High-temperature probe of electron compressibility via asymmetric Coulomb drag

Domain: physics
Published: 2026-02-05
Editorial outcome: minor_revision

Abstract
Lateral charge transport of a two-dimensional (2D) electronic system can be much influenced by feeding a current into another closely spaced 2D conductor, known as the Coulomb drag phenomenon – a powerful probe of electron-electron interactions and collective excitations. Here, we show that Coulomb drag in a deliberately asymmetric van der Waals bilayer can serve as a layer-selective probe of electronic compressibility that remains invisible to standard transport. We devise a MoS2/graphene double layer with large disparity in effective mass and Fermi temperature between them, separated by a ~3 nm hexagonal boron nitride spacer, and operate in the degenerate Fermi liquid regime. The MoS2 drag channel exhibits constant electronic compressibility and acts as a sensitive transducer of graphene’s Landau-level physics at finite magnetic fields. At elevated temperatures and moderate magnetic fields, clear Shubnikov-de Haas-like behaviour in the drag signal tracks the quantum oscillation in compressibility of graphene even when its own magnetotransport remains essentially featureless under the same conditions. Our results establish asymmetric Coulomb drag as a compressibility spectroscopy for 2D systems, enabling access to quantum phenomena that may leave only weak, or even negligible, fingerprints in transport.

Introduction
In two closely spaced low dimensional conductors, charge driving in one active layer is often observed to induce drag characteristics in another passive layer, yielding a current or voltage in the latter. Such effects offer a fundamental yet direct probe for electronic momentum and/or energy exchange via long range Coulomb interactions, as well as many-body physics beyond single-particle transport1. Indeed, Coulomb drag phenomena have been extensively manifested in different regimes, including quantum wells or graphene separated with large distance in the weak coupling limit2,3, and excitonic condensation when interlayer charge carriers are matched in the quantum Hall limit4–12. Morerecently, emerging physical phenomena are also reported in exotic drag between graphene and superconductors, topological insulators, 1D-1D Luttinger liquid, quantum dots and mixed dimensional electrons13–18.

Among those reported, gapped two-dimensional (2D) semiconductors, with inherently large correlations in the massive carriers, have been a missing piece in the jigsaw puzzle of various drag regimes. Especially, a peculiar family of massive-massless double layers has remained largely unvisited. Taking the Wigner-Seitz radius rs (strong correlation when rs > 10) as a measure of interaction strength in 2D electron systems, massless Dirac fermions in monolayer graphene has a density-independent value of rs ~ 0.7-0.819,20. Meanwhile, in bilayer graphene and conventional 2D electron gases in quantum wells, Fermi surfaces are well defined and rs is sufficiently large only when carrier density is remained ultra low (< 10^10 cm^-2), which is manifested in such as an unconventional negative frictional drag in the vicinity of charge neutral in double graphene bilayers21,22. Gapped 2D semiconductors, the transition metal dichalcogenides (TMDs) for instance, host massive fermions and relatively constant magneto-responses in resistivity over a broad range of temperature23,24. Thus, the interplay of these asymmetric drag paradigm, i.e., massive interacting Schrödinger fermions with massless Dirac fermions, is expected to unveil new physical phenomena, yet its experimental access has been rare, so far25–27. This is mainly due to the grand challenge of obtaining Ohmic contacts and maintaining high-mobility charge transport at their low temperature ground states.

In this work, we demonstrate large Coulomb drag responses in a semiconductor-semimetal hybrid, realized in a MoS2-graphene heterostructure separated by an ultrathin 3 nm h-BN dielectric. Using a 2D window contact method, Ohmic contacts are realized in MoS2 throughout the temperature range tested in this study. Unlike conventional drag systems, we observe a drag resistance (Rdrag) as high as several hundred Ω, with a passive-to-active drag ratio (PADR) reaching ~ 0.6, orders of magnitude larger than previously reported values28,29. Furthermore, we identify a well-defined Fermi-liquid like phase of drag responses by examining systematically the temperature dependence, as well as the carrier density dependence of Rdrag. The MoS2 drag layer has an essentially constant electronic compressibility and thereby serves as a high-sensitivity transducer of graphene’s Landau quantization at finite magnetic field. At elevated temperatures (such as above liquid nitrogen temperature) and moderate magnetic fields, the drag signal exhibits clear Shubnikov-de Haas (SdH)-like oscillations that track graphene’s compressibility, even when graphene’s own transport fails to deliver any information at such temperature and magnetic fileds. These observations highlight the crucial role of interlayer correlations of drag in amplifying the electron compressibility in such an asymmetric (in terms of compressibility in MoS2 and graphene) drag system. Our findings of drag-enhanced readout of graphene’s Landau levels using a flat-compressibility MoS2 transducer may offer insights into designing next-generation interaction-driven electronic devices.

Results
Fabrications and characterizations of MLG-MoS2 drag devices
Monolayer graphene, bilayer MoS2 and h-BN flakes were mechanically exfoliated from bulk crystals. As illustrated in Fig. 1a, the van der Waals heterostructure is stacked using the dry transfer method30, and then encapsulated by top and bottom h-BN flakes, with the top h-BN etched into micron-meter sized 2D windows. A windowed contact method is thus employed to achieve Ohmic contacts to the MoS2 channel throughout the temperature range from 0.3 K to 300 K31. This requires MoS2 to be the top layer in the heterostructure to facilitate the fabrication process in this study. The devices were equipped with dual metallic gates and electrodes of Ti/Au via standard lithography and electron-beam evaporation (fabrication details are available in Methods). More detailed fabrication processes can be seen in Supplementary Figs. 1–3. We found that different bottom gate geometry will affect the Coulomb drag measurements (Supplementary Fig. 4), and the main text will focus on the geometric configuration as illustrated in Supplementary Fig. 1.

Figure 1a describes the essential nanostructure in this study: a semiconductor−semimetal drag hybrid, realized in a MoS2-graphene double layer separated by an ultrathin 3 nm h-BN dielectric. Here, considering the low energy physics at the Fermi level within the solid state gate doping range, charge carriers in MoS2 and graphene are massive Schrödinger and massless Dirac fermions, respectively. Figure 1b shows the optical micrograph of a typical drag device (sample-S21), with the corresponding fabrication flow shown in Supplementary Fig. 3. A bright-field scanning transmission electron microscopy (STEM) image and the corresponding electron energy loss spectroscopy (EELS) mapping in Fig. 1e−g clearly reveal the cross-sectional structure of a typical drag device with bilayer MoS2 as the active (or passive) layer and monolayer graphene as the passive (or active) layer. Within the device, carriers in each layer can be tuned independently. For instance, at T = 200 K, typical field effect curves in the MoS2 channel (Fig. 1c) and in the graphene channel (Fig. 1d) can be obtained, respectively. The T-dependent transfer curves of graphene and MoS2 in Supplementary Fig. 5 exhibit intrinsic metallic and semiconduting characteristics, respectively. The linear I-V curves of the MoS2 and graphene channel at different temperatures (Supplementary Figs. 6–7) explicitly show the good Ohmic contact for each layer at low temperatures. The low contact resistance (Supplementary Figs. 8–9) of these two separated layers undoubtedly enables us to execute both the drag and active-layer transport measurements.

Figure 2a–b illustrate the mapping of longitudinal channel resistance of graphene (RGr) and MoS2 (RMoS2) in the same Vbg-Vtg space at T = 200 K in the drag device sample-S21, respectively. In general, as shown in Fig. 2a, RGr is in agreement with the previous observation in a standard dual-gated monolayer graphene device32. However, the charge neutral resistive peak of graphene is partially screened by MoS2 due to the existence of relatively high carrier density in the latter layer, yielding a weak Vtg dependence of RGr at Vtg larger than ~ 1 V. Meanwhile, the band edge of MoS2 in Fig. 2b is squeezed and held almost constant at positive Vtg, which is likely due to the contact part of the MoS2 is not gated by the same gate as its major channel. Notice that the band edge of semiconducting MoS2 in Fig. 2b is highlighted by green dashed line, which is quantitatively extracted from the phase signal in the lock-in measurement, shown in Supplementary Fig. 10.

As a consequence, limited by the screening effect and the contact barriers, the drag response in the current device is confined to the electron-electron regime, making hole drag not accessible. For the drag measurements, we passed a drive current (Idrive) through the active layer and measured the resulting voltage drop (Vdrag) across the passive layer under open-circuit conditions. To eliminate spurious drag signals in the passive layer caused by drive-bias-induced AC gating effects33, we have adopted a balance-bridge setup (comparison between lock-in measurement and the bridge methods can be seen Supplementary Fig. 11)10. Figure 2c shows the drag resistance Rdrag using graphene as the driving layer (i.e., the active layer). Here, the parasitic signals, determined to coincide with the band edge of MoS2 by the phase measurements in Supplementary Figs. 10 and 12, are blanked for visual clarity. The PADR, defined as Idrive/Idrag = Rpassive layer/Rdrag, is usually a direct measure of the interlayer interaction in drag systems. For example, when it comes to a perfect drag in the scenario of exciton condensation, PADR may reach the unity12,34,35. In our system, PADR (Fig. 2d) has a maximum value of ~ 0.6 when MoS2 serves as the active layer, much higher compared to most of the conventional drag systems. This large PADR likely originates from more efficient momentum transfer, which is induced by strong interlayer correlation coupled with weak Coulomb screening within the 2D semiconductor MoS2 19,36,37. Moreover, we notice that maximum Rdrag seems to take place at the onset of the semiconducting MoS2 channel conductance derivative with respect to gate voltage (dG/dVg), as indicated by the red arrows in Fig. 2c–d (also discussed in Supplementary Fig. 13). Line profiles of Rdrag in different directions of drive current (± 160 nA) as a function of Vtg at Vbg = 5 V and T = 200 K are shown in Fig. 2e. The inset in Fig. 2e shows the extracted Vdrag at Vtg = 0 V as a function of Idrive. The linear relationship between Vdrag and Idrive, together with the interlayer leakage current (Supplementary Fig. 14), were carefully checked to confirm the validity of our drag measurements. Furthermore, this observed linear response of Vdrag to Idrive persists at low temperature (as shown in Supplementary Fig. 15), suggesting that the measured drag signal is reliable under these experimental conditions, and the Onsager reciprocity when the drive and drag layers are exchanged (Supplementary Fig. 16) also demonstrates that the system is in the linear response regime, allowing the extraction of the drag resistance from the slope of ΔVdrag/ΔIdrive.

Temperature dependences of the Coulomb drag in the semiconductor-semimetal hybrid
In the following, we investigate the observed drag response in the graphene-MoS2 hybrid at different temperatures. When lowering the temperature, phonon scattering in the MoS2 is known to be largely suppressed and the carrier transport in the system is supposed to be driven from phonon-limited low-mobility regime into the intrinsic high-mobility regime, with the system exhibiting a transition from insulating behavior to a metallic one, known as metal-insulator transition (MIT) when varying from low to high carrier density in the low temperature limit31,38,39. This highly tunable electron transport properties in one of the layers of the drag system may give rise to unique and unconventional drag signals, distinguishing it from previously reported drag systems21,37,40,41.

Figure 2f shows the temperature dependence of drag resistance at different Vtg with the Vbg fixed at 5 V. Rdrag increases monotonically as the temperature increases when MoS2 becomes metallic at large Vtg. A T2 dependence is clearly observed at the base temperatures, which is in good agreement with the theory of frictional drag for Fermi liquid42,43. However, in the high-temperature regime, deviation from the T2 dependence becomes pronounced, eventually evolving into a linear temperature dependence. It is found that the crossover regime from T2 to T dependence broadens as MoS2 becomes more insulating with decreasing Vtg.

