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Excitonic superfluid phase in double bilayer graphene

Abstract

A spatially indirect exciton is created when an electron and a hole, confined to separate layers of a double quantum well system, bind to form a composite boson1,2. Such excitons are long-lived, and in the limit of strong interactions are predicted to undergo a Bose–Einstein condensate-like phase transition into a superfluid ground state1,2,3. Here, we report evidence of an exciton condensate in the quantum Hall effect regime of double-layer structures of bilayer graphene. Interlayer correlation is identified by quantized Hall drag at matched layer densities, and the dissipationless nature of the phase is confirmed in the counterflow geometry4,5. A selection rule for the condensate phase is observed involving both the orbital and valley indices of bilayer graphene. Our results establish double bilayer graphene as an ideal system for studying the rich phase diagram of strongly interacting bosonic particles in the solid state.

Main

In bulk semiconductors, an optically excited electron–hole pair interacts through Coulomb attraction to form a bound quasi-particle, referred to as a spatially direct exciton (Fig. 1a). Such excitons are easily generated but recombine on the nanosecond timescale. By confining the electrons and holes to separate, but closely spaced, two-dimensional (2D) quantum wells, strong attraction is maintained but recombination is blocked, leading to long-lived excitons. These so-called spatially indirect excitons are predicted to exhibit a rich phase diagram of correlated behaviours, including a type of superfluid BEC ground state, at temperatures much higher than for similar phenomena in atomic gases1,2,6.

Figure 1: Double bilayer graphene.
figure 1

a, Optically excited particle hole pairs combine to form, short-lived, spatially direct excitons. Electron–hole pairing across a tunnel barrier prevents recombination, leading to long-lived, spatially indirect excitons. In the QHE regime at large magnetic field, spatially indirect excitons can also result from coupling between partially filled Landau bands. b, Optical image of a double bilayer graphene device, with graphite contact and local graphite back gate. The scale bar is 10 nm. c, Cartoon cross-section of our device construction (spatially indirect excitons are formed between the two BLG layers). FLG, few-layer graphene. d,e, Longitudinal drag resistance for device 19 with a tunnel barrier thickness of d = 5 nm, as a function of filling factors. d, Measured at B = 9 T and T = 20 K. e, Measured at B = 15 T and T = 0.3 K. In e diverging response near zero density for each layer has been removed for clarity.

Realizing the exciton condensate (EC) phase in electron–hole quantum wells (QW) has proved difficult owing to the requirement of fabricating matched electron- and hole-doped layers that are strongly interacting but electrically isolated, while maintaining high mobility7,8. On the other hand, an equivalent condensate state is possible for identically doped (electron–electron or hole–hole) coupled quantum wells under application of a strong magnetic field. In the quantum Hall effect (QHE) regime, tuning both layers to half filling of the lowest Landau level can be viewed as populating the lowest band in each layer with an equal number of electrons and holes, which then couple across the layers, forming an equivalent system of indirect excitons9 (Fig. 1a). Indeed, with this approach, several studies have revealed the existence of the EC in GaAs double layers4,10,11,12,13,14. The EC phase appears at total filling νT = 1 for the balanced case (each layer tuned to ν = 1/2), and remains stabilized when the layer densities are imbalanced as long as νT = 1, since this condition maintains an equal number of electron- and hole-like carriers across the barrier15.

For spatially indirect excitons in a magnetic field B, the energy scale of the condensate is conveniently characterized by the effective interlayer separation, d/B, where is the magnetic length, which describes the carrier spacing within a layer, and d is the thickness of the tunnelling barrier separating the layers. Reducing d increases the interlayer Coulomb interaction, e2/εd (and therefore the exciton binding energy), whereas reducing the magnetic length increases the intralayer Coulomb energy, e2/εℓB (increasing interaction energy between the excitons). Here is the reduced Planck constant, e is the elementary charge and ε is the dielectric constant. For GaAs double layers, a minimum interlayer separation of d 20 nm is required to prevent interlayer tunnelling and maintain sufficiently high mobility, placing a stringent limit on the achievable d/B. Nonetheless, the EC phase in electron-doped GaAs layers is observed to emerge for d/B 2, with a characteristic energy scale of 800 mK (ref. 15).

