Dynamical Rashba Band Splitting in Hybrid Perovskites Modeled by

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Dynamical Rashba Band Splitting in Hybrid Perovskites Modeled by Local Electric Fields Thibaud Etienne, Edoardo Mosconi, and Filippo De Angelis J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09791 • Publication Date (Web): 30 Nov 2017 Downloaded from http://pubs.acs.org on December 2, 2017

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The Journal of Physical Chemistry

Dynamical Rashba band splitting in hybrid perovskites modeled by local electric fields Thibaud Etienne†‡§*, Edoardo Mosconi†‡, Filippo De Angelis†‡ †

Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), CNR-ISTM, via Elce di Sotto, I-06123, Perugia, Italy ‡ D3 Computation, Istituto Italiano di Tencologia, Via Morego 30, 16163 Genova, Italy. § Institut Charles Gerhardt Montpellier, Université de Montpellier, CC 1501, Place Eugène Bataillon, 34095 Montpellier, France

Abstract We report a computational method for evaluating the dynamical Rashba interaction coefficient of tetragonal and cubic methylammonium lead iodide (MAPbI3) perovskite, at various size scales, through Car-Parinello Molecular Dynamics (CPMD) trajectories. This strategy involves the calculation of a time-dependent band structure of the target systems using periodic boundary conditions, and the evaluation of the amplitude of band-splitting due to the Spin-Orbit Coupling (SOC) with the computation of a local electric field. This model, physically motivated by a rewriting of the SOC Hamiltonian according to the heterogeneity of three-dimensional systems and our choice of k-space sampling, involves directly the methylammonium (MA) configuration space sampling. Originally applied to tetragonal and cubic unit cells with static point-charges, this model is further ameliorated in order to take into account the replication of this unit cell through space, and to account for the dynamical nature of charge distribution. Once our protocol has been calibrated based on a toy model, it is exploited for investigating MAPbI3 systems in both tetragonal and cubic phases.

Keywords: perovskites, spin-orbit coupling, Car-Parrinello molecular dynamics, electrostatics. * To whom correspondence should be sent: [email protected]

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I. Introduction Perovskites materials1 are increasingly mentioned as a very promising solution to modern issues in energetics.2-18 Amongst the various advantages inherent to the exploitation of those hybrid materials for photovoltaic applications, one often cites the high photocurrent generation efficiency, which can be referred to, for instance, appropriate optical19-28 and charge generation and transfer properties,1718 or to long carrier lifetimes29-32 and mobility.30, 33 Still, this class of hybrid systems bears vast range of unknown or misconceived features, especially when the influence of the organic part (methylammonium cation) and its dynamics on the whole structural and electronic properties are concerned34-38 or when one seeks an explanation for the low recombination rates.39-45 The large spin-orbit coupling (SOC) of this material, which has been ascribed to the presence of lead (and, to some extent, to iodine),46 is known to be responsible for many properties characteristic of these materials. Among these properties, one finds the tin and lead perovskites band-gap variation,47 the deviation from cubic symmetry48 inducing a band-gap modulation, and reduction of carrier effective masses.49 A possible connection between SOC and spintronics of methylammonium lead iodide perovskite (spin-polarized carrier dynamics, spin-dependent charge recombination) has also recently been reported.50, 51 The so-called Rashba/Dresselhaus effect, related to the intrinsic SOC properties of materials and causing band splitting (vide infra), has gained considerable attention from theoretical communities in the last few years.47-52 Whilst it is established that materials such as MAPbI3 can exhibit a band splitting due to Rashba SOC,31, 38, 53-69 the intrinsic nature of this splitting remains unresolved, and a recent conclusive theoretical demonstration of the dramatic consequences of symmetry breaking36, 70-71 on band structure at the atomic scale has been reported relatively to MAPbI3 perovskites by accounting for the dynamical essence of electronic structure and system geometry.72 Due to the nature of the Rashba interaction, it is required for the band-splitting to occur that the system exhibits local symmetry breaking or a non-centrosymmetric crystal structure.73 Since the MAPbI3 system belongs to the I4/mcm space group, it is unlikely that such material will exhibit, at room temperature, a macroscopic ferroelectric behaviour.74 However, such a behaviour could in principle be observed at lower temperature, for instance at a temperature right above the transition to the orthorhombic phase of MAPbI3.36,75 In addition to the effect of SOC on the reduction of carriers effective masses mentioned above, it has also been reported31 through static and dynamical72 computational strategies that SOC might be related to the enhanced carrier lifetime of MAPbI3 perovskites. In particular, the intrinsic mechanism of lifetime enhancement was proposed to be related to symmetry-forbidden spintransition; i.e. that a given (computed) lifetime enhancement factor is exponentially connected to the Rashba splitting. Though the latter was externally continuously tuned in Ref.31, these results actually place the Rashba SOC paradigm at the center of the metal halide perovskite materials performances, hence paves the way to theoretical developments related to perovskite SOC in order to deepen the understanding of the physical phenomena taking place at the atomic scale in such materials, and being responsible for the important performances of such materials for photocurrent generation. In order to have a physically sound protocol which could retrieve crucial information from the most realistic conditions simulated by first-principles calculations, one should take into due account the fact that these materials experience dynamical structural deformations, and that 2 Environment ACS Paragon Plus

