Spin-Orbital Coupling in All-Inorganic Metal-Halide Perovskites: the Hidden Force that Matters

Highlighted with improved long-term thermal and environmental stability, all-inorganic metal halide perovskites exhibit tunable physical properties, cost-effective synthesis, and satisfactory optoelectronic performance, attracting increasing research interests worldwide. However, a less explored feature of these materials is their strong spin-orbit coupling (SOC), which is the hidden force influencing not only band structure but also properties including magnetoresistance, spin lifetime and singlet-triplet splitting. This review provides an overview of the fundamental aspects and the latest progress of the SOC and debate regarding Rashba effects in all-inorganic metal halide perovskites, providing critical insights into the physical phenomena and potential applications. Meanwhile, crystal structures and photophysics of all-inorganic perovskite are discussed in the context of SOC, along with the related experimental and characterization techniques. Furthermore, a recent understanding of the band topology in the all-inorganic halide perovskites is introduced to push the boundary even further for the novel applications of all-inorganic halide perovskites. Finally, an outlook is given on the potential directions of breakthroughs via leveraging the SOC in halide perovskites.


Introduction: 1.1 Basic physics of SOC and Rashba effects
Spin-orbit coupling (SOC) was first reported in semiconductors with zinc-blende structures. 1 Even though these materials are non-magnetic, they exhibit spin splitting in the conduction band (CB) with no centre of inversion.Paramagnetic semiconductors may exhibit spin-related interactions, such as the spin Hall effect, which was theoretically proposed by D'yaknov and Perel in 1971 and later experimentally proved by Kato et al.. 2,3 The SOC phenomenon is important for the spin Hall effect due to the impurity scattering, which leads to spin accumulation at the edges. 3In semiconductors, an electron with momentum p in an electric field (E) experiences a magnetic field (Beff) in its rest frame of reference: where m is the mass of an electron and c is the speed of light.
In relativistic framework, the SOC manipulates the orbital motion of the electron with the Hamiltonian matrix: In this equation , where μB is the Bohr magneton, m is the electron mass, and c is the speed of light.SOC arises due to the interaction between an electron's intrinsic spin angular momentum (σ) and its orbital angular momentum, which is related to its motion (p) in the presence of an electric field (E).The SOC leads to the momentum-dependent splitting of energy bands and has the unique ability to control the transport properties of semiconductors without any external fields. 4,5 the presence of atomic nuclei or other charged particles, 6 the Hamiltonian of SOC can be written as, where V0 is the Coulomb potential, and σ = (σx, σy, σz) is the vector of Pauli spin matrices.In halide perovskites, SOC becomes significant for electrons near heavy nuclei like Pb, where it depends linearly on ∇V0 and p. 7 The Rashba effect results from the splitting of spin energy levels due to broken inversion symmetry and SOC, specifically stemming from structural inversion asymmetry. 8,9In the case of materials like halide perovskites with even structural symmetry, the breaking of inversion symmetry may occur due to ionic impurities that induce local electric fields. 8The Rashba energy splitting Δ Rashba caused by an electric field (Ê ) is represented by: 10 where αR is the Rashba parameter and ẑ is the unit vector along the z-axis.Clearly, the Rashba effect influences the energy level splitting of charge carriers when an electric field is present. 11,12milar to the Rashba effect, the Dresselhaus effect also results in spin splitting, but with a different spin texture. 7,13Dresselhaus studied this phenomenon in crystals lacking a centre of inversion symmetry, exhibiting bulk inversion asymmetry (BIA). 14SOC is responsible for lifting the spin degeneracy of electronic bulk states in such crystals.The Hamiltonian describing Dresselhaus splitting is represented as follows: 15   = [    (  2 −   2 ) +     (  2 −   2 ) +     (  2 −   2 )]. ( where D is the Dresselhaus coefficient, σx, σy, σz are Pauli matrices to represent spin states of the electron, and   ,   ,   are the components of the wavevector ( ) representing the momentum of charge carriers in the crystal.

SOC in semiconductors: from silicon to perovskites
Semiconductor technology has revolutionized the modern microelectronics and laid the foundation for advanced computing and memory devices. 16,17Semiconductors with appropriate bandgaps have unique properties of transforming light into electric current and vice versa, enabling a wide range of solid-state optoelectronic applications. 18,19The bandgap of semiconductors, namely the energy difference between valence band (VB) and CB edges, directly determines their optical properties, and accordingly, semiconductors are classified into direct or indirect ones.Intrinsic silicon is a prototypical indirect semiconductor with a bandgap of 1.1 eV.As shown in Fig. 1(a), the minimum of the CB, X1 does not coincide with the maximum of the VB, Γ. 20 As a result, photo-excited transition across the bandgap must involve interactions with phonons to keep the total energy and momentum conserved. 21ditionally, the spin degree of freedom plays a crucial role in semiconductor behaviour.
SOC, which couples an electron's spin with its motion, impacts various aspects of semiconductor physics. 22Silicon, with its relatively low atomic number, exhibits weaker SOC compared to elements with higher atomic numbers. 23][34][35][36] Halide perovskites have Rashba-type SOC, 8 which leads to unique spin-related effects and profoundly modifies their properties. 37,38In general, the electronic behaviours of semiconductor materials are largely determined by the extrema points of their CB and VB.The electron dispersion relation, often approximated through the effective-mass approach, follows (E(k) = ħ²k² / 2m*), as shown in Fig. 1(c)(i). 39However, the introduction of SOC in noncentrosymmetric compounds brings about an intriguing departure from this simplification.
Specifically, in the presence of structural "inversion asymmetry," the previously spindegenerate parabolic band structure splits into two distinct spin-polarized bands.The dispersion relationship for electrons and holes in these materials takes on the form, E ± (k) = (ħ²k² / 2m*) ± αR|k|, where the parameter "αR" signifies the Rashba splitting.This modification engenders the appearance of new extremal points characterized by a momentum offset (k0) and a corresponding energy separation (ER), linked by the relationship αR = 2ER / k0, as portrayed in Fig. 1

