gap increases from iodide to bromide to chloride. Further, variations in the A site cation influences the band gap as well as the preferred degree of tetragonal distortion. Smaller A site cations such as K and Rb favor tetragonal structural distortions, resulting in a slightly larger band gap. For variations in the B site in the (Ni, Pd, Pt) group and the (Se, Te) group, the band gap increases with increasing cation size. However, no obsd. chem. trend with respect to cation size for band gap was found for the (Si, Sn, Ge, Pb) group. The findings in this work provide guidelines for the design of halide A2BX6 compds. for potential photovoltaic applications.
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      Xiao, Z.; Zhou, Y.; Hosono, H.; Kamiya, T. Intrinsic Defects in a Photovoltaic Perovskite Variant Cs2SnI6. Phys. Chem. Chem. Phys. 2015, 17, 18900– 18903,  DOI: 10.1039/C5CP03102H
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      22
      Intrinsic defects in a photovoltaic perovskite variant Cs2SnI6
      Xiao, Zewen; Zhou, Yuanyuan; Hosono, Hideo; Kamiya, Toshio
      Physical Chemistry Chemical Physics (2015), 17 (29), 18900-18903CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Cs2SnI6, a rarely studied perovskite variant material, is recently gaining a lot of interest in the field of photovoltaics owing to its nontoxicity, air-stability and promising photovoltaic properties. In this work, we report intrinsic defects in Cs2SnI6 using first-principles d. functional theory calcns. It is revealed that iodine vacancy and tin interstitial are the dominant defects that are responsible for the intrinsic n-type conduction in Cs2SnI6. Tin vacancy has a very high formation energy (>3.6 eV) due to the strong covalency in the Sn-I bonds and is hardly generated for p-type doping. All the dominant defects in Cs2SnI6 have deep transition levels in the band gap. It is suggested that the formation of deep defects can be suppressed significantly by employing an I-rich synthesis condition, which is inevitable for photovoltaic and other semiconductor applications.
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    23. 23
      Maughan, A. E.; Ganose, A. M.; Bordelon, M. M.; Miller, E. M.; Scanlon, D. O.; Neilson, J. R. Defect tolerance to intolerance in the vacancy-ordered double perovskite semiconductors Cs2SnI6 and Cs2TeI6. J. Am. Chem. Soc. 2016, 138, 8453– 8464,  DOI: 10.1021/jacs.6b03207
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      23
      Defect Tolerance to Intolerance in the Vacancy-Ordered Double Perovskite Semiconductors Cs2SnI6 and Cs2TeI6
      Maughan, Annalise E.; Ganose, Alex M.; Bordelon, Mitchell M.; Miller, Elisa M.; Scanlon, David O.; Neilson, James R.
      Journal of the American Chemical Society (2016), 138 (27), 8453-8464CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Vacancy-ordered double perovskites of the general formula A2BX6 are a family of perovskite derivs. composed of a face-centered lattice of nearly isolated [BX6] units with A-site cations occupying the cuboctahedral voids. Despite the presence of isolated octahedral units, the close-packed iodide lattice provides significant electronic dispersion, such that Cs2SnI6 has recently been explored for applications in photovoltaic devices. To elucidate the structure-property relationships of these materials, we have synthesized solid-soln. Cs2Sn1-xTexI6. However, even though tellurium substitution increases electronic dispersion via closer I-I contact distances, the substitution exptl. yields insulating behavior from a significant decrease in carrier concn. and mobility. D. functional calcns. of native defects in Cs2SnI6 reveal that iodine vacancies exhibit a low enthalpy of formation, and that the defect energy level is a shallow donor to the conduction band rendering the material tolerant to these defect states. The increased covalency of Te-I bonding renders the formation of iodine vacancy states unfavorable and is responsible for the redn. in cond. upon Te substitution. Addnl., Cs2TeI6 is intolerant to the formation of these defects, because the defect level occurs deep within the band gap and thus localizes potential mobile charge carriers. In these vacancy-ordered double perovskites, the close-packed lattice of iodine provides significant electronic dispersion, while the interaction of the B- and X-site ions dictates the properties as they pertain to electronic structure and defect tolerance. This simplified perspective based on extensive exptl. and theor. anal. provides a platform from which to understand structure-property relationships in functional perovskite halides.
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    24. 24
      Saparov, B.; Sun, J.-P.; Meng, W.; Xiao, Z.; Duan, H.-S.; Gunawan, O.; Shin, D.; Hill, I. G.; Yan, Y.; Mitzi, D. B. Thin-film deposition and characterization of a Sn-deficient perovskite derivative Cs2SnI6. Chem. Mater. 2016, 28, 2315– 2322,  DOI: 10.1021/acs.chemmater.6b00433
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      24
      Thin-Film Deposition and Characterization of a Sn-Deficient Perovskite Derivative Cs2SnI6
      Saparov, Bayrammurad; Sun, Jon-Paul; Meng, Weiwei; Xiao, Zewen; Duan, Hsin-Sheng; Gunawan, Oki; Shin, Donghyeop; Hill, Ian G.; Yan, Yanfa; Mitzi, David B.
