DOI: 10.5061/dryad.vhhmgqp3h
title: Perfect Coulomb drag and exciton transport in an excitonic insulator

description: Strongly coupled electron-hole bilayers can host quantum states of
 interlayer excitons, such as high-temperature exciton condensates at zero
 magnetic field. This state is predicted to feature perfect Coulomb drag,
 where a current in one layer is accompanied by an equal but opposite
 current in the other. Here we use an optical technique to probe the
 electrical transport of correlated electron-hole bilayers based on
 MoSe2/hBN/WSe2 heterostructures. We observe perfect Coulomb drag in the
 excitonic insulator phase at low temperatures; the counterflow resistance
 of interlayer excitons remains finite. These results indicate the
 formation of an exciton gas that does not condense into a superfluid. Our
 work demonstrates that dynamic optical spectroscopy provides a powerful
 tool for probing exciton transport behavior in correlated electron-hole
 fluids.
description: # Perfect Coulomb drag and exciton transport in an excitonic insulator
 [https://doi.org/10.5061/dryad.vhhmgqp3h](https://doi.org/10.5061/dryad.vhhmgqp3h) ## Description of the data and file structure Dataset for "Perfect Coulomb drag and exciton transport in an excitonic insulator". The dataset contains the underlying data for all the figure panels. The transport properties of an excitonic-insulating electron-hole bilayer are studied using optical spectroscopy. A monolayer MoSe2 and a monolayer WSe2, separated by 3 nm hBN tunneling barrier, are encapsulated in hBN and gated by graphite gates on both sides. An excitation current in the WSe2 layer is induced capacitively by a partial bottom gate modulation. The resulting carrier density responses in the WSe2 layer and the MoSe2 layer are then measured optically, which are proportional to the drive current and the drag current respectively. ### Data from Figure 1 The carrier density in the heterostructure is measured by an optical technique based on reflection spectroscopy. Please see [https://doi.org/10.1038/s41467-023-43799-7](https://doi.org/10.1038/s41467-023-43799-7) for detailed methodology.  * *Fig.1C ne.dat*: The measured electron density n_e, in units of 10^12 cm^-2, in the MoSe2 layer. * *Fig.1D nh.dat*: The measured hole density n_h, in units of 10^12 cm^-2, in the WSe2 layer. * *Fig.1E compressibility.dat*: Charge compressibility (10^12 cm^-2 V^-1) of the electron-hole bilayer. * *Fig.1CDE V_B.dat* and *Fig.1CDE V_G.dat*: The effective interlayer bias voltage and the gate voltage (both in Volts) applied on the heterostructure. * *Fig.1F reflectivity.dat* and *Fig.1F wavelength.dat*: Device reflectivity spectrum as a function of photon wavelength (nm). ### Data from Figure 2 The carrier density modulation induced by the driving voltage on the gate is measured using a monochromatic laser probe. * *Fig.2A Re Deltanh.dat*: The drive layer density response Δn_h (in 10^9 cm^-2). * *Fig.2B Re Deltane.dat*: The drag layer density response Δn_e (in 10^9 cm^-2). * *Fig.2C Drag ratio.dat*: Drag ratio, defined as the drag layer response divided by the drive layer response. * *Fig.2 V_B.dat* and *Fig.2 V_G.dat*: The effective interlayer bias voltage and the gate voltage applied on the heterostructure. ### Data from Figure 3 The behavior of the electron-hole fluids along a constant interlayer bias voltage (1.52 V) is investigated. * *Fig.3ABCH V_G.dat*: The gate voltage used in the following data (Figures 3, A, B, C and H). * *Fig.3BCEFG omega.dat*: The angular frequency of the gate voltage modulation (in Hz) used in the following data (Figures 3, B, C, E, F and G). * *Fig.3A Re Deltanh.dat*: Linecut of the drive layer density response Δn_h in the low-frequency limit. * *Fig.3A Re Deltane.dat*: Linecut of the drag layer density response Δn_e in the low-frequency limit. * *Fig.3A Drag ratio.dat*: Linecut of the drag ratio. * *Fig.3B Re Deltanh.dat*: The drive layer density response Δn_h at different driving frequencies (given in *Fig.3BCEFG omega.dat*). * *Fig.3C Re Deltane.dat*: The drag layer density response Δn_e at different driving frequencies. * Fig.3E Re Deltanh.dat and *Fig.3E Re Deltane.dat*:  Δn_h and  Δn_e as a function of modulation frequency at a representative gate voltage 0.42 V. * *Fig.3F Re Deltanh.dat, Fig.3G Re Deltanh.dat, Fig.3F Re Deltane.dat* and *Fig.3G Re Deltane.dat*: Same as above, but for gate voltages 0.55 V and 0.63 V. * *Fig.3H R_h.dat*, *Fig.3H R_e.dat* and *Fig.3H R_x.dat*: Hole, electron and exciton resistance (in Ohm) extracted by fitting the density response to a effective circuit model. The fitting code is provided (see *EffectiveCircuitModel.m*). ### Data from Figure 4 The behavior of the electron-hole fluids along the net charge neutrality is investigated. * *Fig.4AB omega.dat*: The angular frequency of the gate voltage modulation used in the following data (Figures 4, A and B). * *Fig.4ABC n.dat*: The electron-hole pair density n (in 10^12 cm^-2) used in the following data (Figures 4, A, B and C). * *Fig.4A Re Deltanh.dat*: The drive layer density