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Michele Masseroni, Tingyu Qu, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Thomas Ihn, Klaus Ensslin

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[Evidence of the Coulomb gap in the density of states of <math>  <msub>    <mi>MoS</mi>    <mn>2</mn>  </msub></math>](https://mdr.nims.go.jp/datasets/90e30f3d-735a-4c50-85e1-a581b85020a0)

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Evidence of the Coulomb gap in the density of states of ${\rm MoS}_2$PHYSICAL REVIEW RESEARCH 5, 013113 (2023)Evidence of the Coulomb gap in the density of states of MoS2Michele Masseroni ,1 Tingyu Qu ,2 Takashi Taniguchi,3 Kenji Watanabe ,4 Thomas Ihn ,1 and Klaus Ensslin 11Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland2NUS Graduate School, Integrative Sciences and Engineering Programme (ISEP),National University of Singapore, Singapore 119077, Singapore3International Center for Materials Nanoarchitectonics, 1-1 Namiki, Tsukuba 305-0044, Japan4Research Center for Functional Materials, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 20 December 2022; accepted 26 January 2023; published 14 February 2023)MoS2 is an emergent van der Waals material that shows promising prospects in semiconductor industryand optoelectronic applications. However, its electronic properties are not yet fully understood. In particular,the nature of the insulating state at low carrier density deserves further investigation, as it is important forfundamental research and applications. In this study we investigate the insulating state of a dual-gated exfoliatedbilayer MoS2 field-effect transistor by performing magnetotransport experiments. We observe a positive andnonsaturating magnetoresistance, in a regime where only one band contributes to electron transport. At lowelectron density (∼1.4 × 1012 cm−2) and a perpendicular magnetic field of 7 Tesla the resistance exceeds bymore than one order of magnitude the zero field resistance and exponentially drops with increasing temperature.We attribute this observation to strong electron localization. Both temperature and magnetic field dependencecan, at least qualitatively, be described by the Efros-Shklovskii law, predicting the formation of a Coulomb gapin the density of states due to Coulomb interactions. However, the localization length obtained from fitting thetemperature dependence exceeds by more than one order of magnitude the one obtained from the magnetic fielddependence. We attribute this discrepancy to the presence of a nearby metallic gate, which provides electrostaticscreening and thus reduces long-range Coulomb interactions. The result of our study suggests that the insulatingstate of MoS2 originates from a combination of disorder-driven electron localization and Coulomb interactions.DOI: 10.1103/PhysRevResearch.5.013113I. INTRODUCTIONThe resistivity ρ of some semiconductors shows a metal-insulator transition as a function of the electron density n[1]. For densities larger than a critical value nc the resistiv-ity shows a metallic temperature dependence (dρ/dT > 0),while below nc it shows an insulating temperature dependence(dρ/dT < 0). This metal-insulator transition attracted greatinterest in the late 1990s [2–5]. In two-dimensional (2D) semi-conductors the origin of the metallic phase is controversial[6–8], as it was predicted that any amount of defects wouldinexorably lead to electron localization at zero temperaturein a 2D system [9,10]. The insulating phase at low densitiescan be due to either intriguing correlated states, like Wignercrystals [11], or disorder-induced electron localization [5], aswell as a combination of the two effects.In highly disordered systems, charge transport at low tem-peratures occurs via electron hopping between localized states[12], known as variable-range hopping (VRH). The conduc-tivity in hopping transport at zero magnetic field is usuallyPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.described by an exponential dependence on the temperatureof the formσ (T ) ∝ exp[−(T0T)p],where T0 and p � 1 are constants that depend on the hoppingmechanism. In a noninteracting system, the density of statesclose to the Fermi energy is constant (but finite) and the con-ductivity is described by Mott’s law [13], for which p = 1/3(for two-dimensional systems). When electrons are stronglylocalized, the long-range Coulomb potential