      Chemistry of Materials (2016), 28 (7), 2315-2322CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      In this work, we describe details of a two-step deposition approach that enables the prepn. of continuous and well-structured thin films of Cs2SnI6, which is a one-half Sn-deficient 0-D perovskite deriv. (i.e., the compd. can also be written as CsSn0.5I3, with a structure consisting of isolated SnI64- octahedra). The films were characterized using powder X-ray diffraction (PXRD), SEM (SEM), thermogravimetric anal. (TGA), UV-vis spectroscopy, photoluminescence (PL), photoelectron spectroscopy (UPS, IPES, XPS), and Hall effect measurements. UV-vis and PL measurements indicate that the obtained Cs2SnI6 film is a semiconductor with a band gap of 1.6 eV. This band gap was further confirmed by the UPS and IPES spectra, which were well reproduced by the calcd. d. of states with the HSE hybrid functional. The Cs2SnI6 films exhibited n-type conduction with a carrier d. of 6(1) × 1016 cm-3 and mobility of 2.9(3) cm2/V·s. While the computationally derived band structure for Cs2SnI6 shows significant dispersion along several directions in the Brillouin zone near the band edges, the valence band is relatively flat along the Γ-X direction, indicative of a more limited hole minority carrier mobility compared to analogous values for the electrons. The ionization potential (IP) and electron affinity (EA) were detd. to be 6.4 and 4.8 eV, resp. The Cs2SnI6 films show some enhanced stability under ambient air, compared to methylammonium lead(II) iodide perovskite films stored under similar conditions; however, the films do decomp. slowly, yielding a CsI impurity. These findings are discussed in the context of suitability of Cs2SnI6 for photovoltaic and related optoelectronic applications.
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    25. 25
      Maughan, A. E.; Ganose, A. M.; Candia, A. M.; Granger, J. T.; Scanlon, D. O.; Neilson, J. R. Anharmonicity and Octahedral Tilting in Hybrid Vacancy-Ordered Double Perovskites. Chem. Mater. 2018, 30, 472– 483,  DOI: 10.1021/acs.chemmater.7b04516
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      25
      Anharmonicity and Octahedral Tilting in Hybrid Vacancy-Ordered Double Perovskites
      Maughan, Annalise E.; Ganose, Alex M.; Candia, Andrew M.; Granger, Juliette T.; Scanlon, David O.; Neilson, James R.
      Chemistry of Materials (2018), 30 (2), 472-483CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      The advantageous performance of hybrid org.-inorg. perovskite halide semiconductors in optoelectronic applications motivates studies of their fundamental crystal chem. In particular, recent studies have sought to understand how dipolar, dynamic, and org. cations such as methylammonium (CH3NH3+) and formamidinium (CH(NH2)2+) affect phys. properties such as light absorption and charge transport. To probe the influence of org.-inorg. coupling on charge transport, we prepd. the series of vacancy-ordered double perovskite derivs. A2SnI6, where A = Cs+, CH3NH3+, and CH(NH2)2+. Despite nearly identical cubic structures by powder X-ray diffraction, replacement of Cs+ with CH3NH3+ or CH(NH2)2+ reduces cond. through a redn. in both carrier concn. and carrier mobility. We attribute the trends in electronic behavior to anharmonic lattice dynamics from the formation of hydrogen bonds that yield coupled org.-inorg. dynamics. This anharmonicity manifests as asymmetry of the interoctahedral I-I pair correlations in the X-ray pair distribution function of the hybrid compds., which can be modeled by large atomistic ensembles with random rotations of rigid [SnI6] octahedral units. The presence of soft, anharmonic lattice dynamics holds implications for electron-phonon interactions, as supported by calcn. of electron-phonon coupling strength that indicates the formation of more tightly bound polarons and reduced electron mobilities with increasing cation size. By exploiting the relatively decoupled nature of the octahedral units in these defect-ordered perovskite variants, we interrogated the impact of org.-inorg. coupling and lattice anharmonicity on the charge transport behavior of hybrid perovskite halide semiconductors.
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    26. 26
      Lee, B.; Stoumpos, C. C.; Zhou, N.; Hao, F.; Malliakas, C.; Yeh, C.-Y.; Marks, T. J.; Kanatzidis, M. G.; Chang, R. P. Air-stable molecular semiconducting iodosalts for solar cell applications: Cs2SnI6 as a hole conductor. J. Am. Chem. Soc. 2014, 136, 15379– 15385,  DOI: 10.1021/ja508464w
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      Air-Stable Molecular Semiconducting Iodosalts for Solar Cell Applications: Cs2SnI6 as a Hole Conductor
      Lee, Byunghong; Stoumpos, Constantinos C.; Zhou, Nanjia; Hao, Feng; Malliakas, Christos; Yeh, Chen-Yu; Marks, Tobin J.; Kanatzidis, Mercouri G.; Chang, Robert P. H.
      Journal of the American Chemical Society (2014), 136 (43), 15379-15385CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The authors introduce a new class of mol. iodosalt compds. for application in next-generation solar cells. Unlike tin-based perovskite compds. CsSnI3 and MeNH3SnI3, which have Sn in the 2 + oxidn. state and must be handled in an inert atm. when fabricating solar cells, the Sn in the mol. iodosalt compds. is in the 4+ oxidn. state, making them stable in air and moisture. As an example, using Cs2SnI6 as a hole transporter, the authors can successfully fabricate in air a solid-state dye-sensitized solar cell (DSSC) with a mesoporous TiO2 film. Doping Cs2SnI6 with additives helps to reduce the internal device resistance, improving cell efficiency. In this way, a Z 907 DSSC delivers 4.7% of energy conversion efficiency. By using a more efficient mixt. of porphyrin dyes, an efficiency near 8% with photon confinement was achieved. This represents a significant step toward the realization of low-cost, stable, lead-free, and environmentally benign next-generation solid-state solar cells.