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depending on the size scale considered, these structural deformations might have a different impact on the electronic structure of the material, resulting in a variable SOC magnitude. To this end, we related in a previous report72 the theoretical investigation of the timedependent (i.e. dynamical) Rashba band splitting in bulk MAPbI3 perovskites by considering them at different spatial scales and along Car-Parrinello Molecular Dynamics (CPMD) trajectories,76-77 in order to evaluate how local is the nature of the spin-orbit coupling interaction in these architectures. The central quantity at the core of our investigations was the Rashba interaction coefficient31, 66, 78-80 (RIC, often written ), which amplitude indicates the intensity of the spin-orbit coupling in direction(s) orthogonal to given k-space sampling. In our previous work, we intended to evaluate the magnitude of band-splitting induced by local structural distortion at the sub-picosecond time scale in a globally centrosymmetric structure. We related the resulting Rashba effect to the coupled organic/inorganic degrees of freedom, similarly to what was reported by Azarhoosh et al.81 The Rashba band splitting has also been discussed in experimental reports, where the authors (in)directly involved the Rashba effect in their conclusions for providing an atomistic justification to a target experimental feature. For instance, angle-resolved photoelectron spectroscopy has been reported to be able to probe Rashba effects. Indeed, a “giant Rashba splitting” has recently been observed directly for methylammonium lead bromide.82 It was stated that SOC induced a certain circular dichroism, and the presence of a ring-shaped valence band was observed at low and higher temperature, in the orthorhombic and cubic phases respectively. The Rashba effect amplitude was evaluated to ca. 10 eVÅ for these two phases, to be compared to the theoretical values we reported in a previous study for methylammonium lead iodide.83 Some experimental protocols based on microwave conductivity are able to probe charge carriers generated by light-absorption in volumes having the size of few unit cells, and have provided dynamical photo-conductance data. These data revealed that free mobile charges generation is optimized when the excitation energy is taken right above what has been interpreted as an indirect band-gap, which was ascribed to a Rashba band splitting.84 Rashba-induced indirect band-gap was also invoked for explaining the increase of carrier recombination due to a quenching of the Rashba effect by hydrostatic pressure.85 In this case, the pressure-induced reduction of the Rashba effect amplitude was attributed to a lowering of the local electric field in the vicinity of the lead atom. In the present contribution, the quantity is again computed through CPMD simulations in order to analyze its time-evolution and highlight the importance of structural deformations on the electronic properties of MAPbI3 with periodic boundary conditions under two main forms: the tetragonal and cubic cells. In addition to this direct approach, the -RIC was also approached through a simple model using dynamical point-charges and truncated periodic conditions. This model simply involves the computation of a local electric field at the lead position(s) and the exploitation of the methylammonium configuration space sampling to derive a quantity called which collects structural and electronic features of the whole system into a simple collective metric reproducing the local SOC properties of the material. The relevance of exploiting this descriptor, i.e. the physical motivation for its construction, is demonstrated by a rewriting of the SOC Hamiltonian, adapted to the present three-dimensional heterogenous sytems, according to the kspace samplings we decided to use. The closeness between the time-evolution of these two metrics ( and ) lead us to establish that the local SOC of such hybrid system is driven by the high mobility of its organic part.



