Abstract
Twodimensional (2D) hybrid metal halide perovskites have emerged as outstanding optoelectronic materials and are potential hosts of Rashba/Dresselhaus spinsplitting for spinselective transport and spinorbitronics. However, a quantitative microscopic understanding of what controls the spinsplitting magnitude is generally lacking. Through crystallographic and firstprinciples 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 spinsplitting. This distortion metric can serve as a crystallographic descriptor for rapid discovery of potential candidate materials with strong spinsplitting. 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 spinsplitting in perovskites. New chiral perovskites reported here couple a sizeable spinsplitting with chiral degrees of freedom and offer a unique paradigm of potential interest for spintronics.
Introduction
Twodimensional (2D) metal halide perovskites (MHPs) are natural quantum well (QW) heterostructures of organic and inorganic sublattices with exceptional optoelectronic and photophysical properties^{1,2,3}. They feature rich structural and compositional versatility owing to numerous possible templating organic cations^{4,5} that play a direct role in exciton dielectric confinement^{6}, QW tuning^{7,8,9}, singlettotriplet conversion^{10}, hightemperature ferroelectricity^{11}, nonlinear optical properties^{12}, and chiraloptoelectronic functionality^{13,14,15,16}. Additionally, a complex interplay of electrostatic requirements, steric effects, intermolecular interactions, and hydrogenbonding associated with the organic cations engenders intricate distortions within the inorganic layers that have been previously correlated with exciton energy shifts^{17,18} and exciton selftrapping^{19}. When such distortions break the inversion symmetry of the inorganic layer, Rashba/Dresselhaus spinsplitting can emerge due to spin–orbit coupling (SOC)^{20,21,22,23,24}. Rashba/Dresselhaus spinsplitting is a relativistic quantum phenomenon in condensed matter that leads to important physical manifestations for emerging spinorbitronic devices, including intrinsic spinHall effect^{25}, gatecontrolled spin precession^{26}, (inverse) spin galvanic effects, photogalvanic effects^{27}, and chiroptic effects^{28}, that rely on SOCmediated manipulation of the spin degrees of freedom by electrical, optical, or magnetic means^{22,25}.
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 spinsplitting in MHPs. A large Rashba/Dresselhaus spinsplitting (\(\gg\!{kT}\)) suppresses spinflipping and is a critical requisite for roomtemperature spinbased applications. There have been a few reports with computed spinsplitting values ranging from <<10 meV in some 2D leadiodidebased MHPs^{29,30} to >200 meV in the chiral [S/R11NEA]_{2}PbBr_{4}^{31}. Pb being the same heavymetal constituent in these MHPs, SOC alone cannot justify the wide disparity in the predicted spinsplitting values. Despite a growing interest in spinrelated properties of MHPs, a fundamental understanding of the symmetry breaking and the principles determining the spinsplitting magnitude are generally lacking, posing a major bottleneck in rational discovery of potential MHPs with sizable spinsplitting for prospective spinbased applications.
From a materials discovery perspective, a key question is if there exists a simple structural parameter that controls the spinsplitting in MHPs and, consequently, their spinrelated 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 spinsplitting exhibited in its conduction bands (CBs). Through crystallographic studies combined with densityfunctional theory (DFT), we establish that the computed spinsplitting mainly originates from and scales with inplane asymmetric tilting of adjacent metal halide octahedra, represented by a specific projected bond angle difference \(\Delta {\beta }_{{in}}\) (Fig. 1 and Supplementary Notes 1 and 2), as demonstrated for a broad array of MHPs, including five newly synthesized chiral MHPs. A large \(\Delta {\beta }_{{in}}\) signifies inversion asymmetry within inorganic layers, and in turn, serves as a key local descriptor of spinsplitting, 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 distortioninduced spinsplitting, 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 spinsplitting. 