Dipole Order in Halide Perovskites: Polarization ... - ACS Publications

Article Cite This: J. Phys. Chem. C 2017, 121, 23045-23054

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Dipole Order in Halide Perovskites: Polarization and Rashba Band Splittings Shunbo Hu,† Heng Gao,† Yuting Qi,† Yongxue Tao,† Yongle Li,† Jeffrey R. Reimers,† Menno Bokdam,‡ Cesare Franchini,‡ Domenico Di Sante,§ Alessandro Stroppa,*,†,∥ and Wei Ren*,†

J. Phys. Chem. C 2017.121:23045-23054. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 09/26/18. For personal use only.

†

Department of Physics and International Center of Quantum and Molecular Structures, Shanghai University, Shanghai 200444 China ‡ Faculty of Physics, Center for Computational Materials Science, University of Vienna, A-1090 Vienna, Austria § Institut fuer Theoretische Physik und Astrophysik, Universitaet Wuerzburg, Am Hubland, 97074 Wuerzburg, Germany ∥ CNR-SPIN c/o Università degli Studi dell’Aquila, Via Vetoio 10, I-67010 Coppito (L’Aquila), Italy S Supporting Information *

ABSTRACT: ABX3 (A = organic cation; B = Sn, Pb; and X = halogen) organohalide perovskites have recently attracted much attention for their photovoltaic applications. Such hybrid compounds are derived from the replacement of the inorganic monovalent metal element by an organic cation, for example, methylammonium ion (MA = CH3NH3+) and formamidinium ion (FA= +HC(NH2)2). In particular, since the organic cations are polar, it is interesting to investigate their possible longrange ordering and the corresponding Rashba spin-split bands. In this work, by using density functional theory calculations, we estimate the ferroelectric polarization corresponding to a complete ordering of dipole moments for the optimized structures of 12 perovskite halides, with A = MA, FA; B = Pb, Sn; X = Cl, Br, I. The adiabatic path and functional mode analysis have been discussed for all cases. The calculated values of the polarization may be as high as a conventional inorganic ferroelectric compound, such as BaTiO3. The concomitant inversion symmetry breaking, coupled to the sizable spin−orbit coupling of Pb and Sn, results in a fairly large Rashba spin-splitting effect for both valence and conduction bands. We highlight a rather anisotropic dispersion of spin−orbit split bands which gives rise to different Rashba parameters in different directions perpendicular to the polar axis in k-space. Furthermore, we found a weak and positive correlation between the magnitude of polarization and relevant spin-split band parameters. Since the mechanism for enhanced carrier lifetime in 3D Rashba materials is connected to the reduced recombination rate due to the spin-forbidden transition, our study could aid in the understanding of the fundamental physics of organometal halide perovskites and the optimization and design of materials for better performance.

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INTRODUCTION Hybrid organic−inorganic halide perovskites are a class of materials with ABX3 perovskite topology (A = organic cation; B = Sn, Pb; and X = halogen) where the organic cation often consists of methylammonium (MA = CH 3 NH 3 + ) or formamidinium (FA = +HC(NH 2 ) 2 ) molecules. These materials are usually processed in solution and exhibit excellent optical and electronic properties.1−4 Organohalide lead perovskites have represented a breakthrough in the field of photovoltaics.5,6 Since their first application as sensitizers in mesostructured cells by Kojima and co-workers in 2009,7 showing a power conversion efficiency (PCE) of 3.81%, an amazing growth rate of PCE has been achieved in recent years. In 2011, Park et al. fabricated MAPbI3 perovskite solar cells with a PCE of 6.54%.8 Then, Kim et al. achieved a PCE of up to 9.7% based on spiro-MeOTAD as hole transport materials in 2012.9 In 2013, Noh et al. demonstrated highly efficient solar © 2017 American Chemical Society

cells with a PCE of 12.3% as a result of tunable composition for MAPb(I1−xBrx)3.10 In 2014, Im et al. reported an efficiency of 17.01% by controlling the size of MAPbI3 cuboids during their growth.11 Noh and co-workers achieved a PCE of up to 19% in 201512 and then a certified value of 20.1%.13,14 Recently, Saliba and co-workers have achieved stabilized efficiencies of up to 21.6%.15 These materials have been widely investigated in a large variety of device configurations and have been the subject of an impressive number of experimental and theoretical studies in the past few years.16−50 Since hybrid perovskites contain polar cations, the possible ordering of such electric dipoles has been investigated. In particular, ferroelectricity, i.e., the presence of a permanent and Received: June 16, 2017 Revised: August 28, 2017 Published: September 21, 2017 23045

