Main

The observation of quantum oscillations in magnetization or resistivity in insulators1,2,3,4,5,6,7,8,9,10 has intrigued the condensed-matter physics community because it is long believed that the phenomenon could occur only in metals with a well-defined Fermi surface. Many theoretical proposals have been put forth to explain the observation, including magnetic-field-induced gap oscillations in both single-particle and correlated insulators11,12,13,14,15,16,17,18,19,20,21,22, competing many-body ground states as a function of the magnetic field18,19,21,22 and an exotic fractionalization scenario with neutral fermions and gauge fields23,24. So far, the mechanism responsible for oscillations in different material systems remains under debate, especially for strongly correlated systems. Limits on the ability to control the properties of many of the bulk correlated materials have hindered the development of a cohesive understanding. The emergence of highly tunable correlated insulating states in van der Waals’ heterostructures provides an opportunity to study this problem from a fresh perspective.

In this work, we report the observation of quantum oscillations in the resistivity, exciton binding energy and exciton current in a dipolar excitonic insulator (EI)25,26,27,28,29,30,31,32. This is achieved through a suite of magnetotransport, capacitance and drag counterflow measurements performed on Coulomb-coupled MoSe2/WSe2 electron–hole double layers. The quantum oscillations originate from recurring transitions between competing EI and layer-decoupled quantum Hall (QH) states as a function of the magnetic field and electron–hole pair density19,21,22. Compared with earlier reports on quantum oscillations in EI candidate InAs/GaSb quantum wells6,7,8,33,34, the ability to electrically tune and address the electron and hole layers separately has allowed us to establish the emergence of quantum oscillations in an EI.

Electron–hole double layer

Figure 1a shows a schematic of the cross-section of a dual-gated electron–hole double-layer device. The double layer consists of a natural MoSe2 bilayer and a WSe2 monolayer separated by a thin hexagonal boron nitride (hBN) barrier (5–6 layers thick; Extended Data Fig. 1; the MoSe2 bilayer is more robust for fabrication but behaves practically like a monolayer because electrons are polarized to one layer under a high perpendicular electric field in this study). The MoSe2/WSe2 heterobilayer has a type-II band alignment26,27,28,29,30,32 (Fig. 1b), with the conduction (valence) band minimum (maximum) residing in the Mo layer (W layer). The two gates allow for independent control of the net electron doping density (n) in the double layer and the electric field (E) perpendicular to the sample plane through the symmetric (Vg) and antisymmetric (Δ) parts of the two gate voltages (for symmetric gates), respectively. We apply a constant electric field to reduce the heterobilayer energy gap (from Eg ≈ 1.6 eV (refs. 27,29,32) to Eg ≈ 0.625 eV in device 1). We also apply a bias voltage (Vb) between the Mo and W layers to separate the electron and hole Fermi levels. When the effective charge gap of the double layer (Eg – eVb) becomes smaller than the interlayer exciton binding energy (Eb), the double layer is spontaneously populated by interlayer excitons, forming an EI with a net electric polarization35,36,37,38 (that is, a dipolar EI). Here e is the elementary charge. The electrons in the Mo layer and holes in the W layer are coupled only through the interlayer Coulomb interactions. With negligible single-particle interlayer tunnelling (verified by negligible interlayer tunnelling currents (Extended Data Fig. 2)), the excitons in the double layers are in thermal equilibrium with an exciton chemical potential (eVb – Eg). Note that the EI here refers only to a charge insulating state and does not imply spontaneous interlayer coherence.

