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spin Hall effects in monolayer MoS2
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Effect of trigonal warping on the Berry curvature and valley/spin Hall effects in monolayer MoS2 Wang Chen a , Xiaoying Zhou a,∗ , Pu Liu a , Xianbo Xiao b,∗ , Guanghui Zhou a a
Department of Physics and Key Laboratory for Low-Dimensional Quantum Structures and Manipulation (Ministry of Education), Hunan Normal University, Changsha, 410081, China b School of Computer Science, Jiangxi University of Traditional Chinese Medicine, Nanchang, 330004, China
a r t i c l e
i n f o
Article history: Received 8 November 2019 Received in revised form 6 January 2020 Accepted 16 February 2020 Available online xxxx Communicated by L. Ghivelder Keywords: Berry curvature Orbital magnetic moment Valley and spin Hall effect Monolayer MoS2
a b s t r a c t Recently, the nonlinear optical processes study on monolayer MoS2 has revealed the existence of a trigonal warping in the band structure which distorts the Fermi surface from a circle at low energies to a triangle at higher energies. Here we theoretically demonstrate that the trigonal warping induce important modifications on the Berry curvature, orbital magnetic moment, and valley Hall conductance in monolayer MoS2 . The analytical results show that both Berry curvature and orbital magnetic moment are trigonally warped in contrast to the isotropic case without warping, which is consistent with the numerical calculations. The valley Hall conductance is reduced by the trigonal warping due to the warped Berry curvatures, whereas the spin Hall conductance remains nearly intact. Our findings are useful in explaining the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Monolayer MoS2 has attracted intensive attention due to its applications in valleytronics and spintronics [1–3]. Owing to the inversion symmetry breaking, monolayer MoS2 possesses a direct band gap located at the corners [K and K’ (= −K) points] of the hexagonal first Brillouzin zone (BZ), which has been verified by the first-principle calculations [4–7,9] and photoluminescence experiments [10,11]. The inversion symmetry breaking also lifts the spin degeneracy and induces a large spin splitting of valence bands in the presence of spin-orbit coupling (SOC) [4–7,9]. Interestingly, the spin splitting in two valleys must be opposite in order to fulfill the requirement of time reversal symmetry, which induces a coupled spin and valley feature in the low energy regime [4,5]. Owing to the separated large distance between the K and K’ valley in the momentum space, the intervalley scattering is greatly suppressed under smooth scattering potential [12]. This means that the valley index can be used to encode and manipulate information known as valleytronics [12], much in the same way as the spin index used in spintronic applications. The core problem in valleytronics is to generate and manipulate the valley polarization [12–14]. In monolayer MoS2 , the Berry curvature and orbital magnetic moment in low energy regime
*
Corresponding authors. E-mail addresses: [email protected] (X. Zhou), [email protected] (X. Xiao). https://doi.org/10.1016/j.physleta.2020.126344 0375-9601/© 2020 Elsevier B.V. All rights reserved.
have opposite sign in opposite valley arising from the inversion symmetry breaking, resulting in the valley Hall effect and valley-dependent optical selection rule for inter-band transitions [4]. There is also a spin Hall effect accompanied by the valley Hall effect due to the coupled valley and spin degree of freedom [4]. These intrinsic valley and spin Hall effects make MoS2 to be an ideal platform for the applications of valleytronics and spintronics. Various interesting properties of monolayer MoS2 have been experimentally observed or theoretically predicted, i.e., the field-effect transistor with high off/on ratio [15], valley- and spindependent magneto-capacitance [16] and magneto-transport properties [17], light induced exciton Hall effect [18], and unconventional spin Hall effect [19]. Further, there is a significant trigonal warping which is already observable at ≈ 0.08 eV below the valence band edge of monolayer MoS2 [6,7,20]. This distorts the Fermi surface from a circle at low energies to a triangular shape at higher energies, which has been verified in recent angle-resolved photoemission spectroscopy [21] and third harmonic generation experiments [22]. Importantly, this warping effect plays important role in the electronic properties of monolayer MoS2 , such as the usual magneto-optical selection rules [24] and anomalous large third harmonic optical response [22]. Up to date, the Berry curvature, valley and spin Hall conductances derived from the massive Dirac Hamiltonian to the linear order are well understood [4]. But, how the trigonal warping affects the Berry curvature, orbital magnetic moment, valley and spin Hall conductances is still not clear.
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In this work, we theoretically demonstrate that the trigonal warping induces important modifications on the Berry curvature, orbital magnetic moment, and valley/spin Hall conductances in monolayer MoS2 . The warping effect enhances the density of states, and this enhancement is more prominent at higher energy regime. In contrast to the isotropic case, our analytical results show that both Berry curvature and orbital magnetic moment are trigonally warped in the presence of warping term, which is consistent with the results obtained from the first-principles [5] or tightbinding calculations [23]. The valley Hall conductance is reduced arising from the warped Berry curvatures, whereas the spin Hall conductance remains nearly intact. Our findings provide a further understanding on the electronic states of monolayer MoS2 , which are useful in the explanation of the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. The paper is organized as follows. We describe the Hamiltonian with trigonal warping in Sec. 2 and present the analytical results for Berry curvature and orbital magnetic moment in Sec. 3. We calculate and present some numerical examples for the valley and spin Hall conductances in Sec. 4. Finally, we summarize our results in Sec. 5. 2. Hamiltonian with trigonal warping The effective low-energy Hamiltonian of monolayer MoS2 with trigonal warping around K and K’ valley is [6–8]
H τ ,s z = H i + H 3w ,
(1)
where the isotropic term is
H i = tk · στ +
2
σz + (α + β σz ) k2 +
1 − σz 2
λτ s z ,
(2)
and the trigonal warping contribution can be written as
H 3w = γ k2x − k2y
σ x − 2τ γ k x k y σ y .
