Switchable Rashba Anisotropy in Two-dimensional Hybrid Organic-Inorganic Perovskite by Hybrid Improper Ferroelectricity

Two-dimensional (2D) hybrid organic-inorganic perovskites (HOIPs) are introducing new directions in the 2D materials landscape. The coexistence of ferroelectricity and spin-orbit interactions play a key role in their optoelectronic properties. We perform a detailed study on a recently synthesized ferroelectric 2D-HOIP, (AMP)PbI$_4$ (AMP = 4-aminomethyl-piperidinium). The calculated polarization and Rashba parameter are in excellent agreement with experimental values. We report a striking new 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 k-space are interchanged - a pseudo rotation. A $k \cdot p$ model Hamiltonian and symmetry-mode analysis reveal a quadrilinear coupling between the cation-rotation modes responsible for the Rashba ellipse pseudo-rotation, the framework rotation, and the polarization. These findings may provide new avenues for spin-optoelectronic 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. 1-3 Hybrid organic-inorganic 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 new generation of materials for solar cells and light-emitting diodes. [4][5][6][7][8] The presence of new 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. 9-13 So 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 an inversion symmetry in the crystal structure, can lead to a Rashba effect and thus impact photovoltaic performance. [20][21][22][23] Non-centrosymmetry provides a spin-degenerate band with the possibility to split two reversely spin-polarized states, and the equation of ± ( ) = ℏ 2 2 2 ± | | can be used to clarify the dispersion relation of electrons (or/and holes), in which is the Rashba splitting parameter. 24, 25 The coexistence of ferroelectricity and a Rashba effect is believed to mediate interesting effects, such as the switching of spin-texture chirality with an electric-polarization reversal. 26 These effects are mainly studied in the context of inorganic materials; only a few hybrid-material examples have been explored so far. 24,27,28 In the HOIPs family of compounds, 3D systems have received far more attention than 2D systems, though the 2D counterparts possess exceptional chemical stability and novel and less-explored photovoltaic capabilities. 4, 29 Although the 2D systems are more likely to support an FE polarization, [30][31][32][33] the experimental evidence of the coexistence of FE and Rashba effects in 2D-HOIPs is very rare. Very recently, Sum et al. demonstrated the presence of both ferroelectricity and a Rashba effect in a 2D Dion-Jacobson (DJ) phase HOIP with formula (AMP)PbI 4 , where AMP is the divalent 4-aminomethyl-piperidinium cation (Figure 1, FE state). Its measured value of spontaneous polarization ( ) is 9.8 µC/cm 2 , which is very high amongst HOIPs 34,35 , and more comparable to those of conventional inorganic perovskites such as BaTiO 3 . A measured Rashba splitting energy ( ) of 85 meV and Rashba coefficient ( ) of 2.60 eV·Å suggest real promise for spintronic applications. This exciting new experimental evidence now motivates a theoretical investigation into the microscopic mechanism of ferroelectricity and its interplay with spin-orbit coupling in the 2D-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 a routine work, we focus here on the corresponding anisotropy, which is far much less studied. Our estimate of and 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 spin-polarized 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 spin-texture 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 + to -. Reversing the FE polarization induces an apparent 90° 'rotation' of the anisotropy ellipse, i.e., a pseudo-rotation. This new effect has never been highlighted in the context of usual spin-texture tunability under polarization reversal and may suggest new avenues for spin-optoelectronic devices based on hybrid organic-inorganic perovskites.

Results
Structural analysis: We start our simulations with the experimental ferroelectric phase of (AMP)PbI 4 refined at T = 298 K having space group (#7, 2 ) with monoclinic b-axis in Ref. 36 Two nearby AMP 2+ cations located in the space between [PbI 4 ] 2layers have alternating orientations when adjacent along the or axes, but common orientations when adjacent along diagonal + or − directions, as shown in Figure 1 (FE state), so that the structure lacks a center of inversion, thus paving the way for a possible ferroelectric polarization. It is interesting to note that the cation's center of mass is displaced somewhat from the ideal midpoint between two adjacent [PbI 4 ] 2layers 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 Figure 2. Although the long-range ordering of AMP-cation orientations should be associated with a strong FE polarization along the axis 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 anti-ferroelectric (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 first principles calculations. The AFE model is built from the FE model by simultaneously rotating two AMP 2+ cations adjacent along the b-axis by 180° so that the diagonal AMP cations are related by inversion symmetry. The [PbI 4 ] 2framework is also properly centrosymmetrized 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 Figure 1 (AFE state). The resulting space group symmetry is 2/ (#10, 2ℎ 1 ) with monoclinic b-axis in the same unit cell. framework from AFE (λ = 0) to FE (λ = 1) states. Therefore λ represents the amplitude of the combined roto-displacive distortion connecting the prototype AFE-phase structure and the real FE-phase structure. The intermediate structures along the path (0 < < 1) are only introduced in order to monitor the accidental introduction of a quantum of polarization and to ensure a continuous change of the polarization along the path itself. In practice, has been varied in finite increments, so that continuous AMP-cation 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 ] 2framework.
