Determining Interaction Enhanced Valley Susceptibility in Spin-Valley

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Determining Interaction Enhanced Valley Susceptibility in SpinValley-Locked MoS2 Jiangxiazi Lin,† Tianyi Han,† Benjamin A. Piot,‡ Zefei Wu,† Shuigang Xu,† Gen Long,† Liheng An,† Patrick Cheung,§ Peng-Peng Zheng,§ Paulina Plochocka,∥ Xi Dai,† Duncan K. Maude,∥ Fan Zhang,*,§ and Ning Wang*,†

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†

Department of Physics and Center for Quantum Materials, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China ‡ Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS-UGA-UPS-INSA-EMFL, F-38042 Grenoble, France § Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, United States ∥ Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS-UGA-UPS-INSA-EMFL, F-31400 Toulouse, France S Supporting Information *

ABSTRACT: Two-dimensional transition metal dichalcogenides (TMDCs) are recently emerged electronic systems with various novel properties, such as spin-valley locking, circular dichroism, valley Hall effect, and superconductivity. The reduced dimensionality and large effective masses further produce unconventional many-body interaction effects. Here we reveal strong interaction effects in the conduction band of MoS2 by transport experiment. We study the massive Dirac electron Landau levels (LL) in high-quality MoS2 samples with field-effect mobilities of 24 000 cm2/(V·s) at 1.2 K. We identify the valley-resolved LLs and low-lying polarized LLs using the Lifshitz−Kosevitch formula. By further tracing the LL crossings in the Landau fan diagram, we unambiguously determine the density-dependent valley susceptibility and the interaction enhanced g-factor from 12.7 to 23.6. Near integer ratios of Zeeman-to-cyclotron energies, we discover LL anticrossings due to the formation of quantum Hall Ising ferromagnets, the valley polarizations of which appear to be reversible by tuning the density or an in-plane magnetic field. Our results provide evidence for many-body interaction effects in the conduction band of MoS2 and establish a fertile ground for exploring strongly correlated phenomena of massive Dirac electrons. KEYWORDS: Transition metal dichalcogenide, quantum oscillation, Landau level, Zeeman effect, g-factor, electron−electron interaction hen subjected to an external magnetic field, the degeneracy of electrons with opposite spins is lifted by the Zeeman energy EZ = gμBB, where μB is the Bohr magneton, B is the applied magnetic field, and g is the electron Landé g-factor. In two-dimensional electron gases (2DEGs), the effective g-factor g* was found to be larger than its bare value.1 This is a many-body effect driven by the exchange interactions.2−4 The recently emerged layered transition metal dichalcogenides (TMDCs) have offered a new family of 2D systems with novel electronic and optical properties.5−11 Hampered by the difficulty in obtaining quantum oscillations, the study of such an effect in TMDCs remains scarce. Recently, Xu et al. reported a giant g-factor in p-type few-layer WSe2.12 Movva et al. and Gustafsson et al. observed densitydependent g-factor in WSe2 using different methods.13,14 These interesting findings suggest the presence of strong electron interactions in TMDCs. While these studies focus on the valence band of WSe2, here we choose MoS2 for our

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© XXXX American Chemical Society

experiment, since it is more convenient to reach the conduction band in MoS2.15 For bilayer MoS2, the conduction band minima reside at K and K′ valleys.16 When the bilayer is subjected to an out-ofplane electric field induced by the backgate voltage Vg, the intrinsic inversion symmetry is broken.17,18 This extrinsic symmetry breaking, together with the intrinsic spin−orbital coupling (SOC), lifts the spin degeneracy in opposite directions for the two valleys, causing the spin-valley locking (K↑ and K′↓) for the low-energy electrons19 (Figure 1a). A traditional way to determine the g-factor of a 2DEG is to measure the Shubnikov−de Haas (SdH) oscillations in a tilted magnetic field. The Landau levels (LLs) are spaced by the cyclotron energy Ec = ℏeB⊥/m*, proportional to the perpendicular component B⊥ of the total magnetic field Btot, Received: November 25, 2018 Revised: February 1, 2019 Published: February 5, 2019 A

