Origin of Rashba Spin-Splitting and Strain Tunability in Ferroelectric Bulk

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Origin of Rashba Spin Splitting and Strain Tunability in Ferroelectric Bulk CsPbF3 Preeti Bhumla,* Deepika Gill, Sajjan Sheoran, and Saswata Bhattacharya*

Cite This: J. Phys. Chem. Lett. 2021, 12, 9539−9546 Read Online

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ABSTRACT: Spin−orbit coupling (SOC) in conjunction with broken inversion symmetry acts as a key ingredient for several intriguing quantum phenomena, viz., Rashba−Dresselhaus (RD) effect. The coexistence of spontaneous polarization and the RD effect in ferroelectric (FE) materials enables the electrical control of spin degrees of freedom. Here, we explore the fi FE lead halide perovskite CsPbF3 as a potential candidate in the eld of spintronics by employing state-of-the-art first-principles-based methodologies, viz., density functional theory (DFT) with semilocal and hybrid functional (HSE06) combined with SOC and many-body perturbation theory (G0W0). For a deeper understanding of the observed spin splitting, the spin textures are analyzed using the k.p model Hamiltonian. We find there is no out-of-plane spin component indicating that the Rashba splitting dominates over Dresselhaus splitting. We also observe that the strength of Rashba spin splitting can be substantially tuned on application of uniaxial strain (±5%). More interestingly, we notice reversible spin textures by switching the FE polarization in CsPbF3 perovskite, making it potent for perovskite-based spintronic applications.

ead halide perovskites have evolved as an excellent choice studied as promising functional materials for spin-based − in optoelectronics owing to their exotic properties, viz., applications.29 33 Recently, a significant Rashba effect has L ffi suitable optical band gap, high absorption coe cient, low trap been reported in the tetragonal phase of MAPbI3 (MA = 1−9 + 34,35 ff density, and reasonable manufacturing cost. These alluring CH3NH3 ) due to the rotations of the MA ion. The e ect materials exhibit great applications as absorbers for high power has also been predicted in CsPbBr and MAPbBr perov- − 3 3 conversion efficiency solar cells.10 12 In the past few decades, skites.36,37 Lately, there are several reports on a ferroelectric- ff ff β β consistent research e orts have led to the revolutionary growth coupled Rashba e ect in halide perovskites (viz. -MAPbI3, - ffi 13,14 + of power conversion e ciency to over 25%. Additionally, MASnI3,ortho-MASnBr3,FASnI3 (FA = HC(NH2)2 )), the large spin−orbit coupling (SOC) tied to the presence of opening new possibilities for perovskite-based spin devices.29,38 the heavy element Pb plays a decisive role in determining the However, because of the volatile nature of organic molecules, electronic properties of lead halide perovskites.15,16 Over the these perovskites suffer from poor stability toward heat and years, the influence of SOC is well documented on the band moisture.39 Downloaded via INDIAN INST OF TECH DELHI- IIT on October 1, 2021 at 18:24:37 (UTC). structures of these perovskites.16 Intriguingly, SOC in Therefore, in our present work, we intend to explore See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. conjunction with broken inversion symmetry acts as a key inorganic FE perovskites in terms of RD splitting. Note that ingredient for several exotic phenomena such as persistent spin the interplay between ferroelectricity and the Rashba effect textures,17,18 topological surface states,19 and Rashba− ff − gives rise to electric control of the bulk Rashba e ect, leading Dresselhaus (RD) effects.20 24 to an exciting class of ferroelectric Rashba semiconductors − In the absence of the inversion symmetry, the crystal feels an (FERSCs).40 42 This effect can be theoretically quantified by effective magnetic field due to SOC. This field coupled with employing the k.p perturbation theory. According to this spin moment leads to the momentum-dependent splitting of theory, the lowest-order RD Hamiltonian is given by43,44 bands, known as Rashba−Dresselhaus (RD) splitting. 25 26 ff Dresselhaus and Rashba e ects were originally reported HRD()()()k =−+−ασ Rxykk σ yx ασ D xx kk σ yy (1) in zinc-blende and wurtzite structures, respectively. The primary difference between both these effects resides in the origin of the non-centrosymmetry. In the Rashba effect, there Received: August 8, 2021 is site inversion asymmetry, whereas in the Dresselhaus effect, Accepted: September 23, 2021 bulk inversion asymmetry is present. During the past decade, the RD effect has been the subject of intense research due to its potential applications in the emergent field of spintronics.27,28 In view of this, lead halide perovskites have been