is not efficientlyscreened. Electron correlations result in a Coulomb gap in thedensity of states close to the Fermi energy [14,15]. The mod-ified density of states changes the temperature dependence ofthe hopping conductivity, which is now characterized by theparameter p = 1/2, as described by the Efros-Shklovskii (ES)theory [16].The insulating phase of MoS2 has been experimen-tally studied in monolayers [17] and multilayers [18,19],where both thermally activated transport at intermediate tem-peratures (T ∼ 50 K–100 K) and Mott VRH transport atlower temperatures have been observed. In addition, it isexpected that electron-electron interactions play a major rolein determining the electronic properties due to the large elec-tron effective mass [m∗ ≈ (0.4–0.6)m0] of MoS2, especiallyat low densities. Indeed, signatures of interaction effects havealready been reported in the literature [20,21], among which2643-1564/2023/5(1)/013113(9) 013113-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-1663-8239https://orcid.org/0000-0002-4409-3072https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-5587-6953https://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.5.013113&domain=pdf&date_stamp=2023-02-14https://doi.org/10.1103/PhysRevResearch.5.013113https://creativecommons.org/licenses/by/4.0/MICHELE MASSERONI et al. PHYSICAL REVIEW RESEARCH 5, 013113 (2023)FIG. 1. (a) The top panel shows optical micrographs of the sample before (left) and after (right) depositing the metallic top gate. TheMoS2 flake is outlined in the left figure. The scale bar in the left figure is 4 µm. The dimensions of the device are W = 8 µm, L1 = 12 µm, andL2 = 4 µm. A schematic side view of the device is shown in the lower panel. (b) Current (ISD) versus voltage (VSD) characteristics for differentVTG (step 2V). Inset shows the output characteristics for the lowest voltage VTG = 1.5 V. (c) Four-terminal conductivity (σ ) as a function oftop gate voltage (VTG) at VBG = 0. The dashed and dotted lines are linear fits, which highlight a kink in σ (VTG ) (marked by the number 2).(d) Derivative of the conductivity dσ/dVBG as a function of top and bottom gate voltages. The black dashed lines follow the constant totaldensity condition. The dashed dotted lines mark a local maximum and minimum of the conductivity. The numbers 1, 2, 3, 4 represent the bandedges of the four bands (see discussion in Sec. II A) that contribute to electron transport.there was also the observation of a Wigner crystal in MoSe2[22]. Therefore, MoS2, and in general, semiconducting transi-tion metal dichalcogenides (TMDs), are good candidates forthe observation of the Coulomb gap in the density of states.However, the observation of interaction effects is restrictedto low densities, where the Coulomb energy dominates overthe kinetic energy of electrons. Transport experiments in thisdensity range are challenging in most materials and requirelow defect densities [23]. The observation of the Coulomb gapin MoS2 remains to date elusive [19] due to the large densityof intrinsic defects.Here, we investigate magnetotransport in bilayer MoS2encapsulated in hexagonal boron nitride (hBN). We firstdemonstrate the high quality of our device and a completeunderstanding of the bands that contribute to electron trans-port. Then we tune the density, such that electron transportoccurs in a single (twofold degenerate) band. In this regimewe observe a metal-insulator transition at the electron densitync ≈ 1.7 × 1012 cm−2. Below this transition the zero-field re-sistance and the magnetoresistance show an exponential decaywith increasing temperature, which is qualitatively consistentwith the Efros-Shklovskii law [16], suggesting that electroncorrelations open a gap in the density of states. However,the localization length obtained by fitting the temperaturedependence exceeds by more than one order of magnitudethe one obtained from the magnetic field dependence. Weattribute this discrepancy to the presence of a nearby metallicgate, which provides electrostatic screening and thus reduceslong-range Coulomb interactions.II. RESULTS AND DISCUSSIONIn this study a bilayer MoS2 with dual gated architectureis employed to study electron transport. Figure 1(a) showsoptical images (upper panel) and a schematic side view (lowerpanel) of the device. The fabrication starts by assembling athin hBN and graphite layers onto a silicon/silicon dioxidechip (285 nm of oxide layer). The graphite serves as thebottom gate, while the hBN is the gate dielectric material.The layers are stacked together by means of a dry-transfertechnique (see Refs. [21,24,25] for details). We pattern metal-lic contacts (Cr/Au: 5 nm/10 nm) with standard electronbeam lithography and electron beam evaporation. The regionof the contacts is then cleaned thoroughly with the tip ofan atomic force microscope in contact mode. This step iscrucial to remove the residues of the lithography process. Thebilayer MoS2 is obtained by mechanically cleaving a bulkMoS2 crystal from natural sources (SPI Supply) and identifiedby optical contrast, which has proven to be a very reliablemethod [26–29]. A second stack with a layer of hBN and the013113-2EVIDENCE OF THE COULOMB GAP IN THE DENSITY OF … PHYSICAL REVIEW RESEARCH 5, 013113 (2023)bilayer MoS2 is assembled, then aligned and deposited on topof the contacts. Both exfoliation and assembling take placein a glove box with argon atmosphere (H2O, O2 < 0.1 ppm).The metallic top gate is patterned with standard lithographyprocesses and covers the entire MoS2 flake. In the final stepwe vacuum anneal the sample at 250 ◦C for 4 h to improve thecontact interface. All measurements presented in this work (ifnot explicitly stated otherwise) are performed at a temperatureof 1.3 K at low frequency (∼30 Hz) and low excitation voltage(V rmsSD = 100 µV) with standard lock-in techniques.First, we characterize the contacts to ensure Ohmic behav-ior and low contact resistance. The dc output characteristicsare shown in Fig. 1(b) for different top gate voltages (VTG).The two-terminal resistance changes from <1 k� to ∼1 M�as a function of VTG. The current shows a linear dependence onthe applied source-drain voltage VSD down to the lowest VTG(inset), indicating a vanishing Schottky barrier. The averagecontact resistance of source and drain contacts (1 and 2) is<300 � for VTG � 6 V and does not depend on the appliedbottom gate voltage (VBG). To the best of our knowledge, thisis one of the lowest values reported in the literature and iscomparable to bismuth contacts [30]. The vanishing Schottkybarrier and the low contact resistance allow us to study elec-tron transport at low densities (down to 1.4 × 1012 cm−2) atlow source-drain voltage (100 µV).The four-terminal conductivity (σ ) as a function of VTGis shown in Fig. 1(c). We identify two voltage ranges,where the conductivity features a linear dependence onVTG, which differs by the slope. The two slopes of σ (VTG)yield the mobilities ∼1000 cm2 V−1 s−1 (dotted line) and∼2400 cm2 V−1 s−1 (dashed line). This specific shape of thefour-terminal conductivity is a general property of high-quality single-gated MoS2 devices [see Fig. 1(c) in Ref. [20]and Fig. 1(d) in the supplemental material of Ref. [21] for adirect comparison]. The kink marked with the number 2 isrelated to the population of the upper spin-orbit (SO) splitband, as it will be demonstrated below.Figure 1(d) shows the derivative of the conductivitydσ/dVBG as a function of VTG and VBG. For VBG < 0.7 Vthe VBG dependence of the conductivity is monotonic andresembles its VTG dependence. There are two distinct ratesdσ/dVBG that are separated by a black dashed line (number2). This line follows constant density conditions, separatingthe single band from the two-band regime. From this resultwe conclude that the separation between the bands is not dis-placement field dependent (i.e., the SO gap is not tunable withthe electric field). In contrast, for VBG > 0.7 V the dependenceof the conductivity on VBG is nonmonotonic. The conductivityfeatures a local maximum (number 3) that is almost indepen-dent of VTG. The origin of the negative dσ/dVBG is related tothe population of the bottom layer and will be discussed inSec. II A.So far we have characterized the field effect transistorat zero magnetic field. Understanding the contribution ofthe bands in multilayer MoS2 devices is an essential stepfor interpreting their electronic properties. In the followingwe will demonstrate that all features observed in the con-ductivity can be attributed to the population of differentbands in the MoS2 bilayer. To verify this hypothesis, wenow turn our attention to magnetotransport measurements,from which we determine the electron density and the bandoccupation.A. Band and layer occupationThe aim of this part is determining how the different