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1

Unconventional spin Hall effects in nonmagnetic solids

Arunesh Roy, Marcos H. D. Guimaraes, and Jagoda S lawinskaZernike Institute for Advanced Materials, University of Groningen,

Nijenborgh 4, 9747 AG Groningen, The Netherlands

Direct and inverse spin Hall effects lie at the heart of novel applications that utilize spins ofelectrons as information carriers, allowing generation of spin currents and detecting them via theelectric voltage. In the standard arrangement, applied electric field induces transverse spin currentwith perpendicular spin polarization. Although conventional spin Hall effects are commonly usedin spin-orbit torques or spin Hall magnetoresistance experiments, the possibilities to configure elec-tronic devices according to specific needs are quite limited. Here, we investigate unconventionalspin Hall effects that have the same origin as conventional ones, but manifest only in low-symmetrycrystals where spin polarization, spin current and charge current are not enforced to be orthogonal.Based on the symmetry analysis for all 230 space groups, we have identified crystal structures thatcould exhibit unusual configurations of charge-to-spin conversion. The most relevant geometrieshave been explored in more detail; in particular, we have analyzed the collinear components yield-ing transverse charge and spin current with spin polarization parallel to one of them, as well as thelongitudinal ones, where charge and spin currents are parallel. In addition, we have demonstratedthat unconventional spin Hall effect can be induced by controllable breaking the crystal symmetriesby an external electric field, which opens a perspective for external tuning of spin injection and de-tection by electric fields. The results have been confirmed by density functional theory calculationsperformed for various materials relevant for spintronics. We are convinced that our findings willstimulate further computational and experimental studies of unconventional spin Hall effects.

I. INTRODUCTION

In modern electronic devices that employ spin degreesof freedom, the mechanisms of conversion between spinand charge are essential to ensure all-electric control andlow consumption of energy [1]. The standard spin Halleffect (SHE) causes a transverse spin current (JS) in re-sponse to a charge current (JC) whose spin polarization(s) is perpendicular to both JS and JC . This effect isessential for many spintronics applications [2, 3], such asspin-orbit torques (SOT) which are widely exploited formagnetic memories [4–8], and plays a crucial role in thespin Hall magnetoresistance effect [9, 10]. Because theintrinsic, and often dominant contribution to the SHEdepends only on the electronic structure of a crystal, ithas been intensely explored via first-principles calcula-tions. Several materials with large spin Hall efficiencieshave been revealed, most of them belonging to elemen-tal metals with strong spin-orbit coupling (SOC) [11–13]as well as quantum materials with exotic band features,such as Weyl and Dirac nodal line semi-metals [14–16].Surprisingly, these efforts were mostly focused on findingmaterials with large magnitudes for the spin Hall conduc-tivity (SHC) in conventional configurations, where thespin polarization, spin current and charge current aremutually orthogonal. Nonetheless, the electrical gener-ation of a spin current with out-of-plane spin polariza-tion would be much more efficient for the manipulationof perpendicular magnetic anisotropy ferromagnets, suchas the ones used in modern high-density memory devices.As this effect is forbidden in highly-symmetric conven-tional materials, the exploration of the spin Hall effectswith unusual spin polarization and current directions is apromising field for future spintronic applications [17, 18].

The spin Hall effect can be in general separated be-

tween extrinsic and intrinsic contributions [19]. Whilethe extrinsic component can originate from side-jumpand skew-scattering by impurities, the intrinsic one isdetermined directly from the relativistic electronic struc-ture; all these contributions can be calculated in linearresponse theory [19, 20]. Because the charge and spincurrent are related through a third order spin Hall con-ductivity tensor whose indices may correspond to anyspatial direction (x, y or z), the charge-to-spin conver-sion could in principle occur in 27 different configura-tions. Still, the unconventional analogues of spin Halleffect (USHE) remained completely unexplored until fewyears ago, as they can manifest uniquely in crystals withparticular symmetries [21, 22]. Only recently, spin trans-port experiments performed in MoTe2 revealed compo-nents with unusual spin polarization and spin Hall ef-ficiency comparable with the conventional one [23, 24].While it clearly demonstrated their huge potential forspintronics applications, very few compounds allowingUSHE have been so far proposed. The efficient routeto search or design materials with appropriate character-istics is therefore needed.

In this paper, we provide a symmetry analysis for theallowed components of the SHC tensors for all 230 spacegroups, using the tools implemented on the Bilbao Crys-tallographic Server (BCS) [25, 26]. By revealing the spacegroups for which unconventional charge-to-spin conver-sion configurations are allowed, we make possible thepre-selection of potential candidates for highly efficientmanipulation of perpendicular magnetic anisotropy fer-romagnets. In order to facilitate the discussion and fur-ther research, we have classified these spin Hall effectsinto the following categories; (i) conventional SHE, eitherisotropic or anisotropic, where JC , JS , and s are mutuallyorthogonal; (ii) collinear SHE, where the charge and spincurrents are transverse, but spin polarization is parallel

2

FIG. 1. Classification of the spin Hall effects. We choose the charge current along x direction and illustrate (a) conventionalSHE with spin current along y and spin polarization along z (σz

yx), (b) collinear SHE with spin current and spin polarizationalong z (σz

zx), (c) longitudinal spin Hall effect with spin current along x and spin polarization along z (σzxx).

to one of them, i.e. Js ⊥ JC and JS,C ‖ s; and (iii) lon-gitudinal SHE where the charge current is parallel to thespin current (JS ‖ JC) and the spin polarization is uncon-strained. Based on these considerations, we have iden-tified materials belonging to each class, calculated spinHall conductivities using ab initio methods, and when-ever possible compared the results with the existing ex-perimental data. Importantly, we have also revealed thatunconventional spin Hall effect can be controlled by anexternal electric field, which has been confirmed by firstprinciples calculations in a prototypical material (SnTe).