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This article is organized as follows: after briefly reviewing in section II the basics of spinorbit coupling applied to the scholar case of 2D C2v systems, we introduce in paragraph III.1 the various forms of MAPbI3 that are to be investigated in this contribution. We subsequently present the methodology used for the calculation of the time-dependent band structure of MAPbI3 and the various k-space samplings that are performed through the dynamics to construct the time-dependent local SOC profile. We show then how the classic Bychkov-Rashba Hamiltonian has to be adapted to the case of a three-dimensional heterogeneous system. Paragraph III.4 further introduces the method used for computing the local electric field and the metrics. This protocol is the result of a calibration, which is presented at the second and third points of section IV, while its first paragraph contains information related to the results of time-dependent band structure calculations. Once the theoretical strategy has been calibrated with a ‘toy’ model, it is exploited to characterize the hypothetical cubic and tetragonal phases MAPbI3, where the straight connection between and is more directly pointed. II. Theoretical background - Bychkov-Rashba SOC in a two-dimensional C2v system In most of the cases reported in the literature, the Rashba effect31, 53-55, 66, 78 is depicted as a consequence of the rupture of inversion symmetry orthogonally to a k-points sampling plane. Those effects are usually reported for a two-dimensional system with a C2v symmetry, and are simply resulting in the addition of a term to a given potential-free Hamiltonian in order to obtain the socalled Bychkov-Rashba Hamiltonian (1) where we find the Pauli spin matrix and

(2) i.e., the linear momentum (defining the momentum space sampling, which is orthogonal to the vector - the z direction basis vector in this case) and the Rashba splitting factor, also called the Rashba Interaction Coefficient (RIC, vide supra). The eigenvalues corresponding to this Hamiltonian are

(3) in the expression of which we can isolate the factor

(4) ±k therefore corresponds to the position of the two ± wells vertices, and is defining the energy difference between one energy function and the other, at one vertex position. Scheme 1 of ref72 reproduces this energy band splitting with details related to the various variables aforementioned.


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III. Computational strategy III.1. Preparation of the systems and computational details III.1.1. Car-Parrinello Molecular Dynamics CPMD/GGA-PBE86 simulations have been performed using the Quantum Espresso package.87 Ultrasoft pseudopotentials were used to describe electron-ion interactions, with an explicit inclusion of electrons from O, N and C 2s, 2p; H 1s; Ti 3s, 3p, 3d, 4s; I 5s, 5p; Pb 6s, 6p, 5d shells in the calculations. 25 and 200 Ry were used as plane-wave basis set cutoffs, respectively for the smooth part of the wave functions and the augmented density. For the MAPbI3 cubic 2x2x2 test-model, further named as the TOY model, trajectories of ca. 2.5 ps have been generated with an integration time step of 4 au. 400 au fictitious masses have been used for accounting for the electronic degrees of freedom; atomic masses have been attributed a real value. Initial ions position randomization of the methylammonium cations has been used to reach temperature in the 350-400 K range. No thermostat was further applied. The difference between tetragonal models 1 and 2 discussed in this paper consists in different methylammonium cations relative orientation, leading to different band splitting amplitudes.72 Unlike species 2, species 1 show a sensible amount of Rashba splitting. Tetragonal 2x2x2 MAPbI3 models 1 and 2 dynamics have been simulated with an integration time step of 5 au (total simulation time: 12 ps). Fictious masses of 500 au have been used for modeling the electronic degrees of freedom; again, atomic masses have been set to a real value (excepted for hydrogen atoms for which a 2.00 au is used). Initial ions position randomization has also been used here to reach a temperature in the 350-400 K range; again, no thermostat was further applied. III.1.2. Electronic structure analysis PWscf code has been used for electronic structure analysis, with 25 and 200 Ry plane-wave basis set cutoffs for the smooth part of the wave functions and the augmented density respectively. SOC interaction has been included using ultrasoft pseudopotential together with the GGA-PBE exchange-correlation functional. After preparing the 1x1x1 cubic and tetragonal models (see below), 8x8x8 (4x4x4) k-point grid has been generated for the methylammonium lead iodide 1x1x1 cubic (tetragonal) system, followed by a band structure calculation with a R M and R Г (Г M and Г Z) Brillouin zone sampling. This sampling has been performed with 0.1000 and 0.1018 reciprocal lattice units for the cubic R M and R Г directions, and 0.1414 and 0.2332 reciprocal lattice units for the tetragonal Г M and Г Z directions. Parabola were further generated by performing a threepoint fitting (see below). From the expression of these parabola, the quantities related to the Rashba SOC magnitude were extracted. III.1.3. Cubic (TOY) model preparation The preliminary test-model (TOY-model) was created using the cubic cell parameters from Poglitsch et al.1 25 ps CPMD trajectories were generated, with the inorganic framework kept fixed 5 Environment ACS Paragon Plus


