Strain Tuning of TinâHalide and LeadâHalide Perovskites: A First

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Strain Tuning of Tin-Halide and Lead-Halide Perovskites: A First-Principles Atomic and Electronic Structure Study Christopher Grote, and Robert F. Berger J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b07446 • Publication Date (Web): 18 Sep 2015 Downloaded from http://pubs.acs.org on September 19, 2015

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The Journal of Physical Chemistry

Strain Tuning of Tin-Halide and Lead-Halide Perovskites: A First-Principles Atomic and Electronic Structure Study Christopher Grote and Robert F. Berger∗ Department of Chemistry, Western Washington University, Bellingham, WA E-mail: [email protected]

KEYWORDS: perovskite, photovoltaic, density functional theory

∗

To whom correspondence should be addressed

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Abstract Using density functional theory (DFT)-based calculations, we investigate the effects of biaxial strain and the accompanying structural distortions on the energy landscapes, band gaps, and band edges of the perovskite photovoltaic materials CsSnI3 and CsPbI3 . We show that biaxial strains within ±3% of the respective cubic lattice parameters can alter band gaps by several tenths of an eV mainly through the tuning of antibonding interactions in the valence band maximum, while temperature-controlled octahedral rotations further widen band gaps. Notably, we predict that biaxial strain, particularly tensile strain at low temperature, has the potential to induce ferroelectric polarization in these materials. Throughout this work, we rationalize trends in electronic structure based on the character and symmetry of band-edge crystal orbitals, and discuss their implications with respect to broader classes of materials.

Introduction Solar cells incorporating tin- and lead-halide light absorbers crystallizing in the perovskite structure (ABX3 , as in Fig. 1) have recently emerged as some of the most promising photovoltaic technologies. 1–4 Since their initial use in 2009, 5 improvements in perovskite solar cell architectures have increased their function dramatically, recently eclipsing 20% efficiency. 6 The current state of the art combines methylammonium lead bromide ([CH3 NH3 ]PbBr3 ) with formamidinium lead iodide ([NH2 CHNH2 ]PbI3 ). 7 The success of tin- and lead-halide perovskites as visible light absorbers can be attributed in part to the fact that their band gaps lie near 1.4 eV, the ideal optical gap for solar absorption by Shockley and Queisser’s analysis. 8 For example, a number of tin- and lead-iodide perovskites (A = Cs+ , CH3 NH+ 3; B = Sn2+ , Pb2+ ; X = I− ) have been measured to have optical gaps between 1.2 and 1.7 eV. 9 While photovoltaic efficiency is quite complex and not governed entirely by band gap, band gap is a widely used proxy for a material’s ability to absorb light in the solar spectrum. The continued improvement of perovskite solar cell efficiencies rests not only on advances 2


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in the engineering of solar cell architectures, but also on the development and understanding of novel routes to tune the electronic structure and band gap of tin- and lead-halide perovskites. To achieve this tunability, a variety of approaches can be taken to modify these materials. The effects of elemental substitution at the A, B, and X sites have been demonstrated and rationalized in a number of experimental 5,9,11,12 and computational studies. 13–18 Our previous computational work has looked at the effects of various modes of quantum confinement. 19 To complement this previous work, we examine in this article the effects of biaxial strain and structural distortions. In its most controlled form, strain engineering of perovskite compounds can be achieved via the growth of epitaxial thin films (typically of transition metal oxides), which take up the in-plane lattice parameters of the substrate on which they are grown. 20 More generally, internal strains and distortions can be effected in other ways, such as the growth of superstructures and nanostructures, or the substitution of elements of different sizes. For the prototypical perovskite oxide photocatalyst SrTiO3 , achievable biaxial strains and the accompanying structural distortions have been predicted to alter the band gap by several tenths of an eV. 21 Similar tunability of the band gaps of tin- and lead-halide perovskites would be significant in tailoring their electronic structure for solar cell applications. However, because the character of their band-edge electronic states is fundamentally different from those of SrTiO3 , the extent of this tunability remains an open question for tin- and lead-halide perovskites. In this work, we therefore undertake a detailed study of 1) the structural energetics of distortions in biaxially strained tin- and lead-halide perovskites, and 2) the resulting effects (and explanations thereof) on near-gap electronic structure and band gap.

