Electric field and strain induced Rashba effect in hybrid halide perovskites
EElectric field and strain induced Rashba effect in hybrid halide perovskites
Linn Leppert,
1, 2
Sebastian E. Reyes-Lillo,
1, 2 and Jeffrey B. Neaton
1, 2, 3 Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Department of Physics, University of California Berkeley, Berkeley, California 94720, USA Kavli Energy NanoScience Institute at Berkeley, Berkeley, California 94720, USA ∗ Using first principles density functional theory calculations, we show how Rashba-type energyband splitting in the hybrid organic-inorganic halide perovskites APbX (A=CH NH +3 , CH(NH ) +2 ,Cs + and X=I, Br) can be tuned and enhanced with electric fields and anisotropic strain. In par-ticular, we demonstrate that the magnitude of the Rashba splitting of tetragonal (CH NH )PbI grows with increasing macroscopic alignment of the organic cations and electric polarization, indi-cating appreciable tunability with experimentally-feasible applied fields, even at room temperature.Further, we quantify the degree to which this effect can be tuned via chemical substitution at theA and X sites, which alters amplitudes of different polar distortion patterns of the inorganic PbX cage that directly impact Rashba splitting. In addition, we predict that polar phases of CsPbI and(CH NH )PbI with R c symmetry possessing considerable Rashba splitting might be accessible atroom temperature via anisotropic strain induced by epitaxy, even in the absence of electric fields. Organic-inorganic hybrid halide perovskites have re-ceived considerable attention in the photovoltaic com-munity owing to their high power conversion efficienciesachieved within only a few years of device research . Firstprinciples calculations have played an important role inthe development of these materials, and in particular inthe prediction of a range of novel electronic and structuralphenomena, such as ferroelectric polarization , Rashbaand Dresselhaus energy band splitting and non-trivialtopological phases . The Rashba effect is an energy levelsplitting originating from spin-orbit interactions in sys-tems with broken inversion symmetry, as originally de-scribed by Rashba and Dresselhaus in noncentrosymmet-ric zinc blende and wurtzite semiconductors, respec-tively. It has been confirmed in a wide variety of mate-rials with either interfacial or bulk inversion symmetrybreaking . For example, a "giant" bulk Rashba split-ting, characterized by a Rashba coefficient of 3.8 eVÅ hasbeen found in the layered semiconductor BiTeI . In fer-roelectrics, i.e., systems with a spontaneous macroscopicpolarization which is switchable by an applied electricfield, the inversion symmetry-breaking potential gradientoriginates from the polarization, allowing for the Rashbasplitting to be controlled and switched by an externalelectric field.Recently, a significant Rashba effect of ∼ NH )PbI (MAPbI ), us-ing first-principles calculations , raising hopes thatthe compound might find application as a ferroelec-tric Rashba material in spintronic devices. Thesecalculations, however, rely on structural models forMAPbI that assume polar distortions and, with a fewexceptions , do not account for finite temperature effects.In light of ample experimental evidence and entropicarguments , which have refuted earlier reports of a ferro-electric or on-average polar phase of MAPbI and relatedmaterials , the question remains whether such largeRashba splitting is globally experimentally accessible ormight be tunable at room temperature. In this Letter, we predict with first principles calcula-tions that a Rashba effect can be observed in MAPbI at room temperature with an applied electric field, andquantify how its magnitude is affected by the macroscopicelectronic polarization. The magnitude of the energyband splitting depends on the degree of alignment of theorganic moieties, which can be achieved via polar distor-tions that couple directly to electric fields. We furtherdemonstrate that the displacement patterns, and con-sequently the magnitude of the Rashba splitting at thevalence and conduction band edges, can be controlled bychemical substitution at the A site, e.g., by an organicmolecule with distinct geometry such as formamidinium(FA), CH(NH ) +2 . Finally, we investigate the existenceof novel polar phases of CsPbI and MAPbI and predictthat epitaxial strain can lead to an R c polar phase withsignificant Rashba splitting at room temperature.MAPbI is known to undergo two phase transitionswith decreasing temperature: from cubic ( P m ¯3 m ) totetragonal ( I /mcm ) at T = 327 K, and from tetrago-nal to orthorhombic (