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and the organic methylammonium cations left free to move. A 2.32 ps window was extracted from the total simulation, and we cut single 1x1x1 MAPbI3 unit on which PWscf single point calculations were performed, for 96 selected snapshots. III.1.4. Tetragonal and cubic models preparation A 2.42 ps window has been extracted from the total trajectory, and we have further cut single 1x1x1 tetragonal MAPbI3 unit (1-4 and 2-4) on which we performed a PWscf single point calculation on 800 selected snapshots. From the tetragonal unit, a cubic unit is also cut (1-1 and 2-1), and identical further electronic structure analysis is performed as for the tetragonal unit. All the details related to the structures discussed in this paper are also visually reported in Figure 1 of Ref.72, where we see how tetragonal (1/2-4) and cubic (1/2-1) unit cells are generated from the tetragonal 1/2-32 supercell. III.2. Rashba interaction coefficient assessment The k-space sampling from R in the M or directions for the cubic cell is performed by extracting the energy value at three k-points k(q0), k(q1) and k(q2), defined according to

(5) with = , . Some of the variables are known a priori (6) by observing the geometry of the first cubic Brillouin zone (see also Figure 1). We deduce from our previous assumptions that in this case (7) After this sampling is achieved, a parabola is constructed from these three calculated points. The parabola is further centered in order to evaluate the following parameters, in relation with the Rashba interaction coefficient evaluation discussed above: (8) i.e. the position of the vertices, the depth of the well and the RIC.


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III.3. Adaptation of the SOC Hamiltonian to a 3D heterogeneous system III.3.1. 2D Vs 3D construction scheme for the Rashba interaction coefficient For a system with C2v symmetry, one could invariably sample kx or ky and obtain the RIC including the electric field projected projected in the direction orthogonal to both kx and ky. Indeed, the energy profiles sections in the kx and ky directions are symmetric with respect to permutation between the given coordinates. We are now moving to our 3D heterogenous system and performing a sampling along an axis. For the cubic cell, this axis is defined by the R-M or R- directions. Due to the nature of our system, there exists no direction orthogonal to the sampling axis for which the energy profile section would be identical to the characteristic R-to-M or R-to- sampling axis. We are therefore out of the scholar 2D(∥)-1D(⊥) construction of the harmonic RIC. For the cubic cell we are rather in a 1D(∥)-2D(⊥) scheme, so that we need to define two directions orthogonal to the sampling axis for constructing the appropriate Bychkov-Rashba Hamiltonian and evaluate its eigenvalues, as well as the RIC. These two directions actually define a reference plane , which can have non-vanishing components in two or three directions of space, depending on the given sampling direction. In the case of the R-to- sampling axis, the reference plane, [R- ] orthogonal to the sampling direction [R-], has variable components in the three dimensions of space, while the [R- ] plane has variable components only in the x and y dimensions (see Figure 1). The crucial consequence resulting from this difference lies in the setting of the local electric field components to be included into the expression of RIC. Indeed, as we understood that for a 2D system with C2v symmetry the vector used for the electric field projections to be included into the RIC computation is z, orthogonal to the in-plane wave vector ∥ = kx + ky, here we understand that the RIC evaluation for the cubic cell will include the characteristic components of the relevant reference planes orthogonal to the sampling direction, namely Ex, Ey and Ez for the R- sampling and only Ex and Ey for R-M. It is important to note that since for a given Euclidean space there exists a strict infinity of external combinations of the two vectors orthogonal to the R- and R-M directions, the appropriate combination of the electric field components cannot be rationally constructed from an external scheme, but can only be deduced from an internal geometrical collective variable. III.3.2. MA configuration space for collective variable Given the high mobility of the MA cation, and that for the cubic 1 and 2 systems the three iodine atoms, which are indiscernible, have a grossly constant relative orientation with respect to the three Euclidean axes, only the MA relative configuration can be regarded as a relevant collective variable. Therefore, the most appropriate way to account for this relative configuration is to consider the projection of the MA dipole moment into the x, y and z (R-) or x and y (R-M) which, up to a sign, is simply equal to the projection of the carbon-nitrogen bond vector (noted m in the following) in the respective directions. For example, Figure 2 further illustrates the projection of a given vector parallel to m in the y direction.



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Figure 1 Geometrical identification of the ensemble of vectors orthogonal to the R- and R-M kspace sampling directions in cubic cells.

