Berry curvature in monolayer MoS 2 with broken mirror symmetry
BBerry curvature in monolayer MoS with broken mirror symmetry Kyung-Han Kim and Hyun-Woo Lee
Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea (Dated: November 13, 2018)An ideal 1H phase monolayer MoS has the mirror reflection symmetry but this symmetry isbroken in common experimental situations, where the monolayer is placed on a substrate. By usingthe k · p perturbation theory, we investigate the effect of the mirror symmetry breaking on the Berrycurvature of the material. We find that the symmetry breaking may modify the Berry curvatureconsiderably and the spin/valley Hall effect due to the modified Berry curvature is in qualitativeagreement with a recent experimental result [Science , 1489 (2014)], which cannot be explainedby previous theories that ignore the mirror symmetry breaking. I. INTRODUCTION
In solids of two-dimensional (2D) hexagonal struc-ture, an electron has not only spin but also valley de-gree of freedom, which acts as a pseudospin. Thespin can be used for information storage, transport,and manipulation, and is the central degree of free-dom for spintronics [1, 2]. It was recently realized [3–6] that the valley can play similar roles as the spin,opening the field of valleytronics. How to control thespin/valley degree of freedom is one of fundamental ques-tions in spin/valleytronics, and the spin/valley Hall effect(SHE/VHE) is one possible way to achieve such control.A prototypical material of 2D hexagonal structure isgraphene, which has been studied extensively. Recentlymonolayer transition metal dichalcogenides (TMD) alsohave attracted huge attention as a 2D hexagonal mete-rial. Unlike the graphene, a monolayer TMD may havedirect bandgap with suitable gap size and large spin-orbitcoupling (SOC). It is thus a good candidate material foroptoelectronic and spin/valleytronic devices [4].A 1H phase monolayer MoS [Figs. 1(a),1(b)] is prob-ably most popular among the monolayer TMD materi-als. In this material, the mirror reflection symmetry isrespected but the inversion symmetry is broken intrinsi-cally. Energy bands are split by the SOC with electronspins quantized along the out-of-plane direction. Re-cent experiments on the monolayer MoS investigatedSHE/VHE [7, 8], spin-orbit torque [9, 10], valley magne-toelectric effect [11], valley relaxation [12–15], and spinrelaxation [16–18]. Unfortunately many experimental re-sults remain unexplained, which motivates further theo-retical studies on this material.In this paper, we investigate the SHE and VHE ina monolayer MoS with the broken mirror symmetry.While the mirror reflection with respect to the mirrorplane [Fig. 1(b)] within the monolayer MoS may be agood symmetry for an ideal MoS monolayer suspendedin air, this symmetry is broken in common experimen-tal situations where a MoS monolayer is placed on asubstrate and subject to a gate voltage. We demon-strate that the mirror symmetry breaking may signifi-cantly modify the monolayers Berry curvature, which isan important source of the SHE/VHE [19]. This provides FIG. 1. (Color online) (a) Top and (b) side view of an ideal1H phase monolayer MoS , which consists of Mo atoms (largerdot) and S atoms (smaller dot) and has the mirror symme-try with respect to the middle sublayer plane. (c) Schematicband structure near the K point of an ideal monolayer MoS with the mirror reflection symmetry. Only the two lowest con-duction bands and the two highest valence bands are shown.Band gap energy is ∆ = E c , ↓ (0) − E v , ↓ (0) ≈ .