P nma ) at T = 162 K . All threephases are centrosymmetric and feature corner-sharingPbI octahedra. Neutron scattering experiments havedemonstrated that the MA molecules exhibit four-foldrotational symmetry about their C-N axis and three-foldrotational symmetry around the C-N axis in the P m ¯3 m and I /mcm phases . At room temperature and higher,the dynamics of these rotations are believed to be so facilethat MA can rotate quasi-randomly . Upon decreas-ing the temperature, the rotational motion is dominatedby the molecules’ high-symmetry orientations, accom-panied by a monotonic increase of the rotational angleof the PbI octahedra, which is the order parameter ofthe P m ¯3 m to I /mcm phase transition . Finally, at T (cid:46) K, rotations about the C-N axis freeze out andthe
P nma phase is realized.The rotational dynamics of the organic cation are gen-erally not taken into account in static density functionaltheory calculations of the
P m ¯3 m and I /mcm phasesof MAPbI and other hybrid perovskites. Instead, one a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug or several fixed orientations of the molecules are chosen,and single-point or average properties are computed andreported. Since the interaction of MA with the inorganicPbI cage sensitively depends on the orientation of theC-N axis , the spread of predicted band edge structuresand band gaps can largely be attributed to differencesin the assumed molecular orientation . Furthermore,structural relaxations for fixed MA orientations can leadto spurious distortions of the inorganic cage, which arelargely suppressed at finite temperatures due to the ther-mal motion of the molecules.Our density functional theory (DFT) calculations areperformed within the Perdew-Burke-Ernzerhof (PBE)generalized gradient approximation and the projectoraugmented wave formalism (PAW) as implementedin VASP . We treat 9 valence electrons explicitly forCs (5s ), 14 for Pb (6s ), 7 for I (5s ),7 for Br (4s ), 5 for N (2s ) and 4 for C (2s ).Spin-orbit coupling (SOC) is taken into account self-consistently. Brillouin zone integrations are performedon × × -centered k -point meshes using a Gaus-sian smearing of 0.01 eV and a plane wave cutoff of500 eV such that total energy calculations are convergedto within ∼
10 meV. It has been noted earlier that in Pb-based hybrid halide perovkites, PBE and other gradient-corrected density functionals lead to a fortuitous agree-ment with experimental band gaps as a result of neglect-ing spin-orbit interactions and many-body effects . Wehave confirmed that the band dispersion and magnitudeof the Rashba splitting obtained using PBE+SOC agreewell with those obtained using the screened-exchangefunctional HSE+SOC (see Supporting Information) andhence report PBE results throughout this work.We perform structural optimizations without SOC, re-laxing all ions without imposing symmetry constraintsuntil Hellmann-Feynman forces are less than 0.01 eV/Å.Taking van der Waals (vdW) interactions into accountis important for reaching quantitative agreement withexperimental lattice parameters . In particular, inclu-sion of vdW interactions reduces the unit cell volume ofMAPbI by ∼ to test this effect and find that the Rashbaenergy band splitting can be up to 30% lower in bothCBM and VBM as compared to the values predicted witha PBE-relaxed structure (see Supporting Information fordetails on these calculations). In what follows, we reportPBE results unless otherwise noted.To account for the on-average centrosymmetric struc-ture of MAPbI at room temperature and in order notto introduce artifacts related to a fixed orientation ofthe MA molecules, we start our considerations from theroom temperature experimental I /mcm crystal struc-ture with lattice parameters a = b = 8 . Å and c = 12 . Å , using a √ × √ × unit cell, asshown in Fig. 1a). We align the molecules such thatthey are antiparallel and their net dipole moment is zero,as would be expected on average at room temperature. a) b) c) [001][100] [010][010][001] [100] P o l a r i z a t i on ( µ C / c m ) Distortion (%)
