PHYSICAL REVIEW RESEARCH 3, 043230 (2021) Low-symmetry topological materials for large charge-to-spin interconversion: The case of transition metal dichalcogenide monolayers Marc Vila,1,2 Chuang-Han Hsu,3,4 Jose H. Garcia ,1 L. Antonio Benítez ,1,2,5 Xavier Waintal ,6 Sergio O. Valenzuela ,1,7 Vitor M. Pereira ,4 and Stephan Roche 1,7 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 Department of Physics, Universitat Autònoma de Barcelona, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore 4 Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, Singapore 117546, Singapore 5 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA 6 Université Grenoble Alpes, CEA, IRIG-PHELIQS, 38000 Grenoble, France 7 ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Received 10 June 2020; revised 29 November 2021; accepted 2 December 2021; published 30 December 2021) The spin polarization induced by the spin Hall effect (SHE) in thin films typically points out of the plane. This is rooted on the specific symmetries of traditionally studied systems, not in a fundamental constraint. Recently, experiments on few-layer MoTe2 and WTe2 showed that the reduced symmetry of these strong spin-orbit coupling materials enables a new form of canted spin Hall effect, characterized by concurrent in-plane and out-of-plane spin polarizations. Here, through quantum transport calculations on realistic device geometries, including disorder, we predict a very large gate-tunable SHE figure of merit λs θxy ≈ 1–50 nm in MoTe2 and WTe2 monolayers that significantly exceeds values of conventional SHE materials. This stems from a concurrent long spin diffusion length (λs ) and charge-to-spin interconversion efficiency as large as θxy ≈ 80%, originating from momentum-invariant (persistent) spin textures together with large spin Berry curvature along the Fermi contour, respectively. Generalization to other materials and specific guidelines for unambiguous experimental confirmation are proposed, paving the way toward exploiting such phenomena in spintronic devices. These findings vividly emphasize how crystal symmetry and electronic topology can govern the intrinsic SHE and spin relaxation, and how they may be exploited to broaden the range and efficiency of spintronic materials and functionalities. DOI: 10.1103/PhysRevResearch.3.043230 I. INTRODUCTION (M = Mo, W; X = S, Se, Te), which have been advanced as platforms for realizing exotic phenomena such as topologi- Unconventional manifestations of spin-orbit coupling cal superconductivity [9–12], nonlinear Hall effect [13–17], (SOC) are rapidly extending the ability to generate, control, anisotropic spin Hall transport [18], and out-of-plane spin- and carry spin polarization for applications of spin transport orbit torque [19,20]. When thinned toward the monolayer or spin-driven magnetic torques beyond conventional spin- limit, they transition from the type-II WSM bulk phase to tronic materials [1–4]. Topological materials form a natural the quantum spin Hall regime [21–35] with strain-tunable family to scrutinize in this regard: Their key features often topological gap [36]. derive from a large SOC combined with band inversions, and Recently, large charge-to-spin interconversion (CSI) gen- their topologically protected surface states may prove instru- erated by the spin Hall effect (SHE) has been reported in mental to enable coherent, dissipationless spin currents over multilayers of MoTe2 and WTe2 [37–40]. The CSI efficiency long-distances [5,6]. Three-dimensional (3D) Weyl semimet- is quantified in terms of the spin Hall angle (SHA, θxy ), which als (WSM) are defined by the presence of band degeneracy indicates what fraction of a driving charge current (J c ) can points near the Fermi energy (EF ) with local linear dispersion be converted into spin current (J αs ); θxy depends on the mag- in all directions [7,8]. Layered transition metal dichalco- nitude of SOC and is typically no more than a few percent genides (TMDs) in the 1T (P21 /m) or 1Td (Pmn21 ) phases at room temperature in heavy metals [41]. In traditional SHE accommodate the interesting class of WSM candidates MX2 materials, stronger SOC correlates with shorter spin diffusion length (λs ); consequently, achieving