Abstract
Hybrid organic–inorganic perovskites (HOIPs) are introducing exotic directions in the photovoltaic materials landscape. The coexistence of inversion symmetry breaking and spin–orbit interactions play a key role in their optoelectronic properties. We perform a detailed study on a recently synthesized ferroelectric layered HOIP, (AMP)PbI_{4} (AMP = 4aminomethylpiperidinium). The calculated polarization and Rashba parameters are in excellent agreement with experimental values. Moreover, we report a striking effect, i.e., an extraordinarily large Rashba anisotropy that is tunable by ferroelectric polarization: as polarization is reversed, not only the spin texture chirality is inverted, but also the major and minor axes of the Rashba anisotropy ellipse in kspace are interchanged—a pseudo rotation. A k·p model Hamiltonian and symmetrymode analysis reveal a quadrilinear coupling between the cationrotation modes responsible for the Rashba ellipse pseudorotation, the framework rotation, and the polarization. These findings may provide different avenues for spinoptoelectronic devices such as spin valves or spin FETs.
Introduction
In recent years, there has been an intense research effort surrounding the conversion of solar energy into electric power through photovoltaic cells^{1,2,3}. Hybrid organicinorganic perovskites (HOIPs) with composition ABX_{3} (A = organic cation, such as [CH_{3}NH_{3}]^{+}; B = divalent metal cation, such as Sn^{2+}, Pb^{2+}; X = halogen anion) are proposed as a next generation of materials for solar cells and lightemitting diodes^{4,5,6,7,8}. The increase in structural degrees of freedom from the inclusion of organic cations into a BX_{3} inorganic framework enables structural distortions and columnar shifts which are otherwise forbidden in standard inorganic perovskites. Furthermore, ferroelectricity (FE) originating from structural symmetry breaking is a potentially critical phenomenon in such soft and flexible photovoltaic materials, where the electric field plays an important role in promoting electron–hole pair separation and suppressing charge recombination, inherently breaking the Shockley–Queisser limit^{9,10,11,12,13}. Therefore the FE properties of perovskites have attracted broad attention, though their presence and influence in perovskite solar cells are still a matter of debate^{14,15,16,17,18,19}.
A strong spin–orbit coupling (SOC) involving the heavy Pb atom and its surrounding halogens, combined with the absence of inversion symmetry in the crystal structure, can lead to a Rashba effect which, in turn, can impact photovoltaic performance^{20,21,22,23,24}. Noncentrosymmetry provides a spindegenerate band with the possibility to split two reversely spinpolarized states described by the equation:
which can be used to clarify the dispersion relation of electrons (or/and holes), where α_{R} is the Rashba splitting parameter^{25,26,27}. The coexistence of FE and a Rashba effect is believed to mediate interesting effects, such as the switching of spintexture chirality with an electricpolarization reversal^{28,29}. These effects are mainly studied in the context of inorganic materials; only a few hybridmaterial examples have been explored so far^{25,30,31}.
Very recently, Sum et al. demonstrated the presence of both FE and a Rashba effect in a layered Dion–Jacobson (DJ) phase HOIP with formula (AMP)PbI_{4}, where AMP is the divalent 4aminomethylpiperidinium cation (Fig. 1, FE state)^{32}. Its measured value of spontaneous polarization (P_{s}) is 9.8 μC cm^{−2}, which is very high among HOIPs^{33,34}, and more comparable to those of conventional inorganic perovskites such as BaTiO_{3} (15–26 μC cm^{−2})^{35,36}. A measured Rashba splitting energy (E_{R}) of 85 meV and Rashba coefficient (α_{R}) of 2.60 eV Å suggest real promise for spintronic applications^{30,31}. This exciting experimental evidence now motivates a theoretical investigation into the microscopic mechanism of FE and its interplay with SOC in the HOIPs.
Starting with this aim in mind, we perform a detailed density functional theory (DFT) analysis of (AMP)PbI_{4}, focusing on estimates of the ferroelectric and spin–orbit properties of the band structure. While the estimate of the Rashba splitting is now routine work, we focus here on the corresponding anisotropy, which is far much less studied. Our estimate of P_{s} and α_{R} are 10.72 μC cm^{−2} and 2.39 eV Å, in excellent agreement with the experimental 9.80 μC cm^{−2} and 2.60 eV Å. In addition, the E_{R} of spinpolarized bands is calculated to be 16 meV, which is of the same order of magnitude as the experimental 85 meV value. In addition to a spintexture that can be switched via ferroelectric polarization reversal, we further observe a sizeable anisotropy in the Rashba spin splitting. Remarkably, we show that the axis of major anisotropy can be switched between two orthogonal directions by reversing the ferroelectric polarization from +P to −P. Reversing the FE polarization induces an apparent 90° “rotation” of the anisotropy ellipse, i.e., a pseudorotation. This effect has never been highlighted in the context of usual spintexture tunability under polarization reversal and may suggest interesting avenues for spinoptoelectronic devices based on HOIPs.
