Spin-orbit coupling induced valley Hall effects in transition-metal dichalcogenides

In transition-metal dichalcogenides, electrons in the K-valleys can experience both Ising and Rashba spin-orbit couplings. In this work, we show that the coexistence of Ising and Rashba spin-orbit couplings leads to a special type of valley Hall effect, which we call spin-orbit coupling induced valley Hall effect. Importantly, near the conduction band edge, the valley-dependent Berry curvatures generated by spin-orbit couplings are highly tunable by external gates and dominate over the intrinsic Berry curvatures originating from orbital degrees of freedom under accessible experimental conditions. We show that the spin-orbit coupling induced valley Hall effect is manifested in the gate dependence of the valley Hall conductivity, which can be detected by Kerr effect experiments. The valley Hall effect in transition metal dichalcogenides has been studied as a potential mean to develop new electronic and optoelectronic devices. The authors theoretically demonstrate that valley Hall effect can be derived from spin degrees of freedom, which is distinct from the conventional orbital related type.

V alley degrees of freedom emerge from local extrema in electronic band structures of two-dimensional Dirac materials. When spatial inversion symmetry is broken in such systems, valley-contrasting effective magnetic fields can arise in momentum space, known as Berry curvature fields 1,2 . Upon application of in-plane electric fields, the Berry curvature drives carriers from opposite valleys to flow in opposite transverse directions, leading to valley Hall effects (VHEs) 3,4 . It was first predicted that VHEs can exist in gapped graphene materials, where global inversion breaking is introduced by h-BN substrates 5 or external electric fields 6,7 . More recently, valley Hall phenomena were proposed in monolayer transition-metal dichalcogenides (TMDs) 8 , in which nontrivial Berry curvatures result from intrinsically broken inversion symmetry in the trigonal prismatic structure of their unit cells 9 . Because of its versatility to couple to optical [10][11][12][13][14][15][16][17] , magnetic [18][19][20][21][22] , and electrical 23,24 controls, valley Hall physics in TMD-based materials have been under intensive theoretical and experimental studies in recent years.
In this work, we show that the coexistence of Ising and Rashba SOCs in gated/polar TMDs results in novel valley-contrasting Berry curvatures and a special type of valley Hall effect, which we call spin-orbit coupling induced valley Hall effects (SVHEs). In contrast to conventional Berry curvatures due to inversionasymmetric hybridization of different d-orbitals 8 , the new type of Berry curvatures originates from inversion-asymmetric spin-orbit interactions. To distinguish their physical origins, we refer to the Berry curvature induced by SOCs as spin-type Berry curvatures, and the conventional Berry curvatures/valley Hall effects from orbital degrees of freedom as orbital-type Berry curvatures/orbital VHEs. Importantly, under experimentally accessible gating 33,38 , spin-type Berry curvatures near the conduction band edge can reach nearly ten times of its orbital counterpart. Thus, in gated or polar TMDs the SVHE can dominate over the orbital VHE, which significantly enhances the valley Hall effects in a wide class of TMDs and enriches the valley Hall phenomena in 2D Dirac materials. In addition, the SVHE proposed in this study provides a novel scheme to manipulate the valley degrees of freedom of TMD materials.

Results
Massive Dirac Hamiltonian and spin-type Berry curvatures from Ising and Rashba SOCs. To illustrate the spin-type Berry curvature and spin-orbit coupling induced valley Hall effect (SVHE) in TMDs, we consider gated monolayer MoS 2 as an example throughout this section, but the predicted effects generally exist in the whole class of gated TMDs or polar TMDs. In recent experiments, upon electrostatic gating the conduction band minima near the K-valleys can be partially filled 33,38 , where the electron bands originate predominantly from the 4d z 2 -orbitals of Mo-atoms 30,32 . Under the basis formed by spins of d z 2 -electrons, the effective Hamiltonian near the K-valleys for gated MoS 2 can be written as 33,40,54 : Here, ξ c k ¼ jkj 2 2m Ã c À μ denotes the usual kinetic energy term, m Ã c is the effective mass of the electron band, μ is the chemical potential, k = (k x , k y ) is the momentum displaced from K(−K)-valleys, ϵ = ± is the valley index. The β c so -term refers to the Ising SOC which pins electron spins to out-of-plane directions (depicted by the orange arrows in Fig. 1a). The origin of Ising SOC is the breaking of an in-plane mirror symmetry (mirror plane perpendicular to the 2D lattice plane), as well as the atomic SOC from the transition metal atoms. The α c so -term describes the Rashba SOC which pins electron spins in in-plane directions with helical spin textures (indicated by the golden arrows in Fig. 1a). Rashba SOC will arise when the out-of-plane mirror symmetry (mirror plane parallel to the lattice plane) is broken by gating or by lattice structure (as in the case of polar TMDs 45,46 ). Clearly, H spin has the form of a massive Dirac Hamiltonian (by neglecting the kinetic term which has no contribution to Berry curvatures), and the Ising SOC plays the role of a valley-contrasting Dirac mass, which is on the order of a few to tens of meVs 32 .