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Computational Methods Crystal structures are optimized and electronic properties computed using density functional theory (DFT) within the Perdew-Burke-Ernzerhof (PBE) functional, 22 using the VASP package 23 and PAW potentials. 24 Electrons taken to be valence are 5s2 5p6 6s2 of Cs, 4d10 5s2 5p2 of Sn, 5d10 6s2 6p2 of Pb, and 5s2 5p5 of I. A plane-wave cutoff of 500 eV is used throughout. Unit cells containing a single perovskite formula unit (e.g., Fig. 1a) are treated with a Γ-centered 6 × 6 × 6 k-point mesh. For structures requiring larger unit cells (e.g., Fig. 1b,c), proportionally fewer k-points are used. Zone-center phonon modes and frequencies are computed using density functional perturbation theory. Following previous work on related systems, 21,25–27 biaxial strain is modeled using periodic calculations in which in-plane lattice parameters are fixed at lengths ranging within ±3% of the respective cubic perovskite lattice parameters, while the perpendicular axis and atomic positions are allowed to relax. Further constraints on structural geometries and symmetries are discussed on a case-by-case basis throughout the paper. Space groups are determined using the software FINDSYM. 28 Absolute band-edge energies are computed in two steps. At each value of strain, an undistorted slab of five perovskite layers with unrelaxed SnI2 - or PbI2 -terminated [001] surfaces and 15 Å of vacuum space (an example of which is shown in the Supporting Information) is used to reference a bulk-like semicore Cs 5s state to the vacuum level. A corresponding bulk calculation at the same strain is then used to compute band-edge energies relative to the same semicore state. Using these two calculations, band edges can be determined relative to the fixed vacuum level. As has been well documented, DFT-PBE typically underestimates band gaps relative to experimental optical gaps. Less approximate levels of theory can to some degree alleviate this problem for the classes of compounds studied here. When used together, spin-orbit coupling (which by itself underestimates gaps more severely) and GW quasiparticle corrections (which by themselves overestimate gaps) have proven effective. 29–31 However, we find DFT-PBE satisfactory for this work (and our previous work on tin- and lead-halide perovskites 19 ) 5



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for two reasons. First, standard DFT functionals are capable of qualitatively capturing perovskite band gap trends under strain, which result from changes in the symmetry and character of band-edge orbitals. 21 Second, for tin- and lead-halide perovskites, spin-orbit coupling and GW corrections have been shown to change neither the orbital character of the band-edge states nor the k-points at which the band gaps occur. 29–31

Results and Discussion Compounds and structures of interest We limit our focus in this work to the compounds CsSnI3 and CsPbI3 for several reasons. They have been measured to have nearly ideal optical band gaps for solar absorption, 9 they avoid the geometric complexities of polyatomic A sites, and most importantly, they are representative of broader classes of tin- and lead-halide perovskites varying in their Aand X-site composition. While the quantitative band gaps and energy landscapes reported here depend in part on the size and electronegativity of A- and X-site atoms, the qualitative features of their near-gap electronic band structure (and by extension, their electronic trends with respect to strain and distortion) have proven to be largely independent of A- and X-site identity. 14–16,29 In order to examine the effects of biaxial strain on these compounds, we must first consider their likely structural distortions. The experimentally observed phases of CsSnI3 are shown in Fig. 1. CsSnI3 takes up the cubic perovskite structure (P m3m, Fig. 1a) above 426 K, a tetragonal perovskite structure with Sn–I octahedral rotations (P 4/mbm, Fig. 1b) between 351 and 426 K, and an orthorhombic perovskite structure with additional octahedral tilting (P nma, Fig. 1c) below 351 K. 32 The images on the left side of Fig. 1 emphasize the differences in unit cell size, while those on the right highlight the octahedral rotation patterns. CsPbI3 also takes up the cubic perovskite structure at high temperature (above 602 K). 33 Though its low-temperature phase differs in connectivity somewhat from the perovskite structure, 6


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examination of the tetragonal and orthorhombic perovskite phases of CsPbI3 is relevant nonetheless, as the closely related compound [CH3 NH3 ]PbI3 displays cubic → tetragonal → orthorhombic transitions analogous to those in Fig. 1. 34 The DFT-PBE total energies per formula unit and band gaps of these phases of CsSnI3 and CsPbI3 are shown in Table 1. Their computed and experimental unit cell parameters are shown in the Supporting Information. For CsSnI3 , band gaps are in qualitative agreement with past DFT results, 14,16 and the energetic preference for the orthorhombic phase is consistent with the fact that this phase is observed at low temperature. Computed unit cell axis lengths of the phases of CsSnI3 slightly overestimate the corresponding experimental values 32 (as expected in DFT-PBE), in all cases by less than 3%. Trends in structural energy and band gap are similar for CsSnI3 and CsPbI3 , in that distortions lower the structural energy while widening the band gap. The observation that octahedral rotations widen the band gaps of tin-iodide perovskites has previously been traced to perturbations of the band-edge crystal orbitals that push apart their energies upon lowering of symmetry. 13 These differences in the cubic, tetragonal, and orthorhombic band gaps of each compound suggest that even in the absence of biaxial strain, band gaps can be tuned to some extent via temperature-controlled phase transitions. Table 1: DFT-PBE energies per formula unit and band gaps of the fully relaxed cubic, tetragonal, and orthorhombic phases of CsSnI3 and CsPbI3 . Energies are reported relative to the cubic phase of each compound. Phase Cubic CsSnI3 Tetragonal Orthorhombic Cubic CsPbI3 Tetragonal Orthorhombic

