Thermally driven pure spin and valley current via anomalous Nernst effect in monolayer group-VI dichalcogenides
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Thermally driven pure spin and valley current via anomalous Nernst effect inmonolayer group-VI dichalcogenides
Xiao-Qin Yu , , Zhen-Gang Zhu , , ∗ Gang Su , † and A. -P. Jauho ‡ School of Electronic, Electrical and Communication Engineering,University of Chinese Academy of Sciences, Beijing 100049, China. Sino-Danish center for Education and Research,University of Chinese Academy of Sciences, Beijing 10049, China. Theoretical Condensed Matter Physics and Computational Materials Physics Laboratory,College of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. Center for Nanostructured Graphene (CNG), DTU Nanotech, Department of Micro- and Nanotechnology,Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark.
Spin and valley dependent anomalous Nernst effect are analyzed for monolayer MoS and othergroup-VI dichalcogenides. We find that pure spin and valley currents can be generated perpendicularto the applied thermal gradient in the plane of these two-dimensional materials. This effect providesa versatile platform for applications of spin caloritronics. A spin current purity factor is introducedto quantify this effect. When time reversal symmetry is violated, e.g. two-dimensional materials onan insulating magnetic substrate, a dip-peak feature appears for the total Nernst coefficient. Forthe dip state it is found that carriers with only one spin and from one valley are driven by thetemperature gradient. PACS numbers: 72.20.Pa,75.76.+j,75.70.Tj,73.63.-b
The generation of a spin current is a vital issue inspintronics in which information is carried and storedby spin, rather than charge. A traditional approachto generate a spin current is to drive a spin polarizedcharge current through a spin filter , where a fer-romagnet is used as the source of spin. However, thenet charge current may lead to dissipation in the device,and the introduction of a ferromagnet - semiconductorinterface complicates the applications. To eliminate thedissipation and complexity, pure spin current generationin semiconductors without a net charge current has beenrecently explored. Spin Hall effect has been pro-posed in semiconductors with strong spin-orbit coupling(SOC), which describes the generation of dissipationlessspin current in the perpendicular direction (y-direction)to an applied electric field (x-direction) without breakingtime-reversal symmetry. It can be attributed to a nonva-nishing Berry curvature of energy bands in the presenceof such an external electric field.Recently, a parallel research field to spintronics wasintroduced: spin caloritronics , which is an exten-sion and combination of spintronics and the conventionalthermoelectrics, aiming at increasing the efficiency andversatility of spin-involved thermoelectric devices . Sofar, the spin Seebeck effect was suggested to be away for spin current generation as a consequence of atemperature gradient. Usually a net charge current isstill present in loop circuit use.Monolayer MoS and other transition metal dichalco-genides (TMDC) represent a new class of two-dimensional (2D) materials, intrinsically behaving assemiconductors. Analogous to gapped graphene ,their Berry curvature in each Dirac cone is nonvanishingand leads to a series of anomalous transport phenomena,such as the anomalous Hall effect , valley Hall effect, anomalous Nernst effect (ANE) and valley ANE. How-ever, unlike graphene, inversion symmetry is broken inTMDCs and they exhibit strong SOCs, which lead tocoupling of