We further plot the colormaps of the drag resistance in the T-Vtg phase diagram, as shown in Fig. 2g. The corresponding fitting points for Rdrag in Fig. 2f are featured in the phase diagram. From this comparison, we identify four distinct temperature-dependent drag regions: no drag, T2, T and T2-Tc crossover regions are observed. Coulomb drag resistance is known to be extremely sensitive to temperature, interlayer spacing, carrier density (or density mismatch between the layers) and magnetic field22,44. And drag transport regimes can be defined by the Fermi energy EF, the magnitude of Fermi wave vector kF, interlayer separation d. In the Boltzmann-Langevin theory of Coulomb drag for the Fermi liquid scenario, at low temperatures (T ≪ Td = EF/kBkFd) and in the clean limit (weak disorder or low scattering rate), drag is dominated by the particle-hole continuum and Rdrag is proportional to T245. Thus, we have plotted the estimated characteristic temperature Td in the phase diagram and found that the curve of Td indeed separates the quadratic T2 and linear T dependent drag regimes. The temperature region of a Fermi liquid below Td, in which the drag resistance follows the T2 law, is strongly suppressed as the band edge of MoS2 is approached. At higher temperatures, T > Td, phase-space constraints due to small-angle scattering lead to a linear temperature dependence2,45. While on the insulating side, Rdrag deviates from both T2 and linear temperature dependence, eventually drops to zero as the carrier density in MoS2 decreases and Fermi level moves into the band gap of MoS2 (marked as “no drag” in the diagram).

Drag at the matched density
It is noticed that the carrier density dependent characteristic of Rdrag varies significantly in different kinds of drag system. The relationship between drag resistance and carrier density at the matched density in massive-massless fermion system has been explored theoretically, which is in contrast with that in massive-massive and massless-massless fermion systems25,46. For high density regime (kFd ≫ 1), all three systems exhibit a similar carrier density dependence, specifically following an 1/n3 behavior. For the low density regime (kFd ≪ 1), the carrier density dependence exhibits distinct characteristics for different systems, highlighting their unique properties. Specifically, in the massless-massive case, Rdrag scales as 1/n2, whereas for massive-massive and massless-massless systems, the dependencies are predicted to follow 1/n3 and 1/n, respectively25,43.

In our case, we first estimate the carrier density of the MoS2 and graphene layer (nMoS2 and nGr, respectively) independently from the longitudinal and transverse resistance (Rxx and Rxy, respectively) of the two layers based on measurements of the classical Hall effect at 77 K, as shown in Fig. 3a and b. The Rxy of both MoS2 and graphene varies linearly with the magnetic field, as shown in Supplementary Fig. 17. The equal density line (nMoS2 = nGr) can be easily identified by subtracting the two carrier density colormaps of MoS2 and graphene, as illustrated by the black dash line in Fig. 3c. Subsequently, we plot the drag resistance along the density matched line (nMoS2 = nGr) in logarithmic scale and converges to the expected 1/nα dependence, with α ≈ 2. For our massive-massless fermion system, the range of the equal density line is about 1.2 ~ 3.0 × 10^12 cm^-2. Thus, we estimate that the maximum value of kFd satisfies kFd < 1 using the expression kF = √πn. This result demonstrates the equal density line is at the low density regime with the 1/n2 dependent Rdrag, which agrees well with the theoretical predicts25. Our analysis on the matched-density drag response suggests that, although being at a temperature slightly above Td, the drag response can still be considered as a Fermi liquid at proper doping.

Asymmetric drag as a compressibility spectrometer at high temperature
For a better understanding the charge transport in this asymmetric drag system, we first extracted the Hall carrier density as a function of temperatures and gate voltages, and calculated the Hall mobility for each layer accordingly (Supplementary Fig. 18). The field-effect mobility obtained from the transfer curves of these two layers is also presented in Supplementary Fig. 19 for comparison. We now show the magnetodrag (the longitudinal component) of massive-massless fermion system in the presence of a finite magnetic field B. The Onsager reciprocity relation for magnetodrag resistance remains valid regardless of whether the system is within or outside the Fermi liquid regime (as shown in Supplementary Figs. 20 and 21). It is noteworthy that a slight departure from perfect reciprocity occurs at low temperatures and under magnetic fields. While the microscopic origins of this non-ideal reciprocal Coulomb drag remain unclear, they may stem from a high concentration of disorder in specific devices, which can introduce a small rectified component to the drag signal. Subsequently, we investigate the B-dependent Rdrag at liquid nitrogen temperature (T = 77 K). It is clearly seen in Fig. 4a that MoS2 is absent of magnetoresistance, as it is known to exhibit negligible magneto-resistance and hence flat-compressibility due to the low mobility shown in Supplementary Figs. 18 and 19 at high temperatures47. Figure 4b illustrates the magnetoresistance of the graphene channel itself in the parameter space of B-Vtg. The SdH oscillations within ± 12 T for graphene is essentially featureless at this temperature, which is in agreement with literatures48,49. Remarkably, magnetodrag, i.e., Rdrag-B in Fig. 4c shows well-developed stripped features (indicated by the arrows), which are the quantum oscillations from graphene probed, with a significantly amplified amplitude, via Coulomb drag.

According to the standard perturbative theory, the phenomenological drag resistivity ρD can be written as1,43
ρD = (1/16πS k T σ1 σ2) ∫ q dq ∫ dω |U12(q,ω)|2 ImΠ1(q,ω) ImΠ2(q,ω) / sinh2(ℏω/2kBT), (1)
where S is the cross section area of the layers; σ1 and σ2 denote the Drude conductivities of each layer; U12 is the screened interlayer interaction; Πi is the density-density response of layer i. The magnetic-field dependence at low q, ω enters through the layers’ compressibilities via ImΠi ∝ νi(B)ω/(vFi q). νi is the thermodynamic density of states DOS (compressibility κ = dn/dμ, is proportional to DOS); vFi is the Fermi velocity; d is the interlayer spacing; dominant momenta satisfy q ≲ qT ≡ kBT/ℏvFi.

In a MoS2/graphene bilayer with a 3 nm h-BN spacer, MoS2 has thermally smeared Landau quantization at 100 K (as the cyclotron energy of MoS2 at 12 T is only 2-3 meV, way smaller than kBT ~ 8.6 meV), giving nearly constant ν2(B), whereas graphene retains oscillatory ν1(B) governed by Lifshitz-Kosevich/Dingle factors. Consequently, δρD(B)/T2 ∝ δν1(B) × S(B), (2)
where thermal and disorder damping are encoded in Rxx and Rxy inside δν(B), and S(B) denotes a weakly B-dependent screening factor. Therefore, in this current scenario, Coulomb drag functions as a compressibility transducer of graphene’s SdH oscillations with a growing T2 baseline, and more importantly, an interaction-induced factor of |U12|2 allows clear read out of oscillations at 100 K even when graphene’s own magnetotransport oscillations are muted. The strong interlayer coupling at 3 nm (dominant q ~ kBT/ℏvFi with qd ≈ 0.04 ≪ 1) prevents exponential suppression over spatial distance, and physically, drag tracks the oscillatory compressibility of graphene as
δρD ∝ T2 |U12|2 δκgraphene(B). (3)

We firstly compare the intrinsic magneto-resistance of the graphene layer with magneto-drag response for various Vtg by fixing the temperature and back gate, which is shown in Fig. 4d–f. As Vtg increases, both the values of Rxx of graphene and drag drop due to more carriers enter into the channel. Meanwhile, the increased conductivity of the MoS2 layer gives rise to the proximity screening effect, which in turn effectively suppresses charge inhomogeneity in the graphene layer50. Therefore, the electron mobility in graphene will be greatly enhanced (Supplementary Fig. 18), leading to more evident quantum oscillations. However, when Vtg decreases, the weaken proximity screening effect predictably enhances the electron-electron interactions and |U12|2, amplifying the amplitude of the oscillations in drag signal.

Subsequently, we studied the temperature dependence of the magneto-drag resistance. Here, we included an intuitive schematic illustration in Fig. 5a for better understanding. The low-mobility electrons in MoS2 move randomly between scattering disorders with a magnetic field applied (in the top layer). This diffusive transport model prevents the observation of significant SdH oscillations under 12 T, meaning the electron compressibility in MoS2 is nearly flat as a function of the magnetic field. In stark contrast, the high-mobility electrons in graphene primarily follow cyclotron orbits perpendicular to the magnetic field (in the bottom layer). In other words, the electron states could be easily quantized into distinct Landau Levels. Consequently, the electron compressibility in graphene oscillates with the magnetic field for a fixed carrier density, allowing for the observation of evident SdH oscillations. However, as the temperature increases, transport across the sample occurs via inter-orbit scattering (the dashed lines connecting the orbits), which is a diffusive process, leading to the amplitude of the SdH oscillations to decrease until they vanish.

Indeed, as shown in Fig. 5b, at T = 80 K, it is clearly seen that the transport channel of graphene’s resistance (green) has essentially featureless SdH oscillations, with only a few bumps difficult to distinguish. But for the case of drag system, the Coulomb drag response (red) still exhibit significant oscillations, agreeing well with the above analysis. Figure 5c and d further show the side-by-side comparison of oscillations in lateral transport and drag response at different temperatures. Notice that these drag oscillations versus 1/B are spaced exactly like graphene’s SdH oscillations, as shown in Supplementary Fig. 22. In addition, the flat electron compressibility in MoS2 confirms the origin of the observed oscillations in Rdrag is solely from the graphene layer, suggesting that MoS2 serves as an excellent sensor of probing electron compressibility in high-mobility materials. Figure 5e shows the extracted oscillation amplitudes obtained from lateral transport in graphene (open circles) and from Coulomb drag (solid squares), after subtracting the background from each curve. It is seen that at base temperature lower than 20 K, the SdH oscillations measured from lateral transport in graphene has larger amplitudes than those from the drag response. At elevated temperatures, the amplitudes of drag oscillation rapidly overgrow, functioning as a transducer of κgraphene that effectively amplifies the read out of quantum oscillations. Above liquid nitrogen temperature (indicated by the dashed line in Fig. 5e), the oscillation amplitude difference can be more than one order of magnitude. This kind of result is reproducible in sample-S24, which featured the same device geometry as sample-S21, as shown in Supplementary Fig. 23.