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Figure 1. Band structure, device structure, field-effect conductance, and low-field quantum oscillations of the gated bilayer MoS2. (a) Hexagonal Brillouin zone of the layer-polarized MoS2 and its conduction band minima at K/K′ points, with a splitting ΔSOC between the spin up (red) and spin down (blue) subbands. (b) Cross-section structure and optical image of the device, with a scale bar of 8 μm. (c) Four-probe conductance at 1.2 K. The linear fit (red dashed line) shows a FE mobility of 24 000 cm2/(V·s). (d) Magneto (black) and Hall (blue) resistances measured at 1.2 K at n = 3.15 × 1012 cm−2. The 2-fold degeneracy arises from the coincidence of K↑ and K′↓ LLs. Inset: Fourier transform of Rxx. The carrier densities calculated from Fourier analysis are consistent with those calculated from Hall effect. (e) Temperature-dependent oscillations at n = 3.15 × 1012 cm−2. Inset: oscillation amplitudes at B = 11.8 T fitted with the LK formula.

while the Zeeman energy is proportional to Btot. Thus, the ratio EZ/Ec ∝ g*m*/cos(θ) varies with the tilting angle θ. At certain special angles, EZ/Ec takes integer or half-integer values, where LLs of opposite spins coincide with each other or separate alternately. By identifying these special angles, the spin susceptibility χ* ∝ g*m*, thus g*, can be extracted. For MoS2 K electrons, however, the strong SOC and the mirrorplane symmetry pins the spin to the out-of-plane direction and the LLs are immune to the in-plane Zeeman field. Thus, the commonly used tilted field method is not applicable to MoS2 K electrons.13,20 For a strongly interacting 2DEG, the interaction strength can be characterized by the dimensionless Wigner−Seitz radius rs = 1/( πn aB*),21 where a*B = 4πϵℏ2/(m*e2) is the effective Bohr radius and ϵ is the dielectric constant. For a given m*, as the carrier density n decreases, the interaction gets stronger, resulting in a larger spin susceptibility. Evidently, varying n in a strongly interacting 2DEG can change the EZ/Ec ratio, similar to the effect of varying tilting angle in those 2DEGs without spin pinning. Thus, there are certain special densities where EZ/Ec takes integer or half-integer values, and for large EZ/Ec there can be more than one low-lying polarized LLs. Here we propose to determine the density-dependent g-factor in MoS2 by identifying those special densities and the number of lowlying polarized LLs. To probe the quantum oscillations in TMDCs, the device needs high mobility and Ohmic contact at low temperature, which are challenging especially for thinner samples. Possible solutions include using few-layer graphite to match the work function of the semiconductor22 and applying a large electric field to the contact area in dual-gate configuration.13 Here we use selective etching23 to achieve high-quality bilayer MoS2

samples, where stable contacts and clear quantum oscillations are preserved to extremely low densities where interactions are strong. From n = 4 × 1012 to 1 × 1012 cm−2, g* is found to increase from 12.7 to 23.6. Notably, anticrossings are observed, indicating the formation of exchange-driven quantum Hall ferromagnets. A bilayer MoS2 flake was first exfoliated onto a silicon wafer, then identified by optical contrast, and subsequently encapsulated by two insulating hexagonal boron nitride (hBN) layers. The hBN encapsulation protects the MoS2 from impurity contamination and degradation,23,24 after which atomic force microscopy and Raman spectroscopy were performed to confirm the number of layers of MoS2.20 The sandwich structure was patterned and contacted by Ti/Au electrodes. Figure 1b shows the vertical cross section and optical image of the device. The electrodes are labeled from 1 to 8, among which 1, 5, 2, 4, and 8 were used in the measurement. The encapsulated device lies on a SiO2/Si wafer, to which Vg was applied (Figure 1b). All data were obtained by a standard low-frequency lock-in technique at cryogenic temperatures. We fabricated and measured three samples with the same configuration, labeled A, B, and C. They show consistent behaviors, and only results from sample C are presented here. (Data of the other two samples are provided in Supporting Information.20) Four-probe conductance G of the MoS2 device is shown in Figure 1c. The field-effect (FE) mobility is calculated as μFE = (dG/dVg)l/(wCg), where l = 7.6 μm and w = 2.6 μm are the length and width of the channel, respectively, and Cg is the gate capacitance determined by the linear relation between Vg and n.20 This yields μFE ≈ 24 000 cm2/(V·s) at 1.2 K. The high mobility and low contact resistance allow us to observe Shubnikov-de Haas (SdH) oscillations down to a carrier B