the spin and valley degrees of freedom. Xiao et al . have studied the anomalous Hall effect and therelated spin and valley Hall effect . Mak et al. haveobserved a valley Hall effect in MoS transistors . Thestrong SOC offers a great chance for spintronics and spincaloritronics. Recently, Fang et al. found an extrin-sic mechanism via spin Seebeck effect to produce purespin current through temperature gradient in ferromag-netic/nonmagnetic hybrid metallic system. Here, we pro-pose a new intrinsic mechanism to generate a pure spincurrent via spin Nernst effect (SNE) driven by a temper-ature gradient in monolayer of MoS and other TMDCs.Since the transistor based on MoS has been fabricatedand has high equality of performance, the proposed ef-fect is highly applicable in reality. In addition, the purespin current persists for a wide range of gate voltage (forexample, 0.42 eV for WS ). We believe that this effectis very useful in spin caloritronics.The effective Hamiltonian of MoS around Dirac conesis ˆ H = at ( τ k x ˆ σ x + k y ˆ σ y ) + ∆2 ˆ σ z − λτ ˆ σ z −
12 ˆ s z , (1)where τ = ± λ refers to the spinsplitting at the top of valence band caused by the SOC,ˆ σ denotes the Pauli matrices for the two basis functionsof the energy bands, a is the lattice constant, t is thehopping integral, ∆ is the energy gap, and ˆ s representPauli matrices for spin. The energy eigenvalues are E nτs z = s z λτ n s ( kat ) + (cid:18) ∆ − s z λτ (cid:19) , (2)where s z (= ±
1) indicates the spin index, and n (= ± is deter-mined by Ω nτs z ( k ) = ˆ z · ∇ k × h µ nτs z | i ∇ k | µ nτs z i for 2Dmaterials, where ∇ k means directional derivatives withrespect to the momentum k , and µ nτs z is the periodicpart of the Bloch function. For massive Dirac fermionsdescribed by the effective Hamiltonian in Eq. (1), theBerry curvature isΩ nτs z ( k ) = − τ n a t ∆ ′ [(∆ ′ ) + 4( kat ) ] , (3)where ∆ ′ = ∆ − s z λτ .When a temperature gradient is applied, an electricfield develops in the opposite direction due to the See-beck effect. Besides the parallel effect, in the presence ofthe temperature gradient, the holes (or electrons) expe-rience a Lorentz-like force and thus move in the directionperpendicular to the diffusion current, which is the ANEinduced by the intrinsic non-vanishing Berry curvature .This is also the intrinsic mechanism contributing to theanomalous Hall resistivity . Thus, the velocity mul-tiplied by the entropy density gives rise to the anoma-lous Nernst coefficient (ANC) (details can be foundin Ref. 36) in each Dirac cone with a specified spin α c(v) τs z = 4 πα Z d k (2 π ) Ω nτs z ( k ) S nτs z ( k ) , (4)where c(v) represents the conduction(valence) band, cor-responding to n = 1( − α = ek B h , e isthe electron charge, h is the Planck constant, S nτs z ( k ) = − f nτs z k ln f nτs z k − (1 − f nτs z k ) ln (1 − f nτs z k ) ( k B hasbeen taken into α ) is the entropy density for valley τ , spin s z , and n band, k B is the Boltzmann constant,and f nτs z k is the Fermi distribution function. The en-tropy density develops a peak at E = E f , and is es-sentially zero when the energy is beyond the range of[ E f − k B T, E f + 5 k B T ]. The integration is performedover the neighborhood of one K ( − K ) point in the mo-mentum space. Ω nτs z ( k ) and S nτs z ( k ) can be expressedas functions of the modulus of the wave vector k , andwe can use the Debye model, namely, the integration istaken in a circular region centered at K ( − K ) point andthe area is equal to the half of the first Brillouin zone for K ( − K ) cone. Thus, Eq. (4) can be written as α c(v) τs z = 2 α Z k c Ω nτs z ( k ) S nτs z ( k ) kdk, (5)where k c = √ π / a . The ANC for the τ valley is then α valley nτ = 2 α Z k c [Ω nτ, ↑ ( k ) S nτ, ↑ ( k )+Ω nτ, ↓ ( k ) S nτ, ↓ ( k )] kdk. (6)The spin Nernst coefficient (SNC) reads α spin nτ = 2 α s Z k c [Ω nτ, ↑ ( k ) S nτ, ↑ ( k ) − Ω nτ, ↓ ( k ) S nτ, ↓ ( k )] kdk, (7) -1.4 -1.2 -1.0 -0.8 -0.6 -0.4-0.10.00.1 -1.4 -1.2 -1.0 -0.8 -0.60.00.51.0 E f [eV](b) AN C [ ] E f [eV] K K -K -K
