Tunable spin textures in polar antiferromagnetic hybrid organic inorganic perovskites by electric and magnetic fields

The hybrid organic inorganic perovskites (HOIPs) have attracted much attention for their potential applications as novel optoelectronic devices. Remarkably, the Rashba band splitting, together with specific spin orientations in k space (i.e., spin texture), has been found to be relevant for the optoelectronic performances. In this work, by using first principles calculations and symmetry analyses, we study the electric polarization, magnetism, and spin texture properties of the antiferromagnetic (AFM) HOIP ferroelectric TMCM_MnCl3 (TMCM = (CH3)3NCH2Cl, trimethylchloromethyl ammonium). This recently synthesized compound is a prototype of order disorder and displacement-type ferroelectric with a large piezoelectric response, high ferroelectric transition temperature, and excellent photoluminescence properties [You et al., Science 357, 306 (2017)]. The most interesting result is that the inversion symmetry breaking coupled to the spin orbit coupling gives rise to a Rashba-like band splitting and a related robust persistent spin texture (PST) and/or typical spiral spin texture, which can be manipulated by tuning the ferroelectric or, surprisingly, also by the AFM magnetic order parameter. The tunability of spin texture upon switching of AFM order parameter is largely unexplored and our findings not only provide a platform to understand the physics of AFM spin texture but also support the AFM HOIP ferroelectrics as a promising class of optoelectronic materials.


Introduction
The past few years witnessed the extremely rapid development of hybrid organic-inorganic perovskites (HOIPs), which have been shown to be promising optoelectronic material [1][2][3][4][5] . HOIP materials have several commom features, including the classical ABX perovskite architecture and the presence of organic cation that occupy the A-site. As for the B-site, it can be occupied not only by main group elements, but also by transition metal atoms such as Mn and Fe, thus introducing magnetic degrees of freedom into the compound. As for the X-site, it is usually the halogen element. The HOIP materials have some advantages and in particular, the exceptionally long carrier lifetimes make them very attractive for optoelectronic devices, such as light absorbers and light-emitting diodes [6][7][8][9][10][11] .
To further enhance the optoelectronic performances of HOIP materials, intense research has been directed to explain the microscopic origin of the long lifetimes [9][10][11] . Recently, the presence of Rashba band-splitting has been suggested to be connected with the carrier lifetimes and to improve their optoelectronic performances [12][13][14][15] . When lacking spatial inversion symmetry, the spin-orbit coupling (SOC) effect leads to an effective momentum-dependent magnetic field Ω acting on the spin , and the effective SOC Hamiltonian can be written as = Ω • [16][17] .
In this case, the SOC will split the spin degeneracy with specific spin orientations (i.e. spin texture) in the momentum k-space, as was firstly demonstrated by Rashba 18 and Dresselhaus 19 . The spin texture can often be manipulated and even reversed by switching the electric polarization under external electric field, leading to an all-electric and non-volatile control of spin state [20][21][22][23][24] . Rashba effects were mainly discussed in non-magnetic lead halide perovskites [9][10][11][12][13][14][15][24][25][26][27][28] or non-magnetic ferroelectric semiconductors [20][21][22][23][29][30][31][32][33] . However, to the best of our knowledge, there are no studies on the spin texture in AFM HOIP ferroelectrics. Furthermore, antiferromagnets are very appealing for spintronic applications due to their superior properties, since they produce no stray fields and display intrinsic ultrafast spin dynamics [34][35][36] . In the last few years, intense theoretical and experimental research has shown that it is possible to realize electric-field control of magnetism in multiferroic materials [37][38][39] . The couplings between polarization, magnetism, and spin textures are still largely unexplored but they could have important applications in magneto-optoelectronic devices. Indeed, some recent reviews have pointed out intriguing spin-optotronic properties in HOIP materials [40][41] 42 . In our study, we discuss the interplay among ferroelectric and magnetic orderings, and spin textures by using density-functional theory (DFT). We show that TMCM-MnCl 3 is a prototype of order-disorder and displacement-type ferroelectric whose polarization can be greatly modified by the halogen atom substitutions. The most important result is that a Rashba-like effect in the band structure leads to robust unidirectional persistent spin texture (PST) and/or spiral spin texture 31,43 .