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      Wang, J.; Toby, B. H.; Lee, P. L.; Ribaud, L.; Antao, S. M.; Kurtz, C.; Ramanathan, M.; Von Dreele, R. B.; Beno, M. A. A dedicated powder diffraction beamline at the Advanced Photon Source: Commissioning and early operational results. Rev. Sci. Instrum. 2008, 79, 085105,  DOI: 10.1063/1.2969260
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      A dedicated powder diffraction beam-line at the Advanced Photon Source: Commissioning and early operational results
      Wang, Jun; Toby, Brian H.; Lee, Peter L.; Ribaud, Lynn; Antao, Sytle M.; Kurtz, Charles; Ramanathan, Mohan; Von Dreele, Robert B.; Beno, Mark A.
      Review of Scientific Instruments (2008), 79 (8), 085105/1-085105/7CODEN: RSINAK; ISSN:0034-6748. (American Institute of Physics)
      A new dedicated high-resoln. high-throughput powder diffraction beam-line was built, fully commissioned, and opened to general users at the Advanced Photon Source. The optical design and commissioning results are presented. Beam-line performance was examd. using a mixt. of the NIST Si and Al2O3 std. ref. materials, as well as the LaB6 line-shape std. Instrumental resoln. ≤1.7 × 10-4 (ΔQ/Q) was obsd. (c) 2008 American Institute of Physics.
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      Toby, B. H. EXPGUI, a graphical user interface for GSAS. J. Appl. Crystallogr. 2001, 34, 210– 213,  DOI: 10.1107/S0021889801002242
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      EXPGUI, a graphical user interface for GSAS
      Toby, Brian H.
      Journal of Applied Crystallography (2001), 34 (2), 210-213CODEN: JACGAR; ISSN:0021-8898. (Munksgaard International Publishers Ltd.)
      A description and justification of the EXPGUI program is presented. This program implements a graphical user interface and shell for the GSAS (Generalized Structure and Anal. Software) single-crystal and Rietveld package. Use of the Tcl/Tk scripting language allows EXPGUI to be platform independent. Also included is a synopsis of how the program is implemented.
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      Momma, K.; Izumi, F. VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 2011, 44, 1272– 1276,  DOI: 10.1107/S0021889811038970
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      VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data
      Momma, Koichi; Izumi, Fujio
      Journal of Applied Crystallography (2011), 44 (6), 1272-1276CODEN: JACGAR; ISSN:0021-8898. (International Union of Crystallography)
      VESTA is a 3D visualization system for crystallog. studies and electronic state calcns. It was upgraded to the latest version, VESTA 3, implementing new features including drawing the external morphpol. of crysals; superimposing multiple structural models, volumetric data and crystal faces; calcn. of electron and nuclear densities from structure parameters; calcn. of Patterson functions from the structure parameters or volumetric data; integration of electron and nuclear densities by Voronoi tessellation; visualization of isosurfaces with multiple levels, detn. of the best plane for selected atoms; an extended bond-search algorithm to enable more sophisticated searches in complex mols. and cage-like structures; undo and redo is graphical user interface operations; and significant performance improvements in rendering isosurfaces and calcg. slices.
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      Chupas, P. J.; Qiu, X.; Hanson, J. C.; Lee, P. L.; Grey, C. P.; Billinge, S. J. Rapid-acquisition pair distribution function (RA-PDF) analysis. J. Appl. Crystallogr. 2003, 36, 1342– 1347,  DOI: 10.1107/S0021889803017564
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      Rapid-acquisition pair distribution function (RA-PDF) analysis
      Chupas, Peter J.; Qiu, Xiangyun; Hanson, Jonathan C.; Lee, Peter L.; Grey, Clare P.; Billinge, Simon J. L.
      Journal of Applied Crystallography (2003), 36 (6), 1342-1347CODEN: JACGAR; ISSN:0021-8898. (Blackwell Publishing Ltd.)
      An image-plate (IP) detector coupled with high-energy synchrotron radiation was used for at. pair distribution function (PDF) anal., with high probed momentum transfer Qmax ≤ 28.5 Å-1, from cryst. materials. Materials with different structural complexities were measured to test the validity of the quant. data anal. Exptl. results are presented for cryst. Ni, cryst. α-AlF3, and the layered Aurivillius type oxides α-Bi4V2O11 and γ-Bi4V1.7Ti0.3O10.85. Overall, the diffraction patterns show good counting statistics, with measuring time from one to tens of seconds. The PDFs obtained are of high quality. Structures may be refined from these PDFs, and the structural models are consistent with the published literature. Data sets from similar samples are highly reproducible.
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      PDFgetX2: a GUI-driven program to obtain the pair distribution function from X-ray powder diffraction data
      Qiu, Xiangyun; Thompson, Jeroen W.; Billinge, Simon J. L.
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      There is no expanded citation for this reference.
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      Farrow, C.; Juhas, P.; Liu, J.; Bryndin, D.; Božin, E.; Bloch, J.; Proffen, T.; Billinge, S. PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals. J. Phys.: Condens. Matter 2007, 19, 335219,  DOI: 10.1088/0953-8984/19/33/335219
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      PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals
      Farrow, C. L.; Juhas, P.; Liu, J. W.; Bryndin, D.; Bozin, E. S.; Bloch, J.; Proffen, Th; Billinge, S. J. L.