DOI: 10.1021/acs.jpcc.7b05929 J. Phys. Chem. C 2017, 121, 23045−23054

Article

The Journal of Physical Chemistry C

scale is still an important issue to consider even if a global centrosymmetry may arise at macroscopic scales. However, the theoretical modeling of metal halide perovskites is extremely challenging as it involves the treatment of several subtle, but important, factors that are difficult to compute accurately. Bokdam et al.75 performed an extensive global search for minimum energy structures for ABX3 where A = MA, FA; B = Sn, Pb; X = Cl, Br, I. Furthermore, for each structure, the excitonic properties have been studied.75 In this work, starting from the relaxed unit cell structures reported in ref 75 and under the assumption of complete organic cation ordering, we estimate the largest possible electric polarization and the induced spin−orbit splittings in energy band structures. This is of great importance since the Rashba splittings have been suggested to increase the carrier’s lifetime and to reduce the electron−hole recombination rate.51−55 Therefore, we have estimated the spin−orbit energy-band splittings for all 12 ABX3 relaxed structures, where A = FA, MA; B = Pb, Sn and X = Cl, Br, I. Although clear trends among the magnitude of ferroelectric polarization, the atomic spin−orbit splittings and corresponding Rashba parameters are difficult to infer, and as a result of the complex interplay between electronic structure and the organic cation/framework atomic relaxations, our work demonstrates that relativistic features are common to all investigated systems. Since the spin properties of valence and conduction bands are important for reducing the recombination rate, our estimated spin-splitting parameters for the whole series of halides can help in the understanding of the basic properties of the outstanding functionalities of organohalide perovskite solar cells. Computational Details. Kohn−Sham equations were solved using the projector-augmented-wave (PAW) method with the PBEsol exchange-correlation functional,76 as implemented in VASP.77,78 The energy cutoff for the plane wave expansion was set to 600 eV, and a 4 × 8 × 8 Monkhorst−Pack grid of k-points was used for the 2 × 1 × 1 supercell. The convergence in total energy and Hellmann−Feynman forces was set to 1 μeV and 0.01 eV/Å, respectively. The ferroelectric polarization has been calculated using the Berry phase approach by building an appropriate AFE reference state and considering a suitable path in the configurational space connecting to the FE state. The continuous variation of the electric polarization along the path has been checked carefully. This method avoids the inclusion of quanta of polarization,79 which in turn would lead to misleading results. Spin−orbit coupling (SOC) is selfconsistently included in all band structure calculations. Nevertheless, we neglected SOC when computing the ferroelectric polarization, since we have checked that its inclusion does not affect the final results. In order to obtain very accurate Rashba parameters, many-body effects were included in the electronic structure by calculating the independent quasiparticle energies in the GW0 approximation.80 The PBE81 functional was used to initialize the GW0 calculations. Using Kohn−Sham PBE orbitals, the one-electron energies in G are iterated until the quasiparticle energies are converged while keeping W0 fixed at the DFT−RPA level.82 Full technical details can be found in ref 75. To determine a smooth, interpolated GW0 band structure, we project onto maximally localized Wannier orbitals.83 The Rashba parameters were then obtained by fitting to these band structures. Structural and Ferroelectric Properties. In order to estimate the FE polarization, we need to introduce a reference centrosymmetric structure related to the ferroelectric state

