Multi-terminal transport measurements of MoS2 using a van der Waals heterostructure device platform

inset shows the R C as a function of temperature at different V bg . At high V bg , contact resistance even decreases when

Following the many advances in basic science and applications of graphene, other twodimensional (2D) materials, especially transition metal dichalcogenides (TMDCs), have attracted significant interest for their fascinating electrical, optical, and mechanical properties [1][2][3][4][5][6][7][8] .Among the TMDCs, semiconducting MoS 2 has been the mostly widely studied: it shows a thicknessdependent electronic band structure 3,5 , reasonably high carrier mobility 1,2,[6][7][8][9] , and novel phenomena such as coupled spin-valley physics and the valley Hall effect [10][11][12][13][14] , leading to various applications, such as transistors 1,7,15 , memories 16 , logic circuits 17,18 , light-emitters 19 , and photo-detectors 20 with flexibility and transparency 2,21 .However, as for any 2D material, the electrical and optical properties of MoS 2 are strongly affected by impurities and its dielectric environment 1,2,9,22 , hindering the study of intrinsic physics and limiting the design of 2Dmaterial-based devices.In particular, the theoretical upper bound of the electron mobility of monolayer (1L) MoS 2 is predicted to be from several tens to a few thousands at room temperature (T) and exceed 10 5 cm 2 /Vs at low T depending on the dielectric environment, impurity density and charge carrier density [23][24][25] .In contrast, experimentally measured 1L MoS 2 devices on SiO 2 substrates have exhibited room-T two-terminal field-effect mobility that ranges from 0.1 -55 cm 2 /Vs 1,26,27 .This value increases to 15 -60 cm 2 /Vs with encapsulation by highdielectric materials 1,9 , owing to more effective screening of charged impurities 24 .Due to the presence of large contact resistance from the metal-MoS 2 Schottky barrier, however, these twoterminal measurements underestimate the true channel mobility 7,28,29 .Multi-terminal Hall mobility measurements 8,9 still show mobility substantially below theoretical limits, particularly at low T with best reported values of 174 cm 2 /Vs at 4 K for 1L 9 and 250 cm 2 /Vs and 375 cm 2 /Vs at 5 K for 1L and 2L 8 .Typically, these thin samples exhibit a crossover to non-metallic behaviour at carrier densities below ~10 13 cm -2 8,9,30 , or at smaller carrier densities by engineering of local defect states and improving interfacial quality 31 .The scattering and disorder that leads to this non-metallic behaviour can arise from multiple origins such as lattice defects, charged impurities in the substrate and surface adsorbates, and it has been difficult to identify their separate contributions 1,8,9,23,[30][31][32][33] .

van der Waals heterostructure device platform
We have previously demonstrated that encapsulation of graphene within hBN reduces scattering from substrate phonons and charged impurities, resulting in band transport behaviour that is near the ideal acoustic phonon limit at room T, and ballistic over more than 15 µm at low T 34 .These results were realized with a novel technique to create one-dimensional edge contacts to graphene exposed by plasma-etching a hBN/graphene/hBN stack.Such an approach has not yet proved effective with MoS 2 .However, recent reports that graphene can create a high quality electrical contact to MoS 2 18,35 motivate a hybrid scheme, in which the channel MoS 2 and multiple graphene 'leads' are encapsulated in hBN, and the stack is etched to form graphenemetal edge contacts.This new scheme is distinct from previous approaches, in that the entire MoS 2 channel is fully encapsulated and protected by hBN, and that we achieve multi-terminal graphene contacts without any contamination from device fabrication process.hBN layers, and place the entire stack on a Si/SiO 2 wafer (Supplementary Fig. 1a).The stack was then shaped into Hall bar geometry such that hBN-encapsulated MoS 2 forms the channel.In the contact regions, graphene overlaps the MoS 2 and extends to the edge, where it is in turn contacted by metal electrodes 34 .Details of the fabrication process are described in the Methods section and Supplementary Information S1.High-resolution scanning transmission electron microscopy (STEM) (Fig. 1c; see also Supplementary Fig. 1b for a larger clean interface area of > 3 µm) confirms that the stacking method can produce ultraclean interfaces free of residue that can be seen when an organic polymer film is used for stacking 36 .We note that while Ohmic contacts have also been achieved in metal-MoS 2 contacts by deposition of small work-function metals, vacuum annealing, and electrostatic gating 4,17,18 , top-deposited metal electrodes are not compatible with hBN-encapsulation.
For this study, a series of samples with thickness from 1 -6 layers (1L -6L) was fabricated and measured.The number of layers was identified by Raman and photoluminescence (PL) (see Supplementary Information S2).All samples were obtained by mechanical exfoliation except for the 1L sample, for which we used chemical vapor deposition (CVD) grown monolayer MoS 2 because of the limited size of mechanically exfoliated monolayers.The CVD-grown MoS 2 single crystal has been shown to exhibit high quality from structural, electrical and optical measurements 37 , although the process of transferring it from the growth substrate may introduce more contamination than for mechanically exfoliated flakes.