The paper is organized as follows: Section II gives abrief overview of spin Hall conductivity in the frameworkof linear response theory. In Section III, we describe theemployed methodology, namely symmetry analysis basedon the TENSOR program [27] as well as details of thefirst principles simulations. In Section IV, we discuss theresults of symmetry analysis complemented by numer-ical predictions for various materials relevant for spin-tronics. Section V discusses unconventional spin Hallconductivity induced by applied electric fields. In Sec-tion VI, we formulate the conclusions and suggestionsfor further computational and experimental studies. TheAppendix summarizes allowed configurations for the spinHall effects for all 230 crystallographic space groups.

II. INTRINSIC SPIN HALL CONDUCTIVITY

In the framework of linear response theory, the spincurrent is expressed as:

J = σE × σ ⇒ J ij = σi

jkEk, (1)

which means that an applied electric field Ek induces aspin current along j with spin polarization along i. Inanalogy to the anomalous Hall effect, the SHC tensorcan be expressed by the Kubo formula [19, 20]:

σijk = −

( e

h

)

∫

d3k

(2π)3

∑

n

f(ǫn,k)Ωijk,n(k) (2)

in which f(ǫn,k) is the Fermi-Dirac distribution functionand the spin Berry curvature of the nth band is definedas:

Ωijk,n(k) = h2

∑

m 6=n

−2Im〈nk|J ij |mk〉〈mk|vk|nk〉

(ǫn,k − ǫm,k)2(3)

where vi = 1

h∂H/∂ki denotes the velocity operator and

J ij = vi, σj/2 = (viσj + σj vi)/2 is the spin current

operator. Equation (3) quantitatively describes the de-flection of the electron trajectories caused by spin–orbitinteraction that can intrinsically occur in materials. Itis governed by the strength of the spin-orbit couplingas well as the magnitude of the velocity and spin veloc-ity vectors in a specific direction in momentum space.Equations (2-3) can be evaluated via various approaches,such as Korringa–Kohn–Rostoker (KKR) method [28]and tight-binding (TB) Hamiltonians derived from firstprinciples calculations using either Wannier interpolationor the projections of wave functions on pseudo-atomicorbitals (PAO) [29–32]; the latter, used in this work, isdescribed in detail in Sec. III.

Equation (1) implies that the spin Hallconductivity σi

jk is a third-order tensor with

3 (electric field direction) × 3 (spin current direction) ×3 (spin polarization) = 27 independent componentssince each index can be set along any of the spatialdirections x, y, z. This means that the expression for theintrinsic spin Hall effect, or in a more general case, theKubo formalism describing the spin Hall conductivityin linear response theory [19], can refer to differentconfigurations of spin-to-charge conversion that haveessentially the same origin. However, the presenceor lack of unconventional spin Hall effect is strictlydetermined by the crystal symmetries. As we will showin the next sections, the form of the SHC tensor isentirely governed by the symmetries of a specific spacegroup which may enforce several components to be zero.

3

III. SYMMETRY ANALYSIS AND

COMPUTATIONAL DETAILS

We used the TENSOR program of Bilbao Crystallo-graphic Server (BCS) [27] to find the allowed spin Hallconductivity components σk

ij for all 230 crystallographicspace groups. To this aim, we have started with express-ing the tensor in Jahn’s notation in accordance with theBCS convention. As it is axial and antisymmetric in allthree indices, it can be written as eVVV, wheree denotes axial and symbolizes antisymmetric indicesfor the designated vector V. Then, we have directly usedTENSOR to determine which σk

ij components are allowedby symmetry. As for most space groups these allowedterms are not independent, we have also established thesymmetry-based dependencies. Our analysis can there-fore serve as a useful guide not only for experiments butalso for computational studies, reducing the number oftensor elements that need to be calculated.

Materials simulations were performed using densityfunctional theory (DFT), as implemented in the Quan-tum Espresso package [33, 34]. In all calculations, wetreated the ion-electron interaction with the fully rela-tivistic projected-augmented wave pseudopotentials fromthe pslibrary database [35] and expanded the electronwave functions in a plane-wave basis, setting the cutoffto 50 Ry. The exchange and correlation interaction wastaken into account via the generalized gradient approxi-mation (GGA) parameterized by the Perdew, Burke, andErnzerhof (PBE) functional [36]. The atomic coordinatesof the structures were relaxed with the convergence crite-ria for energy and forces to 10−6 Ry and 10−4 Ry/bohr,respectively. We performed the Brillouin zone (BZ) sam-pling at the DFT level following the Monkhorst-Packscheme [37], and converged the grid sizes separately foreach material.

The intrinsic contributions to the spin Hall conduc-tivity given by Eq. (2)-(3) were calculated in a post-processing step. First, the tight-binding Hamiltonianswere constructed from the projections of ab-initio wave-functions on atomic orbital bases following the implemen-tation in the PAOFLOW code [38, 39]. After interpo-lating the Hamiltonians on the ultra-dense k-points gridsconverged separately for every considered compound, thespin Berry curvatures were computed and integrated overthe BZ using the adaptive smearing method [40]. The ex-ternal electric fields, when applicable, were added at thelevel of the TB Hamiltonians.

IV. RESULTS AND DISCUSSION

The allowed spin Hall conductivity components foundvia symmetry analysis are summarized in the Appendixin the form of separate tables for each space group. Be-fore we discuss them in detail, let us make a few generalremarks on the relationships between the space groupssymmetries and the occurrence of specific components.We have noticed that the space groups possessing fewer

symmetry operations allow more independent compo-nents of the SHC tensor. For example, the simplest spacegroups - SG 1 and 2, allow for all 27 components. Thefurther we progress towards space groups with more sym-metry elements, one or more components get connectedto each other via symmetry operations. The high sym-metry space groups - SG 207 to 230 - are found to possessonly one independent SHC tensor component.