(c)(ii).
A remarkable outcome of this phenomenon is the emergence of two separate Rashba-split branches, each possessing a distinctive spin orientation. 40perimental evidence suggested the presence of Rashba spin splitting in 3D halide perovskites, exemplified by MAPbI3. 82][43] The role of symmetry is pivotal in understanding these effects in perovskite materials.In 2D hybrid perovskites, Maurer et al.
established a connection between lattice distortions and SOC, claiming notable Rashba-Dresselhaus splitting. 7In-plane Pb displacement is claimed to be responsible for the Rashba effect, and specifically, diagonal Pb displacement favours a pure Rashba splitting, while pure Dresselhaus splitting is not observed. 7In another recent theoretical study the existence of the Rashba-Dresselhaus effect was confirmed in 2D halide perovskites. 446][47][48] This effect is accompanied by a phase change in MAPbI3 from I4/mcm to Imm2.The carrier lifetime rapidly falls with pressure as the bandgap becomes more direct, and the photoluminescence quantum yield (PLQY) doubles. 49[52] In 2016, Niesner et al. utilized angle-resolved photoelectron spectroscopy (ARPES) to make a groundbreaking discovery of a massive Rashba splitting in MAPbBr3. 53Their findings confirmed the stronger SOC in the CB than the VB, facilitating direct and isotropic optical transitions.This initial report sparked considerable interest in the scientific community.In 2020, Sajedi et al. used ARPES to re-examine MAPbBr3, but their result did not support the previous findings of Rashba splitting at the VB maximum. 50In a report by Becker et al., magneto-optical photoluminescence (MPL) measurements were used to unveil the Rashba effect in the all-inorganic halide perovskite CsPbBr3 nanocrystals (NCs). 54This Rashba effect is believed to originate from lattice distortions induced by the motion of Cs + ions or surfacerelated influences in CsPbBr3 NCs.Importantly, this Rashba effect induces singlet and triplet energy level splitting without the presence of an external magnetic field.2][43] Their results suggested that the previously observed energy level splitting in weakly confined NCs might predominantly arise from exchange interactions rather than the Rashba effect.In short, more comprehensive experimental efforts on carefully selected samples under well-controlled conditions are warranted to unravel the existence and characteristics of the Rashba effect in halide perovskites.

Effect of SOC on the electronic properties of perovskites
Fig. 1(d) shows the band structure of a prototypical cubic all-inorganic perovskite, CsPbBr3, calculated using the Vienna Ab-initio Simulation Package (VASP). 54The electronic bandgap is located at the R point in the Brillouin zone.The VB maximum stems largely from the antibonding hybridization of Pb 6s and Br np orbitals, while the CB maximum mainly originates from the Pb 6p orbital, with some contribution from the Br np orbitals. 55For halide perovskites, the presence of Pb leads to giant SOC, which is believed to contribute to the low carrier recombination rate. 567][58] It is a general consensus that defects do not significantly influence the recombination rate and carrier lifetime, rendering perovskites highly defect-tolerant.
In hybrid perovskites, B-site ions are known to determine the band structure and influence electronic and optical properties, but A-site ions also introduce a significant impact. 59ecifically, the MA molecule induces the correlation between spin texture and structural deformations.Interestingly, this phenomenon is not static; the orientation of the MA molecules dynamically changes at room temperature, causing variations in the spin texture.Furthermore, the replacement of MA with FA or Cs decreases the presence of Rashba effect due to their reduced polarity. 60Charge recombination in halide perovskites is profoundly influenced by the SOC.In the early scenario depicted in Fig. 1(e), 59 SOC-induced band splitting near the R point, with opposite spin orientations, causes a mismatch between the lowest CB and the highest VB, reducing direct radiative recombination and extending carrier lifetimes.However, Zhang et al.
proposed that direct optical transitions still occur because of the structural distortions and spin texture dynamics induced by the MA molecules, as depicted in Fig. 1(f). 59In other words, the indirect bandgap feature of halide perovskites has a limited influence on the radiative charge recombination.In a recent study, Lu et al. used time-dependent DFT (TD-DFT) and nonadiabatic molecular dynamics (NA-MD) methods to investigate the non-radiative recombination in MAPbI3, both with and without SOC interactions. 61Their results demonstrated that SOC substantially slows non-radiative e-h recombination, extending the charge carrier lifetime.
It is generally believed that all-inorganic perovskite possesses a SOC effect stronger than most other semiconductors. 62,63There are some reviews on the importance and advantages of SOC in organic-inorganic hybrid perovskites. 63,64However, the impact of SOC on all-inorganic halide perovskites, such as CsPbBr3, CsSnBr3 and FAPbBr3, has not been comprehensively reviewed.
This review presents a concise introduction to SOC and its profound impact on the physical properties and optoelectronic applications of all-inorganic halide perovskites.We begin by introducing the fundamental semiconducting and optoelectronic properties of these emerging materials with strong SOC.Subsequently, we discuss the effects of SOC and the Rashba effect on spin lifetime.Moving forward, we explain the experimental techniques used to study spin relaxation and dynamics in all-inorganic halide perovskites.We also discuss how the speculated Rashba effect and quantum confinement influence the light emission in perovskite NCs through modulating the bright and dark states.Finally, we address the recent advancements in understanding the band topology of all-inorganic halide perovskites.

