Structural descriptor for enhanced spin-splitting in 2D hybrid perovskites

Two-dimensional (2D) hybrid metal halide perovskites have emerged as outstanding optoelectronic materials and are potential hosts of Rashba/Dresselhaus spin-splitting for spin-selective transport and spin-orbitronics. However, a quantitative microscopic understanding of what controls the spin-splitting magnitude is generally lacking. Through crystallographic and first-principles studies on a broad array of chiral and achiral 2D perovskites, we demonstrate that a specific bond angle disparity connected with asymmetric tilting distortions of the metal halide octahedra breaks local inversion symmetry and strongly correlates with computed spin-splitting. This distortion metric can serve as a crystallographic descriptor for rapid discovery of potential candidate materials with strong spin-splitting. Our work establishes that, rather than the global space group, local inorganic layer distortions induced via appropriate organic cations provide a key design objective to achieve strong spin-splitting in perovskites. New chiral perovskites reported here couple a sizeable spin-splitting with chiral degrees of freedom and offer a unique paradigm of potential interest for spintronics.

In 2D MHPs, specific conformational and packing characteristics of the organic cation may cause a noncentrosymmetric global space group, while a chiral cation always necessitates a chiral (Sohncke) global space group ( Supplementary Fig. 1). Since strong SOC effects mainly arise in the inorganic component via heavy elements, merely relying on the global space groups without examining the inorganic layer distortions and the resulting local symmetry is, therefore, insufficient to provide a microscopic mechanism of spin-splitting in MHPs. A large Rashba/Dresselhaus spin-splitting () kT) suppresses spin-flipping and is a critical requisite for room-temperature spin-based applications. There have been a few reports with computed spin-splitting values ranging from <<10 meV in some 2D lead-iodide-based MHPs 29,30 to >200 meV in the chiral [S/R-1-1-NEA] 2 PbBr 4 31 . Pb being the same heavy-metal constituent in these MHPs, SOC alone cannot justify the wide disparity in the predicted spin-splitting values. Despite a growing interest in spin-related properties of MHPs, a fundamental understanding of the symmetry breaking and the principles determining the spin-splitting magnitude are generally lacking, posing a major bottleneck in rational discovery of potential MHPs with sizable spin-splitting for prospective spin-based applications.
From a materials discovery perspective, a key question is if there exists a simple structural parameter that controls the spin-splitting in MHPs and, consequently, their spin-related properties in applications. Remarkably, we here identify such a parameter from several possible modes of lattice distortions, providing a quantitative connection between the crystallographic structure of a 2D MHP and the spin-splitting exhibited in its conduction bands (CBs). Through crystallographic studies combined with density-functional theory (DFT), we establish that the computed spin-splitting mainly originates from and scales with in-plane asymmetric tilting of adjacent metal halide octahedra, represented by a specific projected bond angle difference Δβ in (Fig. 1 and Supplementary Notes 1 and 2), as demonstrated for a broad array of MHPs, including five newly synthesized chiral MHPs. A large Δβ in signifies inversion asymmetry within inorganic layers, and in turn, serves as a key local descriptor of spin-splitting, beyond the quantitatively incomplete condition based on noncentrosymmetric global space groups. Moreover, some of the chiral MHPs reported here present a unique combination of chiral degrees of freedom (e.g., enabling chiral induced spin selectivity, CISS 32,33 ) and sizeable distortion-induced spin-splitting, which is of potential interest for spintronic devices.