Fig. 1: Coulomb-coupled MoSe2/WSe2 double layer.
figure 1

a, Device schematic. Interlayer bias voltage Vb (with the W layer grounded), Vg and Δ control the exciton chemical potential, net charge density and out-of-plane electric field in the double layer, respectively. Interlayer excitons (oval) are stabilized at B = 0 T and low electron–hole pair densities. Blue, WSe2 monolayer; orange, MoSe2 bilayer; light blue, hBN; grey, graphite. b, Band alignment under B = 0 T or for the exciton binding energy Eb far exceeding the cyclotron energy Ec (layers not angle aligned). Conduction bands are shown in orange, and valence bands, in blue. Arrows, spin states; dashed lines, Fermi levels. Interlayer excitons spontaneously form when Eb exceeds the effective charge gap (Eg – eVb). c, Same data as a but for high magnetic fields and high pair densities. Layer-decoupled QH states with counterpropagating electron and hole chiral edge states emerge. d, Same data as b but for Eb ≤ Ec. Electron and hole LLs (horizontal solid lines) are spin-split by the Zeeman effect. e,f, Schematic of the open-circuit measurements of Rxx and Rxy in the W layer (e) and drag counterflow measurements (f), where IW and VW in e denote the current and induced voltage, respectively, and Idrive and Idrag in f are the drive current in the Mo layer and the induced drag current in the W layer, respectively. g, Rxx as a function of Vg and Vb at B = 0 T and T = 1.5 K. The dashed lines separate the EI and different doping regions with i, p and n, denoting the intrinsic, hole-doped and electron-doped layers, respectively. In the grey area, Rxx diverges and cannot be reliably determined.

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In the presence of a strong perpendicular magnetic field B, Landau levels (LLs) emerge in the electron and hole layers (Fig. 1c,d). When the electron–hole pair density (np) exceeds the exciton Mott density (nM), the excitons dissociate into an electron–hole plasma27,29,35,36, and layer-decoupled QH states are expected. On the other hand, for low pair densities and low magnetic fields, when the electron–hole cyclotron energy (Ec) is comparable to Eb, EI is expected to dominate (the electron and hole cyclotron energies are comparable in MoSe2/WSe2). In the interesting regime of np ≈ nM and Ec ≈ Eb, it was suggested in refs. 21,22 that recurring transitions between the EI and layer-decoupled QH states could emerge as a function of np and B. Unlike the interlayer-coherent exciton condensates in half-filled LLs in electron–electron or hole–hole Coulomb-coupled bilayers39,40,41,42,43, in which the condensates carry chiral edge states (that is, they are QH ferromagnets) and are unstable in the zero-field limit21, the EI in the electron–hole double layers is stable in the zero-field limit and does not support chiral edge states because interlayer coherence is expected to gap out the counterpropagating edge states21,22.

We probe the double layer under finite magnetic fields by patterning it into a Hall bar device with independent electrical contacts to the Mo and W layers. Because the electron contacts formed via bismuth evaporation in the Mo layer are less reliable, we measure the four-terminal longitudinal (Rxx) and Hall (Rxy) resistances in the W layer and keep the Mo layer in the open-circuit condition (Fig. 1e and Extended Data Fig. 3). As demonstrated by a recent study32, Rxx and Rxy are highly sensitive to interlayer Coulomb interactions. This measurement is supplemented by the drag counterflow measurement (Fig. 1f), which has been shown to directly probe the interlayer Coulomb interactions29,30,41 and the penetration capacitance measurement27, which accesses the exciton binding energy in the EI27,38. Methods provides details on the device fabrication and electrical transport and capacitance measurements. Unless otherwise specified, the sample temperature is T = 1.5 K.

Resistivity oscillations in EI

Figure 1g shows Rxx of the W layer as a function of Vg and Vb at B = 0 T (device 1), where the gate voltage Vg tunes the net doping density n in the double layer and the bias voltage Vb mainly controls the electron–hole pair density np (ref. 27). The electrostatic phase diagram shows i–i, i–n, p–i and p–n regions for the (W, Mo) layers, where i, p and n denote an intrinsic (or charge-neutral), hole-doped and electron-doped layer, respectively. As expected, Rxx diverges at low temperature in the i–i and i–n regions (grey shaded); the W layer is conducting only in the p–i and p–n regions (the p–i|i–i boundary is pinned to Vg = 0 because the W layer is grounded). Rxx also diverges in the EI region (the triangular i–i region that protrudes into the p–n region), in which a hole current in the W layer is not allowed because the electrons and holes are bound and the electron layer is in the open-circuit condition32. At the tip of the EI triangle, the pair density reaches the exciton Mott density, and the interlayer excitons dissociate into an electron–hole plasma27,29. The results are fully consistent with earlier studies27,29,32.