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 cationinduced 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: [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O, [R4ClMBA]_{2}PbBr_{4}, [S2MeBuA]_{2}PbBr_{4}, [S4NH_{3}MBA]PbI_{4}, and [SMHA]_{2}PbI_{4} (Fig. 2) (see “Methods” for details). Table 1 summarizes for all the above MHPs the metrics of interoctahedral tilting distortion leading to equatorial Pb–X–Pb bond angle \(\beta\) < 180°, the disparity in adjacent \(\beta\) angles (\(\Delta \beta\)) (see Fig. 1), as well as the intraoctahedron distortions: \(\Delta d\) and \({\sigma }^{2}\). Here, \(\Delta d\) is the bond length distortion defined as \(\Delta d\,=\,\left(\frac{1}{6}\right)\Sigma {({d}_{i}\,\,{d}_{0})}^{2}/{{d}_{0}}^{2}\) (\({d}_{i}\) denotes the six Pb–X bond lengths and \({d}_{0}\) is the mean M–X bond length), and \({\sigma }^{2}\) is the bond angle variance defined as \({{{\sigma }}}^{2}\,=\,{\Sigma }_{i\,=\,1}^{12}{\left({\theta }_{i}\,\,90\right)}^{2}/11\) (\({\theta }_{i}\) 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 \(\beta\) angles in each MHP, leading to \(\Delta \beta\) ≠ 0 (Figs. 1, 2f–j). We can group the different MHPs in Table 1 based on \(\Delta \beta\) values. High \(\varDelta \beta\): In [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O and [R4ClMBA]_{2}PbBr_{4}, the same type of equatorial Br atom comprises the individual rows along one inplane direction due to 2_{1} screw symmetry. However, two very different equatorial Br atoms alternate along the other inplane direction, leading to \(\Delta \beta\) as large as 12° and 15° in [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O and [R4ClMBA]_{2}PbBr_{4}, respectively (Fig. 2f, g). A similar scenario with a large \(\Delta \beta\) along one of the inplane directions occurs in chiral [S/R11NEA]_{2}PbBr_{4}, as well as noncentrosymmetric achiral [NMA]_{2}PbBr_{4}, [PMA]_{2}PbCl_{4}, and [4AMP]PbBr_{4} (Table 1 and Supplementary Fig. 2). In all these MHPs, the fluctuating local geometry with \(\Delta \beta\) as large as ~12°–15° along either inplane direction breaks the inversion symmetry within the isolated inorganic layer (Table 1) as suggested by PLATON^{34} symmetry analysis (see Methods for details). Low \(\varDelta \beta\): 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 \(\Delta \beta\) (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 \(\beta\) angles (Table 1). The global space group alone is, therefore, insufficient as a criterion to predict inversionasymmetryinduced 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 \(\Delta \beta\) 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 \(\Delta \beta\) and \(\Delta d\) or \({{{\sigma }}}^{2}\) for the MHPs in Table 1 (Supplementary Fig. 5). More importantly, the values of \(\Delta \beta\), \(\Delta d\), and \({\sigma }^{2}\) found in leadiodide MHPs are at least 2.5 times smaller than those found in most lead bromide, and lead chloride MHPs in Table 1 (Supplementary Figs. 5, 6), consistent with the fact that the inorganic layers in leadiodide 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 \(\Delta \beta\) (Supplementary Table 1). In a few centrosymmetric cases, a relatively large \(\Delta \beta\) does occur. However, in two of these cases (Supplementary Fig. 7b, c), equal \(\beta\) 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 \(\beta\) angles on opposite sides of squares defined by Pb. In the case of 2fluoroethylammonium lead chloride ((FC_{2}H_{4}NH_{3})_{2}PbCl_{4}), \(\Delta \beta\) = 11.7° also occurs across opposite sides of the Pbdefined squares (Supplementary Fig. 7a), leading to inversion asymmetric inorganic layers; in this case, an inversion center is found between the inorganic layers.
Spinsplitting 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 allelectron electronic structure code FHIaims^{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 DFTPBE 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 layerstacking direction points along the aaxis while the b and caxes define the two inplane directions of the perovskite layer (Fig. 3a). Accordingly, the ΓX, ΓY, and ΓZ paths in the Brillouin zone coincide, respectively, with a, b, and cdirections 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–f), except for [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O, 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 DFTPBE but does not impact the conclusions for inorganicderived bands in this work. Importantly, the organic and inorganicderived bands are computationally well separated and do not interfere with the analysis of SOC in the inorganicderived bands.
Figure 3d–i shows the electronic band structures of five new chiral MHPs from Fig. 2, as well as previously reported [R4ClMBA]_{2}PbI_{4} (all corresponding to relaxed geometries). The band structures of [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O and [R4ClMBA]_{2}PbBr_{4} exhibit a large spinsplitting in the inorganicderived frontier CB mainly along the ΓY path (Fig. 3d, e). A similar CB spinsplitting occurs in [S11NEA]_{2}PbBr_{4} (Supplementary Fig. 9), as well as in noncentrosymmetric achiral MHPs such as [4AMP]PbBr_{4}, [NMA]_{2}PbBr_{4}, and [PMA]_{2}PbCl_{4} (Supplementary Figs. 11–13). As mentioned before, the ΓY path coincides with the inplane bdirection of the perovskite layer, along which the local geometry