82 eV and spinsplitting energies are ∆ c ≈ v ≈
148 meV for con-duction and valence bands, respectively. About 70 meV above E c , ↓ (0), E c , ↑ ( q ) and E c , ↓ ( q ) become degenerate. an explanation as to why the experimental results [7, 11]on the SHE/VHE deviate from predictions of the previ-ous theoretical studies [20] that assume the mirror sym-metry. We calculate the spin and valley Hall conduc-tivities as a function of the mirror symmetry breakingstrength, which may be continuously modulated in ex-periments by applying a gate voltage. Qualitative agree-ment with recent experiments [7, 11] is found.The paper is organized as follows. In Sec. II A, we in-troduce the k · p perturbed Hamiltonian near the K pointof the monolayer MoS with the mirror symmetry. InSec. II B, we consider the mirror symmetry breaking ef-fect in terms of the effective Hamiltonian. In Sec. III, wecalculate the Berry curvature and orbital magnetic mo-ment using the effective Hamiltonian and compare our re-sult to recent experimental results. In Sec. IV, we discussvarious technical issues related with this paper. Finally,our main results are summarized in Sec. V. a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r II. THEORY
A monolayer MoS has a direct band gap at the Kand K (cid:48) points [21]. Since these two points are the timereversed images of each other, study on one point, saythe K point, is sufficient to understand properties of theboth points. We thus study only the K point whichis highly symmetric and has the property of C pointgroup. From the irreducible representations of C pointgroup [22], K point basis functions of the two lowest con-duction and the two highest valence bands may be writ-ten as | φ c (cid:105) = | d z (cid:105) , | φ v (cid:105) = 1 √ (cid:12)(cid:12) d x − y (cid:11) − i | d xy (cid:105) ) , (1)where the subscript c/v indicates conduction/valencebands. It is useful to introduce the Pauli matrix ˆ σ to dis-tinguish conduction and valence bands with ˆ σ z definedby ˆ σ z (cid:12)(cid:12) φ c/v (cid:11) = ± (cid:12)(cid:12) φ c/v (cid:11) . A. With Mirror Symmetry
In the presence of the mirror symmetry, the k · p per-turbation near the K point results in the effective Hamil-tonian H [20], H = ˆ I + ˆ σ z (cid:16) ε c , ↑ ( q ) ˆ I + ˆ s z ε c , ↓ ( q ) ˆ I − ˆ s z (cid:17) + ˆ I − ˆ σ z (cid:16) ε v , ↑ ( q ) ˆ I + ˆ s z ε v , ↓ ( q ) ˆ I − ˆ s z (cid:17) + α ( − q x ˆ σ x + q y ˆ σ y ) , (2)where ˆ s is the Pauli matrix for spin, q is the Blochmomentum measured with respect to the K point, and ε c/v , ↑ / ↓ ( q ) is quadratic energy-momentum dispersionnear the K point for the conduction/valence band withspin ˆ s z =+1/ −
1. Thus H describes the two lowest con-duction bands and the two highest valence bands nearthe K point [Fig. 1(c)]. The last term in Eq. (2) de-scribes the wavefunction hybridization between | φ c (cid:105) and | φ v (cid:105) as q moves away from the K point [20]. Recall-ing that the Berry curvature arises from the q -dependentchange of the wavefunction, the hybridization is crucialfor the Berry curvature. In principle, the Berry curvaturearises not only by the hybridization within the conduc-tion and valence bands depicted in Fig. 1(c) but also bythe hybridization between (cid:12)(cid:12) φ c/v (cid:11) and higher conductionbands and lower valence bands not shown in Fig. 1(c).However the DFT calculation [23] indicates that whenthe mirror symmetry is present, such hybridization withouter bands has negligible effect on the Berry curvaturesof the two lowest conduction bands and the two highestvalence bands, although it affects ε c/v , ↑ / ↓ ( q ). Thus forthe coefficient α in the last term of H , we take its value α = 3 .
512 eV · ˚A from the previous work [20] that capturesthe hybridization between | φ c (cid:105) and | φ v (cid:105) only. On the FIG. 2. (Color online) (a) Spin-momentum coupling turnsup when the mirror symmetry is broken. (b) The spin ex-pectation value of the lower conduction band (left) withoutthe mirror symmetry breaking and (right) with the mirrorsymmetry breaking. other hand, the quadratic dispersion ε c/v , ↑ / ↓ ( q ) is cho-sen in such a way that the energy eigenvalues E c/v , ↑ / ↓ ( q )of H agrees with the band structure from the DFT cal-culation [24]. B. Without Mirror Symmetry