The final section addresses the dependence of the drag response on the spacer thickness d, a key factor determining the strength of interlayer coupling within the drag system. We have measured the dual-gate mapping of drag resistance for three typical samples with different interlayer spacing, as shown in Supplementary Fig. 24a–c. We could clearly see that the drag response is reproducible in different samples, and the magnitude of drag resistance is inversely proportional to d, which is consistent with previous reports3,27. For better comparison, we plotted the drag resistance as a function of carrier density for samples-S21 and S15 at 77 K in Supplementary Fig. 24d. Moreover, we utilized drag and transport measurements at 77 K to probe the electron compressibility of graphene in two drag systems with different d, as presented in Supplementary Fig. 24(e). The carrier densities for this measurement were fixed at ~ 1.2 × 10^12 cm^-2 for MoS2 and ~ 2.7 × 10^12 cm^-2 for graphene, respectively. The system with the thinner spacer (sample-S21) exhibited stronger interlayer coupling, resulting in a higher magnitude of oscillations in the drag signal compared to the weaker interlayer coupling of the thicker spacer (sample-S15). This significant difference indicates that the thickness of the spacer is critically important for efficiently probing compressibility in the drag system. We have summarized the characteristics of temperature, magnetic field, and carrier density dependence in a collection of experimentally tested Coulomb drag systems3,13–15,18,21,22,27,34,37,40,51–53 in Table 1. The unique role of MoS2 in the observation of pronounced SdH oscillations is of two origins, i.e., a relatively flat electron compressibility in its own, and sufficiently strong Coulomb interaction factor |U12|2. In this regard, if MoS2 in the current system is replaced by other 2D semiconductor with flat electron compressibility-such as MoSe2 and WSe2, provided that Ohmic contacts could be obtained at low temperatures-should theoretically exhibit similar phenomena. Otherwise, for instance, replacing MoS2 with graphene, high screening in both layers will lead to a dramatic reduction of interlayer Coulomb interaction, and their SdH signals tend to be entangled, complicating the decoupling of layer-specific information. Overall, the massless Dirac-massive Schrödinger fermions graphene-MoS2 drag system in this work demonstrates a distinct paradigm where the asymmetric configuration in each constituent layer can lead to selective-&-amplified probe of electron compressibility, thus providing grounds for future investigations in drag electronics.

To conclude, by integrating an Ohmic-contacted semiconducting TMD channel with a semimetal, we realize a strongly coupled graphene/MoS2 bilayer separated by an ultrathin ~ 3 nm h-BN spacer that enables interaction-assisted probe of quantum thermodynamics. Systematic maps of the drag response Rdrag versus temperature, carrier density, and magnetic field place our device in the degenerate Fermi-liquid regime (for T ≲ 100 K with MoS2 in its conduction band). Crucially, the MoS2 layer’s essentially flat compressibility makes it an efficient transducer of graphene’s Landau quantization. At elevated temperatures and moderate fields, the drag signal exhibits clear SdH-like oscillations that persist even when graphene’s own magnetotransport is featureless. The oscillation periodicity of the magnetodrag response follows graphene’s Landau quantization, consistent with a Fermi-liquid perturbative picture in which δRdrag ∝ T2 |U12|2 δκgraphene(B). Beyond extending the drag landscape to mixed massive-massless double layers, these results establish Coulomb drag as a practical compressibility spectroscopy for 2D materials, unlocking access to quantum phenomena that can be suppressed in standard transport. More broadly, the semiconductor-semimetal architecture suggests a route to interaction-engineered sensors and high-temperature quantum-oscillation metrology, with tunable sensitivity via spacer thickness, screening environment, and carrier density.

Methods
Sample fabrication
vdW few-layers of the h-BN/MoS2/h-BN/graphene/h-BN sandwich were obtained by mechanically exfoliating high quality bulk crystals. The vertical assembly of vdW layered compounds was fabricated using the dry-transfer method in a nitrogen-filled glove box. The heterostructures were then transferred onto the pre-fabricated Au or graphite gates. Hall bars of the devices were achieved by reactive ion etching. During the fabrication processes, electron beam lithography was done using a Zeiss Sigma 300 SEM with a Raith Elphy Quantum graphic writer. One-dimensional edge contacts of monolayer graphene were achieved by using the electron beam evaporation with Ti/Au thicknesses of ~ 5/50 nm and the window contacts of bilayer MoS2 were fabricated with a thermal evaporator, with typical Bi/Au thicknesses of ~ 25/30 nm. After atomic layer deposition of about 20 nm Al2O3, big top gate was deposited to form the complete dual gated h-BN encapsulated drag devices as shown in Fig. 1a and b. Note that we always put 2D semiconductor MoS2 on the top of graphene because we adopted window-contact technique as previously reported31. We firstly etched through the windows on top h-BN and use it to pick up MoS2 and other layers afterwards. Then, we adopted Bismuth with low work function as electrodes contacted to MoS2 and achieved Ohmic contact at low carrier density and temperatures. Although the fabrication would be more challenging if the MoS2 and graphene layers were swapped, the drag response would remain identical due to the vertically mirror-symmetric configuration.

Drag measurements
In lock-in measurements, current is typically driven by applying an AC bias voltage Vdrive to one side of the channel while the other side is grounded. However, in Coulomb drag measurements, applying this bias to the drive layer may induce spurious drag signals in the drag layer due to the AC gating effect caused by the drive bias. Here we applied about 0.2 ~ 0.3 V AC bias voltage at 17.777 Hz to drive the active layer through a 1:1 voltage transformer. The transformer was connected to a 10 kΩ potentiometer, which can help to distribute the AC voltage across both ends of the driving layer. This configuration minimizes the AC interlayer potential difference in the channel, thereby reducing the AC coupling between the active and passive layers. We used two 1 MΩ resistors connected with the driving layer and measured the voltage drop across one of the resistors to obtain the driving current. The drag voltages were recorded using low-frequency SR830 lock-in amplifiers. Since the relationship between Vdrag and Idrive is linear, we simply employed a large drive current, such as 100-200 nA, to improve the signal-to-noise ratio in our drag measurements without any other specific intentions. Four-probe measurements were used throughout the transport measurements in an Oxford Teslatron cryostat. Gate voltages on the as-prepared devices were controlled by a Keithley 2400 source meter.