      Journal of Physics: Condensed Matter (2007), 19 (33), 335219/1-335219/7CODEN: JCOMEL; ISSN:0953-8984. (Institute of Physics Publishing)
      PDFfit2 is a program as well as a library for real-space refinement of crystal structures. It is capable of fitting a theor. three-dimensional (3D) structure to at. pair distribution function data and is ideal for nanoscale investigations. The fit system accounts for lattice consts., at. positions and anisotropic at. displacement parameters, correlated at. motion, and exptl. factors that may affect the data. The at. positions and thermal coeffs. can be constrained to follow the symmetry requirements of an arbitrary space group. The PDFfit2 engine is written in C++ and is accessible via Python, allowing it to inter-operate with other Python programs. PDFgui is a graphical interface built on the PDFfit2 engine. PDFgui organizes fits and simplifies many data anal. tasks, such as configuring and plotting multiple fits. PDFfit2 and PDFgui are freely available via the Internet.
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      Neuefeind, J.; Feygenson, M.; Carruth, J.; Hoffmann, R.; Chipley, K. K. The nanoscale ordered materials diffractometer NOMAD at the spallation neutron source SNS. Nucl. Instrum. Methods Phys. Res., Sect. B 2012, 287, 68– 75,  DOI: 10.1016/j.nimb.2012.05.037
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      The Nanoscale Ordered MAterials Diffractometer NOMAD at the Spallation Neutron Source SNS
      Neuefeind, Joerg; Feygenson, Mikhail; Carruth, John; Hoffmann, Ron; Chipley, Kenneth K.
      Nuclear Instruments & Methods in Physics Research, Section B: Beam Interactions with Materials and Atoms (2012), 287 (), 68-75CODEN: NIMBEU; ISSN:0168-583X. (Elsevier B.V.)
      The Nanoscale Ordered MAterials Diffractometer (NOMAD) is neutron time-of-flight diffractometer designed to det. pair distribution functions of a wide range of materials ranging from short range ordered liqs. to long range ordered crystals. Due to a large neutron flux provided by the Spallation Neutron Source SNS and a large detector coverage neutron count-rates exceed comparable instruments by one to two orders of magnitude. This is achieved while maintaining a relatively high momentum transfer resoln. of a δQ/Q∼0.8% FWHM (typical), and a possible δQ/Qof0.24% FWHM (best). The real space resoln. is related to the max. momentum transfer; a max. momentum transfer of 50 Å-1 can be obtained routinely and the max. momentum transfer given by the detector configuration and the incident neutron spectrum is 125 Å-1. High stability of the source and the detector allow small contrast isotope expts. to be performed. A detailed description of the instrument is given and the results of expts. with std. samples are discussed.
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      Ab initio molecular dynamics of liquid metals
      Kresse, G.; Hafner, J.
      Physical Review B: Condensed Matter and Materials Physics (1993), 47 (1), 558-61CODEN: PRBMDO; ISSN:0163-1829.
      The authors present ab initio quantum-mech. mol.-dynamics calcns. based on the calcn. of the electronic ground state and of the Hellmann-Feynman forces in the local-d. approxn. at each mol.-dynamics step. This is possible using conjugate-gradient techniques for energy minimization, and predicting the wave functions for new ionic positions using sub-space alignment. This approach avoids the instabilities inherent in quantum-mech. mol.-dynamics calcns. for metals based on the use of a factitious Newtonian dynamics for the electronic degrees of freedom. This method gives perfect control of the adiabaticity and allows one to perform simulations over several picoseconds.
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      Kresse, G.; Hafner, J. Ab Initio Molecular-Dynamics Simulation of the Liquid-Metal Amorphous-Semiconductor Transition in Germanium. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 49, 14251– 14269,  DOI: 10.1103/PhysRevB.49.14251
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      Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium
      Kresse, G.; Hafner, J.
      Physical Review B: Condensed Matter and Materials Physics (1994), 49 (20), 14251-69CODEN: PRBMDO; ISSN:0163-1829.
      The authors present ab initio quantum-mech. mol.-dynamics simulations of the liq.-metal-amorphous-semiconductor transition in Ge. The simulations are based on (a) finite-temp. d.-functional theory of the 1-electron states, (b) exact energy minimization and hence calcn. of the exact Hellmann-Feynman forces after each mol.-dynamics step using preconditioned conjugate-gradient techniques, (c) accurate nonlocal pseudopotentials, and (d) Nose' dynamics for generating a canonical ensemble. This method gives perfect control of the adiabaticity of the electron-ion ensemble and allows the authors to perform simulations over >30 ps. The computer-generated ensemble describes the structural, dynamic, and electronic properties of liq. and amorphous Ge in very good agreement with expt.. The simulation allows the authors to study in detail the changes in the structure-property relation through the metal-semiconductor transition. The authors report a detailed anal. of the local structural properties and their changes induced by an annealing process. The geometrical, bounding, and spectral properties of defects in the disordered tetrahedral network are studied and compared with expt.
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      Kresse, G.; Furthmüller, J. Efficiency of Ab Initio Total Energy Calculations for Metals and Semiconductors Using a Plane Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15– 50,  DOI: 10.1016/0927-0256(96)00008-0
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      Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
      Kresse, G.; Furthmuller, J.