Gate-tunable graphene-MoS 2 contact
For each sample, we performed temperature-dependent two-probe measurements to examine the quality of the graphene contacts.Figure 2a shows output curves (I ds -V ds ) of a 4L MoS 2 device at back-gate V bg = 80 V.The response is linear at room T and remains linear to low T, indicating an Ohmic contact.Similar behaviour is seen for V bg > 20 V, whereas gapped behaviour corresponding to non-Ohmic contact is seen for V bg < 20 V.This is consistent with previous studies which show a gate-tunable contact barrier between graphene and MoS 2 18,35 .In addition, it establishes the gate voltage range over which multi-terminal measurements can be reliably performed.Figure 2b shows the measured four-terminal resistivity ρ (in log scale) of the same sample from V bg = 20 V to 80 V (corresponding to carrier densities of ~ 4.8 × 10 12 cm -2 to ~ 6.9 × 10 12 cm -2 , respectively), and from room T to 12 K. ρ decreases with increasing V bg , as expected for an n-type semiconductor.With decreasing temperature, ρ drops dramatically over the entire accessible range of V bg , reaching 130 Ω at 12 K.All of the samples studied exhibited similar behaviour: n-type semiconducting behaviour and metallic temperature-dependence in the gate voltage accessible to four-terminal measurements.
By comparing the two-and four-terminal results, the contact resistance can be determined (see Supplementary Information S3).The results for the 4L MoS 2 device, as shown in Fig. 2c, directly demonstrate that the contact resistance can be tuned by back-gate voltage.In fact, a small contact resistance of ~ 2 kΩ⋅µm can be reliably achieved at large gate voltage at room T.
This likely reflects primarily the graphene-MoS 2 junction resistance, since both the graphene resistance and the graphene-metal contact resistance should be substantially less 34 .Below V bg = 20 V, the contact resistance increases upon cooling, indicating activated transport across a contact barrier.However, above V bg = 20 V, the contact resistance decreases upon cooling, reaching a low-T value of ~ 1 kΩ⋅µm above V bg = 50 V.This metallic behaviour directly demonstrates that low-resistance contacts, with no thermal activation, can be achieved at sufficiently high gate voltage.Similar behaviour was observed in all samples (Supplementary Fig. 3), with contact resistance at high V bg ranging from ~ 2 -20 kΩ⋅µm at room T and ~ 0.7 -10 kΩ⋅µm at low T.These values are comparable to room-T values reported previously for graphene 38 and metal [39][40][41] contacts, but larger than the best contacts achieved by MoS 2 phase engineering (0.2 -0.3 kΩ⋅µm) 29 .Due to the increase in band gap with decreasing thickness, the value of V bg required to achieve Ohmic contact is larger for thinner samples.