Importantly, the Pauli matrices σ = (σ x , σ y , σ z ) in Eq. (1) act on spin degrees of freedom. This stands in contrast to the massive Dirac Hamiltonian in ref. 8 : where the Pauli matrices τ = (τ x , τ y , τ z ) act on the subspace formed by different d-orbitals. The V F -term results from electron hopping, and the Dirac mass Δ is generated by the large band gap (~2Δ) on the order of 1−2 eVs in monolayer TMDs 8 . As shown in Fig. 1a, Ising and Rashba SOCs result in nondegenerate spin sub-bands near the conduction band minimum. The energy spectra of upper/lower spin-subbands are given by . The Berry curvatures generated by SOCs in the lower spin-bands with energy E ϵ c;À ðkÞ is given by: Note that Ω c;À spin has valley-dependent signs due to the valleycontrasting Dirac mass generated by Ising SOCs. As a result, under an in-plane electric field, Ω c;À spin can drive electrons in the lower spin-bands at opposite valleys to flow in opposite transverse directions, which leads to transverse valley currents (Fig. 1b). To distinguish this novel phenomenon from the intrinsic VHE in monolayer TMDs 8 , we call this special type of VHE the spin-orbit coupling induced valley Hall effect (SVHE) due to its physical origin in spin degrees of freedom. Likewise, the Berry curvatures generated by Ising and Rashba SOCs are called spin-type Berry curvatures to distinguish it from the orbital-type Berry curvatures due to inversion-asymmetric mixing of different d-orbitals 8 .
We note that for the upper spin-band with energy E ϵ c;þ ðkÞ, we have Ω c;þ spin ¼ ÀΩ c;À spin . Therefore, valley currents from upper and lower spin-bands can partially cancel each other when both of them are occupied. However, non-zero valley currents can still be generated due to the population difference in the spin-split bands.
Based on Eq. (3), Ω c; ± spin has a formal similarity with its orbital counterpart 8 : However, we point out that Ω c; ± spin originates from a very different physical mechanism from Ω c orb and has important implications in valleytronic applications.
On one hand, the magnitude of Ω c orb in TMDs is generally small (~10 Å 2 ) due to the large Dirac mass from the band gap 2Δ 1 − 2 eV 8 . In contrast, for Ω c; ± spin , the Dirac mass β c so is on the order of a few meVs near the conduction band edges 32 . For gated MoS 2 , the Rashba energy can reach α c so k F % 3 meV at the Fermi energy 33 (see Supplementary Note 1 for details), which is comparable to the energy-splitting 2jβ c so j % 3 meV caused by Ising SOCs 32 . In this case, jΩ c; ± spin j near the conduction band minimum can be nearly ten times of jΩ c orb j (Fig. 1c). Therefore, the SVHE is expected to generate pronounced valley Hall signals in gated/polar TMDs.
On the other hand, the strength of Ω c orb is determined by parameters intrinsic to the material, thus can hardly be tuned. However, Ω c; ± spin has a quadratic dependence on the Rashba coupling strength α c so (Eq. 3), which can be controlled by external gating fields. As shown in Fig. 1d, jΩ c; ± spin j can be strongly enhanced by increasing α c so within experimentally accessible gating strength 33 . This suggests that the SVHE can serve as a promising scheme for electrical control of valleys in TMD-based valleytronic devices.
We note that the form of effective Hamiltonian in Eq. (1) also applies to the K-valleys in the valence band (see Supplementary Note 2), thus SVHEs can also occur in the valence band. Unfortunately, as we demonstrate below, the spin-type Berry curvature is much weaker in the valence band due to the giant Ising SOC strength β v so $ 100 meV near the valence band edge 8,[25][26][27][28][29][30]32 .