switchable electric dipole in the unit cell, has been hypothesized to play a key role in reducing the charge recombination.51−55 Despite these preliminary studies, the importance of ferroelectricity on the working mechanism of the ABX3-based solar cell has not been clarified yet, and the existence of ferroelectric (FE) domains is still under debate. Based on macroscopic polarization-electric field (P−E) measurements56−58 or microscopic probing of ferroelectric domains,57,59 some experimental studies have been published. For example, there may exist a local dipole moment at the unit cell scale, but the globally centrosymmetric structure leads to a zero average polarization on a macroscale, and because of that the organic cations are very small and the hydrogen bonding with the inorganic framework is rather weak.26,60 Moreover, MAPbI3 thin films hinder the detection of the residual polarization due to a large conduction current during polarization measurements.56 Piezo force microscopy (PFM) measurements recently revealed the presence of submicrometer ferroelectric domains (nearly 100 nm in size) in β-MAPbI3 samples, and the importance of ferroelectric domain walls on the photovoltaic efficiency of hybrid halide perovskites has been pointed out.59,61 On the contrary, at the macroscale level, MAPbI3 thin films did not show apparent ferroelectric properties at cell-operating temperatures.62 However, the possibility of ferroelectricity at the nanoscale level is still an open question. On the basis of density functional theory (DFT) calculations, several works have been published in an attempt to estimate the ferroelectric polarization in perovskite systems.63−67 Specifically, for halide perovskites, Frost et al. performed DFT calculations on the cubic phase, and they estimated a ferroelectric polarization of ∼38 μC/cm2.55 Zheng et al. have demonstrated that the expected ferroelectric polarization values of halide perovskite33 should be around 4 to 5 μC/cm2. Stroppa et al., by using a combination of DFT simulations and symmetry mode analysis, studied the ferroelectric polarization of MAPbI3.22 It was pointed out that the contribution of the framework to the total polarization can not be neglected in general. Mosconi et al.5,68 studied the tetragonal phase of βMAPbI3 and demonstrated that a ferroelectric long-range alignment is more stable than the one with an antiferroelectric (AFE) order. However, the energy difference is small enough to make both configurations accessible at room temperature. Nevertheless, this would suggest that the ferroelectric phase may be stabilized over the AFE phase in MAPbI3 by applying an external electric field.69 Furthermore, the presence of relatively heavy elements in halide perovskites, spin−orbit coupling (SOC), together with the non-centrosymmetry of ferroelectric materials, might give rise to exotic spin-splitting phenomena, such as Rashba and Dresselhaus effects48−51 seen in the relativistic electronic structure of (nonmagnetic) semiconductors.70−72 Despite the absence of conclusive experiments about the existence of ferroelectric polarization at the nanometer scale, and although it seems clear that there is no ferroelectricity at the macroscale, a new possibility has been proposed. The global centrosymmetry of the compounds may have a local inversion symmetry breaking. Molecular Dynamics simulations reveal a “dynamical Rashba effect”, which implies that even in globally centrosymmetric structures, the coupled inorganic−organic degrees of freedom can produce a spatially modulated Rashba effect, which is characterized by the sub-picosecond time scale of the MA dynamics.6,54,73,74 This suggests that the local noncentrosymmetry due to the local ordering of dipoles at unit cell 23046


Article


Figure 1. Ferroelectric polarization of MAPbI3 and FAPbI3 as a function of the factor λ, which takes into account the molecular rotations and the distortions of the framework. (a, b) Side views of AFE states (λ = 0), intermediate states (λ = 0.5), and final FE (λ = 1) structures. (c) Details of the MA = CH3NH3+ and FA = +HC(NH2)2 cations. (d, e) Polarization of the MAPbI3 and FAPbI3, respectively. The upper, middle, and lower panels show the polarization components along z, y, and x directions, respectively.

normalized amplitude of the roto-displacive atomic distortions connecting AFE (λ = 0) and FE (λ = 1) states (see the Supporting Information). For each λ, we calculate the groundstate electronic structure and the related Berry phase without relaxing the atomic positions except for λ = 1. We caution the reader that only the polarization for λ = 1 has a real physical meaning while the intermediate states, including λ = 0, are only computational states which allow monitoring the continuous evolution of the polarization as a function of λ, thus excluding the inclusion of misleading quanta of polarization. The polarization starts from zero for the AFE structure, and it shows a nonmonotonic behavior as λ increases toward 1. The nonmonotonic behavior as a function of λ is common in the case of organic−inorganic hybrid compounds, as already noted in ref.85 We report all the paths for the different compounds in the Supporting Information. The calculated polarization at the DFT level is summarized in Table 1 with the Cartesian components as well as the modulus of the vectors. In Figure 2 we report the polarizations, where the vertical bars represent the value of the polarization grouped for (A, B)X where X = Cl, Br, I and A = MA, FA; B = Sn, Pb. Each bar has a color corresponding to the different halogen atom. We first discuss the trends in the modulus of polarization Ptot. When fixing the (A, B) groups and considering different halogen atoms X = Cl, Br, I, we see that the average polarization among the different halogen systems decreases from (MA, Sn)X, (MA, Pb)X, (FA,Sn)X, and (FA,Pb)X, respectively. The average polarization is calculated as 1/ 3[P(A,B)Cl + P(A,B)Br + P(A,B)I] for (A, B) = (MA, Sn), (MA, Pb), (FA, Sn), and (FA, Pb), and it is shown as a horizontal

