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Dependence of the Electronic and Optical Properties of Methylammonium Lead Triiodide on Ferroelectric Polarization Directions and Domains: A First Principles Computational Study Jie Jiang,† Ruth Pachter,*,† Yurong Yang,‡ and Laurent Bellaiche‡ †

Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio 45433, United States ‡ Department of Physics and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, United States S Supporting Information *

ABSTRACT: Organic−inorganic perovskites, and in particular methylammonium (MA) lead triiodide, recently emerged as promising for thinfilm solar cell applications and, as a result, have attracted much attention. However, some important phenomena have been less examined in these systems, e.g., effects of ferroelectric domains on the optoelectronic properties. In this work, we investigate the effects of the polarization direction in single domains, and of uncharged and charged ferroelectric domains, on the electronic and optical properties of MA lead triiodide by first principles calculations. Highly accurate quasiparticle band gap calculations enabled characterization of the electronic structure of charged and uncharged domains in comparison to single domains. Additionally, analysis of the effects of a potential on the Born effective charges and respective density of states provided an understanding of changes in the band gap, as dependent on the type of domain, and on the MA moiety direction. Agreement between experimental and calculated optical spectra was achieved by inclusion of electron−hole interactions, also discerning specific transitions. However, due to the flexibility in the MA moiety’s orientation that causes spectral broadening, consideration of a statistical ensemble of configurations is required, which is not taken into account in a single computation. Indeed, our analysis in considering a number of MA directions leads to better agreement with experiment. The calculations predict that the optical response is rather sensitive to the type and size of ferroelectric domains, which implies that such a response could be used for their characterization, thus calling for further experimental exploration.

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INTRODUCTION

However, although effects of ferroelectric domain structures on the band gap were calculated for orthorhombic unit cells at the density functional theory (DFT) level,10 an understanding of the influence of ferroelectric domains on the optical response is still lacking. In this work, we analyzed effects of domain structures on quasiparticle band gaps, optical spectra, and optical constants, considering structures based on optimization of the MAPbI3 cubic phase, including the structure of lowest energy (henceforth single domain, SD), as well as structures with the C−N bond in MA along the [100], [111], and [110] directions, thereafter so-called (100)SD, (111)SD, and (110)SD, which resulted in varying space groups upon optimization. The cubic phase was noted as the phase of practical interest11 occurring at higher temperatures,12 but the orthorhombic phase occurring at lower temperatures13 and the transition from orthorhombic to tetragonal were also previously studied.14 In

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Frost et al. recently suggested that ferroelectric polarization at the nanometric size regime could explain, in part, the high performance of hybrid halide perovskite solar cells, by enabling efficient electron and hole separation pathways, thus reducing recombination of electrons and holes, although the proposition is still under some debate.2,3 The nature of the polar order in these materials is partially unclear4 because of the flexible orientational order of the organic molecular moieties.5 Pecchia et al.6 have shown by two-dimensional Monte Carlo simulations with a dipole−dipole interaction energy that formation of nanodomains occurs with domain periods of about 8−10 nm, while drift diffusion simulations demonstrated different current pathways, which can lead to increased photoconversion efficiencies. At the same time, experimentally, piezoresponse force microscopy measurements demonstrated the presence of ferroelectric domains in methylammonium (MA) lead triiodide (CH3NH3PbI3), which was shown to be reversibly switched by poling with dc biases.7−9 © XXXX American Chemical Society

Received: May 11, 2017 Revised: June 1, 2017

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DOI: 10.1021/acs.jpcc.7b04557 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. Primitive cells for the MAPbI3 SD structure (a), where defining the MA orientation (C−N bond direction) as unit vector τ⃗, the SD structure has an orientation of τ⃗ = [sin(22.5°), 0, cos(22.5°)], i.e., θz = 22.5°, and for the (100)SD, (111)SD, and (110)SD configurations (b)−(d). MAPbI3 two-unit UD (e), two-unit CD (f), four-unit UD (g), and four-unit CD (h) structures. In the two-unit domains, the left cells denote the T1 region and the right cells the T2 region (see discussion in the text). In the four-unit domains, the left two cells comprise the T1 region and the corresponding right two cells the T2 region. The four-unit UD (i) and CD (j) structures after MA rotation were not optimized.

