Spin selectivity in elemental tellurium and other chiral materials

The phenomenon of chirality-induced spin selectivity (CISS), where chiral organic molecules enable the selective transmission of electrons spin-polarized along the direction of electric current, has been studied for nearly two decades. Despite its technological relevance, CISS is not fully understood. Recent studies have expanded the concept of spin selectivity to chiral inorganic crystals, offering promise for magnet-free spintronics and other applications. This Perspective reviews recent developments on spin selectivity in non-magnetic solid-state materials, whereby chirality-dependent charge-to-spin conversion is responsible for transforming electric currents into spin signals, and spin transport within devices. Notably, chiral systems often outperform non-chiral ones in terms of conversion efficiency and facilitate long-range spin transport, which makes them relevant for both fundamental and applied physics. After examining the archetypal example of the chiral crystal, elemental tellurium, and the studies of spin selectivity in Weyl semimetals, we discuss its origin in terms of the unconventional (collinear) Rashba – Edelstein effect. We also explore key factors affecting the conversion efficiency and robustness of spin transport, focusing on persistent spin textures and their influence on spin lifetime. In addition, we discuss the potential impact of band velocities and the role of orbital contributions, as well as the differences associated with reduced dimensionality, providing a roadmap for guiding future theoretical, experimental, and applied studies.


I. INTRODUCTION
Studies of symmetries are invaluable for tailoring materials to achieve specific functionalities in various devices.Interestingly, it is often not the presence, but rather the absence of a particular symmetry that gives rise to unusual phenomena or unveils distinct manifestations of well-known effects.The property of handedness or structural chirality hinges on the existence of left-handed and right-handed versions of the same system, interconnected through inversion or mirror reflection operations. 1 Chirality reduces the overall symmetry, as chiral objects must lack inversion, mirror, and roto-inversion symmetries.This results in effects rarely encountered in high-symmetry systems, expanding the possibilities for innovative applications.Importantly, right-handed and left-handed enantiomers often respond differently to external stimuli, yielding distinct physical and chemical properties.This has a broad impact on medicine and technology; for example, a left-handed molecule can be an effective medication, while a righthanded one may turn out to be toxic.
3][4] While the term CISS can be broad and describe any chirality-dependent effects involving spinselective behavior of electrons, it typically refers to a phenomenon in which chiral molecules enable selective transmission of electrons spinpolarized in a direction of electric current, effectively acting as spin filters.6][7] The theoretical studies suggested that an analog of Rashba-Edelstein effect (REE) causing spin accumulation in response to the charge current could be responsible for chiralitydependent response in such systems. 8Recent investigations highlight an analogous effect in inorganic chiral crystals (e.g., Te), which are a robust platform for magnet-free spintronic devices and various other electronic applications.However, several challenges and gaps in understanding the spin selectivity in chiral materials still need to be addressed before they can be applied in electronic devices.
The elemental Te stands as the archetypal example of a chiral crystal, comprised of helical chains interacting via van der Waals forces, evoking solenoid-like structures.2][13] More recent advancements, such as nuclear magnetic resonance (NMR) measurements, have allowed for a direct estimation of spin accumulation induced by collinear REE. 14,15dditionally, Calavalle et al. employed unidirectional magnetoresistance (UMR) to show chirality-dependent and highly efficient chargeto-spin conversion (CSC) as well as its tunability through gate voltage. 16While theoretical estimations, grounded in the semiclassical Boltzmann theory, affirm the character of the observed gate voltage dependence, they underestimate the magnitude of spin accumulation by an order of magnitude, suggesting the involvement of other factors that contribute to detected signals. 17Last, Te exhibits another intriguing feature-a quasi-persistent spin texture in reciprocal space. 17,18hile this property suggests a possibility of spin transport, its exact understanding and additional experiments are required to validate this hypothesis.
The signatures of spin selectivity have been also experimentally reported in other chiral inorganic materials.For instance, Inui et al. demonstrated chirality-induced spin transport in bulk CrNb 3 S 6 occurring at room temperature without the presence of magnetic fields or permanent magnets. 19The spin accumulation was generated in response to a charge current flowing along the screw axis and observed as an inverse spin Hall signal at the detection electrode, which absorbed the spin density induced in CrNb 3 S 6 crystal.A subsequent study by the same group indicated that the induced magnetization changes linearly with electric current, implying compatibility with the linear response theory. 20Similar experiments were conducted on the semimetallic disilicides TaSi 2 and NbSi 2 , where spin accumulation was once again generated within the chiral material, and linked with the robust spin transport over micrometer-scale distances. 21,22The recent first-principles calculations indicated that the chirality-dependent CSC in these materials could have the same origin as in elemental Te, albeit with weaker spin accumulation. 17,23Moreover, according to theoretical predictions, a similar effect could be found in other chiral materials.For example, cubic FeSi and OsSi representing the so-called B20 materials class may reveal even stronger collinear REE compared to Te. 23 It is worth mentioning that the studies reporting chiralitydependent CSC in crystals often use the term CISS, which indeed can be given a broad sense and refer to converting electric current into spin signals in any system.However, the connection between CISS in chiral molecular junctions, typically discussed in the context of spin current generated in response to the electric field, and the spin accumulation observed in chiral Te and other inorganic materials is not so obvious.Only very recently, Gupta and Droghetti suggested that the current-induced spin accumulation in Te is essentially the same phenomenon as CISS in molecular systems. 24On the other hand, some studies of CISS in chiral molecules report bias-current-free effects, which do not seem to have a counterpart in solid-state materials. 25To avoid any ambiguities, we will call the current-induced spin accumulation in chiral crystals either a collinear REE or spin selectivity or we will use a more general term charge-to-spin conversion.
While the exploration of spin selectivity in chiral inorganic materials is still in its infancy, the field holds great promise for electronic devices, emphasizing the importance of further research.The goal of this Perspective is to explore potential factors that could enhance conversion efficiency and help preserve spin accumulation over large distances.We discuss the role of spin texture, band velocities, contributions from orbital effects, and system's dimensionality, among other considerations.Furthermore, the role of persistent spin texture (PST) in driving long-range spin transport and its quantitative impact, as well as its connection to spin selectivity, hold crucial implications for spintronics and will be thoroughly discussed.Finally, CSC and spin transport within chiral two-dimensional (2D) materials remain largely unexplored, and we will provide an overview of how these effects might be harnessed in thin films of three-dimensional (3D) chiral crystals and van der Waals (vdW) heterostructures that become chiral upon twisting.In sum, our aim is to present the current state of the art of CSC and spin transport in chiral materials and outline a roadmap for future fundamental and applied research.