K (-K )K (-K ) (a)
Spin valley current / K-K xz Mixed spin valley current / K-K yxyz (c) (d) hot hotcold coldK-K
FIG. 1: (a) Illustration of the anomalous Nernst coefficient asa function of the Fermi energy for different spin states in eachDirac cone. The vertical dotted lines indicate the positionsof the maxima of the valence bands with corresponding spinstates and Dirac cones. (b) Spin current purity correspondingto the curves shown in (a). (c) A pure spin current and valleycurrent can be generated when lowering the Fermi level intothe bands of K ↑ ( − K ↓ ). (d) Schematic illustration forthe corresponding mixed spin (valley) current when furtherlowering the Fermi level into K ↓ ( − K ↑ ). The 2D materialis chosen as MoS and the temperature is T=300 K. where α s = α ~ e = k B π .The magnitude of the ANE is determined by the ANC.In the following, we first investigate the properties of theANC for the valley K for a free standing layer, followedby a discussion where a magnetic substrate is considered.Based on the material parameters (Table I) , the ANCwas numerically calculated for the spin Nernst effect andValley Nernst effect. Fig. 1(a) displays the ANC for eachcone and each spin state. The ANCs for the two spinstates in a given valley have the same sign but they areshifted in opposite directions in the energy axis due tothe SOC in valence band ( E f < − K ↓ ( ↑ ) is degenerate to K ↑ ( ↓ ), the correspondingANCs have the same magnitude but opposite signs. Thisgives rise to a striking effect that a pure spin current andvalley current can be generated when the Fermi level islying in the valence band. According to the spin split-ting determined by the energy gap and the SOC of thematerial, this nearly 100% spin current can be generatedin a sizable range of energies (Table I). For example, forMoS , this energy range is around 0.11 eV. This region islarger for other TMDCs, e.g. WS . The pure spin currentand valley current generation in this case are schemati-cally shown in Fig. 1(c). With a further lowering of theFermi level, the purity of the spin and valley current is -1.2 -0.9 -0.60.00.51.0 -1.2 -0.9 -0.6 0.00.51.0 E f [eV] MoSe MoS WSe WS (a) (b) E f [eV] T=100 K T=200 K T=300 K T=400 K
FIG. 2: (a) The spin current purity factor η for the materialsof Table 1 at T = 300 K. (b) The temperature dependence of η for MoS . reduced. To characterize the extent of such mixing, wedefine a spin current purity factor (SCPF) η = ( α v K ↑ + α v − K ↑ ) − ( α v K ↓ + α v − K ↓ ) | α v K ↑ | + | α v − K ↑ | + | α v K ↓ | + | α v − K ↓ | , (8)where α v K ↑ ( ↓ ) is determined by Eq. (4) for K cone invalence band with up (down) spin. When η = ±
1, apure spin current is generated in the y -direction drivenby a temperature gradient in the x -direction. Otherwise,there will be mixing from different valleys and spin states.The variation of η for MoS is shown in Fig. 1(b) anda schematic graph of the mixed spin and valley currentis given in Fig. 1(d). For MoS the η factor becomesill-defined for Fermi level approximately above − . E f > − . TABLE I: Parameters for TMDCs. The energy unit is eV for∆ and λ . The last two columns indicate the range of gatevoltages ∆ E gate = E f − E f , where E f and E f define theupper and lower limit in which spin current purity η > η >