response Δn_h at different driving frequencies (given in *Fig.4AB omega.dat*). * *Fig.4B Re Deltane.dat*: The drag layer density response Δn_e at different driving frequencies. * *Fig.4C Re Deltanh.dat*: Linecut of the drive layer density response Δn_h in the low-frequency limit, as a function of n. * *Fig.4C Re Deltane.dat*: Linecut of the drag layer density response Δn_e in the low-frequency limit, as a function of n. * *Fig.4C Drag ratio.dat*: Linecut of the drag ratio as a function of n. * *Fig.4D Drag Ratio.dat*: Drag ratio along the net charge neutrality line, as a function of n (given in *Fig.4D n.dat*) and temperature T (given in *Fig.4D T.dat*, in units of K). * *Fig.4D n.dat* and *Fig.4D T.dat*: Pair density n and temperature T for Figure 4D. * *Fig.4E R_e.dat* and *Fig.4E ne.dat*: Electron resistance R_e (in Ohm) as a function of electron density n_e in the two-dimensional electron gas in MoSe2. * *Fig.4E R_h.dat* and *Fig.4E nh.dat*: Hole resistance R_h (in Ohm) as a function of hole density n_h in the two-dimensional hole gas in WSe2. * *Fig.4E R_x.dat* and *Fig.4E n.dat*: Exciton resistance R_x (in Ohm) as a function of exciton density n in the excitonic insulator phase. * *Fig.4F R_x.dat*: Exciton resistance R_x as a function of n (given in *Fig.4F n.dat*) and temperature T (given in *Fig.4F T.dat*). * *Fig.4F n.dat* and *Fig.4F T.dat*: Pair density n and temperature T for Figure 4F. ### Data from Figure S1 Interlayer tunneling current in the electron-hole bilayer. * *Fig.S1A Interlayer tunneling current.dat*: The interlayer tunneling current, in units of pA, as a function of gate and bias voltages. * *Fig.S1A V_G.dat* and *Fig.S1AB V_B.dat*: The applied gate and bias voltages.  * *Fig.S1B Interlayer tunneling current.dat*: A linecut of the tunneling current along the net charge neutrality. ### Data from Figure S2 Carrier densities in other regions of the device. * *Fig.S2EF V_G.dat* and *Fig.S2EF V_B.dat*: The applied gate and bias voltages for Figures S2, E and F.  * *Fig.S2E ne.dat and Fig.S2F nh.dat*: The measured electron density n_e and hole density n_h in heterostructure region 1. * *Fig.S2GHIJ V_G.dat* and *Fig.S2GHIJ V_h.dat*: The gate voltage and hole layer voltage used in the capacitance model (Figures S2, G, H, I and J).  * *Fig.S2G nh.dat* and *Fig.S2G nh.dat*: Calculated hole density and electron density (in 10^12 cm^-2) in the region of interest. * *Fig.S2H nh.dat* and *Fig.S2H nh.dat*: Calculated hole density and electron density in the carrier reservoir region with increased interlayer distance. * *Fig.S2I nh.dat* and *Fig.S2I nh.dat*: Calculated hole density and electron density in the contact region that is only gated by the top gate. * *Fig.S2J nh.dat* and *Fig.S2J nh.dat*: Calculated hole density and electron density in the contact region that is only gated by the bottom gate. ### Data from Figure S3 Main results from a second device. All the files starting with *Fig.S3* are the same data as those described in Figures 1-4 but acquired from device D2. All the naming conventions and units are the same as described above. ### Data from Figure S4 Full dataset of density response as a function of V_G, V_B, and modulation frequency. * *Fig.S4 V_G.dat* and *Fig.S4 V_B.dat*: The applied gate and bias voltages for Figure S4.  * *Fig.S4x Re Deltane.dat*, with x=A, B, ..., Q: Real part (in-phase component) of the electron density response (in 10^9 cm^-2) as a function of V_G and V_B. The driving frequency increases from 82 Hz (A) to 159 MHz (Q). * *Fig.S4x Re Deltanh.dat*, with x=A, B, ..., Q: Real part of the hole density response as a function of V_G and V_B. * *Fig.S4x Im Deltane.dat*, with x=A, B, ..., Q: Imaginary part of the electron density response as a function of V_G and V_B. * *Fig.S4x Im Deltanh.dat*, with x=A, B, ..., Q: Imaginary part of the hole density response as a function of V_G and V_B. ### Data from Figure S5 Linear exciton transport at different driving voltages. * *Fig.S5 omega.dat* and *Fig.S5 U.dat*: The modulation frequency (Hz) and the driving voltage (root mean square value, in mV) used for Figure S5. * *Fig.S5A Delta n_h divided by U.dat* and *Fig.S5B Delta n_h divided by U.dat*: Hole density response Δn_h, normalized by the driving voltage U, in units of 10^12 cm^-2 V^-1, for two pair densities 0.15x10^12 cm^-2 (A) and 0.3x10^12 cm^-2 (B). ### Data from Figure S6 Estimation of defect density using optical spectroscopy. * *Fig.S6 V_G.dat*: The applied gate voltage in the following data. * *Fig.S6A Reflectivity.dat* and *Fig.S6A Wavelength.dat*: Gate-dependent reflection spectra as a function of photon wavelength (nm). * *Fig.S6B Reflectivity.dat*: A linecut of reflectivity at WSe2 main exciton wavelength. * *Fig.S6B Residual.dat*: Fitting residual of the reflectivity. * *Fig.S6C Reflectivity.dat* and *Fig.S6C Residual.dat*: Same as above, but at the MoSe2 exciton wavelength. ## Code/software *EffectiveCircuitModel.m*: Matlab code (version R2021b) used to fit the experimental Δn data to the effective circuit model, and extract the charge and exciton resistances.