The symmetry analysis indicates that all space groupspermit the conventional SHE components, which impliesthe universality of the spin Hall effect. It is important tonote that its magnitude depends on the concentration ofcharge carriers as well as the strength of SOC, thus formany materials it might be negligible. While the univer-sal existence of the SHE could be expected, its demon-stration is not straightforward. Here, we deduce it usingsymmetry considerations for any space group, followingthe arguments suggested by Mook et. al. [41]. Based onSijk = viσkvj , which is the symmetry equivalent quan-

tity of the spin Berry curvature in Eq. (3), we determinethe allowed SHC tensor components. The quantity Si

jk isinvariant under any symmetry operation of a particularspace group. Therefore, in the linear regime,

σijk ≡ Si

jk

Operation−−−−−−−−→Symmetry

+Sijk allowed

−Sijk not allowed

(4)

This demonstrates that if Sijk is positive under a cho-

sen symmetry operation (e.g. translation, rotation, mir-ror reflection and inversion operation), the correspondingcomponent of the spin Hall conductivity is allowed; oth-erwise, it will be prohibited by symmetry.

A. Conventional spin Hall effects

In the conventional SHE, the charge current is trans-verse to both spin current and spin polarization. The cor-responding SHC tensor components, σi

jk when i 6= j 6= k,

yields six possible configurations (see Eq. 1) that can beconnected by symmetries. The most efficient spin Hallconductors are often isotropic, and identified among el-emental metals, such as Pt (SG 225) or β-W (SG 223)[11, 42]. Nonetheless, SHC anisotropy could be exploredin spin-logic devices, where, for example, SOTs vary instrength depending on the charge current direction. Low-symmetry crystals can be completely anisotropic withsix independent conventional components, while an in-crease in symmetry leads to a more isotropic behavior,as shown in Table I. Among the known compounds withanisotropic SHE, are bulk Td-WTe2 (SG 31), which hassix independent conventional configurations with valuesranging from -15 to -200 (h/e)(Ω×cm)−1 [43, 44], as wellas metallic rutile oxides, such as IrO2 (SG 136) withthree independent components and magnitudes between10 and -250 (h/e)(Ω×cm)−1 [15]. We have also foundlarge spin Hall conductivity in a pyrite-type structure,PtBi2 (SG 205), whose symmetries allow two indepen-dent SHE components. The values that we estimated

4

TABLE I. Conventional components by the number of inde-pendent components present in all space groups.

Independent Components 6 3 2 1Space groups 1-74 75-194 195-206 207-230

from first principles calculations are σzxy = 975 and σz

yx

= -742 (h/e)(Ω×cm)−1, which yield anisotropy of over30 %. The spin Hall tensors provided in Appendix arethus helpful to rationalize and interpret the existing re-sults as well as to design alternative spin Hall materials.

B. Collinear spin Hall effects

As we defined above, the collinear spin Hall conduc-tivity is the configuration with transverse charge andspin current, and with the spin polarization aligned ei-ther with the charge current or with the spin currentdirection. The components of the SHC tensor are de-noted as σi

jk when j 6= k and i is equal to j or k, e.g.σzzx. In Table II, we summarize the space groups that

allow for collinear spin Hall effects. Note that σiji and

σiij always occur together, although in general they are

not equal. While the components with spin polarizationparallel to the charge current were the first ones reportedexperimentally [23, 24], those with spin currents paral-lel to spin polarization could be even more relevant forapplications, as they can directly contribute to the gen-eration of out-of-plane spin-orbit torques, schematicallyillustrated in Fig. 2 (a). The materials that seem toreveal the largest potential for unconventional SOTs arelow-symmetry transition metal dichalcogenides (TMDs);we will thus analyze them in detail.

Let us first consider a bulk MoTe2 which typicallycrystallizes in a semimetallic monoclinic phase (1T ′ orβ-MoTe2, SG 11) with an inversion symmetry, two-foldscrew axis along the y direction (C2y) and a mirror plane(My), as indicated in Fig. 2 (b). Experiments typi-cally employ slabs consisting of several layers, but wenote that similar symmetry operations will hold for bothodd and even number of layers, therefore the bulk anal-ysis remains valid, as the form of the SHC tensor willnot change. Odd-layered samples still belong to SG 11,while in the case of even number of layers, the crystal willbe described by SG 6 [45]. Both cases will preserve thesame corresponding point group, and in consequence, thesame non-vanishing components of SHC. Our DFT cal-culations confirmed the presence of two pairs of collinearcomponents, namely σx

xy and σxyx as well as σz

zy and σzyz.

Even though their magnitudes are not large, the relatedspin Hall efficiencies (defined as ratios between spin Halland charge conductivity, θkij = σk

ij/σc, listed in Table III)may exceed 1 %, mostly due to the rather low charge con-ductivity (σc ∼ 1800 (Ω×cm)−1) of the semimetal [46].

These results are in a qualitative agreement with themeasurements of σz

yz in MoTe2 [23, 24, 47]; nevertheless,the observed values of spin Hall angles are larger than

TABLE II. Allowed space groups for collinear SHE.

Components Allowed Not allowed

σxxy, σx

yx

1,2, 3-15, 143-149,151,153, 157, 159,

162-163

16-142, 150, 152,154-156, 158,

160-161, 164-230

σxzx, σx

xz

1,2, 75-88,143-148, 168-176

3-74, 89-142,149-167, 177-230

σyxy, σy

yx

1, 2, 143-148, 150,152, 154-156, 158,160–161, 164–167

3-142, 149, 151,153, 157, 159,

162-163, 168-230

σyyz, σy

zy

1, 2, 75 -88,143-148, 168–176

3-74, 89-142,149–167, 177-230

σzyz, σz

zy 1,2, 3-15 16-230

σzzx, σz

xz 1,2 3, 4-230

predicted. The possible reason is that extrinsic contri-butions, which are not included in the calculations, maylead to an additional increase in spin current. Spin ac-cumulation due to the Rashba-Edelstein effect could alsoplay a role but additional studies will be needed in or-der to verify and numerically estimate its contribution.On the other hand, we can also compare the results withthe measurements of unconventional spin-orbit torquesgenerated by 1T ′-MoTe2 crystals [48]. The out-of-planeSOTs were observed for charge currents perpendicularto the mirror plane (My, see Fig. 2 (b)), which in ournomenclature corresponds to the σz

zy component. In con-trast, the electric current flowing along the mirror planedid not yield any unconventional spin-orbit torque, whichis again consistent with the theory, as the correspond-ing σz

zx conductivity is forced to zero by symmetry. Weemphasize that here the estimated spin Hall efficienciesagree quite well with the experiment, suggesting thatUSHE would play a major role in generating SOTs.