Data availability
The Source Data underlying the figures of this study are available at https://doi.org/10.5281/zenodo.18300086. All raw data generated during the current study are available from the corresponding authors upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the manuscript, the author conducted systematic drag experiments in a graphene/MoS2 double layer. The authors observed clear SdH-like oscillations in the drag signal that precisely track the SdH within the graphene layer. As noted by the authors, the MoS2 layer’s nearly flat compressibility facilitates the visualization of graphene’s Landau quantization. While the experimental study is systematic with clear data presentation and highly readable writing, the manuscript lacks a compelling focus. The central claim in the abstract section, i.e. the observation of pronounced SdH oscillations in the drag signal, is only introduced in Figure 4. Figures 1 to 3 appear more as a simple presentation of the drag characteristics in the massless-massive system, without sufficient connection to the main finding. The authors should refine the manuscript to better highlight the key points presented in Figure 4 and strengthen the discussions on the underlying mechanisms and potential impact, which will undoubtedly improve the chances of this work being accepted.\nBelow are some minor comments.\n1.Could the authors provide more detailed discussions on the effect of bottom gate geometry on the measured drag behaviors (Supplementary Figure 4)?\n2.Have the authors checked the linear response of the drag signal to Idrive at low temperatures?\n3.The remarkably high passive-to-active drag ratio is intriguing. Could the authors offer a possible explanation for this observation?\n4.The red arrows referenced in Fig. 1g-h are missing from the actual figures.", "reviewer_2": "Liu et al. report compressibility spectroscopy in two-dimensional systems using an asymmetric Coulomb drag configuration, particularly in the Fermi-liquid regime. The presented results are intriguing and could serve as a powerful methodology for probing subtle quantum phenomena in graphene. However, the fundamental characteristics of the active and passive layers are not clearly defined, and the terminology employed throughout the manuscript lacks clarity and consistency. The following issues should be carefully addressed before the manuscript can be considered for further consideration.\n(1) The completed hBN dual-gated device structure should be characterized using TEM to confirm the thicknesses of each layer (Al2O3, top and bottom hBN, hBN spacer, graphene, and MoS2) and to assess the interface quality. In addition, a cross-sectional device image would greatly improve understanding, including explicit labeling of which layer acts as the active (drive) and passive (drag) layers in all figures.\n(2) Please clearly indicate whether bilayer MoS2 is solely used and consistently serves as the passive layer throughout the manuscript. The current description is confusing.\n(3) The drain current as a function of top- and bottom-gate voltages (Vtg and Vbg) for different temperatures should be provided separately, including both transfer and output characteristics.\n(4) The leakage current between the active and passive layers should be presented for all measurement conditions, particularly as a function of Vtg, Vbg, temperature, and magnetic field.\n(5) Both field-effect mobility and Hall mobility as a function of temperature should be included to provide a clearer understanding of charge transport.\n(6) The Onsager reciprocity relation should be verified and discussed to confirm the presence of genuine Coulomb drag phenomena.\n(7) The bias condition (e.g., I = 160 nA) should be clarified, including why this particular condition is only considered.\n(8) The contact resistance should always be specified, as it directly affects both the drag and active-layer transport measurements.\n(9) The electrical bias conditions and temperatures differ across the figures. Such variations must be discussed comprehensively before presenting the data, to ensure consistent interpretation. Limited fundamental experimental results currently make the discussion difficult to follow.\n(10) Both Vtg and Vbg values should be explicitly annotated in all subfigures (e.g., Fig. 1i).\n(11) The meaning of the y-axis in Fig. 3d should be clearly defined.\n(12) The carrier density measurements in Fig. 3 (at 100 K) and the Coulomb drag data in Fig. 4 (at different temperatures) must be correlated. The Hall mobility and carrier density should be presented as a function of both different temperature and bias for consistency.\n(13) In Fig. 1b, the scale bar indicates 10 mm, please confirm if this is a typo.\n(14) In Fig. 1c, the data points corresponding to GMoS2 for Vtg = -3 V to -1 V appear missing. In addition, including a semi-logarithmic scale would help reveal the n-type semiconducting behavior of bilayer MoS2.\n(15) In Fig. 1f, please define the parameter RMoS. In Fig. 1h, define PVR explicitly. Numerous typographical errors are present throughout the manuscript and should be corrected carefully.\n(16) In Fig. S6c, the Vtg-dependent MoS2 resistance behavior is nearly symmetric and resembles that of graphene, suggesting ambipolar characteristics. Please elaborate on this observation.\n(17) In Fig. S10a, GMoS2 measured at Vbg = 4 V is lower than that at Vbg = 1 V (Fig. 1c), which is counterintuitive for n-type MoS2. Please clarify this discrepancy. If different devices were used, please specify device numbers in all figures.\n(18) The thickness of the hBN spacer is a critical parameter for interlayer coupling at the Gr/hBN/MoS2 interface. The electron compressibility as a function of hBN thickness should be presented and discussed. In addition, please discuss the case where the positions of MoS2 and graphene are swapped. Why is MoS2 always placed on top of graphene? Addressing this question would strengthen the physical reasoning.\n(19) The contact resistance for both MoS2 and graphene as a function of bias and temperature should be presented to ensure accurate analysis of interlayer transport behavior.\n(20) Since the effective mass of MoS2 is roughly an order of magnitude larger than that of graphene, it would be helpful to include an intuitive schematic illustration showing how the Shubnikov–de Haas oscillations in MoS2 could be amplified or modulated through electron compressibility.\nIn summary, while the manuscript presents an interesting approach to probing compressibility in two-dimensional systems via Coulomb drag, it currently lacks structural, electrical, and conceptual consistency. The figures require systematic reorganization, and several critical parameters (bias, temperature, and layer identity) must be clarified before the study can provide a coherent physical picture. If the manuscript provides a more detailed discussion of the fundamental charge transport characteristics in each layer, a deeper and more quantitative review of the Coulomb drag phenomena would be possible.", "reviewer_3": "The manuscript describes experimental measurements of Coulomb drag between MoS2 and monolayer graphene separated by few-nm thick hBN, primarily conducted by driving a current in the graphene and measuring the induced voltage in the MoS2. The authors investigate the dependence of the drag signal on temperature, carrier density and magnetic field and find substantial similarities to the available theoretical predictions. In an applied perpendicular magnetic field, both the graphene layer and the drag resistance show Shubnikov-de Haas oscillations. The magnitude of the oscillations rapidly decreases in the graphene with increasing temperature, while remaining relatively constant in the drag signal, allowing the drag resistance to act as a sensor for otherwise-inconspicuous electronic transport behavior in the graphene.\nThe authors effectively introduce the ideas behind the measurement and the potential significance of their results, without making overstated claims. These experiments are a nice addition to the existing body of work on Coulomb drag, and contribute new information that will be of interest to the field. The fabrication methods are consistent with the current standard for van der Waals heterostructure devices and are well-explained, and they clearly demonstrate their efforts to avoid spurious drag signals in the measurement by using a voltage-balancing bridge. There is sufficient information given in the text and supplemental material to replicate the experiment.\nWhile the authors show Onsager reciprocity (driving one layer and measuring the drag voltage across the other, switching the drive and drag layers) in magnetotransport, it would strengthen the paper to show Onsager reciprocity in other regimes, particularly outside of the Fermi liquid drag regime. It would also be beneficial to show data from additional devices; there are 9 devices described as having been measured, but data from only 2 or 3 of them is shown. The reproducibility of the results would be more convincing with some more extensive characterization of multiple devices.\nA few minor details: There is a typographical error in the legend of Figure 1h, which refers to “PVR” instead of “PADR.” Also, the text refers to red arrows in Figure 1g-h that do not appear to be present.\nOverall, I think this manuscript is suited for publication with minor changes."}, "author_response": "Author Responses to Initial Comments:\nReply to reviewer #1\nGeneral Comment:\nIn the manuscript, the author conducted systematic drag experiments in a\ngraphene/MoS double layer. The authors observed clear SdH-like oscillations in the\n2\ndrag signal that precisely track the SdH within the graphene layer. As noted by the\nauthors, the MoS layer’s nearly flat compressibility facilitates the visualization of\n2\ngraphene’s Landau quantization. While the experimental study is systematic with clear\ndata presentation and highly readable writing, the manuscript lacks a compelling focus.\nThe central claim in the abstract section, i.e. the observation of pronounced SdH\noscillations in the drag signal, is only introduced in Figure 4. Figures 1 to 3 appear more\nas a simple presentation of the drag characteristics in the massless-massive system,\nwithout sufficient connection to the main finding. The authors should refine the\nmanuscript to better highlight the key points presented in Figure 4 and strengthen the\ndiscussions on the underlying mechanisms and potential impact, which will\nundoubtedly improve the chances of this work being accepted.\nResponse:\nWe thank the reviewer a lot for the positive assessment of our work and for the\nsuggestions that have helped to improve our manuscript. We have reorganized the\nmanuscript to further bring out the main findings of this work by adding more figures\nand discussions on the pronounced SdH oscillations in the drag signal. Below are the\npoint-by-point responses to the reviewer’s comments.\nBelow are some minor comments.\nComment 1. Could the authors provide more detailed discussions on the effect of\nbottom gate geometry on the measured drag behaviors (Supplementary Figure 4)?\nResponse:\nWe thank the reviewer a lot for her/his suggestion. Basically, we use the local bottom\ngate to decrease the contact resistance of 2D semiconductor MoS for all the three\n2\ngeometries, as shown in Figure R1. However, there are some small differences between\nthem. In general, we can use silicon local gate to decrease the contact resistance of\nMoS and another bottom gold or graphite gate to tune the carrier density of monolayer\n2\ngraphene independently, as schematically illustrated in Figure R1g. In this geometry,\nthe MoS channel is divided into four distinct regions due to the possible misalignment\n2\nof top and bottom gates: region 1 with both top and bottom gate, region 2 with only\nbottom gate and graphene screening, region 3 with only bottom gate, and region 4 with\nonly silicon-gate tuning. This complicated structure would result in spurious drag\nsignals, which are particularly pronounced near the MoS₂ band edge (Figure R1i).\nTo avoid this problem, we introduce only one local bottom gate to decrease the contact\nresistance and tune the carrier density in graphene simultaneously, as schematically\nshown in Figure R1a and d. However, the area of monolayer graphene in Figure R1d is\nlarger than that of MoS\n2\n. This configuration will prevent tuning the contact resistance\nof MoS efficiently due to the screening induced by graphene and result in chaotic drag\n2\nsignals as shown in Figure R1f. Therefore, we chose the first configuration (Figure R1a)\nwith narrow graphene flake to obtain good Ohmic contacts to MoS channel and\n2\nexclude spurious drag signals.\nWe have replaced Supplementary Figure 4 with Figure R1 and included corresponding\nexplanations in the revised Supplementary Information, which are highlighted in blue\nin Page 8 and quoted below：\n“Overall, the second configuration (d) will prevent tuning the contact resistance of\nMoS efficiently due to the screening induced by graphene and result in chaotic drag\n2\nsignals as shown in (f). For the third configuration (g), the contact of MoS is divided\n2\ninto four distinct regions due to the possible misalignment of top and bottom gates,\nwhich will lead to spurious drag signals presented in (i). Based on the evident\ncomparison of the measured drag signals, we select the first device configuration for\nfurther measurements and analysis.”