Results
Inorganic layer distortions and asymmetry in 2D MHPs. A core objective of this paper is to correlate specific structural distortions in 2D MHPs with sizeable Rashba/Dresselhaus spin-splitting. Importantly, it is not sufficient to merely include chiral spacer cations for the inorganic layers to become chiral (i.e., devoid of inversion and mirror symmetries). A detectable inversion asymmetry depends on the degree and nature of cation-induced distortions within the inorganic layers. To understand how a substantial inversion asymmetry manifests itself within the inorganic layers and how it is related to the use of a chiral spacer cation, we examine the experimentally determined structural properties of nine MHPs with noncentrosymmetric space groups, as well as of eight MHPs with chiral space groups, of which five are specifically synthesized and characterized for this work:  Table 1 summarizes for all the above MHPs the metrics of interoctahedral tilting distortion leading to equatorial Pb-X-Pb bond angle β < 180°, the disparity in adjacent β angles (Δβ) (see Fig. 1), as well as the intraoctahedron distortions: Δd and σ 2 . Here, Δd is the bond length distortion defined as Δd ¼ 1 6 À Á Σðd i À d 0 Þ 2 =d 0 2 (d i denotes the six Pb-X bond lengths and d 0 is the mean M-X bond length), and σ 2 is the bond angle variance defined as (θ i denotes the individual cis X-Pb-X bond angles) 18 . In all MHPs listed in Table 1, regardless of whether the spacer cation is chiral or achiral, there are two inequivalent β angles in each MHP, leading to Δβ ≠ 0 (Figs. 1, 2f-j). We can group the different MHPs in Table 1 Supplementary Fig. 2). In all these MHPs, the fluctuating local geometry with Δβ as large as~12°-15°a long either in-plane direction breaks the inversion symmetry within the isolated inorganic layer (Table 1) as suggested by PLATON 34 symmetry analysis (see Methods for details). Low Δβ: The rest of the chiral and achiral MHPs in Table 1 exhibit a typical situation encountered in known centrosymmetric MHPs with zero or very small Δβ (see Fig. 2h-j). Despite the global chiral space groups, the constituent isolated inorganic layers in these MHPs are nearly centrosymmetric from PLATON analysis due to symmetric disposition of β angles ( Table 1). The global space group alone is, therefore, insufficient as a criterion to predict inversion-asymmetryinduced Rashba/Dresselhaus SOC effects that primarily originate from the inorganic layers. The actual distortions and resulting local symmetry within the inorganic layer are crucial aspects. Specifically, a large Δβ points to a substantial inversion asymmetry in the inorganic layers as realized in specific chiral and noncentrosymmetric achiral MHPs, especially those comprising lead bromide or lead chloride perovskite layers ( Table 1). Note that there is no clear correlation between Δβ and Δd or σ 2 for the MHPs in Table 1 ( Supplementary Fig. 5). More importantly, the values of Δβ, Δd, and σ 2 found in lead-iodide MHPs are at least 2.5 times smaller than those found in most lead bromide, and lead chloride MHPs in    inorganic layers in lead-iodide MHPs are typically nominally centrosymmetric (Table 1). In contrast to Table 1, a survey of 56 2D MHPs with centrosymmetric global space groups reveals that they seldomly exhibit a substantial Δβ (Supplementary Table 1). In a few centrosymmetric cases, a relatively large Δβ does occur. However, in two of these cases ( Supplementary Fig. 7b, c), equal β angles are found on opposite sides of squares formed by Pb atoms in the structure so that an inversion center is retained. In contrast, the above chiral and noncentrosymmetric MHPs have unequal β angles on opposite sides of squares defined by Pb. In the case of 2-fluoroethylammonium lead chloride ((FC 2 H 4 NH 3 ) 2 PbCl 4 ), Δβ = 11.7°also occurs across opposite sides of the Pb-defined squares ( Supplementary Fig. 7a), leading to inversion asymmetric inorganic layers; in this case, an inversion center is found between the inorganic layers.