Interplay between spin-type and orbital-type Berry curvatures. In real gated/polar TMDs, the spin-type Berry curvature Ω spin always coexist with the orbital-type Berry curvature Ω orb . In this section, we demonstrate the interplay between Ω spin and Ω orb near the K-valleys (shown schematically in Fig. 2a, b). Specifically, using monolayer MoS 2 as an example, we study the total Berry curvatures at the K-points based on a realistic tight-binding (TB) model 32 which takes both Ω spin and Ω orb into account. The TB Hamiltonian is presented in the Method section and detailed model parameters are presented in the Supplementary Note 3.
For simplicity, we focus on Berry curvatures at K = (4π/3a, 0), and the physics at −K follows naturally from the requirement imposed by time-reversal symmetry: First, we study the conduction band case where the Ising SOC strength β c so is small. In the absence of gating, the total Berry curvatures Ω c; ± tot: in both spin-subbands at K = (4π/3a,0) consist of orbital-type contributions Ω c; ± orb only, both pointing to the negative z-direction 8 . By gradually turning on the Rashba coupling strength, Ω c; ± spin come into play and change Ω c; ± tot: dramatically. In particular, for the lower spin-subband, Ω c;À spin also points to the negative z-direction (Fig. 2a). This is due to the fact White arrows indicate out-of-plane gating fields/electric polarization labeled by E G . c Magnitudes of spin-type Berry curvature jΩ c; ± spin j near the conduction band edge (red solid curve) and orbital-type Berry curvature jΩ c orb j (black solid curve) near K-points. Rashba coupling strength is set to be α c so ¼ 21:4 meVÁ Å according to α c so k F % 3 meV [33], comparable to 2jβ c so j ¼ 3 meV 31,32 . Parameters for jΩ c orb j are set to be: Δ = 0.83 eV, V F = 3.51 eV ⋅ Å 8 . Clearly, jΩ c; ± spin j is nearly ten times of jΩ c orb j. d jΩ c; ± spin j as a function of α c so at the K-points. Evidently, jΩ c; ± spin j scales quadratically with α c so that β c so < 0 in molybdenum(Mo)-based TMDs 31,32 , which leads to a negative value of Ω c;À spin (Eq. 3). As a result, Ω c;À tot: keeps increasing its magnitude as α c so increases (red solid curve in Fig. 2c). For the upper spin-subband, however, Ω c;þ spin ¼ ÀΩ c;À spin , which is anti-parallel to its orbital counterpart Ω c;þ orb , thus they compete against each other as shown in Fig. 2a. As α c so grows up, Ω c;þ spin becomes comparable to Ω c;þ orb , resulting in a zero total Berry curvature Ω c;þ tot: ¼ 0 at certain α c so (indicated by the intersection between the blue curve and zero in Fig. 2c). As α c so increases further, Ω c; ± spin dominates, leading to different signs of total Berry curvatures in upper and lower spin bands, with Ω c;þ tot: > 0 and Ω c;À tot: < 0. Notably, in tungsten(W)-based TMDs Ising SOCs in the conduction band have a different sign β c so > 0 31,32 . In this case, Ω c;À spin competes with Ω c;À orb , while Ω c;þ spin aligns with Ω c;þ orb . This is contrary to the behaviors in molybdenum(Mo)-based materials. As a result, Ω c;À tot: in W-based materials can change its sign when Ω c;À spin dominates, similar to Ω c;þ tot: in Mo-based case (blue curve in Fig. 2c). As we discuss in the next section, this can reverse the direction of total valley currents. Similar plots as shown in Fig. 2 for W-based TMDs are presented in Supplementary Note 4.
In contrast to conduction bands, valence band edges in TMDs exhibit extremely strong Ising SOC with β v so $ 100 À 200 meV 8,[25][26][27][28][29][30] . This leads to very weak spintype Berry curvatures Ω v; ± spin (shown schematically in Fig. 2b). This is because electron spins near the valence band edges are strongly pinned by the Ising SOCs to the out-of-plane directions, and the Rashba SOCs due to gating or electric polarization cannot compete with Ising SOCs. As a result, the Berry phase acquired during an adiabatic spin rotation driven by Rashb SOC fields becomes negligible. Therefore, in the valence band the orbital-type contribution Ω v; ± orb generally dominates. As shown clearly in Fig. 2d, Ω v; ± tot: are almost insensitive to α v so and remain close to Ω v; ± orb at α v so ¼ 0. It is worth noting that the behavior of total Berry curvatures in Fig. 2c, d can be understood by considering spin-type and orbitaltype contributions separately. This is due to the fact that the total Berry curvature Ω n tot: at the K-points for a given band n can be written as the algebraic sum of Ω n spin and Ω n orb : Ω n tot: ðϵKÞ ¼ Ω n spin ðϵKÞ þ Ω n orb ðϵKÞ. Detailed derivations can be found in Supplementary Note 5.