Similar analysis can be repeated for other TMDs.The experiments performed for WTe2 revealed unconven-tional SOTs occurring in the same configurations as in1T ′-MoTe2 [49, 50]. Although bulk WTe2 crystallizes inan orthorhombic phase, described by SG 31, which yieldsonly six independent conventional components, a few-layer system reduces to SG 6. The symmetry of the SHCtensor will be thus the same as in the case of 1T ′-MoTe2which explains the very similar experimental results. Incontrast, recent measurements performed for hexagonalTMDs, such as WSe2 and MoS2, have not shown anyunconventional spin-orbit torques [51–54]. This can beagain rationalized via a careful analysis of symmetries.The bulk SG 194 reduces to either SG 164 or SG 187 forrespectively even and odd number of layers in the slab[55]. As can be found in Appendix, these space groupsdo not allow for collinear spin Hall effects.

5

FIG. 2. (a) Schematic view of the charge-to-spin conversion generating a spin-orbit torque in the bilayer consisting of MoTe2and the ferromagnet (FM). The charge current (JC) induces a spin current (JS) with a collinear spin polarization in a directionperpendicular to the planes σz

zy. The spin current then exerts a torque on the magnetization (M) of the ferromagnetic layer.(b) Top and side view of bulk 1T’-MoTe2 crystal structure. The mirror plane (My) is denoted by a dashed line and the screwaxis C2y is indicated by a solid line. The parallelogram defines the unit cell.

TABLE III. Spin Hall efficiencies θkij = σkij / σc corre-

sponding to independent SHC components calculated for1T’-MoTe2. We use the experimental value of σc = 1.8× 103

(Ω×cm)−1[46]. All spin Hall angles are expressed in %.

θxxy θyyy θxyx θyzx θxyz θyzz θxzy θzxy θyxx θzyx θyxz θzyz θzzy-1.0 0.4 -0.1 2.2 0.4 0.0 2.0 -7.2 -0.3 10.0 -3.2 0.4 1.14

C. Longitudinal spin Hall effects

In the longitudinal spin Hall effect, the applied electricfield and induced spin current are parallel to each otherwhereas the spin polarization is independent. These SHCcomponents are denoted as σi

jk with j = k, and can befurther classified in two categories. The first one will cor-respond to components with i 6= j, e.g. σy

xx, which are al-lowed in space groups listed in Table IV. The second onewill refer to the special configurations in which i = j = k.They will describe USHE that are simultaneously longi-tudinal and collinear, representing a peculiar setup whereelectrons with spins parallel to the momentum are trans-mitted through a material and those with spins alignedanti-parallel are reflected, or vice versa. The correspond-ing space groups are summarized in Table V.

Longitudinal spin currents have been hardly studied innon-magnetic systems. 1T’-MoTe2 possesses three longi-tudinal components σy

xx, σyzz and σy

yy, but they have notbeen reported in experimental studies so far. This is mostlikely due to the small values for the SHC (see Table III).A recent computational high-throughout study revealedthat a sizable σz

zz component could be found in a metallicP7Ru12Sc2 (SG 174), but again these findings need to beconfirmed by experiments [56]. Moreover, another simul-taneously collinear and longitudinal component σx

xx waspredicted in a ferroelectric GeTe (SG 160) [57]. Conven-tional inverse spin Hall effect was detected in the heavily

TABLE IV. Allowed space groups for longitudinal SHE.

Components Allowed Not allowed

σyxx

1, 2, 3-15, 143-149,151, 153, 157, 159,

162-163

16-142, 150, 152,154-156, 158, 160

-161, 164-230

σzxx, σz

yy

1, 2, 75-88,143-148, 168-176

3-74, 89-142,149-167, 177-230

σxyy

1, 2, 143-148,150, 152, 154-156,158, 160-161, 164-

167

3-142, 149, 151,153, 157, 159, 162-

163, 168-230

σxzz 1,2 3-230

σyzz 1, 2, 3-15 16-230

doped ferroelectric samples yielding θSH ≈ 1 % [58], butthe unusual component has not yet been measured; weexpect the spin Hall efficiency to be lower than the con-ventional one. Further systematic search among semi-metals and (doped) semiconductors is needed in order toidentify other potential candidate materials.

V. UNCONVENTIONAL SPIN HALL EFFECT

INDUCED BY ELECTRIC FIELD

Finally, we will explore the possibility of inducing un-conventional spin Hall effect by modifying crystal sym-metry. Such a control can be achieved in situ using anexternal electric field or strain applied to any materialprovided that the space groups of the crystal withoutand with the stimulus are properly adjusted. Here, wewill consider the two-dimensional semiconductor SnTe, in

6

FIG. 3. Electric field induced unconventional spin Hall effects in 2D-SnTe. (a) Top and side view of SnTe crystal structure.In the upper panel, the dashed line denotes the mirror plane (My) which remains the same in the presence of an out-of-planeelectric field. In the bottom panel, the dashed line corresponds to the glide plane (Mz). Together with the two-fold screwrotation (C2x) the glide plane vanishes upon the electric field, as shown in the right-hand side of the structure. (b) Spin Hallconductivity vs chemical potential with and without electric field. Because σz

yx and σxxy are similar to respectively σz

yx andσxyx, the former have been omitted in the figure. The values of SHC are expressed in bulk units normalized using the effective

thickness of 1ML-SnTe (approximately 10 A) [59]. (c) Scheme of a prototypical device realizing spin injection induced byconstant electric field perpendicular to 2D-SnTe.