observation implies that such a response could practically be used to distinguish ferroelectric domains, thus motivating future experimental investigation.

addition, we investigated uncharged and charged domain (UD and CD, respectively) walls, having varying relative MA polarizations and supercell sizes. DFT and GW (Green’s function approximation with a screened Coulomb interaction W), as well as GW-BSE (Bethe−Salpeter equation), levels of theory were employed here. The accurate calculations, based on careful benchmarking, combined with analysis of the origin of the predicted changes, enabled characterization of band gaps and optical spectra for ferroelectric domains, in comparison to single MAPbI3 domains. Agreement between the calculated (GW-BSE level) and experimental spectra was demonstrated. The computed results predict that the optical response is rather sensitive to the type and size of ferroelectric domains. This

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COMPUTATIONAL DETAILS Calculations were carried out using the Vienna Ab initio Simulation Package VASP 5.415,16 within the DFT framework. The Kohn−Sham equations17,18 are solved using a plane-wave basis set with an energy cutoff of 550 eV, and the projectoraugmented-wave (PAW) potential was applied.16 For comparison with previous work,19 crystal structures were optimized with the PBEsol20 functional (Perdew−Burke−Ernzerhof (PBE)21 revised for solids). The k-point sampling used was 8 B


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consider a so-called zigzag domain, where θz = ϕ in T1, and θz = 180° − ϕ in T2 (see Figure 1f), having polarization vectors τ⃗1 = [sin(ϕ), 0, cos(ϕ)] and τ⃗1 = [sin(ϕ), 0, −cos(ϕ)], respectively. In addition, a parallel pattern will form with θz = ϕ in T1 and θz = 180° + ϕ in T2, with polarization vectors τ⃗1 = [sin(ϕ), 0, cos(ϕ)] and τ⃗1 = [−sin(ϕ), 0, −cos(ϕ)]. For a relatively large charged domain, such as a six-unit domain, a parallel domain was considered,10 while for the smaller two-unit and four-unit parallel domains, the polarization z-components in both domain regions approached zero after structure optimization as θz ≈ 90°. Here we studied two- and four-unit charged domains as zigzag domains. The static dielectric tensor was analyzed for the SD, (100)SD, (111)SD, and (110)SD structures, considering ionic and electronic contributions (εαβ = εαβ(ionic) + εαβ(electronic)). The ionic contribution is derived by calculation of the interatomic force constants using density functional perturbation theory, and the electronic contribution (or ion-clamped static dielectric tensor) is taken from the dielectric function εαβ(ω) at ω = 0, where εαβ(ω) was calculated at the G0W0-BSE +SOC level, as discussed below. The diagonal elements are listed in Table 1 (the small off-diagonal values are not shown).

× 8 × 8 for the single domain structures, 8 × 8 × 4 for two-unit domains, and 8 × 8 × 2 for four-unit domains. Structural parameters were converged to within 0.01 eV/Å for each atom. Pb 5d orbitals were explicitly included in the PAW potential22 in all calculations. Dielectric functions were calculated at the RPA (random phase approximation), TDDFT (time-dependent DFT) including local field effects, and the GW-BSE23 levels of theory. Nonself-consistent G0W0 evaluations were performed on top of the computationally less demanding DFT calculation, as well as fully self-consistent GW (sc-GW) calculations. GW and GWBSE calculations were performed within the PAW framework.24,25 The ENCUTGW parameter (energy cutoff for response function), k-point sampling, and NBANDS (number of bands) were tested for convergence of the quasiparticle band gap to within 10 meV; ENCUTGW = 267 eV, and NBANDS = 160 for the SD, 198 for two-unit domains, and 280 for four-unit domains. k-Point samplings were 4 × 4 × 4 for the single domains, 4 × 4 × 2 for two-unit domains, and 4 × 4 × 1 for four-unit domains. A dense k sampling, e.g., 8 × 8 × 8 for the single domain structures, is necessary for an accurate geometry prediction, but can be reduced for electronic structure calculations. Spin−orbit coupling (SOC) was included in the calculations.