Scattering mechanism in MoS 2
To examine the quality of the hBN-encapsulated devices and determine the scattering mechanisms limiting the carrier mobility of MoS 2 , the Hall mobility µ Hall (T) was derived from ρ(Τ) and the carrier density n(V bg ) (obtained by Hall effect measurements, see Supplementary Information S4). Figure 3a shows µ Hall for the 1L -6L samples as a function of temperature, at carrier densities varying from 4.0 × 10 12 cm -2 to 1.2 × 10 13 cm -2 (see Fig. 3b and Supplementary Table 1).Thinner samples were measured at higher carrier densities required to achieve Ohmic contacts.For all of the samples, mobility increases with decreasing temperature and saturates at a constant value at low T. The low-T mobility in our devices is much higher than previously reported values, and there is no sign of metal-insulator transition as observed at similar carrier densities around 10 13 cm -2 in SiO 2 -supported MoS 2 8,9,30,32,33 .This strongly suggests that extrinsic scattering and disorder (either from SiO 2 or from processing with polymer resists) has been the primary source of non-metallic behaviour in MoS 2 measured to date.
The measured mobility curves can be reasonably fitted to a simple functional form: , where µ imp is the contribution from impurity scattering, and µ ph is the temperature-dependent contribution due to phonon scattering.In all samples, the fitted µ ph (T) is well described by a power law µ ph ~ T -γ above 100 K (Supplementary Fig. 7).This behaviour is consistent with mobility limited by MoS 2 optical phonons, as theoretically predicted to have an exponent of ~ 1.69 in monolayer 23 and ~ 2.5 for bulk MoS 2 42 at T > 100 K.Although this power law behaviour has been observed in experiments by other groups 8,9,30 , a stronger temperature dependence was observed in our devices, with the exponent γ ranging from 1.9 -2.5 (inset table of Fig. 3a), as opposed to 0.55 -1.7 reported previously 8,9,30 .We also note that the room-T mobility, which is dominated by phonon scattering in all of the samples, is seen to vary from 40 -120 cm 2 /Vs.At this point we can find no satisfactory explanation for this variation: there is no discernible trend with thickness, and no variation of the gamma value with carrier density (see Supplementary Fig. 8).Finally, we note that a deviation from the simple form µ ph ~ T -γ in high mobility samples below 100 K may indicate acoustic phonon scattering, although further study is needed to fully explore this regime.
At the lowest temperature, phonon scattering is suppressed, and the residual resistivity is due to dominant long-ranged Coulomb impurities or short-ranged atomic defects [43][44][45] , captured in the measured quantity µ imp .Figure 3b shows the derived values of µ imp as a function of carrier density n.For each sample, µ imp increases with n, with maximum values ranging from 1,020 cm 2 /Vs in the CVD monolayer to 34,000 cm 2 /Vs for 6L, up to two orders of magnitude higher than previously reported values 8,46 (Fig. 3b and Supplementary Table 1).These basic trends allow us to rule out scattering due to impurities or defects located within the MoS 2 itself: bulk charged impurities should give rise to thickness-independent mobility, and short-ranged scatterers due to atomic defects should give rise to a density-independent mobility 45 .On the other hand, interfacial scatterers, including both Coulomb impurities and short-ranged defects, which are limiting scattering mechanisms in high quality conventional 2D electron gas system 44 are promising candidates.Indeed, PDMS transfer, while substantially cleaner than methods involving organic polymers, can potentially introduce adsorbates to the top MoS 2 surface.
To understand the effects of interfacial scattering on samples with different thickness, we model interfacial Coulomb and short-range scattering as a function of carrier density, for samples from 1L to 6L in thickness.For Coulomb scattering, we employed a model based on a perturbative approach by Stern 47 , from which we obtained the screened Coulomb potential used in the mobility calculation.This model has also been commonly used in the context of semiconductor devices (see Supplementary Information S8).Within the model, increasing carrier density enhances screening of the interfacial Coulomb potential, which leads to improved carrier mobility, and increasing the thickness of MoS 2 redistributes the charge centroid further from the interface, also resulting in enhancement of mobility.The calculated mobility is shown in Fig. 3c, assuming the same impurities concentration of 6 × 10 9 cm -2 (chosen to match the 6L data) located at the top MoS 2 interface across devices with different number of layers.Although the qualitative trend of increasing mobility with layer numbers and carrier densities is consistent with the model, the model fails to account for the observed large thickness dependence of more than an order of magnitude between the 1L and 6L devices.The changes in the model calculations including the effects of band structure change with increasing layer thickness is discussed in Supplementary Information S8.
We next consider interfacial short-ranged scatterers with atomically localized scattering potentials, which can be modeled as delta-function potentials within the same framework as used above.Quantum lifetime measurements, to be discussed later, suggest that these scatterers strongly dominate electronic transport in the 1L devices.We therefore set the interfacial shortranged scattering parameter (the product of scattering potential and defect density, see Supplementary Information S8) to fit the mobility of the 1L device.In this case, for the same interfacial scattering the mobility increases rapidly with sample thickness -much more than observed experimentally.
Based on this analysis, we propose that the interfaces in our devices introduce both longranged Coulomb scattering and short-ranged scattering.In this case, we can calculate the total extrinsic mobility using Matthiessen's rule.The combination of long-ranged and short-ranged scatterers provides a better agreement to the observed layer-dependent mobility as shown in Fig. 3e, a salient point which we will revisit again later in quantum oscillations study.Of course, a perfect match to experiment is not expected due to sample-to-sample variation in impurity density.We also note that the long-ranged impurity density of 6 × 10 9 cm -2 is two orders of magnitude smaller than typically obtained for graphene on SiO 2 , and hence accounts for the two orders of magnitude larger mobility we obtained as compared to the best reported devices 8,46 .