Energy per formula unit, relative to cubic (eV) — –0.036 –0.056 — –0.080 –0.117

DFT-PBE band gap (eV) 0.48 0.62 0.81 1.49 1.60 1.82

When considering how biaxial strain interacts with the structural energetics of these distortions, we apply strain perpendicular to the [001] crystallographic direction of the cubic 7



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structure (i.e., one of the cubic perovskite unit cell axes). As a baseline, the compounds are strained without octahedral rotations, analogous to the cubic phase. We then consider structures with octahedral rotations analogous to the tetragonal phase, and rotation/tilt patterns analogous to the orthorhombic phase, in which octahedra rotate around either the [001] or [100] axis. In addition to straining the experimentally observed phases of these compounds, we also consider hypothetical structures with ferroelectric polarization (relative translation of the cation and anion sublattices) along the [001], [100], and [110] directions. Strain-induced ferroelectric polarization has been computationally predicted 25–27 and experimentally demonstrated 20,35,36 for perovskites such as SrTiO3 and BaTiO3 , and represents a topic of significant interest. The possibility of accessing ferroelectric phases of CsSnI3 and CsPbI3 under strain is further motivated by the fact that ferroelectric distortions are seen in the ground-state structures of closely related CsGeX3 perovskites. 37 The biaxially strained perovskite phases we consider in this work, along with their space groups, are enumerated in Table 2. Table 2: The biaxially strained perovskite structures computed in this work, along with their space groups (as determined using the software FINDSYM). 28 Octahedral rotations or ferroelectric polarization occur around or along the crystallographic directions of the cubic structure specified in the table. In all cases, strain is applied perpendicular to [001]. Structure No distortion Rotation [001] Rotation [100] Rotation/tilt [001] Rotation/tilt [100] Polarization [001] Polarization [100] Polarization [110]

Space group P 4/mmm (#123) P 4/mbm (#127) P bam (#55) P nma (#62) P 21 /m (#11) P 4mm (#99) P mm2 (#25) Amm2 (#38)

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Structural energetics and the possibility of ferroelectric polarization The total energies per formula unit of CsSnI3 and CsPbI3 as a function of biaxial strain are shown in Fig. 2a. Each curve represents one of the phases listed in Table 2, geometrically optimized with in-plane lattice parameters ranging from ±3% biaxial strain relative to the optimized cubic unit cell axis. Throughout the range of strain, both compounds are stabilized by octahedral rotations and tilts, ferroelectric distortions, and combinations thereof. Though the qualitative energy landscape is similar for the two compounds, energetic stabilizations associated with octahedral rotations and tilts are larger for CsPbI3 . This can be rationalized based on Goldschmidt tolerance factors. 38 Because Pb2+ is larger than Sn2+ , CsPbI3 has a smaller tolerance factor (defined as t =

√ rA +rX ) 2(rB +rX )

than CsSnI3 , and is there-

fore more prone to octahedral distortions. Consistent with past work on SrTiO3 and related compounds, 21,25–27,39 both octahedral rotations and ferroelectric distortions proceed more favorably around and along the [001] axis under compressive strain, and an axis in the plane of strain under tension. Not surprisingly, rotation/tilt phases (which have more structural degrees of freedom than the undistorted and rotation phases) are lowest in energy across the entire range of biaxial strain. Fig. 2b reports these same structural energies relative to the undistorted phases of CsSnI3 and CsPbI3 at each value of strain, to emphasize the stabilization gained through distortion. Based on these results, if one hopes to induce ferroelectric polarization in these compounds, it is most likely to be seen under tensile strain. Tensile strain suppresses the stabilizing effect of octahedral rotation and tilting, while enhancing the stabilizing effect of ferroelectric distortion. Though the total energies of the rotation/tilt phases of CsSnI3 and CsPbI3 continue to be lower than those of ferroelectric phases even at +3% strain, there is a possibility that ferroelectric distortion can further stabilize the rotation/tilt phases. To check this, we have computed the zone-center phonon modes of the rotation/tilt phases of CsSnI3 and CsPbI3 at +3% strain. Indeed, each of these phases has imaginary phonon frequencies corresponding to ferroelectric distortive modes (59i − 61i cm−1 in CsSnI3 , 37i − 39i cm−1 in CsPbI3 ). When 9