Results and discussion
Structural analysis
We start our simulations with the experimental ferroelectric phase of (AMP)PbI_{4} refined at T = 298 K having space group Pc (#7, \(C_s^2\)) with monoclinic baxis in ref. ^{32}. Two nearby AMP^{2+} cations located in the space between [PbI_{4}]^{2−} layers have alternating orientations when they are adjacent along the b or c axes, but common orientations when adjacent along with diagonal b + c or b − c directions, as shown in Fig. 1 (FE state), so that the structure lacks a center of inversion, thus paving the way for the existence of ferroelectric polarization in the system. It is interesting to note that the cation’s center of mass is displaced somewhat from the ideal midpoint between two adjacent [PbI_{4}]^{2−} layers so as to be slightly closer to one layer or the other. We perform quantum chemical analyses of the isolated AMP^{2+} cation using both the Hirshfield^{37} and Natural population methods^{38} implemented in the Gaussian16 software^{39}. Both methods define an electric dipole moment of the molecule pointing approximately towards the aminomethyl group from the center of the AMP cation, as shown in Supplementary Fig. 2. Although the longrange ordering of AMPcation orientations should be associated with a strong FE polarization along the aaxis at the first glance, this alone is not sufficient for understanding the overall polarization of a hybrid system, which typically has several different contributions to the polarization, as explained in details in ref. ^{32}. According to the modern theory of polarization^{40}, we introduce a centrosymmetric antiferroelectric (AFE) phase by enforcing the existence of an inversion point and define a suitable path connecting the AFE and FE state via atomic rotations or/and displacements in order to estimate the polarization by firstprinciples calculations. The AFE model is built from the FE model by simultaneously rotating two AMP^{2+} cations adjacent along the baxis by 180° so that the diagonal AMP cations are related by inversion symmetry. The [PbI_{4}]^{2−} the framework is also properly centrosymmetric to ensure that the AFE structure (both framework and AMP cations) possesses an inversion symmetry point at (0.5, 0.5, 0.5), as shown in Fig. 1 (AFE state). The resulting space group symmetry is P2/m (#10, \(C_{2h}^1\)) with a monoclinic baxis in the same unit cell.
Ferroelectric properties
We introduce the normalized variable λ to simultaneously parameterize both the rotations of the AMP cations and the atomic displacements of the [PbI_{4}]^{2−} framework from AFE (λ = 0) to FE (λ = 1) states. Therefore λ represents the amplitude of the combined rotodisplacive distortions connecting the prototype AFEphase structure and the real FEphase structure. The intermediate structures along the path (0 ≤ λ ≤ 1) are only introduced in order to monitor the accidental introduction of a quantum of polarization and to ensure a continuous variation of the polarization along the path itself. In practice, λ has been varied infinite increments, so that stepwise AMPcation rotations are approximated by 18 steps of 10° each from 0° (AFE state) to 180° (FE state), together with incremental linear displacements of the [PbI_{4}]^{2−} framework.
In order to understand the origin of the polarization, the separate contributions of the organic AMP^{2+} cations and the inorganic [PbI_{4}]^{2−} a framework to the total polarization are considered in three different ways. Firstly, we rotate AMP cations while keeping the [PbI_{4}]^{2−} framework centrosymmetric. In this way, we obtain the contribution coming from the organic groups (P_{cation}), i.e., a rotational contribution to the polarization. Secondly, the displacive distortions of [PbI_{4}]^{2−} framework are considered while fixing AMP cations in a centrosymmetric configuration. In this way, the polarization only originates from the inorganic framework (P_{frame}), i.e., a displacive contribution to the total polarization. Finally, both AMPcation rotations and [PbI_{4}]^{2−} layer displacements are simultaneously activated in order to recover the total polarization (P_{total}), i.e., the full rotodisplacive contribution to the polarization. In this way, we are able to disentangle the different contributions to the polarization and to observe the couplings between them. For each case, we consider the contribution from the ionic and electronic subsystems to the resulting polarization.