In order to understand the origin of the polarization, the separate contributions of the organic AMP 2+ cations and the inorganic [PbI 4 ] 2framework to the total polarization are considered in three different ways. Firstly, we rotate AMP cations while keeping the [PbI 4 ] 2framework 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 ] 2framework 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 of AMP-cation rotations and [PbI 4 ] 2layer displacements are simultaneously activated in order to recover the total polarization (P total ), i.e., the full roto-displacive 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 Figure 2. In the AFE state, the net polarization is exactly zero, as expected. As λ gradually increases towards 1, the asymmetry with respect to the centric phase and the magnitude of the total polarization both increase steadily along the a-axis but remain near zero for the b and c-axis 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 framework 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 ] 2framework only a little. Because the switchable FE polarization is primarily due to the long range 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 AMP-cation 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 uniquely defined. In the present case, we have fixed a possible AFE ordering, but, according to modern theory of polarization, the final polarization does not depend on the particular path considered for 0 ≤ < 1.  Figure 3a and 3b. The valence band maximum (VBM), conduction band minimum (CBM) and related band gap are localized in k-space around the high-symmetry B point. The VBM and CBM are strongly spin-split due to the spin-orbit interaction and are also shifted away from the B point in k-space. As usual, the "momentum offset" (k R ) corresponds to the distance between the apex of the splitting band and the high-symmetry point in k-space, while E R corresponds to the energy difference between them. In our case, and 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 good agreement with the respective experimental values of 0.067 Å -1 and 85 meV. Indeed, according to the definition α R = 2 / , 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 values for the VBM and CBM are very similar, indicating that they are located essentially at the same point in k-space to form a "direct" band gap, which is beneficial for electronic transitions in photovoltaic-cell applications.
We show the band dispersion of the spin-split bands, i.e., a 3D plot of the VBs around the high-symmetry B point, in Figure 3c. The corresponding spin-textures of the spin-split bands are projected onto the 2D plane perpendicular to the FE polarization (P) in Figure 3d. In the inner and outer bands, the sense of rotation (left-handed or right-handed) of the spin-textures 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 right-handed sense of rotation as the polarization goes from -P to +P, as shown in Figure 3d-3e. The blue curves correspond to the obtained from the VBs while the green curve corresponds to those from the CBs. The red arrows indicate the direction of the polarization, and -P' indicates an alternative -P structure wherein each AMP molecule rotates 180° relative to the +P structure around the axis perpendicular to its maximum-electron-density plane rather than around the b axis. However, this -P' is energetically unfavorable as discussed in text.
While most studies on spin-orbit related properties estimate the Rashba parameter for a given direction (in the plane perpendicular to the polarization) and visualize the spin-texture and its switching properties, here we also 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 reciprocal-space B point within the 2D plane perpendicular to the FE polarization (the yellow plane in Figure 3a).
The calculated parameters are summarized in Figure 4a, where the blue curves correspond to the α R obtained from the VBs and the green curves corresponds to those from the CBs. It can be seen that parameters show a marked anisotropy, both at VBs and CBs. The maximum of the VB at the B point (the blue curve in Figure 4a) is 2.39 eV·Å, located on the -axis, whereas the minimum is 0.94 eV·Å, located on the -axis, which indicates that is not a constant but rather a function of the angle θ with respect to the negative -axis (in the yellow plane in Figure  3a).
Remarkably, we find that when the FE polarization is switched from + to − , the major and minor axis of the Rashba anisotropy are exchanged as shown in Figure 4b, so that the maximum value of the VB (2.22 eV·Å) shifts from the -axis (b-axis) to the -axis (c-axis), and the minimum value (0.87 eV·Å) shifts the opposite way. In other words, while the polarization switches from + to -, the major axis value decreases, while the minor axis 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 pseudo-rotation. To the best of our knowledge, this is the first time that a polarization-switchable Rashba anisotropy has been observed, which is also connected to a very large anisotropy.