(3)
Here, τ = +/ − 1 labels the K/K’ valley, s z = +/ − 1 denotes spinup/down, k = (kx , k y ) is the wavevector, and στ = τ σx , σ y with σi (i = x, y , z) is ordinary Pauli matrix operating on a suitable conduction/valence band basis. Other band parameters are [7] t = 3.378 eV · Å, = 1.633 eV,
2
2
α = 0.672 eV · Å , β = −0.112 eV · Å , 2
λ = 0.075 eV, and γ = −1.252 eV · Å . Solving the secular equation of Hamiltonian (1), we find the energy spectrum is
E ± = h0 ± h,
(4)
where +/-1 denotes the conduction/valence band, h0 =
λτ s z /2, and h =
α k2 +
h2x + h2y + h2z with h x = t τ kx + γ k2x − k2y , h y =
τ sz tk y − 2τ γ kx k y and h z = −λ +β k2 . Then, the explicitly expres2 sion of the correction term is
2τ s z , h= β 2 + γ 2 k4 + β τ s z + t 2 k2 + 2t τ γ k3x − 3kx k2y + 4
Fig. 1. Contour plot of the conduction (a) and valence (b) band in unit of eV for monolayer MoS2 in the presence of trigonal warping. (c) Density of states of monolayer MoS2 with [the (red) solid line] and without [the (black) dash-dotted line] trigonal warping. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Figs. 1(a) and (b) present the contour plots of the conduction and valence bands for monolayer MoS2 with trigonal warping, respectively. The data in each contour line indicates the corresponding energy of the Fermi surface. As shown in Figs. 1(a) and 1(b), in contrast to the isotropic case, we find the Fermi surfaces are significantly trigonal-warped in both the conduction and valence bands. The warping effect becomes more pronounced at higher energy regime, but it is more prominent in the valence band compared to that in the conduction band arising from the electron-hole asymmetry induced by h0 . It is already observable at ≈ 0.04 eV below the valence band edge, which is consistent with the previous firstprinciple calculations [6]. Further, Fig. 1(c) shows the density of states for monolayer MoS2 with [the (red) solid line] and without [the (black) dash-dotted line] trigonal warping. From Fig. 1(c), we see that the density of states is enhanced by the trigonal warping, and this enhancement become more significant at higher energy in both the conduction and valence bands. This means that the trigonal warping will play important role in the electronic properties of monolayer MoS2 . 3. Berry curvature and orbital magnetic moment In this subsection, we present the calculated results for Berry curvature and orbital magnetic moment by using the eigenvalue and wavefunction obtained in Sec. 1. In the presence of inversion symmetry breaking, the charge carriers acquire a valleycontrasting Berry curvature. The Berry curvature n (k) ≡ zˆ · ∇k × ψnk |i ∇k | ψnk at each band is given by [12,13]
∂H ∂H
ψnk ∂ p x ψn k ψn k ∂ p y ψnk − (x ↔ y )
(5) where τ s z = ( − λτ s z ) is the spin-dependent band gap. Notably, in the absence of the trigonal warping (γ = 0), the energy dispersion only depends on the module of k, which reveals an isotropic Fermi surface. The corresponding wavefunctions are
ψ±,k = √
e ik·r 2h (h ∓ h z )
h x − ih y −h z ± h
,
(6)
where ψ+,k (ψ−,k ) is the wavefunction of the conduction (valence) band.
n (k) = i
( E n − E n )2
n =n
,
(7)
where E n and ψn (k) are the eigenvalue and wavefunction of Hamiltonian (1), respectively. After some calculations, we obtain the valley and spin dependent Berry curvature in the valence band given by
τv s z (k) =
τ 2h3
τ s z t 2 2
− β t 2 + 2 τ s z γ 2 k 2
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Fig. 2. The Berry curvatures of valence band for monolayer MoS2 in the 2D k-plane, where a = 3.193 A˚ is the Mo-S bond length. (a)/(b) The Berry curvatures for spinup band around K valley in the presence/absence of trigonal warping. (c) and (d) are the counterpart to (a) and (b) around the K’ valley. All Berry curvatures are in the unit of a2 /π 2 .
+ 2τ β t γ k x
k2x
− 3k2y
.