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Figure 2. SdH oscillations fitted with the valley-resolved LK formula. (a) Upper: quantum oscillation data fitted with the valley-resolved LK formula. Filling factors are labeled. νc = 5 is marked in orange. Lower: two valley components of the fitting. (b) LL configuration corresponding to (a).

Figure 3. Landau level fan diagram with density-dependent valley Zeeman splitting. (a) Mapping of quantum oscillation amplitude in the n−B space. The dark blue regions are Rxx minima, whose corresponding filling factors are marked along the right and top edges. The data below and above 15 T were obtained in a 1.5 K superconducting system and in a 1.2 K resistive high magnetic field system, respectively, leading to the minor color discontinuity at 15 T. The orange dashed line marks the νc, thus, the boundary below which the LLs are fully valley polarized. (b) Rxx (black) and (h/e2)/Rxy (blue) at several special densities corresponding to the labels in (a). The densities of i−iv are 2.25, 1.97, 1.57, and 1.35 (1012 cm−2), respectively. Orange triangles label the νc for each density. (c) Schematics of Landau levels with increasing valley Zeeman splitting. Four vertical dashed lines correspond to the situations in i−iv, respectively.

density as low as 6.3 × 1011 cm−2,20 which is hitherto the lowest density for few-layer TMDCs. Figure 1d shows the symmetrized Rxx and Rxy resistances at n = 3.15 × 1012 cm−2 below 14 T. The LL filling factors are labeled for Rxx by ν = 2πnℏ/eB. The 2-fold degeneracy at low fields demonstrates that the electrons are at K valleys and that the applied backgate voltage breaks the inversion symmetry, polarizing the electrons to one of the two layers. Each doubly degenerated level consists of a K↑ level and a K′↓ level. Moreover, the single frequency obtained from Fourier analysis (inset) excludes the participation of other subbands or carriers

in the other layer. Figure 1e shows the temperature-dependent SdH oscillations at the same n as in Figure 1d. The Lifshitz− Kosevitch (LK) formula is used to fit the oscillation amplitudes (inset). The resulting effective mass gives m* = 0.55 ± 0.08 m0 without an obvious dependence on n or B;20 this value is comparable to the calculated effective mass16 of electrons in the K/K′ valleys. The Figure 2a upper panel shows the SdH oscillations at the density n = 2.25 × 1012 cm−2 up to 28 T, with filling factors labeled. Here the 2-fold degeneracy is fully lifted, indicating the valley Zeeman splitting.25−27 At ν > 5 the peak and flat regions C

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Figure 4. List of density-dependent spin susceptibility and a comparison with other works in various systems. (a) g*m*/m0 versus n. g* and rs are shown on the right and top axes. The horizontal error bars represent the uncertainty in Vg. Fitting proportional to rs is shown. The green line marks the noninteracting limit. The range of n on the bottom axis corresponds to the experiment range. (b) g*m*/m0 versus rs. For comparison, data from all three samples studied in this work and from other works in various systems using different methods are shown, including 1L and 2L WSe2 holes and MoSe2 electrons (extracted from even−odd transitions and quantum Monte Carlo calculations),13,35 1L WSe2 holes (extracted by singleelectron transistor (SET) measurement),14 and AlAs quantum wells (QWs) with different widths (extracted using coincidence in tiled fields).37,38 e and h represent electrons and holes, respectively.

alternate, while at ν < 5 there is no flat region. To understand this anomalous oscillations, we use the valley-resolved LK formula to fit the experiment data:28,29 Nr