The results of the polarization calculations are presented in Fig. 2. In the AFE state, the net polarization is exactly zero, as expected. As λ gradually increases toward 1, the asymmetry with respect to the centric phase and the magnitude of the total polarization both increase steadily along the aaxis but remain near zero for the b and caxis components. For (λ = 1), we extract the estimated value of P_{total}, which is equal to 10.72 μC cm^{−2}, in very good agreement with the experimental value of 9.80 μC cm^{−2}. From our analysis, the contributions from organic cations and inorganic frameworks are 10.34 μC cm^{−2} and 0.59 μC cm^{−2}, respectively. Clearly, the AMP cations dominate the total polarization, so that the framework contribution appears negligible in comparison. An applied electric field then mainly rotates the AMP cations while displacing the [PbI_{4}]^{2−} framework only a little. Because the switchable FE polarization is primarily due to the longrange orientational order of the organic cations, the true AFE state lying between the +P and −P states could be consistent with either ordered or disordered but balanced arrangements of AMPcation orientations, thus supporting a null electric polarization. Indeed, the AFE ordered state of the crystal represents only a computational reference phase to evaluate the final polarization, and in principle may be not exclusively defined. In the present case, we have fixed a possible AFE ordering, but, according to the modern theory of polarization, the final polarization does not depend on the particular path considered for 0 ≤ λ ≤ 1^{41}.
Electronic structure and Rashba SOC effect
We now examine the electronic structure and Rashba SOC effect of ferroelectric (AMP)PbI_{4}. The Brillouin zone (BZ) and corresponding Rashba spinpolarized bands of the FE structure are shown in Fig. 3a, b. The valence band maximum (VBM), conduction band minimum (CBM), and related bandgap are localized in kspace around the highsymmetry B point. The VBM and CBM are strongly spinsplit due to the spin–orbit interaction and are also shifted away from the B point in kspace. As usual, the “momentum offset” (k_{R}) corresponds to the distance between the apex of the splitting band and the highsymmetry point in kspace, while E_{R} corresponds to the energy difference between them. In our case, k_{R} and E_{R} are estimated as 0.0133 Å^{−1} and 16 meV for the VBM and as 0.0132 Å^{−1} and 12 meV for the CBM, which shows rather fairly good agreement with the respective experimental values of 0.067 Å^{−1} and 85 meV. Indeed, according to the definition
α_{R} is calculated to be 2.39 eV Å for the VBM, which agrees with the experimental 2.60 eV Å very well. It must be noted that the k_{R} values for the VBM and CBM are very similar, indicating that they are located essentially at the same point in kspace to form a “direct” bandgap, which is beneficial for electronic transitions in photovoltaiccell applications.
We show the band dispersion of the spinsplit bands, i.e., a 3D plot of the VBs around the highsymmetry B point, in Fig. 3c. The corresponding spintextures of the spinsplit bands are projected onto the 2D plane perpendicular to the FE polarization (P) in Fig. 3d. In the inner and outer bands, the sense of rotation (lefthanded or righthanded) of the spintextures are opposite due to the spin–orbit interaction. As reported in refs. ^{42,43}, the polarization reversal resulting from the application of an external electric field should be able to switch the spin textures. Indeed, we also observe that the spin textures completely switch from left to a righthanded sense of rotation as the polarization goes from −P to +P, as shown in Fig. 3d, e.
While most studies on spinorbit related properties estimate the Rashba parameter for a given direction (in the plane perpendicular to the polarization) and visualize the spintexture and its switching properties, in addition here we focus on the anisotropy of the Rashba parameter and its relationship to the switching of the ferroelectric polarization. To probe this anisotropy, the band structure was calculated on a fine grid of points in the vicinity of the reciprocalspace B point within the 2D plane perpendicular to the FE polarization (the yellow k_{y}k_{z} plane in Fig. 3a). The calculated α_{R} parameters are summarized in Fig. 4a, where we use a polar plot indicating the blue curves corresponding to the α_{R} obtained from the VBs and the green curves corresponding to those from the CBs. It can be seen that α_{R} parameters show a marked anisotropy, both at VBs and CBs. The maximum α_{R} of the VB at the B point (the blue curve in Fig. 4a) is 2.39 eV Å, located on the k_{z}axis, whereas the minimum is 0.94 eV Å, located on the k_{y}axis, which indicates that α_{R} is not a constant but rather a function of the angle θ with respect to the negative k_{z}axis (in the yellow plane in Fig. 3a).
Remarkably, we find that when the FE polarization is switched from +P to −P, the major and minor axis of the Rashba anisotropy are exchanged as shown in Fig. 4b, so that the maximum α_{R} value of the VB (2.22 eV Å) shifts from the k_{z}axis (baxis) to the k_{y}axis (caxis), and the minimum α_{R} value (0.87 eV Å) shifts the opposite way. In other words, while the polarization switches from +P to −P, the major axis α_{R} value decreases, while the minor axis α_{R} value increases. If we compare the +P and −P anisotropy ellipses, the net effect is an apparent 90° rotation of the anisotropy ellipse, i.e., a pseudorotation. To the best of our knowledge, this is the first time that a polarizationswitchable Rashba anisotropy has been observed, which is also connected to a very large anisotropy.