The spin textures have been predicted to support an extraordinarily long spin lifetime which is promising for optoelectronic devices [44][45] . By tuning the ferroelectric or, surprisingly, the antiferromagnetic order parameter, we find that the spin texture can be modified significantly. Our results indicate that not only the electric but also the magnetic field can effectively be used to manipulate the spin textures even in AFM but polar HOIP materials such as TMCM-MnCl 3 . Our results suggest that AFM HOIP ferroelectric is an interesting class of materials which deserves further study.

Results
Structural properties. At [49][50][51] . Note that this dimensionless parameter λ is not the usual linear interpolation for atomic positions but it defines the correlated rotation of cations as well as the displacement of the MnCl 3 framework. Therefore, it represents the normalized amplitude of the roto-displacive path. Since the transition path is artificially assigned to act as a computational tool, the polarization with λ=1 has a real physical meaning. Here we define our convention for the coordinates as the x (y) axis being along a (b) axis, respectively. As for the z axis, it is vertical to the x-y plane and it has an angle about 5° with c axis. In TMCM- properties, but they differ in electronegativity, which, in turn, will effectively change the electric polarization through hydrogen bond network that is responsible for the complex cations and framework interaction. By changing the halogen atoms in the inorganic framework and/or organic cations, we find that the polarization can be significantly modified (see Fig. S2-S3 of SM).
As for the magnetic ground state, we performed collinear calculations showing that TMCM-MnCl 3 has strong AFM interaction within the inorganic MnCl 3 chains. This can be understood in terms of Goodenough-Kanamori (GK) rule which predicts a strong AFM super-exchange interaction between two half-filled (3d 5 ) ions [53][54] . However, the interchain interaction between the inorganic MnCl 3 chains is weak AFM since the distance between neighboring chains is large (more than 9 Å). The energy of different magnetic configurations is shown in Fig. S4  Therefore, considering the tunable ferroelectric and magnetic states, the TMCM-MnCl 3 system provides an ideal platform to investigate the interplay between ferroelectric ordering, magnetic ordering, and spin textures.
Band structure properties. We investigate the electronic properties of valence band maximum (VBM) and conduction band minimum (CBM) by calculating the band structures with/without SOC (see Fig. 2a-2d). Here the conventional cell containing four organic cations and four Mn ions (see Fig. 1). When considering SOC, the spin moment is set to be along y axis. The band structures of G-type AFM and C-type AFM states are shown in Fig. 2b and 2d, respectively. To help understand the spin textures discussed in the following paragraphs, we choose a specific symmetric k-path containing and ,which is perpendicular to the polarization (see Fig. 2e and 2f) 23,[31][32]57 . Here and denote the path from Γ (0,0,0) to Y (0,0.5,0) and Q (0.5,0,0.5), respectively. In order to simplify the illustration of Brillouin zone, we simplify the crystal lattice from slightly monoclinic to orthorhombic (see Fig. 2f). For G-type AFM state (see Fig. 2a), our calculations show that the VBM and CBM are located at Γ point, and the partial density of states (DOS) show that the valence band edge contains contributions from Mn-3d and Cl-2p orbitals, whereas the conduction band edge is mainly composed of Mn-3d orbitals (see Fig. S5). Due to the symmetry (see below for detailed analysis), all eigenstates are at least two-fold degenerate (i.e., spin-up and spin-down states). When taking SOC into account, the Rashba-Dresselhaus effect removes the spin degeneracy into singlets along the symmetry path but it still keeps two-fold degeneracy at Γ point (see Fig. 2b). Interestingly, for the C-type AFM state, the doublet at Γ point splits into two singlets with a sizable spin splitting at VBM about 0.027 eV after inclusion of SOC (see Fig. 2d).
To understand the band degeneracy at Γ point, we perform the symmetry analysis by  Fig. 2b and we use the subscript of (i.e., b) to index the band structure. Using the properties of half-spin system at This represents a new degree of freedom to play with in the spin-texture tuning, which has been very little studied in the literature. The SOC splits the band structure into two branches, which exhibit similar spin textures but with opposite helicity or orientation. Here, we will focus on the inner branches near Γ point, while the spin textures of the outer branches are illustrated in Fig.   S7-S17 of SM. In order to simplify the visualization, we project the spin textures on a specific plane which is perpendicular to the polarization (see Fig. 2f) 23,[31][32]57 .
In the following, we discuss the spin textures in G-type AFM state. We pay attention to the spin texture at CBM, since the spin value at VBM is small due to the weak band splitting. It is useful to introduce the AFM order parameter defined as = ∑ − ∑ , where ( ) is the spin moment along positive (negative) axis, respectively. We use the subscript of L to define different AFM state. For example, ~ y indicates the G-type AFM configuration along y direction. And we use ~ -y to indicate the operation that flip the spin from y to -y direction.