TABLE V. Allowed space groups with SHE that is collinearand longitudinal at the same time.

Components Allowed Not allowed

σxxx

1 - 2, 143 - 148,150, 152, 154 - 156,

158, 160, 161, 164-167

3 - 142, 149, 151,153, 157, 159,162, 168 - 230

σyyy

1, 2, 3 - 15,143-149, 151, 153,157, 159, 162 - 163

16 - 142, 150, 152,154 - 156, 158,

160 - 161, 164 - 230

σzzz

1 - 2, 75 - 88,143 - 148, 168 - 176

3 - 74, 89 - 142,149 - 165, 177 - 230

its monolayer form (1ML-SnTe) consisting of two atomiclayers as illustrated in Fig. 3 (a), which has been exploredin the context of the conventional spin Hall effect [59, 60].The two-dimensional SnTe (2D-SnTe) is described by SG31 and it is invariant under four symmetry operations: (i)identity E, (ii) mirror reflection (My) with respect to the

xz plane, (iii) glide reflection (Mz) combining a mirror

reflection with respect to the xy plane and a fractionaltranslation by a vector τ = (0.5a, 0.5b, 0) where a andb are the lattice constants, and (iv) two-fold screw ro-tation consisting of two-fold rotation around x and thefractional translation by the vector τ . It can be seenthat an electric field applied perpendicular to the plane(along z) will lift both the glide reflection and two-foldscrew rotation, reducing the crystal structure from SG31 to SG 6. In accordance with the tensor forms listed inAppendix, the number of allowed spin Hall componentswould increase from six to thirteen.

Figure 3 (b) shows the spin Hall conductivities calcu-lated for 2D-SnTe in the presence of an out-of-plane elec-

tric field ~E. First, we note that in a two-dimensional sys-tem neither charge nor spin transport can occur along theperpendicular direction (z) due to the lack of electronicdispersion with respect to kz. This eliminates all con-ventional and unconventional SHC tensor elements thatdescribe configurations with out-of-plane currents; thesezero SHCs are thus not shown. Because the componentsσzyx and σx

xy are similar to σzyx and σx

yx, respectively, theyhave also been omitted in Fig. 3 (b). The most importantresult emerging from the plots is that σy

yy, σyxx, σx

yx, σxxy

7

are indeed induced by the electric field, which confirmsour hypothesis derived from the symmetry analysis. No-tably, the component σy

yy can achieve large magnitudes,comparable with the conventional spin Hall conductivityfor energies close to EF .

We also observe that the values of the induced SHCsdepend on the magnitude of the electric field. This be-comes clear by comparing the dark and light blue curvescorresponding to fields of 0.2 and 1.0 V/nm, respectively.The intrinsic spin Hall conductivity is entirely deter-mined by the electronic structure, which is significantlyaltered by the electric field, therefore leading to a changein the SHC values over the entire energy range. Althoughthe conventional components seem to remain robust, we

have found that for the opposite sign of ~E, SnTe may be-come metallic, which means that spin Hall conductivitymust be carefully estimated in each case.

Last, we note that unconventional spin Hall effect in-duced by an electric field could be detected using ferro-magnetic voltage electrodes which probe locally the spinchemical potential in a device similar to the one pre-sented in Fig 3 (c). Since the realization of experimentsusing a two-dimensional material seems more challeng-ing, we emphasize that 1ML-SnTe could be replaced bya multilayer with the AA stacking. This configurationwill preserve SG 31 [59], and should yield essentially thesame effect of unconventional charge-to-spin conversion.

VI. CONCLUSIONS AND OUTLOOK

In summary, we have determined spin Hall conduc-tivity tensors for all 230 crystallographic space groups.While the conventional spin Hall effect is universallypresent in all of them, the unconventional componentsare crystal symmetry selective. We categorized spin Halleffects into conventional, collinear and longitudinal ones,analyzing important examples in each class, and provid-

ing a guide to design compounds suitable for applications.Based on the lists of space groups allowing a specific typeof spin Hall conductivity, further materials candidatescan be easily found using crystal structures databases,such as AFLOWLIB or Materials Project [61, 62].

In addition, we have revealed that unconventional spinHall components can be induced by an external electricfield which breaks symmetries of certain crystals, leadingto a change of the space group. We have verified thisconcept by performing DFT calculations for 2D-SnTe,and we have found that an additional component is aslarge as the conventional ones, suggesting the possibil-ity of the experimental confirmation. Further systematicsearch could reveal several materials with similar or en-hanced properties. We believe that the devices allowingspin injection tuned by the electric field will open excitingperspectives for spintronics.

Finally, we note that the simultaneous presence of sev-eral conventional and unconventional spin Hall compo-nents can be further explored towards the design of novelspintronic devices. One of the most interesting possibili-ties is to use materials with both collinear and collinear-longitudinal components, such as σy

yx and σyyy, in order

to accomplish a more efficient switching mechanism ofspin-orbit torques. Another option is a realization of el-ementary gates in spin logic circuits. We are convincedthat these results and prospects will stimulate furtherresearch from the theoretical and experimental side.

ACKNOWLEDGMENTS

J.S. acknowledges Rosalind Franklin Fellowship fromthe University of Groningen. M.H.D.G. acknowledgesthe support from the Dutch Research Council (NWO -grant STU.019.014). The calculations were carried outon the Dutch national e-infrastructure with the supportof SURF Cooperative, and on the Peregrine high perfor-mance computing cluster of the University of Groningen.