Spin-splitting in 2D MHPs. To investigate the impact of structural distortions on the spin characteristics, we have calculated electronic band structures for the new chiral MHPs, as well as select noncentrosymmetric achiral MHPs in Table 1 (Fig. 3 and Supplementary Figs. 8-13), using DFT and the all-electron electronic structure code FHI-aims 35 (see Methods for details). We here use the semilocal level of DFT, specifically the Perdew-Burke-Ernzerhof (PBE) functional 36,37 . This choice is appropriate since, in contrast to fundamental gaps or energy level alignments between organic and inorganic components of the MHP, SOC itself is a large effect and already accurately captured at the level of semilocal DFT 37 . Regarding the underlying atomic geometries, band structures of two different structures for each material were considered: First, the experimental roomtemperature structures, which average over different thermal motions; second, computationally optimized structures at the DFT-PBE level of theory, amended by the Tkatchenko-Scheffler (TS) van der Waals correction 38 , which correspond to static local minima of the potential energy surface without any averaging over atomic motions. For a consistent comparison, we have aligned the crystal axes for all the MHPs (relaxed and experimental) so that the layer-stacking direction points along the a-axis while the b-and c-axes define the two in-plane directions of the perovskite layer (Fig. 3a). Accordingly, the Γ-X, Γ-Y, and Γ-Z paths in the Brillouin zone coincide, respectively, with a-, b-, and c-directions of perovskite layer in both relaxed and experimental MHP structures (Fig. 3b). In all cases, the calculated frontier CBs and valence bands (VBs) are comprised of inorganicderived states (Fig. 3d- wherein the lowest CBs appear to be derived from the organic component (Fig. 3d). The latter band alignment would have to be validated with a level of theory beyond DFT-PBE but does not impact the conclusions for inorganic-derived bands in this work.  2 PbI 4 (i) along select k-paths. Relaxed atomic geometries were used for band structure calculations. Pb-, Br-, I-and organic-derived electronic states are highlighted in purple, green, yellow, and black colors, respectively. Note that the focus of these plots is the degree of SOC near the inorganic-derived conduction band edges, not the alignment of energy levels or the exact fundamental gap.
Importantly, the organic-and inorganic-derived bands are computationally well separated and do not interfere with the analysis of SOC in the inorganic-derived bands.  (Fig. 3d, e). A similar CB spin-splitting occurs in [S-1-1-NEA] 2 PbBr 4 (Supplementary Fig. 9 [11][12][13]. As mentioned before, the Γ-Y path coincides with the in-plane b-direction of the perovskite layer, along which the local geometry fluctuates due to a large Δβ. The 2 1 -screw translational symmetry common to all the MHPs studied here is absent along the b-direction (Fig. 2f, g, and Supplementary Fig. 2). Note that, for [R-4-Cl-MBA] 2 PbBr 4 with a P2 1 2 1 2 1 space group, the 2 1 -screw symmetry along the b-direction is for Pb atoms rather than equatorial Br atoms. Along the Γ-Z path that coincides with the in-plane c-direction of the perovskite layer, the same type of equatorial halogen atoms and the associated β angles propagate by 2 1 -screw translational symmetry (Fig. 2f, g). The lack of band dispersion and spin-splitting along the Γ-X path (i.e., along the layer-stacking direction) follows from the confinement and localization of inorganic-derived states due to organic cations acting as insulating barriers. A large Δβ along the b-direction renders the isolated inorganic layers inversion asymmetric (  (Fig. 3f-i). Likewise, the noncentrosymmetric achiral lead-iodide MHPs in Table 1 exhibit negligible spin-splitting in their reported band structures 29,30,39 . In these cases, the isolated inorganic layers are nearly centrosymmetric with minimal Δβ (Table 1). A Δβ value of~6°in [S-MBA] 2 PbI 4 leads to marginally inversion asymmetric inorganic layers (Table 1), resulting in a CB splitting ( Supplementary  Fig. 10), which is still 2-4 times smaller compared with the above lead bromide or lead chloride MHPs.