Table 1. Diagonal Components of the Static Dielectric Tensor for the SD Structure, and along Its [100], [111], and [110] Directions

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RESULTS AND DISCUSSION The structures considered are shown in Figure 1 (coordinates, lattice vectors, and space groups are summarized in Table S1 in the Supporting Information). Lattice constants of the optimized SD structure (Figure 1a) are found to be ax = 6.37 Å, ay = 6.24 Å, and az = 6.28 Å, which is consistent with the trend in previous calculations10 having ax = 6.39 Å, ay = 6.36 Å, and az = 6.45 Å, as well as with the experimental value of 6.26 Å.12 Results for structures with the MA moiety along different directions (Figure 1b−d) are ax = 6.30, ay = 6.29, and az = 6.29 Å for the (100)SD structure and ax = 6.28, ay = 6.28, and az = 6.28 Å for the (111)SD structure versus ax = 6.28, ay = 6.28, and az = 6.29 Å for the (110)SD structure. These values are consistent with the work of Motta et al.26 that provide ax = 6.34, ay = 6.34, and az = 6.34 Å for the (111)SD structure and ax = 6.28, ay = 6.38, and az = 6.34 Å for the (110)SD structure, although some deviations are noted. The lattice constants are also consistent with the work of Brivio et al.19 with a = 6.29 Å for the (100)SD structure, a = 6.28 Å for the (111)SD structure, and a = 6.26 Å for the (110)SD structure. Total energies calculated at the PBE+SOC level indicate that the energy relative to the ground state (SD) is 29 meV/cell for the (100)SD structure, 41 meV/cell for the (111)SD configuration, and 49 meV/cell for the (110)SD arrangement. We predict that the (100)SD structure has the lowest energy of the three structures, in agreement with previous work.19 Also, our relative energy order for the three structures agrees with Bechtel et al.27 after consideration of both of MA translation and on-axis rotation. The polarization in the domain structures is shown in Figure 1e−j) with the MA orientation in the xz-plane and the z-axis in the domain direction. The PbI3 network between the T1 and T2 regions is defined as a domain wall. In a UD, θz = ϕ for MA in T1 and θz = −ϕ for MA in T2 (see Figure 1e), where θz is the angle between the C−N bond and the z-axis and ϕ is the value of θz in T1. The polarization unit vectors τ⃗1 and τ⃗2 can be expressed as τ⃗1 = [sin(ϕ), 0, cos(ϕ)] and τ⃗2 = [−sin(ϕ), 0, cos(ϕ)]. At the domain wall, the polarization along the z direction is head-to-tail or tail-to-head for an UD. For a CD, we

ionic contribution

electronic contribution

total

εxx εyy εzz εxx εyy εzz εxx εyy εzz εav

SD

(100)SD

(111)SD

(110)SD

13.84 15.70 24.19 5.64 6.61 6.48 19.48 22.31 30.67 24.15

22.09 19.34 20.36 10.60 6.76 6.07 32.69 26.10 26.43 28.41

30.42 29.60 29.38 17.87 17.88 17.88 48.30 47.48 47.26 47.68

15.37 15.27 22.05 6.94 6.84 11.13 21.11 21.34 26.75 23.06

The average dielectric constant for the SD structure is large, with a value of 24.15. The values of εxx, εyy, and εzz follow the symmetry of polarization, where εxx is different from εyy and εzz for the (100)SD structure and εzz is different from εxx and εyy for the (110)SD structure, while εxx, εyy, and εzz are similar for the (111)SD structure. The average dielectric constant of 47.68 for (111)SD is particularly large, partially resulting from the molecular dipole in the structure, thus having stronger screening of electric fields. Berry phase calculations28,29 for the SD structure resulted in a polarization amplitude of 0.124 C/m2, consistent with previous work,10 and an angle of θz = 19°, where the dipole is almost aligned along the C−N bond (within 3.5°), so that p⃗ = [0.039, 0.0, 0.118] = 0.124 × [sin(19°), 0, cos(19°)] C/m2. The polarization is not quite along the C−N bond in the MA moiety because of the pseudocubic structure. Regarding domain structures, to assess differences in the dynamic polarization, Born effective charge tensors30 were considered. Born charges are defined as the variation of the macroscopic electric polarization with respect to displacement in the atomic sublattice at zero macroscopic field, and are given by C


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The Journal of Physical Chemistry C Zκ*, ij =