Figure 2 Illustration of the projection of m (or, by extension, of any vector parallel to m) in the y direction. The fact that the highly geometrically flexible MA grossly bears a +1 charge, counterbalanced by the heavier and less-flexible PbI3 fully justifies the sole exploitation of MA configuration space for pondering the electric field components into our central collective variables. Moreover, we know that those depend on the sampling axis considered. For instance, in the case of the cubic cell dynamics, one can follow the time-dependent coefficient defined as

(9) as well as

defined as


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(10) Those coefficients are constructed based on the rule stating that the SOC effect is observable in the direction(s) orthogonal to the k-space sampling (see previous considerations). Their construction is illustrated at Figure 1 for the cubic cell with a R- and R-M sampling where we identify an infinity of vectors normal to the R- and R-M sampling directions, belonging to the appropriate plane and defined in the (x,y,z) or (x,y) basis respectively. For the tetragonal cell, if we consider the -M sampling, we see that the coordinates of each point of the -M axis can be reduced to the linear combination of two k directions (kx and ky). One can unequivocally construct a vector with variable components only in the z direction, which would be invariably orthogonal to the -M direction through the dynamics. This assumption reduces the dimensionality of the collective variable to be followed, i.e. , defined as

(11) where the sum over m stands for the four lead atoms included into the tetragonal unit cell. Figure 3 shows how any vector parallel to the z direction is orthogonal to the -M direction in tetragonal cells.

Figure 3 Geometrical identification of the reduced ensemble of vectors orthogonal to the -M kspace sampling direction in tetragonal cells. Note that = ′′.



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III.3.3. Rewriting the Hamiltonian In this paragraph, we show how the SOC Hamiltonian has to be adapted in order to account for the higher dimensionality of the k-space sampling used for the evaluation of the RIC. The general effective Hamiltonian for the MAPbI3 cubic cell writes

(12) where we find the factors and operators (13) Geometrical considerations obviously lead to setting the parameters to (14) and, since we are working within a right-handed Euclidean space, the rotation operators are

(15) where the subscripts define the basis vector around which the system’s referential is rotated to align the basis vectors with the sampling direction. According to these considerations, one can write the eigenvalues of the effective Hamiltonian

(16) The eigenvalues difference accordingly sets the adapted RIC

(17) This actually implies that

(18) where we can identify the obvious connection with the aforementioned collective variable .


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III.4. Computing the local electric field and the Rashba splitting The derivation of point-charge-derived local electric field at lead position(s) is recalled in supporting information. As this electric field is taken to be dynamical, its components at the lead position are computed at the geometry of each frame extracted from the CPMD trajectories. For the first level of approximation considered here, the point-charges used are static, i.e. they are not recomputed for each geometry of our electric field profile. The set of charges are derived from a SIESTA SCF calculation performed on the first frame with a triple-zeta basis set, followed by a Mulliken population analysis, which was carried out at PBE-GGA86 level with SIESTA 3.0 program package88 using a TZP basis set along with pseudo PBE non-relativistic potential for Ti, I, Cl, O, C, N and H atoms. Pb atoms are treated with WC-GGA89 relativistic pseudo-potential. Electrons from I 5s, 5p ; Cl 3s, 3p ; O, N and C 2s, 2p ; H 1s ; Ti 4s, 3d ; Pb 6s, 6p, 5d shells are explicitly included in the calculations. A value of 100 Ry is used as plane-wave cutoff for the grid. Note that SOC interactions are not included in these SIESTA calculations. A first improvement in our model to be introduced is an a posteriori truncature of the periodic conditions: from each frame of a given CPMD trajectory, the geometry and point-charges were replicated in the three dimensions of space to avoid some edge effects. The truncature is achieved according to the replication of the discrete distribution of the charges {q} through space. For instance, the replication truncature scheme for the cubic cell is given by (19) where is pointing the th ponctual charge and is the level of truncature. For instance, the unit cell will be characterized by an equal to zero. In the last equation, is the cubic cell parameter. For the tetragonal cell, the truncature scheme can be adapted by introducing the respective cell parameters for the three dimensions. Finally, we introduced an ultimate improvement in our metric computation strategy: the use of dynamical point-charges, i.e. the computation of atomic charges for each frame of a given trajectory, prior to cell replication and local electric field evaluation. IV. Results and discussion IV.1. Rashba interaction coefficient from time-dependent DFT band structure calculation The DFT dynamical Rashba interaction coefficient profiles are reported for tetragonal 1/2-4/1 in Ref.72. In order to formally confirm the closeness between our metric and the RIC in the case of real-life systems, we need to establish the reliability of our theoretical strategy for computing , based on its evaluation from the geometries extracted from the CPMD trajectory of the TOY model and by comparison with the PBC-DFT value.