long λs concurrently with large SHA is a long-standing challenge for spintronics. To date, the best tradeoff obtained with heavy metals amounts Published by the American Physical Society under the terms of the to λs θxy ≈ 0.1–0.2 nm [41–46]. Creative Commons Attribution 4.0 International license. Further Interestingly, hints of unconventional SHE have been de- distribution of this work must maintain attribution to the author(s) tected in 1T [37] and 1Td phases of MoTe2 multilayers [38], and the published article’s title, journal citation, and DOI. characterized by spin currents carrying spins (S) collinear 2643-1564/2021/3(4)/043230(9) 043230-1 Published by the American Physical Society MARC VILA et al. PHYSICAL REVIEW RESEARCH 3, 043230 (2021) with the charge current, which is unique so far. This is pos- tight-binding representation [49]: sible because, in contrast to bulk crystals, the absence of   the glide mirror symmetry in few-layer slabs allows for ad- H= ( + 4m p + δ)ci,s † ci,s − (m p + md )ci,s † c j,s ditional nonzero components of the spin Hall conductivity i,s i j,s (SHC) tensor [47], thus breaking away from the traditional   + ( − 4md − δ)di,s † di,s − (m p − md )di,s † d j,s constraint imposing a stringent right-hand rule J c ⊥ J s ⊥ S). i,s i j,s Furthermore, a remarkably long spin diffusion length (λs ) has been reported in MoTe2 [38], although its actual magnitude β  − (l̂ i j · ŷ) ci,s † d j,s + ηci,s † di,s is under debate [37,39,40]. These experimental developments 2 i j,s i,s call for an understanding of the mechanisms governing spin dynamics in such systems, especially the large CSI efficiency  i − (ss × l̂ i j ) · (ŷ + ẑ)ci,s † d j,s + H.c. (1) θxy and what enables large λs , given that these are traditionally  2 i j ss anticorrelated quantities. While topological materials could be expected to display This is an effective four-band Hamiltonian generated by two large SHE efficiency, primarily driven by a strong SOC and orbitals (plus spin) per unit cell on a rectangular lattice, one a large intrinsic SHC, here we show that, in addition, the arising from the chalcogen py states and the other from metal reduced symmetry plays a central role by enabling otherwise- dyz orbitals, respectively associated with the ci,s and di,s oper- forbidden persistent (i.e., k-invariant) spin textures (PST) that ators at each unit cell i (s labels the spin projection). The first sustain unusually large spin diffusion lengths. Specifically, four terms in Eq. (1) describe spin-degenerate valence and we show that unique symmetry-induced spin textures of elec- conduction bands with hopping amplitudes set by m p ± md , tronic states in MoTe2 and WTe2 monolayers yield a giant δ parameterizes the degree of band inversion at , and a canted SHE where the spin current polarization lies in the yz constant  is used to match the position of the conduction plane. CSI efficiencies can be as high as 80% and values of λs band and EF with those obtained by DFT. In the fifth term, in the range 10–100 nm, up to one order of magnitude larger β accounts for the x-y crystalline anisotropy, with l̂ i j a unit than in heavy metals with similar spin Hall efficiency [41]. vector pointing from site i to j; the term ∝ η breaks inversion We unveil that these large values arise from the interplay symmetry and determines, for example, whether we are de- of a PST, confirmed by density functional theory (DFT), scribing a monolayer descended from a 1T (η = 0) or 1Td and a large spin Berry curvature (SBC), stemming from the bulk crystal. The last term embodies the SOC, where  ≡ band inversion and hybridization near EF that underlies the ( x σx , − y σy , z σz ), σx,y,z are the spin Pauli matrices and ŷ, nontrivial topology of these monolayers. Moreover, all spin ẑ are unit Cartesian vectors. The parameters are set by fitting transport characteristics are gate tunable, being maximal near the energy dispersion and spin texture to the ones obtained the band edge and allowing a CSI figure of merit of up to by DFT [49]. To be specific, we henceforth concentrate on λs θxy ≈ 50 nm. Our findings hinge on DFT, symmetry con- the case of MoTe2 , as it is the one where experiments have siderations, and a purpose-built effective