Basic properties of all-inorganic halide perovskites
The all-inorganic halide perovskite materials have drawn more scientific attention in photovoltaics and optoelectronics because they are more stable under ambient conditions than their organic-inorganic hybrid counterparts. 65,66CsPbX3 (X = I, Br, Cl) perovskites are one of the promising candidates among all-inorganic halide perovskites due to their stability against moisture, heat, and light.Lasers, solar cells, LEDs, and thermoelectric devices are only a few examples of the many applications based on CsPbX3. 28,54,679][70] This work is not concerned with the general properties of inorganic perovskite materials but instead focuses on the debated Rashba effect and the impact of SOC on their optoelectronic properties.effect.Reproduced with permission.Copyright ©2021 American Physical Society. 75(f) CPTA data, from which the spin relaxation time was estimated.Reproduced with permission. 76pyright © 2021 American Chemical Society.
The remarkable optoelectronic characteristics exhibited by CsPbX3 (X = Cl, Br, I) perovskites are intricately interwoven with their unique crystalline and electronic structures. 77- 80 heir crystal structure is underpinned by a fundamental parameter known as the Goldschmidt tolerance factor. 81Particularly, CsPbX3 has three distinct crystal structures, i.e., cubic, tetragonal and orthorhombic (Fig. 2(a)).The perovskite framework allows for structural tilting of the octahedra.The tilting behaviour involving halide ions situated exclusively in the equatorial plane (Xeq) leads to a tetragonal structural modification (P4/mbm).Alternatively, the tilting phenomenon can encompass both the equatorial plane and the normal axial direction (Xax), instigating an orthorhombic structural modification (Pbnm or equivalently Pnma).These structural deviations, linked to the Goldschmidt tolerance factor, underlie the unique optoelectronic properties of CsPbX3 perovskites.
3][84][85][86] Inorganic halide perovskite thin films have been extensively exploited for photovoltaic applications. 87The low-dimensional nanostructures, such as nanosheets, nanowires, and NCs, are explored because of their quantum confinement effect, which significantly boosts the exciton binding energy and enhances the PLQY for lighting applications. 88,89Due to their versatility in modifying their composition, shape, and optical properties, NCs shine out among nanostructures.NCs are typically found to have high surface-to-volume ratios and short radiative lifetimes, making them excellent candidates for catalytic and optical applications. 65,90Most notably, as demonstrated in Fig. 2(b), by adjusting the composition of Cl, Br, and I ions, the optical bandgap of CsPbX3 NCs may be easily controlled, covering the entire visible spectrum from 410 nm to 700 nm.Even under harsh circumstances such as intense X-ray excitation, the colour-tunable emission of CsPbX3 NCs can still be retained, indicating the robustness of these materials. 91

Importance of SOC in all-inorganic halide perovskites
In Fig. 2(c), the band dispersion of a generic material with strong SOC is depicted, where the presence of SOC modifies the band degeneracy, forming inner (E+) and outer (E-) branches. 73Furthermore, Rashba and Dresselhaus effects lead to different spin textures.The intriguing properties of all-inorganic halide perovskites related to the SOC effect have recently garnered substantial attention.Remarkably, despite being non-magnetic materials, these perovskites exhibit the potential for generating spin-polarized current, thereby promising for future explorations in spin-orbitronics and spintronics. 92th the breaking of inversion symmetry, the Rashba effect in perovskites can be either static or dynamic.The static Rashba effect occurs when the Pb framework deviates from its ideal position, leading to the creation of a non-centrosymmetric phase.On the other hand, the dynamic Rashba effect is related to the interaction between the A-site cation and the Pb framework.Fig. 2(d) illustrates the static and dynamic Rashba effects at various temperatures ranging from 20 K to 300 K. 93 It is acknowledged that MAPbBr3 experiences a notable static Rashba effect due to the centrosymmetry-breaking MA cations. 94In CsPbBr3, surface or interface distortions are hypothesized to result in a static Rashba splitting of approximately 26 meV at low temperatures of less than 50 K. 93Above 60 K, the polar cage of the octahedron undergoes distortion, giving rise to dynamic Rashba splitting.