IV.2. Impact of the cell replication on the closeness between () and () As discussed in Section III.4, we truncated the periodic conditions for approaching the electronic properties of MAPbI3 without experiencing border effects by replicating the unit cells through space for each frame extracted from the CPMD trajectory. In order to assess the impact of such replication, we performed the evaluation of () for the toy model considering the R-M sampling direction, with and without using the replication truncature. When replica were considered, different levels of truncature have been used to establish up to which level a convergence on the () value is achieved. It appears (see Figure S1) that the deviations experienced when moving from = 2 to = 3 are negligible, which means that the convergence is achieved for = 2. This level of truncature will therefore be used in section IV.4 (and for the related data reported in SI) for the 1 and 2 systems. Figure S2 also reports the comparison between () and () when the second level of replication of the cubic cell is used, together with static Mulliken charges. This figure shows that there already exists an obvious correlation between our simple electrostatic model and the DFT profile. In order to achieve a better matching between the two central quantities, we introduce in the next paragraph an additional improvement to our model by considering dynamical charge distributions for the calculation of ( ). IV.3. Including the time-dependence of charge distribution The second variable in the calibration of our theoretical strategy is the nature (static vs dynamical) of the point-charges used for the computation of the local electric field and (). Figure 4 reports the joint computation of () from the electric field derivation based on cell replication and using dynamical Mulliken charges and () from time-dependent DFT band structure.


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Figure 4 Plot of scaled () (electric field-derived splitting metric – red curve, “Elec”) and Rashba () (blue curve, “DFT”) for the toy model (R-M sampling). For the computation of () the second level of truncature has been used for the replication of unit cell through space, and the Mulliken charges have been re-computed for each of the 96 frames. A first look at the Figure 4 immediately suggests that, despite the limitations inherent to the simplicity of our splitting metric, this model is able to reliably predict the variation in SOC amplitude through time. Indeed, one clearly identifies the peaks in DFT to be unequivocally assigned to the corresponding fluctuations in the () profile. For the sake of comparison, the joint computation of RIC and unit cell-derived splitting metric derived with dynamical Mulliken charges is reported in Figure S3. These preliminary results indicate that our theoretical strategy has been appropriately calibrated and might be exploited for further calculations on fully relaxed hybrid systems. IV.4. Local Rashba interaction coefficient in cubic and tetragonal MAPbI3 Now that the reliability of our model has been confirmed, we are able to exploit the local electric field calculations to correlate them with the DFT () calculations in the case of global hybrid MAPbI3 systems with originally parallel (1) and anti-parallel (2) relative configurations of the methylammonium bond vectors in the tetragonal cell. We respectively report in Figures 5 and 6 the joint plot of () and () for the cubic (1-1) and tetragonal (1-4) cells, where the splitting descriptor () has been evaluated by a postprocessing of the CPMD trajectory involving the re-computation of Mulliken charges and replication of the unit cell in the three dimensions of space (with = 2). In order to demonstrate the



full transferability of our electrostatic model to more general geometrical problems of higher dimensionality, we considered here the R- direction for the cubic cell, for which the reference plane orthogonal to the sampling axis has three variable dimensions. For the sake of completeness, identical computations for systems 2-1/4 are reported in SI (Figures S4 and S5).

Figure 5 Joint plot of CPMD-derived scaled () - red curve (Elec) - and () - blue curve (DFT) for the cubic 1-1 system (rot) with a R- sampling. Dynamical charge computation and cell replication (level 2) are implied in the evaluation of the time-dependent splitting metric (), whose average value through the dynamics trajectory is 6.24 V/m.