[1] J. Puebla, J. Kim, K. Kondou, and Y. Otani, Communi-cations Materials 1, 24 (2020).

[2] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,and T. Jungwirth, Reviews of Modern Physics 87, 1213(2015).

[3] T. Jungwirth, J. Wunderlich, and K. Olejnık, Nature Ma-terials 11, 382 (2012).

[4] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna,Y. Lyanda-Geller, and L. P. Rokhinson, Nature Physics5, 656 (2009).

[5] P. Gambardella and I. M. Miron, Philosophical Transac-tions of the Royal Society A: Mathematical, Physical andEngineering Sciences 369, 3175 (2011).

[6] K. Jabeur, G. Di Pendina, F. Bernard-Granger, andG. Prenat, IEEE Electron Device Letters 35, 408 (2014).

[7] G. Prenat, K. Jabeur, P. Vanhauwaert, G. Di Pend-ina, F. Oboril, R. Bishnoi, M. Ebrahimi, N. Lamard,O. Boulle, K. Garello, et al., IEEE Transactions on Multi-

Scale Computing Systems 2, 49 (2015).[8] N. Sato, F. Xue, R. M. White, C. Bi, and S. X. Wang,

Nature Electronics 1, 508 (2018).[9] S.-Y. Huang, X. Fan, D. Qu, Y. Chen, W. Wang, J. Wu,

T. Chen, J. Xiao, and C. Chien, Physical Review Letters109, 107204 (2012).

[10] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss,A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys-ical Review Letters 108, 106602 (2012).

[11] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa,Physical Review Letters 100, 096401 (2008).

[12] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, andR. A. Buhrman, Science 336, 555 (2012).

[13] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, andR. A. Buhrman, Applied Physics Letters 101, 122404(2012).

[14] Y. Sun, Y. Zhang, C. Felser, and B. Yan, Physical Review

8

Letters 117, 146403 (2016).[15] Y. Sun, Y. Zhang, C.-X. Liu, C. Felser, and B. Yan,

Physical Review B 95, 235104 (2017).[16] P. K. Das, J. S lawinska, I. Vobornik, J. Fujii, A. Regoutz,

J. M. Kahk, D. O. Scanlon, B. J. Morgan, C. McGuin-ness, E. Plekhanov, D. Di Sante, Y.-S. Huang, R.-S.Chen, G. Rossi, S. Picozzi, W. R. Branford, G. Panac-cione, and D. J. Payne, Physical Review Materials 2,065001 (2018).

[17] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A.Buhrman, J. Park, and D. C. Ralph, Nature Physics 13,300 (2017).

[18] L. Liu, C. Zhou, X. Shu, C. Li, T. Zhao, W. Lin, J. Deng,Q. Xie, S. Chen, J. Zhou, R. Guo, H. Wang, J. Yu, S. Shi,P. Yang, S. Pennycook, A. Manchon, and J. Chen, NatureNanotechnology 16, 277 (2021).

[19] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back,and T. Jungwirth, Reviews of Modern Physics 87, 1213(2015).

[20] M. Gradhand, D. V. Fedorov, F. Pientka, P. Zahn, I. Mer-tig, and B. L. Gyorffy, Journal of Physics: CondensedMatter 24, 213202 (2012).

[21] S. Wimmer, M. Seemann, K. Chadova, D. Koedderitzsch,and H. Ebert, Physical Review B 92, 041101 (2015).

[22] M. Seemann, D. Kodderitzsch, S. Wimmer, and H. Ebert,Physical Review B 92, 155138 (2015).

[23] P. Song, C.-H. Hsu, G. Vignale, M. Zhao, J. Liu, Y. Deng,W. Fu, Y. Liu, Y. Zhang, H. Lin, V. M. Pereira, and K. P.Loh, Nature Materials 19, 292 (2020).

[24] C. K. Safeer, N. Ontoso, J. Ingla-Aynes, F. Herling, V. T.Pham, A. Kurzmann, K. Ensslin, A. Chuvilin, I. Ro-bredo, M. G. Vergniory, F. de Juan, L. E. Hueso, M. R.Calvo, and F. Casanova, Nano Letters 19, 8758 (2019).

[25] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova,S. Ivantchev, G. Madariaga, A. Kirov, and H. Won-dratschek, Zeitschrift fur Kristallographie - CrystallineMaterials 221, 15 (2006).

[26] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato,and H. Wondratschek, Acta Crystallographica Section A62, 115 (2006).

[27] S. V. Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci,and J. M. Perez-Mato, Acta Crystallographica SectionA: Foundations and Advances 75, 438 (2019).

[28] M. Gradhand, D. V. Fedorov, F. Pientka, P. Zahn, I. Mer-tig, and B. L. Gyorffy, Phys. Rev. B 84, 075113 (2011).

[29] J. H. Ryoo, C.-H. Park, and I. Souza, Physical Review B99, 235113 (2019).

[30] J. Qiao, J. Zhou, Z. Yuan, and W. Zhao, Physical ReviewB 98, 214402 (2018).

[31] L. A. Agapito, A. Ferretti, A. Calzolari, S. Curtarolo, andM. Buongiorno Nardelli, Physical Review B 88, 165127(2013).

[32] L. A. Agapito, S. Ismail-Beigi, S. Curtarolo, M. Fornari,and M. Buongiorno Nardelli, Physical Review B 93,035104 (2016).

[33] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ-cioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris,G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis,A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello,L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P.Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcov-itch, Journal of Physics: Condensed Matter 21, 395502(2009).

[34] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau,M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz-zoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carn-imeo, A. Dal Corso, S. De Gironcoli, P. Delugas, R. A.DiStasio Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fu-gallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni,J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Kucuk-benli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L.Nguyen, H.-V. Nguyen, A. Otero-de-la Roza, L. Paulatto,S. Ponce, D. Rocca, R. Sabatini, B. Santra, M. Schlipf,A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser,P. Umari, N. Vast, X. Wu, and S. Baroni, Journal ofPhysics: Condensed Matter 29, 465901 (2017).