ΔR xx = 2R 0 ∑

∑

r = 1 σ = K,K′

depth; when EZ*/Ec is an integer, LLs of different valleys coincide at ν > νc, producing a pronounced filling factor sequence at ν = 1, 2, ..., νc, νc + 2, νc + 4, .... Four out of all observed consecutive special densities are marked in the map with i−iv, the line cuts of which are shown in Figure 3b. Note that the anomalous dip at ν = 6 in ii arises from the anticrossing gap at high magnetic fields, which will be discussed later. Figure 3c shows the schematics of LLs with increasing valley Zeeman splitting. The vertical dashed lines correspond to the situations in i−iv. Orange triangles marked the positions of νc. In Figure 3a, the orange dashed line connects the νc’s of all densities, marking the boundary below which the LLs are fully valley polarized. From the relation νc = ⌈EZ*/Ec⌉, where ⌈⌉ is the ceiling function, the EZ*/Ec ratios corresponding to i−iv can be easily extracted to be 4.5, 5, 5.5, and 6, respectively. In the massive Dirac Fermionic model, the anomalous zeroth LL (thick red line in Figure 3c) is present in valley K but not in K′.6 As a consequence, the splitting between the lowest LLs in different valleys E*Z and the splitting between the LLs of the same orbital index in different valleys EZ are related by EZ* = EZ + Ec. Since in the large n (noninteracting) limit, EZ = 2 μBB, the noninteracting effective g-factor for the conduction-band valley Zeeman effect is g* = 2 + 2m0/m* = 5.64, where 2m0/m* = 3.64 arises from the valley magnetic moment of massive Dirac bands.30,31 (We have assumed the K↑ N = 0 LL shifts down in energy. Otherwise, the noninteracting limit of g* would be 1.64.) Starting from the four special densities shown in Figure 3b, we trace the evolution of the LLs to higher and lower density ranges to identify other consecutive special densities. Within the accessible density range, we also find the densities corresponding to E*Z /Ec = 4, 3.5, and 6.5. In Figure 4a, we list all observed densities corresponding to integer and halfinteger EZ*/Ec ratios. The left and top axes are the calculated effective valley susceptibility ∝g*m*/m0 and Wigner−Seitz radius rs, using the dielectric constant ϵ ≈ 4ϵ0.32 The best fitting proportional to n−1/2 (or rs) gives 0.55 + 13.1n−1/2, which may be improved in future experiments that can take into account the neglected electron−hole asymmetry31 and minority orbitals.19 The noninteracting limit is marked by the green line. The giant valley susceptibility and its enhancement

ij −rπ ℏ yz rλ zz cos(rϕ ) expjjj σ j Ecτσ zz sinh(rλ) k {

(1)

where λ = 2π2KBT/Ec is the thermal damping and τK,K′ are the valley-resolved scattering times. Given the anomalous LL structure for massive Dirac electrons in MoS2,6 ϕK = 2πBF/B + πEZ*/Ec + π and ϕK′ = 2πBF/B − πEZ*/Ec − π. The effective valley Zeeman energy EZ* = g*μBB is defined as the energy difference between the lowest LLs in different valleys, as labeled in Figure 2b. The purple curve in Figure 2a shows the fitted ΔRxx using eq 1 with Nr = 10. The best fitting yields EZ*/ Ec = 4.5; thus, g* = 16.4. The lower panel plots the individual contributions to the fitted ΔRxx from each valley; while both become stronger with increasing B, those of valley K′ electrons are relatively weaker and cease at ∼18 T or ν = 5. These findings suggest that the peaks of Rxx correspond to LLs in valley K and that the flat regions correspond to LLs in valley K′, as a result of a valley-dependent scattering time τK > τK′.20 τK,K′ obtained by this method has the same order of magnitude with that obtained by a temperature-dependent analysis.20 A critical filling factor νc can be defined (marked in orange), as the largest ν below which the LLs are fully valley polarized. The LL configuration corresponding to Figure 2a is drawn in Figure 2b. The configuration of LLs evolves as n varies, indicating a change in the E*Z /Ec ratio. We measured the SdH oscillations in different densities to obtain the LL fan diagram, i.e., the mapping of oscillation amplitudes in the n−B space (Figure 3a). Note that the dark blue regions are Rxx minima, corresponding to energy gaps in the density of states, whose corresponding filling factors are marked along the right and top edges. Since the LLs in valley K/K′ enjoy sharper/flatter Rxx peaks, they show a brighter/dimmer color in the diagram. We identify several special densities by the following signature oscillation patterns: when EZ*/Ec is a half-integer, LLs of different valleys are alternating and equally spaced at ν > νc, yielding Rxx minima at all integer filling factors with similar D