through some atomic distortions. As discussed in detail in refs 22 and 84, the distortion connecting the AFE and FE structure is of displacive and roto-displacive type, i.e., involving small atomic displacements of the BX3 framework and displacements and rotations of the organic cations. As a reference structure, we adopt the AFE configurations shown in Figure 1a (left) and b (left) (for MAPbI3 and FAPbI3), where the MA and FA cations (see (I) and (II) in Figure 1c) have been appropriately rotated. The MA and FA cations are polar with a dipole moment parallel to the C−N bond and the C−H bond, respectively (see Figure 1c). The construction of the AFE reference state starting from a generally relaxed unit cell is not straightforward. In fact, it requires centro-symmetrization of the framework and rotation of the organic cations to the AFE configuration. For each composition, the AFE reference structure has been carefully checked to ensure the presence of the inversion symmetry point. The details of the construction of the AFE reference unit cell starting from an arbitrary relaxed unit cell is given in the Supporting Information. Analysis of the FE Polarization. The AFE structures possess inversion symmetry by construction, and therefore, there is no net polarization apart from a possible quantum of polarization, easily recognizable and removable. As soon as some atomic distortions are included in order to reach the FE state, the inversion symmetry is lost and the electric polarization arises. In the following, we consider the total polarization, that is, the sum of ionic and electronic contributions. We construct the path from the AFE to FE structures by introducing a parameter λ which represents the 23047


Article

The Journal of Physical Chemistry C Table 1. Calculated Ferroelectric Polarization of ABX3 Perovskite Halides Compounds system

Px (μC/cm2)

Py (μC/cm2)

Pz (μC/cm2)

Ptot (μC/cm2)

MASnCl3 MASnBr3 MASnI3 MAPbCl3 MAPbBr3 MAPbI3 FASnCl3 FASnBr3 FASnI3 FAPbCl3 FAPbBr3 FAPbI3

−8.68 8.03 −3.78 4.01 −3.38 2.68 3.30 5.57 −5.93 0.34 1.22 −3.60

0.94 8.67 0.00 4.59 −3.60 2.68 −4.64 −6.72 −5.96 −0.97 −0.13 1.86

9.58 −15.08 15.63 −11.48 11.70 13.52 −13.85 4.37 0.00 −5.05 −1.61 −3.60

12.96 19.16 16.08 13.00 12.70 14.03 14.98 9.76 8.41 5.16 2.02 5.43

Figure 3. Rashba parameters of MAPbI3. (a) BZ highlighting the highsymmetry points and high-symmetry directions. (b) GW0 band dispersion around the R point, which is obtained from the Hamiltonian based on Wannier orbitals from previous GW 0 calculation. (c) Rashba parameters αR for VBM of MAPbI3 are shown by means of a polar plot to represent the evolution along different directions around the R point on the plane perpendicular to the ferroelectric polarization. The numerical values are represented by a color code, and k0 is indicated by concentric circles labeled by 0.02,0.04, ..., 0.1 Å−1. (d) 3D plot of spin−orbit split valence bands near the R point on the plane perpendicular to electric polarization. The spin texture is also shown. The color code corresponds to the negative (valence) band eigenvalues as in (b). (e, f) Spin textures projected on the 2D plane perpendicular to the polarization showing the characteristic Rashba topology. The color code represents the sz spin component.

Figure 2. Magnitudes of the ferroelectric polarization of ABX3 perovskite halides compounds. The average polarization is shown as a horizontal blue segment.