However, Schlipf et al. challenged this interpretation by demonstrating that dynamic
Rashba-Dresselhaus splitting in MAPbI3 halide perovskites can only manifest in the presence of an out-of-equilibrium phonon population, such as that generated by coherent terahertz radiation illumination. 75The Rashba-Dresselhaus effect is symmetry-forbidden, but the electron-phonon interactions can induce a phonon-assisted, dynamic Rashba-Dresselhaus spin splitting.As shown in Fig. 2(e), the dynamic Rashba-Dresselhaus effect generates a pronounced peak-dip-hump structure in the photoluminescence (PL) signal of MAPbI3. 75In the case of CsPbBr3, the origin of Rashba splitting is still subject to debate due to the centrosymmetry and the relative position of the triplet state. 41,74u et al. extensively investigated the Rashba effects in CsPbBr3 and MAPbBr3 single crystals, employing various techniques. 94In particular, they unclosed the temperaturedependent behavior of the dynamic Rashba effect, which becomes pronounced at elevated temperatures. 94Intriguingly, they identified the static Rashba effect in MAPbBr3, manifesting at temperatures below 90 K.This static Rashba effect was attributed to surface reconstruction, induced by the ordered arrangement of MA cations, which introduced structural asymmetry.
Conversely, CsPbBr3, characterized by Cs cations and a more symmetric configuration, displayed an absence of Rashba splitting. 94In a complementary study, Schlipf et al. explored the dynamic Rashba effect in MAPbI3, elucidating its intricate relationship with temperature and lattice vibrations. 75Their investigation was conducted below 160 K, a regime where MAPbI3 crystallizes into an orthorhombic structure with a centrosymmetric space group (Pnma).Notably, this crystalline system exhibits 20 infrared-active optical phonons, intimately linked to the deformation of the PbI6 octahedra.The large Rashba energy associated with the CB's spin splitting indicates a significant phonon assisted Rashba effect.Schlipf et al.'s work emphasized the role of phonons in influencing the Rashba effect. 75e long spin lifetime of all-inorganic halide perovskites has recently attracted lots of attention.As shown in Fig. 2(f), CsSnBr3 NCs demonstrated a spin lifetime of 18 ps at ambient temperature, 76 and its SOC strength is roughly one-third that of its Pb equivalent.showed spin relaxation of 32 ps at 50 K and 3 ps at 250 K. 99 Recently, Zhao et al. reported the exciton spin relaxation of CsPbBr3 is about 20 ps at room temperature, which was the longest among metal halide perovskite materials. 95The properties such as long exciton spin lifetime and high spin injection rate are considered essential factors in future optoelectronics and spintronic device applications.
For all-inorganic halide perovskite materials, spin-related studies, such as spin relaxation and spin manipulation, are still at the primary stage.In inorganic semiconductors, the spin relaxation mechanisms can be classified into three types: Elliot−Yafet (E-Y), 101  spin by scattering, while in the D-P mechanism, the Rashba effect from structural inversion asymmetry induces spin precession. 2,96,101,102The understanding of these mechanisms and their interplay with phonon scattering is crucial for manipulating spin states. 96cently, Zhou et al. measured the spin lifetime as a function of temperature in APbI3 and APbBr3 (A = CH3NH3, Cs) halide perovskites. 76The spin lifetime of CsPbBr3 is reduced from 3.8 ps to 3.2 ps when the temperature decreases from 295 K to 77 K, indicating that the D-P mechanism plays a more significant role in the Br-based halide perovskite.On the other hand, in the case of CsPbI3, the spin lifetime increases from 1.3 ps to 6 ps as the temperature decreases from 300 K to 77 K, indicating that for I-based halide perovskite, the contribution of the E-Y mechanism is dominant. 96These results provide a promising opportunity to tune the spin-based optoelectronic properties in halide perovskites.

Bright and dark state splitting in all-inorganic halide perovskites
In all-inorganic halide perovskite NCs, the exceptionally efficient light emission is attributed to a distinctive excitonic fine structure mainly composed of triplet states. 103The presence of a highly emissive triplet state with a significant oscillator strength can be attributed to the complex interaction between electron-hole exchange and SOC of the J = ½ conduction band levels.Fig. 3(a) illustrates the band diagram of the exciton state of cubic CsPbBr3 NCs, and the CB and VB feature Γ4 -and Γ1 + symmetry, respectively.Due to SOC, the CB is split into Γ8-and Γ6states, and the VB transforms to Γ6 + .The exchange interaction between electron and hole results in the formation of a higher bright triplet state (J=1) and a lower dark singlet state (J=0). 54,104It was reported that in large NCs of CsPbBr3, the Rashba effect changes the order of bright and dark exciton levels, while the bright-dark level inversion effect is suppressed in small NCs.exciton Bohr radius in a variety of halide perovskites.Reproduced with permission.Copyright © 2020 Springer Nature Limited. 427,108 The short-range exchange in CsPbX3 NCs positions the singlet state below the triplet, but a significant Rashba effect can potentially alter the fine structure, placing the dark singlet exciton above the bright triplet exciton, as shown in Fig. 3(b). 54Only at cryogenic temperatures is the presence of a bright triplet state clearly observed.In contrast, at room temperature there is a mixing of PL excitonic signals due to the small energy difference between the dark singlet state and the bright triplet. 54Moreover, the emission energy can be impacted by various factors, including composition, phase transition, anisotropy, and lattice disorder. 109,110wever, Tamarat et al. contradicted the claims in Fig. 3(b) and provided an alternative view on the energy level arrangement. 43The PL spectra from individual FAPbBr3 NCs, particularly sharp zero-phonon lines (ZPLs), revealed bright triplet sublevels, as illustrated in Fig. 3(c)(i).The structural variations in Fig. 3(c)(ii-iv), featuring one, two, and three lines at 0 T, corresponded to different crystalline phases (cubic, tetragonal, or orthorhombic).An additional emission line emerged from longitudinal optical (LO) phonon sidebands, positioned just below the bright triplet state.This indicated a magnetic brightening phenomenon linked to a low-lying dark exciton state.Importantly, the spectral footprint of the dark singlet exciton also manifested in CsPbI3 under a 7 T external magnetic field. 42These findings underscore the significant impact of exchange interaction on the fine structure splitting, influenced by quantum confinement, NC shape anisotropy, and electron-hole interaction screening effects.