      Computational Materials Science (1996), 6 (1), 15-50CODEN: CMMSEM; ISSN:0927-0256. (Elsevier)
      The authors present a detailed description and comparison of algorithms for performing ab-initio quantum-mech. calcns. using pseudopotentials and a plane-wave basis set. The authors will discuss: (a) partial occupancies within the framework of the linear tetrahedron method and the finite temp. d.-functional theory, (b) iterative methods for the diagonalization of the Kohn-Sham Hamiltonian and a discussion of an efficient iterative method based on the ideas of Pulay's residual minimization, which is close to an order N2atoms scaling even for relatively large systems, (c) efficient Broyden-like and Pulay-like mixing methods for the charge d. including a new special preconditioning optimized for a plane-wave basis set, (d) conjugate gradient methods for minimizing the electronic free energy with respect to all degrees of freedom simultaneously. The authors have implemented these algorithms within a powerful package called VAMP (Vienna ab-initio mol.-dynamics package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semi-conducting surfaces, phonons in simple metals, transition metals and semiconductors) and turned out to be very reliable.
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      Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 11169– 11186,  DOI: 10.1103/PhysRevB.54.11169
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      Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set
      Kresse, G.; Furthmueller, J.
      Physical Review B: Condensed Matter (1996), 54 (16), 11169-11186CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      The authors present an efficient scheme for calcg. the Kohn-Sham ground state of metallic systems using pseudopotentials and a plane-wave basis set. In the first part the application of Pulay's DIIS method (direct inversion in the iterative subspace) to the iterative diagonalization of large matrixes will be discussed. This approach is stable, reliable, and minimizes the no. of order Natoms3 operations. In the second part, we will discuss an efficient mixing scheme also based on Pulay's scheme. A special "metric" and a special "preconditioning" optimized for a plane-wave basis set will be introduced. Scaling of the method will be discussed in detail for non-self-consistent and self-consistent calcns. It will be shown that the no. of iterations required to obtain a specific precision is almost independent of the system size. Altogether an order Natoms2 scaling is found for systems contg. up to 1000 electrons. If we take into account that the no. of k points can be decreased linearly with the system size, the overall scaling can approach Natoms. They have implemented these algorithms within a powerful package called VASP (Vienna ab initio simulation package). The program and the techniques have been used successfully for a large no. of different systems (liq. and amorphous semiconductors, liq. simple and transition metals, metallic and semiconducting surfaces, phonons in simple metals, transition metals, and semiconductors) and turned out to be very reliable.
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      Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758– 1775,  DOI: 10.1103/PhysRevB.59.1758
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      From ultrasoft pseudopotentials to the projector augmented-wave method
      Kresse, G.; Joubert, D.
      Physical Review B: Condensed Matter and Materials Physics (1999), 59 (3), 1758-1775CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      The formal relationship between ultrasoft (US) Vanderbilt-type pseudopotentials and Blochl's projector augmented wave (PAW) method is derived. The total energy functional for US pseudopotentials can be obtained by linearization of two terms in a slightly modified PAW total energy functional. The Hamilton operator, the forces, and the stress tensor are derived for this modified PAW functional. A simple way to implement the PAW method in existing plane-wave codes supporting US pseudopotentials is pointed out. In addn., crit. tests are presented to compare the accuracy and efficiency of the PAW and the US pseudopotential method with relaxed-core all-electron methods. These tests include small mols. (H2, H2O, Li2, N2, F2, BF3, SiF4) and several bulk systems (diamond, Si, V, Li, Ca, CaF2, Fe, Co, Ni). Particular attention is paid to the bulk properties and magnetic energies of Fe, Co, and Ni.
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      Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406,  DOI: 10.1103/PhysRevLett.100.136406
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      Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces
      Perdew, John P.; Ruzsinszky, Adrienn; Csonka, Gabor I.; Vydrov, Oleg A.; Scuseria, Gustavo E.; Constantin, Lucian A.; Zhou, Xiaolan; Burke, Kieron
      Physical Review Letters (2008), 100 (13), 136406/1-136406/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Popular modern generalized gradient approxns. are biased toward the description of free-atom energies. Restoration of the first-principles gradient expansion for exchange over a wide range of d. gradients eliminates this bias. We introduce a revised Perdew-Burke-Ernzerhof generalized gradient approxn. that improves equil. properties of densely packed solids and their surfaces.
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      Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868,  DOI: 10.1103/PhysRevLett.77.3865
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      Generalized gradient approximation made simple
      Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
      Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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      Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106,  DOI: 10.1063/1.2404663
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      Influence of the exchange screening parameter on the performance of screened hybrid functionals
      Krukau, Aliaksandr V.; Vydrov, Oleg A.; Izmaylov, Artur F.; Scuseria, Gustavo E.
      Journal of Chemical Physics (2006), 125 (22), 224106/1-224106/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      This work reexamines the effect of the exchange screening parameter ω on the performance of the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional. We show that variation of the screening parameter influences solid band gaps the most. Other properties such as mol. thermochem. or lattice consts. of solids change little with ω. We recommend a new version of HSE with the screening parameter ω = 0.11 bohr-1 for further use. Compared to the original implementation, the new parametrization yields better thermochem. results and preserves the good accuracy for band gaps and lattice consts. in solids.
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      Ganose, A. M.; Savory, C. N.; Scanlon, D. O. Electronic and defect properties of (CH3NH3)2Pb(SCN)2I2 analogues for photovoltaic applications. J. Mater. Chem. A 2017, 5, 7845– 7853,  DOI: 10.1039/C7TA01688C
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      Electronic and defect properties of (CH3NH3)2Pb(SCN)2I2 analogues for photovoltaic applications
      Ganose, Alex M.; Savory, Christopher N.; Scanlon, David O.