Graphene double layers possess several advantages for realizing the EC phase, including wide tunability of carrier density across electrons and holes by field effect gating, single atomic layer thickness allowing interlayer spacing down to a few nanometres without significant tunnelling16, and the possibility of the EC phase transition exceeding cryogenic temperatures17. However, while Coulomb drag measurements of double monolayer graphene (MLG) heterostructures have successfully probed the regime of strong interactions (small d/B), no evidence of the EC phase has been reported16,18. Here, we report measurement of double bilayer graphene (BLG) structures in the QHE regime for interlayer separations spanning d = 2.5 to 5 nm, where d is the thickness of the hexagonal boron nitride (hBN) tunnel barrier. In addition to the potentially more favourable electronic dispersion in comparison with MLG19,20,21, the zeroth Landau level (ZLL) of BLG is eightfold degenerate, with the spin and valley isospin degeneracy supplemented by an accidental orbital degeneracy22. This multitude of broken symmetry states further expands the phase diagram of possible superfluid states.

Correlation between the layers in the QHE regime is probed by a combination of Coulomb drag and magnetoresistance measurements in both counterflow and parallel flow geometries (see Methods). At B = 9 T and T = 20 K, the longitudinal drag shows conventional behaviour (Fig. 1d), namely a finite response at partial Landau level (LL) filling that drops to zero when either layer is tuned to a QHE gap. As T decreases and B increases, we observe complete LL symmetry breaking with fully developed QHE gaps for all integer filling fractions. The overall drag signature diminishes at B = 15 T and T = 0.3 K, but apparently remains robust at certain filling fractions, as shown in Fig. 1e. Labelling regions of the plot by the coordinates of the bottom and top layer filling fraction, (νbot, νtop), an electron–hole asymmetry is apparent. In the electron–electron (e–e) quadrant, magnetodrag is observed whenever there is partial filling of both the ν = 1 and 3 LLs [(1, 1), (1, 3) (3, 1) and (3, 3)], whereas in the hole–hole (h–h) quadrant it is partially filled ν = 2 and 4 LLs [(−2, −2), (−2, −4), (−4, −2) and (−4, −4)].

Based on recent understanding of how the eightfold degeneracy of the ZLL in BLG lifts at large B (refs 23,24), we can assign a spin, valley, and orbital index to the symmetry broken states of each layer (see Supplementary Information), and observe that strong magnetodrag in Fig. 1e appears only where both layers are in a zero orbital state. Magnetodrag due to momentum or energy coupling19,25,26 is expected to vanish in the zero-temperature limit, whereas the 0.3 K response in Fig. 1e exceeds 1 kΩ in some regions, suggesting a different origin. One possibility is the formation of indirect excitons between the layers that are not yet phase coherent, resembling the EC precursor reported in GaAs double layers13. This interpretation would suggest a selection rule where EC formation is limited to the zero orbital ground states only.

Figure 2a–c shows the longitudinal magnetodrag (Rxxdrag), Hall drag (Rxydrag), and drive layer Hall conductance (σxydrive) for a device with interlayer separation d = 3.6 nm, measured at B = 18 T and T = 20 mK (for simplicity we focus our discussion on the e–e quadrant only, but a complete mapping of the ZLL can be found in the Supplementary Information). A large response is observed in Rxxdrag Rxydrag and Rxydrive following a diagonal line corresponding to total filling fraction νT = 1, 3 and 5 (νT = νtop + νbot). Figure 2d shows Rxydrag and Rxydrive for varying magnetic field measured along a line of varying drive layer density. Rxydrive shows conventional behaviour with well-defined QHE plateaux observed at νdrive = 1 and 2, while the magnetodrag is near zero at this sample temperature over most of the density range. However, when the drive and drag layer densities sum to νT = 1, Rxydrive deviates strongly from its single-layer value and exhibits re-entrant behaviour with quantized magnitude h/e2. At the same total filling, Rxydrag takes on this same quantized value. The amplitude of Rxxdrag first rises dramatically in the vicinity of νT = 1 and then dips rapidly to zero at exact filling. Quantization of both Rxydriveand Rxydrag at integer total filling, concomitant with a local zero-valued Rxxdrag, provides strong evidence of the formation of an EC phase11.