(8)
Therefore, according to Eq. (7), in the conduction band, we have
cτ sz (k) = − τv sz (k). Note that the Berry curvature has opposite sign in opposite valleys, which means that electrons from different valleys acquire opposite anomalous velocity, leading to the valley Hall effect [12]. Fig. 2 depicts the Berry curvatures in the 2D k-plane of valence band for spin-up electron states in monolayer MoS2 . The counterpart for spin-down electron states can be obtained by usτ ,s −τ ,−s z (k) required by time reversal symmeing v z (−k) = − v try. As shown in the figures, we can see that τ ,↑ (k) are significantly peaked at both K and K valleys but with opposite signs, and decay rapidly, and eventually vanish away from the two valleys in Figs. 2(a)-(d), which can be derived from Eq. (8) directly. Further, as expected in Eq. (8), we find the Berry curvatures are prominently warped in the presence of the warping effect H 3w in comparison with Figs. 2(a) and 2(c), which is consistent with the results obtained from the first-principle [5] or tight-binding calculations [23]. In the absence of trigonal warping, the Berry curvatures only depend on the module of k [see Eq. (8)]. In this case, the Berry curvatures are isotropic in the 2D k-plane as shown in Figs. 2(b) and 2(d). Setting k = 0 in Eq. (8), we obtain the Berry curvatures at valleys K and K’ as 2τ t 2 /2τ ,s z , reducing to the results in Ref. [4], which are independent on the trigonal warping and the electron-hole asymmetric effect. In order to show the anisotropy of the Berry curvatures quantitatively, we present the Berry curvatures as a function of the wavevector angle θ = arctan(k y /kx ) in Fig. 3(a) by calculating the integral over k. From Fig. 3(a), we find the Berry curvatures are also spin polarized since the valley and spin are coupled due to the time reversal symmetry. In the absence of trigonal warping, the Berry curvatures are independent on the wavevector-angle, but have opposite signs at opposite valleys. In contrast, the Berry curvatures oscillate as the wavevector angle θ with a period of 2π /3 in the presence of warping effect, which clearly shows the Berry curvatures are trigonally warped. Meanwhile, the trigonal warping also provides significant corrections to the magnitude of the Berry curvatures (see the solid and dashed lines with the same color). Those corrections to Berry curvature also modify the valley and spin Hall conductance as shall be discussed later. Fig. 3(b) presents the Berry curvatures as a function of wavevector angle near the band edges which are fre-
3
Fig. 3. (a) The Berry curvatures of the valence band as the wavevector-angle θ = arctan(k y /k x ) after calculating the integral over k. The solid (dashed) lines denote the valley and spin resolved Berry curvature with (without) trigonal warping. (b) The Berry curvatures as a function of wavevector-angle near the band edges with the solid (dashed) lines for E F = 0.85 (−0.7) eV. (c) The Orbital magnetic moments as a function of the module of k for the K valley, where the solid (dashed) lines for the results with (without) trigonal warping.
quently measured experimentally, where the solid (dashed) lines for E F = 0.85 (−0.7) eV. From the figure, we find the Berry curvatures near band edges also oscillate as the wavevector angle θ with a period of 2π /3 but with different magnitude arising from the correction induced by the Fermi-Dirac distribution f ( E k ). Near the conduction band minimum, there is a phase difference of π /3 in the oscillation pattern in comparison with that in Fig. 3(a) because the Berry curvatures have opposite sign in the conduction compared with that in the valence band [see Eq. (7)]. Near the valence band maximum, the Berry curvatures vanish for τ s z = −1 bands due to the large spin splitting in the valence band. Next, we study the trigonal warping effect on the orbital magnetic moment. The Bloch electrons in a periodic crystal exhibit orbital magnetic moment in the presence of electromagnetic field due to the self-rotation of the Bloch electron wave packet in the semi-classic picture [12,25,26]. For eigenstates ψnk , the orbital magnetic moment in 2D system mn (k) = −i 2eh¯ ∇k ψnk | × [ H (k) − E (k)]|∇k ψnk · zˆ is given by
mn (k) = im0
∂H ∂H
ψnk ∂ p x ψn k ψn k ∂ p y ψnk − (x ↔ y )
n =n
( E n − E n )
, (9)
where m0 = e /2h¯ . In our present case, a direct calculation yields τs
m v z (k) =
− β t 2 + 2 τ s z γ 2 k 2 2 + 2τ β t γ kx k2x − 3k2y .
τ m 0 τ s z t 2 2h2
(10)
Similar to that of Berry curvatures, the orbital magnetic moments also have opposite signs at opposite valley, which will induce a valley-zeeman effect and asymmetric Landau levels under perpendicular magnetic fields [23,27]. In the absence of trigonal warping, i.e., γ = 0, the orbital magnetic moments only depend on the module of k according to Eq. (10), which means the orbital magnetic moment is isotropic in the 2D k-plane. In contrast, in the presence of trigonal warping, the orbital magnetic moments depend on not only the module of k but also the wavevector angle θ = arctan(k y /kx ), resulting in an anisotropic feature. Actually, the orbital magnetic moments are also trigonally warped like that in Berry curvatures and we don’t present them in the 2D k-plane
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here for space limitations. In order to show the corrections to the orbital magnetic moments from the warping effect, we present the orbital magnetic moments as a function of the module of k around K valley in Fig. 3(b). We find the orbital magnetic moments are peaked at the K valley and gradually decayed away from the valley. Owing to the valley and spin coupled band structure and large spin splitting [4], the orbital magnetic moments are spin polarized at K valley. In comparison, the warping effect decreases the orbital magnetic moments away from the K valley (see the red solid and dashed lines). Setting k = 0 in Eq. (10), we obtain the Berry curvatures at K and K’ valley as 2τ m0 t 2 /τ ,s z reducing to the results in Ref. [4], which are independent on the trigonal warping and the electron-hole asymmetric effect. This means the magnitudes of orbital magnetic moments at K or K’ valley remain unchanged in the presence of warping effect. The orbital magnetic moments versus k at K’ valley is the simply opposite to that at K valley, and we do not present them here. 4. Valley and spin Hall effect In this subsection, we discuss the intrinsic valley and spin Hall effect induced by the finite Berry curvature. In the presence of non-zero n (k), the equation of motion for the wavepackage is [12,25,26]
r˙ =
∂ E nk ˙ e − k × n (k) , k˙ = − (E + r˙ × B) , ∂ h¯ k h¯
from which we know that the Berry curvature drives an anomalous transverse velocity in the presence of an electric field E even in the absence of magnetic field B, which is
e v a = − E × n (k) . h¯
(11)
This anomalous transverse velocity v a is responsible for the intrinsic contribution to the anomalous Hall effect. The spin and valley resolved intrinsic Hall conductance can be calculated from the Berry curvature directly, which is τ sz σxy =
e2
2π h¯ n=c , v
τ sz
f ( E n ) n
(k) dk,
(12)
−1
where f ( E n ) = 1 + e ( E n − E F )/k B T is the Fermi-Dirac distribution. Here, it is worth to note that the valley and spin Hall conτs ductance σxy z are independent on the crystal orientation although the Berry curvatures are trigonally warped, because the orientation freedom in the Berry curvatures has been wiped out when calculating the Hall conductance by performing the integration over the Brillouin zone. Owing to the time reversal symmetry, we have τ ,s z −τ ,−s z σxy = −σxy . Therefore, a direct conclusion is that the total (charge) Hall conductance is zero in our systems because the charge carriers in the two valleys have opposite transverse velocities due to the opposite signs of the Berry curvatures constrained by the time reversal symmetry. However, there is finite valley and spin Hall conductance due to the inversion symmetry breaking. v s The valley σxy and spin σxy Hall conductance are given by
K↑ K↓ K ↑ K ↓ K↑ K↓ v σxy = σxy + σxy − σxy − σxy = 2 σxy + σxy ,
K↑ K ↑ K↓ K ↓ K↑ K↓ s σxy = σxy + σxy − σxy − σxy = 2 σxy − σxy .