Until now, we have considered the ideal monolayerMoS which has the mirror symmetry with respect toits 2D plane. However, the mirror symmetry is bro-ken when the monolayer MoS is placed on a substrate,which may generate an atomic scale potential gradient(or electric field) and modify the effective onsite andhopping energies of the S atoms in the bottom sub-layer of MoS . These effects can lead to the couplingbetween the spin ˆ s and the Bloch momentum q (spin-momentum coupling) [25–31]. Figure 2(a) illustrates themicroscopic process by which the spin-momentum cou-pling may emerge. n ↑ / ↓ denotes one of the four conduc-tion or valence bands shown in Fig. 1(c) whereas n (cid:48)↑ / ↓ de-notes outer bands not shown in Fig. 1(c), both with thegiven spin. The left panel in Fig. 2(a) illustrates the effectof atomic SOC, which induces the spin-orbit interactionbetween n ↑ and n (cid:48)↓ and between n ↓ and n (cid:48)↑ . The mid-dle panel illustrates the effect of the mirror-symmetry-breaking, which induces the hybridization between n ↑ and n (cid:48)↑ , and between n ↓ and n (cid:48)↓ . The right panel sum-marizes the combined effect of the atomic SOC and themirror symmetry breaking; n ↑ and n ↓ now couple to eachother through the virtual transitions to n (cid:48)↑ and n (cid:48)↓ .As a result, the effective Hamiltonian of the monolayerMoS with the broken mirror symmetry becomes H = H + H [27], where H = ˆ I + ˆ σ z (cid:2) β i ( q × ˆ s ) · ˆ z + β r q · ˆ s (cid:3) . (3)Here β i and β r are the spin-momentum coupling con-stants which depend on the degree of mirror symme-try breaking. Note that in addition to the conventionalRashba spin-momentum coupling q × ˆ s · ˆ z , the Weyl spin-momentum coupling q · ˆ s coexists. The correspondingcouplings for the valence bands are ignored for the rea-son specified below.Both the Rashba and Weyl couplings induce q -dependent spin character change of the wavefunction.Thus the couplings can affect the Berry curvature. Forthe valence bands, however, this effect is strongly sup-pressed (and thus ignored) since the intrinsic spin-dependent splitting of the valence bands, ∆ v ≈
148 meV[Fig. 1(c)], which exists even without the mirror sym-metry breaking, is much stronger than the Rashba andWeyl couplings. For the conduction bands, on the otherhand, this effect can be significant since the intrinsicspin-dependent splitting of the conduction bands, ∆ c ≈ β i and β r only through the combination β ≡ (cid:112) β + β . Here we thus estimate β . A recent exper-iment on electron-doped MoS placed on a SiO sub-strate [32] reports that the product β i k F ranges 1 ∼ k F is the Fermiwavelength. Considering that the electron density n = k / π in the experiment is of the order of 10 cm − , weconclude that β i is of the order of 10 meV · ˚A. For β r , wedo not have any direct estimation but we expect β r tobe comparable to or smaller than β i . This leads to theestimation of β ∼
10 meV · ˚A. Here we remark that thisestimation is at odds with the freestanding monolayermodel. A recent DFT calculation [27] examines the effectof a perpendicular electric field E z on a suspended idealmonolayer MoS and finds β = 0 . E z eV · ˚A, where E z is in units of V/˚A. Combined with E z ∼ .
03 V/˚A [33]estimated from gate voltages in experiments [7, 11], thiscalculation leads to β ∼ · ˚A, which is one or-der of magnitude smaller than the above estimation ob-tained from the experiment [32]. We attribute this differ-ence to the neglect of interatomic hopping between MoS and substrates (SiO ) in the freestanding monolayermodel [27]. Recent studies [28, 29, 31] report that in-teratomic hopping with environment atoms can enhancethe spin-momentum coupling strength more than one or-der of magnitude than estimated from the electric fieldstrength. As a reference for estimation of this hoppingeffect, we use results on a monolayer MoS –monolayergraphene heterostructure [34–37], for which it is reportedthat the graphene acquires the spin-momentum couplingstrength of ∼