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Figure 5. Signatures of Landau level anticrossings. (a) Rxx versus ν at carrier densities around EZ*/Ec = 4. Red and blue dashed lines mark the Landau levels in valley K and K′, respectively. The magenta arrow marks the anticrossing gap at ν = 7. (b) Temperature dependence of the anticrossing at n = 3.24 × 1012 cm−2, offset for clarity. Filling factors are marked. (c) Tilt angle dependence of the anticrossing at ν = 7 at n = 3.24 × 1012 cm−2, offset for clarity. The dotted traces are guides to the eye for the positions of enhanced peaks in Rxx. Inset: the nonoffset data featuring the unchanged minima at ν = 6 and 8.

with decreasing n are most likely due to the electron−electron interactions in our system.4,21,33 Our studied densities yield rs ∼ 10, which is indeed in the strong interaction regime.21 We stress that the trend of enhancing susceptibility with decreasing density is observed in all three samples examined, despite the large difference in their mobilities.20 This implies that the susceptibility enhancement is not sensitive to impurities. Another factor that could lead to a density-dependent susceptibility is the nonparabolicity in the dispersion. This effect only becomes appreciable away from the conduction band minimum and in the weak interacting regime34 and thus can be neglected in our study. During the preparation of this manuscript, we noticed similar reports of the density-dependent g-factor in the conduction band of TMDCs. Larentis et al. studied monolayer and bilayer MoSe235 and Pisoni et al. studied monolayer MoS2.36 In Figure 4b, we compare the spin susceptibility extracted in our three samples with several other reports in various systems, including TMDCs and AlAs 2DEGs. Data from the references are read from their figures; the values are converted on the basis of our definition and are calculated from n or g* if rs or g*m*/m0 is not directly available. Our result of MoS2 K-valley electrons is consistent with studies in various systems, which indicates the exchange interaction enhancement of spin susceptibility is a universal phenomenon among different 2D systems. At those densities with integer E*Z /Ec, LL anticrossings are also observed, e.g., a secondary gap opens when two LLs come close in energy. Figure 5a highlights a strong anticrossing at ν = 7 (magenta arrow) at n = 3.24 × 1012 cm−2 (EZ*/Ec = 4): the two LLs from K and K′ valleys (dotted traces) swap their relative positions abruptly with varying density, without losing the Rxx dip at ν = 7. Such LL anticrossings are most likely due to the formation of quantum Hall Ising ferromagnets driven by the exchange interaction.3,39,40 This picture is consistent with the observation that the anticrossings at ν = 9 and 11 are less pronounced, as the exchange interaction strength decreases with decreasing B or increasing the LL orbital index. Note that LL anticrossing is also observed for another density n = 1.97 × 1012 cm−2 (E*Z /Ec = 5), as the dip at ν = 6 (Figure 3b-ii).