      Journal of Materials Chemistry A: Materials for Energy and Sustainability (2017), 5 (17), 7845-7853CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
      In the past 5 yr, hybrid halide perovskites have emerged as a class of highly efficient photovoltaic (PV) absorbers, with excellent electronic properties and low cost synthesis routes. Unfortunately, despite much research effort, their long-term stability is poor and presents a major obstacle toward commercialisation. The layered perovskite (CH3NH3)2Pb(SCN)2I2 (MAPSI) has recently been identified as a promising PV candidate material due to its enhanced stability and favorable electronic properties. We demonstrate, using relativistic hybrid d. functional theory, that the MAPSI structural motif can be extended to include a range of other metals, halides and even pseudohalides. In this way, the electronic structure of MAPSI can be tuned without affecting its stability with respect towards decompn. These results indicate the possibility of Pb-free MAPSI analogs, with suitable properties for photovoltaic top cells in tandem devices.
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      Biswas, D.; Ganose, A. M.; Yano, R.; Riley, J.; Bawden, L.; Clark, O.; Feng, J.; Collins-Mcintyre, L.; Sajjad, M.; Meevasana, W.; Hoesch, M.; Rault, J.; Sasagawa, T.; Scanlon, D.; King, P.; Kim, T. K. Narrow-band anisotropic electronic structure of ReS2. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, 085205,  DOI: 10.1103/PhysRevB.96.085205
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      Narrow-band anisotropic electronic structure of ReS2
      Biswas, D.; Ganose, Alex M.; Yano, R.; Riley, J. M.; Bawden, L.; Clark, O. J.; Feng, J.; Collins-Mcintyre, L.; Sajjad, M. T.; Meevasana, W.; Kim, T. K.; Hoesch, M.; Rault, J. E.; Sasagawa, T.; Scanlon, David O.; King, P. D. C.
      Physical Review B (2017), 96 (8), 085205/1-085205/7CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      We have used angle-resolved photoemission spectroscopy to investigate the band structure of ReS2, a transitionmetal dichalcogenide semiconductor with a distorted 1T crystal structure. We find a large no. of narrow valence bands, which we attribute to the combined influence of structural distortion and spin-orbit coupling. We further show how this leads to a strong in-plane anisotropy of the electronic structure, with quasi-one-dimensional bands reflecting predominant hopping along zigzag Re chains. We find that this does not persist up to the top of the valence band, where a more three-dimensional character is recovered with the fundamental band gap located away from the Brillouin zone center along kz. These expts. are in good agreement with our d.-functional theory calcns., shedding light on the bulk electronic structure of ReS2, and how it can be expected to evolve when thinned to a single layer.
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      Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 2003, 118, 8207– 8215,  DOI: 10.1063/1.1564060
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      Hybrid functionals based on a screened Coulomb potential
      Heyd, Jochen; Scuseria, Gustavo E.; Ernzerhof, Matthias
      Journal of Chemical Physics (2003), 118 (18), 8207-8215CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Hybrid d. functionals are very successful in describing a wide range of mol. properties accurately. In large mols. and solids, however, calcg. the exact (Hartree-Fock) exchange is computationally expensive, esp. for systems with metallic characteristics. In the present work, we develop a new hybrid d. functional based on a screened Coulomb potential for the exchange interaction which circumvents this bottleneck. The results obtained for structural and thermodn. properties of mols. are comparable in quality to the most widely used hybrid functionals. In addn., we present results of periodic boundary condition calcns. for both semiconducting and metallic single wall carbon nanotubes. Using a screened Coulomb potential for Hartree-Fock exchange enables fast and accurate hybrid calcns., even of usually difficult metallic systems. The high accuracy of the new screened Coulomb potential hybrid, combined with its computational advantages, makes it widely applicable to large mols. and periodic systems.
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      Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Erratum: “Hybrid functionals based on a screened Coulomb potential”[J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 2006, 124, 219906,  DOI: 10.1063/1.2204597
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      Hybrid functionals based on a screened Coulomb potential. [Erratum to document cited in CA139:042043]
      Heyd, Jochen; Scuseria, Gustavo E.; Ernzerhof, Matthias
      Journal of Chemical Physics (2006), 124 (21), 219906/1CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The screening parameter quoted in this paper (ω=0.15) is not the one that was actually used in the code. In order to reproduce our results, two ω values are needed: ω =0.15 2≈0.106 for Hartree-Fock and ω=0.15x21/3 ≈0.189 for the PBE part. This erratum applies to all of our ωPBEh and HSE calcns. published to date. The authors emphasize that all the results obtained with this functional are reproducible using the two ω values quoted. The validity of all the published HSE values is not affected. Optimization of a single value of ω for a functional of the type reported in this paper will be presented elsewhere.
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      Hobbs, D.; Kresse, G.; Hafner, J. Fully unconstrained noncollinear magnetism within the projector augmented-wave method. Phys. Rev. B: Condens. Matter Mater. Phys. 2000, 62, 11556– 11570,  DOI: 10.1103/PhysRevB.62.11556
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      Fully unconstrained noncollinear magnetism within the projector augmented-wave method
      Hobbs, D.; Kresse, G.; Hafner, J.