Observation of Shubnikov-de Haas (SdH) oscillations in MoS 2
Figure 4 shows the longitudinal (R xx ) and Hall resistance (R xy ) of the monolayer (Fig. 4a), 4L MoS 2 (Fig. 4b) and 6L MoS 2 (Fig. 4c) samples as a function of applied magnetic field.We observe pronounced SdH oscillations in MoS 2 for the first time, providing additional strong evidence of high quality and homogeneity in the heterostructure devices.In the highest-mobility (6L) sample (Fig. 4c), the onset of SdH oscillations is close to 1 Tesla (T), further confirming its ultra-high mobility.Encouragingly, the high-field Hall resistance (blue curve, R xy ) begins to reveal plateau-like structures at high magnetic fields coinciding with R xx minima.These emerging features were similarly observed in early studies of graphene samples with moderate mobility 48 , giving hope that fully developed quantum Hall states can be observed with further improvements in sample quality.The periodicity of the SdH oscillations can be used to estimate the carrier density, or equivalently to measure the level degeneracy g for a known density (for details see Supplementary S7).For the 1L samples, we observe 2 < g < 4, indicating that the bands may be partially valley-spin split (the multi-band nature of the 6L sample complicates this analysis).This is consistent with extra SdH oscillations that begin to emerge at high fields, but more detailed study is required to explore this splitting in detail.
The quantum scattering time τ q which is limited by both small and large angle scattering that destroys quantized cyclotron orbits, can be estimated from the magnetic field corresponding to the onset of SdH oscillations, following the relation µ q = e τ q /m* ~ 1/B q 49 , where e is the electron charge and m* is the effective mass obtained from ab initio bandstructure calculations 50 .This yields quantum mobilities for 1L, 4L and 6L MoS 2 of ~ 1,400 cm 2 /Vs, ~ 3,100 cm 2 /Vs and ~ 10,000 cm 2 /Vs, respectively.A more accurate estimate of τ q in MoS 2 can be obtained using a Dingle plot (Supplementary Fig. 9), a well-established method in conventional 2D electron gas systems (2DEGs) (for details see Supplementary S6).The inset of Fig. 4a shows SdH oscillations of 1L MoS 2 , after subtraction of a magnetoresistance background, as function of 1/B.The red dashed line is the fitted envelope, from which we estimate a quantum scattering time of τ q = 176 fs.We show the values of τ q obtained using both methods (oscillation onset and Dingle plots) in Supplementary Fig. 9c.The ratio of transport to quantum scattering time can provide additional evidence of predominant scattering sources.In our samples a ratio near 1 in 1L devices indicates predominantly short-ranged scattering, and an increase in τ t /τ q with increasing thickness indicates a crossover to long-range scattering.This trend consistent with our previous physical picture of low-T electronic transport dominated by a mix of short-and long-ranged interfacial impurities.