(13) (14)
Fig. 4 shows the valley (a) and spin (b) Hall conductances as a function of Fermi energy for four kinds of monolayer transition metal dichalcogenides MX2 , where M = Mo, W, and X = S, Se. The solid (dashed) lines denote the results with (without) trigonal
Fig. 4. The valley (a) and spin (b) Hall conductance as a function of Fermi energy for four kinds of monolayer transition metal dichalcogenides MX2 , where M = Mo, W, and X = S, Se. The solid (dashed) lines denote the results with (without) trigonal warping. The inset in (b) is an amplification of the spin Hall conductance when the E F lies in the band gap. The band parameters for MoSe2 , WS2 and WSe2 are adopted from Ref. [8]. All conductances are in unit of σ0 = e 2 /2π h¯ , and the temperature is set as 77 K.
warping. As shown in Fig. 4(a), we find that the valley Hall conductances are quantized when the Fermi level E f lies in the band gap, i.e., the pristine samples, and gradually decrease for both the hole and electron doped samples with the increasing of doping concentration. Notably, the valley Hall conductances are reduced by the trigonal warping because this warping effect reduces the volume between the hook face of the valley Berry curvatures and the 2D k-plane in comparison with that in the isotropic case. From Fig. 4(b), we find that the spin Hall conductance is also quantized when the Fermi level lies in the band gap. For electron doped sample, the spin Hall conductance in MoS2 and MoSe2 are nearly zero due to the tiny spin splitting in their conduction bands (see the black and red lines), whereas the spin Hall conductances are finite in WS2 and WSe2 arising from the relatively large spin splitting in the conduction bands. For hole doped sample, there is a prominent spin Hall conductance when E f lies between the two spin split valence-band tops. In the presence of trigonal warping, in contrast to the valley Hall conductance, the spin Hall conductance is slightly enhanced because the trigonal warping enhances the spin polarization of the Berry curvatures due to its breaking on the valley-spin locked feature in the band structures [see Eq. (8)] and consequently the spin Hall conductance [see Eq. (14)]. But this modification is quite small and therefore negligible, we can conclude that the spin Hall conductance remain intact in the presence of trigonal warping. Further, there are various other monolayer transition metal dichalcogenides sharing similar band structures with that of monolayer MoS2 . Here, the conclusions for the valley and spin Hall conductances reported here are also applicable to them. 5. Summary In summary, we studied the trigonal warping on the Berry curvature, valley and spin Hall effect in monolayer MoS2 . Our results showed that the warping effect implies important modifications to the density of states, Berry curvature, orbital magnetic moment, and valley Hall conductances in monolayer MoS2 . In contrast to the isotropic case, the analytical results for Berry curvature and orbital magnetic moment show that both of them are trigonally warped in the presence of warping term, which is consistent with the results obtained from the first principle [5] or tight-binding calculations [23]. The valley Hall conductance was found to be reduced arising from the warped Berry curvature, whereas the spin Hall conductance remain intact. Our findings provide a further un-
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derstanding on the electronic states of monolayer MoS2 , which are useful in the explanation of the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 11804092 and 11664019), and Project funded by China Postdoctoral Science Foundation (Grant Nos. BX20180097 and 2019M652777). References [1] Qing Hua Wang, Kourosh Kalantar-Zadeh, Andras Kis, Jonathan N. Coleman, Michael S. Strano, Nat. Nanotechnol. 7 (2012) 699. [2] Xiaodong Xu, Wang Yao, Di Xiao, Tony F. Heinz, Nat. Phys. 10 (2014) 343. [3] Gui-Bin Liu, Di Xiao, Yugui Yao, Xiaodong Xude, Wang Yao, Chem. Soc. Rev. 44 (2015) 2643. [4] Di Xiao, Gui-Bin Liu, Wanxing Feng, Xiaodong Xu, Wang Yao, Phys. Rev. Lett. 108 (2012) 196802. [5] Wanxiang Feng, Yugui Yao, Wenguang Zhu, Jinjian Zhou, Wang Yao, Di Xiao, Phys. Rev. B 86 (2012) 165108. [6] A. Kormányos, V. Zólyomi, Neil D. Drummond, Péter Rakyta, Guido Burkard, Vladimir I. Fal’ko, Phys. Rev. B 88 (2013) 045416. [7] Gui-Bin Liu, Wen-Yu Shan, Yugui Yao, Wang Yao, Di Xiao, Phys. Rev. B 88 (2013) 085433.