Figure 1. (a) Optimized crystal structure of CsPbF3 in the rhombohedral R3c phase. Cs, Pb, and F atoms are indicated by red, gray, and yellow colors, respectively. (b) The first hexagonal Brillouin zone showing the high symmetry path for band structure calculations in the R3c phase of CsPbF3. Electronic band structure of CsPbF3 for the R3c phase, calculated using (c) PBE and (d) PBE+SOC. The conduction and valence bands considered in the discussion are indicated by an orange color. (e) Schematic representation of bands showing Rashba splitting. (f) Splitting of conduction band minimum (CBm) and valence band maximum (VBM) of the chosen bands along the M-Γ-K path. The inset shows the enlarged view of CBm. (g) Projected density of states (pDOS) in the R3c phase of CsPbF3 calculated using HSE06+SOC. The Fermi energy is set to zero in the energy axis. σ σ Here, x and y are Pauli spin matrices, ki is the crystal using the climbing image nudged elastic band (CINEB) α α momentum (i = x, y, z), and R and D represent the Rashba method. ̅ and Dresselhaus coefficients, respectively. On solving the free- CsPbF3 mainly manifests in the cubic (Pm3m)and electron Hamiltonian with these terms, we get two spin-split rhombohedral (R3c) phases.52,53 The Pm3̅m phase is polarized states having opposite spin polarization. Though centrosymmetric, i.e., contains an inversion center. On the Rashba and Dresselhaus exhibit a similar type of splitting, other hand, the rhombohedral R3c phase (see Figure 1a) is projection of spin in momentum space, i.e., spin texture, gives non-centrosymmetric and exhibits FE behavior due to the 53 the idea of the nature of the splitting. Striving toward sizable distortion of cations away from anionic polyhedra. The calculated change in FE polarization of the rhombohedral Rashba-type splitting, CsPbF3 perovskite may be a desirable candidate. To the best of our knowledge, the available (R3c) phase relative to the centrosymmetric structure is 34 μ 2 literature does not encompass a detailed quantitative study C/cm along the [0001] direction in a hexagonal setting of the RD effect in this material. The presence of Pb and (along the [111] direction in a rhombohedral setting). The ferroelectricity in the non-centrosymmetric phase of CsPbF details of optimized crystal structures are provided in section I 3 of Supporting Information. Next, we have investigated the indicates the possibility of the RD effect in this material. electronic band structures of the Pm3̅m and R3c phases. The Motivated by this idea, we have undertaken a theoretical band structure of the Pm3̅m phase in the presence of SOC study based on the perturbative k.p formalism and backed by reveals that there is no momentum-dependent splitting owing first-principles calculations. In the present work, we have ̅ to its centrosymmetric structure (band structure and pDOS of studied the Pm3m and R3c phases of CsPbF3. First, we have the Pm3̅m phase are given in section II of Supporting examined the electronic band structures in the above- Information). In addition, the cubic Pm3̅m phase is not mentioned phases. After that, we have estimated the band dynamically stable (see section III for phonon band gap of these phases using first-principles-based approaches 45,46 structures). Therefore, we have explored RD splitting in the combined with SOC, viz., density functional theory (DFT) latter phase (i.e., R3c). We have plotted the band structure of ϵ 47 with semilocal exchange-correlation ( xc) functional (PBE ), 48,49 50,51 the FE R3c phase along the high symmetry path using a hybrid DFT with HSE06, and single-shot GW (G0W0) hexagonal setting (as shown in Figure 1b). First, we have under the many-body perturbation theory (MBPT). Sub- performed non-spin-polarized calculations. After that, we have sequently, we have analyzed the electronic band structure of considered SOC in the calculation of the electronic band the R3c phase in terms of Rashba splitting under the combined structure. Figure 1c,d shows the electronic band structures framework of DFT and perturbative k.p formalism. We have calculated using PBE and PBE+SOC, respectively. A direct also investigated the effect of strain on the electronic band gap band gap of 3.26 eV is observed without SOC, whereas upon and Rashba parameters of the R3c phase. Finally, we have including SOC, the band gap is reduced to 2.42 eV (indirect) determined the minimum energy pathway of the FE transition around the Γ point. This change in the band gap is attributed