tight-binding model recently reported in-plane SHE [37,38]. Nonetheless, because deployed in spintronic simulations using state-of-the-art quan- the Hamiltonian model (1) captures equally well the case of tum transport methodologies [48]. Importantly, we discuss WTe2 and similar low-symmetry TMDs [49], qualitatively how similarly large spintronic figures of merit can be antic- comparable results can be expected in those monolayers as ipated, based on the same key physical ingredients, among well. Additionally, we found very similar results between the several other classes of materials that would be interesting to 1Td and 1T phases and therefore we focus here on the former explore. This paves the way to uncovering systems endowed while the latter is reported in the Supplemental Material [49]. with both large θxy and large λs , breaking free from a chal- We favoured the conduction band in the fits and will focus lenging constraint that has hindered spintronic applications. exclusively on cases where EF lies in the conduction band. II. THEORETICAL MODEL III. NUMERICAL SPIN DYNAMICS CALCULATIONS We computed the bandstructures of 1Td -derived monolay- In Fig. 1(a), we plot the band structure of the aforemen- ers of MoTe2 and WTe2 within DFT (see the Supplemental tioned four-band model near EF and one of the time-reversal- Material [49]). Effective Hamiltonians based on maximally symmetric Q points, where the valence and conduction localized Wannier functions were subsequently extracted, al- extrema occur. The small splitting arises from the small lowing straightforward computation of the SHC and spin inversion-symmetry breaking (η = 0) that occurs in mono- textures with no intervening approximations. Yet, such a layers derived from the Td bulk structure. Since this work Hamiltonian is still too complex to be efficiently deployed focuses on the scenario where EF lies in the conduction in large-scale transport calculations on system sizes involving band, the figure shows a closeup of the conduction electron millions of unit cells. We therefore built a k · p Hamiltonian pockets; the full DFT band structure and the k · p fit are dis- to describe the two conduction and two valence bands nearest cussed in the Supplemental Material [49]. Figure 1(b) shows EF which, at the  point, transform according to the repre- the spin texture at the Fermi energy, sα EF , with two cru- sentations Bu (valence) and Ag (conduction) of the C2h point cial observations: the existence of an approximate persistent group [23] (see the Supplemental Material [49] for further spin texture through the whole Fermi contour [50–52] and discussion). Extension of the symmetry-allowed k · p terms to canted spins with sy EF  sz EF sx EF , consistent with the full Brillouin zone yields the following nearest-neighbor prior studies [15,53]. 043230-2 LOW-SYMMETRY TOPOLOGICAL MATERIALS FOR LARGE … PHYSICAL REVIEW RESEARCH 3, 043230 (2021) regime, we add Anderson disorder to the Hamiltonian and extract statistics only within the appropriate scaling region of the device conductance [49,58]. α Figure 2 shows Rnl (L) for the three spin orientations at EF . We see clear differences in the magnitude of the nonlocal signals and their relaxation distances for different orientations of the injected spins, ranging from tens of nanometers to the α subnanometer scale. By fitting the length dependence of Rnl to the solution of the spin diffusion equations (dashed lines in Fig. 2) [58], we obtained λys ≈ 30 nm and λzs ≈ 10 nm, while λxs has a negligible value. These values are comparable FIG. 1. (a) Closeup of the band structure near EF for the four- with strong-SOC metals such as Pt, β-W or β-Ta [41,43]. It band model [Eq. (1)] of Td -MoTe2 monolayer near one of the is significant that the spin diffusion lengths follow the trend time-reversal-symmetric Q points. The blue-shaded region indicates λys  λzs λxs , in correspondence to that of the spin texture the energy range covered in the spin transport calculations. Inset: around the Fermi contour at equilibrium. The upper inset of monolayer crystal structure. (b) Spin texture of one of the bands of Fig. 2 shows that this hierarchy holds over the entire range the electron pocket near Q at EF (Fermi