We propose a structural mechanism for this highly unusual polarization-switchable pseudo rotation of the Rashba anisotropy. As the AMP-cation orientations evolve between the + and states, the roughly 180 ∘ rotation (approximately around the b-axis) of each cation not only switches the aminomethyl group between the + and − sides of the cation, but also rotates its maximum-electron-density plane (MEDP) by about 90 ∘ in the bc plane, which is dual to the plane of the BZ (can be seen in Supplementary Figure 3). To verify our hypothesis that this polarization-reversal-induced 90 ∘ rotation of the AMP cation's MEDP is responsible for the rotation of the Rashba-anisotropy, we construct an alternative -′ structure from the + 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 + andsides, 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 (Figure 4c and 4f). Furthermore, compared to the isoenergetic +P and normalstructures, the alternative -′ structure has a 0.48 eV higher energy than + , and is separated from + by an energy barrier of greater than 1.20 eV, demonstrating that the -′ structure would be very difficult to obtain at room temperature.
We derive the Because the rotation of the AMP cation in the bc plane effects a 90 ∘ rotation of the crystal fields at the cation site, 1 and α 1 keep the same but 2 and 2 change sign. The anisotropy of Rashba parameter at -P state is: − = � 1 − 2 cos (2 ) = � 1 + 2 cos �2� + 2 �� = �3.30 + 2.41 cos �2� + 2 ��, which is rotated by 90 ∘ relative to + . The proposed • model is thus nicely consistent with our DFT results in terms of a 90° pseudo-rotation.
We introduce parameter = � It is interesting to note that the pseudo-rotation of the Rashba anisotropy induced by polarization reversal occurs not only at a single point in k-space, 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 Figure 4d-f above. At both of these points, the pseudo-rotation of the anisotropy ellipse occurs when reversing the polarization perpendicular to the framework layers.

Symmetry mode analysis:
The + and -FE structures can be viewed as distorted "children" of an idealized "parent" structure, which allows us to characterize their symmetry-breaking structural variables (i.e., symmetry modes) in terms of the irreducible representations (irreps) of the parent space-group symmetry. We construct this idealized parent from the centrosymmetric AFE structure in the left panel of Figure. 1a by regularizing the PbI6 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 ± direction, resulting in centrosymmetric space group (#51, 2ℎ 5 ) on an orthorhombic (a, b/2, c) unit-cell 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 -axis, which belongs to parent irrep 3 + , (2) the ferroelectric displacements along the a-axis, which belong to FE parent irrep Γ 2 − , and (3) the large AMP-cation rotations around the b-axis, which are achieved through the cooperative action of two parent irreps, ferrorotational (FR) irrep Γ 4 + and anti-ferrorotational (AFR) irrep 1 − .
Using the INVARIANTS software package, we find that these four primary order parameters provide a quadralinear invariant term in the free energy expansion, so that simultaneously invoking the framework ( 3 + ) and the cation (Γ 4 + , 1 − ) rotational modes breaks the inversion symmetry of parent, thus allowing them to couple to a secondary (i.e., improper) ferroelectric moment (Γ 2 − ).
Because the PbI 4 framework is only slightly disturbed in switching between the + andstates, it is convenient to treat the 3 + framework rotation as a large preexisting structural feature in a slightly less-symmetric parent structure with space group (#59, 2ℎ 13 ) and unit-cell basis and origin identical to those of the child. For this this less-symmetric parent structure, the FR and AFR cation rotations form a simple trilinear invariant with the ferroelectric moment.
The + cation arrangement is obtained from the parent via the superposition of a 90° FR motion (Γ 4 + ) and a 90° AFR motion ( 1 − ), so that two of the AMP cations are rotated by a full 180° while the other two remain stationary (as shown in Figure. 1). Thecation 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 Γ 4 + and 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., Γ 4 + and 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 mode, allows to switch the polarization giving rise to a hidden pseudo-rotation in k-space, as highlighted in this work with the switchable Rashba anisotropy ellipse. It is intriguing that the switchable Rashba anisotropy occurs coherently at different relevant point in k-space.

Conclusions
2D-HOIPs are emerging as a new 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 ( ) and the anisotropy of the Rashba parameter ( ) for a 2D-HOIP, As previously reported for Rashba ferroelectrics, the spin-texture sense of rotation can be reversed by a reversal 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 ∘ pseudo-rotation of the Rashba anisotropy, which we have also been confirmed from a theoretical ⋅ model. The same effect is observed in both the VBs and CBs at multiple points in the Brillouin Zone where the Rashba splitting is significant. A structural mechanism for this novel effect is presented and explained in terms of a quadrilinear coupling of order parameters involving two large-amplitude AMP-cation 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 spin-split bands. Spin-optoelectronic devices based on hybrid organic-inorganic trihalide perovskites like spin-LEDs have been recently discussed for the parent compound MAPbBr3, 44 circularly polarized light detections, 45