9540 https://doi.org/10.1021/acs.jpclett.1c02596 J. Phys. Chem. Lett. 2021, 12, 9539−9546 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter to SOC arising from the presence of Pb-6p orbitals in the consequence, the CBm and VBM are located slightly off the Γ conduction band (see pDOS in Figure 1g). Hence, SOC is point. The CBm shifts from Γ toward M and K by 0.075 Å−1 indispensable and duly considered in all further calculations. and 0.061 Å−1, respectively. The VBM shifts from Γ toward M ϵ −1 −1 Note that the PBE xc functional is well-known to under- and K by 0.022 Å and 0.012 Å , respectively (see Figure 1f). estimate the band gap due to its inability to capture the This shift from the Γ point in either direction is known as electron’s self-interaction error. Therefore, we have also offset momentum (δk) (see Figure 1e). The difference of ϵ Γ calculated the band gap with the hybrid xc functional energies at the and extremum point is known as the Rashba HSE06 (with SOC), which comes out to be 3.57 eV. To spin-splitting energy (δE). date, there is no theoretical or experimental report on the band To grasp the overall nature of splitting, the constant energy − gap of the R3c phase. In view of this, for a better estimation of 2D contour plots of spin texture are plotted in the kx ky Γ the band gap, we have employed G0W0 approximation on top plane centered at the point (see Figure 3a) for schematic of HSE06+SOC orbitals, which results in a band gap of 5.01 representation of spin textures). Figure 3b,c shows x, y, and z eV (see Figure 5a). Subsequently, we have also plotted the components of spin at constant energies around CBm and imaginary part (Im(ε)) of the dielectric function with the VBM, respectively. As we can see from spin textures of CBm converged k-grid (for k-grid convergence, see section IV in and VBM, the in-plane spin components (Sx and Sy) are Supporting Information) for the R3c phase of CsPbF3 using perpendicular to the crystal momentum, and the out-of-plane HSE06+SOC and G0W0@HSE06+SOC (see Figure 5b). The component (Sz) is completely absent. This results in the helical first peak of Im(ε) corresponds to the electronic band gap, shape of spin texture with inner and outer bands having an which is the same as we get from band structure calculations. opposite orientation, which confirms the existence of dominant For the imaginary part (Im(ε)) of the dielectric function of the Rashba splitting. In order to have a quantitative study of the Pm3̅m phase, see section V in Supporting Information. The RD effect, we have considered the k.p Hamiltonian. The model values of band gaps are listed in Table 1. It is worth noting that Hamiltonian of the R3c structure possessing C3v point group symmetry near the Γ point in the presence of SOC can be − Table 1. Band Gap (in eV) of the Pm3̅m and R3c Phases written as56 58 Using Different ϵxc Functionals HΓ()kk=++Hkk0 () ασyx βσ xy PBE G0W0@ structure PBE +SOC HSE06 HSE06+SOC HSE06+SOC 33 2 2 +[γσzx()3()kk + y − kkkk xyyx + ] (2) Pm3̅m 2.62 1.69 3.6154,55 2.71 4.38 k R3c 3.26 2.42 4.34 3.57 5.01 Here H0( ) is the Hamiltonian of the free electrons with k α β ffi eigenvalues E0( ). and are the coe cients of the linear term, and γ is the coefficient of the higher-order term in the HSE06+SOC/G0W0@HSE06+SOC only enhances the band gap without any notable change in the nature of the band Hamiltonian. Since we have not noticed an out-of-plane spin structure and the strength of Rashba splitting.27,34 To confirm component in the spin texture, we have neglected the higher- this, we have compared the band structures of the Pm3̅m phase order terms in the Hamiltonian (see section VII in Supporting using PBE+SOC, HSE06+SOC, and G W @HSE06+SOC and Information for more details). Hence, the Hamiltonian for the 0 0 Γ have found that the band profile remains the same (see section point (considering only linear terms) can now be written as VI in Supporting Information for details). Therefore, in view of ασ βσ the computational cost, PBE+SOC is considered to compute HΓ()kHkk=++=00 ()xy k yx Hk () RD parameters. +−+−ασRD()()xykk σ yx ασ xx kk σ yy (3) The energy eigenvalues corresponding to this Hamiltonian is 22 ℏ22 ℏ k kx y * * ff given as Ek0()=+* * and mx , my are the e ective 22mx my masses in the x and y directions, respectively. Rashba and ffi fi α α − β α Dresselhaus coe cients are de ned as R =( )/2 and D =(α + β)/2, respectively.59 On diagonalizing eq 3, we get energy eigenvalues as