FIG. 1. a) Nonpolar I /mcm reference structure using exper-imental lattice parameters and atomic positions and MA unitsaligned anti-parallely. b) Fully relaxed P mm polar structurewith MA units aligned in parallel. c) Polarization as a func-tion of % distortion along the path from centrosymmetric topolar structure. The inset schematically shows that the po-larization is a consequence of polar displacements of Pb (redarrows) and equatorial I (blue arrows) associated with theparallel alignment of MA. However, even at finite temperature, MA molecules canbe oriented, and polar distortions can be induced, by asufficiently large electric field. In our calculations, wesimulate this situation by performing a structural relax-ation starting from the experimental I /mcm phase withthe MAs aligned in parallel. This fully relaxed structurewith approximate P mm symmetry is shown in Fig. 1b),and is characterized by vanishing octahedral rotationsand polar distortions that are dominated by displace-ments of the Pb and the equatorial I atoms relative tothe I /mcm reference structure. We construct a struc-tural pathway between the centrosymmetric and the fullypolarized structure that consists of a rigid rotation of twoof the MA molecules, a decrease of the octahedral rota-tion amplitude, and an increase in amplitude of the polardistortions.Using the Berry phase approach within the moderntheory of polarization , we calculate the macroscopicpolarization of the fully polarized P mm structure tobe 12.6 µ C/cm , in very good agreement with previousDFT results . Fig. 1c) quantifies the polarization in-crease along the structural pathway discussed above from I /mcm to P mm . In P mm , Pb and apical I atoms aredisplaced by 0.1 Å and 0.01 Å along the [001] direction.The equatorial I atoms experience a displacement of 0.2 Åalong the same axis in the opposite direction (see Sup-porting Information for a full list of atomic displacementsand Born effective charges).We now turn to the evolution of the electronic struc-ture of MAPbI along the same structural pathway, fo-cusing on the Rashba splitting of the energy bands in k -space. The band structures of centrosymmetric and polarMAPbI are shown in Fig. 2a) and b), respectively. As iswell known, the conduction band minimum (CBM) is pri-marily comprised of Pb p -like states, whereas the orbitalcharacter of the valence band maximum (VBM) is I p andPb s . Fig. 2b) demonstrates that breaking the inversion -0.3-0.2-0.10.0 Z G R -10123 Γ A X Z Γ R M E - E F ( e V ) a) b) Γ A X Z Γ R M I pPb sPb p
FIG. 2. a) Band structure of MAPbI in centrosymmetric I /mcm symmetry, calculated using PBE including SOC. Allbands are two-fold degenerate. The colors signify the dom-inant orbital character of each band. The VBM is predom-inantly of I p character, but the inset shows that the VBMadditionally has Pb s (circles, red color scale) character. b)Band structure of fully polarized MAPbI . Breaking the in-version symmetry by aligning the molecules, leads to a Rashbasplitting of the bands in the directions perpendicular to thedistortion (see text). symmetry of the structure, lifts the degeneracy of thesebands, as has been discussed in previous studies . Themagnitude of this band splitting is k -dependent, and canbe approximately understood from quasi-degenerate per-turbation theory , where the Rashba Hamiltonian H R = λ ( k ) · σ with λ ( k ) = (cid:104) φ n k | (cid:126) m c ( ∇ Φ × ( (cid:126) k + p )) | φ n k (cid:105) istreated as a perturbation to a zero-order model Hamil-tonian without spin-orbit interactions. Here, Φ is thecrystal potential, σ the Pauli spin matrices, p the mo-mentum operator, and φ n k are Kohn-Sham states, m areelectron effective masses, and c is the speed of light. Ina 3D system with a polar distortion along the [001] di-rection, k c = (0 , , πc ) = k (cid:107) , where c is the [001] latticeparameter, defines the quantization axis along which thedegeneracy of the bands is maintained. This can be seenin Fig. 2b), where the Rashba splitting from Γ(0 , , to Z (0 , , πc ) is negligibly small. Along the directions inthe plane perpendicular to k (cid:107) , which in our case can bespanned by the vectors k a = ( πa , , and k b = (0 , πb , ,the Rashba splitting takes on the largest values.We quantify the magnitude of the Rashba effect usingthe parameter α R = 2 E R /k R , where k R is the distance in k -space between the crossing point of the spin-split bandsand the CBM or VBM, and E R is the respective energydifference as shown in Fig. 3a). Since the Rashba split-ting is isotropic in the (001) plane, we calculate the bandstructure from Γ (0,0,0) to X ( πa ,0,0) for five structuresalong the path specified in Fig. 1 and plot k CBM R , k VBM R , α CBM R and α VBM R as a function of polarization. Fig. 3b)and c) demonstrate that the Rashba effect in the CBMincreases with increasing polarization and reaches a max-imum value of α CBM R =2.3 eVÅ, roughly 60% of the valuereported for BiTeI and in good agreement with previ- a) b) k R E R -0.50.00.51.0-0.05 0 0.05 E − E F ( e V ) k-point distance (Å − ) c) k R ( Å − ) Polarization ( µ C / cm )0123 α R ( e VÅ ) Polarization ( µ C / cm )CBMVBM FIG. 3. a) The Rashba parameter α R is defined using thek-space splitting k R and the energy splitting E R . b) TheRashba splitting k R increases monotonically for both CBMand VBM. c) α CBM R and α VBM R as a function of polarizationfor MAPbI . ous reports . α VBM R behaves similarly, and reaches avalue about half the size of α CBM R . The Rashba wavevec-tor k R increases monotonically for both the CBM andthe VBM with polarization. However, the specific choiceof structural pathway leads to E V BMR , and consequently α VBM R , having a maximum at P ≈ µ C/cm .We can estimate the electric field, E c , necessary toalign the MA molecules and to induce the polar distor-tions of the P mm phase to first order as E c ≈ ∆ EV P ,where the unit cell volume V and the polarization P are obtained from our first-principles calculations and ∆ E is the energy barrier for the alignment of the MAmolecules; for ∆ E , we use a recently measured value of70 meV , which is slightly higher than computed val-ues of between ∼
20 meV and ∼
50 meV for the room-temperature phase of MAPbI . The resulting criticalfield is E c ≈ V/cm, a large value, corresponding to20 V across a MAPbI film of ∼
200 nm thickness. How-ever, since the Rashba effect increases with increasing po-larization, partial MA alignment at smaller fields shouldbe sufficient to observe Rashba splitting in MAPbI . As-suming an experimentally feasible bias of 4 V , that cor-responds to a polarization of about 3 µ C/cm followingthe above considerations, we predict a Rashba effect of α CBM R ≈ eVÅ, a smaller but not insignificant value.In Refs. 15 and 8 it was shown that the spin textures ofVBM and CBM can be controlled by realizing differentdistortion amplitudes and patterns of the perovskite lat-tice, and in particular by inducing different relative dis-placements of Pb and apical and equatorial halide atoms.Here we demonstrate that the relative magnitude of the a)b) c)d)MAPbBr FAPbBr [001][100] [010][010][001] [100] α R ( e VÅ ) Polarization ( µ C / cm )MAPbBr FAPbBr -0.2-0.10.00.10.2 MAPbI MAPbBr FAPbBr D i s p l a c e m en t ( Å ) PbX apX eq ∆ k ( Å − ) FIG. 4. Fully polarized, relaxed structure of a) MAPbBr andb) FAPbBr c) Displacement of Pb and apical and equatorialhalide atoms X=I,Br. d) α total R = α CBM R + α VBM R as a functionof polarization for MAPbBr and FAPbBr . The inset shows ∆ k = k CBM R − k VBM R which is close to zero for FA and increaseswith increasing polarization for MA. Rashba splitting in VBM and CBM can be controlled inthe same way, and that such an effect can be achieved inpractice by chemical substitution at the A and X sites.To show this we first replace I by Br and then addi-tionally MA by FA, which has been a commonly usedsubstitute for MA in recent work . ComparingMAPbBr and FAPbBr is rather straightforward, since,contrary to MAPbI and FAPbI , both are cubic at roomtemperature. For these two compounds we use structuraldistortion pathways analogous to those used for MAPbI ,in both cases starting from a centrosymmetric structurewith experimental lattice parameters and P m ¯3 m symme-try as reference . The fully relaxed structures withparallely aligned MA and FA molecules are shown inFig. 