II. FUNDAMENTALS OF CHARGE-TO-SPIN CONVERSION AND SPIN TRANSPORT IN CHIRAL CRYSTALS A. Chirality-dependent charge-to-spin conversion in crystals
Let us start with a brief discussion of charge-to-spin conversion in chiral crystals.By charge-to-spin conversion, we mean the generation of spin signals in response to applied electric current or vice versa.There are two well-known conversion mechanisms: the spin Hall effect (SHE) and the Rashba-Edelstein effect (REE), both widely utilized in spintronics devices.In the former, a charge current induces a spin current, leading to the accumulation of spins with contrary signs at the opposite edges of the sample.Typically, electric and spin currents, as well as the polarization of the spin current, are mutually orthogonal.Although low-symmetry crystals, including chiral ones, offer rich possibilities for unconventional configurations, where electric currents induce, for example, longitudinal spin currents, SHE is not a chiralitydependent phenomenon as it remains invariant under inversion.][30][31] In the Rashba-Edelstein effect, the electric current locally induces spin accumulation.Due to the applied electric field, the Fermi surface of a crystal shifts, which causes a variation of the electronic distribution.If the electronic bands are spin-polarized, this typically leads to the spin imbalance at the Fermi level, manifesting in the real space as an induced net spin density [see Fig. 1(a)].This effect, also called current-induced spin accumulation, was mostly associated with two-dimensional electron gas (2DEG) or surfaces and materials with Rashba spin textures.However, it can also occur in systems with any other spin textures, including Dresselhaus and Weyl types [see Fig. 1(b)], which can give rise to unconventional (collinear) REE with the induced spins parallel to the electric current.In addition, since REE directly depends on the spin polarization of electronic states, opposite signs of spin textures in different chiral enantiomers would lead to opposite sign of the induced spin accumulation.This is in line with the chirality-dependent CSC observed in bulk Te and attributed to REE. 16,17 Similar effect was found in Weyl semimetals TaSi 2 and NbSi 2 showing the sign reversal of spin signals for opposite chiralities.Surprisingly, there are no reports of collinear REE in other non-magnetic inorganic crystals.
Our recent research addressed the relationship between the crystal symmetry, type of spin texture, and the presence of conventional or unconventional configurations of REE. 26 The most general conclusions can be derived from the symmetry analysis of the response tensor.In the linear response theory, the Rashba-Edelstein effect is described as a spin accumulation ds induced by a charge current j, namely, ds k ¼ v ki j i , where the REE susceptibility tensor v determines the allowed configurations based on the crystal symmetry.Not coincidentally, the REE tensor has the symmetry of the gyration tensor that determines the optical activity, linking the spin accumulation with optical properties.Overall, the current-induced spin accumulation can occur in materials that are non-centrosymmetric and gyrotropic; namely, the lack of inversion symmetry is not a sufficient condition for the presence of REE.An interesting example of a non-centrosymmetric crystal where REE is forbidden by is monolayer MoS 2 and other transition metal dichalcogenides (TMDs) that share the same crystallographic space group (SG 187). 26ifferent crystal symmetries yield diverse forms of the susceptibility tensor.The collinear REE similar to the one found in Te arises from diagonal elements v ii that describe the accumulation of spins (anti-)parallel to the electric current.Diagonal components are allowed for all materials described by 65 chiral space groups as well as for an additional eight that are non-chiral [see the scheme in Fig. 1(c)].Although it would be more intuitive to establish a clear correspondence between spin texture and spin selectivity, it is not straightforward.Logically, materials with purely Rashba spin textures will exhibit a conventional response, and those with radial-like spin textures can demonstrate collinear REE.However, in some cases, only the presence of radial components in the spin texture is sufficient for an unconventional response.For example, Dresselhaus spin texture can also yield collinear REE, as it seems to be the case in Pd 4 Se. 26Thus, while spin textures can serve as an indicator, whether a particular response is permitted can only be determined based on the symmetry operations of the crystal.
These considerations raise important questions.With numerous (73) space groups allowing for chirality-dependent CSC, why has it not been detected in a broader range of materials until now?Are there additional conditions necessary to make it more detectable?The conventional REE has primarily been observed in 2D systems, and not in bulk crystals.Does this suggest that detecting it in 3D bulk materials is inherently more challenging, given that spin accumulation is induced locally?What makes bulk tellurium more special, and is it its peculiar spin texture that is simultaneously radial and quasi-persistent along the screw axis, that enhances spin transport robustness?Conversely, could chirality-dependent CSC be more easily detectable in 2D materials?

