Article pubs.acs.org/JPCC
Ab Initio Calculations of Band Gaps and Absolute Band Positions of Polymorphs of RbPbI3 and CsPbI3: Implications for Main-Group Halide Perovskite Photovoltaics Jakoah Brgoch,† Anna J. Lehner,† Michael Chabinyc,‡,† and Ram Seshadri*,‡,† †
Materials Research Laboratory and Materials Department and ‡Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, United States ABSTRACT: Lead halide perovskites have attracted great interest because of rapid improvements in the efficiency of photovoltaics based on these materials. To predict new related functional materials, a good understanding of the correlations between crystal chemistry, electronic structure, and optoelectronic properties is required. Describing the electronic structure of these materials using density functional theory provides a choice of exchange-correlation functionals, including hybrid functionals, and inclusion of spin-orbit coupling, which is critical for the correct description of band gap and absolute band positions (ionization energy). Here, various computational schemes that employ different choices of exchange-correlation and hybrid functionals, and include or exclude spin-orbit coupling were implemented to examine these effects. Using PbI2 as an initial structural model, it is found that standard exchange correlation functionals (PBE) in conjunction with spin-orbit coupling suffice to locate ionization energies efficiently through the use of slab calculations. Band gaps require the use of hybrid functionals carried out on single unit cells and spin-orbit coupling. Polymorphs of alkali metal lead halides, APbI3 (A = Rb, Cs) are examined in the cubic perovskite structure and the reduced dimensional NH4CdCl3/Sn2S3 structure with quasi-two-dimensional connectivity. The somewhat elevated Born effective charges computed for these structures suggest that while the Pb2+ 6s lone-pairs are stereochemically inert, the presence of proximal instabilities could have implications for the functional properties of these materials.
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INTRODUCTION Lead halide-based compounds with the general composition APbX3 (A = CH3NH+3 , or HC(NH2)+2 ; X = halogen) have attracted immense interest because of the rapid improvement in photovoltaic efficiency for devices incorporating organic− inorganic hybrid compounds with the perovskite-type crystal structure.1−6 A key advantage of these materials lies in the various synthetic routes such as solution processing7,8 or vapor deposition9,10 that yield highly crystalline films.11 The ease of large scale preparation and wide range of potential optoelectronic applications highlight the importance of this unique material class. Focusing specifically on the crystallographically tractable inorganic alkali metal lead halides (RbPbI3 and CsPbI3) provides a great opportunity to understand the electronic structure of these materials using density functional theory (DFT). Both compositions crystallize at room temperature in the NH4CdCl3/Sn2S3-type structure, which contains ribbons of edge-connected PbI6 octahedra and thus quasi-two-dimensional connectivity.12−14 Interestingly, CsPbI3 has a reversible phase transition to the cubic perovskite structure occurring between ≈560 and 600 K,12,14 whereas the Rb+ analogue does not undergo a transformation. Nevertheless, a majority of current computational reports rely on CsPbI3 in the high-temperature cubic structure to act as a model for the hybrid alkylammonium compounds.15−20 To the best of our knowledge, only one © 2014 American Chemical Society
published report on the CsPbI3 electronic structure considers the room temperature (orthorhombic) phase20 and none explores RbPbI3, regardless of structure. Comparing these computational studies to the experimental results, it is clear that DFT within the generalized gradient approximation (GGA) struggles to accurately reproduce the electronic structure. For example, using GGA as the exchange and correlation functional, the cubic CsSnI3 is calculated to have a band gap that underestimates the experimental value by at least 1 eV.16,21 There is a better correspondence of the calculated and experimental band gap in the lead halide materials. However, they still differ by ≈0.5 eV.22 Advanced computational methods such as the screened hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE)18 and many-body perturbation theory (GW method)19,21,23 have been employed to achieve better correspondence with the experimental data. Regardless of the exchange-correlation functional, some computational reports also omit spin-orbit coupling (SOC), which is critical in 6s, 6p systems. The sizable effects of SOC were experimentally observed in these compounds more than a decade ago24,25 but have only recently been included in the studies.19,20,26 Splitting the unoccupied degenerate p-orbitals Received: September 2, 2014 Revised: October 30, 2014 Published: November 7, 2014 27721
dx.doi.org/10.1021/jp508880y | J. Phys. Chem. C 2014, 118, 27721−27727
The Journal of Physical Chemistry C
Article
PBE as well as a screened hybrid functional (HSE06).38 Additionally, the importance of SOC in these heavy lead halides has been identified previously.20 Thus, SOC implemented in conjunction with the two exchange and correlation functionals, e.g., PBE+SOC and HSE06+SOC, is required to describe the electronic structure. After achieving a convergence criteria of 0.01 meV, absolute band positions were determined using the macroscopic averaging technique of Baldereschi and coworkers.39 The calculated ion-clamped (high frequency) dielectric tensors (ϵ∞ ij ) including local field effects were determined from the change in polarization (Pi) in response to an applied finite electric field (ϵj = 0.005 eV/Å). The values of ϵ∞ ij are calculated following eq 1:
through SOC increases the band dispersion of these compounds, leading to further issues describing the electronic structure via DFT.20 Accounting for SOC along with manybody effects through the GW approximation is a formidable task due to the high computational cost associated with these methods.23,27 Thus, an examination of the most reliable and computationally affordable calculation method to reproduce band gaps is crucial for enabling the rational design of new, efficient optoelectronic materials. In addition to the correct estimation of the band gaps for these compounds, the absolute band position with respect to a vacuum is essential. To the best of our knowledge, an investigation of the influence of hybrid functionals on the absolute band positions has been largely overlooked for these compounds. Here, we commence with studying the binary lead(II) iodide PbI2 as a model compound to identify the best computational scheme. The methods identified using PbI2 are then employed to examine the more structurally complex alkali metal lead iodides, RbPbI3 and CsPbI3. The two experimentally reported crystal structures, a room temperature orthorhombic structure (Pnma; called δ)8,14 and the high-temperature cubic (perovskite) structure (Pm3̅m; called α),14 allow the complex relationship between crystal structure, composition, reducing dimensionality, and electronic structure to be examined. Additionally, the presence of these polymorphs along with the potential stereochemical activity of Pb2+ lone-pair electrons may indicate a structural instability. The dielectric response including the Born effective charge tensors (Z*) and dielectric tensors (ϵ∞) of these materials are presented to identify potential (incipient) instabilities that impact optoelectronic properties. The importance of computational method is highlighted throughout this work to aid future theoretical and experimental efforts covering this family of technologically important compounds.