4a) and b), respectively. Both the displacementsof Pb and Br atoms (see Fig. 4c)) in MAPbBr , and theBorn effective charges (see Supporting Information), arevery similar in magnitude to MAPbI . The displacementof Pb in MAPbBr is about 70% of that of MAPbI ,which can be attributed to the smaller unit cell volumeof the Br compound. The Rashba splitting in CBM andVBM shows a similar trend as a function of polariza-tion, but with a maximum of only α CBM R =1.9 eVÅ and α total R = α CBM R + α VBM R =2.5 eVÅ (Fig. 4d)), as the po-larization is smaller (8.2 µ C/cm ) and the SOC in Br isweaker than in I.Replacing MA with FA changes the picture consider-ably. Firstly, in the fully polarized structure, the in-plane lattice vectors perpendicular to [001] increase from8.5 Å in the nonpolar experimental structure to a = 9 . Åand b = 8 . Å in the fully polarized structure, owing tothe two-dimensional geometry of FA. Furthermore, thealignment of FA leads to a small relative Pb atom dis-placement of -0.03 Å, whereas the dominant contribution to the distortion along [001] arises from the apical Bratoms, resulting in a polarization of 9.1 µ C/cm . TheRashba splitting reaches a maximum of α total R =2.5 eVÅ,i.e., the same value as in MAPbBr . Note however, thatunlike for MAPbBr , where the splitting occurs mainly inthe CBM, both CBM and VBM exhibit similar amountsof Rashba splitting for FAPbBr . This is demonstratedin the inset of Fig. 4d) which shows the calculated trendin ∆ k = k CBM R − k VBM R with polarization. ∆ k increaseswith increasing polarization for MAPbBr , because theinversion symmetry breaking field along the path changesmainly due to the displacement of the Pb and the equa-torial Br atoms. This in turn leads to stronger Rashbasplitting for the CBM due to its predominant Pb 6 p char-acter. Conversely, in FAPbBr , the displacement of Pb,and both the equatorial and apical Br atoms, results ininversion symmetry breaking that affects both the CBMand the VBM (predominantly Br 5 p and Pb 5 s orbitalcharacter).We now turn to evaluating the possibility of stabiliz-ing a polar phase at room temperature that would al-low the observation of Rashba splitting without the needfor strong electric fields. In what follows, we investi-gate low-energy polar phases of CsPbI and MAPbI .From a computational perspective, replacing MA withCs avoids complications related to the molecular orien-tation and provides an approximate way of assessing theeffect of biaxial strain on MAPbI . Our approach is mo-tivated by the well-studied effects of anisotropic straindue to epitaxial growth on the phase stability in particu-lar ferroelectric phases in traditional oxide perovskites .Previous computational studies have considered the ef-fect of hydrostatic pressure and biaxial strain . Fur-thermore, the experimental stabilization of the cubic P m ¯3 m phase of CsPbI at room temperature has beenattributed to strain . However, no studies thus far haveconsidered polar halide perovskites under biaxial strain.For MAPbI , we use PBE-TS and a plane wave cut-off energy of 600 eV to obtain accurate lattice param-eters for the experimentally observed centrosymmetricphases P nma , I /mcm , as well as the polar R c phase(see Supporting Information). Tab. I lists the energeticsfor each phase and compares them with the correspond-ing phases of CsPbI , for which we additionally con-sider polar P mm , Amm and R m structures, as wellas the non-perovskite P nma room-temperature phase ofCsPbI (n- P nma ). We calculate that R c is the onlyenergetically relevant polar phase for both compounds,with P mm , Amm , and R m being only ∼ , antiparallel alignment of the MAunits in the R c structure suppresses the polar Γ − modeand results in R ¯3 c structural symmetry. The alignmentof the molecules is associated with a small energy cost of ∼