segment in Figure 2. For (MA, Sn)X and (MA, Pb)X there is no clear trend by changing the halogen atoms from Cl over Br and I, i.e., in order of decreasing electronegativity of the halogens. For instance, in (MA, Sn)X when changing X = Cl, Br, I the polarization first increases and then decreases, while in (MA, Pb)X first decreases and then increases. For (FA, Pb)X, it has the same trend as (MA, Pb)X. For (FA, Sn)X on the other hand, a definite trend arises: the polarization tends to decrease when the electronegativity of the halogen atom decreases, i.e., from Cl, over Br, to I. Among all the structures, the largest polarization is about 19.16 μC/cm2 for MASnBr3 compound. In order to gain more insights into the origin of the different polarization, we separated the contributions from the framework and the organic cations. This is discussed in the Supporting Information. Electronic Structure: Rashba Parameters and Spin Textures. Now we focus on the electronic structures of halide family compounds. Figure 3a shows the Brillouin zone (BZ) of cubic halide compounds. Here we take the MAPbI3 as an example since the other halide compounds have similar properties. To study the electronic structure close to the valence band maximum (VBM) and the conduction band minimum (CBM) of MAPbI3, we calculate the band structure around the R point as shown in Figure 3b. The Rashba spinsplittings with VBM and CBM are not located at highsymmetry points of the BZ (R point) but instead slightly shifted. Moreover, the CBM and VBM are not located at the same points, having different Rashba momentum offsets. This results in the formation of an indirect band gap, which limits the radiative recombination rate. On the other hand, since the

indirect band gap is generally only a few tens of a meV smaller than the direct optical transitions gap, the absorption spectrum is not largely affected by the presence of such indirect gap. The interplay of a low recombination rate and strong absorption has been proposed as an explanation for the high efficiencies of hybrid perovskite solar cells.6,86,87 The Rashba effect is the consequence of the breaking of inversion symmetry in the crystal in a direction orthogonal to a k-point sampling plane, and it is described by the so-called Bychkov−Rashba Hamiltonian71,72,88 HR =

ℏ2 (kx2 + k y2)σ0 + αR (σ × k)z ̂ 2m *

(1)

with αR known as the Rashba parameter, σ0 and σ = (σx,σy,σz) are the identity and spin Pauli matrices, respectively, and ẑ is the polar direction. To obtain the maximum Rashba parameters for a given compound, we extracted the low energy Hamiltonian using a set of maximally localized Wannier 23048


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The Journal of Physical Chemistry C functions.89 For each case, around the R point in the BZ, the Rashba parameters along all the directions on the plane perpendicular to the polarization direction have been calculated. Taking MAPbI3 as an example, Rashba parameters of valence band are plotted in Figure 3c by means of a polar plot to show the evolution along different directions around the R point. The polar plot clearly highlights that the band structure and corresponding Rashba splittings are highly anisotropic. It is therefore important to consider this anisotropy when reporting the Rashba parameters. For example, for MAPbI3, the momentum offset k0 in valence band can vary from 0.06 to 0.08 Å−1 while the corresponding αR can vary from 1.35 to 1.85 eV/Å. We can obtain the maximum Rashba parameter and its direction on this plane from this polar plot. To more clearly present the anisotropic band structure and Rashba parameter, we computed the 3-dimensional (3D) dispersion of the spin−orbit split valence bands for k-points near R points on the plane perpendicular to the electric polarization (Figure 3d). Moreover, we show the spin-texture on each point on the 3D band dispersion. Clearly, the spintexture is circulating counterclockwise on the upper split band and clockwise in the lower split band. When projected on the 2dimensional (2D) plane, it clearly shows up the characteristic Rashba topology (see Figure 3e,f). The color on the 2D spin texture represents the z-component of the mean value of the spin, i.e., sz. Our simulations predict a Rashba spin-splitting for all of the 12 MA/FA-based organohalide perovskites investigated in this work. For each compound, we report the maximum bandsplitting ER and the corresponding momentum offset. The values are summarized in Table 2, while in Figure 4 we

Figure 4. Rashba parameters αR of the conduction (a) and valence (b) bands for ABX3 perovskite halides compounds (see also Table 2). Different colors correspond to different halogen atoms.