We propose a structural mechanism for this highly unusual polarizationswitchable pseudo rotation of the Rashba anisotropy. As the AMPcation orientations evolve between the +P and −P states, the roughly 180° rotation (approximately around the baxis) of each cation not only switches the aminomethyl group between the +a and −a sides of the cation but also rotates its maximumelectrondensity plane (MEDP) by about 90° in the bc plane, which is dual to the k_{y}k_{z} plane of the BZ (can be seen in Supplementary Fig. 3). To verify our hypothesis that this polarizationreversalinduced 90° rotation of the AMP cation’s MEDP is responsible for the rotation of the Rashbaanisotropy, we construct an alternative −P′ structure from the +P structure by rotating each AMP cations by 180° around the axis perpendicular to its MEDP, so that its aminomethyl group is still moved between the +a and −a sides, but without allowing the MEDP to rotate in the bc plane. When the structure is thus evolved, we see no rotation of the Rashba anisotropy in its band structures (Fig. 4c, f). Furthermore, compared to the isoenergetic +P and normal −P structures, the alternative −P′ structure has a 0.48 eV higher energy than −P and is separated from +P by an energy barrier of greater than 1.20 eV, demonstrating that the −P′ structure would be very difficult to obtain at room temperature.
To understand the Rashba anisotropy and spin texture under polarization switching, we construct the effective k·p model near the B point. The little group of the B point is Cs with only mirror symmetry operator My. Under the mirror transformation \(My:\left( {x,y,z} \right) \to \left( {x,  y,z} \right),(k_x,k_y,k_z) \to (k_x,  k_y,k_z)\) and \((\sigma _x,\sigma _y,\sigma _z) \to (  \sigma _x,\sigma _y,  \sigma _z)\). Then the twoband Hamiltonian for the VBM or CBM at k_{x} = 0 near the B point can be written as
Here m_{y} and m_{z} are the effective masses along the k_{y} and k_{z} directions near the B point, and σ_{x}, σ_{y}, and σ_{z} are Pauli matrices. α_{1}, α_{2}, and α_{3} are three Rashba parameters. The last three terms are SOC terms that include the Rashba and Dresselhaus SOC effects, which can be rewritten in the form
where
and
are the Dresselhaus and Rashba parameters, respectively^{44,45}. The last term k_{y}σ_{x} leads to the emergence of the s_{x} spin component on the k_{y}k_{z} plane which is consistent with our firstprinciples calculation of the spin textures. The eigenvalues of the twoband Hamiltonian reads as
To understand the Rashba anisotropy under polarization switching, the eigenvalues can be written as
in polar coordinates. Here k_{r} is defined as the distance of k from the B point in the k_{y}k_{z} plane, and θ is the angle between \(\vec k_r\) and \(\vec k_z\). One can define the general “Rashba parameter” with different directions on the k_{y}k_{z} plane in polar coordinates as
where the arrow ↑ means the polarization along with the +x direction corresponding to the +P state. The reorientation of the AMP cation, which involves the combination of a M_{x} mirror operation and a C_{4x} rotation operation, effects a 90° rotation of the crystal fields at the cation site such that \((x,y,z) \to (  x,z,y)\) and \(\left( {k_x,k_y,k_z} \right) \to (  k_x,k_z,k_y)\). The eigenvalue of effective Hamiltonian of the −P state should then be
in polar coordinates. So the anisotropy of the Rashba parameter of the −P state is
where the arrow ↓ means the polarization along the −x direction. It is easy to see that
which means the anisotropy of the Rashba parameter is rotated by 90° under polarization switching, which is equivalent to a 180° rotation of polarization. The proposed k·p model is thus nicely consistent with our DFT results in terms of a 90° pseudorotation.
We introduce parameter
to quantify the strength of the anisotropy where α_{R}(k_{z}) and α_{R}(k_{y}) are the numerical values of the Rashba parameter along the k_{z} and k_{y} axes of the BZ, respectively. The larger the value of µ, the more pronounced the anisotropy of α_{R} is. In the case of the −P and +P polarizations, µ has quite similar values of 1.7 and 1.6, respectively, reflecting the robustness of the Rashba anisotropy in this system.
