Absence of inter-layer tunnel coupling of K -valley electrons in bilayer MoS 2
Riccardo Pisoni, Tim Davatz, Kenji Watanabe, Takashi Taniguchi, Thomas Ihn, Klaus Ensslin
AAbsence of inter-layer tunnel coupling of K -valley electrons in bilayer MoS Riccardo Pisoni, Tim Davatz, Kenji Watanabe, Takashi Taniguchi, Thomas Ihn, and Klaus Ensslin Solid State Physics Laboratory, ETH Z¨urich, 8093 Z¨urich, Switzerland National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan (Dated: April 22, 2019)In Bernal stacked bilayer graphene interlayer coupling significantly affects the electronic band-structure compared to monolayer graphene. Here we present magnetotransport experiments onhigh-quality n -doped bilayer MoS . By measuring the evolution of the Landau levels as a functionof electron density and applied magnetic field we are able to investigate the occupation of conductionband states, the interlayer coupling in pristine bilayer MoS , and how these effects are governed byelectron-electron interactions. We find that the two layers of the bilayer MoS behave as two inde-pendent electronic systems where a two-fold Landau level’s degeneracy is observed for each MoS layer. At the onset of the population of the bottom MoS layer we observe a large negative com-pressibility caused by the exchange interaction. These observations, enabled by the high electronicquality of our samples, demonstrate weak interlayer tunnel coupling but strong interlayer electro-static coupling in pristine bilayer MoS . The conclusions from the experiments may be relevant alsoto other transition metal dichalcogenide materials. Of the multitude of two-dimensional (2D) host materi-als, transition metal dichalcogenides (TMDs) are promis-ing candidates for exploring quantum correlated elec-tronic phases and electron-electron interaction effects dueto their intrinsic 2D nature, large spin-orbit interactionand large effective mass carriers. Molybdenum disulfide(MoS ) is one of the most widely studied TMDs andstill most of its fundamental quantum electronic prop-erties have thus far been elusive. Contrary to monolayerMoS , in pristine bilayer MoS inversion symmetry is re-stored [1–3]. As a result, the orbital magnetic momentand the valley-contrasting optical dichroism vanish [1, 4].A potential difference between the two layers breaks theinversion symmetry [5, 6]. The influence of a perpendic-ular electric field on bilayer MoS has been extensivelyprobed by optical excitation [5, 6]. Very little is knownabout the electronic transport properties of bilayer MoS when electric and magnetic fields are both applied per-pendicular to the sample plane [7]. Magnetotransportstudies of 2D holes have been recently performed in bi-layer WSe revealing the presence of two subbands, eachlocalized in the top and bottom layer, and demonstrat-ing an upper bound of the interlayer tunnel coupling of19 meV [8, 9]. A thorough study of the interlayer cou-pling in the conduction band of bilayer transition metaldichalcogenides is still missing [7, 10].Here we report a magnetotransport study of electronsin the conduction band of dual-gated bilayer MoS . Allstudied bilayer samples exhibit Shubnikov-de Haas (SdH)oscillations with a twofold Landau level degeneracy at T = 1 . K and K (cid:48) valley in each layer. By tuning the Fermienergy in each layer individually we are able to populatelower and upper spin-orbit split bands in both layers. The exchange interaction in a single layer yields a pro-nounced negative compressibility visible in occupation ofthe states detected via the Landau fan diagram. In addi-tion, we observe an intricate interplay between spin- andvalley-polarized Landau levels originating from the twodecoupled MoS layers. We do not observe any obvioussignature in the Landau level spectrum when the electro-static potential difference between the two layers vanishesand the structural inversion symmetry is expected to berestored.Figure 1(a) shows the schematic cross-section of thedual-gated bilayer MoS device under study. The MoS isencapsulated between two hBN dielectrics with graphitelayers as top and bottom gates. We fabricate pre-patterned Au bottom contacts below the bilayer MoS .Ohmic behavior of these contacts is achieved by applyinga sufficiently positive top gate voltage ( V TG ). The het-erostructure is assembled using a dry pick-up and transfermethod [11–13]. We fabricated and measured three bi-layer MoS samples, labeled A, B, and C, which show thesame behavior. We will mainly discuss samples A and Bhere.The samples use commercial bulk MoS crystal (SPIsupplies) mechanically exfoliated on SiO /Si substrates.Using a combination of optical contrast, photolumines-cence spectroscopy and atomic force microscopy bilayerMoS flakes are identified. Figure 1(b) shows the opticalmicrograph of sample A. The bilayer MoS flake is out-lined with a white dashed line. Top and bottom graphitegates are outlined in purple and top and bottom hBN areoutlined in blue and cyan, respectively. In the inset ofFig. 