B. Spin textures and spin transport in chiral crystals
Spin textures seem relevant not only for inducing spin accumulation via REE but also for ensuring robust spin transport, a key factor in its detectability.While chiral crystals can in principle host any type of real-space representation of conventional REE generated in a 2D system.Bottom panel: conventional REE induced in a 3D system.Here, a 3D Fermi surface with a Rashbalike spin texture, such as the one of bulk GeTe, is needed. 26(b) The same as panel (a) but showing an idealized radial spin texture, which gives rise to a collinear REE.(c) Flow chart categorizing materials allowing for collinear REE.Within non-centrosymmetric space groups, only the gyrotropic ones may exhibit REE.All 65 Sohncke groups, including chiral groups and achiral ones allowing for chiral structures, have diagonal components of v. 27 Among non-chiral structures, we identified eight space groups that still permit collinear REE, specifically SG 81-82, 111-114, 121-122. 26pin texture across the Brillouin zone (BZ), the symmetry restrictions related to both crystallographic and wave-vector point groups make the radial spin textures most favorable.32 The recent study of Gos albez-Martínez et al. highlights various types of radial spin textures, 33 different from standard hedgehog configurations where spins align parallel to the momentum for all directions, schematically illustrated in Fig. 2(a).Particularly intriguing among these sub-types are quasi-persistent spin textures, characterized by predominantly uniform spin distribution in momentum space, which remains compatible with collinear REE.23 This recent classification enables a more detailed understanding of spin textures and their implications for spin transport.
Interestingly, Te was the first crystal to reveal the radial spin texture in angle-resolved photoemission spectroscopy (ARPES) experiments. 34n addition to the parallel spin-momentum locking, another notable result is the detection of a nearly uniform spin polarization aligned with the principal axis, in agreement with the theoretical predictions. 17,18The spin texture of Te has the peculiar property of being radial and quasipersistent [see Fig. 2(b)], as non-radial components may appear along directions different from the rotation axes, which is allowed by D 3 point group symmetry. 33The persistent spin texture could be potentially related to the significantly larger UMR in Te as compared to other materials. 16A similar quasi-persistent spin texture was predicted via density functional theory in TaSi 2 and NbSi 2 , 17,23 though its ARPES validation is still pending.We note, however, that spin distribution is more isotropic than in Te and it is limited to only specific bands around the Fermi level.To prove a connection between the quasi-persistent spin texture and the reported micrometer-range spin transport, 21,22 quantitative studies based on realistic models are needed.