fluctuates due to a large \(\Delta \beta\). The 2_{1}screw translational symmetry common to all the MHPs studied here is absent along the bdirection (Fig. 2f, g, and Supplementary Fig. 2). Note that, for [R4ClMBA]_{2}PbBr_{4} with a P2_{1}2_{1}2_{1} space group, the 2_{1}screw symmetry along the bdirection is for Pb atoms rather than equatorial Br atoms. Along the ΓZ path that coincides with the inplane cdirection of the perovskite layer, the same type of equatorial halogen atoms and the associated \(\beta\) angles propagate by 2_{1}screw translational symmetry (Fig. 2f, g). The lack of band dispersion and spinsplitting along the ΓX path (i.e., along the layerstacking direction) follows from the confinement and localization of inorganicderived states due to organic cations acting as insulating barriers. A large \(\Delta \beta\) along the bdirection renders the isolated inorganic layers inversion asymmetric (Table 1), thus leading to a strong CB spinsplitting. In contrast with the above MHPs, there is negligible spinsplitting in [S2MeBuA]_{2}PbBr_{4}, [S4NH_{3}MBA]PbI_{4}, [R4ClMBA]_{2}PbI_{4}, and [SMHA]_{2}PbI_{4} (Fig. 3f–i). Likewise, the noncentrosymmetric achiral leadiodide MHPs in Table 1 exhibit negligible spinsplitting in their reported band structures^{29,30,39}. In these cases, the isolated inorganic layers are nearly centrosymmetric with minimal \(\Delta \beta\) (Table 1). A \(\Delta \beta\) value of ~6° in [SMBA]_{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 timereversal 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 highsymmetry points, and modifies the dispersion relation of electron/hole bands near the Γpoint to assume the effective form^{22,40,41,42}:
where \({E}^{+}\) and \({E}^{}\) are the energies of spinsplit subbands for the given direction in kspace, and \({\alpha }_{{{\mathrm{eff}}}}\,=\,\Delta {E}^{\pm }/2{{{{{k}}}}}_{0}\) (\({{{{{k}}}}}_{0}\) is the characteristic momentum offset) is the effective spinsplitting coefficient (Fig. 3c). For 2D electron gases respecting C_{nv} point group symmetry, Rashba spinsplitting occurs along the inplane wavevector \({{{{{{\bf{k}}}}}}}_{{{}}}\,=\,({{{{{{\bf{k}}}}}}}_{{{{{{\bf{x}}}}}}},{{{{{{\bf{k}}}}}}}_{{{{{{\bf{y}}}}}}})\) owing to structural inversion asymmetry perpendicular to the 2D plane (\({{{{{{\bf{k}}}}}}}_{{{{{{\boldsymbol{\perp }}}}}}}={{{{{{\bf{k}}}}}}}_{{{{{{\bf{z}}}}}}}\)), i.e., along the stacking direction^{22}. The highsymmetry C_{n} rotational axis is parallel to \({{{{{{\bf{k}}}}}}}_{{{{{{\bf{z}}}}}}}\), and the twofold spin degeneracy is thus maintained along \({{{{{{\bf{k}}}}}}}_{{{{{{\bf{z}}}}}}}\). 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 highsymmetry C_{2} axis coincides with the inplane ΓZ path (\({k}_{z}\) direction), while the CB spinsplitting occurs dominantly along the other inplane ΓY path (\({k}_{y}\) direction) due to significant \(\Delta \beta\) (Fig. 3d, e). The much smaller but nonzero spinsplitting 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 (\(\left\langle \sigma \right\rangle\)) values are calculated from the DFT spinor wave functions as the expectation values of the Pauli spin matrices (\({\sigma }_{i}\))^{31}. The DFT calculated \(\left\langle \sigma \right\rangle\) values for [S4NO_{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 \(\Gamma Y\) path. Moreover, the outofplane \(\langle {\sigma }_{x}\rangle\) component is dominant, with the inplane \(\,\langle {\sigma }_{z}\rangle\) and \(\langle {\sigma }_{y}\rangle\) components being much smaller along the \(\Gamma Y\) path (Fig. 4a). The spin polarization mapped onto the reciprocal 2D plane of the perovskite layer is essentially characteristic of the Dresselhaustype spintexture arising from BIA and captures the spin polarization anisotropy in the spinsplit 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 \(\Gamma\) to linear order in inplane 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 \({\alpha }_{{ij}}\) denote band specific SOC coefficients. Upon diagonalizing \({H}_{{BIA}}\), we find the energy correction to the band dispersion as:
Along the \(\Gamma Y\) and \(\Gamma Z\) paths, this leads to a band splitting of the form in Eq. (1) with the respective effective SOC coefficients of \({\alpha }_{{{\mathrm{eff}}}}\,=\,\sqrt{{\alpha }_{{xy}}^{2}\,+\,{\alpha }_{{yy}}^{2}}\,\)and \({\alpha }_{{{\mathrm{eff}}}}\,=\,{\alpha }_{{zz}}\). This band splitting is equivalent to Zeeman splitting under an effective spin–orbit magnetic field, \({{{\mathbf{B}}}}_{{{\mathbf{eff}}}}\)^{25,28}
where \({\mu }_{b}\) is the Bohr magneton. Analysis shows that, if \({{\mathbf{{B}}}_{{eff}}}\) is predominantly along one principal axis, the spin polarization is directed parallel or antiparallel to \({{\mathbf{{B}}}_{{eff}}}\) with a magnitude proportional to the effective gfactor of the Bloch function.