Ω ∂pi e ∂uκj

although there are small variations for the UD structures, larger differences are noted for the CDs as compared to UDs, for example, for the PbI3 group. For the SD structure, the isotropic components of the Born charges for −CH3 and −NH3 are +0.465 and +0.786, respectively. By taking the C−N bond length of 1.48 Å as the dipole length, we obtain a molecular dipole of 4.43 D, with similar results for the two-unit UD and CD structures, for both T1 and T2. Band gaps (Eg) for MAPbI3, calculated at the PBE, PBE +SOC, and sc-GW+SOC levels of theory, as dependent on the MA orientation and domain structures, are summarized in Table 3. The PBE results demonstrate that inclusion of SOC decreases Eg in all cases. The fundamental band gap for the SD structure was calculated as 1.67 eV by sc-GW, including Pb 5d orbitals, where their exclusion results in a band gap of 1.46 eV. Considering measured exciton binding energies of 20−60 meV,32 the band gap agrees well with the experimental optical gaps of ca. 1.57−1.65 eV,11,33−38 and with a previously calculated value of 1.67 eV.19 The sc-GW+SOC band gaps are 1.50, 1.27, and 1.74 eV for the (100)SD, (111)SD, and (110)SD structures, respectively, thus quite sensitive to the MA orientation. Although the PBE+SOC results significantly underestimate the band gaps, the ordering is still consistent among the PBE+SOC, G0W0+SOC, and sc-GW+SOC levels of theory. In comparing band gap shifts for UD and CD structures versus the SD structure, i.e., Eg − ESD in Table 3, where ESD denotes the band gap for the lowest energy SD structure, a negligible shift is noted for UDs at the DFT level, but there is a larger corresponding response for CDs. The band gap shift in UDs is not sensitive to the MA direction and domain size, unlike for CDs. To explain these results, we analyzed the local potential and layer-by-layer local density of states (LDOS) along the domain direction at the PBE+SOC level for four-unit domains, as shown in Figure 2. For UDs (Figure 2a) there is no induced charge, and the potential and LDOS are the same for the different layers. However, for the CD (negative charges on the tail-to-tail domain wall and positive on the head-to-head domain wall), the induced dipole causes the potential to drop from the positive to the negative wall (see dashed line in Figure 2b, middle panel), and the LDOS to upshift in energy from the positive to the negative walls (see dashed lines in Figure 2b, right panel). Thus, the negatively charged domain wall (bottom in Figure 2b) has the valence band maximum (VBM) and the positively charged wall has the conduction band minimum (CBM). Relative to the SD structure, the band gap (VBM−

where κ are atoms in the unit cell, i and j indicate Cartesian directions, Ω is the unit cell volume, e is the electron charge, p⃗ is the macroscopic polarization (dipole per unit cell) and u⃗k is an atomic displacement from equilibrium.31 The polarization P⃗ = P⃗ el + P⃗ion has electronic and ionic contributions, where e Im ∑ dk ⃗ ⟨unk ⃗|∇k ⃗ |unk ⃗⟩ Pel⃗ = (2π )3 n

∫

and ⃗ = Pion

e Ω

∑ Zsionrs⃗ s

with unk⃗ the cell-periodic part of the Kohn−Sham Bloch function at band n and at the k point, and Zion the nuclear s charge at rs⃗ . The isotropic average is defined as 1 Z̅ * = ∑ Zii* 3 i and is dimensionless. Results of Born effective charges for the structures considered here are summarized in Table 2, noting Table 2. Calculated Isotropic Born Effective Charges Z̅ * Pb I C N HC HN PbI3 CH3NH3

SD unit cell

two-unit UD T1 (T2)

two-unit CD T1 (T2)

4.57 −1.94 0.20 −0.71 0.088 0.50 −1.25 1.25

4.52 (4.52) −1.92 (−1.93) 0.20 (0.20) −0.71 (−0.72) 0.088 (0.088) 0.50 (0.50) −1.25 (−1.26) 1.26 (1.26)

4.64 (4.64) −1.97 (−1.95) 0.20 (0.20) −0.70 (−0.70) 0.091 (0.091) 0.49 (0.49) −1.27 (−1.23) 1.25 (1.25)

that similar values are obtained for the T1 and T2 domains in most cases. For the two-unit CD, the values for PbI3 in T1, which has tail-to-tail polarization, decreased from −1.25 to −1.27, while in T2, which has head-to-head polarization, is increased to −1.23, consistent with an induced negative charge on the tail-to-tail arrangement but a positive value for the headto-head arrangement. Comparing the Born effective charges listed by atom for the SDs, UDs, and CDs, it is noted that,