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the rotation/tilt phases at +3% strain are relaxed with ferroelectric distortions along their soft phonon modes to identify the lowest-energy structures, their total energies are stabilized by an additional 0.035 and 0.014 eV per formula unit for CsSnI3 and CsPbI3 , respectively. This suggests that, especially at low temperatures, tensile biaxial strain can likely induce polarization in tin- and lead-halide perovskites.

Band edges and band gaps under strain Before rationalizing the band gaps of biaxially strained CsSnI3 and CsPbI3 , it is worthwhile for two main reasons to consider the character and energy of their band-edge electronic states. First, the effects of biaxial strain on band gaps (and hence the solar absorption behavior) can be traced to the manner in which strain alters band-edge orbitals. Second, the tunability of the absolute band-edge energies, not only the band gap, is critical to the design and optimization of photovoltaic device configurations. 40,41 The computed band-edge crystal orbitals of cubic CsSnI3 , which are nearly identical to those of cubic CsPbI3 , are shown in Fig. 3a (band structures are shown in the Supporting Information). The band gaps of both compounds are direct, as both band edges reside at k-point R = ( 12 , 12 , 12 ). The valence band maximum (VBM) consists of an antibonding combination of Sn 5s and I 5p states, while the conduction band minimum (CBM) consists of nonbonding, threefold degenerate Sn 5p states. 16 Fig. 3b illustrates how the absolute energies of these band edges change relative to the vacuum level (computed as described in the Methods section) when the cubic perovskite unit cell is biaxially reshaped without further distortions. As one might expect, the VBM energy is raised under compressive strain, when the antibonding interactions between Sn2+ or Pb2+ and four of its nearest-neighboring I− anions are forced to shorter distances, and is lowered under tensile strain. The CBM is less affected by strain because changes in bond lengths have little bearing on nonbonding orbitals. When the computed band gaps of these same structures (i.e., the energy differences between VBM and CBM) are plotted 11




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Conclusions This article is intended to serve two main purposes. Fundamentally, the work explores and rationalizes geometric routes to tune the electronic structure of a broad class of perovskite materials. In addition, our results carry implications for the design of photovoltaic materials. Fig. 4 suggests that, by engineering the geometric environment to produce biaxial strain and utilizing temperature-controlled phase transitions, the band gaps of tin- and lead-halide perovskites can be tuned to essentially any value in the solar spectrum. In combination with other modifications to these materials that have been explored previously, such as doping and substitution at the A and X sites and quantum confinement, synthetic chemists and materials scientists can gain increasing levels of control and optoelectronic tunability in these classes of materials.

Acknowledgments We gratefully acknowledge Western Washington University and the Research Corporation for Scientific Advancement for financial support, the latter through a Cottrell College Science Award.

Supporting Information Available Representative structural images, computed and experimental geometric parameters, and electronic band structures can be found in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org/.

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(38) Goldschmidt, V. M. "Die Gesetze der Krystallochemie". Naturwissenschaften 1926, 14, 477–485. (39) Pertsev, N. A.; Tagantsev, A. K.; Setter, N. "Phase Transitions and Strain-Induced Ferroelectricity in SrTiO3 Epitaxial Thin Films". Phys. Rev. B 2000, 61, R825–R829. (40) Brgoch, J.; Lehner, A. J.; Chabinyc, M.; Seshadri, R. "Ab Initio Calculations of Band Gaps and Absolute Band Positions of Polymorphs of RbPbI3 and CsPbI3 : Implications for Main-Group Halide Perovskite Photovoltaics". J. Phys. Chem. C 2014, 118, 27721– 27727. (41) Butler, K. T.; Frost, J. M.; Walsh, A. "Band Alignment of the Hybrid Halide Perovskites CH3 NH3 PbCl3 , CH3 NH3 PbBr3 and CH3 NH3 PbI3 ". Mater. Horiz. 2015, 2, 228–231.

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Strain Tuning of TinâHalide and LeadâHalide Perovskites: A First

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