bandsare filled with electrons as we change the gate voltages. Forthis purpose we measure the magnetoconductivity σxx, apply-ing a perpendicular magnetic field B. The conductivity σxx isobtained by tensor inversion of the two-dimensional resistivitywith components ρxx = WV3,4/(IL2) and ρxy = V3,5/I . Fromthe Shubnikov–de Haas oscillations (SdHO) we determine thedensity ni of band i as we did in previous works [21,24,25].The total density is given byntot = 1e(CT VTG + CBVBG), (1)where CT = 146 nF/cm2 and CB = 225 nF/cm2. The totaldensity is related to the band densities via ntot = ∑i ni. We donot determine the total density from the Hall effect, becausethe contacts extend across most of the conducting channel,leading to an overestimation of the density.Figure 2(a) shows the derivative of the magnetoconduc-tivity dσxx/dVBG as a function of B and VBG for VTG =11.5 V. The conductivity features oscillations periodic in B−1,as we expect for SdHO. From the Fourier spectrum of thedσxx/dVBG(1/B) (see Ref. [24,25] for details) we determinethe densities ni shown in Fig. 2(b). As in Ref. [24], we at-tribute a twofold valley degeneracy to each band i, whichaccounts for the two K valleys (see schematics in the lowerright panel of Fig. 2). In the regime VBG < 0.7 V, where wefind only two frequencies in the Fourier spectrum, there isremarkable agreement between the calculated ntot and theexperimentally defined density n1 + n2. In our interpretation,the densities n1 and n2 belong to the top MoS2 layer. Inthe regime VBG > 0.7 V, the density n1 + n2 saturates, be-cause the bottom MoS2 layer becomes conducting, screeningthe field effect of the bottom gate. This behavior is in completeagreement with the behavior found and explained in previousstudies [24,25].The layer occupation can be controlled in dual-gated de-vices. For VTG > 0 and VBG < 0, only the top layer is occupiedby electrons and we can tune between single- and double-band transport. The onset of n2 (marked by the number 2)corresponds to the population of the upper SO split band ofthe top layer (see schematics of the band structure in Fig. 2).To support this interpretation we determine the density nSO ≈3.5 × 1012 cm−2 that is required to start filling the upper SOsplit band. In an effective mass approximation this densitycorresponds to an energy �SO ∼ 14 meV, in agreement withthe value reported in Ref. [24].Figure 2(c) shows dσxx/dVBG at B = 7 T as a function ofthe voltage applied to the top and bottom gates. This mea-surement defines a phase diagram for the bilayer MoS2 thatis divided into five different regimes by the dashed (dotted)lines. Each line indicates the onset of a specific band (from 1to 4 increasing VBG). The bilayer is tuned from an insulatinginto a conducting phase, where up to four bands contributeto electron transport. In the remaining part of this paper, we013113-3MICHELE MASSERONI et al. PHYSICAL REVIEW RESEARCH 5, 013113 (2023)FIG. 2. (a) Numerical derivative of the conductivity (dσxx/dVBG) as a function of B and VBG for VTG = 11.5 V [for color scale see (c)].(b) Density as a function of VBG. The total density ntot is obtained by Eq. (1), n1,2,3 are determined from the Fourier spectrum of thetransconductivity in (a), and n4 = ntot − n1 − n2 − n3. (c) dσxx/dVBG at B = 7 T as a function of VTG and VBG. Lower right panels (1–4)show the schematic of the conduction band and the layer occupancy of the single-layer (1 and 2) and bilayer (3 and 4) regimes. The color ofthe bands encodes the spin polarization.focus on the narrow single-band transport regime enclosed bythe dashed lines in Fig. 2(c).B. Strong electron localizationWe now turn our attention to the magnetoresistance in thesingle-band transport regime. According to the Drude model,the longitudinal resistance does not depend on magnetic field.However, we observe a nonsaturating positive magnetore-sistance in the low-density range (<3.5 × 1012 cm−2). Themagnetoresistance increases by more than one order of mag-nitude at B = 7 T and ntot = 1.4 × 1012 cm−2, as shown inFigs. 