98% can be generated, respectively.∆ 2 λ ∆ E gate ( η > E gate ( η > Fig. 2 shows the SCPF for different materials andtemperatures. An almost 100 % spin current can be gen-erated in a wider range of gate voltage for materials withlarger spin splitting. For example, in the energy scalefrom -0.6 eV to -1.0 eV for WS (about 0.4 eV, which isquite large), the SCPF is almost 100%. At lower tem-peratures the SCPF is enhanced [Fig. 2(b)] as the ther-mal activation is suppressed. Another feature is that theSCPF changes its sign when the Fermi level is shifted -0.150.000.15 -0.040.000.040.08-1.0 -0.5 0.0 0.5 1.0-0.150.000.15 -1.0 -0.5 0.0 0.5 1.00.000.040.08 N e r n s t c o e ff i c i e n t [ ] MoS MoSe WS WSe (d)(b) (c) S p i n N e r n s t c o e ff i c i e n t [ s ] MoS MoSe WS WSe (a) E f [eV] T=100 K T=200 K T=300 K T=400 K E f [eV] T=100 K T=200 K T=300 K T=400 K
FIG. 3: The ANCs [(a), (b)] and SNC [(c), (d)] for K coneare calculated as a function of the Fermi energy for differentmaterials and temperatures, respectively. The temperaturein (a) and (c) is taken at 300 K. MoS is fixed in (b) and (d). sufficiently deep in the valence band, which means thegenerated spin current is reversed. In the deep valenceband case, the W-based materials have larger | η | (butnegative, and can be as large as -0.3) because of theirstronger SOC.The valley ANC is a sum of the two spin componentsand the SNC is a difference between them. Therefore,a two-peak feature of the valley ANC [Fig. 3(a)] and adip-peak feature of the SNC of the valence band [Fig.3(c)] can be observed due to the energy shifts of thetwo spin states. The SNC undergoes a sign-change whenlowering the Fermi level sufficiently. The value of thevalley ANC (SNC) for MoS can reach 0.14 α (0.07 α )at room temperature, which is comparable to the val-ley ANC in graphene . The different behavior betweenMoS and graphene is that the Nernst coefficients are nolonger spin degenerate, leading to a large SNC for MoS .Nonzero SNC reflects that there is a spin imbalance at theopposite edges of the sample (open circuit case), whichcould be a source of spin injection and spin current gen-eration (loop circuit case) in future applications of spincaloritronics.The valley ANC are strongly affected by the metal el-ements in the TMDCs, as shown in Fig. 3(a). When theFermi level lies in the valence band, the two-peak fea-ture becomes more distinct with increasing of the SOC,as one moves through MoS , MoSe , WS to WSe . Thetwo peaks in the valence band can be explained by thespin splitting and the strong SOC. For instance, the en-ergy difference between the two peaks of WSe (WS )[Fig. 3(a)] is 0.454 eV ( 0.43 eV), compatible to the spinsplitting of the valence band 0.46 eV (0.45 eV). Thus,measuring the separation of the two peaks provides amethod to estimate the effect of the spin splitting andthe SOC. The two-peak feature of the valley ANC be-comes more distinct at low temperature since the peakof the entropy density around the Fermi level is sharper[Fig. 3(b)]. Raising the temperature enhances the mag- J s J c +++++++++--------------- ISHE SNE D V ISHE
HotColdTL y z x
Hole doped TMDCs systems D T T + FIG. 4: A proposed H-shape detector for detecting the spinNernst effect in TMDCs. The spin current, generated in theright leg by a temperature gradient, is injected into the leftleg through a horizontal bridge, which can be converted intoa detectable charge voltage drop ∆ V ISHE by the inverse spinHall effect. nitude of the peak of the SNC but does not affect thepositions (Fig. 3(c)).To make an estimation, α ≈ .