It is interesting to note that the pseudorotation of the Rashba anisotropy induced by polarization reversal occurs not only at a single point in kspace, but at several points where the Rashba splitting is effective. Our observations of the Rashba anisotropy in the VB and CB near both the B and G points of the BZ are all qualitatively consistent with the description and explanation of Fig. 4d–f above. At both of these points, the pseudorotation of the anisotropy ellipse occurs when reversing the polarization perpendicular to the framework layers.
Symmetry mode analysis
The +P and −P FE structures can be viewed as distorted “children” of an idealized “parent” structure, which allows us to characterize their symmetrybreaking structural variables (i.e., symmetry modes) in terms of the irreducible representations (irreps) of the parent spacegroup symmetry. We construct this idealized parent from the centrosymmetric AFE structure in the left panel of Fig. 1a by regularizing the PbI_{6} octahedra and zeroing the PbI_{6} octahedral rotations in order to straighten out the framework, and by replacing each AMP cation with a polar vector (attached to a dummy atom at the molecular centroid location) that points along the appropriate ±a direction, resulting in centrosymmetric space group Pcmm (#51, \(D_{2h}^5\)) on an orthorhombic (a, b/2, c) unitcell basis with zero origin shift relative to the child. We also idealize the FE children by replacing each AMP cation with an appropriate polar vector in order to apply the symmetry mode analysis consistently.
The key order parameters of the idealized parent structure that characterize our FE child structures are (1) the PbI_{6} octahedral framework rotation around the aaxis, which belongs to parent irrep \(Y_3^ + \), (2) the ferroelectric displacements along the aaxis, which belong to FE parent irrep \({{\Gamma }}_2^  \), and (3) the large AMPcation rotations around the baxis, which are achieved through the cooperative action of twoparent irreps, ferrorotational (FR) irrep \({{\Gamma }}_4^ + \) and antiFR (AFR) irrep \(Y_1^  \), as shown in Fig. 5. Using the INVARIANTS software package, we find that these four primary order parameters provide a quadrilinear invariant term in the free energy expansion, so that simultaneously invoking the framework (\(Y_3^ + \)) and the cation (\({{\Gamma }}_4^ + \), \(Y_1^  \)) rotational modes breaks the inversion symmetry of parent, thus allowing them to couple to a secondary (i.e., improper) ferroelectric moment (\({{\Gamma }}_2^  \)).
Because the PbI_{4} framework is only slightly disturbed in switching between the +P and −P states, it is convenient to treat the \(Y_3^ + \) framework rotation as a large preexisting structural feature in a slightly lesssymmetric parent structure with space group Pnmm (#59, \(D_{2h}^{13}\)) and unitcell basis and origin identical to those of the child. For this lesssymmetric parent structure, the FR and AFR cation rotations form a simple trilinear invariant with the ferroelectric moment.
The +P cation arrangement is obtained from the parent via the superposition of a 90° FR motion \(({\Gamma}_4^ + )\) and a 90° AFR motion \((Y_1^  )\), so that two of the AMP cations are rotated by a full 180° while the other two remain stationary (as shown in Fig. 1). The −P cation arrangement is achieved instead by merely reversing the sense of the AFR contribution. Due to the unfavorable energy at the 90° midpoint of rotation, it makes no sense to view the FR and AFR rotations as separate processes; they must occur simultaneously and cooperatively so that the \(\Gamma _4^ + \) and \(Y_1^  \) order parameters are tightly coupled to have equal amplitudes. The product of two such large order parameters facilitates a strong trilinear coupling to the FE polarization. It remarkable that two centric modes, i.e., \(\Gamma _4^ + \) and \(Y_1^  \), combine together to give rise a hybrid mode, which in turn, breaks inversion symmetry and allows the polarization to arise in this system. Moreover, the coherent switching of both of them, i.e., the switching of the hybrid model, allows switching the polarization giving rise to a hidden pseudorotation in kspace, as highlighted in this work with the switchable Rashba anisotropy ellipse. It is intriguing that the switchable Rashba anisotropy occurs coherently at different relevant points in the kspace.
In summary, HOIPs are emerging as a class of photovoltaic materials. In this work, DFT methods have been used to study the ferroelectric properties and Rashba spin–orbit effect as well as the coupling between the FE polarization (P_{s}) and the anisotropy of the Rashba parameter (α_{R}) for a layered HOIP, (AMP)PbI_{4}, where AMP is 4(aminomethyl)piperidinium. We calculate P_{s} and α_{R} to be 10.72 μC cm^{−2} and 2.39 eV Å, respectively, which are nicely consistent with the experimentally reported values of 9.8 μC cm^{−2} and 2.60 eV Å. To gain insight into the origin of the polarization, the contributions of the organic AMP^{2+} cations (10.34 μC cm^{−2}) and inorganic [PbI_{4}]^{2−} framework (0.59 μC cm^{−2}) have been disentangled, demonstrating that the AMP cations contribute almost all of the total polarization. This clearly suggests that a proper choice of organic cations could possibly enhance the ferroelectric polarization, by exploiting both the molecular dipole moment and their relative tilting, giving rise to an uncompensated electric polarization.