3(a)–3(c). A qualitatively similar positive magnetoresis-tance was observed in monolayer MoS2 [31], where a negativemagnetoresistance (due to weak localization) gradually turnedinto a positive magnetoresistance as the temperature and den-sity were lowered. This effect was attributed to the transitionfrom weak localization to weak antilocalization, despite theauthors realizing that the shape of the magnetoresistance wasnot well described by the theory of weak antilocalization.Also, in other bilayer [24,32] and three-layer MoS2 devices[25] a similar behavior has been observed, but was not furtherinvestigated. The observation of a positive magnetoresistancein samples prepared and studied in different research groupssuggests that there is a common origin for this effect. Owingto the relatively low electron mobility in MoS2 and the largeintrinsic defect density [33], we consider the role of disorder[34] and electron localization to describe the nonsaturatingpositive magnetoresistance.Figures 3(a)–3(c) show the four-terminal longitudinal re-sistivity ρ as a function T for three different densities (from1.4 × 1012 cm−2 to 3.5 × 1012 cm−2) and various magneticfields. At zero magnetic field (violet curve) we observe ametal-insulator transition as we lower the density. It is re-markable that this transition occurs at relatively high densities∼1.7 × 1012 cm−2, where the ratio between Coulomb andkinetic energy is still moderate rs = EC/Ekin ∼ 7. Therefore,the transition is more likely to be a manifestation of strongelectron localization when the Fermi energy approachesthe bottom of the conduction band, rather than an insulat-ing state due to correlations. At the lowest temperature, weestimate the product (kF × �e) between the Fermi wave vec-tor (kF) and electron mean free path (�e). Strong electronlocalization occurs for kF × �e � 1 [as shown in Fig. 3(a)],while band conduction occurs for kF × �e � 1. Applying aperpendicular magnetic field breaks this condition, inducing ametal-insulator transition even for kF × �e > 1 [see Fig. 3(b)].The resistivity is plotted on a logarithmic scale as a func-tion of T −1/2 in Fig. 3(d) for the lowest density and variousmagnetic fields. At low temperature (<10 K) all the curvesfollow a linear dependence, as predicted by the ES theory.The slope of the curves increases with increasing magnetic013113-4EVIDENCE OF THE COULOMB GAP IN THE DENSITY OF … PHYSICAL REVIEW RESEARCH 5, 013113 (2023)FIG. 3. (a)–(c) Resistivity ρ as a function of temperature T for three electron densities, from 1.4 × 1012 to 3.5 × 1012, and variousmagnetic fields. Color bar in (d). (d) Resistivity plotted on a logarithmic scale versus T −1/2 for n = 1.4 × 102 cm2 and various magneticfields. The plot shows that the temperature dependence is in agreement with ES law for any magnetic field between 0 and 7 T. (e) Similar to(d) but at B = 7T and various densities. The plot shows the transition from insulating to metallic temperature dependence. (f) The parameterT 1/2∗ obtained from (d) for two exemplar densities. The solid red lines are linear fits. (g) The parameter T 1/2∗ obtained from (e) for B = 0 andB = 7 T.field. We therefore define a parameter T∗(B) that depends onthe magnetic field, such that the resistivity is described byρ(T, B) = ρ0 exp[(T∗(B)T)1/2].This function seems to capture the temperature dependenceof the resistivity for T < 10 K, while above this tempera-ture the resistance shows a weaker temperature and magneticfield dependence. The parameter T 1/2∗ obtained from the fit isshown in Fig. 3(f). This parameter depends roughly linearly(at least above B ≈ 1 T) on the applied magnetic field and theslope decreases with increasing electron density. Similarly, weplot the resistivity for different densities [see Fig. 3(e)] andextrapolate the parameter T 1/2∗ as a function of densities. Thedensity dependence of T 1/2∗ is shown in Fig. 3(g), where wesee T 1/2∗ increasing rapidly for decreasing density.We considered other models to describe our data, suchas Mott’s law (p = 1/3) and thermally activated nearest-neighbor hopping (p = 1). While thermally activated hoppingclearly fails in describing the temperature dependence of theresistivity, we obtain reasonable fits also with Mott’s law,which yields a slightly larger least-mean-square error. Basedon the temperature dependence of the resistivity, we cannotreliably distinguish between the two models, because the re-sistivity does not change by several orders of magnitude. Forthis reason we also consider the role played by the magneticfield.At finite magnetic fields the tails of the electron wavefunctions decay faster and the overlap between the localizedwave functions decreases. This leads to a reduction of thetunneling probability and thus to an increase of the resistance.This