\nFigure R1. (a) The schematic of a typical MoS -graphene drag device (used in the\n2\nmanuscript). This device configuration consists of a large local Au (or graphite) back\ngate and top gate. Bi/Au electrodes are used for Ohmic contact for MoS and Ti/Au\n2\nelectrodes are used for one dimensional Ohmic contact for graphene. (b) and (c) show\nthe corresponding zoomed-in schematic top view of the Bi/Au electrode and measured\ndrag signals as a function of V and V , respectively. (d) The schematic illustrates a\nbg tg\ndrag device with the graphene layer larger in size than the MoS\n2\nlayer. The dashed line\nrepresents graphene. (e) and (f) are the corresponding zoomed-in schematic top view\nof the Bi/Au electrode and measured drag signals as a function of V and V ,\nbg tg\nrespectively. (g) The schematic depicts a drag device utilizing a silicon gate for tuning\nthe carrier density in the contact region of MoS , while the overlapped area represents\n2\nthe top and bottom gates. (h) and (i) show the corresponding zoomed-in schematic top\nview of the Bi/Au electrode and measured drag signals as a function of V and V ,\nbg tg\nrespectively. Overall, the second configuration (d) will prevent tuning the contact\nresistance of MoS efficiently due to the screening induced by graphene and result in\n2\nchaotic drag signals as shown in (f). For the third configuration (g), the contact of MoS\n2\nis divided into four distinct regions due to the possible misalignment of top and bottom\ngates, which will lead to spurious drag signals presented in (i). Based on the evident\ncomparison of the measured drag signals, we select the first device configuration for\nfurther measurements and analysis.\nComment 2. Have the authors checked the linear response of the drag signal to I at\ndrive\nlow temperatures?\nResponse:\nWe thank the reviewer for the question. We have checked the response of drag signal\nto I as a function of V at V = 5 V and T = 77 K, as shown in Figure R2. From\ndrive tg bg\nFigure R2, we can clearly see that the linear relationship between V and I still\ndrag drive\nholds at low temperatures for different V .\ntg\nFigure R2. (a) The drag response as a function of V for various applied drive current\ntg\nat V = 5 V and T = 77 K. (b) The relationship between drag response and drive current\nbg\nfor different V . The colored circles represent the drag voltage data extracted from (a),\ntg\nand the corresponding colored lines show the linear fittings for each represent V .\ntg\nIn this revision, we have added Fig. R2 in the revised Supplementary Information as\nSupplementary Figure 15, highlighted in blue in Page 20.\nComment 3. The remarkably high passive-to-active drag ratio is intriguing. Could the\nauthors offer a possible explanation for this observation?\nResponse:\nWe thank the reviewer a lot for the question. Typically, the passive-to-active drag ratio\n(PADR) is small due to the strong screening of Coulomb interactions by the electron\ngases. However, if the electron-electron correlation is strong or the screening of\nCoulomb interaction is weak in the drag system, the drag ratio would be large. For\nexample, in some strong correlated systems such as 2D excitonic insulator (Science,\n388, 274-278, 2025; Science, 388, 278-283, 2025) and double graphene or quantum\nwells under perpendicular magnetic field (Nature, 637, 327-332, 2025; Nature Physics,\n13, 746-750, 2017; Nature, 488, 481-484, 2012), a large PADR or even \"perfect\" drag\n(approaching 1) can occur due to the formation of electron-hole pairs. But this is quite\ndifferent from our case because there are no excitons in our systems.\nOne possible reason for the observed PADR is strong interlayer correlation induced by\n2D semiconducting MoS . The Wigner–Seitz radius r correspond to the ratio between\n2 s\nthe Coulomb interactions to the kinetic energy, which can describe well the electronic\ncorrelation strength in the system. Since r\ns\n= m∗e2/(4πϵ\n0\nϵℏ2√𝜋𝑛\ne\n) with ϵ, n\ne\nand m∗\ndenoting the dielectric constant, electron density and effective electron mass, large m∗,\nlow n and small ϵ could be used to achieve the large r values (Nature 595, 53-57,\ne s\n2021). For instance, the effective electron mass m∗ in 2D MoS is 0.4 - 0.6 (Phys. Rev.\nlarger than that in semiconducting GaAs, and much larger than that in graphene. Thus,\nr ＞ 10 can be easily obtained for an electron density of n ~ 1011 cm-2 in MoS , giving\ns e 2\nrise to stronger electronic correlation and more efficient momentum transfer in our\nhybrid 2D semiconductor/semimetal drag structure than conventional double quantum\nwells and (semi)metal/(semi)metal systems.\nAnother possible reason for this large PADR is the weak Coulomb screening in 2D\nMoS . In general, Thomas–Fermi wavevector q is used to characterize how efficiency\nthe material screens. For two-dimensional electron gas, the simple Thomas-Fermi\ntheory leads to the long-wavelength Thomas-Fermi screening wavevector (see Reviews\nof modern physics, 83, 407-470, 2011),\n𝑞2𝐷 =\n2𝜋𝑒2𝐷(𝐸𝐹).\n𝑇𝐹\n𝜀0𝜀𝑟\nThe density states of MoS at the Fermi level is，\n2\n𝐷(𝐸 ) =\ng𝑠𝑔𝑣𝑚∗\n.\n𝐹 2𝜋ℏ2\nTherefore, the Thomas–Fermi wavevector q of MoS is，\nTF 2\n𝑀𝑜𝑆\n2𝑚∗𝑒2\n𝑞 2 = ,\n𝑇𝐹 𝜀\n0\n𝜀 𝑟ℏ2\nwhere g and g are spin and valley degeneracy, and 𝜖 is background dielectric\ns v 𝑟\nconstant.\nBut for the monolayer graphene, the density states at the Fermi level is，\n𝐷(𝐸 ) =\n2√𝑛\n,\n𝐹 √𝜋ℏ𝑣\n𝐹\nThus, the Thomas–Fermi wavevector q of monolayer graphene is,\nTF\n𝑞𝑀𝐺 =\n4√𝜋𝑛𝑒2\n.\n𝑇𝐹 𝜀0𝜀𝑟 ℏ𝑣 𝐹\nFrom these equations, we can see that q of graphene increases quickly with n and\nTF\nbecomes much larger than the nearly constant q of MoS , demonstrating the Coulomb\nTF 2\nscreening in 2D MoS is much weaker than that in monolayer graphene. The weak\n2\ntransfer and larger PADR in our hybrid structure when compared with other\nconventional systems. (Advanced Electronic Materials, 9.3: 2201105,2023)\nWe have added these points in the revised main text, highlighted in blue in Page 5, and\nquoted below:\n“This large PADR likely originates from more efficient momentum transfer, which is\ninduced by strong interlayer correlation coupled with weak Coulomb screening within\nthe 2D semiconductor MoS 19,36,37.”\nResponse:\nWe thank the reviewer for pointing this out. We have added the red arrows in the revised\nFig. 2c-d (Fig. 1g-h for old version) in the main text.\nIn the end, we sincerely appreciate the very helpful comments given by Referee #1. The\nrevisions implemented based on the reviewer's suggestions have substantially improved\nthe quality of our manuscript. We are deeply grateful for her/his support in the\nacceptance of this work by Nature Communications.\n----------------------------------------\nReply to reviewer #2\nGeneral Comment:\nLiu et al. report compressibility spectroscopy in two-dimensional systems using an\nasymmetric Coulomb drag configuration, particularly in the Fermi-liquid regime. The\npresented results are intriguing and could serve as a powerful methodology for\nprobing subtle quantum phenomena in graphene. However, the fundamental characteristics of\nthe active and passive layers are not clearly defined, and the terminology employed\nthroughout the manuscript lacks clarity and consistency. The following issues should\nbe carefully addressed before the manuscript can be considered for further\nconsideration.\nResponse:\nWe greatly appreciate the reviewer for the positive feedback on the novelty and\nsignificance of our study. We have added some fundamental characteristics of both\ngraphene and MoS layer and refined our manuscript to improve the clarity and\n2\nconsistency of our findings. Below are the point-by-point responses to the reviewer’s\ncomments.\nComment 1: The completed hBN dual-gated device structure should be characterized\nusing TEM to confirm the thicknesses of each layer (Al O , top and bottom hBN, hBN\n2 3\nspacer, graphene, and MoS ) and to assess the interface quality. In addition, a cross-\n2\nsectional device image would greatly improve understanding, including explicit\nlabeling of which layer acts as the active (drive) and passive (drag) layers in all figures.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. Figure R3 presents the bright-field\nscanning transmission electron microscopy (STEM) image and the corresponding\nelectron energy loss spectroscopy (EELS) mapping, clearly revealing the cross-\nsectional structure of the typical drag device with bilayer MoS as the active (or passive)\n2\nlayer and monolayer graphene as the passive (or active) layer.\nFigure R3. (a) A bright-field scanning transmission electron microscopy (STEM) image\nand (b) EELS mappings of the cross section of a typical graphene-MoS drag device.\n(HAADF-STEM) image of the region where bilayer MoS and monolayer graphene are\n2\nseparated by a thin h-BN layer. All of the scale bars are 20 nm.\nIn this revision, we have added this image in Fig. 1 in the main text for better display\nof our drag device configuration.\nComment 2: Please clearly indicate whether bilayer MoS is solely used and\n2\nconsistently serves as the passive layer throughout the manuscript. The current\ndescription is confusing.\nResponse:\nWe thank the reviewer a lot for this comment. The drag response in our system is\nreciprocal regardless of which layer (MoS or graphene) is donated as the drive or\n2\npassive layer, as shown in Fig. R4. To prevent any confusion, we have now clearly\nlabeled the drive and passive layers in each relevant figure of the revised manuscript.\nFigure R4. (a) The schematic of Onsager relation with the active (passive) layer\nalternated in the drag system. (b) The drag resistance as a function of V at different\ntg\nV , measured at T = 200 K. The red and blue circles represent serving MoS and\nbg 2\ngraphene as the active layer, respectively.\nWe have added Figure R4 as Supplementary Figure 16 to explicitly demonstrate this\nOnsager reciprocity relation.\nComment 3: The drain current as a function of top- and bottom-gate voltages (V and\ntg\nV ) for different temperatures should be provided separately, including both transfer\nbg\nand output characteristics.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. We have measured the output and\ntransfer characteristics of both MoS and graphene at different temperatures. As\n2\ndepicted in Figures R5 and R6, the linear I-V curves of MoS and graphene with\n2\ndifferent gate voltages and temperatures suggest that good Ohmic contacts are achieved\nin all conditions.\nFigure R5. (a) The color map of the drain current in MoS as a function of V and\n2 ds\ntemperature at V = 1V and V = 5 V. (b) The I-V curves of MoS at different\ntg bg 2\ntemperatures extracted from (a). The I-V curves of MoS with different V at (c) T =\n2 tg\nThe I-V curves of MoS with different V at (f) T = 100 K, (g) T = 50 K and (e) T = 20\n2 bg\nK, respectively. The applied top voltage is 1 V.\nFigure R6. (a) The I-V curves of monolayer graphene at different temperatures for V\ntg\n= 1 V and V = 5 V. The I-V curves of monolayer graphene at T = 20 K for (b) V = 5\nbg bg\nV and (c) V = 1 V, respectively. The I-V curves of monolayer graphene at T = 200 K\ntg\nfor (b) V\nbg\n= 5 V and (c) V\ntg\n= 1 V, respectively.\nThe transfer curves of both graphene and MoS at different temperatures are shown in\nintrinsic metallic conducting characteristics. In contrast, MoS displays two distinct\n2\nbehaviors: its resistance decreases with increasing temperature at low V , but decreases\ntg\nwith decreasing temperature at high V . This behavior clearly demonstrates the metal-\ntg\ninsulator transition characteristic of n-type semiconductor MoS , consistent with\n2\nprevious reports.\nFigure R7. The transfer curves of (a) monolayer graphene and (b) MoS as a function\n2\nof gate voltages for different temperatures, respectively.\nIn the updated Supplementary Information (changes highlighted in blue), we have\nreplaced the original Supplementary Figure 5 with Figure R7. Additionally, Figure R5\nand Figure R6 have been added as the new Supplementary Figures 6 and 7.\nComment 4: The leakage current between the active and passive layers should be\npresented for all measurement conditions, particularly as a function of V , V ,\ntg bg\ntemperature, and magnetic field.\nResponse:\nWe appreciate this very insightful comment from the reviewer. We have measured the\nleakage current between the active and passive layers at different temperatures, gate\nvoltages and magnetic field, as shown in Figure R8. We could confirm that the leakage\ncurrent is negligible during our drag measurements.\nFigure R8. The leakage current I between the active and passive layers as a function\ng\nof temperatures at V = 1V and V = 5 V when the magnetic field is held for 0 T (yellow\ntg bg\nline) and 12 T (blue dots), respectively. The insets show I as a function of V , V and\ng tg tg\nmagnetic field.\nWe have added Figure R8 as Supplementary Figure 14, highlighted in blue in Page 19,\nto verify the validity of drag measurements.\nComment 5: Both field-effect mobility and Hall mobility as a function of temperature\nshould be included to provide a clearer understanding of charge transport.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. We have plotted the field-effect\nmobility of graphene and MoS as a function of temperature at V = 1 V and V = 5 V\n2 tg bg\nin Figure R9. The field-effect mobility is calculated from the transfer curves in Figure\nR7. Note that we couldn’t determine the accurate geometry factor L/W from the special\ngeometry of our drag devices. However, we have found the geometry factor is about 1\nfor a similar device from the previous report (UPB Sci. Bull. Ser. A, 72, 257-271, 2010).\nTherefore, we adjust this approximate value for calculating the mobility of devices.\nFigure R9. The field-effect mobility of (a) MoS and (b) graphene for various\n2\ntemperatures at V = 1 V and V = 5 V, respectively.