Fig. 3(d)
illustrates a substantially stronger exchange interaction in CsPbX3 than the FA and MA counterparts, 42 and the bright triplet splitting suggests a strong crystal field splitting in the orthorhombic crystal structure.Experiments on CsPbBr3 NCs under magnetic fields smaller than 7 T did not reveal the expected 'dark state' due to the narrow energy gap between the dark and bright states.However, exposure to magnetic fields exceeding 10 T induced significant changes in PL, indicating the presence of long-lived energy sublevels. 42,104,111It is now generally believed that the lowest-energy exciton dark states are located just a few meV below the bright triplet state, resolving the debate on the dark singlet state's position in inorganic lead halide perovskites. 112

Spectroscopic Techniques for Investigating the Spin-Related Properties of Halide Perovskites:
4][115] Techniques such as DTS are primarily used in probing carrier spin relaxation lifetime.Magneto-photocurrent, magneto-electroluminescence (MEL), 7][118] The ultrafast spectroscopy technique has been widely applied for studying quantum well structure in the past, 119 which possesses an excellent time resolution of 100 fs.
Spin mixing will modulate the spin population, resulting in different recombination rates and changes in PL/electroluminescence emission intensity. 117Other techniques can also be combined with TRPL to study the spin-related dynamics at low and room temperatures.This section will extensively cover these techniques applied in probing the spin dynamics of halide perovskites.

Optical techniques
Optical techniques are commonly utilized because they are fast and non-invasive.[123][124][125][126]  Reproduced with permission. 99Copyright © 2020 American Chemical Society.(e) Schematic of two e-h spin-pair configurations, singlet (S) and triplet (T0), confined by a magnetic field (B) along the z-direction.Panels (i) and (ii) show the scenarios when the spin relaxation rate is lower and higher than the intersystem crossing (ISC) rate between S and T 0 , respectively.A high spin relaxation rate causes incoherence of spin precession, diminishing the magnetic field effect.Reproduced with permission. 117Copyright © 2015 Springer Nature Limited As shown in Fig. 4(a), circular excitation generates left (σ+) or right (σ-) circularly polarized emission, while linear excitation combines excitons with opposite Jz = +1 and Jz =−1 states.In 3D bulk MAPbI3, it has been reported that coherent SOC-induced momentum splitting of the CB and VB can align spins for circular excitation. 59Although both CB and VB exhibit linear-k momentum splitting at the R point, the CB's splitting is more pronounced due to Pb's heavier mass than I atom and stronger SOC effect.This results in a slight indirect bandgap but predominantly behaves as a direct bandgap.However, in the case of linear excitation, it consistently leads to a direct optical transition.This coherent spin texture behaviour challenges previous understandings and suggests that both circular and linear excitations can induce direct optical transitions.
Furthermore, the SOC effect in CsPbBr3 in the non-linear optical regime was investigated by Xu et al., wherein the generation of transient spin-polarised electrons by two-photon absorption (TPA) using circularly polarized light excitation was demonstrated. 128However, detecting high-order non-linear responses generated by TPA is unsuitable for analysis due to the poor detection limit.Therefore, THz emission spectroscopy with high sensitivity was employed to detect transient photocurrent in CsPbBr3.The degree of spin polarisation of up to ~21.3 % was obtained at room temperature due to an asymmetric charge distribution in the Bloch states of CB and VB.
Spin-dephasing dynamics in halide perovskite materials is important due to its prominent function in quantum information processing applications.In a recent report, spin-dephasing dynamics in CsPbBr3 were studied by time-resolved Faraday rotation (TRFR). 129The optical spin polarisation and coherent spin precession at cryogenic and room temperatures were observed in CsPbBr3 NCs.Two distinct spin-dephasing mechanisms were identified.At low temperatures, spin dephasing was mainly triggered by inhomogeneous hyperfine fields, which were suppressed by applying magnetic fields.At elevated temperatures, spin dephasing was caused primarily by thermally activated LO phonons.
The pump-probe techniques have become a prominent approach for time-resolved studies, widely applied in characterizing semiconductors, including halide perovskite materials.1][132][133]  A balanced photodiode bridge was used to measure the Faraday rotation, with a lock-in amplifier and an oscilloscope simultaneously monitoring the lock-in.The above circularly polarized 'pump' and linearly polarized 'probe' are combined in an ultrafast optical measurement of TRFR.As a function of time, the quantum beating between the quantized states causes an oscillatory Faraday rotation.The TRFR measurement utilizes a circularly polarized 'pump' (right or left-handed) to induce interband transition and generate spinoriented e -and h + (spin-polarized exciton). 134The linearly polarized 'probe' pulse is applied normal to the sample surface with a similar or different frequency to the pump.The magnetic field leads to the precession of the e -/h + spin about B, with the Larmour frequency Ω = geμB B/ ħ.The induced spin coherence can be monitored from the Faraday rotation angle of the probe beam.
The optical Hanle effect is a steady-state 'pump-probe' measurement based on the Faraday rotation. 127The Hanle effect was first observed in 1924. 135Since then, it has been used to study spin relaxation, and lifetime. 136Fig. 4(c) shows the setup of optical Hanle effect measurement, in which the samples were measured using CW lasers serving as 'pump' and 'probe'.The wavelength of the probe laser was tuned across the exciton band of the halide perovskite, while the pump laser had a fixed 635 nm wavelength.The measurements are set up in a Voigt geometry with a split coil electromagnet supplying the magnetic field.For measuring the Faraday rotation, the probe beam is adjusted to be at normal incidence on the sample, and the pump beam has a small angle of incidence through the same focusing lens as the probe beam.At the focal point of the pump beam, a polariser is positioned to allow or block any light based on its polarisation.A Wollaston polarising beam splitter and a half-wave plate are used to study and quantify the Faraday rotation.The optical Hanle effect measurement allows the probe beam wavelength to be precisely and independently tuned through the material's absorption edge with high energy resolution.The Faraday rotation signal is a time integration over all past time in this steady-state measurement, and it reduces with the application of a transverse magnetic field (B) because spin beating might repeal the time-integrated signal.The optical Hanle effect measurements use a circularly polarized 'pump' for the optical orientation of spin-polarised excitons and a linearly polarized 'probe' to measure exciton spin polarisation via Faraday rotation.Finally, a transverse applied magnetic field (B) diminishes the spin polarization.The Hanle curve obtained from this measurement gives general information about spin lifetime and recombination lifetime that can be extracted from the Lorentzian or Voigt function if the g factor is known.
The spin dynamics of charged carriers are usually probed using circularly polarized light.