effect results in an exponential increase of the resistanceof the form [12]ρ(B) ∝ exp(Bm),where m depends on the range of magnetic field and theassumptions made in the theoretical model [12,35,36].First, we focus on the low magnetic field range (�B � a,�B being the magnetic length and a the localization length),where the effect of B can be treated as a small correctionto T0. The low magnetic field correction of the percolationparameter ξ is given by [37]�ξ (B) = ξ (B) − ξ (0) = C2a4�4B(T0T)3/2= A(T )B2,where C2 = 0.002 is a numerical parameter, and �B = √h/eB.This expression yields the magnetoresistivity�ρ(B)ρ(0)= exp[A(T )B2] − 1 ≈ A(T )B2, (2)which can be expanded in a quadratic expression forA(T )B2   1.Figures 4(a) and 4(b) show the low magnetic fieldrange of the resistivity at the density 1.54 × 1012 cm−2 (i.e.,013113-5MICHELE MASSERONI et al. PHYSICAL REVIEW RESEARCH 5, 013113 (2023)FIG. 4. (a and b) Magnetoresistivity �ρ(B)/ρ(B = 0) as a function of B for T = 10 K and T = 1.3 K. Inset: same measurement but shownin a larger magnetic field range. (c) The curvature A(T ) of �ρ(B)/ρ(B = 0) as a function of T on a log-log scale. The dashed (dotted) lineshows the expected temperature dependence according to ES (Mott) law. (d) ρ(B) in a logarithmic scale at the lowest temperature (1.3 K). Thedashed line is a linear fit and serves as a guide for the eyes. The dotted line is the parabolic fit shown in (b).below nc) and for two exemplary temperatures. At T = 10 Kthe resistivity shows a quadratic dependence on B, as pre-dicted by Eq. (2). At this temperature the model fits thedata well in the entire magnetic field range probed in ourexperiments, as shown by the inset of Fig. 4(a). The rangeof magnetic field for which this model is able to describethe data shrinks with lowering the temperature, because theparameter �ξ (B) grows with lowering the temperature (∝T −3/2), thus limiting the magnetic field range for whichour approximation is valid. Therefore, it is not surpris-ing that at T = 1.3 K [Fig. 4(b)] the model deviates fromthe data at Bc ≈ 1 T, where �ξ/ξ0 ≈ 0.5 becomes a largecorrection.Fitting the magnetoresistance at different temperatures pro-vides the temperature dependence of the curvature, which isthe only fitting parameter. The result of the fit is shown inFig. 4(c) on a log-log scale for the density 1.54 × 1012 cm−2.The data follow the temperature dependence T −(1.6±0.1). Forcomparison we show the temperature dependence (T −1) pre-dicted by the Mott theory, which clearly deviates from thetrend of our data, while the data are in good agreement withES theory (T −3/2). Based on this observation, we concludethat, in “diluted” MoS2, long-range Coulomb interactions leadto the formation of a gap in the density of states at the Fermienergy. This conclusion is further supported by the interac-tion parameter rs ∼ 7, which is modest but indicates that theCoulomb energy is significantly larger than the kinetic energyof free electrons.At this point we would like to discuss our quantitativeresults. By fitting the data with ES law we can estimate thelocalization length a according tokBT0 = C1e24πε0εra, (3)where εr ≈ 7 is the relative dielectric constant of MoS2,and C1 = 6.2 is a numerical constant [37]. This equation,however, provides localization lengths of a few microme-ters, which is surprisingly large considering the interparticlespacing (n−1/2 ∼ 10 nm). To verify the validity of this re-sult, we estimate the localization length from the curvatureof the magnetoresistance and compare the two results. Byinserting T0 (obtained from the temperature dependence) inthe definition of A(T ) we obtain a ≈ 100 nm at a density of1.54 × 1012 cm−2. This result differs by more than one orderof magnitude from the one obtained by Eq. (3).Here, we offer an argument that might explain the discrep-ancy between these two results. We note that in our devicethe metallic gate is only separated from the 2D electronsystem by a thin hBN layer (d = 13 nm). This distance iscomparable to the interparticle distance n−1/2 ∼ 10 nm. As aconsequence, the metallic gate screens Coulomb interactionsat large distance (r � d), where the Coulomb potential be-comes dipole-like (∝ r−3). The presence of the metal platepartially suppresses the Coulomb gap due to its screeningeffect [38–43]. For small hopping distances the density ofavailable states still has a gap, while for long hopping