The polarization (P) is along [101 ] direction while -P is along [1 01] direction. As we can see in Here we discuss the interplay between ferroelectric ordering, magnetic ordering, and spin texture. In Fig. 3b, we fix the magnetic ordering but reverse the ferroelectric polarization from P In Fig. 4c and 4d, we fix the magnetic ordering but flip the ferroelectric polarization from P After the space inversion, it can be (− ) = ( ). Thus, the helicity of spiral spin texture is reversed (see Fig. 4d). It is interesting about the different tunability of VBM and CBM under same external field. It is also important to note that this compound has been recently synthesized and the switching of polarization has been realized with a well-defined P-E loop 42 , therefore we expect that the manipulation of spin textures by the external electric field could be easily verified by experiments. In Fig. 4e and 4f, we fix the ferroelectric order but flip the AFM ordering from ~ y to -y to see the variation of spin texture. We find the PST of VBM (see We also investigate the spin textures with other magnetic configurations (see Fig. S7-S17 of SM). By manipulating the magnetic order parameter with different orientation and different magnetic state, the corresponding spin-texture will change accordingly and it is the origin of magneto-crystalline anisotropy 59 . This property is dual of the spin-texture electric-anisotropy first discussed in the HOIP material (NH 2 CHNH 2 )SnI 3 60 where it has been shown that the shown that AFM materials can be manipulated by applying magnetic fields 35,[62][63] . The magnetic moments can be appreciably rotated in a quasi-static manner within the Stoner-Wohlfarth model 64 .
In this picture, the ordered magnetic state is preserved when the magnetization is reversed and a spin-flop field can rotate the magnetic moments by 90° 65 . Besides, the AFM state could be reoriented by optical excitation [66][67] , exchange bias [68][69] , strain [70][71] , and other different approaches 35,63 . We note that the manipulation of ferroelectric polarization and magnetic configuration was realized in the classical multiferroic material TbMnO 38 and BiFeO 39 .
Therefore, the ferroelectric and magnetic orderings in a polar AFM HOIP system could be tuned and the spin texture can be manipulated at the same time, thus leading to interesting magneto-optoelectronic applications.
Our study shows the possibility of tuning spin textures by electric and magnetic fields in AFM HOIP ferroelectrics and enhancing its optoelectronic performance, although there remain some challenges such as the wide bandgap and low magnetic ordering temperatures. In our TMCM-

Conclusions
In this work, we propose the manipulations of spin textures in the AFM HOIP ferroelectric TMCM-MnCl 3 . By using first-principles calculations, we identify a Rashba-like splitting in the band structure. The symmetry analyses based on magnetic space group are used to explain the band degeneracy. We find robust PST in G-type AFM state and it can be effectively manipulated by switching not only polarization but also magnetic ordering. We also find the coexistence of PST and typical spiral spin texture, depending of the relevant band electronic states, in C-type AFM state. To the best of our knowledge, this is the first case of coexistence of PST and spiral spin texture in the same compound. By manipulating the ferroelectric, and, interestingly, the magnetic order parameter, the spin texture can be modified significantly. Our work introduces new directions in the field of spin-texture manipulation by external fields, which goes beyond the usual electric-field control of Rashba effect in non-magnetic materials. Considering that TMCM-MnCl 3 belongs to the important class of HOIPs, which is relevant to optoelectronic research, we expect that, this study could suggest new magneto-optoelectronic properties in HOIPs. Since the switching of polarization in TMCM-MnCl 3 has been experimentally demonstrated 42 , we hope to stimulate new experiments to verify manipulations of spin-textures in TMCM-MnCl 3 by electric and/or magnetic fields. We expect that AFM HOIP ferroelectrics have the potential to improve the optoelectronic performance and give a new strategy to design new multifunctional materials. k-point mesh is used for the Brillouin integration. The electric polarization is calculated by using the Berry phase method 48,78 . In this approach, we first define a centrosymmetric reference phase which shows an antiferroelectric (AFE) alignment of dipoles in the unit cell and then we continuously rotate and translate the organic cations to reach the ferroelectric (FE) phase by defining a roto-displacive path in the configuration space. In our work, we take the Van der Waals interactions into account by DFT-D3 correction method [79][80] as implemented in the VASP software.