In nonmagnetic solids, the combination of time-reversal and crystal inversion symmetries leads to a twofold spin degeneracy of energy bands. When the crystal inversion symmetry is broken, SOC lifts the spin degeneracy for a general k, except at special Kramer points and high-symmetry points, and modifies the dispersion relation of electron/hole bands near the Γ-point to assume the effective form 22,[40][41][42] : where E þ and E À are the energies of spin-split subbands for the given direction in k-space, and α eff ¼ ΔE ± =2k 0 (k 0 is the characteristic momentum offset) is the effective spin-splitting coefficient (Fig. 3c). For 2D electron gases respecting C nv point group symmetry, Rashba spin-splitting occurs along the in-plane wavevector k jj ¼ ðk x ; k y Þ owing to structural inversion asymmetry perpendicular to the 2D plane (k ¼ k z ), i.e., along the stacking direction 22 . The high-symmetry C n rotational axis is parallel to k z , and the twofold spin degeneracy is thus maintained along k z . In contrast, in the present 2D MHPs for which the local symmetry of the inorganic layer corresponds to either C 2 or C 2v point groups (i.e., bulk inversion asymmetry, BIA), the high-symmetry C 2 axis coincides with the in-plane Γ-Z path (k z direction), while the CB spin-splitting occurs dominantly along the other in-plane Γ-Y path (k y direction) due to significant Δβ (Fig. 3d, e). The much smaller but non-zero spin-splitting in the VB, on the other hand, can be understood from the fact that the VB states are comprised principally from 4p (5p) atomic orbitals of the lighter Br (I) atoms in contrast with the CB states which originate from the 6p orbitals of the heavier Pb atoms (Fig. 3d-i).
The spin polarization ( σ h i) values are calculated from the DFT spinor wave functions as the expectation values of the Pauli spin matrices (σ i ) 31 . The DFT calculated σ h i values for [S-4-NO 2 -MBA] 2 PbBr 4 •H 2 O with C 2 point group, as an example, reveal opposite signs of spin polarization for the upper and lower spinsplit frontier CBs (derived from the lead bromide framework) along the Γ À Y path. Moreover, the out-of-plane hσ x i component is dominant, with the in-plane hσ z i and hσ y i components being much smaller along the Γ À Y path (Fig. 4a). The spin polarization mapped onto the reciprocal 2D plane of the perovskite layer is essentially characteristic of the Dresselhaus-type spin-texture arising from BIA and captures the spin polarization anisotropy in the spin-split CBs (Fig. 4b). This can be understood from the theory of invariants in conjunction with the strongly anisotropic character of 2D MHPs (see Supplementary Notes 3 and 4). For C 2 point group, the BIA related Hamiltonian H BIA is written near Γ to linear order in in-plane wavevector components k y and k z as: where, J i denote Pauli operators representing the components of the total angular momentum (note that spin is not a good quantum number due to strong SOC) and α ij denote band specific SOC coefficients. Upon diagonalizing H BIA , we find the energy correction to the band dispersion as: Along the Γ À Y and Γ À Z paths, this leads to a band splitting of the form in Eq. (1) with the respective effective SOC coefficients of α eff ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi α 2 xy þ α 2 yy q and α eff ¼ α zz . This band splitting is equivalent to Zeeman splitting under an effective spin-orbit magnetic field, B eff 25,28 where μ b is the Bohr magneton. Analysis shows that, if B eff is predominantly along one principal axis, the spin polarization is directed parallel or anti-parallel to B eff with a magnitude proportional to the effective g-factor of the Bloch function. The effective g-factor along a given direction i is defined as are the expectation values of the spin and total angular momentum components, respectively. For the spin-split CBs, the effective g-factor is anisotropic owing to associated P-type Bloch functions, in conjunction with a strong anisotropy between the in-plane and out-of-plane directions of the lead halide framework. The latter can be described within multiband K.P theory in terms of a crystal field effect