[35] A. D. Corso, Computational Materials Science 95, 337(2014).

[36] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re-view Letters 77, 3865 (1996).

[37] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188(1976).

[38] M. Buongiorno Nardelli, F. T. Cerasoli, M. Costa, S. Cur-tarolo, R. De Gennaro, M. Fornari, L. Liyanage, A. R.Supka, and H. Wang, Computational Materials Science143, 462 (2018).

[39] F. T. Cerasoli, A. R. Supka, A. Jayaraj, M. Costa, I. Siloi,J. S lawinska, S. Curtarolo, M. Fornari, D. Ceresoli, andM. Buongiorno Nardelli, Computational Materials Sci-ence 200, 110828 (2021).

[40] J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys.Rev. B 75, 195121 (2007).

[41] A. Mook, R. R. Neumann, A. Johansson, J. Henk, andI. Mertig, Physical Review Research 2, 023065 (2020).

[42] X. Sui, C. Wang, J. Kim, J. Wang, S. H. Rhim, W. Duan,and N. Kioussis, Physical Review B 96, 241105 (2017).

[43] J. Zhou, J. Qiao, A. Bournel, and W. Zhao, PhysicalReview B 99, 060408 (2019).

[44] B. Zhao, B. Karpiak, D. Khokhriakov, A. Johansson,A. M. Hoque, X. Xu, Y. Jiang, I. Mertig, and S. P. Dash,Advanced Materials 32, 2000818 (2020).

[45] R. Beams, L. G. Cancado, S. Krylyuk, I. Kalish,B. Kalanyan, A. K. Singh, K. Choudhary, A. Bruma,P. M. Vora, F. Tavazza, A. V. Davydov, and S. J. Stran-ick, ACS Nano 10, 9626 (2016).

[46] H. P. Hughes and R. H. Friend, Journal of Physics C:Solid State Physics 11, L103 (1978).

[47] A. M. Hoque, D. Khokhriakov, K. Zollner, B. Zhao,B. Karpiak, J. Fabian, and S. P. Dash, CommunicationsPhysics 4, 1 (2021).

[48] G. M. Stiehl, R. Li, V. Gupta, I. E. Baggari, S. Jiang,H. Xie, L. F. Kourkoutis, K. F. Mak, J. Shan, R. A.Buhrman, and D. C. Ralph, Physical Review B 100,184402 (2019).

[49] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, N. D.Reynolds, R. A. Buhrman, and D. C. Ralph, PhysicalReview B 96, 054450 (2017).

[50] D. MacNeill, G. Stiehl, M. Guimaraes, R. Buhrman,J. Park, and D. Ralph, Nature Physics 13, 300 (2017).

[51] J. Hidding, S. H. Tirion, J. Momand, A. Kaverzin,M. Mostovoy, B. J. V. Wees, B. J. Kooi, and M. H. D.Guimaraes, Journal of Physics: Materials 4, 04LT01(2021).

[52] Q. Shao, G. Yu, Y.-W. Lan, Y. Shi, M.-Y. Li, C. Zheng,X. Zhu, L.-J. Li, P. K. Amiri, and K. L. Wang, Nanoletters 16, 7514 (2016).

[53] W. Zhang, J. Sklenar, B. Hsu, W. Jiang, M. B.Jungfleisch, J. Xiao, F. Y. Fradin, Y. Liu, J. E. Pear-

9

son, J. B. Ketterson, et al., APL Materials 4, 032302(2016).

[54] S. Novakov, B. Jariwala, N. M. Vu, A. Kozhakhmetov,J. A. Robinson, and J. T. Heron, ACS Applied Materials& Interfaces 13, 13744 (2021).

[55] J. Ribeiro-Soares, R. M. Almeida, E. B. Barros, P. T.Araujo, M. S. Dresselhaus, L. G. Cancado, and A. Jorio,Physical Review B 90, 115438 (2014).

[56] Y. Zhang, Q. Xu, K. Koepernik, J. Zelezny, T. Jung-wirth, C. Felser, J. van den Brink, and Y. Sun,arxiv:1909.09605 (2019).

[57] H. Wang, P. Gopal, S. Picozzi, S. Curtarolo, M. Buon-giorno Nardelli, and J. S lawinska, npj ComputationalMaterials 6, 7 (2020).

[58] S. Varotto, L. Nessi, S. Cecchi, J. S lawinska, P. Noel,S. Petro, F. Fagiani, A. Novati, M. Cantoni, D. Petti,E. Albisetti, M. Costa, R. Calarco, M. BuongiornoNardelli, M. Bibes, S. Picozzi, J.-P. Attane, L. Vila,R. Bertacco, and C. Rinaldi, Nature Electronics (2021).

[59] J. S lawinska, F. T. Cerasoli, P. Gopal, M. Costa, S. Cur-tarolo, and M. B. Nardelli, 2D Materials 7, 025026(2020).

[60] J. S lawinska, F. T. Cerasoli, H. Wang, S. Postorino,A. Supka, S. Curtarolo, M. Fornari, and M. BuongiornoNardelli, 2D Materials 6, 025012 (2019).

[61] S. Curtarolo, W. Setyawan, S. Wang, J. Xue, K. Yang,R. H. Taylor, L. J. Nelson, G. L. Hart, S. Sanvito,M. Buongiorno Nardelli, N. Mingo, and O. Levy, Com-putational Materials Science 58, 227 (2012).

[62] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder,and K. Persson, APL Materials 1, 011002 (2013).

[63] J. F. Nye et al., Physical properties of crystals: their rep-

resentation by tensors and matrices (Oxford UniversityPress, 1985).