\ntg bg\nIn order to obtain the Hall mobility of graphene and MoS we firstly measured the Hall\n2,\ncarrier densities of these two layers as a function of temperature at V = 1 V and V =\ntg bg\n2\n(V = 1 V and V = 5 V) carrier concentration increases significantly with temperature\ntg bg ,\nbecause thermal energy excites more electrons to cross the energy gap. After we obtain\nthe Hall carrier density in both graphene and MoS layer, we are able to calculate the\n2\ncorresponding Hall mobility of these two layers for various temperatures by using the\nformula 𝜎 = 𝑛𝑒𝜇 and plot them in Figure R10 (c) and (d), respectively.\nFigure R10. The Hall carrier density as a function of temperature in (a) MoS and (b)\n2\ngraphene at V = 1 V and V = 5 V, respectively. The Hall mobility in (c) MoS and\ntg bg 2\n(d) graphene at V = 1 V and V = 5 V for different temperatures, respectively.\ntg bg\nNote that the Hall and field-effect mobility values exhibit a minor difference, a\nconsequence of the distinct measurement principles employed for each technique. We\nhave added Figures R9 and R10 as Supplementary Figures 19 and 18 in the revised\nversion, highlighted in blue in Pages 24 and 23, respectively.\nComment 6: The Onsager reciprocity relation should be verified and discussed to\nconfirm the presence of genuine Coulomb drag phenomena.\nResponse:\nWe thank the reviewer for the suggestion that have helped to improve our manuscript.\nWe agree with the reviewer on the importance of this verification. We have also\ndiscussed this in Comment 2. Our measurements have confirmed that the system's\nresponse is fully consistent with the Onsager reciprocity relation. The corresponding\nreciprocal relation is presented in Figure R11 (same as Fig. R4) and Supplementary\nFigure16, which strongly supports the genuine Coulomb drag.\nFigure R11. (a) The schematic of Onsager relation with the active (passive) layer\nalternated in the drag system. (b) The drag resistance as a function of V at different\ntg\nV , measured at T = 200 K. The red and blue circles represent serving MoS and\nbg 2\ngraphene as the active layer, respectively.\nWe have incorporated Figure R11 into the Supplementary Information as the new\nSupplementary Figure 16, highlighted in blue in Page 21.\nComment 7: The bias condition (e.g., I = 160 nA) should be clarified, including why\nthis particular condition is only considered.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. As displayed in Figure 1i, the drag\nvoltage is proportional to the drive current at 200 K, demonstrating that the drag\nresistance is I - independent. For other reviewer’s concern, we also employed the\ndrive\nI - V measurements as a function of gate voltages at low temperature to confirm\ndrive drag\nthe linear drag resistance, as shown in Fig. R2. To improve the signal-to-noise ratio in\nour measurements, we simply employ a large drive current, such as 160 nA, without\nany other specific intentions. For the gate bias, we typically select high bottom gate\nvoltages, such as 4 or 5 V, to minimize the contact resistance in 2D semiconductor\nMoS (Fig. R12) and ensure our measured drag response is reliable. Finally, we could\n2\nobserve the drag resistance variation as a function of carrier density (or V when the\ntg\nV is fixed), from which we were able to extract the corresponding drag resistance.\nbg\nIn this revision, we have added some explanation for the measured conditions in the\nMethods part of the manuscript, highlighted in blue in Page 10, quoted below:\n“Since the relationship between V and I is linear, we simply employed a large\ndrag drive\ndrive current, such as 100 - 200 nA, to improve the signal-to-noise ratio in our drag\nmeasurements without any other specific intentions.”\nComment 8: The contact resistance should always be specified, as it directly affects\nboth the drag and active-layer transport measurements.\nResponse:\nWe thank the reviewer for the suggestion that have helped to improve our manuscript.\nIt is true that the contact resistance in 2D semiconductor MoS is crucial for the drag\n2\ntransport measurements. We have performed both the two-probe and four-probe\nmeasurements of the MoS and graphene layer to extract the contact resistance as a\n2\nfunction of temperature and gate voltages, as shown in Figures R11 and R12. Note that\nthe contact resistance unit is not normalized by multiplying channel width due to the\nunconventional drag device geometry. The low contact resistances (V = 1 V, V = 5\ntg bg\nV) at low temperatures together with linear I-V curves in Figures R5 and R6 (Comment\n3) suggest that good Ohmic contacts for both MoS\n2\nand graphene are achieved, enabling the reliable execution of both the drag and\ntransport measurements.\nFigure R11. (a) Two-terminal resistance R , four-terminal resistance R and the\n2p 4p\nextracted contact resistance R for graphene layer in sample-S21 as a function of\nc\ntemperature. The contact resistance R for graphene layer in sample-S21 as a function\nc\nof V at (b) T = 20 K, (c) T = 50 K, and (d) T = 100 K, respectively. The contact\ntg\nresistance R for graphene layer in sample-S21 as a function of V at (e) T = 20 K, (f)\nc bg\nT = 50 K, and (g) T = 100 K, respectively.\nFigure R12. (a) Two-terminal resistance R , four-terminal resistance R and the\n2p 4p\nextracted contact resistance R for MoS layer in sample-S21 as a function of\nc 2\ntemperature. The contact resistance R for MoS layer in sample-S21 as a function of\nc 2\nV at (b) T = 20 K, (c) T = 50 K, and (d) T = 100 K, respectively. The contact resistance\ntg\nR for MoS layer in sample-S21 as a function of V at (e) T = 20 K, (f) T = 50 K, and\nc 2 bg\n(g) T = 100 K, respectively.\nWe have added Figures R11 and R12 as Supplementary Figures 8 and 9 in the revised\nversion, highlighted in blue in Pages 12 and 13, respectively.\nComment 9: The electrical bias conditions and temperatures differ across the figures.\nSuch variations must be discussed comprehensively before presenting the data, to\nensure consistent interpretation. Limited fundamental experimental results currently\nmake the discussion difficult to follow.\nResponse:\nWe thank the reviewer for the suggestion that have helped to improve our manuscript\na lot. To gain a better understanding of the distinct temperature-dependent behaviors of\nR and R observed when probing the electron compressibility in Figures 4 and 5,\nxx drag, xx\nwe remeasured the drag response as a function of temperature for various V values at\ntg\na fixed V = 5 V (consistent with the conditions in Figures 4 and 5 of the manuscript),\nbg\nwithout an applied magnetic field (as plotted in Figure R13). In consistent with the\nobservations at V = 4 V, the drag resistance follows the predicted T2 dependence of a\nbg\nFermi liquid at low temperatures, as shown in Figure R13a. As temperature increases,\nthis behavior transitions, becoming distinctly non- T2 and ultimately approaching a\nlinear T dependence. This crossover range is found to be widen as MoS is becoming\n2\ninsulating by reducing V . These various kinds of temperature-dependent behaviors are\ntg\nsummarized in the T - V - R phase diagram (Figure R13b). By featuring the fitting\ntg drag\npoints from the data in Figure 13a, we are able to define four distinct regimes within\nthe phase space: the T2 region, the T region, a T2 - T crossover region, and a region\nwhere no drag is observed.\nFigure R13. (a) Temperature dependence of R (colored open symbols) at V = 5 V\ndrag bg\nfor different V . The black dashed lines represent the fits to the low-temperature data\ntg\nwith a quadratic temperature dependence, while the black solid lines correspond to the\nfits to the high-temperature data, assuming a linear temperature dependence. (b) The\nblue, yellow, and purple filled areas show the R (T) responses with T-linear, T - T2\ndrag\ncrossover, and T2 behavior, respectively. Boundaries (blue and purple solid circles) are\nobtained by fitting the lines in (a). The open square symbols are the critical temperature\nT , which is defined as T = E /k k d with k , k and d being the Boltzmann constant,\nd d F B F B F\nFermi vector, and the interlayer distance, respectively.\nFor the consistency of temperature condition during the experiments, we remeasured\nthe carrier density as a function of V and V at T = 77 K (Figure R14), which is the\ntg bg\nsame as the temperature condition in Figures 4 and 5 in the main text.\nWe have added Fig. R13 to the new Fig. 2 in the revised main text, highlighted in blue\nin Page 4.\nComment 10: Both V and V values should be explicitly annotated in all subfigures\ntg bg\n(e.g., Fig. 1i).\nResponse:\nWe thank the reviewer a lot for the suggestion. The corresponding V and V values\ntg bg\nfor all measurement conditions has been annotated in the revised manuscript.\nComment 11: The meaning of the y-axis in Fig. 3d should be clearly defined.\nResponse:\nWe thank the reviewer for the suggestion. The y-axis in Fig. 3d is log\n10\n(|𝑅\ndrag\n|). To\nmake sure the consistent interpretation as discussed in comment 9 and 12, we measured\nthe carrier density and the corresponding drag response at 77 K, as shown in Figure\nR14. Meanwhile, in order to improve the clarity of the y-axis in Figure R14d, we have\nconverted both x-axis and y axis to a logarithmic scale.\nFigure R14. (a)-(b) The carrier density n of MoS and graphene in the drag system as a\n2\nfunction of V and V at T = 77 K. Data are obtained by using the formula n = B/eR ,\ntg bg H\nwhere e is the elemental charge and B/R is obtained by extracting the slope of Hall\nH\nresistance at B = 1 T and 0 T. (c) The differential carrier density δn plotted by\nsubtracting the color map (b) with (a). Notice that black dashed line indicates the\nscenario of matched-density between the graphene and MoS layer. (d) The map of\n2\ncorresponding drag resistance as a function of V and V at T = 77 K. (e) R plotted\ntg bg drag\nalongside the black dashed line in (d), which shows 1/n2 dependence in the matched-\ndensity drag, suggesting the validity of massive-massless Fermions drag in strong\ncoupling regime.\nFigure R14 is the modified version of the original Figure 3, which is highlighted in blue\nin Page 5 of the revised manuscript.\nComment 12: The carrier density measurements in Fig. 3 (at 100 K) and the Coulomb\ndrag data in Fig. 4 (at different temperatures) must be correlated. The Hall mobility and\ncarrier density should be presented as a function of both different temperature and bias\nfor consistency.\nResponse:\nWe thank the reviewer a lot for the suggestion that have helped to improve our\nmanuscript. To better understand the drag data in Fig. 4 and ensure consistency between\nFig. 3 and Fig. 4. We remeasured the carrier density as a function of gate voltage bias\nat T = 77 K, as shown in Figure R14(a) and (b). The carrier density of MoS and\n2\ngraphene for V = 1 V and V = 5 V at T = 77 K extracted from Fig. R14(a) and (b)\ntg bg\nare about 1.18 × 1012 cm-2 and 2.74 × 1012 cm-2, respectively, which agree well with\nthose from Fig. R10(a) and (b). The Hall mobility of these two layers as a function of\ntemperature and gate voltages are shown in Figures R10 and R15, respectively. The\nsignificant difference in Hall mobility - less than 103 cm2/V∙s for MoS and higher than\n2\n104 cm2/V∙s for graphene at low temperatures (below 77 K) under V = 1 V and V =\ntg bg\nSdH oscillations are easily detected in the graphene layer, yet the electron\ncompressibility of the MoS layer cannot be probed via transport measurements with a\n2\nhigh-mobility graphene.\nFigure R15. The Hall mobility as a function of V in (a) MoS and (b) graphene at V\ntg 2 bg\n= 5 V and T = 77 K.\nFigure R15 has been included in the revised Supplementary Figure 18 (highlighted in\nblue) to improve the consistency of presentation across Figures 4 and 5 in the main text.\nComment 13: In Fig. 1b, the scale bar indicates 10 mm, please confirm if this is a typo.\nResponse:\nWe thank the reviewer a lot for pointing out this mistake. The scale bar in Fig. 1b is\nindeed a typo and has been corrected to 10 μm in the revised manuscript.\nComment 14: In Fig. 1c, the data points corresponding to G for V = -3 V to -1 V\nMoS2 tg\nappear missing. In addition, including a semi-logarithmic scale would help reveal the\nn-type semiconducting behavior of bilayer MoS .\nin transport method to measure the resistance (or conductance) of the MoS layer, which\n2\nis different from a dc measurement. Due to the semiconducting nature of MoS ,\n2\nits\nresistance increases rapidly as the Fermi level approaches the band gap, as presented in\nFig. R16(a) and (b). This characteristic causes the lock-in amplifier’s phase to rise\nabove 10°, and in some cases even change sign (Fig. R16c). Therefore, we regard\nsignals acquired under such large phase shifts (＞±10°) as unreliable and treat them as\ninaccurate measurements of the intrinsic MoS resistance.\ntg\nof lock-in amplifier for the corresponding conductance signals in (a) or (b). The colored\narrows in these figures indicate the turning points beyond which the intrinsic\nconductance of MoS can be measured accurately.\nnew Supplementary Figure 10, highlighted in blue in Page 15.\nComment 15: In Fig. 1f, please define the parameter R . In Fig. 1h, define PVR\nMoS2\nexplicitly. Numerous typographical errors are present throughout the manuscript and\nshould be corrected carefully.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. R is defined as the lateral\nMoS2\nresistance of MoS layer. The PVR is one typo error, and should be the passive-to-\n2\nactive drag ratio (PADR), which is defined as I\ndrag\n/I\ndrive\n= R\ndrag\n/R\npassive layer\n. We have\ncarefully modified our manuscript to make it more accurate.