Figure 5b shows the temperature dependence of the anticrossing at n = 3.24 × 1012 cm−2. The anticrossing gap at ν = 7 disappears at ∼3.3 K, whereas the single-particle gap at ν = 6 disappears at a much higher temperature ∼10.1 K. This provides further evidence for the many-body origin of the anticrossings. Small enhancements in Rxx are found associated with the anticrossing gap at ν = 7. Such enhancement is more obvious at a higher field near the gap at ν = 5, although the swapping of LLs is not fully revealed at this filling factor limited by the magnetic field strength. The enhancement is reminiscent of those observed in the AlAs quantum well at much lower densities38,41 and can be related to the charge transport along the domain wall loops near the first-order Ising transition.42 Possible evolution and hysteresis41,43 of such a peak in future experiments at lower temperatures may completely decipher the anticrossings. Note that there may be a slight variation of anticrossing densities among different filling factors ν = 5, 7, 9, and 11, of which a more detailed study is limited by the accuracy of the current experiment. Figure 5c reveals the Rxx behavior under tilted magnetic fields at n = 3.24 × 1012 cm−2. Away from the anticrossing, the Rxx features near ν = 6 and 8 remain virtually unchanged for all accessible tilt angles (Figure 5c inset). This insensitivity arises from the spin splitting and pinning to the out-of-plane direction, due to the SOC and mirror-plane symmetry. By sharp contrast, on the two sides of the ν = 7 anticrossing, the Rxx peaks do evolve with the tilt angle. The two peaks exchange their relative positions abruptly without eliminating the ν = 7 minimum (dotted traces), corresponding to the reversal of Ising polarization. The enhanced resistance spikes also evolve with the angle (dotted traces). Such a tilt angle dependence implies that an in-plane magnetic field can couple the two LLs near their anticrossing, modify the exchange interaction, and tune the valley polarization of quantum Hall Ising ferromagnet. In conclusion, we fabricate high-quality bilayer n-type MoS2 devices and study the valley-resolved SdH oscillations relevant to the spin-valley locked massive Dirac electron LLs. Using the valley-resolved Lifshitz−Kosevitch formula and the LL crossings in the Landau fan diagram, we determine the special densities corresponding to integer and half-integer ratios of E

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(4) Zhu, J.; Stormer, H. L.; Pfeiffer, L. N.; Baldwin, K. W.; West, K. W. Spin susceptibility of an ultra-low-density two-dimensional electron system. Phys. Rev. Lett. 2003, 90, 056805. (5) Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 2012, 7, 699−712. (6) Li, X.; Zhang, F.; Niu, Q. Unconventional quantum Hall effect and tunable spin Hall effect in Dirac materials: application to an isolated MoS2trilayer. Phys. Rev. Lett. 2013, 110, 066803. (7) Xu, X.; Yao, W.; Xiao, D.; Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nat. Phys. 2014, 10, 343− 350. (8) Mak, K. F.; McGill, K. L.; Park, J.; McEuen, P. L. The valley Hall effect in MoS2transistors. Science 2014, 344, 1489−1492. (9) Zhang, Y. J.; Oka, T.; Suzuki, R.; Ye, J. T.; Iwasa, Y. Electrically switchable chiral light-emitting transistor. Science 2014, 344, 725− 728. (10) Lu, J. M.; Zheliuk, O.; Leermakers, I.; Yuan, N. F. Q.; Zeitler, U.; Law, K. T.; Ye, J. T. Evidence for two-dimensional Ising superconductivity in gated MoS2. Science 2015, 350, 1353−1357. (11) Wang, Z.; Shan, J.; Mak, K. F. Valley- and spin-polarized Landau levels in monolayer WSe2. Nat. Nanotechnol. 2016, 12, 144− 149. (12) Xu, S.; Shen, J.; Long, G.; Wu, Z.; Bao, Z.-q.; Liu, C.-C.; Xiao, X.; Han, T.; Lin, J.; Wu, Y.; Lu, H.; Hou, J.; An, L.; Wang, Y.; Cai, Y.; Ho, K. M.; He, Y.; Lortz, R.; Zhang, F.; Wang, N. Odd-integer quantum Hall states and giant spin susceptibility in p-type few-layer WSe2. Phys. Rev. Lett. 2017, 118, 067702. (13) Movva, H. C. P.; Fallahazad, B.; Kim, K.; Larentis, S.; Taniguchi, T.; Watanabe, K.; Banerjee, S. K.; Tutuc, E. Densitydependent quantum Hall states and Zeeman splitting in monolayer and bilayer WSe2. Phys. Rev. Lett. 2017, 118, 247701. (14) Gustafsson, M. V.; Yankowitz, M.; Forsythe, C.; Rhodes, D.; Watanabe, K.; Taniguchi, T.; Hone, J.; Zhu, X.; Dean, C. R. Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe2. Nat. Mater. 2018, 17, 411−415. (15) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, i. V.; Kis, A. Single-layer MoS2 transistors. Nat. Nanotechnol. 2011, 6, 147−150. (16) Cheiwchanchamnangij, T.; Lambrecht, W. R. L. Quasiparticle band structure calculation of monolayer, bilayer, and bulk MoS2. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 205302. (17) Wu, S.; Ross, J. S.; Liu, G.-B.; Aivazian, G.; Jones, A.; Fei, Z.; Zhu, W.; Xiao, D.; Yao, W.; Cobden, D.; Xu, X. Electrical tuning of valley magnetic moment through symmetry control in bilayer MoS2. Nat. Phys. 2013, 9, 149−153. (18) Lee, J.; Mak, K. F.; Shan, J. Electrical control of the valley Hall effect in bilayer MoS2 transistors. Nat. Nanotechnol. 2016, 11, 421− 425. (19) Liu, G.-B.; Shan, W.-Y.; Yao, Y.; Yao, W.; Xiao, D. Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 085433. (20) See the Supporting Information. (21) Spivak, B.; Kravchenko, S. V.; Kivelson, S. A.; Gao, X. P. A. Colloquium: Transport in strongly correlated two dimensional electron fluids. Rev. Mod. Phys. 2010, 82, 1743. (22) Cui, X.; Lee, G.-H.; Kim, Y. D.; Arefe, G.; Huang, P. Y.; Lee, C.-H.; Chenet, D. A.; Zhang, X.; Wang, L.; Ye, F.; Pizzocchero, F.; Jessen, B. S.; Watanabe, K.; Taniguchi, T.; Muller, D. A.; Low, T.; Kim, P.; Hone, J. Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform. Nat. Nanotechnol. 2015, 10, 534−540. (23) Xu, S.; Wu, Z.; Lu, H.; Han, Y.; Long, G.; Chen, X.; Han, T.; Ye, W.; Wu, Y.; Lin, J.; Shen, J.; Cai, Y.; He, Y.; Zhang, F.; Lortz, R.; Cheng, C.; Wang, N. Universal low-temperature Ohmic contacts for quantum transport in transition metal dichalcogenides. 2D Mater. 2016, 3, 021007. (24) Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watanabe, K.; Campos, L. M.; Muller, D. A.; Guo, J.;