Next, we examine Rxx and σxy (Hall conductivity) as a function of Vg and Vb at B = 12 T (Fig. 2a,b). Results at other magnetic fields up to 31.5 T are shown in Extended Data Fig. 4. Hole LLs are observed in the p–i and p–n regions. Each fully filled LL is characterized by an Rxx dip and a nearly quantized σxy at \(\frac{{\nu }_{\rm{h}}{e}^{2}}{2\uppi \hslash }\) (see the line cuts in Fig. 2d). Here Vb(e) is the hole (electron) LL filling factor and ℏ is the reduced Planck constant. In the p–i region, the hole LLs form vertical stripes because the hole density is independent of Vb. In the p–n region, they form diagonal stripes because the doping density in the W layer (and the Mo layer) changes with both Vg and Vb. The electron LLs in the Mo layer are nearly invisible in Rxx and σxy of the W layer; they are better resolved in the drag counterflow (Fig. 5) and capacitance (Extended Data Fig. 5) measurements.

Fig. 2: Quantum oscillations in the EI phase.
figure 2

a,b, Rxx (a) and σxy (b) of the W layer (with the Mo layer in the open-circuit configuration) as a function of Vg and Vb at B = 12 T and T = 1.5 K. The black and white dashed lines are the phase boundaries from the data for B = 0 in Fig. 1a. The EI phase is shifted and expanded by the magnetic field. Quantized \({\sigma }_{{xy}}=\frac{{\nu }_{\rm{h}}{e}^{2}}{h}\) for νh = 1–6 is identified in b. In the grey area, Rxx and σxy cannot be reliably measured because of the diverging Rxx. c–e, Rxx (left axis) and σxx and σxy (right axis) along the solid yellow (c), cyan (d) and purple (e) lines in a. The QH states are characterized by Rxx and σxx dips and quantized σxy (horizontal dashed lines). The EI states are characterized by Rxx peaks and suppressed σxx and σxy. Blue arrows identify strong EI states between two integer QH states.

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The most interesting features in the Rxx and σxy maps are located near the EI triangle of the phase diagram. At the bottom of the triangle (0.60 ≲ Vb ≲ 0.64 V), where the exciton binding energy is large, the diagonal LL stripes are interrupted by the EI with diverging Rxx and vanishing σxy (only at the highest magnetic fields (Extended Data Fig. 4), the LL stripes penetrate through the EI). Near the tip of the triangle (0.64 ≲ Vb ≲ 0.66 V), where the exciton binding energy is small, a fully filled LL can nearly penetrate through the EI without interruption even under moderate magnetic fields; on the other hand, a half-filled LL (one with an Rxx peak) transitions to the EI with much higher Rxx and strongly suppressed σxy.

Figure 2c–e shows three representative line cuts along the solid lines shown in the maps (the extracted longitudinal conductivity σxx is also included). Only a series of QH states are observed away from charge neutrality at νe − νh = 3 (Fig. 2d). At charge neutrality (Fig. 2c), the transition from the νh = 6 QH state (Vb ≈ 0.662 V) to the νh = 5 QH state (Vb ≈ 0.645 V) is interrupted by a weak EI near Vb ≈ 0.653 V; it shows enhanced Rxx and suppressed σxy and σxx (on top of the wide peak). Only the EIs with diverging Rxx and vanishing σxx and σxy are stable (Vb ≲ 0.64 V). In Fig. 2e, an EI with a sharp dip in σxx near Vg ≈ –0.01 V, which interrupts the transition from the νh = 5 QH state (Vg ≈ –0.05 V) to the νh = 4 QH state (Vg ≈ 0.03 V), is also illustrated by a line cut at constant Vb ≈ 0.64 V.

We examine the magnetic-field dependence of Rxx at charge neutrality in Fig. 3a (along the yellow solid line (Fig. 2a)) up to B = 31.5 T at 0.3 K. With increasing field, the EI region expands. On one hand, the exciton Mott transition or the metal–insulator transition (black dashed line) moves to higher Vb in a hyperbolic fashion due to exciton diamagnetic shift44. On the other hand, the onset for exciton injection (white dashed line) shifts linearly to lower Vb due to the exciton Zeeman effect. We obtain an interlayer exciton g-factor of ~20, in agreement with earlier optical studies45. LLs emerge in the electron–hole plasma region for B ≳ 3T. They disperse with Vb slightly nonlinearly because the interlayer capacitance depends on Vb (ref. 27). At high fields, the EI phase with large Rxx is interrupted by LLs with Rxx dips (νe = νh = 1, 2, 3 and 4). The onset field for the LLs decreases with increasing νe or νh. The distinct horizontal stripes in the phase diagram are caused by the LLs in the graphite gates, which are capacitively coupled to the electron–hole double layer9,10; they are independent of Vb because the W layer is grounded.