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Figure 6 Joint plot of CPMD-derived scaled () - red curve (Elec) - and () - blue curve (DFT) for the tetragonal 1-4 system (rot) with a -M sampling. Dynamical charge computation and cell replication (level 2) are implied in the evaluation of the time-dependent splitting metric (). The average of () through the simulation is 2.56 V/m. Again, one clearly recognizes the joint time-evolution of the two quantities for both cubic and tetragonal cells. We can therefore conclude that there locally exists a strong correlation between the methylammonium configuration space in this type of lead halide perovskites and the dynamical amplitude of its Rashba spin-orbit coupling. Indeed, one can state that the most significant contribution to the symmetry breaking in the dynamically relaxed MAPbI3 arises from the sparse configuration space of its organic subsystem. V. Conclusion We have reported a computational study of hybrid MAPbI3 dynamical solid-state electronic properties. More especially, we investigated the local spin-orbit coupling observed when considering structural distortions in this organometal halide perovskite at the atomic scale. The CarParrinello Molecular Dynamics methodology has been exploited to retrieve the structural deformations of the hybrid system through time. Based on the time-evolution of the perovskites geometry we computed time-dependent band structure and evaluated the dynamical band splitting induced by Rashba spin-orbit interaction for cubic and tetragonal MAPbI3. This TD-SOC profile has been correlated to a novel simple electrostatic model allowing one to predict the change in the Rashba interaction amplitude through time from a local electric field interaction and by taking into



account the sampling of methylammonium configuration space. The interesting agreement between these two approaches lead us to conclude that the local SOC in MAPbI3 perovskites is driven by the mobility of the organic part of this hybrid system. Acknowledgements Ms Francesca Cavazzini is gratefully acknowledged for her support and for fruitful discussions. The authors gratefully acknowledge the project PERSEO-“PERrovskite-based Solar cells: towards high Efficiency and lOng-term stability” (Bando PRIN 2015-Italian Ministry of University and Scientific Research (MIUR) Decreto Direttoriale 4 novembre 2015 n. 2488, project number 20155LECAJ) for funding. References

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Table of Contents Graphic

10.0 9.0 8.0 7.0 α [eV Å]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6.0 5.0 4.0 3.0 2.0 1.0 0.0

DFT Elec 0.5

1.0 1.5 Time [ps]


2.0


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In the case 1 of a 2D system with C2v symmetry, we were talking in terms of in-plane wave vector (kk ), 2

and the derivation direction for V was defined quasi-unequivocally (±ˆ z ? kk ). 3

4 For kM k and kk , there exists an infinity of directions orthogonal to our sampling axis, and the reference 5

mathematical6 object for the derivation of the electrostatic potential must no longer be a vector but a plane. 7

We are therefore in a 1D(k)-2D(?) perspective because of the nature of our sampling. Indeed, any two 8

9 directions orthogonal to our sampling axis define a plane. 10

In the case11of the R !

axis, this plane (⇡ ) has variable components in the three euclidian dimensions,

ACS Paragon Plus Environment 12 unlike ⇡M , which has variable components only in the x and y dimensions. 13 14 As we understood that for a 2D system with C2v symmetry the direction of the electric field to be 15 Figure III.1 Geometrical of the to ensemble of vectors orthogonal R-' and R-M included in ↵16computation is identification the one orthogonal the in-plane wave vector, here to wethe understand that the k-space sampling directions in cubic cells. 17 ↵ evaluation 18 will only include the components (x, y and z for the R ! axis, only x and y for the R ! M 19 axis) characteristic to the respective ⇡ plane. 20 Since for 21 a given euclidian space there exists a strict infinity of external combinations of two vectors 22 orthogonal to23the R! or R! M, the appropriate combination of electrostatic potential gradient com24 be constructed from an external scheme, but can only be deduced from internal geometriponents cannot 25 26 collective variable. cal/electrostatic 27 Given that 28 for the ROT/ISO system, the three iodine atoms, which are indiscernible, have a grossly 29 constant relative orientation with respect to the three euclidian axes (see figures 1 and 2 below), only the 30 31 methylammonium relative configuration can be regarded as the collective variable. In this context, the most 32 Figure III.2 Illustration of the projection of m (or, by extension, of any vector parallel to m) in the appropriate way 33 to account for this relative configuration is to take the projection of the MA dipole in the y direction.34 x, y and z (R! ) or x and y (R! M) which, up to a sign, is simply equal to the projection of the CN 35 36 the respective directions. Together with the fact that the highly geometrically flexible MA bond vector in The fact that the highly geometrically flexible MA grossly bears a +1 charge, counterbalanced by 37 (i.e. configuration the balanced space electrostatic bears +138 charge counterbalanced by the heavier PbI the the heavier and less-flexible PbI3 fully justifies theand soleless-flexible exploitation of3 MA for 39 thegeometrical electric field components central variables. Moreover, know the a series, but also we amongst butpondering unbalanced variability of MAinto withour respect to collective PbI3 – through 40 that those depend on the sampling axis considered. For instance, in the case of the cubic cell configurations41of a single trajectory), this justifies the implication for the results of E/↵ for cubic cells of C dynamics, 42 one can follow the time-dependent coefficient ' (t) defined as (m being the43 CN vector) 9 8 44 = < X 45 [E |m| 1 (t) / ↵ (t) (12) [1] (t) · ⇠][⇠ · m(t)] Pb 46 ; : ⇠=x,y,z 47 (9) in the case of48R! direction, 49 9 8 C = as well as '50M (t) defined as < X 51 [EPb[1] (t) · ⇠][⇠ · m(t)] |m| 1 (t) / ↵M (t) (13) ; : 52 ⇠=x,y 53 in the case of54R! M direction, and 55 4 X 56 (14) [EPb[m] (t) · ⇠][⇠ · m(t)]|m| 1 (t) / ↵M (t) 57 8 m=1 58 59 60