As previously reported for Rashba ferroelectrics, the sense of rotation of the spintexture can be reversed by a switching of the FE polarization direction along the polar axis^{40}. However, in (AMP)PbI_{4}, we calculate a large and robust anisotropy in the Rashba parameter. Remarkably, we find that the major and minor axes of the ellipse can be exchanged under reversal of the electric polarization, causing a 90° pseudorotation of the Rashba anisotropy, which we have also been confirmed from a theoretical k·p model. The same effect is observed in both the VBs and CBs at multiple points in the BZ where the Rashba splitting is significant. A structural mechanism for this effect is presented and explained in terms of a quadrilinear coupling of order parameters involving two largeamplitude AMPcation rotation modes, a large octahedral framework rotation, and the ferroelectric polarization. To the best of our knowledge, this is the first report of a coupling of ferroelectric polarization and switchable highly anisotropic Rashba spinsplit bands and it may be possible that the same property can be found in other layered hybrid perovskites. This deserves further investigation. Indeed, spinoptoelectronic devices based on hybrid organic–inorganic trihalide perovskites like spinLEDs have been recently discussed for the parent compound MAPbBr_{3}^{46}, circularly polarized light detections have been considered for chiral hybrid perovskite^{47}, spinFETs. This strong coupling may provide a potential probe of the spin degrees of freedom in photovoltaic materials and a different avenue for developing spinoptoelectronic devices based on HOIPs materials.
Methods
Computational details
The starting point for our calculations is the experimental crystallographic data reported in ref. ^{33}, where we relax all of the atoms of the ferroelectric structure until the Hellmann–Feynman forces are smaller than 0.001 eV Å^{−1}. The projector augmented wave (PAW)^{48} method is used to solve Kohn–Shan equations with the PBE exchangecorrelation functional^{49}, as implemented in the Vienna ab initio simulation package. The energy cutoff for the plane wave expansion is set to 550 eV, and a 2 × 2 × 2 Monkhorst–Pack grid of kpoint is used after systematic convergence tests. The van der Waals interactions are taken into account in calculations using the DFTD3 method^{50}. The Berry phase approach is used to evaluate the ferroelectric polarization by constructing a properly AFE reference state and a suitable path connecting FE with AFE states^{40,51,52}. Details of this approach are discussed in the results section. Spin–orbit interaction is selfconsistently considered in all band structure calculations, but not in the calculations of polarization. Van der Waals interactions are included in all calculations^{53}.
Data availability
The data that support the findings of this study are available from the authors on reasonable request, see author contributions for specific data sets.
References
Cui, Y. et al. Over 16% efficiency organic photovoltaic cells enabled by a chlorinated acceptor with increased opencircuit voltages. Nat. Commun. 10, 2515 (2019).
Hong, L. et al. Ecocompatible solventprocessed organic photovoltaic pells with pver 16% efficiency. Adv. Mater. 31, e1903441 (2019).
Aguiar, A. et al. Simple BODIPY dyes as suitable electrondonors for organic bulk heterojunction photovoltaic cells. Dyes Pigments 172, 107842 (2020).
Tsai, H. et al. Highefficiency twodimensional RuddlesdenPopper perovskite solar cells. Nature 536, 312–316 (2016).
Wu, T. & Wang, J. Global discovery of stable and nontoxic hybrid organicinorganic perovskites for photovoltaic systems by combining machine learning method with first principle calculations. Nano Energy 66, 104070 (2019).
Ghosh, D., Aziz, A., Dawson, J. A., Walker, A. B. & Islam, M. S. Putting the squeeze on lead iodide perovskites: pressureinduced effects to tune their structural and optoelectronic behavior. Chem. Mater. 31, 4063–4071 (2019).
Ferdani, D. W. et al. Partial cation substitution reduces iodide ion transport in lead iodide perovskite solar cells. Energy Environ. Sci. 12, 2264–2272 (2019).
Brenes, R., Eames, C., Bulovic, V., Islam, M. S. & Stranks, S. D. The impact of atmosphere on the local luminescence properties of metal halide perovskite grains. Adv. Mater. 30, e1706208 (2018).
Kahmann, S. & Loi, M. A. Hot carrier solar cells and the potential of perovskites for breaking the ShockleyQueisser limit. J. Mater. Chem. C 7, 2471–2486 (2019).