1 (b) we sketch the contact geometry where contacts1 and 4 are used for current injection and extraction andcontacts 2 and 3 serve as voltage probes.Figure 1(c) shows the four-terminal resistance, R , ,as a function of V BG and V TG at T = 1 . V TG a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r − V BG (V)7891011 V T G ( V ) R , (Ω) 4 5 6 7 8 B (T) − ∆ R , ( Ω ) . . n (10 cm − ) 0 . . . m ∗ ( m e ) (a) (b) V TG V BG GraphiteGraphiteh-BNh-BN
MoS μ m (d) ν = 25 23 21 (c) FIG. 1. (a) Schematic of the device. Bilayer MoS isencapsulated between two hBN layers and Au contacts arepre-patterned on the bottom hBN before the MoS layer istransferred. Graphite flakes serve as bottom and top gates.(b) Optical micrograph of sample A. The bilayer MoS flakeis highlighted with a white dashed line. Inset: contact ge-ometry, current is injected to contact 1 and extracted fromcontact 4, voltage is measured between contacts 2 and 3 (scalebar is 2 µ m). (c) Four-terminal resistance R , as a func-tion of V TG and V BG at T = 1 . V BG values at which the bottom MoS layer starts to be pop-ulated. Solid and dotted lines represent V TG and V BG val-ues at which the upper spin-orbit split bands are occupied inthe top and bottom MoS layers, respectively. (d) Temper-ature dependence of the SdH oscillations at V BG = − . V TG = 9 V, n ≈ . × cm − . An odd filling factorsequence ν = 21 , , , ..., is observed. Inset: electron effec-tive masses ( m ∗ ) extracted for the three different samples asa function of electron density. Black, blue, and red markersrepresent samples A, B, and C, respectively. and V BG . At fixed V TG we observe a sudden increase in R , at V BG ≈ layer. As a result,for the V BG values on the left side of the white dashedline we probe the electron transport only through thetop MoS layer. At V BG ≈ . V TG and V BG values along the white solid line, weobserve additional resistance kinks that we attribute tothe occupation of the upper spin-orbit split bands in thebottom and top MoS layers, respectively.We investigate magnetotransport phenomena in bi-layer MoS using lock-in techniques at 31 . R , , the four-terminal linear resis-tance with a smooth background subtracted, as a func-tion of magnetic field B at various temperatures ranging from 1 . . n = 3 . × cm − .For T = 1 . ≈ T = 100 mK the onset of SdH oscillations moves to yetlower magnetic fields yielding a lower bound for the quan-tum mobility of ≈ / Vs (see Fig. 4). The elec-tron density is calculated from the period of the SdHoscillations in 1 /B considering valley degenerate Landaulevels at the K and K (cid:48) conduction band minima [13].At T = 1 . n = 3 . × cm − we observe thesequence of odd filling factors ν = 21 , , , ... . Thetwofold Landau level’s valley degeneracy is lifted at lowertemperatures (see below).We determine the electron effective mass m ∗ from thetemperature dependence of the SdH oscillations by fit-ting ∆ R , to ε/ sinh( ε ), where ε = 2 π k B T / (cid:126) ω c and (cid:126) ω c = eB/ m ∗ is the cyclotron frequency [14–16]. Theinset of Fig. 1 (c) shows the extracted m ∗ for the threedifferent samples. We obtain a density-averaged mass of m ∗ ≈ . m e which does not show any obvious depen-dence neither on n nor on B [17, 18].In Fig. 1 (d) we extract the m ∗ of the K and K (cid:48) elec-trons localized in the top MoS layer, thus effectivelycalculating the effective mass of a monolayer MoS [7].The effective masses we extract in our bilayer samples aresystematically 10 −
20% lower compared to the ones mea-sured in monolayer MoS [13]. In bilayer MoS the topMoS layer is encapsulated between hBN and the bottomMoS layer, which is devoid of electrons. We speculatethat the higher dielectric constant ( (cid:15) ≈ .