160 meV · ˚A [34]. Efficient electrical gatecontrol of spin current is also demonstrated [35, 36]. Toobtain the estimation of β for our problem, one should take into account the fact that the hopping effect isinverse quadratically proportional to the energy spac-ing (Fig. 2) between bands connected by the hopping,and the energy spacing ( | . − . | / . (energy gap 1 . (8 . | . − | / . and graphene (0 eV) bands. This consid-eration implies that β for the MoS2-SiO system is aboutfactor 16 smaller than the corresponding value ∼ · ˚A for the MoS -graphene structure. One thus ob-tains β ∼
10 meV · ˚A for the MoS -SiO system, whichagrees with the above estimation obtained from the ex-periment [32].We remark that the estimated value of β ∼
10 meV · ˚Ais about two orders of magnitude smaller than the cor-responding value of ∼ · ˚A in strong spin-momentumcoupling systems such as Bi/Ag(111) [38]. Thus it is rea-sonable to expect that H in Eq. (3) would generate onlyweak effects. One example is the spin-conservation vio-lation. Whereas H in Eq. (2) conserves ˆ s z , H does not.But since β is very small, the spin conservation is violatedonly weakly. Figure 2(b) shows that for most values of q , the expectation value (cid:104) ˆ s z (cid:105) of the lowest conductionbands stay close to +1 or −
1, confirming the weaknessof the spin-conservation violation. The spin-conservationinduces sizable deviation of (cid:104) ˆ s z (cid:105) from ± q in which the lowest and the second lowest con-duction bands become degenerate. However our study inthe next section is focused on electronic states very closeto the K point ( | q | (cid:28) . − ), so the spin-conservationviolation is a weak effect and we use a spin index s todenote eigenstates. III. RESULT
In this section, we demonstrate that H can inducesizable correction to the Berry curvature even though β is small. This becomes possible since the two spinbranches of the lowest conduction bands are separatedby a small energy spacing of 3 meV. Its demonstrationgoes as follows.The Berry curvature Ω n,s ( q ) = ∇ q × i (cid:104) ns q | ∇ q | ns q (cid:105)· ˆ z at each band [39, 40] is given byΩ n,s ( q ) = i (cid:48) (cid:88) n (cid:48) s (cid:48) (cid:34) (cid:104) ns q | ∂H∂q x | n (cid:48) s (cid:48) q (cid:105) (cid:104) n (cid:48) s (cid:48) q | ∂H∂q y | ns q (cid:105) [ E n,s ( q ) − E n (cid:48) ,s (cid:48) ( q )] − (cid:18) ∂∂q x ↔ ∂∂q y (cid:19)(cid:35) , (4)where | ns q (cid:105) and E n,s ( q ) are respectively the energyeigenstate and the corresponding energy eigenvalue of H , and the summation over ( n (cid:48) , s (cid:48) ) runs over the fourbands of H excluding the case ( n (cid:48) , s (cid:48) ) = ( n, s ). Theprime symbol ( (cid:48) ) above Σ is introduced to denotethis exclusion. The second line of Eq. (4) contains ∂H/∂q x,y = ∂H /∂q x,y + ∂H /∂q x,y and allows one to FIG. 3. (Color online) Calculated values of the Berrycurvature near the K point (a) with the mirror symme-try ( β = 0 meV · ˚A) and (b) without the mirror symmetry( β = 20 meV · ˚A). Note the scale difference in vertical axes of(a) and (b). (c) Berry curvature (left) and (d) orbital mag-netic moment (right) at the K point as a function of β . seperate Ω n,s ( q ) into three contributions. When ∂H/∂q x and ∂H/∂q y are replaced by ∂H /∂q x and ∂H /∂q y , oneobtains the first contribution, which depends on β i / r only implicitly through | ns q (cid:105) , | n (cid:48) s (cid:48) q (cid:105) , E n,s ( q ), E n (cid:48) ,s (cid:48) ( q ), andcaptures the hybridization effect between (c, ↑ / ↓ ) and (v, ↑ / ↓ ) bands (orbital hybridization) due to the last termof H [Eq. (2)]. When ∂H/∂q x and ∂H/∂q y are replacedby ∂H /∂q x and ∂H /∂q y , one obtains the second con-tribution, which depends explicitly on β i / r and capturesthe hybridization effect between (c, ↑ ) and (c, ↓ ) bands(spin hybridization) [see Eq. (3)]. “Mixed” contributionscoming from ∂H /∂q x and ∂H /∂q y , for instance, vanishsince ∂H /∂q x and ∂H /∂q y induce completely differenttypes of hybridization. Thus only two contributions sur-vive. When ( n, s ) denotes (c, ↓ ), for instance, straight-forward calculation producesΩ c , ↓ ( q ) = Ω cvc , ↓ ( q ) + Ω ccc , ↓ ( q ) , (5)Ω cvc , ↓ ( q ) = 2 α ∆ (cid:2) E c , ↓ ( q ) − E v , ↓ ( q ) (cid:3) , (6)Ω ccc , ↓ ( q ) = 2 β ∆ c (cid:2) E c , ↓ ( q ) − E c , ↑ ( q ) (cid:3) , (7)where Ω cv n,s ( q ) and Ω cc n,s ( q ) are the Berry curvaturesfrom the orbital and the spin hybridizations, respectively.