ϵ∞ ij = δij +
4π ∂Pi , ϵ0 ∂ϵj
i, j = x, y, z (1)
To compare the effect of the exchange-correlation functional as well as the inclusion of SOC on the dielectric response, ϵ∞ ij was determined using the same four computational schemes described above. The Born effective charge tensor (Zij*), which is a signature of a ferroelectric instability, was also calculated using the four computational schemes following eq 2: Zij* =
Ω ∂Pi , e ∂uj
i, j = x, y, z (2)
where Ω is the primitive cell volume, Pi is the macroscopic polarization per unit cell, and u is the displacement along the direction j.40
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RESULTS AND DISCUSSION The Electronic Structure of Lead Halides. The atomic positions and unit cell volumes of all compounds described in the Methods section were optimized, ensuring they are in their lowest-energy (DFT) ground-states. The lattice parameters and unit cell volumes (Table 1) of the alkali metal lead iodides are larger than the experimentally measured volumes by 4% to 8%. These differences are within the acceptable errors produced by DFT and are similar to a number of previous structural optimizations of similar compounds.16,18PbI2, however, differs with an optimized volume that is nearly 13% larger than the experimental structure. The discrepancy is almost entirely due
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METHODS DFT calculations were performed using the Vienna ab initio Simulation Package (VASP)28−31 with the wave functions described by a plane-wave basis set and the ionic potential described by the projector augmented wave (PAW) potentials of Blöchl.32,33 The compounds investigated here include PbI234 in space group P3̅m1 as well as both RbPbI3 and CsPbI3 in the α phase (Pm3̅m) and the δ phase (Pnma).8,14 All of the compounds have been experimentally reported previously except for α-RbPbI3; thus, α-RbPbI3 was modeled here as a hypothetical structure. Additionally, TiO2 (anatase) was calculated to compare absolute band positions among all the materials examined. All of the compounds were optimized using the exchange and correlation described by the Perdew− Burke−Ernzerhof generalized gradient approximation (PBE)35 until the residual forces were less than 0.01 eV/Å on each atomic site. The energy cutoff of the plane wave basis set was 400 eV. A Γ center k-mesh grid with a minimum of 100 kpoints was employed with the convergence criteria set at 0.001 meV. Crystal structures were visualized using the software VESTA.36 Absolute band positions were obtained by constructing slab models of the respective structures using a half-filled 1 × 1 × 8 supercell with four optimized unit cells containing the structure with four additional vacuum (empty) unit cells.37 The slab models were examined using four computational schemes to ensure correct assignment of the absolute band positions. Exchange and correlation for the supercells was described by
Table 1. Optimized (PBE-Level) Unit Cell Volumes for PbI2, RbPbI3, and CsPbI3 in the α and δ Structure8,14,a vol. (Å3)
compound PbI2
P3̅m1
α-RbPbI3
Pm3m ̅
α-CsPbI3
Pm3̅m
δ-RbPbI3
Pnma
δ-CsPbI3
Pnma
opt. exp. opt. exp. opt. exp. opt. exp. opt. exp.
143.97 125.51 260.42 260.42 248.79 915.88 854.23 963.44 892.71
difference
ref.
12.8%
34
4.46%
14
6.73%
14
7.34%
14
The hypothetical α-RbPbI3 was optimized using the Cs+ analogue as a starting point. The experimental unit cell volumes and %-difference are presented with associated references. a
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dx.doi.org/10.1021/jp508880y | J. Phys. Chem. C 2014, 118, 27721−27727
The Journal of Physical Chemistry C
Article
to the optimized c-lattice parameter, which is 9% larger than experiment. Because PbI2 is a layered structure (P3̅m1; CdI2type), illustrated in Figure 1a, the overestimation is anticipated.
structures, ensuring the electronic structure is a representative model. To compare the electronic structure of the multiple compositions explored here, the valence band maximum (VBM) was normalized by the average electrostatic potential (Φel), which is a combination of the ionic and Hartree potentials. The four methods described in the experimental section were implemented to calculate Φel and confirm the absolute band positions. Figure 2 indicates Φel is nearly
Figure 1. Crystal structures of (a) PbI2, and APbI3 (A = Rb, Cs) in the (b) δ-form (Pnma) and (c) α-form (Pm3̅m). The large spheres represent Rb+ or Cs+, spheres inside polyhedra are Pb2+, and the small spheres are I−.
Layered compounds held together by van der Waals forces are not well described by conventional DFT. The addition of semiempirical dispersion terms, e.g., DFT+D2,41 could improve the optimization, although these corrections were not accounted for here. Employing the HSE06 hybrid functional also slightly improves (