that causes mixing of P 1=2 and P 3=2 CB states. Analysis using tetragonal crystal field parameters (sinθ = 0.2-0.32) previously determined from experiments on a related 2D lead bromide perovskite 43,44 reveals that the computed effective g-factors are dominant in the out-ofplane direction (γ x ¼ Àcos 2θ), but much smaller in magnitude in the in-plane directions (γ k ¼ Àsin 2 θ), explaining why hσ x i >> hσ y i and hσ z i (Fig. 4). For a more detailed discussion, including for the VB, see Supplementary Note 4. B eff has only a z-component along k z , while it has both x-and y-components along k y (Eq. (4)). The dominant CB spin-splitting along the Γ À Y path suggests a dominant x-component of B eff . For a system with C 2v point group symmetry with C 2 axis pointing along k z , spin-splitting is exclusively along k y with x-directed B eff , while k z -related terms are strictly forbidden by symmetry (Supplementary Table 7). Therefore, while the local inversion asymmetry (i.e., owing to a large Δβ) induces spin-splitting, the global symmetry governs the specifics of k-dependent spin-splitting and spin polarizations. To further exemplify this point, (FC 2 H 4 NH 3 ) 2 PbCl 4 (Pnma global space group) crystallizes with a center of inversion in-between the two inorganic layers comprising the unit cell 45 . Each inorganic layer is noncentrosymmetric (nominal Pmc2 1 layer group) owing to a large Δβ. The local inversion asymmetry of the inorganic layer leads to CB splitting, but the global inversion symmetry leads to a zero net To further establish a possible quantitative relationship between spin-splitting and structural characteristics, we plot ΔE ± , k 0 , and α eff estimated for the series of relaxed and unrelaxed MHPs studied here (Fig. 3, Supplementary Figs. 8-13, and Supplementary Table 2) as a function of various distortion metrics (Fig. 5). Remarkably, all three parameters strongly correlate with Δβ. Upon decomposing β into in-plane (β in ) and out-of-plane (β out ) components (see Supplementary Note 1), the in-plane disparity (Δβ in ) values are found to correlate with the spin-splitting parameters most strongly, while there is barely a correlation with the out-of-plane disparity (Δβ out ) values. The latter apparent lack of correlation is plausibly because the Δβ out values are clustered in a low-distortion regime except for [R-4-Cl-MBA] 2 PbBr 4 , which exhibits the largest Δβ out values (~12°and 14°) found in the series (Fig. 5). Secondary weaker correlations are found with the maximum in-plane distortion (D in ¼ 180 À β in ) and σ 2 , since they both are angular distortions related to Δβ in (Fig. 5). Other distortion parameters correlate less clearly or not at all with computed spin-splitting.
To make the empirical correlation between spin-splitting and Δβ unambiguous, we have performed simulations on idealized 2D Cs 2 PbBr 4 models using a ð ffiffi ffi 2 p ffiffi ffi 2 p Þ À R45 (per Wood's notation 46,47 ) perovskite lattice, by systematically varying exclusively either Δβ in or Δβ out from 0°to 20°along the bdirection (Fig. 6, Supplementary Fig. 16, and Supplementary Table 3). Idealized Cs 2 PbX 4 models have also been used in the past for conceptual studies of structure-property relationships in 2D perovskites 48 . The intraoctahedral distortions (Δd and σ 2 ) induced by Δβ in or Δβ out are relatively small in the present Cs 2 PbBr 4 models as compared with the experimental MHPs and are similar between both sets of models, thereby allowing us to isolate and compare the dominant effects of Δβ in versus Δβ out . The calculated band structures reveal that, while the lowest CB subband energy width along the spin-splitting k-path decreases very similarly with Δβ in and Δβ out , k 0 and ΔE ± increase steeply with Δβ in , but less significantly with Δβ out (for details, see Supplementary Note 2). Both Δβ in and Δβ out create an inversion asymmetric local Br environment around Pb, meeting the conditions for the creation of spin-splitting in principle. According to our findings, the quantitative difference of spinsplittings caused by Δβ in vs. Δβ out is an intrinsic consequence of the distortion direction.