Ω c , ↑ ( q ), Ω v , ↓ ( q ), and Ω v , ↑ ( q ) can also be evaluated in asimilar way.The Berry curvatures Ω c/v , ↑ / ↓ ( q ) are evaluated for β = 0 meV · ˚A [Fig. 3(a)] and 20 meV · ˚A [Fig. 3(b)] as afunction of q x with q y = 0. Note that Ω c , ↑ / ↓ ( q ) is morethan one order of magnitude enlarged due to the mirror-symmetry breaking whereas Ω v , ↑ / ↓ ( q ) is only very weaklyaffected. This difference between Ω c , ↑ / ↓ ( q ) and Ω v , ↑ / ↓ ( q ) stems from H , which affects the wavefunction charac-ter only for the conduction bands. In case of Ω c , ↓ ( q ),Ω ccc , ↓ ( q ) is responsible for the enlargement of Ω c , ↓ ( q ). Weemphasize that Ω ccc , ↓ ( q ) is much larger than Ω cvc , ↓ ( q ) notbecause of its numerator but because of its denominator[Eq. (6)]; At the K point, for β = 20 meV · ˚A, its numer-ator 2 β is about (175) times smaller than the numer-ator 2 α of Ω cvc , ↓ ( q ), but its denominator ∆ ∼ (3 meV) is about (600) times smaller than the corresponding de-nominator ∆ ∼ (1 .
82 eV) . Their combined effect ismore than one order of magnitude enlargement. Fig-ure 3(c) shows the β dependence of the Berry curvatureat the K point. While Ω v , ↑ / ↓ ( q ) remains almost indepen-dent of β , Ω c , ↑ / ↓ ( q ) grows quadratically as β grows. Wealso calculate the orbital magnetic moment [41] m n,s ( q ) = i e (cid:126) (cid:48) (cid:88) n (cid:48) s (cid:48) (cid:34) (cid:104) ns q | ∂H∂q x | n (cid:48) s (cid:48) q (cid:105) (cid:104) n (cid:48) s (cid:48) q | ∂H∂q y | ns q (cid:105) [ E n,s ( q ) − E n (cid:48) ,s (cid:48) ( q )] − (cid:18) ∂∂q x ↔ ∂∂q y (cid:19)(cid:35) . (8)The result is shown in Fig. 3(d). Note that the mir-ror symmetry breaking barely affects the orbital mag-netic moment in contrast to its significant effects on theBerry curvature. This difference arises since m n,s ( q ) isinversely proportional to the energy difference in contrastto the difference square in case of Ω n,s ( q ). This differ-ence in the energy denominator makes the β effect muchweaker.Next we compare our calculation results with exper-iments [7, 11]. In the recent experiment [7], right/leftcircularly polarized light is used to selectively excite elec-trons near the K/K (cid:48) point from (v, ↑ / ↓ ) to (c, ↑ / ↓ ) andthe Hall conductivity is measured for the optically ex-cited states as a function of the gate voltage and thelight intensity, which controls the number of excited car-riers. In the degenerate limit, the Hall conductivity fromthe intrinsic and side-jump contributions [7, 41] is σ H ≈ − e (cid:126) Ω c , ↓ (0) · n Kc , ↓ , (9)where n Kc , ↓ is the photocarrier density under the assump-tion that excitation occurs only to the (c, ↓ ) band near theK point. The minus sign in Eq. (9) arises since the side-jump contribution is two times bigger and of oppositesign to the intrinsic contribution. The result is shown inFig. 4(a). Note that (i) σ H is essentially linear in n Kc , ↓ and(ii) the slope of the σ H vs n Kc , ↓ curve grows roughly as β .In comparison, the experimental data (Fig. 3 in Ref [7])indicates that (iii) σ H grows linearly with the density∆ n ph of the photoexcited carriers and (iv) the slope ofthe linear dependence varies with the gate voltage V g .Here ∆ n ph and n Kc , ↓ are related by n Kc , ↓ = ∆ n ph P , wherethe ratio P is reported to be much smaller than 1 sinceit is suppressed by valley relaxation [12–15] and moresignificantly by the spin relaxation within the conduc-tion bands [16, 17]. Such spin relaxation is crucial since FIG. 4. (Color online) (a) Valley Hall conductivity as afunction of the photocarrier density n Kc , ↓ for various values of β . (b) Spin/Valley Hall conductivity in a monolayer MoS . Ω c , ↑ ( q ) and Ω c , ↓ ( q ) tend to cancel each other [Fig. 3(b)].If we assume that P remains constant during the exper-iment, the calculation result (i) agrees with the experi-mental result (iii). Regarding (ii) vs (iv), we first notethat the slope in the experiment is roughly proportionalto ( V g + 20) . Thus if the effective β in the experiment iszero at V g = −