larger in the conduction band than in the valence band. This suggests that the SO coupling is stronger in the conduction band than in valence band near the R point. This is confirmed by inspection on the s and p character of the electronic orbit: the ratio of p character and s character is generally larger in the conduction states than in valence states.90 The largest value is predicted for the conduction band of FAPbI3, where a large spin-splitting of ∼0.11 eV and a momentum offset k0 = 0.11 Å−1 give rise to a sizable Rashba coupling αR = 3.77 eVÅ. Recently, Niesner et al.91 observed a giant Rashba splitting in MAPbBr3 organic−inorganic perovskite of about 10 eV/Å.91 Our calculations predict a rather smaller value, about 1.28 eV/Å and 3.01 eV/Å on valence bands and on conduction bands, respectively. In our results, MASnBr3, MASnI3, MAPbBr3, MAPbI3, FASnBr3, FAPbBr3, and FAPbI3 perovskites have fairly large Rashba parameters αR in conduction bands with values ranging from 3 to 4 eV/Å. For comparison, GeTe has αR = 4.8 eV/Å.92 Finally, we want to note that large αR can also be found for Sn systems, despite the fact that that the atomic SO strength is larger in Pb than in Sn, since it scales as Z4, where Z is the atomic number (Z = 50, 82 for Sn and Pb). Last but not least, the anisotropy of the Rasbha parameters in the plane perpendicular to the polar direction should be taken into account. In fact, our calculations (not reported here) show that αR strongly depends on the direction along which it is calculated (anisotropy of Rasbha parameters). For example, for a given compound, αR can show variation up to 1.15 eV/Å in the SO split valence bands; in the conduction bands it can vary up to 2 eV/Å. In our discussions, we have always reported the maximum αR after proper scanning on the plane perpendicular to polarization (see 3D band structures in Figure 3d). To summarize, we predict fairly large Rashba spin-splittings for different systems, and we highlight a strong dependence of the Rasbha parameters not only on the specific compound but

Table 2. Estimated Rashba Parameters for Organohalide Perovskitesa ER (meV) MASnCl3 MASnBr3 MASnI3 MAPbCl3 MAPbBr3 MAPbI3 FASnCl3 FASnBr3 FASnI3 FAPbCl3 FAPbBr3 FAPbI3

k0 (Å−1)

αR (eV/Å)

VB

CB

VB

CB

VB

CB

7.19 18.87 28.44 7.81 25.88 72.62 8.62 26.26 12.31 22.09 10.33 40.79

182.38 143.52 99.45 113.41 120.30 136.01 142.97 192.69 37.08 96.77 120.95 110.24

0.04 0.03 0.04 0.02 0.04 0.08 0.03 0.04 0.02 0.03 0.02 0.04

0.16 0.09 0.06 0.09 0.08 0.08 0.13 0.12 0.04 0.07 0.07 0.11

0.37 1.28 1.56 0.65 1.28 1.86 0.49 1.31 1.22 1.33 0.83 1.68

2.29 3.29 3.12 2.56 3.01 3.50 2.18 3.21 1.88 2.66 3.31 3.77

a Since ER shows anisotropy in the plane perpendicular to the polar direction in k-space, we report the maximum value of ER (see Figure 3). From eq 1, the momentum offset of the split bands is given by k0 = m*αR/ℏ2, while the Rashba energy of the split band minimum is ER = m*α2R/(2*ℏ2). The eigenvalues difference at a particular k-point (k0) defines αR = 2ER/k0.

summarized them as a bar graph, where the vertical bars represent the values ER grouped for (A, B)X where X = Cl, Br, I and each bar has a color corresponding to the different halogen atoms. Also, in this case, there is no clear trend as a function of the halogen atom, when studying each group (A, B) where A = MA, FA and B = Pb, Sn. However, some general considerations can be drawn. First of all, from Figure 4 it is evident that αR is 23049


Article

The Journal of Physical Chemistry C Table 3. Calculated Polarization of ABX3 Perovskite Halide Compounds and Rashba Parameters k0 (Å−1)

ER (meV) 2

Sn class MASnCl3 MASnBr3 MASnI3 FASnCl3 FASnBr3 FASnI3 Pb class MAPbCl3 MAPbBr3 MAPbI3 FAPbCl3 FAPbBr3 FAPbI3

αR (eV/Å)

Ptot (μC/cm )

VB

CB

VB

CB

VB

CB

12.96 19.16 16.08 14.98 9.76 8.41

7.19 18.87 28.44 8.62 26.26 12.31

182.38 143.52 99.45 142.97 192.69 37.08

0.04 0.03 0.04 0.03 0.04 0.02

0.16 0.09 0.06 0.13 0.12 0.04

0.37 1.28 1.56 0.49 1.31 1.22

2.29 3.29 3.12 2.18 3.21 1.88

13.00 12.70 14.03 5.16 2.02 5.43

7.81 25.88 72.62 22.09 10.33 40.79

113.41 120.30 136.01 96.77 120.95 110.24

0.02 0.04 0.08 0.03 0.02 0.04

0.09 0.08 0.08 0.07 0.07 0.11

0.65 1.28 1.86 1.33 0.83 1.68

2.56 3.01 3.50 2.66 3.31 3.77

Figure 5. Scatter plots for Sn compounds in (a), (b), and (c) and for Pb compounds in (d), (e), and (f) for ABX3 perovskite halides compounds. The Pearson correlation coefficient is also reported. See text for further details.