Table 3. Band Gaps (in eV) for the Structures Summarized in Figure 1 Eg − ESDa

Eg PBE SD (θz = 22.55°) (100)SD (111)SD (110)SD two-unit UD (θz = 19.46°) four-unit UD (θz = 19.04°) two-unit CD (θz = 74.56°) four-unit CD (θz = 75.01°) four-unit UD (θz = 70.00°)b four-unit CD (θz = 20.00°)b a

1.61 1.53 1.42 (1.4226) 1.58 (1.6326) 1.65 1.65 1.49 1.42 1.69 0.93

PBE+SOC 0.65 0.54 0.33 (0.1226) 0.69 (0.4426) 0.66 0.66 0.52 0.45 0.69 0.21

G0W0+SOC 19

1.26 (1.27 ) 1.09 0.82 1.26 1.35 1.44 1.16 1.22 1.45 0.86

sc-GW+SOC 1.67 1.50 1.27 1.74

PBE

PBE+SOC

G0W0+SOC

sc-GW+SOC

−0.08 −0.19 −0.03 0.04 0.04 −0.12 −0.19 0.08 −0.68

−0.11 −0.32 0.04 0.01 0.01 −0.13 −0.20 0.03 −0.44

−0.17 −0.44 0.00 0.09 0.18 −0.10 −0.04 0.19 −0.40

−0.17 −0.40 0.07

19

(1.67 ) (1.6034) (1.5234) (1.4634)

ESD denotes the band gap for the SD structure. bStructure not optimized. D


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Figure 2. Total potential and layer-by-layer resolved LDOS (Fermi energy is at zero) calculated at the PBE+SOC level, along the domain directions in the four-unit UD (a) and CD (b) structures. The dashed lines in (b), right panel, denote the VBM (left line) and the CBM (right line).

CBM difference) decreases, with the shift given as −2eEd, where E is the induced field and d is the distance between the positively and negatively charged domain walls. For a larger domain, d is larger, and the band gap will decrease further. This is evident for the CDs, specifically the −0.13 and −0.20 eV shifts for the two-unit and four-unit domains, respectively. In addition, if the MA moiety is rotated toward the domain direction, i.e., decreasing θz, the dipole z-component will increase, and the charges on domain walls will increase. The band gap is expected to be decreased further in this case. Motivated by this analysis, MA was rotated in the four-unit CD structure, decreasing θz from 70 to 20°, increasing the dipole zcomponent by

negligible value to about 0.2 eV (0.18 eV for the four-unit UD) is demonstrated with G0W0+SOC, but not sensitive to the domain polarization direction, as expected, i.e., having values of 0.18 and 0.19 eV with θz ≈ 20° and θz ≈ 70°, respectively. On the other hand, the G0W0+SOC results further confirm that the band gap reduction in CDs is sensitive to the domain polarization direction. The band gap shifts are −0.40 and −0.04 eV for four-unit CDs with θz ≈ 20° and θz ≈ 70°, respectively, indicating an order of magnitude change upon MA rotation by 50°. The effect of MA orientation on band gaps in CDs is due to charge effects, hence different than for a SD, where there are no induced charges. In a single domain structure, the band gap change with MA rotation in the xzplane is within 0.2 eV, having values of 1.50 eV for the (100)SD structure (θz = 90°) and 1.67 eV for the SD structure (θz = 22.55°). Next, we benchmark calculation of the dynamical dielectric function. The imaginary part of the frequency-dependent dielectric function (ε = ε(1) + iε(2)) is given by the Cartesian tensor

⎤ ⎡ cos(200) 1 − ⎥ = 1.75 ⎢ ⎦ ⎣ cos(700)