4) reportedfor monolayer MoS [19–22] compared to hBN ( (cid:15) ≈ . m ∗ value.The interaction strength is characterized by the di-mensionless Wigner-Seitz radius r s = 1 / ( √ π na ∗ B ), where a ∗ B = a B ( κm e /m ∗ ) is the effective Bohr radius and κ thedielectric constant. For the considered electron densityrange we estimate r s = 1 . −
10, placing the system in aregime where interaction effects are important [23–25].Figure 2(a) shows ∆ R , (color scale) as a functionof B applied perpendicular to the sample and V BG , at V TG = 9 V and T = 1 . V BG < . V BG resembles the one of monolayer MoS [13]. For V BG < − . K ↑ and K (cid:48)↓ bands are seen. As the electrondensity increases in this regime, we observe an alternat-ing parity of the filling factor sequence [see filling factorsequences in Fig. 2 (a)]. These results can be explainedin an extended single-particle picture where the valley g factor is density dependent, following the interpretationof previous works [7–10, 26].At V BG = − . K ↓ and K (cid:48)↑ valleys. Where the slopechanges, the electron density is n = 3 . × cm − . As-suming two-fold valley-degeneracy and using the experi- − − − V BG (V)02468101214 n ( c m − ) n t , lb n t , ub n b , lb n b , ub n tot n t n b − − − V BG (V)45678 B ( T ) − ∆ R , (Ω) (a)(b) ν = 182022 212325151317 FIG. 2. (a) Sample A. Four-terminal resistance ∆ R , as a function of V BG and magnetic field at T ≈ . V TG = 9 V. For V BG < . layer. The slope change at V BG = − . V BG = 1 . layer appears. At V BG = 3 . K and K (cid:48) valleys. (b) Electron densities in the bilayer MoS bandsas a function of V BG at T ≈ . V TG = 9 V. Green,orange, blue, and cyan dashed lines correspond to electrondensities in the lower ( n t , lb , n b , lb ), upper ( n t , ub , n b , ub ) spin-orbit split bands in the top and bottom layer, respectively.Green and blue solid lines represent the total carrier densityin the top ( n t ) and bottom ( n b ) layer, respectively. Blacksolid line corresponds to the total electron density ( n tot ) inthe bilayer MoS . At V BG = 5 . layer is achieved. mentally determined electron effective mass, we calculatethe Fermi energy to be E F = 14 meV, in good agreementwith the intrinsic spin-orbit interaction measured previ-ously for K -valley electrons in monolayer MoS [13]. Wewould like to note that our results justify the assump-tions in [27] that bilayer MoS investigated in the rightregime behaves as single-layer MoS with the caveat thatthe effective mass is different because of the dielectric en-vironment.The measured Landau level structure for V BG < . re-sults [13]. For V BG crossing the voltage 1 . V BG that existed below this threshold changes sign from positive to nega-tive. Second, an additional set of Landau levels appears[blue dashed lines in Fig. 2 (a)]. At V BG = 3 . V BG . To determinethe electron density of the individual layers and bandsfrom the Landau fan diagram we generate a Fouriertransform map of ∆ R , vs. 1 /B for each V BG valuein Fig. 2 (a) (see supplemental information). The Fouriertransform of the SdH oscillations shows multiple peaks inthe amplitude spectrum as we increase V BG . From thesepeaks we extract the electron density of the various spin-orbit split bands in bilayer MoS using n = ( g v e/h ) × f ,where f is the frequency of the Fourier transform peaksand g v = 2 accounts for the valley degeneracy. Theresults of this procedure are shown in Fig. 2 (b). For V BG < . layerwhere they occupy the lower (green dashed line) and up-per (orange dashed