also on the directions in k-space along which they are calculated. To the best of our knowledge, the Rasbha anisotropy for perovskite halides has not been discussed in the literature yet. Since it may affect the photovoltaic performances, we expect that other organic−inorganic perovskite could have practical performances provided that the energy gap is suitable for photovoltaic application. In particular, by considering a previous study on energy band gaps and excitonic binding energies by Bokdam et al.75 one expects that FAPbI3 whose gap is 1.5 eV or FASnI3 whose gap is 1.3 eV can also have functional photovoltaic performance. Indeed, both have a low exciton binding energy of ∼30 meV and both of them are close to this Shockley−Queisser limit, where the optimal band gap is ∼1.3 eV. Note that FAPbI3 has the largest Rashba parameter αR ∼ 3.77 eV/Å among all the considered compounds, while MAPbI3 has αR ∼ 3.50 eV/Å. Polarization and Rashba Parameters: Is There Any Correlation? Since the emergence of ferroelectric polarization due to the inversion symmetry breaking leads to split energy bands through the spin−orbit coupling, and since the property

of these bands, like the spin texture, is strongly linked to the polarization,70,92 it is advisable to investigate if the magnitude of the polarization can influence the magnitude of the Rashba parameters, such as ER, k0, and αR. Indeed, all of them change significantly along the ABX3 series. In order to highlight and quantify possible correlations between Ptot and the Rashba spinsplittings, we adopt a statistical approach by considering the scatter plots and by calculating the Pearson correlation coefficient. This coefficient measures the linear correlation between two variables X and Y. First of all, in Table 3 we report (X, Y) where X = Ptot and Y= ER, k0, and αR in VB and CB for the different compounds. Since Sn and Pb are expected to give different SO splitting, we group the data according to Sn and Pb separately so that for each group the SO strength is kept fixed. We show the different scatter plots in Figure 5. Each scatter plot is represented by a Cartesian plane, where X corresponds to the module of the polarization and Y to the different Rashba parameters for the different compounds. The correlation coefficient is given according to the following formula93 23050


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The Journal of Physical Chemistry C Table 4. Polarization and Rashba Parameters in ABX3 Halides As Obtained in This Work and in Previous Studies αR (eV/Å) (this work) Ptot (μC/cm2) (this work) MASnCl3 MASnBr3 MASnI3 FASnCl3 FASnBr3 FASnI3 MAPbCl3 MAPbBr3 MAPbI3 FAPbCl3 FAPbBr3 FAPbI3 a

Ptot (μC/cm2) (other calculations)

12.96 19.16 16.08 14.98 9.76 8.41 13.00 12.70 14.03 5.16 2.02 5.43 b

5.35e 8.2b 13.42,a 12.6,b 8,c 5,d 13.8e 9.1b

c

d

e

f

VB

CB

0.37 1.28 1.56 0.49 1.31 1.22 0.65 1.28 1.86 1.33 0.83 1.68

2.29 3.29 3.12 2.18 3.21 1.88 2.56 3.01 3.50 2.66 3.31 3.77

αR (eV/Å) (other calculations) VB

CB

1.0f 1.2f

1.9f

1.10e

0.5f

0.6b 0.9,b 1.4f

1.9b 2.3,b,g 1.5f

2.6g

g

Reference 96. Reference 58. Reference 47. Reference 33. Reference 85. Reference 97. Reference 87. 6

r=

We have disentangled the contributions of the organic MA or FA cations and of the inorganic framework to the total polarization. In conclusion, we have shown that the calculated ferroelectric polarizations among the whole series reach values as high as conventional inorganic ferroelectrics. The concomitant breaking of the inversion symmetry, coupled to the sizable SOC of Pb and Sn, results in a fairly large Rashba spin-splitting effects both for the valence and conduction bands. This is promising since the presence of Rashba spin-splittings has been proposed as a possible route to explain the large photovoltaic efficiency of halides-based solar cells, thus suggesting that also other halide perovskites with comparable Rasbha splitting as the prototypical case of MAPbI3 may have interesting photovoltaic performances. In particular, we suggest that FAPbI394,95 and FASnI3, due to combination of optimal band gaps, exciton binding energies, and relatively large αR Rasbha parameters, may represent interesting candidates for photovoltaic applications.