broadened with T = 300 K); of energies analyzed, from E = 30 meV to the band edge the solid line marks the Fermi contour, arrows depict the in-plane at ≈ −110 meV (we measure energies relative to EF of un- spin projection, and the color indicates the spin projection along z. doped MoTe2 ). Details of how the (persistent) spin texture impacts λαs and its scaling with energy are discussed in the We explored the spin transport properties using linear Supplemental Material [49]. Both λys and λzs increase about response theory and the Landauer-Büttiker formalism as im- threefold as EF moves toward the band edge (dot-dashed plemented in KWANT [54]. We simulated the nonlocal spin line). Moreover, we found that λys at EF = −140 meV (in the valve illustrated in the inset of Fig. 2, where contacts 2 and band gap) increases up to 156 nm, while deeper into the 3 are ferromagnetic (FM) to allow injection and detection of gap (E = −320 meV) we see no decay in the spin signal, spin-polarized currents [55–57]: FM electrode 2 injects a spin- consistent with the onset of ballistic regime where spin is polarized current I0α with spins polarized along α ∈ {x, y, z}; transported by topologically protected surface states [49]. this creates a spin accumulation that diffuses along the chan- The SHE was investigated by computing the spin accu- nel and is detected as a nonlocal voltage Vnl at electrodes 3 and mulation and analyzing its polarization, sα , along the three 4, located a distance L from the source and far from the path Cartesian directions α. Charge current along y generates a of charge current between electrodes 1 and 2. This effect is transverse spin current parallel to x by the SHE, which results α quantified by the nonlocal resistance Rnl ≡ Vnl /I0α . The spin in spin accumulation at the open lateral boundaries. The effi- diffusion length for α-pointing spins, λαs , is obtained from ciency of CSI is characterized by the SHA, defined as the ratio the decay of Rnl α with L in the diffusive regime (mean free θiαj ≡ Js,i α /Jc, j , where h̄J αs /2e (J c ) is the spin (charge) current path shorter than L). To ensure our results reflect the diffusive density, e is the electron charge, and i, j ∈ {x, y} denote the respective current directions. To numerically determine the SHA, we calculated the spin accumulation response function per unit of current applied to the lead and fit it to the solution of the spin drift-diffusion equations:   α α sinh w−2x sα (x) θxy λs 2λαs =−  , (2) Jc,y |e|Ds cosh 2λwα s where w is the device width and Ds is the spin diffusion coefficient (see the Supplemental Material [49] for details). Figure 3 shows the averaged spin accumulation along the channel cross section, sα (x), for each spin orientation. In a typical SHE scenario, the electrical current, spin current, and the spin polarization are all mutually orthogonal (for this geometry, that would generate a finite sz only); however, we observe a nonzero sy as well due to the absence of a glide mirror symmetry in 1Td -MoTe2 monolayer. In fact, |sz | ∼ |sy |, α FIG. 2. Rnl (solid lines) against the channel length, L, for spins implying that the accumulated spins point obliquely in the polarized along x, y, and z. Error bars result from averaging over yz plane, with significant projection parallel to the electrical 150 disorder configurations (w = 50 nm). Dashed lines are fits to current. Interestingly, note that the spin accumulation displays Eq. (10) in Ref. [49]. Left inset: Scheme of the nonlocal spin valve. the hierarchy sy  sz sx , echoing the trend seen above for Black (red) regions denote the device (leads), with leads 2 and 3 the spin texture and spin diffusion lengths. α being ferromagnetic. Current I0α flows from lead 2 to 1 and Vnl is We determine θxy by fitting the numerically calculated spin measured between leads 3 and 4. Right inset: Energy dependence of accumulation to Eq. (2), using the values of Ds extracted λy,z s . The dot-dashed line marks the conduction band minimum. from the two-terminal conductance of this device and λαs from 043230-3 MARC VILA et al. PHYSICAL REVIEW RESEARCH 3, 043230 (2021) yield a CSI figure of merit λs θxy ≈ 1–50 nm, with the largest values attained at the band edge and for y-pointing