line) spin-orbit split bands as we in-crease V BG . At V BG = 1 . layerstarts to be populated (blue dashed line). The secondaryLandau fan that appears at V BG = 1 . layer. Beyond V BG = 1 . decrease . This density decrease isdirect experimental evidence for the negative compress-ibility of the bottom layer at low densities [28–31]. At V BG > . based on the data in Fig. 2.To this end, we consider the electrostatic model schemat-ically displayed in the inset of Fig. 3 consisting of threelayers of different dielectric constants, in which electricdisplacement fields exist due to the applied voltages V TG and V BG . The MoS bilayer is modeled as two groundedconducting planes of finite density of states with a ge-ometric capacitance C BL and a displacement field D BL between them. It is our goal to express dD BL /dD B , i.e.,the change in D BL upon a change in the displacementfield D B between back gate and MoS , at constant topgate voltage in terms of the measured V BG -dependentchanges of the layer densities. This quantity allows usto directly compare the strength of the effect with theresults obtained by Eisenstein et al [28] in the case ofa GaAs double quantum well, and with the numericalresults of Tanatar and Ceperley [30].The model (see supplemental material for details) re-sults in dD BL dD B (cid:12)(cid:12)(cid:12)(cid:12) V TG = C BL C B × − − − V BG (V) − . − . . . . . . d D B L / d D B C BL = 0 .
01 Fm − C BL = 0 .
65 Fm − n B (10 cm − ) 234510 r S V TG = 9 . d T d BL d B D BL D T D B hBNGraphite ϵϵ BL ϵ V TG V BG FIG. 3. Sample A. Ratio of the electric displacement fields asa function of V BG at T ≈ . V TG = 9 V. The top greenaxis represents the electron density in the bottom layer ( n B ).The blue axis denotes the r S parameter that accounts forintralayer interactions in the bottom layer assuming in-planedielectric constant ≈ . . m e . Inset: electrostatic model of our dual-gatedbilayer MoS device. × C B (cid:18) C T + e dn t dV BG (cid:12)(cid:12)(cid:12) V TG (cid:19) − C T e dn b dV BG (cid:12)(cid:12)(cid:12) V TG C BL (cid:18) C T + e d ( n t + n b ) dV BG (cid:12)(cid:12)(cid:12) V TG (cid:19) + C T e dn b dV BG (cid:12)(cid:12)(cid:12) V TG , (1)where C T and C B are the geometric capacitances per unitarea between MoS and top- and bottom-gate, respec-tively. The quantities n t and n b are the measured totalelectron densities in the two layers shown in Fig. 2 (b). Inthe case of V BG < . n b and its V BG -derivativeare zero, the displacement field ratio in eq. (1) is exactlyone. Negative compressibility in the region V BG > . dD BL /dD B < C BL takes on two plausible ex-treme values (see supplemental information for details).This shows that the result depends very little on the exactvalue of this parameter. A strong negative compressibil-ity with dD BL /D B ≈ − . V BG = − n b = 1 × cm − , roughly ten times stronger than theeffect observed in Ref. [28]. To compare the value tothe numerical results of Ref. [30], we have added an esti-mated scale bar of r s -values to the top axis in Fig. 3. Thenegative compressibility values measured in our sampleagree fairly well with the predictions of the numericalcalculations at these r s -values.Resolving individual layer electron densities in Fig. 2indicates that the two MoS layers are weakly coupled.This observation is in contrast to Bernal stacked bi-layer graphene, where the interlayer coupling of ≈ . reported an upper boundfor the interlayer tunnel coupling of ≈