20 V and proportional to ( V g + 20 V), thecalculation result (ii) also agrees with the experimentalresult (iv). As a passing remark, we mention that thetheoretical curve in Fig. 3 in Ref. [7] is incorrect by fac-tor 4 due to an error in the Supplementary Material ofRef. [7]. Once this error is corrected, the curve becomes 4times steeper and agrees with our curve for β = 0 meV · ˚Ain Fig. 4(a) [except for minor difference due to materialparameter difference].On the other hand, another experiment [11] measuredthe VHE of an n -doped monolayer MoS in a differ-ent way, which does not use the optical excitations butinstead utilizes the optical detection (Kerr rotation) ofmagnetic moment accumulation at edges. The measureddependence on the gate voltage seems to arise mainlyfrom the chemical potental variation and the mirror-symmetry breaking effect appears to be weak. In thelow-energy limit, the corresponding valley Hall conduc-tivity is given by σ valleyH = 2 e (cid:126) (cid:88) s = ↑ / ↓ (cid:20)(cid:90) d q (2 π ) f c ,s ( q )Ω c ,s ( q ) − q F ,s π Ω c ,s ( q F ,s ) (cid:21) , (10)where the integration is performed near the K point, thefactor 2 is introduced to take into account both the Kand K (cid:48) points, and f c , ↑ / ↓ ( q ) and q F , ↑ / ↓ are the Fermidistribution function and Fermi momentum for the up-per/lower conduction band. The last term is the side-jump contribution. Figure 4(b) shows the β dependenceof σ valleyH in the n -doped monolayer MoS . Here we as-sume that the Fermi energy is fixed (regardless of β ) at10 meV above the lower conduction band bottom andthe temperature 10 K. Interestingly the calculated ratio σ valleyH /n , where n is the carrier density in the n -dopedsystem, is significantly smaller than the corresponding ra-tio σ H /n Kc , ↓ from Fig. 4(a); the two ratios are − . × − nS · cm and − . × − nS · cm for β = 10 meV · ˚A, and − . × − nS · cm and − . × − nS · cm for β = 20meV · ˚A. This deviation, which gets stronger for larger β , arises since the both spin-split conduction bands arepopulated in the optical-detection-based [11] scheme andthey tend to generate contributions of opposite sign dueto the spin hybridization, whereas only one conductionbands are preferrably populated in the optical-excitation-based [7] scheme and there is no cancellation. Thus thetwo detection schemes of VHE are not equivalent whenthe mirror symmetry is broken.We also calculate the spin Hall conductivity σ spinH ,which can be obtained from Eq. (10) by multiplying eachterm by the proper spin expectation value. Note that forlarge β , σ spinH is significantly larger than σ valleyH [Fig. 4(b)]since the two spin-split conduction band contributionsnow add up.Another interesting implication of our study is theRashba-Edelstein effect [42]. When the mirror sym-metry is broken, in-plane chiral spin component arisesfrom the Rashba and Weyl spin-momentum coupling be-tween the two lowest conduction bands in monolayerMoS [Eq. (3)]. The in-plane spin component and theBerry curvature are maximized at the degenerate points[Fig. 1(c)] with completely hybridized eigenstates. Insuch a situation, an electric field can generate spin accu-mulation (Rashba-Edelstein effect). A similar enhance-ment of the Berry curvature has been theoretically pro-posed [43, 44] for graphene in the context of the quan-tum anomalous Hall effect. There is also an experi-mental paper which reported the change of the inverseRashba-Edelstein effect depending on the Fermi energyin 2D Rashba system [45]. A recent experiment [8] ona heterostructure made of a n -doped monolayer MoS and a ferromagnet Co reported large inverse Rashba-Edelstein effect, which may be related to the strong spin-momentum coupling effects near the degenerate points.This relation may be tested experimentally through thematerial variation since the monolayer MoX (X=S, Se,Te) all exhibits the degenerate points whereas WX doesnot [24]. IV. DISCUSSION
Here we discuss a few related issues. The first issueis the substrate effect. In Sec. II B, the substrate effectwas taken into account through H . But substrates maygenerate other types of perturbations as well, which areignored in this paper. Here we argue that when the cou-pling with the substrate is weak, the neglect of otherperturbations can be a good approximation and H de-scribes the most important perturbation in terms of theBerry curvature correction. This point can be seen fromthe general expression of the Berry curvature in Eq. (4).When a perturbation is weak, it usually induces onlyminor corrections to the Berry curvature. But an excep-tional situation can occur when a perturbation inducesa hybridization