In Supplementary Tables 4-6, we further investigated a possible correlation between spin-splitting strength and the presence of a formal local dipole (often invoked in model description of spin-splitting) in the structure. Whereas Δβ in necessarily generates a formal dipole on the Pb site, Δβ out can be introduced with or without a resulting formal dipole on the Pb site in the idealized Cs 2 PbBr 4 models (Supplementary Fig. 16 and Supplementary Tables 4-6). Consistently, Δβ in models exhibit a noncentrosymmetric polar Pmc2 1 space group, whereas the Δβ out models with and without a formal dipole exhibit noncentrosymmetric polar Pma2 and nonpolar P222 1 space groups, respectively (Supplementary Table 3). We find that the spin-splitting associated with Δβ out does not change significantly regardless of whether a formal local dipole is present in the selected geometry ( Fig. 6e and Supplementary Fig. 20). Notably, the strong correlation of spin-splitting parameters with Δβ in overlaps with the confidence intervals of the fitting shown in Fig. 5, implying that the empirical correlation in observed MHPs is quantitative. This direct structure-property correlation enables a fast route to an informed discovery/screening of promising spin-selective MHPs, simply based on crystal structure information that can be readily accessed in perovskite databases 49 .  (Table 1). Data points corresponding to X = Cl, Br, and I are denoted with diamonds, circles, and squares, respectively. The filled and open symbols denote, respectively, the experimental and relaxed MHP atomic configurations. The solid lines are fits to a linear regression model, and the corresponding R 2 values and 95% confidence intervals (shaded regions) are shown in each subplot. Δβ, Δβ in , and Δβ out denote the asymmetric interoctahedral distortions, whereas Δd and σ 2 denote the intraoctahedral distortions (see the main text for details). "max. D in " and "max. D out " denote, respectively, the maximum in-plane (D in ¼ 180 À β in ) and out-of-plane (D out ¼ 180 À β out ) distortion values corresponding to smaller β in and β out angles within the inorganic layer.  50 . In 2D MHPs, however, noncentrosymmetric global space groups often arise from the molecular organic sublattice and do not necessarily imply strong spin-splitting. As the SOC effects are principally derived from the inorganic sublattice, the local distortions and the ensuing local asymmetry within the inorganic layers must underlie and control the spin-splitting in 2D MHPs. Our study establishes a generic local structural descriptor (Δβ in ), which signifies the local inversion asymmetry and quantitatively correlates with the DFTpredicted spin-splitting in a broad array of MHPs. Using this single descriptor, one can rapidly screen promising spin-splitting candidates from ever-growing perovskite libraries, with DFT or appropriately parametrized tight binding approaches 51,52 available for a posteriori validation if desired.
Note that low-frequency, high-amplitude optical phonons are known to arise from the lead halide framework below 100 cm −1 (<<kT at room temperature) 53,54 . The room-temperature experimental X-ray structures, therefore, represent a time average of all possible distortions induced by these soft phonons under thermal equilibrium. Nevertheless, the key structural attribute strongly correlating with the spin-splitting is still Δβ in (Fig. 5 and Supplementary Table 2). This distortion prevails in the thermal time average structures, apparently connected to symmetry-breaking local minimum-energy structures on the Born-Oppenheimer surface, as corroborated by DFT (Supplementary Table 2). While instantaneous structural fluctuations due to phonon modes may quantitatively increase or decrease the spin-splitting of the adiabatic electronic band structure at any given time, the average spin-splitting is expected to retain a clearly defined trend, imparted by the thermal average structure.