dis-tances the density of states is constant. At lower temperatures,long-distance hopping becomes favorable and VRH is deter-mined by the constant density of states (like in the Mott VRH).Therefore, it is expected that in the presence of a nearby013113-6EVIDENCE OF THE COULOMB GAP IN THE DENSITY OF … PHYSICAL REVIEW RESEARCH 5, 013113 (2023)metallic gate, there is a critical temperatureTc = e24πε0εrkBad2below which the system shows a transition from ES to the“screened” Mott VRH [41]. This VRH is different from theusual Mott VRH, which is characterized by a different densityof states.We estimate this temperature to be around 20 K by assum-ing a ≈ 100 nm, placing our system just below the transitiontemperature. In contrast, the hopping distancerh ∼ 14a(T0T)1/2∼ 10 nmis of the order of d , thus still in the Coulomb gap regime.Therefore, it is not clear from these characteristic quantitiesin which regime our sample is. As we argued above, wecannot, based on the temperature dependence, clearly dis-tinguish between ES and Mott VRH. Under this conditionEq. (3) may not be valid, as we are close to the transitionbetween the two hopping transport regimes. On the otherhand, the magnetic field promotes electron localization andreduces the probability of long-range hopping (less overlapof wave function tails). As shown in Fig. 3(f), the magneticfield dependence of the fitting parameter T∗(B) ∝ B2 is com-patible with the wave-function shrinking effect. In addition,the curvature A(T ) clearly establishes that at finite field VRHfollows ES theory. For this reason we consider our estimationof the localization length from the analysis shown in Fig. 4 tobe valid, from which we estimate a ∼ 100 nm.Finally, we consider the asymptotic limit at high mag-netic fields (a � �B). The typical hopping distance is stronglyreduced by the effect of the magnetic field and the typicalhopping energy increases. As a consequence, the Coulombgap might be negligible at high B fields, as proposed byNguyen [37]. The resistivity is expected to follow the asymp-totic behavior described by Eq. (4),�ρ(B)ρ(0)∝ exp(C√B), (4)with C ∝ T −1/2 being a temperature-dependent coefficient.This equation captures the temperature dependence of theresistivity at high magnetic fields. However, the resistivityseems to follow exp(CB) instead of Eq. (4), as shown inFig. 4(d). Given the divergent susceptibility of MoS2 at lowdensity [20], we speculate that the spin polarization mightcontribute in increasing the magnetoresistance. As it was re-ported in Ref. [44] the mechanism that provides an enhancedmagnetoresistance is related to the blocking of hopping dueto spin polarization. Disentangling spin from orbital effectsrequires in-plane magnetic fields. However, the in-plane mag-netic field does not couple to the electron spins in the Kvalleys of MoS2, because the spin is locked out-of-plane dueto SO coupling [20,45].III. CONCLUSIONThe low-temperature resistivity of bilayer MoS2 undergoesa transition from metallic to insulating temperature depen-dence at a critical density nc ≈ 1.7 × 1012 cm−2. This densityis one order of magnitude larger than in silicon metal-oxide-semiconductor field-effect transistors [46] and three ordersof magnitude larger than in GaAs quantum wells [47,48].We attribute this transition to a disorder-induced transition,in agreement with other metal-insulator-transition observedin disordered materials [46]. In fact, our observation is inline with the proposal of Klapwijk and Das Sarma [5], whichstates that the transition should be observed when the electrondensity is close to a few electrons per ionized impurity. Theionized impurities may be sulfur vacancies, which are knownto have an inhomogeneous distribution with average densityof 1 × 1012−1 × 1013 cm−2 [49], thus the same order of mag-nitude as nc in our sample.In the insulating phase, the resistance drops exponentiallywith increasing temperature, compatible with variable-rangehopping. The limited temperature range considered in ourexperiments does not provide a conclusive distinction be-tween Mott and Efros-Shklovskii laws. On the other end, themagnetic field dependence at low fields closely follows anEfros-Shklovskii law. The Coulomb gap is likely to appearin the density of states of MoS2, as the interaction parameter(rs ∼ 7) suggests that Coulomb energy is the dominant en-ergy scale. 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