The effective gfactor along a given direction i is defined as \({\gamma }_{i}\,=\,\left\langle {\sigma }_{i}\right\rangle /\left\langle {J}_{i}\right\rangle\), where \(\left\langle {\sigma }_{i}\right\rangle\) and \(\left\langle {J}_{i}\right\rangle\) are the expectation values of the spin and total angular momentum components, respectively. For the spinsplit CBs, the effective gfactor is anisotropic owing to associated \(P\)type Bloch functions, in conjunction with a strong anisotropy between the inplane and outofplane 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 (\({\sin }\theta\) = 0.2–0.32) previously determined from experiments on a related 2D lead bromide perovskite^{43,44} reveals that the computed effective gfactors are dominant in the outofplane direction (\({\gamma }_{x}\,=\,{{\cos }}\,2\theta\)), but much smaller in magnitude in the inplane directions (\({\gamma }_{\parallel }\,=\,{{{\sin }}}^{2}\theta\)), explaining why \(\langle {\sigma }_{x}\rangle\) >> \(\langle {\sigma }_{y}\rangle\) and \(\langle {\sigma }_{z}\rangle\) (Fig. 4). For a more detailed discussion, including for the VB, see Supplementary Note 4. \({{\mathbf{{B}}}_{{eff}}}\) has only a zcomponent along \({k}_{z}\), while it has both x and ycomponents along \({k}_{y}\) (Eq. (4)). The dominant CB spinsplitting along the \(\Gamma Y\) path suggests a dominant xcomponent of \({{\mathbf{{B}}}_{{eff}}}\). For a system with C_{2v} point group symmetry with C_{2} axis pointing along \({k}_{z}\), spinsplitting is exclusively along \({k}_{y}\) with xdirected \({{\mathbf{{B}}}_{{eff}}}\), 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 \(\Delta \beta\)) induces spinsplitting, the global symmetry governs the specifics of kdependent spinsplitting 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 inbetween the two inorganic layers comprising the unit cell^{45}. Each inorganic layer is noncentrosymmetric (nominal Pmc2_{1} layer group) owing to a large \(\Delta \beta\). The local inversion asymmetry of the inorganic layer leads to CB splitting, but the global inversion symmetry leads to a zero net spin polarization of the resulting 2fold degenerate upper and lower CBs (Supplementary Note 5).
To further establish a possible quantitative relationship between spinsplitting and structural characteristics, we plot \(\Delta {E}^{\pm }\), \({{{{{\bf{k}}}}}}_{{{{{0}}}}}\), and \({\alpha }_{{{\mathrm{eff}}}}\) 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 \(\Delta \beta\). Upon decomposing \(\beta\) into inplane (\({\beta }_{{in}}\)) and outofplane (\({\beta }_{{out}}\)) components (see Supplementary Note 1), the inplane disparity (\(\Delta {\beta }_{{in}}\)) values are found to correlate with the spinsplitting parameters most strongly, while there is barely a correlation with the outofplane disparity (\(\Delta {\beta }_{{out}}\)) values. The latter apparent lack of correlation is plausibly because the \(\Delta {\beta }_{{out}}\) values are clustered in a lowdistortion regime except for [R4ClMBA]_{2}PbBr_{4}, which exhibits the largest \(\Delta {\beta }_{{out}}\) values (~12° and 14°) found in the series (Fig. 5). Secondary weaker correlations are found with the maximum inplane distortion (\({D}_{{in}}\,=\,180\,\,{\beta }_{{in}}\)) and \({\sigma }^{2}\), since they both are angular distortions related to \(\Delta {\beta }_{{in}}\) (Fig. 5). Other distortion parameters correlate less clearly or not at all with computed spinsplitting.
To make the empirical correlation between spinsplitting and \(\Delta \beta\) unambiguous, we have performed simulations on idealized 2D Cs_{2}PbBr_{4} models using a \((\sqrt{2}\times \sqrt{2})\,\,R45^\circ\) (per Wood’s notation^{46,47}) perovskite lattice, by systematically varying exclusively either \(\Delta {\beta }_{{in}}\) or \(\Delta {\beta }_{{out}}\) 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 structureproperty relationships in 2D perovskites^{48}. The intraoctahedral distortions (\(\Delta d\) and \({\sigma }^{2}\)) induced by \(\Delta {\beta }_{{in}}\) or \(\Delta {\beta }_{{out}}\) 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 \(\Delta {\beta }_{{in}}\) versus \(\Delta {\beta }_{{out}}\). The calculated band structures reveal that, while the lowest CB subband energy width along the spinsplitting kpath decreases very similarly with \(\Delta {\beta }_{{in}}\) and \(\Delta {\beta }_{{out}}\), \({{{\bf{k}}}}_{0}\) and \(\Delta {E}^{\pm }\) increase steeply with \(\Delta {\beta }_{{in}}\), but less significantly with \(\Delta {\beta }_{{out}}\) (for details, see Supplementary Note 2). Both \(\Delta {\beta }_{{in}}\) and \(\Delta {\beta }_{{out}}\) create an inversion asymmetric local Br environment around Pb, meeting the conditions for the creation of spinsplitting in principle. According to our findings, the quantitative difference of spinsplittings caused by \(\Delta {\beta }_{{in}}\) vs. \(\Delta {\beta }_{{out}}\) is an intrinsic consequence of the distortion direction.