Finally, in the discussed MHP systems, the predicted anisotropic spin polarization of the CB is reminiscent of a persistent spin texture 25,55 , which is posited to enable longer spin lifetimes. For chiral systems, the spin polarization is opposite for opposite organic cation chirality 31 , and the ensuing chiral degrees of freedom coupled with a large spin-splitting predicted for some of the current chiral MHPs represent a promising avenue for spin manipulation in prospective spintronic devices. Characterization. Single-crystal X-ray diffraction was carried out at room temperature on a Rigaku XtaLAB Synergy-S diffractometer, using Mo-Kα radiation (λ = 0.710 Å) and X-ray tube operating at 50 kV and 30 mA. Structure solutions were obtained by SHELXS direct methods and refined using the SHELXL leastsquares method within the Olex 2 package. Symmetry analysis for just the inorganic framework/layers (i.e., after excluding the organic cations from the unit cell) was analyzed post refinement using PLATON's ADDSYM tool 31 . Default values of Fig. 6 First principles simulations on 2D Cs 2 PbBr 4 models. a, b Representative (√2 × √2) − R45°Cs 2 PbBr 4 models with symmetrical in-plane (a) and out-of-plane (b) tilting of adjacent PbBr 6 octahedra, i.e., with Δβ ¼ 0. c, d Representative (√2 × √2) − R45°Cs 2 PbBr 4 models with asymmetrical in-plane (c) and out-of-plane (d) tilting of adjacent PbBr 6 octahedra with Δβ in;out ¼ 20 . e Plots showing the evolution of lowest conduction subband energy width, k 0 , and ΔE ± along the spin-splitting k-path as a function of Δβ in and Δβ out in the series of Cs 2 PbBr 4 models. See Supplementary Fig. 16 and Supplementary Note 2 for details. The legends Δβ out 1 and Δβ out 2 in (e) correspond to the models with and without formal dipole on the Pb site, respectively. The shaded regions in (e) are 95% confidence intervals of fitting as a function of Δβ in in Fig. 5.

Methods
angle (0.3°for metrical lattice symmetry) and distance (0.25 Å for coinciding atoms for inversion, translational, and rotational symmetry elements) criteria were used in PLATON's symmetry analysis.
Computational methods. The all-electron electronic structure code FHI-aims 35 was used to carry out the DFT calculations. All calculations are based on numeric atom-centered orbital (NAO) basis sets. Calculations were carried out both for unrelaxed experimental geometries, as well as with relaxed geometries (i.e., local minima of the Born-Oppenheimer potential energy surface). Crystal axes for all the MHPs (relaxed and experimental geometries) have been aligned for consistent comparison so that the layer-stacking direction points along the a-axis while the band c-axes define the two in-plane directions of the perovskite layer. Relaxed geometries were obtained with the semilocal PBE functional 36 plus the TS pairwise dispersion scheme for van der Waals (vdW) interactions 38 . FHI-aims "tight" numerical defaults and k-space grid (2 × 4 × 4) were employed (except for [4AMP] PbBr 4 , whose k-space grid was set to (3 × 2 × 5) due to its specific unit cell). DFT-PBE+TS has proven to give a precise description of complex MHPs in the authors' past work 5,7,8,31,32,56 , with typical deviations between experimental and computational lattice parameters of~1% or less and with bond angle deviations of a few degrees. Electronic band structures suitable for analysis of the impact of SOC were obtained at the PBE level of theory, with a well-benchmarked secondvariational non-self-consistent SOC approach 37 , and employing FHI-aims "tight" numerical defaults and k-space grid (2 × 4 × 4). Spin-texture calculations were carried out as shown in ref. 31 . For more general band structure characteristics, such as the fundamental gap or energy level alignments between the organic and inorganic components, higher-level methods such as GW 57 or at least hybrid density-functionals 5,7,8,31,56,58,59 would be required, as was also done in past work by the authors 5,7,8,31,32,56 . However, we here focus exclusively on the SOC effects on the inorganic-related frontier bands in various MHPs, rather than the levelalignment of the organic and inorganic band edges. SOC is a large effect that is qualitatively well captured already at the PBE level of theory, as shown in ref. 37 . We therefore rely here on the computationally more affordable semilocal DFT-PBE approach as a useful tool for such SO splittings in semiconductors with sufficiently large gaps.