In the case of a 2D system with C2v symmetry, we were talking in terms of in-plane wave vector (kk ), and the derivation direction for V was defined quasi-unequivocally (±ˆ z ? kk ).

For kM k and kk , there exists an infinity of directions orthogonal to our sampling axis, and the reference

mathematical object for the derivation of the electrostatic potential must no longer be a vector but a plane. We are therefore in a 1D(k)-2D(?) perspective because of the nature of our sampling. Indeed, any two directions orthogonal to our sampling axis define a plane. In the case of the R !

axis, this plane (⇡ ) has variable components in the three euclidian dimensions,

unlike ⇡M , which has variable components only in the x and y dimensions. As we understood that for a 2D system with C2v symmetry the direction of the electric field to be

FigureinIII.1 Geometricalis identification of the to ensemble of vectors orthogonal R-' and R-M included ↵ computation the one orthogonal the in-plane wave vector, here to wethe understand that the k-space sampling directions in cubic cells. ↵ evaluation will only include the components (x, y and z for the R !

axis, only x and y for the R ! M

axis) characteristic to the respective ⇡ plane. Page 25 of 29


Since for a given euclidian space there exists a strict infinity of external combinations of two vectors orthogonal to the R!

or R! M, the appropriate combination of electrostatic potential gradient com-

1 ponents cannot be constructed from an external scheme, but can only be deduced from internal geometri2 3 cal/electrostatic collective variable. 4 ACS Paragon Plus Environment 5 for the ROT/ISO system, Given that the three iodine atoms, which are indiscernible, have a grossly 6 constant relative orientation with respect to the three euclidian axes (see figures 1 and 2 below), only the 7 8 methylammonium relative configuration can be regarded as the collective variable. In this context, the most 9 Figure III.2 projection of m (or, by is extension, of any vector of parallel to m) in the appropriate 10 wayIllustration to account of forthe this relative configuration to take the projection the MA dipole in the 11 y direction. x, y and z (R! ) or x and y (R! M) which, up to a sign, is simply equal to the projection of the CN 12 13 bond in the directions. Together withgrossly the factbears that athe geometrically flexible Thevector fact 14 that therespective highly geometrically flexible MA +1highly charge, counterbalanced byMA 15 charge (i.e. configuration the balanced space electrostatic bears +1 counterbalanced by the heavier PbI the the heavier and less-flexible PbI3 fully justifies theand soleless-flexible exploitation of3 MA for 16 the electric field components into our central collective variables. Moreover, we know butpondering unbalanced 17 geometrical variability of MA with respect to PbI3 – through a series, but also amongst the that those18depend on the sampling axis considered. For instance, in the case of the cubic cell configurations of a single trajectory), this justifies the implication for the results of E/↵ for cubic cells of C dynamics,19one can follow the time-dependent coefficient ' (t) defined as 20CN vector) (m being the 21 9 8 22 = < X 23 [EPb[1] (t) · ⇠][⇠ · m(t)] |m| 1 (t) / ↵ (t) (12) ; : 24 ⇠=x,y,z (9) 25 in the case of 26 R! direction, 9 8 27 C = as well as28'M (t) defined as < X [EPb[1] (t) · ⇠][⇠ · m(t)] |m| 1 (t) / ↵M (t) (13) 29 ; : ⇠=x,y 30 31 R! M direction, and in the case of 32 4 X 33 (14) [EPb[m] (t) · ⇠][⇠ · m(t)]|m| 1 (t) / ↵M (t) 34 8 m=1 35 36 37

8 < X in the case of R!

:

⇠=x,y,z

direction, 8

Dynamical Rashba Band Splitting in Hybrid Perovskites Modeled by

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