Conclusion
We demonstrate a vdW heterostructure device platform in which an atomically thin MoS analyzed the data and wrote the paper.

S2. Identification of MoS 2 flakes on PDMS
We used Raman spectroscopy and photoluminescence (PL) measurements (in Via, Renishaw, 532 nm laser) to identify the number of layers of MoS 2 flakes on PDMS 3,4 .Figure S2a

S3. Contact resistance and Schottky barrier of graphene-MoS 2 contacts
We estimate the contact resistance as Furthermore, we calculated the Schottky barrier height of graphene-MoS 2 contact using the 2D thermionic emission relation 5 , where , A, T, , q, , and !" are drain current, the effective Richardson constant, temperature, the Boltzmann constant, electronic charge, the Schottky barrier height, and sourcedrain bias (50 mV), respectively.Here, is an ideality factor, which is related with tunneling -20

S6. Quantum scattering time analysis in MoS 2
We calculate the quantum scattering time in MoS 2 from Shubnikov-de Haas (SdH) oscillations.
Comparison of the ratio of the transport scattering time to quantum scattering time by number of layer of MoS 2 will provide the deep understanding about scattering sources in MoS 2 .First, transport scattering time !!= !* !/! are obtained from Hall mobility of MoS 2 .And, to obtain the quantum scattering time, we used the Ando formula, , where !!is Dingle term, !! is the cyclotron frequency and !! is the quantum scattering time.We also calculate the quantum scattering time from quantum mobility, which are estimated from the onset of SdH oscillation magnetic field.We note that band structure calculations of MoS 2 predict that there are two bands (at the K and Λ points) near the Fermi level, and that for 1L to 3L, the Κ band (m * ~ 0.5m 0 ) is lowest in energy, meanwhile the Λ band (m * ~ 0.6m 0 ) is lowest for > 4L 10 .Figure S9d exhibits the !!/! !ratio increase as increase of number of layer, and it is due to the long-range scattering origins (! !/! !!> 1) such as charged impurities and adsorbents become the dominant source of scattering in few-layer MoS 2 which lead dominant small angle scattering that destroy cyclotron orbit motions, while the short-range scattering origins (! !/! ! ~1) such as vacancies, ripple and cracks are dominant at the monolayer MoS 2 by the nature of 2D materials.The problem consists of a thin layer of semiconductor of thickness t s , MoS 2 in this case, sandwiched between two dielectrics layers.Let z denotes the direction normal to these layers, and m z be the quantization mass of MoS 2 .We uses a triangular well approximation as an initial estimate to the electrostatics of MoS 2 in the confinement direction, where F s is its electric field, where the eigen-solutions are known 8 .The eigen-energies E j must satisfy, where Ai and Bi are the Airy functions, and The eigen-functions are then given by, where v = eF s t s /⌘, ⌘ = ~2⇡ 2 /2m z t 2 s and = E j /⌘.Within this triangular model approximation, the electric fields in the di↵erent regions are related via The carrier densities can be computed from, where i and v denotes the subbands and valleys.g v is the valley degeneracy.m d is the density of states mass.The Fermi energy E f is determined from Eq. 3, by imposing that n = C ox V g .
The solutions to the triangular well approximation provides an initial guess to the electrostatics.With this, we solved the multilayers MoS 2 electrostatics by solving the Poisson and Schrödinger equation self-consistently within the e↵ective mass framework.In this work, we include both the K and ⇤ valleys, with band edge energies that are close to one another in multilayers MoS 2 .With reference to calculations obtained from density functional theory 11 14 , we extract the in-plane and out-of-plane masses: m d,K = 0.5m 0 , m d,⇤ = 0.6m 0 , m z,K = 1.5m 0 , m z,K = 1.0m 0 .Due to their di↵erent m z , the two valleys have di↵erent band edge energy o↵set depending on the MoS 2 thickness, and is taken to be zero when t s = 4 nm.Their valley degeneracies are g v,K = 2 and g v,⇤ = 6.