spins. The upper limit exceeds that of traditional SOC materials (Pt, β-W, β-Ta, or Au) for which λs θxy ≈ 0.1–0.2 nm [41,43,61], and is up to two to three times larger than that induced by prox- imity in graphene [62–64]. Such remarkable figure of merit stems from the combination of large SBC and the persistent spin texture near the MoTe2 band edges [49]. These results represent the expected behavior in the monolayer limit of recent experiments performed on few-layer MoTe2 [37,38] and WTe2 [39,40]. IV. EXPERIMENTAL DETECTION Such a peculiar spin response should become mani- fest in suitably designed nonlocal spin-precession experi- FIG. 3. Spin accumulation (solid lines) as a function of posi- ments [37,63,65,66]. To probe this canted SHE, we propose tion across the channel width, of spins along x, y, and z. Error the device concept pictured in the insets of Fig. 4, which bars result from averaging over 200 disorder configurations (w = relies on the reciprocal/inverse SHE (ISHE) [41]. It consists 50 nm). Dashed lines are fits to Eq. (2). Bottom inset: Scheme of the of a Hall bar comprising a graphene channel and a trans- two-terminal device, where a current flowing along y creates a spin versely aligned monolayer TMD crystal. A nonequilibrium accumulation in the x direction. Top inset: Energy dependence of the spin accumulation is induced in the graphene channel through spin Hall angles, with the conduction band minimum marked by a a FM electrode whose magnetization direction determines that dot-dashed line. of the spin density injected into graphene underneath. This generates a pure-spin current that diffuses toward—and is absorbed by—the remote TMD. It is assumed that the spin Fig. 2 [49]. The results are displayed in the inset of Fig. 3. current is absorbed by the TMD at its edge and continues (We note that while the charge conductivity along x and y is to follow the diffusion direction, given that the spin resis- slightly anisotropic, resulting in an equally anisotropic SHA, tance in the TMD is two orders of magnitude lower than in α α |θxy | and |θyx | are still very similar [49].) At EF , the SHA for graphene for λys = 30 nm or λzs = 10 nm [49]. By ISHE, a spins pointing along y and z is ≈10% (with opposite sign). transverse voltage VISHE develops on the TMD, which can be Remarkably, both increase substantially when approaching measured along its length as illustrated in Fig. 4. In experi- y the band edge, at which point |θxy | slightly overcomes |θxy z | ments, the diffusing spins can be controlled by spin precession y with values as large as |θxy | ≈ 80%. We also computed the in a noncollinear magnetic field B. To capture this situation, SHC and the SHA with the Kubo formula and obtain the we generalized the Bloch diffusion equations to account for same result both qualitatively and quantitatively (supplemen- anisotropic spin diffusion and calculated VISHE (B) using the tary Fig. 11). The increase of θxy is attributed to hot spots of approach described in Ref. [63] (which accurately reproduces SBC near the bottom of the electron pockets [15,38] (Supple- CSI in real devices). Figure 4 shows the precession response mentary Figs. 4–6), which directly determine the SHC/SHA for two selected orientations of the TMD crystal in the magnitude [18,49,59,60]. Importantly, our combined results limit of full absorption (RISHE ≡ VISHE /I0y ) [49]. We observe 12 8 By Bx RISHE RISHE Bz Bz 8 y 4 z y x z RISHE (:) x RISHE (:) 4 0 0 -4 -8 -4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 B (T) B (T) FIG. 4. Simulated response of the inverse SHE (RISHE ) to spin precession for two orientations of the TMD crystal (coordinate axes in the insets). The device geometry is shown in the insets, with the TMD depicted in yellow and the FM injector in red (magnetization indicated by an arrow). The polarization of the spin current reaching the TMD (J αs ) is controlled externally with a magnetic field, B, oriented either along the graphene channel (dashed lines) or out-of-plane (solid lines). Typical experimental device dimensions were used in the simulation [49]. 