Zeeman-to-cyclotron energies, revealing the enhancement of the valley Zeeman effect by electron−electron interactions. We also discover persistent LL anticrossings that indicate the formation of quantum Hall ferromagnets. Our results provide compelling evidence for strong many-body interaction effects in the conduction bands of MoS2 and establish a fertile ground for exploring strongly correlated phenomena of massive Dirac electrons.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.8b04731.

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Additional data of sample C, including AFM and Raman characterizations, basic electrical characterization, and more quantum oscillation data; experimental data of two other samples (sample A and B) including optical photos, FET conductance, carrier density, effective masses, and oscillation mapping; comparison between three samples including Raman characterization, E*Z /Ec ratios, and conductivity plots and a table of mobilities; and details of the valley-resolved LK formula used in the fitting (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Jiangxiazi Lin: 0000-0002-7283-5619 Gen Long: 0000-0002-2004-5455 Paulina Plochocka: 0000-0002-4019-6138 Ning Wang: 0000-0002-4902-5589 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Roman Gorbachev, Vladimir Fal’ko, Ding Pan, Junwei Liu, and Denis Maryenko for fruitful discussions. This work is supported by the Research Grants Council of Hong Kong (Project No. 16300717, SBI17SC16) and by FB417UoM-HKUST. J.L. acknowledges the financial support of Hong Kong Ph.D. Fellowship on her Ph.D. study. P.C., P.Z., and F.Z. are supported by Army Research Office under Grant Number W911NF-18-1-0416. Part of this work has been supported by the graphene flagship project (604391). We also acknowledge the technical support from the Raith-HKUST Nanotechnology Laboratory for the electron-beam lithography facility at MCPF and the support of the LNCMI-CNRS, member of the European Magnetic Field Laboratory (EMFL).

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REFERENCES

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DOI: 10.1021/acs.nanolett.8b04731 Nano Lett. XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.nanolett.8b04731 Nano Lett. XXXX, XXX, XXX−XXX