B. Screened Interfacial Coulomb potential
To calculate the scattering rate due to Coulomb centers, we must first find the scattering potential induced by a point charge.This Coulomb potential is governed by the following Poisson equation; where r is the 2D position vector describing the plane perpendicular to the gate confinement direction.The presence of the external point charge resulted in a Coulomb potential (r, z, z 0 ) which also induced charge ⇢ ind .Next, we need to obtain an expression for ⇢ ind .
Here, we employed a perturbative approach by Stern 9 commonly used in the context of semiconductor devices.The presence of the perturbation potential (z) results in correction of the eigen-energies E The charge induced ⇢ ind is the change in the amount of charge due to the change in eigen-energies E i,v ; Hence, Eq.4 can be approximated by the following; This is the key result by Stern 9 .
It renders the problem easier to express the in-plane part of thr potential (r, z, z 0 ) in its 2D Fourier representation, Thus we have the Poisson equation in the semiconductor region; Here, ✏ sc = 7.4 is the out-of-plane dielectric constant of MoS 2 10 .Multiply by 1/q•exp( q|z z 1 |) with 0 < z < t s and integrating for 0 < z 1 < t s , we arrived at, Note we assume the semiconductor is sandwiched by dielectrics, i.e. no metal gates.
Otherwise, we have additional terms given by A(q)exp( qz) + B(q)exp(qz) to account for screening by the metals.

C. Mobility
With 0 (q, z, 0), one uses Fermi Golden rule to obtain the scattering probability; where k ki and ✓ is the angle between the initial and final wave vector.N im is the impurities concentration.To calculate the relaxation time, we write, where m c is the transport mass.The above can be re-expressed into an equivalent matrix form as follows; which can easily be solved by inverting the matrix above.The relaxation time can be similarly computed for the other valley.
The mobility for each subband j and valley v then follows from, where f (E, E f ) is the Fermi-Dirac distribution function.The total e↵ective mobility is then where n j,v is the carrier densities in subband j and valley v.

D. Short-range scattering potential
In the main manuscript, we discussed two tpyes of short-range scattering potential i.e.
bulk and interfacial.The former could be due to atomic defects such as S vacancies in MoS 2 itself, which is present in its natural crystal form.The latter, interfacial short-range scatterers, could be due to presence of chemical adsorbates that was introduced to the top interface due to subsequent processing steps after MoS 2 transfer onto the SiO 2 substrate.
As explained in the main manuscript, we found the latter model to agree better to the experiments.We model the short-range scatterers at the top MoS 2 interface (i.e.z = 0) within the same framework describe above by simply making the replacement to the scattering potential in Eq. 9 i.e. 0 (q, z, 0) !v sc (z) and N sc denotes the short-range scatterers concentration (cm 2 ).On the other hand, the scattering potentials are distributed across the di↵erent layers of MoS 2 when modeling of bulk short-range scatterings.In the case of S vacancies in MoS 2 , the total short-range scatterers concentration (cm 2 ) increases with layer thickness and one expects a decrease in electron mobility with layer thickness contrary to what is observed in our experiments.We therefore focus only on interfacial short-range scattering potential in this work.