19 meV [8]. In ourresults the interlayer coupling in the conduction band ofbilayer MoS is not observable. We achieve same electrondensities in both layers (red circle in Fig. 2 (b)) for threedifferent samples with no experimental evidence for in-terlayer coupling. Band structure calculations [33] revealthat strong interlayer hybridization in the conductionband of MoS occurs predominantly from orbitals whichare responsible for the minima at the Q -point, which arenot occupied in our samples. Conversely, weak interlayerhybridization is expected from the orbitals forming the K -valleys, which is consistent with our experimental ob-servations.At lower temperatures finer details of the Landau levelstructure are resolved. In Fig. 4 (a) we show ∆ R , as afunction of B and V BG at V TG = 13 . T = 100 mKfor sample B. For V BG < layer is devoid of electrons and we only ob-serve the Shubnikov-de Haas oscillations of the top layer.At T = 100 mK we are able to resolve valley-spin polar-ized Landau levels originating from the lowest conductionband minima in the top layer. The Landau level struc-ture of the spin-valley coupled bands in the bottom layerappears for V BG ≥ T ≈
100 mK, testifying to the high-mobility of our dual-gated bilayer MoS devices. Weare able to measure spin-valley polarized LLs originat-ing from the lower and upper spin-orbit split bands of K -valley electrons populating the top and bottom MoS layers. Our observations demonstrate that electrons inbilayer MoS behave like two independent electronic sys-tems. The exchange interaction at the turn on of thetwo-dimensional electron gas in the bottom layer leadsto the observation of a large negative compressibility.Our work demonstrates fundamental electronic transportproperties as well as the importance of interaction effectsin pristine bilayer MoS . These results bear relevancefor understanding electronic transport in twisted bilayerTMDs.We thank Guido Burkard, Vladimir Falko, AndorKorm´anyos, Mansour Shayegan and Peter Rickhaus forfruitful discussions. We thank Peter M¨arki as wellas the FIRST staff for their technical support. Weacknowledge financial support from ITN Spin-NANO V BG (V)345678 B ( T ) − R , (Ω) . . . . . . V BG (V)6 . . . . . B ( T ) − R , (Ω) . . . . . V BG (V)6 . . . . . B ( T ) − R , (Ω) (b) (c)(a) K K’ K K’Top layer Bottom layer | K’ > | K > | K’ > | K > E F FIG. 4. (a) Sample B. Four-terminal resistance ∆ R , as afunction of V BG and magnetic field at T ≈
100 mK and V TG =13 . V BG = 2 . layer is filled withelectrons and SdH oscillations appear (white dashed lines).Inset: conduction band minima sketch at the K and K (cid:48) pointfor top and bottom MoS layer. The horizontal black dashedline corresponds to the highest Fermi energy reached in thetop layer before the bottom layer is occupied. (b) Avoidedcrossing patterns between spin-valley coupled LLs originatingfrom the lower and upper spin-orbit split bands in the topMoS layer. (c) Crossings between LLs in the lower spin-orbitsplit bands originating from top and bottom MoS layers. Marie Sklodowska-Curie grant agreement no. 676108,the Graphene Flagship and the National Center of Com-petence in Research on Quantum Science and Technol-ogy (NCCR QSIT) funded by the Swiss National ScienceFoundation. Growth of hexagonal boron nitride crystalswas supported by the Elemental Strategy Initiative con-ducted by the MEXT, Japan and JSPS KAKENHI GrantNumbers JP15K21722. [1] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys-ical Review Letters , 196802 (2012).[2] K. F. Mak, K. He, J. Shan, and T. F. Heinz, NatureNanotechnology , 494 (2012).