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Effect of trigonal warping on the Berry curvature and valley/spin Hall effects in monolayer MoS2 Wang Chen a , Xiaoying Zhou a,∗ , Pu Liu a , Xianbo Xiao b,∗ , Guanghui Zhou a a
Department of Physics and Key Laboratory for Low-Dimensional Quantum Structures and Manipulation (Ministry of Education), Hunan Normal University, Changsha, 410081, China b School of Computer Science, Jiangxi University of Traditional Chinese Medicine, Nanchang, 330004, China
a r t i c l e
i n f o
Article history: Received 8 November 2019 Received in revised form 6 January 2020 Accepted 16 February 2020 Available online xxxx Communicated by L. Ghivelder Keywords: Berry curvature Orbital magnetic moment Valley and spin Hall effect Monolayer MoS2
a b s t r a c t Recently, the nonlinear optical processes study on monolayer MoS2 has revealed the existence of a trigonal warping in the band structure which distorts the Fermi surface from a circle at low energies to a triangle at higher energies. Here we theoretically demonstrate that the trigonal warping induce important modifications on the Berry curvature, orbital magnetic moment, and valley Hall conductance in monolayer MoS2 . The analytical results show that both Berry curvature and orbital magnetic moment are trigonally warped in contrast to the isotropic case without warping, which is consistent with the numerical calculations. The valley Hall conductance is reduced by the trigonal warping due to the warped Berry curvatures, whereas the spin Hall conductance remains nearly intact. Our findings are useful in explaining the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. © 2020 Elsevier B.V. All rights reserved.
1. Introduction Monolayer MoS2 has attracted intensive attention due to its applications in valleytronics and spintronics [1–3]. Owing to the inversion symmetry breaking, monolayer MoS2 possesses a direct band gap located at the corners [K and K’ (= −K) points] of the hexagonal first Brillouzin zone (BZ), which has been verified by the first-principle calculations [4–7,9] and photoluminescence experiments [10,11]. The inversion symmetry breaking also lifts the spin degeneracy and induces a large spin splitting of valence bands in the presence of spin-orbit coupling (SOC) [4–7,9]. Interestingly, the spin splitting in two valleys must be opposite in order to fulfill the requirement of time reversal symmetry, which induces a coupled spin and valley feature in the low energy regime [4,5]. Owing to the separated large distance between the K and K’ valley in the momentum space, the intervalley scattering is greatly suppressed under smooth scattering potential [12]. This means that the valley index can be used to encode and manipulate information known as valleytronics [12], much in the same way as the spin index used in spintronic applications. The core problem in valleytronics is to generate and manipulate the valley polarization [12–14]. In monolayer MoS2 , the Berry curvature and orbital magnetic moment in low energy regime
*
Corresponding authors. E-mail addresses: [email protected] (X. Zhou), [email protected] (X. Xiao). https://doi.org/10.1016/j.physleta.2020.126344 0375-9601/© 2020 Elsevier B.V. All rights reserved.
have opposite sign in opposite valley arising from the inversion symmetry breaking, resulting in the valley Hall effect and valley-dependent optical selection rule for inter-band transitions [4]. There is also a spin Hall effect accompanied by the valley Hall effect due to the coupled valley and spin degree of freedom [4]. These intrinsic valley and spin Hall effects make MoS2 to be an ideal platform for the applications of valleytronics and spintronics. Various interesting properties of monolayer MoS2 have been experimentally observed or theoretically predicted, i.e., the field-effect transistor with high off/on ratio [15], valley- and spindependent magneto-capacitance [16] and magneto-transport properties [17], light induced exciton Hall effect [18], and unconventional spin Hall effect [19]. Further, there is a significant trigonal warping which is already observable at ≈ 0.08 eV below the valence band edge of monolayer MoS2 [6,7,20]. This distorts the Fermi surface from a circle at low energies to a triangular shape at higher energies, which has been verified in recent angle-resolved photoemission spectroscopy [21] and third harmonic generation experiments [22]. Importantly, this warping effect plays important role in the electronic properties of monolayer MoS2 , such as the usual magneto-optical selection rules [24] and anomalous large third harmonic optical response [22]. Up to date, the Berry curvature, valley and spin Hall conductances derived from the massive Dirac Hamiltonian to the linear order are well understood [4]. But, how the trigonal warping affects the Berry curvature, orbital magnetic moment, valley and spin Hall conductances is still not clear.
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In this work, we theoretically demonstrate that the trigonal warping induces important modifications on the Berry curvature, orbital magnetic moment, and valley/spin Hall conductances in monolayer MoS2 . The warping effect enhances the density of states, and this enhancement is more prominent at higher energy regime. In contrast to the isotropic case, our analytical results show that both Berry curvature and orbital magnetic moment are trigonally warped in the presence of warping term, which is consistent with the results obtained from the first-principles [5] or tightbinding calculations [23]. The valley Hall conductance is reduced arising from the warped Berry curvatures, whereas the spin Hall conductance remains nearly intact. Our findings provide a further understanding on the electronic states of monolayer MoS2 , which are useful in the explanation of the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. The paper is organized as follows. We describe the Hamiltonian with trigonal warping in Sec. 2 and present the analytical results for Berry curvature and orbital magnetic moment in Sec. 3. We calculate and present some numerical examples for the valley and spin Hall conductances in Sec. 4. Finally, we summarize our results in Sec. 5. 2. Hamiltonian with trigonal warping The effective low-energy Hamiltonian of monolayer MoS2 with trigonal warping around K and K’ valley is [6–8]
H τ ,s z = H i + H 3w ,
(1)
where the isotropic term is
H i = tk · στ +
2
σz + (α + β σz ) k2 +
1 − σz 2
λτ s z ,
(2)
and the trigonal warping contribution can be written as
H 3w = γ k2x − k2y
σ x − 2τ γ k x k y σ y .