22 22 Ek±()=± Ekoxy () αβ k + k (4) The band structures obtained from DFT and the model Hamiltonian around the Γ point are shown with dashed lines Figure 2. (a) Band structure of the R3c phase using G0W0@ HSE06+SOC. (b) Imaginary part (Im(ε)) of the dielectric function and dots in Figure 3, panels d and e, respectively. The good ﬁ for the R3c phase of CsPbF3 calculated using HSE06+SOC and agreement between both band structures justi es the reliability G0W0@HSE06+SOC. of the chosen Hamiltonian. The DFT calculations for CBm give δE = 43.5 meV, δk = 0.075 Å−1 along the Γ-M direction, where k = 0. Hence, the value of α = 2δE = 1.16 eV Å. Along Inclusion of SOC leads to the splitting of bands in the kz = y δk π/c plane, which is perpendicular to the direction of Γ δ δ the -K direction, where kx = ky, E = 43.5 meV, k = 0.061 polarization (see Figure 1d). On the contrary, splitting is Å−1, and the calculated value of αβ22+ = 2δE = 1.43 eV Å. completely absent along Γ-A, i.e., in a direction parallel to the δk polarization axis, indicating that the momentum-dependent On putting the value of α, we get β = −0.84 eV Å, which Γ α α splitting around the point is the Rashba-type splitting. As a results in R = 1.00 eV Å and D = 0.16 eV Å. Similarly, for

9541 https://doi.org/10.1021/acs.jpclett.1c02596 J. Phys. Chem. Lett. 2021, 12, 9539−9546 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter

Figure 3. (a) Schematic representation of spin textures in Rashba splitting. Spin-projected constant energy contour plots of spin texture calculated − Γ in the kx ky plane centered at the point. The upper and lower panels represent the spin textures calculated at a constant energy: (b) E = EF + 2.5 − Γ eV and (c) E = EF 0.2 eV, respectively. Electronic band structures showing spin splitting of (d) CBm and (e) VBM around the point, respectively. Band structure is plotted along the (2π 0.25, 0, 0) − (0, 0, 0) − (22ππ0.16, 0.16, 0) direction of momentum space, which is the M-Γ-K a ab direction. DFT and k.p band structures are plotted with dashed lines and dots, respectively. In the color scale, red depicts spin-up, while blue depicts spin-down states.

VBM, δE = 3.4 meV and δk = 0.022 Å−1 along the Γ-M calculations. The calculated values of RD parameters also 2δE fi Γ direction. So, the value of α = = 0.31 eV Å. Along the Γ-K con rm that the splitting around the point is mainly δk ff δ δ −1 dominated by the Rashba e ect. direction, E = 3.4 meV and k = 0.012 Å . Therefore, After thorough analysis of RD spin splitting, we have αβ22+ = 2δE = 0.57 eV Å, which gives β = −0.48 eV Å. investigated the effect of strain on the band structure of the δk α β α α R3c phase. For this, we have applied uniaxial strain in the z- Using and parameters, the estimated values of R and D α direction, which is defined as are 0.40 eV Å and D = 0.09 eV Å, respectively. The values are summed up in Table 2. As we can see from the values of RD cc− x = 0 × 100% c (5) Table 2. Rashba Parameters for Band Splitting at the Γ 0 Point in the R3c Phase where c0 is the equilibrium lattice constant, and c is the strained −1 −1 − position δE (meV) δkΓ− (Å ) δkΓ− (Å ) α (eV Å) α (eV Å) lattice constant. The lattice vector c is varied from 5% to M K R D “ ” “−” CBm 43.5 0.075 0.061 1.05 0.15 +5%, where + and are used to denote tensile and VBM 3.4 0.022 0.012 0.41 0.08 compressive strain, respectively. After optimizing the structures at a given strain, we have plotted the band structures along a high symmetry path. Figure 4a shows the band structure of the parameters, the Rashba effect dominates in the conduction R3c phase along the Γ-M-K-Γ-A-L path, under a uniaxial strain band. The origin of a large Rashba effect in CBm can be of ±5%. To clearly examine the shift, we have also plotted the attributed to the stronger SOC stemming from a higher band structures along M-Γ-K (see Figure 4b). Here, we have contribution of the Pb-6p orbital in the conduction band (see focused on CBm as there is large Rashba splitting in Figure 1g). This large contribution of SOC in CBm compared comparison to VBM. From the band structures in Figure 4b, to VBM is in good agreement with the previous findings.60 On we infer that strain causes a shift in momentum (δk) on either fitting the DFT band structure around the Γ point for CBm, side of the Γ point. The arrows represent the direction of shift. fi α − β α we nd = 0.90 eV Å and = 1.20 eV Å. This gives R = We have taken the equilibrium band structure (i.e., with 0% α fi 1.05 eV Å and D = 0.15 eV Å. Similarly, tting the band strain) as the reference for all the strained band structures. structure for VBM gives α = −0.33 eV Å and β = 0.49 eV Å, Interestingly, under compressive strain, the bands shift off from α α Γ and hence R = 0.41 eV Å and D = 0.08 eV Å. These values on either side toward M and K. On the contrary, under are in good agreement with the predicted values based on DFT tensile strain, the bands shift toward Γ from either side. This in