III. RECENT DEVELOPMENTS AND FUTURE DIRECTIONS A. Efficiency of charge-to-spin conversion
One of the important prerequisites for employing chiral crystals in spintronic devices is CSC efficiency, which measures how effectively an electric current is converted into a spin signal.Thus, it is crucial to identify the factors that enhance the spin accumulation induced by an electric current and make its detection more feasible.Our recent computational study aimed to address these questions through the analysis of spin-resolved electronic structures and calculated Rashba-Edelstein susceptibility tensors. 23Contrary to the common belief that the strength of spin-orbit coupling (SOC) is the sole determinant of high conversion efficiency, our findings reveal a more nuanced relationship.The comparison of chiral Te and Se, which share the same crystal structure, demonstrated that Te, due to its higher SOC, indeed exhibits a stronger REE.However, the studies of OsSi and FeSi showed a contrary result; FeSi with the weaker SOC manifests a higher REE than OsSi, which we attributed to the lower band velocities in FeSi, as they yield a lower charge current in response to a given electric field.
Moreover, we identified other factors that can enhance the efficiency of REE.The presence of bands with one particular type of spin texture near the Fermi level, particularly Weyl-type texture in the case of chiral materials, will lead to a higher spin accumulation, as compared to mixed or undefined spin patterns.Also, a high spin polarization along a particular direction will enhance REE generated by the electric current flowing along that same direction.In this sense, the quasi-persistent spin texture in Te directly enhances the induced spin density along the helical chains, and this is not its only role; it can also potentially increase the spin lifetime, improving the detectability of the generated spin accumulation.
Another mechanism that could enhance the generated spin accumulation is the potential contribution from the current-induced orbital magnetization. 35However, this possibility was not yet explored in the context of chiral crystals and collinear REE.It is noteworthy that in the case of Te, the spin accumulation, as estimated from NMR shift, is approximately one order of magnitude lower than the value derived from DFT and TB-based calculations. 14,17In principle, the orbital component could complement the spin part, potentially contributing to the shift. 36However, the challenge lies in experimentally distinguishing these two influences, and this aspect has yet to be investigated.A similar analysis could be extended to Se, where orbital effects might play a more substantial role compared to Te due to its lower SOC.Furthermore, the mechanisms related to the relaxation of orbital magnetization remain elusive, and they could significantly differ from those leading to spin relaxation.The rapid progress in the field of orbitronics observed in the last months [37][38][39][40] suggests that these effects will be studied soon in chiral materials.
Last, we note that, in contrast to the properties of magnetic systems, the efficiency of the collinear REE does not seem to strongly depend on temperature, which makes it relevant for applications requiring room temperature operability.Even though our calculations for Te show that the induced spin accumulation may drop by a factor of 2 at room temperature at the low hole concentrations, 17 the order of its magnitude remains the same.Moreover, the temperature dependence becomes less pronounced at higher values of doping. 17,24Other materials considered using the same approach, TaSi 2 and NbSi 2 , show   no temperature dependence at all, 17,23 which seems to be in line with the experimental studies reporting spin signals at room temperature. 21hese results are promising, but temperature dependence should be carefully taken into account in future studies aimed at the enhancement of CSC efficiency.