Fig. 3: Magnetic-field phase diagram at charge neutrality.
figure 3

a,b, Rxx of the W layer (device 1; a) and drag ratio \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) (device 3; b) as a function of Vb and B at charge neutrality and T = 0.3 K. The white dashed line denotes the band edge, and the black dashed line separates the EI and the electron–hole plasma phases. In the grey area (left of the white line), Rxx diverges and cannot be reliably measured. The EI phase with diverging Rxx and large \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) expands with the magnetic field. The fully filled LLs with vanishing Rxx and \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) and (νe, νh) = (1, 1), (2, 2), (3, 3), (4, 4) protrude into the EI phase. c, Mean-field phase diagram in exciton chemical potential (eVb – Eg) and B at charge neutrality (Methods). White region, EI; grey, layer-decoupled QH states.

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We further study the temperature dependence of the quantum oscillations at constant B = 12 T. Figure 4a shows Rxx as a function of Vb and temperature. Figure 4c displays line cuts at the representative temperatures. Figure 4c,d also includes σxx and σxy (Extended Data Fig. 6 shows additional data). Recurring transitions between the EI and QH states are observed as Vb increases. As the temperature decreases, Rxx increases, and σxx and σxy decrease for the EI states; Rxx and σxx decrease, and σxy approaches a quantized value for the QH states. At the boundary separating the two states, these quantities are nearly temperature independent. The temperature dependence of the oscillation amplitude of Rxx is analysed in Extended Data Fig. 7. It follows the Lifshitz–Kosevich formula for the QH states and substantially deviates from this formula for the EI states. Quantum oscillations are also observed in the penetration capacitance, from which we extract the exciton binding energy as a function of Vb (Methods and Extended Data Fig. 5). The results from two different devices (devices 1 and 2) in Fig. 4b are similar. The binding energy oscillates as a function of Vb on a monotonic decreasing background; the local minimum corresponds to a QH state.

Fig. 4: Oscillating exciton binding energy (B = 12 T).
figure 4

a, Rxx of the W layer as a function of Vb and T at charge neutrality. b, Exciton binding energy as a function of Vb (bottom axis for device 1 and top axis for device 2). It oscillates on a monotonically decreasing background. The horizontal dashed line marks the cyclotron energy Ec (comparable for electrons and holes). c,d, Rxx (left axis; c), σxx (right axis; c) and σxy (d) versus Vb at varying temperatures (1.5 K to 14.5 K in 1-K step) for device 1. The dashed lines in d denote the quantized values. As Vb increases, the EI state (with increasing Rxx and decreasing σxx and σxy) transitions into the QH states (with decreasing Rxx and σxx and quantized σxy) at low temperatures. The fully filled LLs (νe and νh) are labelled in a and denoted by the vertical purple lines in b–d.

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The results above demonstrate transitions tuned by the magnetic field and pair density (controlled by Vb) between the EI and QH states, which give rise to the observed quantum oscillations in the EI region of the phase diagram. Eb decreases continuously with pair density due to screening27,29,46, whereas Ec increases with B. When Eb becomes comparable to or smaller than Ec, the EI transitions to layer-decoupled QH states when both layers are in the fully filled LLs21,22. The EI phase returns when the LLs are partially filled until the pair density exceeds the Mott density. Beyond the Mott density, excitons dissociate into an electron–hole plasma and only layer-decoupled LLs are present. In this picture, the frequency of the quantum oscillations in 1/B is set by the size of the electron–hole Fermi surface (Extended Data Fig. 8).

We perform mean-field calculations for the ground-state energy density of the electron–hole double layer21 to compare with experiment (Methods). The theoretical phase diagram (Fig. 3c) qualitatively captures the experimental phase diagram (Fig. 3a). A fan of layer-decoupled QH states at integer νe (=νh) merges at high magnetic fields, and their onset field decreases with increasing νe or νh. However, the scale of the magnetic field in the mean-field result is substantially higher because the density-dependent exciton binding energy due to screening and the associated exciton Mott transition have not been taken into account. Future studies including these effects are required to quantitatively describe the experimental results.