In Supplementary Tables 4–6, we further investigated a possible correlation between spinsplitting strength and the presence of a formal local dipole (often invoked in model description of spinsplitting) in the structure. Whereas \(\Delta {\beta }_{{in}}\) necessarily generates a formal dipole on the Pb site, \(\Delta {\beta }_{{out}}\) 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, \(\Delta {\beta }_{{in}}\) models exhibit a noncentrosymmetric polar Pmc2_{1} space group, whereas the \(\Delta {\beta }_{{out}}\) 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 spinsplitting associated with \(\Delta {\beta }_{{out}}\) 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 spinsplitting parameters with \(\Delta {\beta }_{{in}}\) overlaps with the confidence intervals of the fitting shown in Fig. 5, implying that the empirical correlation in observed MHPs is quantitative. This direct structureproperty correlation enables a fast route to an informed discovery/screening of promising spinselective MHPs, simply based on crystal structure information that can be readily accessed in perovskite databases^{49}.
Discussion
Zhang et al. put forth important symmetry criteria based on global space groups and site point groups to observe either conventional or compensated Dresselhaus and/or Rashba effects in nonmagnetic inorganic compounds^{50}. In 2D MHPs, however, noncentrosymmetric global space groups often arise from the molecular organic sublattice and do not necessarily imply strong spinsplitting. 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 spinsplitting in 2D MHPs. Our study establishes a generic local structural descriptor (\(\Delta {\beta }_{{in}}\)), which signifies the local inversion asymmetry and quantitatively correlates with the DFTpredicted spinsplitting in a broad array of MHPs. Using this single descriptor, one can rapidly screen promising spinsplitting candidates from evergrowing perovskite libraries, with DFT or appropriately parametrized tight binding approaches^{51,52} available for a posteriori validation if desired.
Note that lowfrequency, highamplitude optical phonons are known to arise from the lead halide framework below 100 cm^{−1} (<<kT at room temperature)^{53,54}. The roomtemperature experimental Xray 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 spinsplitting is still \(\Delta {\beta }_{{in}}\) (Fig. 5 and Supplementary Table 2). This distortion prevails in the thermal time average structures, apparently connected to symmetrybreaking local minimumenergy 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 spinsplitting of the adiabatic electronic band structure at any given time, the average spinsplitting 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 spinsplitting predicted for some of the current chiral MHPs represent a promising avenue for spin manipulation in prospective spintronic devices.
Methods
Materials
(S)4nitroαmethylbenzylamine hydrochloride [“S4NO_{2}MBA∙HCl”, 97%] (also called (S)αmethyl4nitrobenzylamine hydrochloride); (S)(+)1methylhexyamine [“SMHA”, 99%] (also called (S)(+)1aminoheptane); (S)(−)2methylbutylamine [“S2MeBuA”, 95%]; aq. HI (57 wt. % in H_{2}O, distilled, stabilized, 99.95%), aq. HBr (48 wt.% in H_{2}O, ≥99.99%) and aq. H_{3}PO_{2} (50 wt.% in H_{2}O) were purchased from Sigma Aldrich. (R)4chloroαmethylbenzylamine [“R4ClMBA”, 97%] was purchased from Fischer Scientific.
Synthesis
Singlecrystals of all the chiral MHPs studied here, except for [S4NO_{2}MBA]_{2}PbBr_{4}·H_{2}O and [S4NH_{3}MBA]PbI_{4}, were grown by slowly cooling over 60 h a hot (95 °C) aq. HI or aq. HBr solution of PbI_{2} or PbBr_{2} and chiral amine in a 1:2 molar ratio in a sealed vial with N_{2}. In a typical synthesis, 0.122 mmol of PbI_{2} or PbBr_{2} and 0.244 mmol of neat chiral amine were used. For growing singlecrystals of [S4NO_{2}MBA]_{2}PbBr_{4}·H_{2}O, a solution of 0.122 mmol of PbBr_{2} and 0.244 mmol of S4NO_{2}MBA∙HCl in 0.5 ml aq. HBr and 1.7 ml methanol was slowly evaporated at room temperature in the ambient atmosphere. For growing singlecrystals of [S4NH_{3}MBA]PbI_{4}, a hot solution of 0.245 mmol of PbI_{2,} and 0.49 mmol of S4NO_{2}MBA∙HCl in 0.2 ml of aq. HI and 0.1 ml of aq. H_{3}PO_{2} was slowly cooled from 85 °C to room temperature over 5 h. The asobtained singlecrystals were filtered, washed thoroughly with diethyl ether, and vacuumdried.