(3)
Here, τ = +/ − 1 labels the K/K’ valley, s z = +/ − 1 denotes spinup/down, k = (kx , k y ) is the wavevector, and στ = τ σx , σ y with σi (i = x, y , z) is ordinary Pauli matrix operating on a suitable conduction/valence band basis. Other band parameters are [7] t = 3.378 eV · Å, = 1.633 eV,
2
2
α = 0.672 eV · Å , β = −0.112 eV · Å , 2
λ = 0.075 eV, and γ = −1.252 eV · Å . Solving the secular equation of Hamiltonian (1), we find the energy spectrum is
E ± = h0 ± h,
(4)
where +/-1 denotes the conduction/valence band, h0 =
λτ s z /2, and h =
α k2 +
h2x + h2y + h2z with h x = t τ kx + γ k2x − k2y , h y =
τ sz tk y − 2τ γ kx k y and h z = −λ +β k2 . Then, the explicitly expres2 sion of the correction term is
2τ s z , h= β 2 + γ 2 k4 + β τ s z + t 2 k2 + 2t τ γ k3x − 3kx k2y + 4
Fig. 1. Contour plot of the conduction (a) and valence (b) band in unit of eV for monolayer MoS2 in the presence of trigonal warping. (c) Density of states of monolayer MoS2 with [the (red) solid line] and without [the (black) dash-dotted line] trigonal warping. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Figs. 1(a) and (b) present the contour plots of the conduction and valence bands for monolayer MoS2 with trigonal warping, respectively. The data in each contour line indicates the corresponding energy of the Fermi surface. As shown in Figs. 1(a) and 1(b), in contrast to the isotropic case, we find the Fermi surfaces are significantly trigonal-warped in both the conduction and valence bands. The warping effect becomes more pronounced at higher energy regime, but it is more prominent in the valence band compared to that in the conduction band arising from the electron-hole asymmetry induced by h0 . It is already observable at ≈ 0.04 eV below the valence band edge, which is consistent with the previous firstprinciple calculations [6]. Further, Fig. 1(c) shows the density of states for monolayer MoS2 with [the (red) solid line] and without [the (black) dash-dotted line] trigonal warping. From Fig. 1(c), we see that the density of states is enhanced by the trigonal warping, and this enhancement become more significant at higher energy in both the conduction and valence bands. This means that the trigonal warping will play important role in the electronic properties of monolayer MoS2 . 3. Berry curvature and orbital magnetic moment In this subsection, we present the calculated results for Berry curvature and orbital magnetic moment by using the eigenvalue and wavefunction obtained in Sec. 1. In the presence of inversion symmetry breaking, the charge carriers acquire a valleycontrasting Berry curvature. The Berry curvature n (k) ≡ zˆ · ∇k × ψnk |i ∇k | ψnk at each band is given by [12,13]
∂H ∂H
ψnk ∂ p x ψn k ψn k ∂ p y ψnk − (x ↔ y )
(5) where τ s z = ( − λτ s z ) is the spin-dependent band gap. Notably, in the absence of the trigonal warping (γ = 0), the energy dispersion only depends on the module of k, which reveals an isotropic Fermi surface. The corresponding wavefunctions are
ψ±,k = √
e ik·r 2h (h ∓ h z )
h x − ih y −h z ± h
,
(6)
where ψ+,k (ψ−,k ) is the wavefunction of the conduction (valence) band.
n (k) = i
( E n − E n )2
n =n
,
(7)
where E n and ψn (k) are the eigenvalue and wavefunction of Hamiltonian (1), respectively. After some calculations, we obtain the valley and spin dependent Berry curvature in the valence band given by
τv s z (k) =
τ 2h3
τ s z t 2 2
− β t 2 + 2 τ s z γ 2 k 2
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Fig. 2. The Berry curvatures of valence band for monolayer MoS2 in the 2D k-plane, where a = 3.193 A˚ is the Mo-S bond length. (a)/(b) The Berry curvatures for spinup band around K valley in the presence/absence of trigonal warping. (c) and (d) are the counterpart to (a) and (b) around the K’ valley. All Berry curvatures are in the unit of a2 /π 2 .
+ 2τ β t γ k x
k2x
− 3k2y
.