Fig. 4(d)
shows the optical transition between the VB and CB using circularly polarised light. 99en the pump pulse remains in positive helicity (σ + ), promoting carrier transition from VB to CB with angular momentum change from MJ, VB = −1/2 to MJ, VB = +1/2 (angular momentum conservation).The probe pulse is applied with a delay time, either in positive or negative helicity.TRPL and integrated PL have been utilized to study radiative recombination.Recently, Strohmair et al. explored the spin-related dynamics using DTS, which was utilized to examine the relaxation mechanisms, recombination dynamics, and spin lifetime of CsPbI3 perovskite NCs at both room temperature and cryogenic temperatures. 99

Magneto-optical techniques
Due to the magnetic nature of materials and the development of spintronic devices, the magnetic field effect (MFE) has been investigated to characterize magnetic materials. 137sides, MFE has also been explored to study various semiconducting materials, including halide perovskites. 117,138[141][142][143] As illustrated in Fig. 4(e), the magnetic field (B) can modify the spin states by changing the inter-sublevel spin-mixing rates and in turn the spin sublevels in the spin-pair manifold.In MPL measurements, MFE can be observed when the spin relaxation rate is slower than the intersystem crossing (ISC), as shown in panel (i) of Fig. 4(e).On the other hand, the MFE diminishes when the spin relaxation is larger than the ISC rate [panel (ii) of Fig. 4(e)].In halide perovskites, the spin-lattice relaxation rate is much slower than in typical semiconductors because of multiple factors, such as hyperfine interaction, SOC, and exchange interaction, leading to the observation of MFE.In the spin-pair species, the spintronic behaviour follows Δωp = μBΔgB/ℏ (μB is the Bohr magneton); therefore, if the spin precession frequency difference between the spin-1/2 electron (e) and hole (h) is significant, the spin pair species promote ISC from the singlet, S, to triplet, T 0 state, which may occur several times throughout the spin-pair lifetime until the spin coherence is lost before achieving a steady state.The magnetic field B can alter the inter-sublevel spin mixing rates and recombination and dissociation rates for spin pairs on all sub-level spin configurations, leading to changes in PL/EL emission intensity and photocurrent.The presence of spin pair species leads to a spin mixing process in the spin sublevel, a necessary condition for MFE. 117

Topological Insulating Phase in all-Inorganic Halide Perovskites
In 2012, through a high-throughput search, K. Yang et al. identified ternary halides, Cs (Sn,Pb,Ge)(Cl,Br,I)3, as one of the previously unrealized families of topological insulator materials. 1446][147][148][149][150] The origin of nontrivial topological insulating character in the proposed all-inorganic metal-halide perovskites 145,[147][148][149][150] is the same as that for prototypical three-dimensional topological insulator family X2Y3 (X = Bi, Sb; Y = Te, Se), [151][152][153][154][155][156][157] where topological bulk band gap is associated with the mass term in the Bernevig-Hughes-Zhang (BHZ) model. 158,159In all-inorganic halide perovskite materials ABX3, first-principles calculations 145 show that B 6p orbital states form the CB, while the VB maximum is formed by a linear combination of B 6s and X 5p orbital states with quenched orbital degrees of freedom in a singlet s-orbital symmetry.With even/odd parity of the basis sets constituting the VB/CB, band inversion and parity exchange lead to the nontrivial band topology.However, such a nontrivial topological insulating phase is not intrinsic like X2Y3 family and is generally induced rather by external stimuli.
As shown in Fig. 5(a), for all-inorganic metal halide perovskites ABX3, the effective 4 × 4 continuum Hamiltonian, up to quadratic power in momentum k in the vicinity of R-point, can be written in the subspace spanned by spin doublets (s = ½) forming VB and total angular momentum doublets (j = ½) forming CB as 145 where Pauli matrices  surface spectrum (lower row) for 51-layer slab geometry with topologically protected surface states.(e) Three-dimensional band dispersion for gapless surface states and its isoenergy contour map.Reproduced with permission. 145Copyright ©2012 American Physical Society.
The constraint set by microscopic parameters on the nontrivial character of band topology suggests that a topologically nontrivial insulating phase can be achieved in ABX3 halide perovskite materials with large SOC and large hopping parameters.The nontrivial insulating phase can be realized either with heavy metal elements on B-sites and heavy halogen elements on X-site or via an externally controlled environment, such as pressure and straininduced enhancement in the bandwidth by decreasing the lattice constants.For instance, through various independent studies on CsBI3 (B = Pb, Sn) 145 , CsBX3 (B = Pb, Sn; X = Cl, Br, I) 148 , and CsSnX3 (X = l, Br, Cl) 149 , it has been predicted that all-inorganic ABX3 halide perovskite compounds display topologically nontrivial insulating phase under reasonable hydrostatic pressure.For instance, pressure-induced variation in the band gap of CsBX3 (B = Pb, Sn; X = Cl, Br, I), red-shift in the trivial band gap while blue-shift in the nontrivial band gap with increase in pressure, demonstrates a topological phase transition from trivial to a nontrivial phase. 148Furthermore, topological insulating phase in CsPbI3 with ferroelectric response 147 and CsPbBrxI3−x (x = 0, 1, 2, and 3) mixed halide perovskite compound 160 has been discussed under accessible strain.Consistent with ferroelectric all-inorganic metal halide perovskite CsPbI3, 147 a similar structure of low energy valence and conduction Rashba bands, formed by spin and total angular momentum doublets S = 1/2 and J = 1/2, has been found in organic-inorganic hybrid metal halide perovskites, AMX3 where A=CH3NH3, i.e., methylammonium (MA); M = Pb and Sn; and X = I and Br. 161It indicates that an electric field switchable S = 1/2 and J = 1/2 Rashba bands, as predicted for organic-inorganic hybrid metal halide perovskites 161 , may also be a key ingredient for spintronic applications in all-inorganic metal halide perovskites with nontrivial band topology and ferroelectric response.
Unique electrical and optical properties in the topological insulating phase, due to the surface states intertwining with the bulk band topology, give rise to substantial implications in the field of electronics, spintronics, plasmonics, and optoelectronics.For instance, topological insulators materials have the ability to overcome Boltzmann tyranny and lead to low-power topological quantum electronic device applications owning to the Rashba effect 162 , negative capacitance effect 163 , and quantum confinement effects 164 .[167][168][169][170][171][172][173] Furthermore, owing to the fact that surface/edge states in the topological insulators exhibit an exotic electronic response to light, significant progress has also been made in exploring the optoelectronic/plasmonic properties and device applications with topological insulator materials.Notable examples include but are not limited to, topological quantum phase transitions 174 , surface plasmons for energy-harvesting applications 175,176 , wide bandwidth photodetection from terahertz to the infrared 177 , transparent conductive electrodes with topological insulator nanostructures 178 , ultrafast photo-response with topological insulator Bi2Te3 nanowires 179 , high light responsivity, high detectivity, and a fast response speed in the topological insulator Bi2Se3/Si heterostructure 180 , and control over spin-polarized photocurrents with helicity or linear polarization of light, 181,182 leading to novel spinoptoelectronic or opto-spintronic devices 183 .
Since the origin and the tunability of band topology in all-inorganic halide perovskites is like that in the topological insulator family X2Y3 (X = Bi, Sb; Y = Te, Se), such intriguing optoelectronic properties displayed by the X2Y3 family provide a hint of band topology driven high-performance optoelectronic device applications in all-inorganic halide perovskites.Along with excellent power conversion efficiency, tunable physical properties, cost-effective synthesis, and long-term thermal and environmental stability, strong SOC and band topology may push the boundary even further for the novel applications of all-inorganic halide perovskites.