Characterization
Singlecrystal Xray diffraction was carried out at room temperature on a Rigaku XtaLAB SynergyS diffractometer, using MoK\(\alpha\) radiation (\(\lambda\) = 0.710 Å) and Xray 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 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 allelectron electronic structure code FHIaims^{35} was used to carry out the DFT calculations. All calculations are based on numeric atomcentered 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 layerstacking direction points along the aaxis while the b and caxes define the two inplane 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}. FHIaims “tight” numerical defaults and kspace grid (2 × 4 × 4) were employed (except for [4AMP]PbBr_{4}, whose kspace grid was set to (3 × 2 × 5) due to its specific unit cell). DFTPBE+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 wellbenchmarked secondvariational nonselfconsistent SOC approach^{37}, and employing FHIaims “tight” numerical defaults and kspace grid (2 × 4 × 4). Spintexture 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, higherlevel methods such as GW^{57} or at least hybrid densityfunctionals^{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 inorganicrelated 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 DFTPBE approach as a useful tool for such SO splittings in semiconductors with sufficiently large gaps.
Data availability
Additional data supporting the findings of this work are provided in Supplementary Information. The singlecrystal Xray structures of the new chiral MHPs, namely, [R4ClMBA]_{2}PbBr_{4}, [S1MeHA]_{2}PbI_{4}, [S4NO_{2}MBA]_{2}PbBr_{4}∙H_{2}O, [S2MeBuA]_{2}PbBr_{4}, and [S4NH_{3}MBA]PbI_{4} have been deposited in The Cambridge Crystallographic Data Center (CCDC) database under deposition numbers 20954822095486. These data can be obtained free of charge via https://www.ccdc.cam.ac.uk/structures/ and also from the Hybrid^{3} perovskite database via https://materials.hybrid3.duke.edu/materials/search using the search terms: “R4ClMBA2PbBr4”, “S1MeHA2PbI4”, “S4NO2MBA2PbBr4.H2O”, “S2MeBuA2PbBr4”, and “S4NH3MBAPbI_{4}”. Atomic coordinates for the experimental and relaxed structures of MHPs and Cs_{2}PbBr_{4} models used for band structure calculations in this study have been provided in Supplementary Note 6. Relevant DFT band structure data is available in the NOMAD repository (https://doi.org/10.17172/NOMAD/2021.07.171). Other relevant data can be obtained from the corresponding authors upon reasonable request.
References
Saparov, B. & Mitzi, D. B. Organic–inorganic perovskites: structural versatility for functional materials design. Chem. Rev. 116, 4558–4596 (2016).
Smith, M. D., Connor, B. A. & Karunadasa, H. I. Tuning the luminescence of layered halide perovskites. Chem. Rev. 119, 3104–3139 (2019).
Quarti, C., Katan, C. & Even, J. Physical properties of bulk, defective, 2D and 0D metal halide perovskite semiconductors from a symmetry perspective. J. Phys. Mater. 3, 042001 (2020).
Mao, L., Stoumpos, C. C. & Kanatzidis, M. G. Twodimensional hybrid halide perovskites: principles and promises. J. Am. Chem. Soc. 141, 1171–1190 (2018).
Jana, M. K. et al. A directbandgap 2D silverbismuth iodide double perovskite: the structuredirecting influence of an oligothiophene spacer cation. J. Am. Chem. Soc. 141, 7955–7964 (2019).
Hong, X., Ishihara, T. & Nurmikko, A. V. Dielectric confinement effect on excitons in PbI_{4}based layered semiconductors. Phys. Rev. B 45, 6961–6964 (1992).
Liu, C. et al. Tunable semiconductors: control over carrier states and excitations in layered hybrid organicinorganic perovskites. Phy. Rev. Lett. 121, 146401 (2018).
DunlapShohl, W. A. et al. Tunable internal quantum well alignment in rationally designed oligomerbased perovskite films deposited by resonant infrared matrixassisted pulsed laser evaporation. Mater. Horiz. 6, 1707–1716 (2019).
Mitzi, D. B., Chondroudis, K. & Kagan, C. R. Design, structure, and optical properties of organic–inorganic perovskites containing an oligothiophene chromophore. Inorg. Chem. 38, 6246–6256 (1999).
Ema, K., Inomata, M., Kato, Y., Kunugita, H. & Era, M. Nearly perfect triplettriplet energy transfer from wannier excitons to naphthalene in organicinorganic hybrid quantumwell materials. Phys. Rev. Lett. 100, 257401 (2008).
Yang, C.K. et al. The first 2D homochiral lead iodide perovskite ferroelectrics: [R and S1(4Chlorophenyl)ethylammonium]_{2}PbI_{4}. Adv. Mater. 31, 1808088 (2019).
Yuan, C. et al. Chiral lead halide perovskite nanowires for secondorder nonlinear optics. Nano Lett. 18, 5411–5417 (2018).
Long, G. et al. Spin control in reduceddimensional chiral perovskites. Nat. Photon. 12, 528–533 (2018).
Long, G. et al. Chiralperovskite optoelectronics. Nat. Rev. Mater. 5, 423–439 (2020).
Lu, H. et al. Spindependent charge transport through 2D chiral hybrid leadiodide perovskites. Sci. Adv. 5, eaay0571 (2019).