(8)
Therefore, according to Eq. (7), in the conduction band, we have
cτ sz (k) = − τv sz (k). Note that the Berry curvature has opposite sign in opposite valleys, which means that electrons from different valleys acquire opposite anomalous velocity, leading to the valley Hall effect [12]. Fig. 2 depicts the Berry curvatures in the 2D k-plane of valence band for spin-up electron states in monolayer MoS2 . The counterpart for spin-down electron states can be obtained by usτ ,s −τ ,−s z (k) required by time reversal symmeing v z (−k) = − v try. As shown in the figures, we can see that τ ,↑ (k) are significantly peaked at both K and K valleys but with opposite signs, and decay rapidly, and eventually vanish away from the two valleys in Figs. 2(a)-(d), which can be derived from Eq. (8) directly. Further, as expected in Eq. (8), we find the Berry curvatures are prominently warped in the presence of the warping effect H 3w in comparison with Figs. 2(a) and 2(c), which is consistent with the results obtained from the first-principle [5] or tight-binding calculations [23]. In the absence of trigonal warping, the Berry curvatures only depend on the module of k [see Eq. (8)]. In this case, the Berry curvatures are isotropic in the 2D k-plane as shown in Figs. 2(b) and 2(d). Setting k = 0 in Eq. (8), we obtain the Berry curvatures at valleys K and K’ as 2τ t 2 /2τ ,s z , reducing to the results in Ref. [4], which are independent on the trigonal warping and the electron-hole asymmetric effect. In order to show the anisotropy of the Berry curvatures quantitatively, we present the Berry curvatures as a function of the wavevector angle θ = arctan(k y /kx ) in Fig. 3(a) by calculating the integral over k. From Fig. 3(a), we find the Berry curvatures are also spin polarized since the valley and spin are coupled due to the time reversal symmetry. In the absence of trigonal warping, the Berry curvatures are independent on the wavevector-angle, but have opposite signs at opposite valleys. In contrast, the Berry curvatures oscillate as the wavevector angle θ with a period of 2π /3 in the presence of warping effect, which clearly shows the Berry curvatures are trigonally warped. Meanwhile, the trigonal warping also provides significant corrections to the magnitude of the Berry curvatures (see the solid and dashed lines with the same color). Those corrections to Berry curvature also modify the valley and spin Hall conductance as shall be discussed later. Fig. 3(b) presents the Berry curvatures as a function of wavevector angle near the band edges which are fre-
3
Fig. 3. (a) The Berry curvatures of the valence band as the wavevector-angle θ = arctan(k y /k x ) after calculating the integral over k. The solid (dashed) lines denote the valley and spin resolved Berry curvature with (without) trigonal warping. (b) The Berry curvatures as a function of wavevector-angle near the band edges with the solid (dashed) lines for E F = 0.85 (−0.7) eV. (c) The Orbital magnetic moments as a function of the module of k for the K valley, where the solid (dashed) lines for the results with (without) trigonal warping.
quently measured experimentally, where the solid (dashed) lines for E F = 0.85 (−0.7) eV. From the figure, we find the Berry curvatures near band edges also oscillate as the wavevector angle θ with a period of 2π /3 but with different magnitude arising from the correction induced by the Fermi-Dirac distribution f ( E k ). Near the conduction band minimum, there is a phase difference of π /3 in the oscillation pattern in comparison with that in Fig. 3(a) because the Berry curvatures have opposite sign in the conduction compared with that in the valence band [see Eq. (7)]. Near the valence band maximum, the Berry curvatures vanish for τ s z = −1 bands due to the large spin splitting in the valence band. Next, we study the trigonal warping effect on the orbital magnetic moment. The Bloch electrons in a periodic crystal exhibit orbital magnetic moment in the presence of electromagnetic field due to the self-rotation of the Bloch electron wave packet in the semi-classic picture [12,25,26]. For eigenstates ψnk , the orbital magnetic moment in 2D system mn (k) = −i 2eh¯ ∇k ψnk | × [ H (k) − E (k)]|∇k ψnk · zˆ is given by
mn (k) = im0
∂H ∂H
ψnk ∂ p x ψn k ψn k ∂ p y ψnk − (x ↔ y )
n =n
( E n − E n )
, (9)
where m0 = e /2h¯ . In our present case, a direct calculation yields τs
m v z (k) =
− β t 2 + 2 τ s z γ 2 k 2 2 + 2τ β t γ kx k2x − 3k2y .
τ m 0 τ s z t 2 2h2
(10)
Similar to that of Berry curvatures, the orbital magnetic moments also have opposite signs at opposite valley, which will induce a valley-zeeman effect and asymmetric Landau levels under perpendicular magnetic fields [23,27]. In the absence of trigonal warping, i.e., γ = 0, the orbital magnetic moments only depend on the module of k according to Eq. (10), which means the orbital magnetic moment is isotropic in the 2D k-plane. In contrast, in the presence of trigonal warping, the orbital magnetic moments depend on not only the module of k but also the wavevector angle θ = arctan(k y /kx ), resulting in an anisotropic feature. Actually, the orbital magnetic moments are also trigonally warped like that in Berry curvatures and we don’t present them in the 2D k-plane
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here for space limitations. In order to show the corrections to the orbital magnetic moments from the warping effect, we present the orbital magnetic moments as a function of the module of k around K valley in Fig. 3(b). We find the orbital magnetic moments are peaked at the K valley and gradually decayed away from the valley. Owing to the valley and spin coupled band structure and large spin splitting [4], the orbital magnetic moments are spin polarized at K valley. In comparison, the warping effect decreases the orbital magnetic moments away from the K valley (see the red solid and dashed lines). Setting k = 0 in Eq. (10), we obtain the Berry curvatures at K and K’ valley as 2τ m0 t 2 /τ ,s z reducing to the results in Ref. [4], which are independent on the trigonal warping and the electron-hole asymmetric effect. This means the magnitudes of orbital magnetic moments at K or K’ valley remain unchanged in the presence of warping effect. The orbital magnetic moments versus k at K’ valley is the simply opposite to that at K valley, and we do not present them here. 4. Valley and spin Hall effect In this subsection, we discuss the intrinsic valley and spin Hall effect induced by the finite Berry curvature. In the presence of non-zero n (k), the equation of motion for the wavepackage is [12,25,26]
r˙ =
∂ E nk ˙ e − k × n (k) , k˙ = − (E + r˙ × B) , ∂ h¯ k h¯
from which we know that the Berry curvature drives an anomalous transverse velocity in the presence of an electric field E even in the absence of magnetic field B, which is
e v a = − E × n (k) . h¯
(11)
This anomalous transverse velocity v a is responsible for the intrinsic contribution to the anomalous Hall effect. The spin and valley resolved intrinsic Hall conductance can be calculated from the Berry curvature directly, which is τ sz σxy =
e2
2π h¯ n=c , v
τ sz
f ( E n ) n
(k) dk,
(12)
−1
where f ( E n ) = 1 + e ( E n − E F )/k B T is the Fermi-Dirac distribution. Here, it is worth to note that the valley and spin Hall conτs ductance σxy z are independent on the crystal orientation although the Berry curvatures are trigonally warped, because the orientation freedom in the Berry curvatures has been wiped out when calculating the Hall conductance by performing the integration over the Brillouin zone. Owing to the time reversal symmetry, we have τ ,s z −τ ,−s z σxy = −σxy . Therefore, a direct conclusion is that the total (charge) Hall conductance is zero in our systems because the charge carriers in the two valleys have opposite transverse velocities due to the opposite signs of the Berry curvatures constrained by the time reversal symmetry. However, there is finite valley and spin Hall conductance due to the inversion symmetry breaking. v s The valley σxy and spin σxy Hall conductance are given by
K↑ K↓ K ↑ K ↓ K↑ K↓ v σxy = σxy + σxy − σxy − σxy = 2 σxy + σxy ,
K↑ K ↑ K↓ K ↓ K↑ K↓ s σxy = σxy + σxy − σxy − σxy = 2 σxy − σxy .