[64] C. P. Brock, T. Hahn, H. Wondratschek, U. Muller,U. Shmueli, E. Prince, A. Authier, V. Kopsky, D. Litvin,E. Arnold, et al., International tables for crystallographyvolume a: Space-group symmetry (2016).

APPENDIX

To obtain spin Hall conductivity tensors, we followthe convention from the Bilbao Crystallographic Server(BCS) and Physical Properties of Crystals by John Nye(Appendix B) [63]. Calculation of any physical ten-sor using the TENSOR program requires a well definedorthogonal basis. An orthogonal basis (a′, b′, c′) requires

a′ ‖ a, c′ ‖ c∗, b′ ‖ c′ × a (5)

where (a, b, c) are conventional crystal lattice vectors[64] in a reference frame (Ox‖a, Oy‖b, Oz‖c). Anorthogonal reference frame (x, y, z) can be obtainedsimilarly following Eq. (5) which is required to obtainthe symmetry allowed spin Hall conductivity tensors.

Below we list SHC tensors calculated for all 230 crys-tallographic space groups.

SG 1 - SG 2

σx =

σxxx σx

xy σxxz

σxyx σx

yy σxyz

σxzx σx

zy σxzz

27 components

σy =

σyxx σy

xy σyxz

σyyx σy

yy σxxy

σyzx σy

zy σyzz

27 independent σz =

σzxx σz

xy σzxz

σzyx σz

yy σzyz

σxzx σz

zy σzzz

SG 3 - SG 15

σx =

0 σxxy 0

σxyx 0 σx

yz

0 σxzy 0

13 components

σy =

σyxx 0 σy

xz

0 σyyy 0

σyzx 0 σy

zz

13 independent σz =

0 σzxy 0

σzyx 0 σz

yz

0 σzzy 0

SG 16 - SG 74

σx =

0 0 00 0 σx

yz

0 σxzy 0

6 components σy =

0 0 σyxz

0 0 0σyzx 0 0

6 independent σz =

0 σzxy 0

σzyx 0 00 0 0

SG 75 - SG 88

σx =

0 0 σxxz

0 0 σxyz

σxzx σx

zy 0

13 components,7 independents

σxzx = σy

zy, σxxz = σy

yz,σy =

0 0 σyxz

0 0 σyyz

σyzx σy

zy 0

σyzx = −σx

zy, σyxz = −σx

yz,σzxx = σz

yy, σzxy = −σz

yx σz =

σzxx σz

xy 0σzyx σz

yy 00 0 σz

zz

10

SG 89 - SG 142 σx =

0 0 00 0 σx

yz

0 σxzy 0

6 components,3 independents,σyxz = −σx

yz

σy =

0 0 σyxz

0 0 0σyzx 0 0

σzxy = −σz

yx, σyzx = −σx

zy σz =

0 σzxy 0

σzyx 0 00 0 0

SG 143 - SG 148

σx =

σxxx σx

xy σxxz

σxyx σx

yy σxyz

σxzx σx

zy 0

21 components,9 independent,

σxxx = - σx

yy = −σyyx =- σy

xy,σxzx = σy

zx,σy =

σyxx σy

xy σyxz

σyyx σy

yy σxxy

σyzx σy

zy 0

σxxz = σy

yz, σyzx = −σx

zy,σyxx = σx

yx = −σyyy = σx

xy,σyxz = σx

yz, σzxx = σz

yy,σzxy = σz

yy

σz =

σzxx σz

xy 0σzyx σz

yy 00 0 σz

zz

SG 149, 151, 153, 157,

159, 162, 163 σx =

0 σxxy 0

σxyx 0 σx

yz

0 σxzy 0

10 components,4 independent

σyxx = σx

xy = σxyx = −σy

yy

σy =

σyxx 0 σy

xz

0 σyyy 0

σyzx 0 0

σyzx = −σx

zy, σyxz = −σx

yz,σzxy = −σz

yxσz =

0 σzxy 0

σzyx 0 00 0 0

SG 150, 152, 154 - 156,

158, 160, 161, 164 - 167

σx =

σxxx 0 00 σx

yy σxyz

0 σxzy 0

10 components,4 independent

σxxx =−σx

yy = −σyyx = −σy

xy,σy =

0 σyxy σy

xz

σyyx 0 0

σyzx 0 0

σyzx = −σx

zy, σyxz = −σx

yz,σzxy = −σx

yzσz =

0 σzxy 0

σzyx 0 00 0 0

SG 168 - SG 176

σx =

0 0 σxxz

0 0 σxyz

σxzx σx

zy 0

13 components,7 independent

σxzx = σy

zy, σxxz = σy

yz,σy =

0 0 σyxz

0 0 σyyz

σyzx σy

zy 0

σyzx = −σx

zy, σyxz = −σx

yz,σzxx = σz

yy, σzxy = −σz

yxσz =

σzxx σz

xy 0σzyx σz

yy 00 0 σz

zz

SG 177 - SG 194 σx =

0 0 00 0 σx

yz

0 σxzy 0

6 components, 3independent σy =

0 0 00 0 σx

xy

0 σyzy 0

σyzx = −σx

zy,σyxz = −σx

yz,σzxy = −σz

yx

σz =

0 σzxy 0

σzyx 0 00 0 0

SG 195 - SG 206

σx =

0 0 00 0 σx

yz

0 σxzy 0

6 components, 2independent

σy =

0 0 σyxz

0 0 0σyzx 0 0

σzxy = σx

yz = σyzx

σyxz = σx

zy = σzyx

σz =

0 σzxy 0

σzyx 0 00 0 0

SG 207 - SG 230

σx =

0 0 00 0 σx

yz

0 σxzy 0

6 components, 1independent

σy =

0 0 σyxz

0 0 0σyzx 0 0

σzxy = σx

yz = σyzx =

−σzyx = −σx

zy = −σyxz

σz =

0 σzxy 0

σzyx 0 00 0 0