Summary and Outlook
In summary, the SOC effect in all-inorganic halide perovskite has been systematically discussed, along with its influence on the band structure, charge recombination, light emission, spin lifetime and other properties.While possessing better stability compared to the hybrid counterparts, all-inorganic halide perovskites retain heavy Pb atoms, which leads to a strong SOC effect that exerts a significant influence over their physical properties and device performance.In addition, the inorganic cation Cs possesses a relatively high symmetry compared to the organic ones, which underlies the composition-property correlation in this class of halide perovskite materials.However, both internal and external factors may still induce structural inversion asymmetry and slightly distort the metal-halide octahedra, thereby inducing the Rashba effect and the band splitting.
Recent investigations in the domain of all-inorganic halide perovskites have unveiled several intriguing developments.7,108 Furthermore, a debate persists regarding the existence of the Rashba effect in these materials, prompting a quest for definitive experimental evidence. 50This ongoing debate highlights the need for further research to elucidate the intricate interplay between inversion asymmetry, lattice dynamics, defect characteristics, and SOC, offering prospects for advancing Rashba engineering and the control of quantum materials.
The research exploring the correlation between spin lifetime and SOC is still in its infancy.
Different techniques such as transient absorption spectroscopy, TRFR, TPA, DTS, and TRPL were used to measure all-inorganic halide perovskites with different compositions under different conditions, thus it is challenging to consolidate the results.All-inorganic halide perovskite is hypothesized to potentially exhibit a static Rashba effect at temperatures below 60 K and a dynamic Rashba effect at higher temperatures. 93It is of pivotal importance to explore the dynamic Rashba contribution arising from the thermally activated polar distortion of PbX6 octahedra, as this might offer a means to control spin lifetime and other properties at application-relevant temperatures.
Topological insulating phases, with unique electrical and optical properties, hold the promise of advancing electronics, spintronics, and optoelectronics, enabling low-power quantum devices.The adaptable band topology in all-inorganic halide perovskites offers highperformance optoelectronic potential, driven by efficiency, tunability, stability, SOC, and suggests that the band topology could be the next frontier for the research on all-inorganic halide perovskites.It has been predicted that the halide perovskite compounds exhibit a red shift in trivial bandgap while a blue shift in nontrivial bandgap by increasing pressure. 148The halide perovskite-based 3D topological insulators can offer a new avenue for diverse potential applications by introducing a new degree of freedom, the so-called topological order.The topological surface states protected by time-reversal symmetry in 3D cubic halide perovskites might offer new physics, and the topological perovskite/perovskite heterostructure interface could serve as a playground where diverse types of quasiparticles interact with each other in the confined 2D region and may exhibit exotic properties.Consequently, further experimental, and theoretical research should be carried out to explore the topological order in other halide perovskite systems with different A-and B-site cations.
The exciting properties enabled by the hidden force of SOC will propel all-inorganic halide perovskite materials to become a rising star in exploring novel applications.Metal halide perovskites hold great promise in revolutionizing emerging disciplines such as spintronics and opto-spintronics due to their unique features, like low-cost fabrication, tunable band gaps, and exceptional charge carrier properties.From their robust SOC to their applications in spintronic devices like vertical spin-valve configurations, spin-LEDs, and spin photovoltaic devices, these materials have already demonstrated their potential, and have open pathways for future novel devices.The ongoing exploration of inorganic metal halide perovskites and refinement of their properties offer opportunities for improved device performance and the potential to advance next-generation optoelectronics, spintronics, opto-spintronics, and related fields.