[3] T. Cao, G. Wang, W. Han, H. Ye, C. Zhu, J. Shi, Q. Niu,P. Tan, E. Wang, B. Liu, and J. Feng, Nature Commu-nications , 887 (2012).[4] W. Yao, D. Xiao, and Q. Niu, Physical Review B ,235406 (2008).[5] S. Wu, J. S. Ross, G.-B. Liu, G. Aivazian, A. Jones,Z. Fei, W. Zhu, D. Xiao, W. Yao, D. Cobden, and X. Xu,Nature Physics , 149 (2013).[6] J. Lee, K. F. Mak, and J. Shan, Nature Nanotechnology , 421 (2016).[7] J. Lin, T. Han, B. A. Piot, Z. Wu, S. Xu, G. Long,L. An, P. K. M. Cheung, P.-P. Zheng, P. Plochocka, D. K.Maude, F. Zhang, and N. Wang, arXiv:1803.08007 [cond-mat] (2018), arXiv: 1803.08007.[8] B. Fallahazad, H. C. Movva, K. Kim, S. Larentis,T. Taniguchi, K. Watanabe, S. K. Banerjee, and E. Tu-tuc, Physical Review Letters , 086601 (2016).[9] H. C. Movva, B. Fallahazad, K. Kim, S. Larentis,T. Taniguchi, K. Watanabe, S. K. Banerjee, and E. Tu-tuc, Physical Review Letters , 247701 (2017).[10] S. Larentis, H. C. P. Movva, B. Fallahazad, K. Kim,A. Behroozi, T. Taniguchi, K. Watanabe, S. K. Banerjee,and E. Tutuc, Physical Review B , 201407 (2018).[11] R. Pisoni, Y. Lee, H. Overweg, M. Eich, P. Simonet,K. Watanabe, T. Taniguchi, R. Gorbachev, T. Ihn, andK. Ensslin, Nano Letters , 5008 (2017).[12] R. Pisoni, Z. Lei, P. Back, M. Eich, H. Overweg, Y. Lee,K. Watanabe, T. Taniguchi, T. Ihn, and K. Ensslin,Applied Physics Letters , 123101 (2018).[13] R. Pisoni, A. Korm´anyos, M. Brooks, Z. Lei, P. Back,M. Eich, H. Overweg, Y. Lee, P. Rickhaus, K. Watanabe,T. Taniguchi, A. Imamoglu, G. Burkard, T. Ihn, andK. Ensslin, Physical Review Letters , 247701 (2018).[14] T. Ando, A. B. Fowler, and F. Stern, Reviews of ModernPhysics , 437 (1982).[15] A. Isihara and L. Smrcka, Journal of Physics C: SolidState Physics , 6777 (1986).[16] V. M. Pudalov, M. E. Gershenson, and H. Kojima, Phys-ical Review B , 075147 (2014).[17] Y. Zhang and S. Das Sarma, Physical Review B ,075308 (2005).[18] C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B.Bachelet, Physical Review Letters , 256601 (2002).[19] T. Cheiwchanchamnangij and W. R. L. Lambrecht, Phys-ical Review B , 205302 (2012).[20] A. Ramasubramaniam, Physical Review B , 115409(2012).[21] A. Molina-S´anchez and L. Wirtz, Physical Review B ,155413 (2011).[22] A. Laturia, M. L. V. d. Put, and W. G. Vandenberghe,npj 2D Materials and Applications , 6 (2018).[23] T. Okamoto, K. Hosoya, S. Kawaji, and A. Yagi, Phys-ical Review Letters , 3875 (1999).[24] A. A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov,and T. M. Klapwijk, Physical Review Letters , 086801(2001).[25] K. Vakili, Y. P. Shkolnikov, E. Tutuc, E. P. De Poortere, and M. Shayegan, Physical Review Letters , 226401(2004).[26] M. V. Gustafsson, M. Yankowitz, C. Forsythe,D. Rhodes, K. Watanabe, T. Taniguchi, J. Hone, X. Zhu,and C. R. Dean, arXiv:1707.08083 [cond-mat] (2017),arXiv: 1707.08083.[27] J. Lin, T. Han, B. A. Piot, Z. Wu, S. Xu, G. Long, L. An,P. Cheung, P.-P. Zheng, P. Plochocka, X. Dai, D. K.Maude, F. Zhang, and N. Wang, Nano Letters , 1736(2019).[28] J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Physical Review Letters , 674 (1992).[29] M. S. Bello, E. I. Levin, and B. I. Shklovskr, , 8 (1981).[30] B. Tanatar and D. M. Ceperley, Physical Review B ,5005 (1989).[31] S. V. Kravchenko, D. A. Rinberg, S. G. Semenchinsky,and V. M. Pudalov, Physical Review B , 3741 (1990).[32] L. M. Zhang, Z. Q. Li, D. N. Basov, M. M. Fogler, Z. Hao,and M. C. Martin, Physical Review B , 235408 (2008).[33] A. Korm´anyos, V. Z´olyomi, V. I. Fal’ko, and G. Burkard,Physical Review B98