9542 https://doi.org/10.1021/acs.jpclett.1c02596 J. Phys. Chem. Lett. 2021, 12, 9539−9546 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter

± Γ Γ Figure 4. Band structures of CsPbF3 (R3c phase) under a uniaxial strain of 5%, calculated using PBE+SOC for path (a) -M-K- -A-L and (b) M- Γ δ ff δ α α -K. (c) Rashba spin-splitting energy ( E), (d) o set momentum ( k), and (e) Rashba parameters ( R and D) as a function of strain. The values are calculated for CBm in the R3c phase. Note that δE is the same along both M-Γ and Γ-K directions. turn changes δk and δE as a function of strain, and the overall In summary, using first-principles-based methodologies, viz. ff α α e ect of these parameters will change R and D accordingly. PBE, HSE06, and many-body perturbation theory (G0W0), we To quantify the effect of strain on Rashba parameters, we have have systematically studied the electronic structure of the ̅ calculated their values at a given strain within the framework of Pm3m and R3c phases of CsPbF3. Note that SOC is duly − δ δ α α DFT. Figure 4c e shows E, k, R, and D as a function of considered in all the calculations. The R3c phase being non- strain (for details, see section VIII in Supporting Information). centrosymmetric is investigated in terms of RD splitting. The From Figure 4c, we have observed that under compressive DFT calculations are well corroborated by the symmetry α fi strain R is signi cantly enhanced from 1.05 to 1.48 eV Å adapted k.p model Hamiltonian. The helical nature of in-plane making the material tunable for spintronics application. Also, spin components in spin textures indicates the Rashba-type we have seen notable change in the electronic band gap on the splitting. The calculated values of RD parameters suggest that application of strain. The band gap values are mentioned in the Rashba effect is predominant in our material, while the Figure 4b (for more quantitative details, see section IX in Dresselhaus effect has a negligible contribution. In addition, a Supporting Information). larger Rashba effect is observed in CBm than VBM of the R3c Lastly, we have explored the possibility of polarization phase owing to the larger contribution of the Pb-6p orbital in ff switching in FE CsPbF3 using strain e ects. To predict the the conduction band. Further, we have noticed that the Rashba feasibility of this phenomenon, we have analyzed the minimum δ δ α α parameters, viz., k, E, R, and D, almost linearly increase energy pathway of the FE transition using the CINEB 61,62 under compressive strain. The band gap of CsPbF3 is method. For this, we have chosen the structure significantly tuned with the application of strain. Also, we deformation path between two FE states with opposite have observed reversible spin textures via FE switching. These spontaneous polarization through a centrosymmetric reference. results indicate that FE CsPbF3 has enough potential as an As shown in Figure 5, the Eb denotes the activation barrier for excellent amenable material in the field of spintronics. the polarization switching, which comes out to be 0.75 eV. This suggests that a switchable FE polarization is plausible in COMPUTATIONAL METHODS the material. The strain tunability and reversible spin textures ■ fi owing to FE switching in CsPbF3 give electrical control of The DFT-based rst-principles calculations have been spins and provide an all-semiconductor design for spintronic performed using the Vienna ab initio simulation package devices such as in spin-field effect transistors.63,64 (VASP).65,66 The ion-electron interactions in all of the

9543 https://doi.org/10.1021/acs.jpclett.1c02596 J. Phys. Chem. Lett. 2021, 12, 9539−9546 The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letter 1 ⟨S⟩= ⟨Ψ||Ψσ ⟩ ikik2 (6) σ Ψ where i are the Pauli matrices, and k is the spinor eigenfunction obtained from noncollinear spin calculations. The spin texture is calculated with a closely spaced 12 × 12 k- grid around high symmetry points. The PyProcar code is used to calculate the constant energy contour plots of the spin texture.74