For comparison, for the four-unit UD, θz was increased from 20 to 70°, with the dipole z-component decreased by 1.75 (see structures in Figure 1i,j). For these structures, calculated PBE +SOC band gaps (0.69 eV for an UD and 0.21 eV for a CD) demonstrate that the rotation increases the band gap by 0.03 eV in the UD and decreases it by 0.24 eV in the CD. Considering that the band gap for the SD structure by PBE +SOC is 0.65 eV, the band gap for the four-unit CD after MA rotation is significantly decreased by 0.44 eV. In comparing DFT predictions with G0W0+SOC level results (see Table 3), an increase in the band gap shift in UDs from a

(2) (ω) = εαβ

4π 2e 2 1 limq → 0 2 Ω q

∑ 2wk ⃗δ(wk ⃗εck ⃗ − εvk ⃗ − w) c ,v ,k ⃗

× ⟨uck ⃗ + eα⃗ q ⃗|uvk ⃗⟩⟨uck ⃗ + eβ⃗ q ⃗|uvk ⃗⟩*

where ω is in units of energy; Ω is the volume of the primitive cell; ec⃗ k⃗ are unit vectors for the three Cartesian directions; c and E


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Figure 3. Comparison of the calculated dielectric functions ε(1) (a) and ε(2) (b) for the SD structure at the RPA (black), TDDFT (red), and G0W0BSE+SOC (green) levels. Comparison of ε(1) (c) and ε(2)(d) dielectric functions calculated at the G0W0-BSE+SOC level (green), corrected by the sc-GW band gap, to measured spectra by Löper et al.35 (solid black). G0W0-BSE+SOC exciton oscillator strengths are also given in (d). The PBE +SOC band structure of SD is shown in (e). The high symmetry k points in the cubic BZ in (e) were described by Fujiwara et al.41

assigned to the first transitions from the first two valence bands to the first conduction bands (counting from the Fermi level) at the R k point.41 Peak 1 is assigned to transitions in three regions, specifically from the first two valence bands to the first two conduction bands near the M1 and M2 k points, and from the third and fourth valence bands to the first two conduction bands near the R k point. Although Fujiwara et al.41 assigned the peak to the transitions near the M2 k point only, we found that transitions near the M1 and R k points also contribute. Peak 3 results from more valence bands to the first two conduction bands and involves more k regions in the BZ, not assigned specifically here. Turning to the exciton picture, peak 0 originates from the first exciton (doubly degenerate), peak 1 is from the second to fourth excitons with a large oscillator strength, and peak 2 is dominated by a doubly degenerate exciton at 3.5 eV. Absorption spectra were calculated using

v refer to conduction and valence states, respectively; and uck⃗ are the cell-periodic orbitals at k.⃗ 39 In the calculation of ε(2) αβ (ω), k⃗ is restricted to the irreducible wedge of the first Brillouin zone (BZ). The real part of the dielectric tensor obtained from Kramers−Kronig transformation is given by (1) εαβ (ω) = 1 +

2 P π

∫0

∞

(1) εαβ (ω′)ω′

ω′2 − ω 2 + iη

dω′

where P denotes the principal value of the integral. Calculations of ε(1) and ε(2) for the SD structure performed at the RPA, TDDFT, and G0W0-BSE levels, including SOC in all cases, are summarized in Figure 3a,b. The RPA and TDDFT results are similar, although inclusion of local field effects somewhat suppresses ε(1). G0W0-BSE results indicate smaller ε(1) and ε(2) values, and reduction in the third and fourth peaks at higher energy. Notably, these differences result in better agreement with experiment for the higher level of theory (see Figure 3c,d), providing a quantitative description, after the optical gap is corrected by sc-GW. The dielectric constants around 1.0 eV were calculated as 9.7, 8.7, and 6.3 by RPA, TDDFT, and G0W0-BSE+SOC, respectively, where the measured ionclamped value is 6.5.40 To gain an understanding of the ε(2) spectrum calculated at the G0W0-BSE level (peaks labeled by 0, 1, and 2 in Figure 3d), we compare to the DFT results (Figure 3e), where peak 0 is