B. Persistent spin texture and long spin lifetimes
The role of persistent spin textures (PST) in increasing the spin relaxation length is one of the most important issues to be unveiled in connection with spin selectivity and spin transport in chiral materials.Although quasi-persistent spin textures arising as a form of radial spin texture are a rather new concept, a unidirectional spin polarization of bands in different non-chiral materials was previously studied.Originally, a similar concept was proposed and experimentally realized in quantum wells, where the system's parameters can be adjusted to balance the strengths of Rashba and Dresselhaus interactions. 41,42Such tuning generates a peculiar spin wave mode called a persistent spin helix, which leads to ideally infinite spin lifetime and protection of propagating spins against scattering. 43Subsequently, Tao and Tsymbal theoretically predicted an analogous effect in orthorhombic crystals in which equal strengths of Rashba and Dresselhaus parameters are enforced by symmetry. 44Even though the existence of PST was confirmed via DFT calculations in several materials, including SnTe, BiInO 3 , CsBiNb 2 O 7 , and other oxides, [44][45][46][47] the mechanism of spin protection against scattering and its quantitative impact on spin lifetime still need to be verified.
][50][51] The spin texture is aligned out-of-plane, and it originates from the unique crystal field arising from the prismatic ligand coordination of the metal atom characteristic for TMDs.In some of these crystals, the long spin lifetimes are already experimentally confirmed. 524][55] We note, however, that Zeeman-type spin texture will contribute to neither conventional nor unconventional REE in freestanding TMDs, as it is forbidden by symmetry.In contrast, the advantage of quasi-persistent spin textures in chiral crystals is that it can give rise to collinear REE and help maintain the induced spin accumulation over large distances, as the spins are parallel to the electrons' motion.

C. Chiral two-and one-dimensional materials
The conventional REE was broadly studied in 2D systems, and it was detected despite the absence of PST.The unconventional REE was explored in a few 3D crystals, but its analogs in two-or onedimensional (1D) materials received less attention.Models were developed to study spin transport in chiral metal surfaces, 56 but some of the known chiral materials can be exfoliated, giving rise to thin films with intriguing properties.For instance, tellurene, a 2D allotrope of Te, has recently emerged as a promising semiconductor suitable for various electronic applications. 57The so-called a phase, which seems to be most stable beyond a monolayer's thickness, resembles the bulk system with chiral chains running within the 2D plane [see Fig. 3(a)].Although the spin texture appears quite uniform, 58 further research is needed to understand its impact on charge-to-spin conversion and spin transport.This field is rapidly advancing, and thin films of chiral bulk structures like quartz, as well as the chiral surfaces of achiral crystals, hold promise for the exploration of spin selectivity.
Another class of 2D crystals that can be made chiral are vdW materials and their heterostructures.Even though vdW monolayers often possess mirror symmetries that exclude chiral structure, twisting layers with respect to each other seems a feasible way to generate chirality. 59As sketched in Fig. 3(b), left-handed and right-handed enantiomers can be generated by twisting the upper layer clockwise or counterclockwise, respectively.The easiest to analyze is twisted bilayer graphene (TBLG); if we start from the parallel AA stacking and twist the upper layer around the carbon atom, the original point group symmetry D 6h (SG 191) will typically change to either D 3 (SG 150) or D 6 (SG 177).According to point group symmetry, TBLGs should host radial-like spin textures at the K-points and support collinear REE. 32owever, DFT calculations revealed a more complex spin polarization pattern, and the existence of REE needs to be verified. 60Other potential candidates for unconventional REE would include graphene/TMD bilayers, 61 but also this direction requires further exploration and experimental confirmation.Alternatively, 1D materials, such as inorganic polymers crystallizing in the form of helical chains, could be even more intriguing to investigate, offering a bridge between the studies on CISS in molecules and analogous solid-state phenomena.In some cases, nanowires can be structurally simpler, enabling the development of toy models that provide a more intuitive understanding of concepts related to spin selectivity.One such model was introduced by Yoda et al. who suggested an analogy between the current-induced orbital magnetization and a classical solenoid. 36The model is aimed for the design of materials that may manifest a large orbital magnetization.In a more recent study, Kim et al. analyzed a specific example of Se chains, which are structurally identical to Te chains.It emphasizes the significant contribution of induced orbital magnetization in response to an external electric field, in addition to the spin magnetization resulting from collinear REE. 62It is worth noting that recently, over 100 potentially exfoliable 1D materials have been proposed, and many of them might be worth exploring in the context of chirality and spin selectivity. 63