E. Calculation results
Here we show results of our calculation for both interfacial Coulomb impurities and short-range scatterers located at the top MoS 2 interface.The long-range Coulomb limited moblity, µ LR , predicts a layer dependence weaker than the experiments of less than an order of magnitude within the range of carrier densities considered, while the short-range limited mobility, µ SR , predicts a much stronger thickness dependence larger than 4 orders of magnitude.In sum, their total contribution according to Matthiessen rule would allow for better quantitative agreement with the experiments.

Figure 1a and 1b
Figure 1a and 1b show a schematic diagram and optical micrograph of a Hall bar device structure.We employed a 'PDMS (Polydimethylsiloxane) transfer' technique 2 to place few- 2 layer is encapsulated by hBN and contacted by graphene.The vdW heterostructure provides a standard device platform that enables us to measure intrinsic electrical transport of 2D materials and achieve high mobility 2D devices for studying the unique transport properties and novel quantum physics.By forming robust and tunable electrical contacts and dramatically reducing interfacial impurities, intrinsic electron-phonon scattering can be observed at high T, and substantially improved mobility can be achieved at low T.This enables the first observation of Shubnikov-de Haas oscillations in MoS 2 .Modeling and quantum lifetime analysis suggest that a combination of short-ranged and long-ranged interfacial scattering limits the low-T mobility, indicating that further improvements should be possible.MethodsDevice fabrication.The hBN/MoS 2 /graphene/hBN stacks were fabricated using the 'PDMS transfer' 2 technique on 285 nm SiO 2 /Si substrates.The transfer techniques are described in detail in the Supplementary Information S1.The stacks were then shaped to the desired Hall bar structure through electron-beam patterning and reactive ion etching (RIE) with a mixture of CHF 3 and O 2 .Finally, metal leads were patterned by e-beam lithography and subsequent deposition of metals (Cr 1nm/Pd 20nm/Au 50nm).The metal leads make edge-contact to graphene electrodes as reported previously34 .TEM sample preparation.For high-resolution imaging, we fabricated a cross-sectional TEM lift-out sample from the finished encapsulated devices, using a FEI Strata 400 dual-beam Focused Ion Beam.STEM imaging was conducted in a FEI Tecnai F-20 STEM operated at 200kV, with a 9.6 mrad convergence semiangle and high-angle annular dark field detector.False coloring was added by hand.Electrical measurements and magneto-transport measurements.Two-terminal transport characteristics were measured by applying DC bias (Keithley 2400) to the source and gate electrodes and measuring the drain current using a current amplifier (DL 1211).For fourterminal measurements, a standard lock-in amplifier (SR830) measured voltage drop across the channel with constant current bias.Magneto-transport measurements were performed in a Physical Property Measurement System (PPMS) (Fig.4c) and a He 3 cryostat at the National High Magnetic Field Laboratory (NHMFL) (Fig.4a and b).G.H.L. and Y.D.K. performed device measurements under supervision of P.K. and J.H.. X.C., G.H.L., G.A., X.Z.performed optical spectroscopy and data analysis.D.A.C. grew and prepared the CVD MoS 2 sample.T.L. performed the theoretical calculations.K.W. and T.T. prepared hBN samples.P.Y.H. and D.A.M. performed TEM analyses.X.C., G.H.L., Y.D.K. and J.H.

Figure 3 |
Figure 3 | Temperature, carrier density dependence of Hall mobility and scattering

Supplementary Fig. 2 |
shows the PL spectra of a CVD-grown monolayer MoS 2 .Bright-field optical images of few-layer MoS 2 exfoliated on PDMS are shown in Fig. S2b.The number of layers of these flakes was confirmed by the Raman spectra in Fig. S2c.We confirmed that MoS 2 flakes on PDMS have the correlation between Raman peak position difference of E 1 2g and A 1g modes and number of layers, which is consistent to our previous report 4 .Photoluminescence and Raman spectra of MoS 2 on PDMS a, PL spectra of CVD 1L MoS 2 on PDMS.The inset shows bright field optical image of CVD 1L MoS 2 on PDMS.b, Bright-field optical images of 2L to 6L MoS 2 flakes exfoliated on PDMS.Scale bar: 10 µm.c, Raman spectra of 2L to 6L MoS 2 on PDMS.As guided by the dashed lines, Raman peak position difference between E 1 2g and A 1g modes increases with MoS 2 thickness.