Quantum oscillations in Coulomb drag

Finally, we perform the drag counterflow measurements to further examine the EI-to-QH transitions. We drive an a.c. bias current (Idrive) in the Mo layer and measure the drag current (Idrag) in the W layer29 (Fig. 1f), keeping interlayer tunnelling negligible (Extended Data Fig. 3). The drag ratio \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) at low temperature, as shown for device 3 in Fig. 3b, provides a measure for the interlayer Coulomb coupling29. In particular, near-perfect Coulomb drag (\(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\approx 1\)) is observed for the EI phase; near-zero drag is observed for the layer-decoupled QH states (νe = νh = 2, 3 and 4) and the electron–hole plasma. The results are fully consistent with the phase diagram inferred from Rxx in Fig. 3a.

We further examine Coulomb drag in the presence of an electron–hole density imbalance. Figure 5a,b shows Idrive and Idrag in the Mo and W layers, respectively, as a function of Vg and Vb for device 2 at B = 12 T. The Mo layer is turned on electrically only in the i–n and p–n regions. A drag current in the W layer with nearly identical magnitude but opposite sign with respect to Idrive is observed in the triangular EI region. Quantum oscillations are observed in Idrag near the EI phase boundary and beyond. Figure 5c shows line cuts for the currents together with their ratio along the νe = νh and νe − νh = –1 lines (yellow and blue, respectively; Fig. 5a). Near-perfect Coulomb drag is observed at νe = νh for Vb ≲ 0.55 V; the drag ratio drops sharply and exhibits quantum oscillations with increasing Vb. A large but imperfect drag (with ratio up to about 0.5) is observed at νe − νh = –1 near the onset of electron–hole injection; the drag ratio also drops sharply and shows quantum oscillations with increasing Vb.

Fig. 5: Finite-angular-momentum excitonic correlations.
figure 5

a,b, Drive current (Idrive) in the Mo layer (a) and drag current (Idrag) in the W layer (b) as a function of Vg and Vb at B = 12 T and T = 1.5 K. Near-perfect Coulomb drag is observed in the EI region shown schematically by the black dashed lines (weaker states above the triangle are not included at this temperature). Substantial Coulomb drag is also observed outside the EI region with partially filled LLs, indicating the presence of excitonic correlations with finite angular momentum. c, Line cuts in a and b (left axis) and drag ratio \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) (right axis) along the solid yellow line with νe = νh (top) and cyan line with νe − νh = –1 (bottom) in a. An enhanced drag ratio indicates stronger interlayer excitonic interactions. The layer-decoupled QH states exhibit a substantial drop in the drag ratio. d, Phase diagram for the QH states in each layer (orange and blue lines for the Mo and W layers, respectively) constructed from b (see the main text). The regions with Δν = νe − νh = –1, 0, 1 and 2 are shaded in cyan, yellow, purple and green, respectively. The EI state dominates the Δν = 0 region. The dashed lines in a, b and d denote the boundaries of different doping regions from electrostatics and the EI from Coulomb drag.

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We construct a phase diagram in Vg and Vb for layer-decoupled QH states (νe, νh) by tracing zero drag (Fig. 5d). The regions with νe − νh = –1, 0, 1 and 2 are shaded. The EI with near-perfect Coulomb drag dominates the νe = νh region, except that near the tip of the triangle with νe − νh = 4, 5 and 6. The presence of substantial Coulomb drag (with ratio up to about 0.5) at partially filled LLs in the νe − νh = –1, 1 and 2 regions suggests the presence of interlayer excitonic correlations with angular momentum quantum of –ℏ, ℏ and 2ℏ, respectively21. The result suggests a tendency to stabilize EIs with finite-angular-momentum excitons, as suggested by mean-field calculations21. However, the imperfect Coulomb drag shows the presence of ionized electrons and holes possibly from thermal and/or quantum fluctuations. Further studies under higher magnetic fields and/or with stronger excitonic binding are required to fully establish the finite-angular-momentum EIs.