665 nAK − . Thus, α spin n ≈ . × α (cid:0) ~ e (cid:1) (Fig. 3(d), 2 for valley degen-eracy) for MoS at room temperature corresponds to0.54 nA charge current for ∆ T = 2K, which should bedetectable. Specifically, to measure the SNE, we pro-pose an H-shape detector that is made of TMDCs (seeFig. 4). A temperature drop ∆ T in x direction is in-troduced in the right leg to generate spin current viathe SNE: J s = α spin n ( − ∂ x T ) e y . The spin current isinjected into the left leg through a bridge that is sup-posed to be short (in ballistic regime) and narrow. Onaccount of strong SOC in TMDCs, the electric field E ISHE = ρθ SHE ( e ~ ) J s × ˆ s will be generated perpendicu-lar to the spin direction ˆ s (z direction here) via the in-verse spin Hall effect (ISHE), resulting in a charge voltagedrop (in open circuit case) ∆ V ISHE = − σ SH σ ( e ~ ) α spin n ∆ T ,where θ SHE = σ SH /σ is spin Hall angle, σ SH is spin Hallconductivity, and σ is conductivity estimated by n c µe .Mobility µ of MoS is quite different in experiments.The early reported mobilities range from 0 . V − s − for a MoS transistor. It has been raisedto 200 cm V − s − in a MoS transistor with halfniumoxide gate recently . Although MoS p-type transis-tors have been fabricated successfully , there are stilllack of experimental data for σ . We exploit µ = 400cm v − s − and carrier density n c = 10 cm − fromRef. 42. The σ SH = − . π × − e h . Therefore, σ ≈ . × − e h . Thus | ∆ V ISHE | ≃ . µ V at roomtemperature for MoS with ∆ T =2K, which should bemeasurable . Lower σ (low µ and n c ) of the left legleads to a larger voltage drop. The voltage drop is evenlarger for other TMDCs with stronger SOC.Moreover, because of the time-reversal symmetry, nonet ANE is produced. It is instructive to consider a sit-uation where MoS layer is placed on a magnetic insu-lating substrate with a perpendicular moment to resolve more intrinsic information of the spin and valley Nernst -1.0 -0.5 0.0-8-4048 K cone dominant in peaks T o t a l AN C [ - ] E f [eV] MoS MoSe WS WSe -K dominant in dips FIG. 5: The total ANC as function of the Fermi energy fordifferent materials at 300K with the magnetization of magnet M z = 6 T . effect. An insulating ferromagnetic yttrium iron garnet(YIG=Y Fe O ) film could serve as such a substrate, inwhich an inplane or a perpendicular magnetization can be realized experimentally. A large magnetic mo-ment may be induced by a weak magnetic field so thatthe Landau level structure may be ignored. The Zeemanterm, i.e. − gµ B M z s z , should be added to Eq. (1). Ithas no impact on the eigenfunctions and the Berry curva-ture. Nonetheless it gives rise to corrections to the eigen-values, resulting in an asymmetry of the valence bandsin different valleys.In this circumstance, the time-reversal symmetry isbroken so that the total ANC is nonvanishing, shownin Fig. 5. The typical feature is the dip-peak profile ofthe total ANC. At the K point the down (up) spin bandis shifted upward (downward); while in the − K valley,the up (down) spin band is shifted upward (downward).The dips are dominated by − K ↓ states and the contri-bution from the K ↑ state is vanishing. The measuredcurrent originates thus from a single valley with singlespin component. For the peak of the total ANC, thecontribution from the K valley is dominating. Therefore,with lowering the Fermi level, single spin current (downspin) carrying single valley information can be generatedby a temperature gradient, which hints at possible appli-cations in spin caloritronics and valleytronics.This work is supported by Hundred Talents Programof The Chinese Academy of Sciences. G. S. is sup-ported in part by the MOST (Grant Nos. 2012CB932900,2013CB933401), the NSFC (Grant No. 11474279) andthe CAS (Grant No.XDB07010100). The Center forNanostructured Graphene (CNG) is sponsored by theDanish National Research Foundation, Project DNRF58. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] S. A. Wolf et al. , Science , 1488 (2001). I. ˇZuti´c et al. , Rev. Mod. Phys. , 323 (2004). A. Fert, Rev. Mod. Phys. , 1517 (2008). D. D. Awschalom, and N. Samarth, Physics , 50 (2009). A. Aharony et al. , Phys. Rev. B , 125328 (2008). Z. -G. Zhu, Phys. Lett. A , 695 (2008). J. E. Hirsch, Phys. Rev. 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