Supplementary Fig. 4 |
high charge carrier concentration and at low temperature.To estimate the Schottky barrier height of graphene-MoS 2 with different back gate voltage (V bg ), we employed the Arrhenius plot, ln( / !/! ) as a function of 1/ as shown in Fig. S4.Because the slope of Arrhenius plot is the − + !" / , we can extract the Schottky barrier heights for different MoS 2 thickness.Here, we assume the ideality factor as 2 < !!< 20, and we check the availability of ideality factor from the Ohmic behavior in 2 probe output curve at low temperature.Fig. S5 exhibits the calculated Schottky barrier heights in hBN-encapsulated MoS 2 devices with different MoS 2 thickness.Large modulation of graphene's Fermi energy allows for the high tunability of the Schottky barrier height in graphene-MoS 2 contact, resulting in the small Schottky barrier height with relatively high V bg .The Schottky barrier height becomes close to zero at V bg of ~ 80V even for monolayer MoS 2 , which enable us to form the Ohmic contact at very low temperature.Arrhenius plots of hBN-encapsulated MoS 2 devices for calculations of the Schottky barrier heights.a, 1L (CVD) b, 2L c, 3L d, 4L MoS 2 .

S6. Modeling MoS 2
electron mobility limited by interfacial Coulomb impurities A. Electrostatics

Fig. S8 :
Fig. S8: Calculated electron mobility due to interfacial Coulomb impurities (µ LR ) and shortrange scatterers (µ SR ) located at the top MoS 2 interface as function of gate bias and for di↵erent number of MoS 2 layers.We consider two cases, one where only K valleys are considered in (ac), and where both K and ⇤ valleys are included in (d-f).We also displayed the experimentally measured low temperature mobilities for 1L and 6L devices.We found that the calculated µ LR predicts a layer dependence weaker than the experiments, while the calculated µ SR predicts a much

Table 1 .
The solid fitting lines are drawn by the model in the main text.All the fitting parameters are listed in Supplementary Table 1.For a visual guideline, a dashed line of power law µ ph ~ T -γ is drawn and fitted values of γ for each device are listed in the inset table.b, Impurity-limited mobility (µ imp ) as a function of carrier density of MoS 2 .For comparison, the previously reported values from MoS 2 on SiO 2 substrates (Ref.8, 46) are plotted.c to e, The solid lines show the theoretically calculated long-ranged (LR) impurity limited mobility (c), short-ranged (SR) impurity limited mobility (d) and mobility including both LR and SR based on Matthiessen's rule 1/µ = 1/µ LR +1/µ SR as a function of carrier density for 1L to 6L MoS 2 (e).The experimental data from 1L and 6L are shown in dots (c-e).

Supplementary Fig. 3 | Gate-tunable contact resistance of graphene-MoS 2 contact. Contact
3here !!! is two-probe resistance, !!! !is the four-probe resistance of MoS 2 , L is the two-probe length and l is fourprobe length.The calculated contact resistance as a function of temperature and back gate voltage is shown in Fig.S3.Due to increase of the MoS 2 band gap with decreasing thickness from few-layers to monolayer3, it is more difficult to form Ohmic contact, in other words lower A large gate-tunability of Fermi energy of graphene enables us to move graphene's Fermi level close to conduction band of MoS 2 by increasing back-gate voltage, resulting in reliable and stable Ohmic contact even for monolayer MoS 2 as shown in Fig.S3a.At reasonably high charge carrier concentration, contact resistance from ~ 0.7 kΩ⋅µm to 10 kΩ⋅µm can be reliably achieved across all samples at low temperature.resistance as a function of back-gate voltage (V bg ) and temperature for 1L (CVD), 2L, 3L and 4L.