∑i = 1 (xi − x ̅ )(yi − y) 6

6

∑i = 1 (xi − x ̅ )2 ∑i = 1 (yi − y ̅ )2 1

6

1

6

where x ̅ = 6 ∑i = 1 xi and y ̅ = 6 ∑i = 1 yi where i = 1, ..., 6 for MASnCl3, MASnBr3, MASnI3, FASnCl3, FASnBr3, and FASnI3 for the Sn class and i = 1, ..., 6 for MAPbCl3, MAPbBr3, MAPbI3, FAPbCl3, FAPbBr3, and FAPbI3 for the Pb class. The r coefficient is a dimensionless number such that −1 ≤ r ≤ 1: if r = 0 there is no (linear) correlation between the variables, while as long as r approaches 1(−1) there is an increasing positive (negative) correlation between the variables: if one variable increases the other increases or decreases for positive or negative correlation, respectively. In Figure 5a−c we show the scatter plots of Sn compounds for X = Ptot and Y = ER, k0, and αR, respectively, both in VB and CB. In Figure 5d−f, the same quantities for Pb compounds. For Sn compounds, Figure 5 (top), we see that r goes from 0.03 to 0.49: there is a positive correlation, meaning that if Ptot increases then the Rashba parameters tend to increase. However, the correlation is rather weak, especially for the valence bands, where it tends to be negligible. For the Pb compounds, the r coefficient is positive and slightly larger than previous compounds, except in (f). However, the correlation is rather weak and generally positive. Therefore, our results suggest that, somewhat counterintuitively, there is no significant correlation between the magnitude of the polarization and the magnitude of the Rasbha splitting. Finally, in Table 4, our calculated electric polarization and Rasbha parameters are compared with previous studies. Although they are in general similar, some differences arise due to different unit cells considered, inclusion or not of relaxations, and due to the possible anisotropy of Rasbha parameters.

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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.7b05929. Construction of the AFE reference structure, ferroelectric polarization, Figures S1-S5, and Tables S1 and S2 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*: Tel: +39 0862 433759. Fax: +39 0862 433033. E-mail: [email protected]. *Tel: + 86 21 66132812. Fax: +86 21 66134208. E-mail: [email protected]. ORCID

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Shunbo Hu: 0000-0003-0472-0999 Jeffrey R. Reimers: 0000-0001-5157-7422 Wei Ren: 0000-0001-7317-3867

CONCLUSIONS ABX3 organohalide perovskites, such as MAPbI3, have recently emerged as promising candidates for photovoltaic applications, although there are still many open questions about their basic photophysical properties. By means of DFT calculations, we have presented an extended study of the FE polarization of the ABX3 hybrid perovskites (A = MA, FA; B = Sn, Pb; and X = Cl, Br, I) focusing on simple units cells containing one organic cation fully relaxed through molecular dynamics simulations.75

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 51672171 and 11274222), the National Key Basic Research Program of China (Grant No. 23051


Article


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2015CB921600), the Eastern Scholar Program from the Shanghai Municipal Education Commission, and the fund of the State Key Laboratory of Solidification Processing in NWPU (SKLSP201703). The Special Program for Applied Research on Super Computation of the NSFC−Guangdong Joint Fund (the second phase), the supercomputing services from AM-HPC, and Shanghai Supercomputer Center are also acknowledged. M.B. gratefully acknowledges funding by the Austrian Science Fund (FWF): P 30316-N27. D.D.S. acknowledges support by the German Research Foundation (DFG-SFB 1170 ToCoTronics) and ERC-StG-336012-Thomale-TOPOLECTRICS. The authors thank I. Baburin (TU Dresden, Theoretische Chemie), G. Kresse, and S. Picozzi for useful discussions and also thank B. Campbell and H. Stokes for discussions on rotational mode analysis. We thank M. Aroyo and M. Nespolo for interesting discussions during the “Shanghai International Crystallographic School working with the Bilbao Crystallographic Server” held at Shanghai University (Jun 11−17, 2017).

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