Conclusion

We observe resistivity and Coulomb drag quantum oscillations in a dipolar EI based on Coulomb-coupled MoSe2/WSe2 electron–hole double layers. The oscillations originate from phase transitions between the competing EI and layer-decoupled QH states21,22. Our results demonstrate a highly tunable platform for studying quantum oscillations in correlated insulators and pave the way for explorations in several directions. To name a couple, the finite-angular-momentum excitonic correlations present a possibility to realize the QH effect for excitons. Studying how the collective excitations in the EI would evolve in a phase transition from the EI to QH states, in which there are only single-particle excitations, will provide insights into the nature of the phase transition.

Methods

Device design and fabrication

The device and contact geometry have been described in refs. 27,29,32. The optical images of devices 1 and 2 are shown in Extended Data Fig. 1a,b, respectively. The top and bottom graphite gates are outlined by the dashed and solid red lines, respectively. The WSe2 monolayer (green) and MoSe2 bilayer (blue) are separated by a thin hBN barrier (1.5–2 nm) in the transport channel (shaded red) and by a thick hBN barrier (10 nm) in the exciton contact regions (shaded pink). A natural MoSe2 bilayer is chosen because it does not crack as easily as a monolayer during the fabrication process. Under a high perpendicular electric field as used in this study, electrons in the bilayer are polarized to one layer and the bilayer functions effectively as a monolayer. To achieve high doping densities at the metal–semiconductor contacts for ohmic contacts at low temperatures, Pt–WSe2 contacts are gated only by the top gate; similarly, Bi–MoSe2 contacts are gated only by the bottom gate. During the measurements, Δ was maintained constant (3 V for device 1 and 5.5 V for device 2). Here \(\varDelta \equiv \frac{{V}_{{{\rm{bg}}}}-{V}_{{{\rm{tg}}}}}{2}\) is the antisymmetric part of the top- and bottom-gate voltages Vtg and Vbg, respectively; it is proportional to the perpendicular electric field (the symmetric part of the gate voltages, \({V}_{{\rm{g}}}\equiv \frac{{V}_{{{\rm{tg}}}}+{V}_{{{\rm{bg}}}}}{2}\), controls the net charge density in the channel).

The devices were fabricated by the layer-by-layer dry transfer method27,29,32,47. In short, all the individual layers were first exfoliated by Scotch tape onto 285-nm SiO2/Si substrates and screened by an optical microscope for appropriate thickness and geometry before stacking. Layers were picked up sequentially using a polymer stamp made of a thin layer of polycarbonate on a polypropylene-carbonate-coated polydimethylsiloxane block. The completed stack was released onto pre-prepatterned Pt electrodes on SiO2/Si substrates to form contacts to the W layer and to the gate electrodes. The polymer residue was removed using chloroform and isopropanol. Next, separate contacts for the W layer were defined by etching the top graphite electrode using electron-beam lithography patterning (Nabity) and oxygen plasma reactive ion etching (Oxford PlasmaLab 80 Plus). Finally, contacts to the Mo layer were fabricated by another electron-beam lithography patterning followed by Bi evaporation in a thermal evaporator48. The main results have been reproduced in three devices. Extended Data Fig. 9 shows additional data from device 3.

Electrical measurements

The transport properties of the electron–hole double layers were examined using two measurement configurations: the open-circuit Rxx and Rxy measurements (Fig. 1e) and the drag counterflow measurement (Fig. 1f), which reflect the charge and exciton transport properties, respectively29,32. Extended Data Fig. 2a,b shows the corresponding circuit diagrams. In the open-circuit geometry (device 1; Figs. 1–3), the Mo layer is kept in the open-circuit configuration, that is, no electron current (and, therefore, no exciton current) can flow; four-terminal Rxx and Rxy measurements are performed on the W layer using the standard lock-in technique. Here Rxx and Rxy measure the transport properties of the unbound holes in the W layer32. There are a total of eight contacts in the W layer defined by the etched top graphite gate (pins 1–8). To obtain Rxx, a small a.c. bias voltage of 0.15 mV (root mean square (r.m.s.)) was applied between pins 2 and 6 at 11.33 Hz, and the bias current out of pin 6 and the longitudinal voltage drop between pins 1 and 5 were measured simultaneously. To obtain Rxy, a small a.c. bias voltage of 0.15 mV (r.m.s.) was applied between pins 1 and 4 at 11.33 Hz, and the bias current out of pin 4 and the voltage drop between pins 2 and 6 were measured simultaneously. Other longitudinal and Hall measurement configurations gave similar results.