Detecting spin-orbit coupling induced valley Hall effects. In this section, we discuss how to detect unique experimental signatures of SVHEs in n-type monolayer TMDs using Kerr effect measurements. In particular, we study the cases of molybdenum (Mo)-based and tungsten(W)-based TMDs separately.
As demonstrated in the previous section, for Mo-based materials the total Berry curvature in the lower spin-band Ω c;À tot: can be significantly enhanced by Ω c;À spin under gating (Fig. 2c). Therefore, when only the lower spin-bands are filled, the extra contribution from SVHEs can strongly enhance the total valley Hall conductivity σ V xy in gated/polar TMDs, which is expected to far exceed the intrinsic σ V xy from orbital VHEs. Moreover, when Ω c spin dominates, Ω c;À tot: has a different sign (Fig. 2c). When both spin-bands are filled, valley currents from upper and lower spin bands partially cancel each other, with finite valley currents generated from their population difference. In this case σ V xy is expected to increase at a lower rate as doping level increases. This behavior is very different from the orbital valley Hall effect for electron-doped samples: since Ω c; ± orb have the same   Fig. 2(c) at α c so ¼ 0), when both spin sub-bands are filled, σ V xy is expected to increase at a higher rate as a function of doping level.
To study this unique signature of σ V xy due to SVHEs, we calculate σ V xy for n-type monolayer MoS 2 using the tight-binding model 32 presented in the Method section. The σ V xy for electrondoped TMDs is given by (see Supplementary Note 6 for details): 2 f c;þ ðkÞΩ c;þ tot: ðkÞ þ f c;À ðkÞΩ c;À tot: ðkÞ Here, the integral is calculated near the K-point, and f c; ± ðkÞ ¼ f1 þ exp½ðE ϵ¼þ c; ± ðkÞ À μÞ=k B Tg À1 are the Fermi functions associated with the upper/lower spin-bands near the conduction band edge. In the limit T → 0, the calculated σ V xy as a function of chemical potential μ for gated (red solid curve) and pristine (black solid curve) monolayer MoS 2 are shown in Fig. 3a. The chemical potential μ is measured from the conduction band minimum.
When μ < 2jβ c so j, only the lower spin-band is occupied, i.e., f c, + (k) = 0. It is evident from Fig. 3a that as μ increases, the net σ V xy for gated MoS 2 (red solid curve in Fig. 3a) grows much more rapidly than the intrinsic σ V xy (black solid curve in Fig. 3a). For μ > 2jβ c so j, the intrinsic σ V xy starts increasing at a slightly higher rate, while the σ V xy for gated sample increases at a lower rate. To detect this distinctive flattening behavior in the σ V xy À μ curve due to SVHEs, we propose polar Kerr effect experiments ( Fig. 3b) which can directly map out the spatial profile of net magnetization in a 2D system 24,55 . In the steady state, valley currents J v ¼ σ V xy E ẑ generated by the electric field E (green arrows in Fig. 3b) are balanced by valley relaxations at the sample boundaries, which establishes a finite valley imbalance n V / σ V xy near the sample edges. Due to valley-contrasting Berry curvatures, the valley imbalance n V induces a nonzero out-of-plane orbital magnetization M z ∝ n V 1 , which can be measured by the Kerr rotation angle θ K , with θ K / n V / σ V xy 24 . Therefore, θ K as a function of doping level for intrinsic/gated monolayer TMDs are expected to exhibit similar features as the σ V xy À μ curves in Fig. 3a. In previous Kerr measurements on MoS 2 , the Kerr angle due to valley imbalance generated by orbital VHE is roughly θ K ≈ 60 μrad when both subbands are filled 24 . According to Fig. 3a, the SVHE can enhance σ V xy by nearly 4-5 times when μ > 2jβ c so j, hence we expect θ K ≈ 200-300 μrad at the same doping level for gated MoS 2 . Now we discuss the distinctive signature of SVHE in n-type tungsten(W)-based TMDs. As we pointed out in the last section, Ω c;À spin competes with Ω c;À orb in W-based materials due to β c so > 0. Therefore, when Ω c;À spin dominates, Ω c;À tot: has an opposite sign to Ω c;À orb . When only the lower spin-band is filled (i.e., μ < 2jβ c so j and f c, + (k) = 0), this changes the sign of σ V xy (Eq. 5), and the direction of valley currents is reversed.