(13) (14)
Fig. 4 shows the valley (a) and spin (b) Hall conductances as a function of Fermi energy for four kinds of monolayer transition metal dichalcogenides MX2 , where M = Mo, W, and X = S, Se. The solid (dashed) lines denote the results with (without) trigonal
Fig. 4. The valley (a) and spin (b) Hall conductance as a function of Fermi energy for four kinds of monolayer transition metal dichalcogenides MX2 , where M = Mo, W, and X = S, Se. The solid (dashed) lines denote the results with (without) trigonal warping. The inset in (b) is an amplification of the spin Hall conductance when the E F lies in the band gap. The band parameters for MoSe2 , WS2 and WSe2 are adopted from Ref. [8]. All conductances are in unit of σ0 = e 2 /2π h¯ , and the temperature is set as 77 K.
warping. As shown in Fig. 4(a), we find that the valley Hall conductances are quantized when the Fermi level E f lies in the band gap, i.e., the pristine samples, and gradually decrease for both the hole and electron doped samples with the increasing of doping concentration. Notably, the valley Hall conductances are reduced by the trigonal warping because this warping effect reduces the volume between the hook face of the valley Berry curvatures and the 2D k-plane in comparison with that in the isotropic case. From Fig. 4(b), we find that the spin Hall conductance is also quantized when the Fermi level lies in the band gap. For electron doped sample, the spin Hall conductance in MoS2 and MoSe2 are nearly zero due to the tiny spin splitting in their conduction bands (see the black and red lines), whereas the spin Hall conductances are finite in WS2 and WSe2 arising from the relatively large spin splitting in the conduction bands. For hole doped sample, there is a prominent spin Hall conductance when E f lies between the two spin split valence-band tops. In the presence of trigonal warping, in contrast to the valley Hall conductance, the spin Hall conductance is slightly enhanced because the trigonal warping enhances the spin polarization of the Berry curvatures due to its breaking on the valley-spin locked feature in the band structures [see Eq. (8)] and consequently the spin Hall conductance [see Eq. (14)]. But this modification is quite small and therefore negligible, we can conclude that the spin Hall conductance remain intact in the presence of trigonal warping. Further, there are various other monolayer transition metal dichalcogenides sharing similar band structures with that of monolayer MoS2 . Here, the conclusions for the valley and spin Hall conductances reported here are also applicable to them. 5. Summary In summary, we studied the trigonal warping on the Berry curvature, valley and spin Hall effect in monolayer MoS2 . Our results showed that the warping effect implies important modifications to the density of states, Berry curvature, orbital magnetic moment, and valley Hall conductances in monolayer MoS2 . In contrast to the isotropic case, the analytical results for Berry curvature and orbital magnetic moment show that both of them are trigonally warped in the presence of warping term, which is consistent with the results obtained from the first principle [5] or tight-binding calculations [23]. The valley Hall conductance was found to be reduced arising from the warped Berry curvature, whereas the spin Hall conductance remain intact. Our findings provide a further un-
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derstanding on the electronic states of monolayer MoS2 , which are useful in the explanation of the valley and spin Hall data on monolayer MoS2 and other transition metal dichalcogenides. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 11804092 and 11664019), and Project funded by China Postdoctoral Science Foundation (Grant Nos. BX20180097 and 2019M652777). References [1] Qing Hua Wang, Kourosh Kalantar-Zadeh, Andras Kis, Jonathan N. Coleman, Michael S. Strano, Nat. Nanotechnol. 7 (2012) 699. [2] Xiaodong Xu, Wang Yao, Di Xiao, Tony F. Heinz, Nat. Phys. 10 (2014) 343. [3] Gui-Bin Liu, Di Xiao, Yugui Yao, Xiaodong Xude, Wang Yao, Chem. Soc. Rev. 44 (2015) 2643. [4] Di Xiao, Gui-Bin Liu, Wanxing Feng, Xiaodong Xu, Wang Yao, Phys. Rev. Lett. 108 (2012) 196802. [5] Wanxiang Feng, Yugui Yao, Wenguang Zhu, Jinjian Zhou, Wang Yao, Di Xiao, Phys. Rev. B 86 (2012) 165108. [6] A. Kormányos, V. Zólyomi, Neil D. Drummond, Péter Rakyta, Guido Burkard, Vladimir I. Fal’ko, Phys. Rev. B 88 (2013) 045416. [7] Gui-Bin Liu, Wen-Yu Shan, Yugui Yao, Wang Yao, Di Xiao, Phys. Rev. B 88 (2013) 085433.
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