IV. CONCLUSIONS
In summary, we discussed the recent progress on charge-to-spin conversion and spin transport in chiral inorganic crystals.Even though the topic is old dating back to early studies of chiral Te, it was not given enough attention being eclipsed by the extensive research on CISS in molecules.Therefore, the field warrants further exploration, with several open questions that need to be answered.Why are there so few crystals revealing a collinear REE? Are additional factors required for its detection in 3D materials?What is the influence of persistent spin textures on spin lifetime, and could the current-induced orbital magnetization enhance the induced spin signals?Furthermore, collinear REE has not been studied in exfoliated chiral thin films or vdW heterostructures, and the studies on 1D nanowires were mostly limited to models.While our discussion was focused on non-magnetic systems, it is worthwhile to remark that magnetic interfaces and chiral antiferromagnets are still largely unexplored in the context of unconventional CSC and thus warrant further studies. 64,65We believe that this Perspective will stimulate further theoretical and computational studies, as well as experimental endeavors, ultimately leading to the detection of spin selectivity in various inorganic materials.

FIG. 1 .
FIG. 1.(a) Top panel: Schematic illustration of a Fermi surface with a Rashba-type spin texture.The band shifts from the equilibrium in response to the charge current, which induces spin accumulation perpendicular to the current.The filled arc indicates the induced spin imbalance.Note that only one band is sufficient to generate REE.Middle panel:real-space representation of conventional REE generated in a 2D system.Bottom panel: conventional REE induced in a 3D system.Here, a 3D Fermi surface with a Rashbalike spin texture, such as the one of bulk GeTe, is needed.26(b) The same as panel (a) but showing an idealized radial spin texture, which gives rise to a collinear REE.(c) Flow chart categorizing materials allowing for collinear REE.Within non-centrosymmetric space groups, only the gyrotropic ones may exhibit REE.All 65 Sohncke groups, including chiral groups and achiral ones allowing for chiral structures, have diagonal components of v. 27 Among non-chiral structures, we identified eight space groups that still permit collinear REE, specifically SG 81-82, 111-114, 121-122.26

FIG. 2 .
FIG. 2. (a) Idealized spherical Fermi surface and the corresponding hedgehog radial spin texture.(b) Schematic illustration of a hole pocket in doped Te and its projection onto a plane-parallel to the screw axis with the superimposed spin texture.The results of first-principles calculations for Te are reported in Refs.17and 23.

17
FIG. 2. (a) Idealized spherical Fermi surface and the corresponding hedgehog radial spin texture.(b) Schematic illustration of a hole pocket in doped Te and its projection onto a plane-parallel to the screw axis with the superimposed spin texture.The results of first-principles calculations for Te are reported in Refs.17and 23.
FIG. 2. (a) Idealized spherical Fermi surface and the corresponding hedgehog radial spin texture.(b) Schematic illustration of a hole pocket in doped Te and its projection onto a plane-parallel to the screw axis with the superimposed spin texture.The results of first-principles calculations for Te are reported in Refs.17and 23.

FIG. 3 .
FIG. 3. (a) Schematic view of the bulk crystal structure of trigonal Te. 2D thin films can be obtained through exfoliation along the indicated plane.A single chain corresponds to a one-dimensional system.(b) Conceptual illustration of generating chiral materials via twisting; left-handed and right-handed enantiomers are obtained for twisting angles 6h, respectively.