\nComment 16: In Fig. S6c, the V -dependent MoS resistance behavior is nearly\ntg 2\nsymmetric and resembles that of graphene, suggesting ambipolar characteristics. Please\nelaborate on this observation.\nResponse:\nWe thank the reviewer for the comment. As we have already discussed in Fig. R16, the\n“ambipolar-like” transfer curves observed in MoS are mainly caused by spurious phase\n2\nsignals that appear when the Fermi level approaches the band edge of the two-\ndimensional semiconductor MoS₂ during the ac lock-in measurements (the same data\nis plotted in Fig. R17). Therefore, In Fig. R17(a), the accurate resistance of MoS is\n2\nrepresented solely by the signals on the right side of the resistance peak, as marked by\nthe colored arrows.\nFigure R17. (a) The field effect curves of MoS measured by the ac lock-in method at\ncorresponding resistance signals in (a). Since the data with a phase exceeding ± 10° are\ntypically considered unreliable, the intrinsic conductance of MoS can only be obtained\n2\nbeyond the colored arrows in (a).\nWe have incorporated Figure R17 in original Supplementary Figure 6 as the new\nSupplementary Figure 10, highlighted in blue in Page 15.\nComment 17: In Fig. S10a, GMoS measured at V = 4 V is lower than that at V = 1\n2 bg bg\nV (Fig. 1c), which is counterintuitive for n-type MoS . Please clarify this discrepancy.\nResponse:\nWe appreciate the reviewer a lot for pointing this out. The y-axis tick values contain a\ntypographical error in Supplementary Figure 10a. The number 80 should be 100, so that\nthe conductance of MoS at V = 4 V is larger than that at V = 1 V. The conductance\n2 bg bg\nof MoS versus V for different V is plotted in Figure R18, which clearly exhibits n-\n2 tg bg\ntype conduction.\nFigure R18. The comparison of the V\ntg\ndependence of GMoS2 , dG/dV\ng\nand R\ndrag\n. (a) The\nconductance of MoS\n2\n(GMoS2 ), (b) the MoS\n2\nchannel conductance derivative with\nrespect to gate voltage (dG/dV ), and (c) the drag resistance (R ) of the device as a\ng drag\nfunction of V\ntg\nat T = 200 K and V\nbg\n= 4 V. (d) GMoS2 , (e) dG/dV\ng\n, and (f) R\ndrag\nas a\nfunction of V at T = 200 K and V = 6 V. The green regions in the figures indicate the\ntg bg\nonset of dG/dV , within which the maximum value of R is found.\ng drag\nWe have already corrected this typographical error in the revised Supplementary Figure\n13, highlighted in blue in Page 18.\nComment 18: The thickness of the hBN spacer is a critical parameter for interlayer\ncoupling at the Gr/hBN/MoS interface. The electron compressibility as a function of\n2\nhBN thickness should be presented and discussed. In addition, please discuss the case\nwhere the positions of MoS and graphene are swapped. Why is MoS always placed\n2 2\non top of graphene? Addressing this question would strengthen the physical reasoning.\nResponse:\nWe thank the reviewer a lot for the suggestion that have helped to improve our\nmanuscript. We totally agree with the reviewer that the thickness of the hBN spacer d\nis a critical parameter for drag response. Accordingly, we have measured the dual-gate\nmapping of drag resistance for three typical samples with different interlayer spacing,\nas shown in Fig. R19(a) to (c). We could clearly see that the magnitude of drag\nresistance is inversely proportional to d, which is consistent with previous report\n(Nature Physics, 8, 896-901, 2012; Nature communications, 5, 5824, 2014). For better\ncomparison, we plotted the drag resistance as a function of carrier density for samples\nS21 and S15 at 77 K in Fig. R19(d). Note that we couldn’t extract the exact values of\nHall carrier density for sample S24 due to the insufficient electrodes. Moreover, we\nutilized drag and transport measurements at 77 K to probe the electron compressibility\nof graphene in two drag systems with two interlayer spacing d, as presented in Fig.\nR19(e). The carrier densities were fixed at 1.2×1012 cm-2 for MoS and 2.7×1012 cm-2\n2\nfor graphene, respectively. The system with the thinner spacer (sample S21) exhibited\nstronger interlayer coupling, resulting in a higher magnitude of oscillations in the drag\nsignal compared to the weaker interlayer coupling of the thicker spacer (sample S15).\nThis significant difference indicates that the thickness of the spacer is critically\nimportant for efficiently probing compressibility in the drag system.\nFigure R19. The drag resistance as a function of V and V for three samples, labeled\ntg bg\n(a) S21, (b) S24 and (c) S15, which feature different interlayer h-BN spacer thicknesses.\n(d) The drag resistance as a function of carrier density in MoS for samples S21 and\nresistance of graphene and drag resistance as a function of magnetic field between\nsamples S21 and S15. The data is measured at 77 K with the same fixed carrier density\nin both the graphene and MoS layer.\ndiscussion could be found in the revised manuscript, highlighted in blue in Page 9,\nquoted below:\n“The final section addresses the dependence of the drag response on the spacer\nthickness d, a key factor determining the strength of interlayer coupling within the drag\nsystem. We have measured the dual-gate mapping of drag resistance for three typical\nsamples with different interlayer spacing, as shown in Supplementary Figure 24(a) to\n(c). We could clearly see that the drag response is reproducible in different samples,\nand the magnitude of drag resistance is inversely proportional to d, which is consistent\nwith previous report3, 27. For better comparison, we plotted the drag resistance as a\nfunction of carrier density for samples S21 and S15 at 77 K in Supplementary Figure\n24(d). Moreover, we utilized drag and transport measurements at 77 K to probe the\nelectron compressibility of graphene in two drag systems with different d, as presented\nin Supplementary Figure 24(e). The carrier densities for this measurement were fixed\nat ∼ 1.2 × 1012 cm−2 for MoS and ∼ 2.7 × 1012 cm−2 for graphene, respectively. The\n2\nsystem with the thinner spacer (sample S21) exhibited stronger interlayer coupling,\nresulting in a higher magnitude of oscillations in the drag signal compared to the weaker\ninterlayer coupling of the thicker spacer (sample S15). This significant difference\nindicates that the thickness of the spacer is critically important for efficiently probing\ncompressibility in the drag system.”\nNow, we discuss why we always put MoS on the top of graphene. In order to achieve\n2\nwe have adopted window-contact technique as previously reported. We firstly etched\nthrough the windows on top h-BN and use it to pick up MoS and other layers\n2\nafterwards. Then, we adopted Bismuth with low work function as electrodes contacted\nto MoS and achieved Ohmic contact even at low carrier density and temperatures.\n2\ngraphene, as illustrated in Supplementary Figures 1 and 3. Although the fabrication\nwould be super challenging if the MoS and graphene layers were swapped, the drag\n2\nresponse would remain identical owing to the vertically mirror-symmetric\nconfiguration.\nWe have added some explanation in the revised main text and Methods part to address\nthis question, highlighted in blue in Pages 2 and 10, respectively, quoted below:\n“This requires the MoS to be the top layer in the heterostructure to facilitate the\n2\nfabrication process in this study.”\n“Note that we always put 2D semiconductor MoS on the top of graphene because we\n2\nadopted window-contact technique as previously reported31. We firstly etched through\nthe windows on top h-BN and use it to pick up MoS and other layers afterwards. Then,\n2\nwe adopted Bismuth with low work function as electrodes contacted to MoS and\n2\nachieved Ohmic contact at low carrier density and temperatures. Although the\nfabrication would be more challenging if the MoS and graphene layers were swapped,\n2\nthe drag response would remain identical due to the vertically mirror-symmetric\nconfiguration.”\nComment 19: The contact resistance for both MoS and graphene as a function of bias\n2\nand temperature should be presented to ensure accurate analysis of interlayer transport\nbehavior.\nResponse:\nWe thank the reviewer for the comment. We have performed both the two-probe and\nfour-probe measurements of the MoS and graphene layer to extract the contact\n2\nresistance as a function of temperature and gate voltages, as shown in Figures R11 and\nR12. The low contact resistances (V = 1 V, V = 5 V) at low temperatures together\ntg bg\nwith linear I-V curves in Figures R5 and R6 suggest that good Ohmic contacts for both\nMoS\n2\nand graphene are achieved, enabling the reliable execution of both the drag and\ntransport measurements.\nWe have added Supplementary Figures 8 and 9 in the revised Supplementary\nInformation to discuss the contact resistance of the both layers.\nComment 20: Since the effective mass of MoS is roughly an order of magnitude larger\n2\nthan that of graphene, it would be helpful to include an intuitive schematic illustration\nshowing how the Shubnikov–de Haas oscillations in MoS could be amplified or\n2\nmodulated through electron compressibility.\nResponse:\nWe thank the reviewer a lot for the suggestion. The MoS layer exhibits low electron\n2\nmobility due to its high electron effective mass and high concentration of defects and\ndisorders. As schematically illustrated in Fig. R20, the low mobility establishes a\ndiffusive transport model in MoS with a magnetic field applied. Electrons move\n2\nrandomly between scattering disorders, characterized by non-circular, jagged paths in\nthe top layer. This diffusive conducting model prevents the observation of significant\nSdH oscillations under 12 T, meaning the electron compressibility in MoS is nearly\n2\nflat as a function of the magnetic field.\nIn stark contrast, high-mobility graphene enters the quantum regime under low\ntemperatures and high magnetic fields. Electrons primarily follow cyclotron orbits\nperpendicular to the magnetic field (in the bottom layer). In other words, the electron\nstates could be easily quantized into distinct Landau Levels (LLs). Consequently, the\nelectron compressibility in graphene oscillates with the magnetic field for a fixed carrier\ndensity, allowing for the observation of evident SdH oscillations. However, as the\ntemperature increases, transport across the sample occurs via inter-orbit scattering (the\ndashed lines connecting the orbits), which is a diffusive process, leading to the\namplitude of the SdH oscillations to decrease until they vanish.\nBut for the case of drag system, the amplitude of magneto-drag resistance oscillations\nis significantly amplified by the screening interlayer interaction |𝑈 |2, and the\n12\ntemperature dependence is extremely different from single-layer transport, allowing the\noscillations to remain visible even at high temperatures. Furthermore, the flat electron\ncompressibility in MoS confirms the origin of the observed oscillations in R is\n2 drag\nsolely from the graphene layer, suggesting that MoS serves as an excellent sensor of\n2\nprobing electron compressibility in high mobility materials.\nFigure R20. The schematic illustration of diffusive conductive model in low mobility\nMoS layer with flat electron compressibility and quantized electron states in high\n2\nmobility graphene layer with oscillatory electron compressibility.\nIn this revision, we have incorporated Figure R20 in Figure 5, and added the\ncorresponding description of this schematic for better understanding the asymmetric\ndrag system, highlighted in blue in Page 9.\nIn summary, while the manuscript presents an interesting approach to probing\ncompressibility in two-dimensional systems via Coulomb drag, it currently lacks\nstructural, electrical, and conceptual consistency. The figures require systematic\nreorganization, and several critical parameters (bias, temperature, and layer identity)\nmust be clarified before the study can provide a coherent physical picture. If the\nmanuscript provides a more detailed discussion of the fundamental charge transport\ncharacteristics in each layer, a deeper and more quantitative review of the Coulomb\ndrag phenomena would be possible.\nResponse:\nWe are very grateful for Referee#2’s positive assessment and kind comments for our\nmanuscript. By adding a more detailed discussion of the fundamental charge transport\ncharacteristics in each layer, the revised manuscript indeed provides a deeper\nunderstanding of the novel drag phenomenon. Her/his constructive suggestions have\nstrengthened the overall quality of our work. We sincerely appreciate the reviewer’s\npositive evaluation regarding the novelty and significance of our work.\n----------------------------------------\nReply to reviewer #3\nGeneral Comment:\nThe manuscript describes experimental measurements of Coulomb drag between MoS2\nand monolayer graphene separated by few-nm thick hBN, primarily conducted by\ndriving a current in the graphene and measuring the induced voltage in the MoS2. The\nauthors investigate the dependence of the drag signal on temperature, carrier density\nand magnetic field and find substantial similarities to the available theoretical\npredictions. In an applied perpendicular magnetic field, both the graphene layer and the\ndrag resistance show Shubnikov-de Haas oscillations. The magnitude of the oscillations\nrapidly decreases in the graphene with increasing temperature, while remaining\nrelatively constant in the drag signal, allowing the drag resistance to act as a sensor for\notherwise-inconspicuous electronic transport behavior in the graphene.