To demonstrate the reversal of valley current directions due to SVHEs in W-based TMDs, we compare the σ V xy of a pristine monolayer tungsten-diselenide (WSe 2 ) and a polar TMD tungsten-selenide-telluride (WSeTe) where strong band-splitting . For Mo-based TMDs, SVHEs strongly enhance σ V xy in the regime μ $ 2jβ c so j comparing to the intrinsic value. This creates a significant valley imbalance n V and valley magnetization at the sample boundaries, which can be signified by a large Kerr angle θ K . c Total σ V xy versus chemical potential μ for polar TMD WSeTe (red curve) and pristine WSe 2 (black curve). Clearly, in the regime μ < 2jβ c so j the sign of σ V xy in WSeTe is reversed due to SVHEs. d Schematics for polar Kerr experiments to detect SVHEs in tungsten-based polar TMD WSeTe. The reversed valley current is signified by the sign reversal of θ K induced by Rashba SOCs is predicted 50 . Using fitted values of α c so and β c so for WSeTe (details presented in Supplementary Note 4), we calculate the σ V xy À μ curves for WSe 2 and WSeTe as shown in Fig. 3c. Clearly, for μ < 2jβ c so j, σ V xy of WSeTe has a different sign from that of WSe 2 . As a result, the valley currents in WSe 2 and WSeTe under applied electric field flow in opposite transverse directions (Fig. 3d). This leads to opposite valley magnetization on the same boundaries, which can be signified by the sign difference in θ K correspondingly 24 .
We note that the relation θ K ∝ n V discussed above relies on the fact that the valley orbital magnetic moments remain almost unaffected by Ω spin . This is explained in details in Supplementary Note 7.

Discussion
We discuss a few important aspects on SVHEs studied above. First, the novel SVHE, as well as its unique signatures studied above applies in general to the whole class of monolayer TMDs. In particular, for molybdenum-based materials, strong Ising and Rashba SOC effects in the conduction band, such as MoSe 2 and MoTe 2 32,50 , have sizable band-splitting of 20-30 meV near the band edge and exhibit pronounced signals of SVHEs. Detailed calculations of SVHEs in MoTe 2 are presented in Supplmentary Note 8.
Second, a strong gating field is not necessarily required to induce strong Rashba SOCs in TMDs. As we mentioned above, in polar transition-metal dichalcogenides MXY (M = Mo,W; X,Y = S, Se, Te and X ≠ Y) 45,46,50,51 , out-of-plane electric polarizations are built-in from intrinsic mirror symmetry breaking in the crystal structure. Thus, Ising and Rashba SOCs naturally coexist in these polar TMD materials without any further experimental design. This is very different from graphene-based devices where valley currents are generated by inversion breaking from substrates 5-7 or strains 56 . On the other hand, in heterostructures formed by TMD and other materials, interfacial Rashba SOCs can also emerge. For example, strong Rashba SOC has been reported recently in graphene/TMD hybrid structures 57 . In the cases mentioned above, one can use moderate gating to tune the Fermi level in the range μ $ 2jβ c so j and study the unique σ V xy À μ curve due to SVHEs (Fig. 3).
Third, the Berry phase in Eq. (5) for one K valley can also be generated in two-dimensional Rashba systems with large perpendicular magnetic field if orbital effects are ignored 58 . However, to generate a Zeeman splitting of a few meVs, an external magnetic field on the order of 100 T is needed 33 . In such a strong magnetic field, the orbital effects cannot be ignored. Therefore, the family of TMDs with large Ising SOCs is very unique for demonstrating the novel SVHE.
Lastly, due to Ising SOC, valley Hall effects are generically accompanied by spin Hall effects in TMDs 8 , which can also establish finite out-of-plane spin imbalance near the edges and contribute to polar Kerr effects. However, spin magnetic moments ≲μ B (μ B : Bohr magneton) are generally small compared to the orbital valley magnetic moments~3-4 μ B in TMDs. Thus, polar Kerr effects are expected to be dominated by orbital magnetization.

Methods
where refer to the spin-independent term and the atomic spin-orbit coupling term, respectively. μ is the chemical potential, and I 6×6 is the 6 × 6 identity matrix. The Rashba SOC term is given by And the Ising SOC in the conduction band is given by Details of the matrix elements above can be found in Supplementary Note 3.

Data availability
The data supporting the findings of this study are available within the paper, its Supplementary Information files, and from the corresponding author upon reasonable request.