043230-4 LOW-SYMMETRY TOPOLOGICAL MATERIALS FOR LARGE … PHYSICAL REVIEW RESEARCH 3, 043230 (2021) magnitudes of RISHE nearly three orders of magnitude larger terials are preferred for a targeted pursuit. This is because than the values reported for graphene/TMDs [62–64,67] and SOC combined with the underlying band inversions of, e.g., graphene/bulk-WSMs [37,39]. This is a direct consequence topological insulators or topological semimetals, invariably of the extremely large SHA predicted here for MoTe2 [68]. leads to SBC hot spots [22,71,72]. Since space groups have The essence of the experiment is that the precession also been recently classified according to their compatibility response depends strongly on the crystal orientation. As evi- with different topological phases [73,74], one may systemat- denced in Fig. 2, the spin relaxation in the TMD is anisotropic, ically select those topological classes whose symmetries are and the CSI depends crucially on both the majority spin ori- simultaneously compatible with PST and canted SHC. entation and crystalline orientation. In Fig. 4(a), the TMD’s To be more specific, we provide materials that satisfy the crystallographic y axis is transverse to the spin propagation. qualities mentioned above, thus making them potential candi- A magnetic field parallel to z causes spins to precess in the dates for showing large CSI figures of merit. We focused on graphene plane but, according to Fig. 3, only the y spin pro- materials from the space groups reported in Ref. [69] as they jection contributes to the ISHE signal with an efficiency of present PST. Although they only comprise orthorhombic crys- y |θxy |; RISHE is symmetric with respect to the sign of B because tals with no inversion symmetry, we note that extending the the magnetization at the FM injector is parallel to y, resulting symmetry analysis of Ref. [69] to other crystal systems may in the maximum signal at B = 0. When the field is parallel to provide greater number of potential SHE materials with PST. x, the spins acquire a z component (in addition to that in y), We searched these space groups in the Topological Material which is asymmetric with respect to B and adds a contribution Database [74–77], and among the several topological insula- y to the ISHE with an efficiency of |θxy z |; because |θxy |  |θxy z |, tors and semimetals, we searched which of those had already the signal remains roughly symmetric. In Fig. 4(b), the crys- been experimentally characterized. At least three topological tallographic y axis is parallel to the spin propagation. As the y materials were found: the type-II Weyl semimetal Ta3 S2 (SG and z directions are now orthogonal to the FM magnetization, 39) [78] and AuSn4 (SG 41), which is a topological nodal-line the line shapes are antisymmetric. The signal is zero at B = 0 semimetal [79]. According to the symmetry requirements for and, by sweeping B from negative to positive along z (x), multicomponent SHE [47], the space groups from Ref. [69] the spin component along y (z) changes sign. Therefore, two in their bulk form cannot host a canted SHE. However, these combined observations in this proposed experiment represent restrictions are lifted in systems with glide mirrors and screw a “smoking gun” demonstration of the intrinsic canted SHE axis with vertical translation when going to the monolayer predicted in this work: (i) RISHE (B) should display a different limit, as in 1Td -MoTe2 . This is indeed the case for TaIrTe4 line shape under different field orientations for a fixed TMD (SG 31), a van der Waals material being a Weyl semimetal crystal; and (ii) rotation of the crystal converts the line shapes in three dimensions but a topological insulator in monolayer from predominantly symmetric to antisymmetric. form [80–82]. Overall, we have identified Ta3 S2 , AuSn4 , and TaIrTe4 as potential compounds for large CSI, with monolayer TaIrTe4 also being compatible with multicomponent SHE. V. GENERALIZATION TO OTHER SYSTEMS Our numerical calculations show that the 1Td and 1T phases of monolayer TMDs exhibit a canted spin Hall effect with large spin diffusion lengths and spin Hall angles. We precisely pinpoint this to the concurrence of a PST (which VI. POTENTIAL APPLICATIONS naturally enhances λs ) and hot spots of SBC near the band edge. Additionally, the reduced symmetry has two fundamen- Discovering materials