■ ASSOCIATED CONTENT *sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpclett.1c02596. Optimized structures of Pm3̅m and R3c phases; band structure and projected density of states (pDOS) of cubic Pm3̅m phase; phonon band structures of Pm3̅m and R3c phases, k-grid convergence in Pm3̅m and R3c phases; imaginary part (Im(ε)) of the dielectric function for cubic Pm3̅m phase; comparison of PBE+SOC, HSE06+SOC and G0W0@HSE06+SOC band structures of cubic Pm3̅m phase; higher order term contribution in SOC Hamiltonian of R3c phase; RD parameters as a function of strain in R3c phase; band gap as a function of strain in R3c phase (PDF) Figure 5. (a) Climbing image nudged elastic band (CINEB) calculation for the polarization switching process in CsPbF 3 ■ AUTHOR INFORMATION perovskite. Two ferroelectric (FE) structures in the ground state with the opposite directions of electric polarization are shown. Eb is Corresponding Authors the activation barrier energy for the polarization switching process. Preeti Bhumla − Department of Physics, Indian Institute of Reversible in-plane spin textures calculated at constant energy E = EF Technology Delhi, New Delhi, India 110016; orcid.org/ + 2.5 eV with opposite spin polarization: (b) −P, (c) + P. 0000-0002-4973-2347; Email: Preeti.Bhumla@ physics.iitd.ac.in elemental constituents are described using the projector Saswata Bhattacharya − Department of Physics, Indian augmented wave (PAW)67,68 method as implemented in Institute of Technology Delhi, New Delhi, India 110016; VASP. We have considered pseudopotentials with the orcid.org/0000-0002-4145-4899; Phone: +91-11-2659 following valence states: Cs, 5s25p66s1; Pb, 6s25d106p2;F, 1359; Email: [email protected] 2s22p5. The structures are optimized using Perdew−Burke− Ernzerhof generalized gradient approximation (PBE-GGA), Authors relaxing all ions until Hellmann−Feynman forces are less than Deepika Gill − Department of Physics, Indian Institute of 0.001 eV/Å. The cutoff energy of 520 eV is used for the plane- Technology Delhi, New Delhi, India 110016 wave basis set such that the total energy calculations are Sajjan Sheoran − Department of Physics, Indian Institute of converged within 10−5 eV. The Γ-centered 6 × 6 × 6 and 9 × 9 Technology Delhi, New Delhi, India 110016; orcid.org/ × 4 k-grids are used to sample the irreducible Brillouin zones 0000-0002-6816-3944 ̅ ofthecubicphasewiththePm3m space group and Complete contact information is available at: rhombohedral phase with the R3c space group of bulk https://pubs.acs.org/10.1021/acs.jpclett.1c02596 CsPbF3, respectively. The phonon calculations are performed for 3 × 3 × 3 and 2 × 2 × 2 supercells in the Pm3̅m and R3c 69,70 Notes phases using the PHONOPY package. In order to predict The authors declare no competing financial interest. the band gap, single-shot GW (G0W0) calculations have been performed on top of the orbitals obtained from the ϵ ■ ACKNOWLEDGMENTS HSE06+SOC xc functional [G0W0@HSE06+SOC]. For this, we have used 6 × 6 × 6 and 4 × 4 × 2 k-grids in the Pm3̅m and P.B. acknowledges UGC, India, for the senior research R3c phases, respectively. The number of bands is set to four fellowship [Grant No. 1392/(CSIR-UGC NET JUNE times the number of occupied bands. The polarizability 2018)]. D.G. acknowledges UGC, India, for the senior calculations are performed on a grid of 50 frequency points. research fellowship [Grant No. 1268/(CSIR-UGC NET FE polarization is evaluated in the framework of the Berry- JUNE 2018)]. S.S. acknowledges CSIR, India, for the junior phase theory of polarization.71,72 The minimum energy research fellowship [Grant No. 09/086(1432)/2019-EMR-I]. pathways of FE transitions are determined through the S.B. acknowledges the financial support from SERB under the climbing image nudged elastic band (CINEB) method.73 core research grant (Grant No. CRG/2019/000647). We Spin textures are plotted by calculating expectation values of acknowledge the High Performance Computing (HPC) facility spin operators Si (i = x, y, z), given by at IIT Delhi for computational resources.

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