      Physical Review B: Condensed Matter and Materials Physics (2000), 62 (17), 11556-11570CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      Spin-polarized calcns. in solids were generally confined to a global quantization axis to simplify both the theor. model and its implementation in self-consistent codes. This approxn. is justified as many materials exhibit a collinear magnetic order. However, in recent years much interest was directed towards noncollinear magnetism in which the magnetization d. is a continuous vector variable of position. We develop the all-electron projector augmented-wave (PAW) method for noncollinear magnetic structures, based on a generalized local-spin-d. theory. The method allows both the at. and magnetic structures to relax simultaneously and self-consistently. The algorithms were implemented within a powerful package called VASP (Vienna ab initio simulation package), which was used successfully for a large variety of different systems such as cryst. and amorphous semiconductors, simple liqs., and transition metals. The approach was used to study small clusters of Fe and Cr; some of these clusters show noncollinear magnetic arrangements.
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      Ganose, A. M.; Butler, K. T.; Walsh, A.; Scanlon, D. O. Relativistic electronic structure and band alignment of BiSI and BiSeI: candidate photovoltaic materials. J. Mater. Chem. A 2016, 4, 2060– 2068,  DOI: 10.1039/C5TA09612J
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      Relativistic electronic structure and band alignment of BiSI and BiSeI: candidate photovoltaic materials
      Ganose, Alex M.; Butler, Keith T.; Walsh, Aron; Scanlon, David O.
      Journal of Materials Chemistry A: Materials for Energy and Sustainability (2016), 4 (6), 2060-2068CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
      Bi-based solar absorbers are of interest due to similarities in the chem. properties of Bi halides and the exceptionally efficient Pb halide hybrid perovskites. While they both experience the same beneficial relativistic effects acting to increase the width of the conduction band, Bi is non-toxic and non-bioaccumulating, meaning the impact of environmental contamination is greatly reduced. We use hybrid d. functional theory, with the addn. of spin orbit coupling, to examine 2 candidate bismuth contg. photovoltaic absorbers, BiSI and BiSeI, and show that they possess electronic structures suitable for photovoltaic applications. We calc. band alignments against commonly used hole transporting and buffer layers, which indicate band misalignments are likely to be the source of the poor efficiencies reported for devices contg. these materials. We suggest alternative device architectures expected to result in improved power conversion efficiencies.
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      Savory, C. N.; Ganose, A. M.; Scanlon, D. O. Exploring the PbS-Bi2S3 series for next generation energy conversion materials. Chem. Mater. 2017, 29, 5156– 5167,  DOI: 10.1021/acs.chemmater.7b00628
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      As photovoltaics become an ever more important part of the global energy economy, the search for inexpensive, earth-abundant solar absorbers has grown rapidly. The binary compds. PbS and Bi2S3 have both seen success in previous photovoltaic studies; however, bulk PbS has a small band gap, restricting its efficiency, and Bi2S3, while strongly absorbing, can be limited by its layered structure. The mixed PbS-Bi2S3 series has previously been the focus of mostly structural studies, so in this article, we examine the electronic structure of the known members of this series using hybrid d. functional theory. We find that the lead bismuth sulfides are able to retain optimal properties, such as low carrier effective masses and strong absorption, from both parent phases, with band gaps between 0.25 and 1.32 eV. PbBi2S4 emerges from our computational screening as a possible earth-abundant solar absorber, with a predicted max. efficiency of 26% at a film thickness of 0.2 μm and with the retention of the three-dimensional connectivity of lead and bismuth polyhedra.
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      Lead halide perovskite semiconductors are soft, polar materials. The strong driving force for polaron formation (the dielec. electron-phonon coupling) is balanced by the light band effective masses, leading to a strongly-interacting large polaron. A first-principles prediction of mobility would help understand the fundamental mobility limits. Theories of mobility need to consider the polaron (rather than free-carrier) state due to the strong interactions. In this material we expect that at room temp. polar-optical phonon mode scattering will dominate and so limit mobility. We calc. the temp.-dependent polaron mobility of hybrid halide perovskites by variationally solving the Feynman polaron model with the finite-temp. free energies of ‾Osaka. This model considers a simplified effective-mass band structure interacting with a continuum dielec. of characteristic response frequency. We parametrize the model fully from electronic-structure calcns. In methylammonium lead iodide at 300K we predict electron and hole mobilities of 133 and 94 cm2V-1s-1, resp. These are in acceptable agreement with single-crystal measurements, suggesting that the intrinsic limit of the polaron charge carrier state has been reached. Repercussions for hot-electron photoexcited states are discussed. As well as mobility, the model also exposes the dynamic structure of the polaron. This can be used to interpret impedance measurements of the charge-carrier state. We provide the phonon-drag mass renormalization and scattering time consts. These could be used as parameters for larger-scale device models and band-structure dependent mobility simulations.
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      Measurements of the temp. and pressure variation of the 35Cl nuclear quadrupole resonance (NQR) frequency in Rb2PtCl6 and Cs2PtCl6 are reported. The resonance frequency is measured at atm. pressure for temps. from 4 to 500°K and at 4 temps. between 290 and 380°K for pressures to 5000 kg cm-2. Previously published data for K2PtCl6 are also included in the anal. Static lattice NQR frequencies are deduced. The differences between the static lattice frequencies are compared with the calcns. of Smith and Stoessiger. Thermal averaging of the elec. field gradient at a Cl site is assumed to be dominated by the Q3, Q4, Q5, and Q6 internal modes of the PtCl6 octahedra and by the rotary lattice mode. The rotary model frequencies are deduced; they are of similar magnitude and increase in the same sequence as the frequencies deduced from ir and Raman data. An anal. of the pressure dependence of the NQR frequencies leads to pressure coeffs. for the rotary mode frequencies. The influence of the cage of R atoms surrounding a PtCl6 octahedron increases through the series K2PtCl6, Rb2PtCl6, Cs2PtCl6. A thermodynamic anal. of the NQR data is presented which shows the importance of taking sp. vol. effects into account.