35 meV.In the case of CsPbI , the polar phases P mm and Amm exhibit negligibly small Rashba energy band split-ting of less than 0.005 Å − . The Rashba splitting of TABLE I. Energy gain and estimated equilibrium strain ofselected phases of CsPbI with respect to the high tempera-ture cubic phase. The room temperature phase of CsPbI is anon-perovskite structure with P nma symmetry, here denotedas n-
P nma . Cs: space group ∆ E (meV/f.u.) σ ab σ bd R c
76 — -1.0 I /mcm
84 -2.2 -0.4
P nma
120 -1.4 -1.3n-
P nma
170 — —MA: space group ∆ E (meV/f.u.) σ ab σ bd R c
49 — -1.0 I /mcm
97 -2.0 -0.6
P nma
183 -2.5 -1.2the CBM of R m and R c is k CBM R =0.012 Å − , signif-icantly larger. In R c -MAPbI , the size of the Rashbasplitting approximately doubles compared with the cor-responding CsPbI phase, with k CBM R =0.023 Å − and α CBM R =1.6 eVÅ obtained for the R c structure, highlight-ing the crucial role of the MA molecule for large Rashbasplitting.To investigate whether the polar R c phase canbe accessed with biaxial strain, we calculate the epi-taxial strain diagram of CsPbI using "strained-bulk"calculations . In both the I /mcm and the P nma structure, there are two symmetry-inequivalent epitaxialmatching planes, as illustrated in Fig. 5a). The ab-plane(blue) is spanned by the lattice vectors t a and t b , whereasthe bd-plane (violet) is spanned by t b and t d = t a + t b .For R c , where t d = t b − t a − t c (Fig. 5c)), R c is re-duced to Cc symmetry under strain. Fig. 5b) shows thatthe polar R c phase is stabilized at about 1% compres-sive strain, and energetically competes with the I /mcm phase throughout a range of strains. Above 3% ten-sile strain, R c becomes lower in energy than the P nma phase due to the suppression of halide octahedral rota-tions at tensile strain. This suggests that the R c phasemight be realized at room temperature under epitaxialor other forms of large anisotropic strain .Since an explicit calculation of the epitaxial strain di-agram of MAPbI is complicated by the presence of theMA moieties, we follow Ref. 51 to estimate the strain cor-responding to the energy minimum of a structural phaseas σ j = 100 · (cid:80) i ( | t i | − | t i | ) / | t i | . Here j denotes therespective epitaxial matching planes j = ab and j = bd . The t i refer to the reference lattice vectors constructedfrom the cubic reference phase. The values reported inTable I for CsPbI are close to the respective energy min-ima in Fig. 5, demonstrating that our method of estimat-ing the equilibrium strain is reliable (see Supporting In-formation for details). With the exception of the P nma phase, we find that the σ j of CsPbI and MAPbI arevery similar, suggesting a rather similar energy vs. straindiagram for MAPbI and thus the possibility of accessing R c -MAPbI with biaxial strain.In conclusion, we have investigated routes by whichRashba splitting can be observed in room temperatureMAPbI and related halide perovskites. Due to the ro-tational freedom of the organic cation, electric fields canbreak inversion symmetry in I /mcm -MAPbI and leadto Rashba splitting. Since the magnitude of the splittingincreases with increasing polarization, we expect that theeffect will be observable at moderate electric fields thatlead to partial MA alignment. We further propose thatthe band edge characteristics of the splitting can be tunedby inducing different distortive patterns in the Pb-halidecage, and we have considered two examples, substitutingMA by FA and anisotropic strain. We predict that un-der the effect of moderate to high biaxial tensile strain,a polar R c phase with significant Rashba splitting isaccessible for CsPbI and MAPbI , suggesting an alter-native experimental route to observe the Rashba effectin halide perovskites. ACKNOWLEDGMENTS
Work at the Molecular Foundry was supported by theOffice of Science, Office of Basic Energy Sciences, of theU.S. Department of Energy, and Laboratory Directed Re-search and Development Program at the Lawrence Berke-ley National Laboratory under Contract No. DE-AC02-05CH11231. LL acknowledges financial support by theFeodor-Lynen program of the Alexander-von-Humboldtfoundation.