Wang, Z., Song, Z., Yan, Y., Liu, S. & Yang, D. Perovskite—a perfect top cell for tandem devices to break the S–Q limit. Adv. Sci. 6, 1801704 (2019).
Fang, H.H., Adjokatse, S., Shao, S., Even, J. & Loi, M. A. Longlived hotcarrier light emission and large blue shift in formamidinium tin triiodide perovskites. Nat. Commun. 9, 243 (2018).
Polman, A., Knight, M., Garnett, E. C., Ehrler, B. & Sinke, W. C. Photovoltaic materials: present efficiencies and future challenges. Science 352, aad4424 (2016).
Tayebjee, M. J., McCamey, D. R. & Schmidt, T. W. Beyond ShockleyQueisser: molecular approaches to highefficiency photovoltaics. J. Phys. Chem. Lett. 6, 2367–2378 (2015).
Shi, C. et al. Twodimensional organic–inorganic hybrid rareearth double perovskite ferroelectrics. J. Am. Chem. Soc. 142, 545–551 (2020).
Pandey, R. et al. Mutual insight on ferroelectrics and hybrid halide perovskites: a platform for future multifunctional energy conversion. Adv. Mater. 31, e1807376 (2019).
Balachandran, P. V., Kowalski, B., Sehirlioglu, A. & Lookman, T. Experimental search for hightemperature ferroelectric perovskites guided by twostep machine learning. Nat. Commun. 9, 1668 (2018).
Hua, X.N. et al. A roomtemperature hybrid lead iodide perovskite ferroelectric. J. Am. Chem. Soc. 140, 12296–12302 (2018).
Pan, Q. et al. A threedimensional molecular perovskite ferroelectric: (3ammoniopyrrolidinium)RbBr3. J. Am. Chem. Soc. 139, 3954–3957 (2017).
Xu, W.J. et al. A molecular perovskite with switchable coordination bonds for hightemperature multiaxial ferroelectrics. J. Am. Chem. Soc. 139, 6369–6375 (2017).
Lee, H., Im, J. & Jin, H. Emergence of the giant outofplane Rashba effect and tunable nanoscale persistent spin helix in ferroelectric SnTe thin films. Appl. Phys. Lett. 116, 022411 (2020).
Lin, Z., Si, C., Duan, S., Wang, C. & Duan, W. Rashba splitting in bilayer transition metal dichalcogenides controlled by electronic ferroelectricity. Phys. Rev. B 100, 155408 (2019).
He, J. et al. Tunable metalinsulator transition, Rashba effect and Weyl Fermions in a relativistic chargeordered ferroelectric oxide. Nat. Commun. 9, 492 (2018).
Djani, H. et al. Rationalizing and engineering Rashba spinsplitting in ferroelectric oxides. npj Quantum Mater. 4, 51 (2019).
Kepenekian, M. et al. Rashba and Dresselhaus effects in hybrid organic–inorganic perovskites: from basics to devices. ACS Nano 9, 11557–11567 (2015).
Zhai, Y. et al. Giant Rashba splitting in 2D organicinorganic halide perovskites measured by transient spectroscopies. Sci. Adv. 3, e1700704 (2017).
Picozzi, S. Ferroelectric Rashba semiconductors as a novel class of multifunctional materials. Front. Phys. 2, 10 (2014).
Vajna, S. et al. Higherorder contributions to the RashbaBychkov effect with application to the Bi/Ag(111) surface alloy. Phys. Rev. B 85, 075404 (2012).
Maaß, H. et al. Spintexture inversion in the giant Rashba semiconductor BiTeI. Nat. Commun. 7, 11621 (2016).
Kim, M., Im, J., Freeman, A. J., Ihm, J. & Jin, H. Switchable S = 1/2 and J = 1/2 Rashba bands in ferroelectric halide perovskites. Proc. Natl. Acad. Sci. USA 111, 6900–6904 (2014).
Rinaldi, C. et al. Ferroelectric control of the spin texture in GeTe. Nano Lett. 18, 2751–2758 (2018).
Krempaský, J. et al. Entanglement and manipulation of the magnetic and spinorbit order in multiferroic Rashba semiconductors. Nat. Commun. 7, 13071 (2016).
Park, I. H. et al. Ferroelectricity and Rashba effect in a twodimensional DionJacobson hybrid organicinorganic perovskite. J. Am. Chem. Soc. 141, 15972–15976 (2019).
Zheng, F., Takenaka, H., Wang, F., Koocher, N. Z. & Rappe, A. M. Firstprinciples calculation of the bulk photovoltaic effect in CH_{3}NH_{3}PbI_{3} and CH_{3}NH_{3}PbI_{3}_{x}Cl_{x}. J. Phys. Chem. Lett. 6, 31–37 (2015).