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      Direct Observation of Dynamic Symmetry Breaking above Room Temperature in Methylammonium Lead Iodide Perovskite
      Beecher, Alexander N.; Semonin, Octavi E.; Skelton, Jonathan M.; Frost, Jarvist M.; Terban, Maxwell W.; Zhai, Haowei; Alatas, Ahmet; Owen, Jonathan S.; Walsh, Aron; Billinge, Simon J. L.
      ACS Energy Letters (2016), 1 (4), 880-887CODEN: AELCCP; ISSN:2380-8195. (American Chemical Society)
      Lead halide perovskites such as methylammonium lead triiodide (CH3NH3PbI3) have outstanding optical and electronic properties for photovoltaic applications, yet a full understanding of how this soln.-processable material works so well is currently missing. Previous research has revealed that CH3NH3PbI3 possesses multiple forms of static disorder regardless of prepn. method, which is surprising in light of its excellent performance. Using high energy resoln. inelastic X-ray (HERIX) scattering, we measure phonon dispersions in CH3NH3PbI3 and find direct evidence for another form of disorder in single crystals: large-amplitude anharmonic zone edge rotational instabilities of the PbI6 octahedra that persist to room temp. and above, left over from structural phase transitions that take place tens to hundreds of degrees below. Phonon calcns. show that the orientations of the methylammonium (CH3NH3+) couple strongly and cooperatively to these modes. The result is a noncentrosym., instantaneous local structure, which we observe in at. pair distribution function (PDF) measurements. This local symmetry breaking is unobservable by Bragg diffraction but can explain key material properties such as the structural phase sequence, ultralow thermal transport, and large minority charge carrier lifetimes despite moderate carrier mobility. From the PDF we est. the size of the fluctuating symmetry broken domains to be between 1 and 3 nm in diam.
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      Yang, R. X.; Skelton, J. M.; da Silva, L.; Frost, J. M.; Walsh, A. Spontaneous Octahedral Tilting in the Cubic Inorganic Caesium Halide Perovskites CsSnX3 and CsPbX3 (X= F, Cl, Br, I). J. Phys. Chem. Lett. 2017, 8, 4720– 4726,  DOI: 10.1021/acs.jpclett.7b02423
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      Spontaneous Octahedral Tilting in the Cubic Inorganic Cesium Halide Perovskites CsSnX3 and CsPbX3 (X = F, Cl, Br, I)
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      Journal of Physical Chemistry Letters (2017), 8 (19), 4720-4726CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      The local crystal structures of many perovskite-structured materials deviate from the av. space-group symmetry. The authors demonstrate, from lattice-dynamics calcns. based on quantum chem. force consts., that all of the Cs-lead and Cs-Sn halide perovskites exhibit vibrational instabilities assocd. with octahedral tilting in their high-temp. cubic phase. Anharmonic double-well potentials are found for zone-boundary phonon modes in all compds. with barriers ranging from 108 to 512 meV. The well depth is correlated with the tolerance factor and the chem. of the compn., but is not proportional to the imaginary harmonic phonon frequency. The authors provide quant. insights into the thermodn. driving forces and distinguish between dynamic and static disorder based on the potential-energy landscape. A pos. band gap deformation (spectral blue shift) accompanies the structural distortion, with implications for understanding the performance of these materials in applications areas including solar cells and light-emitting diodes.
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      Laurita, G.; Fabini, D. H.; Stoumpos, C. C.; Kanatzidis, M. G.; Seshadri, R. Chemical tuning of dynamic cation off-centering in the cubic phases of hybrid tin and lead halide perovskites. Chem. Sci. 2017, 8, 5628– 5635,  DOI: 10.1039/C7SC01429E
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      Chemical Science (2017), 8 (8), 5628-5635CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
      Hybrid halide perovskites combine ease of prepn. and relatively abundant constituent elements with fascinating photophys. properties. Descriptions of the chem. and structural drivers of the remarkable properties have often focused on the potential role of the dynamic order/disorder of the mol. A-site cations. We reveal here a key aspect of the inorg. framework that potentially impacts the electronic, thermal, and dielec. properties. The temp. evolution of the X-ray pair distribution functions of hybrid perovskites ABX3 [A+ = CH3NH3 (MA) or CH(NH2)2 (FA); B2+ = Sn or Pb; X- = Br, or I] in their cubic phases above 300 K reveals temp.-activated displacement (off-centering) of the divalent group 14 cations from their nominal, centered sites. This symmetry-lowering distortion phenomenon, previously dubbed emphanisis in the context of compds. such as PbTe, is attributed to Sn2+ and Pb2+ lone pair stereochem. Of the materials studied here, the largest displacements from the center of the octahedral sites are found in tin iodides, a more moderate effect is found in lead bromides, and the weakest effect is seen in lead iodides. The A-site cation appears to play a role as well, with the larger FA resulting in greater off-centering for both Sn2+ and Pb2+. Dynamic off-centering, which is concealed within the framework of traditional Bragg crystallog., is proposed to play a key role in the remarkable defect-tolerant nature of transport in these semiconductors via its effect on the polarizability of the lattice. The results suggest a novel chem. design principle for future materials discovery.
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