\nThe authors effectively introduce the ideas behind the measurement and the potential\nsignificance of their results, without making overstated claims. These experiments are\na nice addition to the existing body of work on Coulomb drag, and contribute new\ninformation that will be of interest to the field. The fabrication methods are consistent\nwith the current standard for van der Waals heterostructure devices and are well-\nexplained, and they clearly demonstrate their efforts to avoid spurious drag signals in\nthe measurement by using a voltage-balancing bridge. There is sufficient information\ngiven in the text and supplemental material to replicate the experiment.\nResponse:\nWe thank the reviewer a lot for her/his positive assessment of our work. Below, we\nprovide detailed responses to each of the reviewer's comments in a point-by-point\nmanner.\nComment 1: While the authors show Onsager reciprocity (driving one layer and\nmeasuring the drag voltage across the other, switching the drive and drag layers) in\nmagnetotransport, it would strengthen the paper to show Onsager reciprocity in other\nregimes, particularly outside of the Fermi liquid drag regime.\nResponse:\nWe thank the reviewer a lot for the suggestion. We present here additional data\ncomparing the drag response by switching the drive and drag layers. Fig. R21(a) and\n(b) show the color plots of magneto-drag resistance versus magnetic field at different\ngate voltages and T = 200 K, using the MoS as the drive layer and graphene as the drag\n2\nlayer respectively. The Onsager reciprocity has been confirmed valid even though the\ndrag transport is in the non-Fermi liquid regime. In addition, we have measured the\nmagneto-drag response for reciprocal drag circuit configurations at the same gate\nvoltages, as presented in Fig. R21(c) and (d). The general degree to which the two data\nsets coincide measurements provide strong evidence that Onsager reciprocity is valid\nfor our devices in either Fermi liquid or non-Fermi liquid regime.\nFigure R21. The drag resistance versus magnetic field for varying V when (a) MoS\ntg 2\nand (b) graphene serving as the drive layer, respectively. The drag resistance as a\nfunction of magnetic field at the same gate voltage (V = 4.1 V and V = 1 V) but\nbg tg\ndifferent temperatures, for reciprocal layer configurations: (c) MoS drag graphene and\n2\n(d) graphene drag MoS . The curves in (c) and (d) is vertically shifted for better\n2\nvisualization.\nWe have added Figure R21 as Supplementary Figure 20 in the revised Supplementary\nInformation, highlighted in blue in Page 25.\nComment 2: It would also be beneficial to show data from additional devices; there\nare 9 devices described as having been measured, but data from only 2 or 3 of them is\nshown. The reproducibility of the results would be more convincing with some more\nextensive characterization of multiple devices.\nResponse:\nWe thank the reviewer a lot for her/his valuable suggestion. Note that not all these\ndevices survive the process or retain sufficient functional electrodes for subsequent\nmeasurements. The reproducible drag data of new samples could be found in the Fig.\nR19. As for the results of probing electron compressibility by drag measurements, Fig.\nR22 shows the longitudinal channel resistance of graphene and the magneto-drag\nresistance data from another typical device sample S24 for an additional comparison.\nThe geometry of sample S24 is the same as sample S21. In order to obtain good Ohmic\ncontact of MoS in the devices, we applied a large V (7.5 V), and measure the R of\n2 bg xx\ngraphene as a function of magnetic field for different temperatures. One could observe\nevident oscillations of R in graphene at low temperatures in Fig. R22(a). Apparently,\nxx\nthe amplitude of these Shubnikov–de Haas oscillations diminishes as the temperature\nrises, leading to a smooth resistance background when the temperatures exceed 100 K.\nFor comparison, we have plotted the R for reciprocal drag circuit configurations\ndrag, xx\nunder the same measurement conditions as the graphene channel, as shown in Fig.\nR22(b) and (c). Notably, the oscillation amplitudes in the drag response do not vanish\neven as the temperature increases to 140 K, confirming our main finding: the\nreproducible persistence of SdH-like oscillations in the drag signal even when\ngraphene’s own transport is featureless. The results are fully consistent with those of\nsample S21 in the main manuscript.\nFigure R22. (a) R of graphene as a function of magnetic field for different\nxx\ntemperatures. R as a function of magnetic field for different temperatures, where\ndrag, xx\n(b) MoS and (c) graphene serve as the drive layer. All data are taken at V = 7.5 V and\n2 bg\nV = 1 V.\ntg\nIn this revision, we have added Figures R19 and R22 as Supplementary Figure 24 and\n23, respectively, in the updated Supplementary Information. These figures present the\nreproducible results obtained from samples other than Sample S21, which is primarily\ndiscussed in the main text.\nComment 3: A few minor details: There is a typographical error in the legend of Figure\n1h, which refers to “PVR” instead of “PADR.” Also, the text refers to red arrows in\nFigure 1g-h that do not appear to be present.\nResponse:\nWe appreciate the reviewer for pointing these out. We have corrected these\ntypographical errors and modified the figures in the revised manuscript.\nOverall, I think this manuscript is suited for publication with minor changes.\nResponse:\nWe sincerely thank Referee#3 for the thorough and insightful comments. The revisions\naccording to the reviewer’s suggestions have improved the quality of our manuscript a\nlot. We appreciate her/his time in evaluating our study and support of publication in\nNature Communications.\nAuthor Responses to Initial Comments:\nReply to reviewer #1\nComment:\nThe authors have adequately addressed my comments. However, the unique role of\nMoS in the observation of pronounced SdH oscillations in the drag signal is not\n2\nsufficiently elaborated. It is recommended to strengthen the relevant discussion. For\nexample, if MoS layer is replaced with graphene or other 2D materials, can similar\n2\nphenomena be observed?\nResponse:\nWe thank the reviewer a lot for the comment that have helped to improve our\nmanuscript. It is a great question that whether similar phenomena could be observed\nwhen the MoS is replaced by other 2D materials.\n2\nconsequence of replacing MoS with other 2D materials in the Page 9 of the main text,\n2\nhighlighted in blue, quoted below:\n“The unique role of MoS in the observation of pronounced SdH oscillations is of two\n2\norigins, i.e., relatively flat electron compressibility in its own, and sufficiently strong\nCoulomb interaction factor |𝑈 |. In this regard, if MoS in the current system is\n12 2\nreplaced by other 2D semiconductor with flat electron compressibility - such as MoSe\n2\nand WSe , provided that Ohmic contacts could be obtained at low temperatures - should\n2\ntheoretically exhibit similar phenomena. Otherwise, for instance, replacing MoS with\n2\ngraphene, high screening in both layers will lead to a dramatic reduction of interlayer\nCoulomb interaction, and their SdH signals tend to be entangled, complicating the\ndecoupling of layer-specific information.”\nWhile that SdH oscillations in the drag signal more evident than intrinsic transport at\nelevated temperatures.\nReply to reviewer #2\nGeneral Comment:\nI appreciate the authors' substantial efforts to address my previous concerns, ranging\nfrom material characterization to in-depth analysis. While a few minor points remain,\nthey do not affect the overall quality or validity of the work. I therefore agree that the\nmanuscript can be accepted for publication in its present form.\nResponse:\nWe are deeply grateful for her/his support in the acceptance of this work by Nature\nCommunications.\nReply to reviewer #3\nGeneral Comment:\nThe authors have made substantial revisions to their manuscript to address the\ncomments and concerns of the reviewers. In this paper, the authors describe\nexperimental signatures of Coulomb drag between monolayer graphene and bilayer\nMoS separated by a few-nanometer-thick flake of hexagonal boron nitride. They argue\n2\nthat the MoS acts as a sensor of the electronic compressibility of graphene,\n2\ndemonstrated by the drag signal exhibiting oscillations that track the Shubnikov de\nHaas oscillations of the graphene layer in a finite perpendicular magnetic field. The\nexperimental results are interesting and generally support the claims made in the paper.\nResponse:\nWe greatly appreciate the reviewer for the positive assessment of our study. Below are\nthe point-by-point responses to the reviewer’s comments.\nComment 1:\nIn the revised manuscript, the authors have satisfactorily addressed my specific\ncomments regarding typos and the importance of showing data from additional devices.\nThey have also provided significantly more characterization of the transport\ncharacteristics of individual layers and some proof of Onsager reciprocity. It would be\nmore convincing to also show a full map of drag resistance as a function of bottom and\ntop gate voltages for both graphene and MoS as the drive layers, rather than just three\n2\nlinecuts at constant bottom gate voltage shown in Figure R4/S16 and the separated\ncurves of drag resistance versus magnetic field in Figure R21/S20. Similar data are\nalready shown in Figure S21 and are useful in seeing the correspondence between the\ntwo measurement configurations.\nResponse:\nWe thank the reviewer a lot for her/his suggestion. Here we have included the full maps\nof drag resistance of sample-S21 as a function of bottom and top gate voltages as shown\nin Figure R1, with graphene and MoS serving as the drive layer, respectively.\nthe graphene layer and (b) the MoS layer serving as the active layer, respectively.\nthe lock-in amplifier, as seen in Supplementary Figure 12. (c) The schematic of Onsager\nrelation with the active (passive) layer alternated in the drag system. (d) The drag\nresistance as a function of V at different V , measured at T = 200 K. The red and blue\ntg bg\ncircles represent serving MoS and graphene as the active layer, respectively.\nresistance as a function of bottom gate and magnetic field with graphene and MoS\n2\nserving as the drive layer, respectively. The general high degree of correspondence\nbetween the R maps for different drive layers (Figures R1 and R2) provides strong\ndrag\nevidence that Onsager reciprocity is respected in this regime.\nFigure R2. Dual-gate maps of the drag resistance of sample-S24 at T = 200 K with (a)\nthe graphene layer and (b) the MoS layer serving as the drive layer. The drag resistance\n2\nversus magnetic field for varying V when (c) graphene and (d) MoS serving as the\nbg 2\ndrive layer, respectively. The drag resistance as a function of magnetic field at the same\ngate voltage (V = 4.1 V and V = 1 V) but different low temperatures, for reciprocal\nbg tg\nlayer configurations: (e) graphene drag MoS and (f) MoS drag graphene. The curves\n2 2\nin (e) and (f) is vertically shifted for better visualization. Note that the curves in (e) and\n(f) for reciprocal configurations do not overlap perfectly; this deviation may be induced\nby disorder within the device.\nFor better showing the Onsager reciprocal relation in our drag systems, we have added\nthe full maps of drag response for reciprocal configurations in Supplementary Figures\n16 and 20, highlighted in blue in Pages 21 and 25.\nComment 2:\nFurthermore, there are clear differences between the graphene-drive and MoS -drive\n2\ncurves in Figure R21(c/d) at several temperatures. The curves are qualitatively similar,\nbut this discrepancy should be addressed.\nResponse:\nWe thank the reviewer a lot for her/his comment that have helped to improve our\nmanuscript. The reviewer correctly notes that while the trends in Figure R21 (c) and (d)\nare highly similar, the curves do not overlap perfectly. While the precise origin of this\ndiscrepancy remains under investigation, the deviation from ideal reciprocity may stem\nfrom disorder within specific devices, which can contribute a small rectified component\nto the drag signal under low temperatures and magnetic fields.\nWe have addressed the discrepancy regarding the reciprocal relation in the main text\n(Page 8) and the caption of Figure S20 in the revised Supplementary Information (Page\n26), highlighted in blue.\n“The Onsager reciprocity relation for magnetodrag resistance remains nearly valid\nregardless of whether the system is within or outside the Fermi liquid regime (as shown\nin Supplementary Figure 20 and 21). It is noteworthy that a slight departure from perfect\nreciprocity occurs at low temperatures and under magnetic fields. While the\nmicroscopic origins of this non-ideal reciprocal Coulomb drag remain unclear, they\nmay stem from a high concentration of disorder in specific devices, which can introduce\na small rectified component to the drag signal.”\n“Note that the curves in (e) and (f) for reciprocal configurations do not overlap perfectly;\nthis deviation may be induced by disorder within the device.”\nOverall, I think there are still a few changes that should be made, but the scientific work\nseems solid and the manuscript is suitable for publication with some modifications.\nResponse:\nWe are very grateful for the reviewer’s positive assessment and kind comments for our\nmanuscript. Her/his constructive suggestions have strengthened the overall quality of\nour work. We sincerely appreciate the reviewer’s positive evaluation regarding the\nnovelty and significance of our work."}]

Decision letter