Figure 2: Superfluid exciton condensate.
figure 2

ac, Magnetodrag (Rxxdrag), Hall drag (Rxydrag) and drive layer Hall conductance (σxy) in the e–e quadrant for device 37 with a tunnel barrier thickness of d = 3.6 nm, measured at B = 18 T, T = 20 mK and Vbias = 0 V. d, Line cut of Rxydrag, Rxydrive and the amplitude of Rxxdrag near νT = 1 at different magnetic fields for device 45 with a tunnel barrier thickness of d = 2.5 nm. Inset shows the schematic of the Coulomb drag measurement. e, Line cut of counterflow Hall resistance RxyCF longitudinal resistance RxxCF, and parallel flow Hall resistance Rxy near νT = 1 measured from the top BLG at B = 18 T. Inset shows the schematic of the counterflow measurement, which enables transport of charge-neutral excitons through the system. The linecuts shown in d and e are taken with non-zero density imbalance between the two BLG layers at νT = 1, where the condensate phase is fully developed as described in Fig. 3.

Confirmation that a superfluid phase of charge carriers has truly formed is provided by magnetotransport in the counterflow geometry9, in which charge current is carried through the double-well system by excitons generated (and then annihilated) at the contacts (inset Fig. 2e). Charge-neutral excitons feel no Lorentz force even under very large B, and zero Hall resistance is expected4,15. Indeed, Fig. 2e shows vanishing counterflow Hall resistance when νT = 1. The dissipationless nature of the EC is revealed by simultaneous observation of zero longitudinal resistance RxxCF. Figure 2e also plots Hall resistance in the parallel flow configuration, which is a linear combination of drag and counterflow measurements. The Hall resistance in the parallel flow geometry Rxy shows a prominent peak at νT = 1, approaching the quantized value of 2h/e2. (This doubling of the quantization is due to the fact that current flows through the double BLG system twice, and Rxy is defined as Vxy/I instead of Vxy/2I.) The stark difference between RxyCF and Rxy provides further evidence and confirmation the origin of the νT = 1 state lies in the strong correlation and interlayer phase coherence between the two BLG layers.

In Fig. 3a we plot the magnitude of Rxydrag, Rxydrive, Rxxdrag and RxyCF versus d/B. For device 37 (d = 3.6 nm), quantized Rxydrive and Rxydrag together with zero-valued RxyCF persist only over a narrow range, effectively establishing both an upper and lower critical value for d/B. The upper bound is understood by the requirement to be in the so-called strongly interacting regime (that is, achieve a minimum effective interlayer interaction). We note that the critical value d/B 0.6 is approximately 30% that was reported for GaAs4,14. Reducing the interlayer spacing from 3.6 nm to 2.5 nm results in a decrease of the lower critical d/B (Fig. 3a). However, we note that this boundary corresponds to approximately the same absolute magnetic field value of approximately 18 T. This may relate to the minimum magnetic field required to fully lift the ZLL degeneracy (set by sample disorder, which is approximately the same between these two devices). Alternatively this could be signal of a transition to a new, as yet unidentified, phase as d/B tends towards zero.

Figure 3: Tunability of the condensate phase.
figure 3

a, Rxydrag, Rxydrive, |Rxxdrag| and RxyCF measured at Δν = −0.3 as a function of effective interlayer separation d/B for device 37 (d = 3.6 nm) and 45 (d = 2.5 nm). Device 45 shows a much smaller lower critical value of d/B above which full quantization of Rxydrag and Rxydrive are observed. b, RxyCF measured at B = 18 T, T = 20 mK from device 37, as a function of filling factors for the νT = 1 state. Points along νT = 1 can be parametrized by the interlayer density imbalance Δν = νbotνtop. c, Temperature dependence of RxyCF for device 37 measured at B = 18 T, plotted in an Arrhenius scale, revealing that the energy gap Δ varies with Δν. Inset, the activation gap Δ obtained from c, (open circles) appears symmetric with Δν, and fits well to a parabola.

Figure 3b shows the counterflow Hall resistance RxyCF plotted as a function of filling fractions νtop and νbot. The EC state, as evidenced by a zero-valued RxyCF, again follows a diagonal line corresponding to νT = 1. Along this diagonal the state is described by an interlayer density imbalance, which we parametrize as Δν = νbotνtopν = 0 only for νtop = νbot = 1/2). To understand the effect of this layer imbalance, we examine the temperature dependence of the νT = 1 state over a large range of Δν.