with largest λs θxy has been a tal consequences: It allows the extra nonzero components in long-standing challenge. This is partly because traditional the SHC tensor (hence a canted SHE) and it allows the PST understanding of SHE and spin diffusion posits that, while (hence a large λs ). large SOC boosts the generation of spin current via SHE It is important to note that, though we focused here on (quantified by θxy ), it detrimentally reduces the spin diffusion the MoTe2 family in order to quantitatively demonstrate all length [41]. Our extensive and realistic quantum simulations the above features, the concurrence of PST and large SBC directly demonstrate that materials hosting low-symmetry- may now be expected in precisely identifiable space groups enabled PST (even if only approximate) break free from that (SG). On the one hand, Tao and Tsymbal [69] have re- adverse compromise while, at the same time, displaying a cently shown that robust PSTs (i.e., not relying on fine-tuned new canted SHE, which greatly increases the geometrical parameters [52]) likely arise in crystals belonging to some flexibility of possible SHE-based devices. These results come nonsymmorphic space groups; this symmetry-based approach at a time of impressive achievements in using 2D materials can be extended to enumerate all compatible crystal families to carry spin currents over long distances and controlling their and the features of their allowed PSTs. On the other hand, flow by electrostatic gating [83–85]. This should allow prompt crystal families compatible with SHC components other than exploration beyond our proof-of-principle system, MoTe2 , σxy z have been also enumerated [47,70], thereby identifying thereby accelerating the potential delivery of low-power spin- all the possible material platforms for canted, multicomponent electronic devices and circuits [86]. For example, the spin SHE. Finally, the magnitude of intrinsic SHE depends directly polarization generated by the canted SHE can exert an out- on the strength of the SOC and on the existence of nonzero of-plane antidamping torque in magnets with perpendicular SBC (see our Supplemental Material’). Although both might magnetic anisotropy [19,87,88], which are essential for next- be serendipitously large in some materials, topological ma- generation, high-density spintronic applications [2,89]. 043230-5 MARC VILA et al. PHYSICAL REVIEW RESEARCH 3, 043230 (2021) VII. CONCLUSIONS nontrivial electronic topology, respectively. We finally men- tion that a much larger range of possible interesting materials Our models and quantum transport calculations of the spin- should be available by engineering proximity effects and tronic response of 1Td and 1T MoTe2 monolayers reveal the interfacial symmetries, as discussed for van der Waals het- origin of a novel, canted SHE with long spin diffusion lengths, erostructures [93–95]. which reflects the unconventional spin textures allowed by their reduced symmetry and strong SOC. The obtained CSI figure of merit λs θxy ≈ 1–50 nm is superior to that of tra- ACKNOWLEDGMENTS ditional spintronic materials (Pt, Au, W, and Ta) [41,43] by up to two orders of magnitude. Given the similar electronic M.V. acknowledges support from “La Caixa” Founda- structures of MoTe2 and WTe2 , including the persistent canted tion and the Centre for Advanced 2D Materials at the spin texture [90], comparable performance is expected in the National University of Singapore for its hospitality. X.W. ac- latter. Our findings also call for a careful analysis of SHE mea- knowledges the Agence National pour la Recherche Flagera surements, since the interpretation of all-electrical detection in GRANSPORT funding. ICN2 authors were supported by Hall bars [38,91,92] usually ignores the possibility of multiple the European Union Horizon 2020 research and inno- spin Hall components. We show how the presence of canted vation programme under Grant Agreements No. 881603 SHE can be experimentally identified by reciprocal SHE and (Graphene Flagship) and No. 824140 (TOCHA, H2020- how the different SHC contributions may be isolated in a spin FETPROACT-01-2018). ICN2 is funded by the CERCA precession setup. 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