SUPPORTING INFORMATION
Comparison of Rashba splitting, effective masses andband gaps of
P m ¯3 m -MAPbI using PBE, PBE-TS andHSE. Displacements, Born effective charges and atomiccoordinates of MAPbI , MAPbBr and FAPbBr . Fur-ther discussion of the epitaxial strain diagram of CsPbI . ∗ [email protected] S. D. Stranks and H. J. Snaith, Nat. Nanotechnol. , 391(2015). A. Stroppa, D. Di Sante, P. Barone, M. Bokdam,G. Kresse, C. Franchini, M.-H. Whangbo, and S. Picozzi,Nat. Comm. , 5900(1 (2014). A. Stroppa, C. Quarti, F. De Angelis, and S. Picozzi, J.Phys. Chem. Lett. , 2223 (2015). C. Grote, B. Ehrlich, and R. F. Berger, Phys. Rev. B ,205202 (2014). F. Brivio, K. T. Butler, A. Walsh, and M. van Schilf-gaarde, Phys. Rev. B , 155204 (2014). b) t d t b t c t a t d t b t c t a a) c) -4 -3 -2 -1 0 1 2 3 4 ene r g y ga i n ( m e V ) epitaxial strain (%) ab-I4/mcmbd-I4/mcmbd-R3cab-Pnmabd-Pnma FIG. 5. a) Lattice vectors of the
P nma phase. The plane spanned by the vectors t a and t b (denoted by j =ab) is shown inblue, whereas the plane spanned by t b and t d ( j =bd) is shown in violet. For the I /mcm phase the epitaxial matching planesare defined in the same way. b) Epitaxial phase diagram of CsPbI . c) Lattice vectors of the R c phase. The matching planeis spanned by t b and t d . J. Even, L. Pedesseau, J. M. Jancu, and C. Katan, Phys.Stat. Sol. RRL , 31 (2014). R. Robles, C. Katan, D. Sapori, L. Pedesseau, J. Even,S. Chimiques, C. Uab, and F. Umr, ACS Nano , 11557(2015). F. Zheng, L. Z. Tan, S. Liu, and A. M. Rappe, Nano Lett. , 7794 (2015). T. Etienne, E. Mosconi, and F. De Angelis, J. Phys. Chem.Lett. , 1638 (2016). S. Liu, Y. Kim, L. Z. Tan, and A. M. Rappe, Nano Lett. , 1663 (2016). G. Dresselhaus, Phys. Rev. B , 580 (1955). E. Rashba, Sov. Phys. Solid State , 1109 (1960). A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.Duine, Nat. Mat. , 871 (2015). K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano,T. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miya-hara, A. Kimura, K. Miyamoto, T. Okuda, H. Namatame,M. Taniguchi, R. Arita, N. Nagaosa, K. Kobayashi, Y. Mu-rakami, R. Kumai, Y. Kaneko, Y. Onose, and Y. Tokura,Nat. Mater. , 521 (2011). M. Kim, J. Im, a. J. Freeman, J. Ihm, and H. Jin, Proc.Nat. Acad. Sci. , 6900 (2014). A. Amat, E. Mosconi, E. Ronca, C. Quarti, P. Umari,M. K. Nazeeruddin, M. Gratzel, and F. D. Angelis, NanoLett. , 3608 (2014). J. Beilsten-Edmands, G. E. Eperon, R. D. Johnson, H. J.Snaith, and P. G. Radaelli, Appl. Phys. Lett. , 173502(2015). Z. Fan, J. Xiao, K. Sun, L. Chen, Y. Hu, J. Ouyang, K. P.Ong, K. Zeng, and J. Wang, J. Chem. Phys. Lett. , 1155(2015). S. Govinda, P. Mahale, B. P. Kore, S. Mukherjee, M. S.Pavan, C. De, S. Ghara, A. Sundaresan, A. Pandey, T. N.Guru Row, and D. D. Sarma, J. Phys. Chem. Lett. , 2412(2016). ACS Energy Lett. , 142 (2016). A. Filippetti, P. Delugas, M. I. Saba, and A. Mattoni, J.Phys. Chem. Lett. , 4909 (2015). K. Gesi, Ferroelectrics , 249 (1997). C. C. Stoumpos, C. D. Malliakas, and M. G. Kanatzidis,Inorg. Chem. , 9019 (2013). Y. Kutes, L. Ye, Y. Zhou, S. Pang, B. D. Huey, and N. P.Padture, J. Phys. Chem. Lett. , 3335 (2014). A. Poglitsch and D. Weber, J. Chem. Phys. , 6373(1987). T. Chen, B. J. Foley, B. Ipek, M. Tyagi, J. R. D. Copley,C. M. Brown, J. J. Choi, and S. H. Lee, Phys. Chem.Chem. Phys. , 31278 (2015). A. Mattoni, A. Filippetti, M. I. Saba, and P. Delugas, J.Phys. Chem. C , 17421 (2015). Y. Kawamura, H. Mashiyama, and K. Hasebe, J. Phys.Soc. Jap. , 1694 (2002). J.-H. Lee, N. Bristowe, P. Bristowe, and T. Cheetham,Chem. Commun. (2015). C. Quarti, E. Mosconi, and F. De Angelis, Chem. Mater. , 6557 (2014). P. E. Blöchl, Phys. Rev. B , 17953 (1994). G. Kresse and D. Joubert, Phys. Rev. B , 1758 (1999). G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993). G. Kresse and J. Furthmüller, Phys. Rev. B , 11169(1996). C. Elsässer, M. Fähnle, C. T. Chan, and K. M. Ho, Phys.Rev. B , 13975 (1994). D. A. Egger and L. Kronik, J. Phys. Chem. Lett. , 2728(2014). A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. ,073005 (2009). Y. Li, J. K. Cooper, C. Giannini, Y. Liu, F. M. Toma, andI. D. Sharp, J. Phys. Chem. Lett. , 493 (2015). R. D. King-Smith and D. Vanderbilt, Phys. Rev. B ,1651 (1993). J. H. Lee, J.-H. Lee, E.-H. Kong, and H. M. Jang, Sci.Rep. , 21687 (2016). S. Y. Leblebici, L. Leppert, Y. Li, S. E. Reyes-Lillo,S. Wickenburg, E. Wong, J. Lee, M. Melli, D. Ziegler, D. K.Angell, D. F. Ogletree, P. D. Ashby, F. M. Toma, J. B.Neaton, I. D. Sharp, and Weber-Bargioni, Nat. Energy ,16093 (2016). G. E. Eperon, S. D. Stranks, C. Menelaou, M. B. Johnston,L. M. Herz, and H. J. Snaith, Energy Environ. Sci. , 982(2014). W. Rehman, R. L. Milot, G. E. Eperon, C. Wehrenfennig,J. L. Boland, H. J. Snaith, M. B. Johnston, and L. M.
Herz, Adv. Mater. , 7938 (2015). S. A. Kulkarni, T. Baikie, P. P. Boix, N. Yantara, N. Math-ews, and S. Mhaisalkar, J. Mater. Chem. A , 9221 (2014). K. M. Rabe, M. Dawber, C. Lichtensteiger, C. H. Ahn, andJ.-M. Triscone, in
Phys. Ferroelectr. - A Mod. Perspect. ,edited by K. M. Rabe, C. H. Ahn, and J.-M. Triscone(Springer-Verlag, Berlin, Heidelberg, 2007) pp. 10–11. C. Grote and R. F. Berger, J. Phys. Chem. C , 22832(2015). G. E. Eperon, G. M. Paterno, R. J. Sutton, A. Zampetti,A. A. Haghighirad, F. Cacialli, and H. J. Snaith, J. Mater. Chem. A , 19688 (2015). N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev,Phys. Rev. Lett. , 1988 (1998). O. Diéguez, K. M. Rabe, and D. Vanderbilt, Phys. Rev.B , 144101 (2005). L. Protesescu, S. Yakunin, M. I. Bodnarchuk, F. Krieg,R. Caputo, C. H. Hendon, R. X. Yang, A. Walsh, andM. V. Kovalenko, Nano Lett. , 3692 (2015). S. E. Reyes-Lillo and K. M. Rabe, Phys. Rev. B88