Fig. 1 (
Fig. 1(b) depicts the crystal structure of three-dimensional metal halide perovskites, a class

FIG. 1 .
FIG. 1. Band structures and SOC of several typical material systems.(a) Partial band structure of silicon.Reproduced with permission. 20Copyright ©1974 American Physical Society.(b) Schematic representation of a typical halide perovskite crystal structure with formula ABX3.Reproduced with permission.Copyright 2022 Royal Society of Chemistry. 25(c) (i) Schematic electron dispersion relation of a regular CB that shows a doubly spin-degenerate parabolic band having a single minimum at k=0. (ii) Same as in (i) but subjected to Rashba splitting; two parabolic branches having oppo-site spin sense are formed.The Rashba energy (ER) and momentum offset (k0) are denoted.Reproduced with permission.Copyright 2017 American Association for the Advancement of Science, under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.39(d) Calculated band structure of cubic CsPbBr3 perovskite with a bandgap between R6 + (VB maximum) and R6 − symmetry (CB minimum).The inset is a cubic crystallite representing the first Brillouin zone.54Copyright © 2018 Springer Nature Limited.(e) Schematic of the spin-forbidden transition model.59(f) Schematic of the spinallowed transition obtained from first principles.The green and orange arrows represent the helical spin orientations in different bands.Blue solid and open circles denote photoexcited electrons (e -) and holes (h + ), respectively.Reproduced with permission.Copyright © 2018 American Chemical Society.59

FIG. 2 .
FIG. 2. Basic properties and Rashba effect of inorganic metal halide perovskites.(a) Illustration of the crystal structures of CsPbX3 (X = Cl, Br, I), highlighting their varying phases with bond angle.Reproduced with permission.Copyright © 2017 American Chemical Society. 71(b) PL spectra of CsPbX3 (X=Cl, Br, I).Reproduced with permission.Copyright © D'yakonov-Perel (D-P), 2 and Bir-Aronov-Pikus (BAP). 102E-Y and D-P spin relaxation mechanisms are relevant for metal halide perovskite materials.The E-Y mechanism flips the charge carrier's

FIG. 3 .
FIG. 3. (a) Band diagram illustrating exciton states of cubic CsPbBr3 NCs.(i) CB and VB with Γ4 -and Γ1 + symmetry, respectively.Due to SOC, the CB splits into Γ8-and Γ6states.(ii) shows the lowest dark (Γ1-, J = 0) state and the bright (Γ4-, J = 1) exciton state.Reproduced with permission. 104Copyright © 2017 American Chemical Society.(b) Schematic of the exciton band edge highlighting the electron-hole exchange and the Rashba effect.Reproduced with permission.Copyright © 2018 Springer Nature Limited. 54(c) (i) Illustration of the band-edge exciton fine structures for three distinct crystal anisotropies, among which the lowest state

FIG. 4 .
FIG. 4. (a) Schematic of the PL kinetics of halide perovskites under circular and linear polarized light excitation. 59(b) Illustration of Faraday rotation measurement with the transversely applied magnetic field.Circularly polarized pump pulses produce spin-polarized excitons and quantized spin states.An exciton population imbalance (N±) causes different light absorptions between RCP and LCP, and refraction indices (η), i.e., Faraday rotation θF ∝ (η+ − η−) ∝ (N+ − N−).Reproduced with permission. 127Copyright © 2017 Springer Nature Limited.(c) Schematics of the Hanle experimental setup.Hanle measurements use the Voigt geometry and a split-coil electromagnet.'Pump' optically orients spin-polarized excitons, while 'probe' analyses spin polarization via Faraday rotation.Half-wave plate and Wollaston beam splitter were used to detect the Faraday rotation.Reproduced with permission. 127Copyright © 2017 Springer Nature Limited.(d) Illustration of optical transitions caused by

Fig. 4 (
b) illustrates Faraday rotation measurement with the transversely applied magnetic field.A circularly polarised pump pulse populates excitons that are spin polarised along the beam path and in a coherent superposition of quantized spin states along the magnetic field (B).A population imbalance in the two exciton states (N±) leads to different absorptions between left/right-handed circularly polarized (LCP/RCP, σ + /σ -) light and generates different indices of refraction (η), i.e., Faraday rotation θF ∝ (η+ − η−) ∝ (N+ − N−).

FIG. 5 .
FIG. 5. (a) Basic electronic structure, band inversion, and topological phase transition in allinorganic metal-halide perovskites ABX3.SOC and hydrostatic pressure operation induce band inversion, and the system enters a topologically nontrivial insulating phase.(b) Dependence of the band gap value sign on the lattice constants.(c) Total energy dependence of the unit-cell volume for CsPbI3 and CsSnI3.The positive band gap values (negative mass term; ( ) 0 m k  ) represent the topologically trivial phase, whereas the negative band gap values (positive mass term; ( ) 0 m k  ) represent the topologically nontrivial phase.(d) Tight binding calculations showing the topological phase transition in the bulk band spectrum (upper row) and (0,0,1)

Table I .
Summary of spin lifetimes measured in halide perovskites.
100Since spin relaxation is a sensitive process, it can vary due to the sample's preparation condition and quality.According to published literature, all-inorganic perovskite CsPbI3