Li, L. et al. Twodimensional hybrid perovskitetype ferroelectric for highly polarizationsensitive shortwave photodetection. J. Am. Chem. Soc. 141, 2623–2629 (2019).
von Hippel, A. Ferroelectricity, domain structure, and phase transitions of Barium Titanate. Rev. Mod. Phys. 22, 221–237 (1950).
Shieh, J., Yeh, J. H., Shu, Y. C. & Yen, J. H. Hysteresis behaviors of barium titanate single crystals based on the operation of multiple 90° switching systems. Mater. Sci. Eng. B 161, 50–54 (2009).
Fonseca Guerra, C., Handgraaf, J.W., Baerends, E. J. & Bickelhaupt, F. M. Voronoi deformation density (VDD) charges: assessment of the Mulliken, Bader, Hirshfeld, Weinhold, and VDD methods for charge analysis. J. Comput. Chem. 25, 189–210 (2004).
Reed, A. E., Weinstock, R. B. & Weinhold, F. Natural population analysis. J. Chem. Phys. 83, 735–746 (1985).
Frisch, M. J. et al. Gaussian 16, Revision A.03 (Wallingford, CT, 2016).
Spaldin, N. A. A beginner’s guide to the modern theory of polarization. J. Solid State Chem. 195, 2–10 (2012).
Rabe, K. M., Ahn, C., & Triscone, J.M. Physics of Ferroelectrics: A Modern Perspective (Springer, Berlin, Heidelberg, 2007).
Di Sante, D., Barone, P., Bertacco, R. & Picozzi, S. Electric control of the giant Rashba effect in bulk GeTe. Adv. Mater. 25, 3625–3626 (2013).
Stroppa, A. et al. Tunable ferroelectric polarization and its interplay with spinorbit coupling in tin iodide perovskites. Nat. Commun. 5, 5900 (2014).
Tao, L. L., Paudel, T. R., Kovalev, A. A. & Tsymbal, E. Y. Reversible spin texture in ferroelectric HfO2. Phys. Rev. B 95, 245141 (2017).
Tao, L. L. & Tsymbal, E. Y. Persistent spin texture enforced by symmetry. Nat. Commun. 9, 2763 (2018).
Wang, J. et al. Spinoptoelectronic devices based on hybrid organicinorganic trihalide perovskites. Nat. Commun. 10, 129 (2019).
Chen, C. et al. Circularly polarized light detection using chiral hybrid perovskite. Nat. Commun. 10, 1927 (2019).
Blöchl, P. E. Projector Augmentedwave method. Phys. Rev. B 50, 17953–17979 (1994).
Perdew, J. P. et al. Restoring the densitygradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFTD) for the 94 elements HPu. J. Chem. Phys. 132, 154104 (2010).
Resta, R. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Rev. Mod. Phys. 66, 899–915 (1994).
KingSmith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 4 (1993).
Grimme, S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J. Comput. Chem. 25, 1463–1473 (2004).
Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłodowskaCurie grant agreement No. 713679 and from the Universitat Rovira i Virgili (URV). J.M.P. and C.d.G thank the Spanish Ministry of Science (grants CTQ201787269P and CTQ201783566P) and the Generalitat de Catalunya (grant 2017SGR629) for support. J.M.P. also thanks ICREA foundation for an ICREA ACADEMIA award. H.G. acknowledges support from the National Postdoctoral Program for Innovative Talents (No. BX20190361) and Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515110965). B.J.C. acknowledges helpful conversations with Harold T. Stokes and John Colton. B.J.C. and A.S. thank Z. V. Vardeny for useful discussions. A.S. acknowledges A. Cassinese, F. Chiarella, and M. Barra for useful discussions. F.W. acknowledges the kind hospitality by CNRSPIN c/o Department of Chemical and Physical Science of University of L’Aquila (Italy) where this project was discussed during the vising period from November 2, 2019 to December 14, 2019.
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A.S. conceived and designed the study; A.S., C.G., and J.M.P. supervised the project; F.W. performed the DFT calculations and drafted the paper; H.G. constructed the k·p model and assisted the analysis of Rashba anisotropy. B.J.C. conducted the symmetrymode analysis. All authors reviewed and approved the paper.
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Wang, F., Gao, H., de Graaf, C. et al. Switchable Rashba anisotropy in layered hybrid organic–inorganic perovskite by hybrid improper ferroelectricity. npj Comput Mater 6, 183 (2020). https://doi.org/10.1038/s4152402000450z
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DOI: https://doi.org/10.1038/s4152402000450z
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