Ma, J. et al. Chiral 2D perovskites with a high degree of circularly polarized photoluminescence. ACS Nano 13, 3659–3665 (2019).
Knutson, J. L., Martin, J. D. & Mitzi, D. B. Tuning the band gap in hybrid tin iodide perovskite semiconductors using structural templating. Inorg. Chem. 44, 4699–4705 (2005).
Du, K.z et al. Twodimensional Lead(II) halidebased hybrid perovskites templated by acene alkylamines: crystal structures, optical properties, and piezoelectricity. Inorg. Chem. 56, 9291–9302 (2017).
Smith, M. D., Jaffe, A., Dohner, E. R., Lindenberg, A. M. & Karunadasa, H. I. Structural origins of broadband emission from layered Pb–Br hybrid perovskites. Chem. Sci. 8, 4497–4504 (2017).
Liu, X. et al. Circular photogalvanic spectroscopy of Rashba splitting in 2D hybrid organic–inorganic perovskite multiple quantum wells. Nat. Commun. 11, 323 (2020).
Martin Schlipf, F. G. Dynamic RashbaDresselhaus Effect. Preprint at https://arxiv.org/abs/2004.10477 (2020).
Kepenekian, M. et al. Rashba and Dresselhaus effects in hybrid organic–inorganic perovskites: from basics to devices. ACS Nano 9, 11557–11567 (2015).
Wang, F. et al. Switchable Rashba anisotropy in layered hybrid organic–inorganic perovskite by hybrid improper ferroelectricity. npj Comput. Mater. 6, 183 (2020).
Niesner, D. et al. Structural fluctuations cause spinsplit states in tetragonal (CH_{3}NH_{3})PbI_{3} as evidenced by the circular photogalvanic effect. Proc. Natl Acad. Sci. U.S.A. 115, 9509–9514 (2018).
Manchon, A., Koo, H. C., Nitta, J., Frolov, S. M. & Duine, R. A. New perspectives for Rashba spin–orbit coupling. Nat. Mater. 14, 871–882 (2015).
Datta, S. & Das, B. Electronic analog of the electro‐optic modulator. Appl. Phy. Lett. 56, 665–667 (1990).
Wang, J. et al. Spindependent photovoltaic and photogalvanic responses of optoelectronic devices based on chiral twodimensional hybrid organic–inorganic perovskites. ACS Nano 15, 588–595 (2021).
Sercel, P. C., Vardeny, Z. V. & Efros, A. L. Circular dichroism in nonchiral metal halide perovskites. Nanoscale 12, 18067–18078 (2020).
Yin, J. et al. Layerdependent Rashba band splitting in 2D hybrid perovskites. Chem. Mater. 30, 8538–8545 (2018).
Schmitt, T. et al. Control of crystal symmetry breaking with halogen substituted benzylammonium in layered hybrid metalhalide perovskites. J. Am. Chem. Soc. 142, 5060–5067 (2020).
Jana, M. K. et al. Organictoinorganic structural chirality transfer in a 2D hybrid perovskite and impact on RashbaDresselhaus spinorbit coupling. Nat. Commun. 11, 4699 (2020).
Lu, H. et al. Highly distorted chiral twodimensional tin iodide perovskites for spin polarized charge transport. J. Am. Chem. Soc. 142, 13030–13040 (2020).
Naaman, R., Paltiel, Y. & Waldeck, D. H. Chiral induced spin selectivity gives a new twist on spincontrol in chemistry. Acc. Chem. Res. 53, 2659–2667 (2020).
Spek, A. L. Structure validation in chemical crystallography. Acta Cryst. D65, 148–155 (2009).
Blum, V. et al. Ab initio molecular simulations with numeric atomcentered orbitals. Comput. Phys. Commun. 180, 2175–2196 (2009).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Huhn, W. P. & Blum, V. Onehundredthree compound bandstructure benchmark of postselfconsistent spinorbit coupling treatments in density functional theory. Phys. Rev. Mater. 1, 033803 (2017).
Tkatchenko, A. & Scheffler, M. Accurate Molecular Van Der Waals Interactions from GroundState Electron Density and FreeAtom Reference Data. Phys. Rev. Lett. 102, 073005 (2009).
Mao, L. et al. Hybrid Dion–Jacobson 2D lead iodide perovskites. J. Am. Chem. Soc. 140, 3775–3783 (2018).
Bychkov, Y. A. & Rashba, E. I. Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39, 78–81 (1984).
Rashba, E. I. Properties of semiconductors with an extremum loop. 1. cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop. Sov. Phys. Solid State 2, 1224–1238 (1960).
Dresselhaus, G. SpinOrbit Coupling effects in zinc blende structures. Phys. Rev. 100, 580–586 (1955).
Tanaka, K. et al. Electronic and excitonic structures of inorganic–organic perovskitetype quantumwell crystal (C_{4}H_{9}NH_{3})_{2}PbBr_{4}. Jpn. J. Appl. Phys. 44, 5923–5932 (2005).