In the drag counterflow geometry (devices 2 and 3; Figs. 3 and 4), both layers are kept in the closed-circuit configuration so that excitons can flow in the transport channel. An a.c. bias current is driven in the Mo layer and the drag current is measured in the W layer (Fig. 1f). The drag ratio \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}\) at low temperature reflects the exciton population fraction in the system29. In particular, \(\frac{{I}_{\rm{drag}}}{{I}_{\rm{drive}}}=1\) and 0 correspond to pure exciton and pure charge transport, respectively. An a.c. bias voltage of 10 mV (r.m.s.) at 7.33 Hz was applied to the Mo layer through a 1:1 voltage transformer. The voltage transformer was connected to a 10-kΩ potentiometer to distribute the a.c. voltage on the two ends of the Mo layer; this step minimized the a.c. coupling between the Mo and W layers42,43. The d.c. interlayer bias voltage Vb was applied to the middle of the potentiometer; this kept both contacts in the Mo layer at the same d.c. potential and maintained a 10-mV a.c. voltage drop between them29. The drive current Idrive was measured by monitoring the voltage drop across a 150-kΩ resistor connected in series with the Mo layer; the drag current Idrag was measured by an ammeter connected in series with the W layer.

All the electrical measurements were carried out in a closed-cycle 4He cryostat (Oxford TeslatronPT) with temperature down to 1.5 K and magnetic field up to 12 T. All the currents and voltages were measured by lock-in amplifiers (Stanford Research Systems SR830 and SR860). In the voltage measurements, the signals were first sent to a preamplifier (Ithaco DL1201) with an input impedance of 100 MΩ before being measured by the lock-in amplifier. The measurement results are largely independent of the excitation amplitude (3–20 mV) and frequency (7–37 Hz).

Capacitance measurements

Details of the capacitance measurements have been described in ref. 27. The penetration capacitance (CP) of the dual-gated devices 1 and 2 was measured under B = 12 T by applying an a.c. bias voltage of 5 mV (r.m.s.) at 737 Hz to the top gate and measuring the induced charge carrier density on the bottom gate by a GaAs high-electron-mobility transistor through a low-temperature capacitance bridge49. The resulting CP as a function of Vg and Vb is shown in Extended Data Fig. 5. The exciton binding energy for each Vb (Fig. 4c) was obtained by integrating the normalized CP with respect to Vg over a small window centred at charge neutrality, Eb ≈ ∫(CP/Cgg)dVg, where Cgg is the gate-to-gate geometrical capacitance27.

Mean-field calculations

To obtain the phase diagram shown in Fig. 3c, we performed mean-field calculations using the self-consistent Hartree–Fock approximation described in ref. 21, and assumed a uniform carrier distribution in our solutions. The single-particle part of the Hamiltonian consists of LLs from the two semiconductor layers; for simplicity, we assumed equal electron and hole cyclotron energies (a good approximation given the similar electron and hole masses of ~0.4m0, where m0 is the free electron mass). The LL energy in the conduction (valence) band increases (decreases) with the level index. The bias voltage Vb controls the effective energy difference between the conduction band minimum and the valence band maximum. We considered the screening effects from the gate electrodes on both intralayer and interlayer Coulomb interactions between charge carriers. In momentum (q) space, the intralayer and interlayer Coulomb interactions are \({V}_{{\rm{A}}}\left({\bf{q}}\right)=\frac{2\uppi {e}^{2}}{\epsilon q}\tanh ({qD})\) and VE(q) = VA(q)e−qd, respectively. The factor tanh(qD) comes from screening from the gate electrodes. Here d = 1.3d0 and D = 1.3D0 are the effective distances between the two semiconductor layers and between the two gates, respectively (d0 = 1.8 nm and D0 = 8 nm denoting the geometrical distances). The factor of 1.3 accounts for the anisotropic dielectric constant of hBN (ε ≈ 5.2). We also increased the dielectric constant by 50% to account for the correlated screening effect. Finally, we evaluated the Hartree contribution to the interaction as the potential energy of the capacitor and the exchange integrals following ref. 21.