between energy bands with small energyspacing since Eq. (4) contains the square of the energydifference in its denominator. In case of MoS , the twospin branches of the lowest conduction bands are sepa-rated by only 3 meV and thus most important pertur-bations by a substrate are those that hybridize the twospin branches. Considering that the two spin brancheshave opposite signs of ˆ s z , perturbations should be ableto flip ˆ s z to induce the hybridization between the twospin branches. Thus they should contain ˆ s x or ˆ s y . Alsoconsidering that ∂H/∂q x,y appears in the numerator ofEq. (4) instead of H , perturbations should depend on q .Thus near the K (or K’) point, most important perturba-tions are those that are linear in q and depend on ˆ s x orˆ s y . Note that the two perturbation terms of H [Eq. (3)]are exactly of this type, which shows that they are themost important perturbations. This justifies the neglectof other types of perturbations.The second issue is on the detection of the valley andspin Hall currents. Unlike charge Hall currents, spin Hallcurrents are not directly observable [46] and as far as weare aware of, there is no direct way to observe valley Hallcurrents either. One possible option is to infer the val-ley and spin Hall currents from the accumulated valleyand spin densities at side edges of a system. Althoughthis method is adopted in many experiments, the valleyand spin Hall conductivities calculated from bulk states[Eq. (10) for instance] may not match quantitatively withthe valley and spin accumulations at edges since valleyand spin are not strictly conserved. Relaxation approx-imations may be used to quantify the non-conservationeffect but this provides only a phenomenological descrip-tion. In a fundamental level, this issue has not beenclearly understood and goes beyond the scope of thispaper. We just remark that in case of spin, Ref. [47]proposed an alternative definition of a spin current andargued that the modified definition can improve the con-nection between the bulk spin Hall conductivity and theedge spin accumulation. In Sec. III, we defined a spincurrent in a conventional way (spin current ∝ spin timesvelocity) since this definition is more commonly used andalso the modified definition becomes identical to the con-ventional definition at the K and K’ points ( q = ).Lastly we discuss possible effects of the skew scatter-ing briefly. In case of the anomalous Hall effect, it iswell known [48] that the skew scattering is the dominantmechanism in very clean systems whereas the intrinsicBerry curvature is dominant in relatively dirty systems. In case of the MoS experiment [7], the measured Hallconductivity σ H is in rough agreement with the predic-tion of the Berry curvature theory [20] that neglects themirror symmetry breaking effect. Thus we suspect thatthe intrinsic Berry curvature is more important in this ex-periment. This has also motivated us to investigate thedeviation between the measured σ H and the theory [20]in terms of the Berry curvature modification by the mir-ror symmetry breaking. However there is a possibilitythat the skew scattering contributes to the deviation asinferred in Ref. [7], although the skew-scattering-basedtheory of the deviation has not developed yet. Refinedexperiments are needed to determine whether the mirrorsymmetry breaking or the skew scattering is the mainreason of the deviation. V. SUMMARY
In summary, we calculated the Berry curvature in amonolayer MoS in realistic situations where the mirrorsymmetry is broken. Our focus was not the well-knownRashba spin momentum coupling at the Γ point, but thespin momentum coupling at the K point which is a directband gap point. We found that the symmetry-breakingcontribution to the Berry curvature, which varies withthe gate voltage, may be larger than the previouslyknown Berry curvature in an ideal monolayer MoS withthe symmetry. However we estimated that the symme-try breaking barely affects the orbital magnetic moment.This provides an explanation to the gate voltage depen-dence of VHE in the recent experiment on VHE [7]. Italso provides an explanation as to why the two recentexperiments [7, 11] show different results with regards tothe gate voltage dependence of the VHE. 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