Ema, K. et al. Huge exchange energy and fine structure of excitons in an organicinorganic quantum well material. Phys. Rev. B 73, 241310 (2006).
Lermer, C. et al. Toward fluorinated spacers for MAPIderived hybrid perovskites: synthesis, characterization, and phase transitions of (FC_{2}H_{4}NH_{3})_{2}PbCl_{4}. Chem. Mater. 28, 6560–6566 (2016).
Hermann, K. Crystallography and Surface Structure: An Introduction for Surface Scientists and Nanoscientists (Wiley‐VCH Verlag GmbH & Co. KGaA, 2017).
Wood, E. A. Vocabulary of surface crystallography. J. Appl. Phys. 35, 1306–1312 (1964).
Pedesseau, L. et al. Advances and promises of layered halide hybrid perovskite semiconductors. ACS Nano 10, 9776–9786 (2016).
HybriD³ materials database, https://materials.hybrid3.duke.edu/.
Zhang, X., Liu, Q., Luo, J.W., Freeman, A. J. & Zunger, A. Hidden spin polarization in inversionsymmetric bulk crystals. Nat. Phys. 10, 387–393 (2014).
BoyerRichard, S. et al. Symmetrybased tight binding modeling of halide perovskite semiconductors. J. Phys. Chem. Lett. 7, 3833–3840 (2016).
Cho, Y. & Berkelbach, T. C. Optical properties of layered hybrid organic–inorganic halide perovskites: a tightbinding GWBSE study. J. Phys. Chem. Lett. 10, 6189–6196 (2019).
Quan, L. N. et al. Vibrational relaxation dynamics in layered perovskite quantum wells. Proc. Natl Acad. Sci. 118, e2104425118 (2021).
Thouin, F. et al. Phonon coherences reveal the polaronic character of excitons in twodimensional lead halide perovskites. Nat. Mater. 18, 349–356 (2019).
Anshory, M. & Absor, M. A. U. Straincontrolled spinsplitting in the persistent spin helix state of twodimensional SnSe monolayer. Phys. E Low. Dimens. Syst. Nanostruct. 124, 114372 (2020).
Jana, M. K. et al. Resolving rotational stacking disorder and electronic level alignment in a 2D oligothiophenebased lead iodide perovskite. Chem. Mater. 31, 8523–8532 (2019).
Golze, D., Dvorak, M. & Rinke, P. The GW compendium: a practical guide to theoretical photoemission spectroscopy. Front. Chem. 7, 377 (2019).
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 8207–8215 (2003).
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: “Hybrid functionals based on a screened Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 124, 219906 (2006).
Acknowledgements
All authors acknowledge funding from the Center for Hybrid OrganicInorganic Semiconductors for Energy (CHOISE), an Energy Frontier Research Center funded by the Office of Basic Energy Sciences, Office of Science within the U.S. Department of Energy (DOE) through contract number DEAC3608G028308. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paidup, irrevocable, worldwide license to publish or reproduce the published form of this work or allow others to do so, for U.S. Government purposes. R.Z. was supported by the National Science Foundation under Award Number DMR1729297. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DEAC0206CH11357. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy (DOE) Office of Science User Facility operated under Contract No. DEAC0205CH11231. Authors also thank Tianyang Li for useful discussions.
Author information
Authors and Affiliations
Contributions
M.K.J. and R.S. contributed equally to the work. M.K.J., R.S., P.S., V.B., and D.B.M. conceived the idea and designed the work. M.K.J. led and coordinated the work. M.K.J., Y.X., and D.B.M. carried out the synthesis and crystallographic investigations of chiral MHPs. R.S., R.Z., and V.B. carried out first principles DFT calculations. P.S. carried out the analysis of spinsplitting using multiband k.p theory and the theory of invariants. All authors analyzed the results and jointly prepared the paper.
Corresponding authors
Ethics declarations
Competing interests
V.B. is a member of the executive board of MS1P e.V., the nonprofit which licenses the FHIaims electronic structure code used in this work. V.B. does not receive any financial gains from this position. The remaining authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Ron Naaman and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jana, M.K., Song, R., Xie, Y. et al. Structural descriptor for enhanced spinsplitting in 2D hybrid perovskites. Nat Commun 12, 4982 (2021). https://doi.org/10.1038/s41467021251497
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021251497
This article is cited by

Revealing the impact of organic spacers and cavity cations on quasi2D perovskites via computational simulations
Scientific Reports (2023)

Mixedcation chiral perovskites displaying warmwhite circularly polarized luminescence
Science China Chemistry (2023)

Mixed perovskites (2D/3D)based solar cells: a review on crystallization and surface modification for enhanced efficiency and stability
Advanced Composites and Hybrid Materials (2023)

Elucidating the origin of chiroptical activity in chiral 2D perovskites through nanoconfined growth
Nature Communications (2022)

The chemistry and physics of organic—inorganic hybrid perovskite quantum wells
Science China Chemistry (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.