αij(E) =

2 E ch

εij(1)(E)2 + εij(2)(E)2 − εij(1)(E)

where i, j = x, y, z. The optical constants n and k were calculated by

nij(E) = F

εij(1)(E) +

εij(1)(E)2 + εij(2)(E)2 2 DOI: 10.1021/acs.jpcc.7b04557 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Figure 4. Optical spectrum of the SD structure calculated at the G0W0-BSE+SOC level (green) in comparison to experimental data (solid black for data by Löper et al.35 and dashed black line for data by Jiang et al.38) (a), and for the (100)SD (dashed red), (111)SD (dashed blue), (110)SD (dashed purple) structures (b). Boltzmann averaged calculated optical spectrum for the SD, (100)SD, (111)SD, and (110)SD structures is in solid green (b). Calculated optical constants n (c) and k (d) for the SD structure (green) in comparison to experimental data (solid black line35). Comparison of G0W0-BSE+SOC calculated spectra for the pseudocubic SD (red) vs two-unit UD (blue) and two-unit CD (green) structures (e) and two-unit (red) vs four-unit (blue) domains for UDs (f) and CDs (g).

ensemble of configurations, which is not taken into account in a single calculation. The calculated exciton binding energy of 0.20 eV for the SD domain is in agreement with a previously calculated value of 0.15 eV,42 but larger than measurements.32 Absorption spectra for structures with MA along the [100], [111], and [110] directions are shown in Figure 4b. The exciton binding energies are 0.18 eV for the (100)SD structure, 0.12 eV for (111)SD, and 0.17 eV for (110)SD. Despite similarities, some differences are observed for the three structures, particularly for the (111)SD structure. A Boltzmann averaged structure indeed broadens the spectrum around 2.5 eV. The optical constants n and k for the SD structure (shown in Figure 4c,d) also demonstrate agreement with experiment.

and

kij(E) =

−εij(1)(E) +

εij(1)(E)2 + εij(2)(E)2 2

The average absorption coefficient is defined by α = (αxx + αyy + αzz)/3, and the average optical constants are defined by n = (nxx + nyy + nzz)/3 and k = (kxx + kyy + kzz)/3. Results are summarized in Figure 4. The spectrum for the SD structure (Figure 4a) is in good agreement with measurements, although the spectral broadening observed experimentally is not reproduced, partially due to the flexibility in the MA moiety orientation that likely requires consideration of a statistical G


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acknowledge ONR Grant N00014-12-1-1034 and DARPA Grant HR0011-15-2-0038 (under the MATRIX program).

Interestingly, the spectra of the two-unit UD and CD structures are blue- and red-shifted, respectively, relative to the results for the SD (see Figure 4e), while with a size increase from two- to four-unit domains, the spectra are blue-shifted in both UDs and CDs (Figure 4f,g). In addition, for CDs, additional peaks emerge between the first and second peaks, specifically at 1.3 eV for the two-unit domain and at 1.3 and 1.6 eV for the four-unit domain. Such results imply that UDs and CDs can be potentially distinguished via their optical responses.

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CONCLUSIONS In this work, we analyzed the effects of ferroelectric domains and of the polarization direction in single domains, on the electronic and optical properties of CH3NH3PbI3 by first principles calculations. We note that inclusion of SOC and Pb d orbitals were found to be essential for accurate prediction of the electronic structures, and of the properties of charged and uncharged domains in comparison to single domains. Analysis of the effects of a potential on the Born effective charges and respective density of states at the DFT level provided an understanding of variations in the band gap for these structures, as dependent on the type of domain wall, and on the MA moiety direction. The band gap decrease in charged domains can be quite large when the polarization direction aligns with the domain wall direction, and also, the band gap tends to decrease with domain size. The GW calculations found that the quasiparticle band gap in UDs is increased by inclusion of many-body effects, demonstrating the importance of this level of theory to correctly predict electronic structures of MAPbI3 domains. Based on benchmarking of dielectric function calculations, agreement with experiment was demonstrated for G0W0-BSE results after correcting by the band gap from self-consistent GW calculations. However, due to the flexibility in the MA moiety’s orientation, causing spectral broadening in the experimental spectra, consideration of a statistical ensemble of configurations is required, which is not taken into account here. In addition, the transitions were discerned in comparison to the DFT band structure. Finally, based on the calculations, we may possibly assume that, when inducing large ferroelectric domains, the optical response could distinguish between them, thus motivating further experimental exploration.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b04557. Coordinates, lattice vectors, lattice constants, and space groups for the structures considered (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ruth Pachter: 0000-0003-3790-4153 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the computational resources and helpful assistance provided by the AFRL DSRC. Y.Y. and L.B. H


Article

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