The minimum value of the RxyCF shows activated behaviour with varying temperature (Fig. 3c), allowing us to deduce an associated gap4 as a function of the layer imbalance. In the inset of Fig. 3c, we plot the activation gap versus Δν. The data are fitted well by a parabolic dependence27 with a minimum of Δ 0.6 K near zero density imbalance. The behaviour of the activation energy suggests that an interlayer density imbalance strengthens the interlayer correlation. Similar observations were previously reported for the νT = 1 phase in GaAs double quantum wells28 and may have the same origin. Activated behaviour is observed also for the EC states at νT = 3 and 5; however, they exhibit much smaller energy gaps, and are therefore in general less developed compared to the νT = 1 state. A description of the features observed at these fillings, as well as the equivalent in the e–h quadrant, is provided in the Supplementary Information. However, a full analysis of these states is beyond the present manuscript and will be discussed elsewhere.

Finally, we study the stability of the νT = 1 state against perpendicular electric field. A voltage bias, Vbias, is applied to one of the BLG layers (the bottom BLG in this case) to induce the displacement field, D. The Hall drag signal shows multiple transitions with varying displacement field (Fig. 4a). The value of the displacement field at each critical point, open circles in Fig. 4b, shows good correspondence with D values for which we expect a transition of the valley order in at least one of the bilayers23,24. Moreover, it appears that the condensate phase is stabilized (finite drag) when the layers have opposite valley ordering, but suppressed (zero drag) for same ordering (see Supplementary Information). Since valley and layer are approximately equivalent for BLG in the lowest Landau level29, the valley (layer) dependence of the exciton condensate could suggest that interlayer coherence occurs mainly between the two adjacent single-layer graphenes30; however, further work will be necessary to understand the precise role of the valley ordering.

Figure 4: Interlayer bias.
figure 4

a, Rxydrag measured for device 37, as a function of interlayer bias, Vbias. The νT = 1 Rxydrag peak vanishes and reappears as the EC state displays four different transitions with varying Vbias. b, Calculated displacement field D for the two BLG layers as a function of interlayer bias Vbias (see Supplementary Information). The dashed and solid lines mark critical D-field values for transitions between different valley polarizations, |K + > and |K − >. The grey shaded area corresponds to opposite valley polarization in the double BLG, |K + K − > and |K − K + >, whereas the white area indicates the same valley polarization, |K + K + > and |K − K − >. White circles in lower panel mark transitions in the relative valley order between the layers23,24.

This work marks the beginning of a systematic study of excitonic superfluidity in graphene double-layer heterostructures. The capability of engineering and studying the superfluid state in the quantum Hall regime paves the way for realizing such condensates at higher temperature and possibly zero magnetic field.

Methods

Our devices are assembled using the van der Waals transfer technique31. The device geometry includes a local graphite bottom gate, an aligned metal top gate and graphite electrical leads, as shown in Fig. 1b and described in ref. 32. The two BLG are separated by a thin layer of hBN. Even for the thinnest hBN used (2.5 nm) the interlayer tunnelling resistance is measured to be larger than 109 Ω. Correlation between the layers in the QHE regime can be probed by a combination of Coulomb drag33 and magnetoresistance measurements in both counterflow and parallel flow geometries4,9,15. In the drag measurement, a current Idrive is sent through the drive BLG layer, while the longitudinal and Hall voltage (Vxx and Vxy) of the drive and drag layers are measured simultaneously. We define the magneto- and Hall drag resistance as Rxxdrag = Vxxdrag/Idrive and Rxydrag = Vxydrag/Idrive. Except where indicated, both BLG layers are grounded with no interlayer bias applied across the hBN tunnelling barrier. In the counterflow (parallel flow) measurement, equal current is sent through both layers, flowing in the opposite (same) direction, while measuring longitudinal and Hall resistance in each layer15 (see Supplementary Information for schematics of each configuration).

Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors thank A. Levchenko for helpful discussions. This work was supported by the National Science Foundation (DMR-1507788). C.R.D. acknowledges partial support by the David and Lucille Packard Foundation. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida.

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J.I.A.L., J.H. and C.R.D. designed the experiment. Experimental work and analysis was carried out by J.I.A.L., advised by J.H. and C.R.D. All authors contributed to writing the manuscript.

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Correspondence to J. I. A. Li or C. R. Dean.

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Li, J., Taniguchi, T., Watanabe, K. et al. Excitonic superfluid phase in double bilayer graphene. Nature Phys 13, 751–755 (2017). https://doi.org/10.1038/nphys4140

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