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 boson^{1,2}. Such excitons are longlived, and in the limit of strong interactions are predicted to undergo a Bose–Einstein condensatelike phase transition into a superfluid ground state^{1,2,3}. Here, we report evidence of an exciton condensate in the quantum Hall effect regime of doublelayer 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 geometry^{4,5}. 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 quasiparticle, 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, twodimensional (2D) quantum wells, strong attraction is maintained but recombination is blocked, leading to longlived excitons. These socalled 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 gases^{1,2,6}.
Realizing the exciton condensate (EC) phase in electron–hole quantum wells (QW) has proved difficult owing to the requirement of fabricating matched electron and holedoped layers that are strongly interacting but electrically isolated, while maintaining high mobility^{7,8}. 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 excitons^{9} (Fig. 1a). Indeed, with this approach, several studies have revealed the existence of the EC in GaAs double layers^{4,10,11,12,13,14}. The EC phase appears at total filling ν_{T} = 1 for the balanced case (each layer tuned to ν = 1/2), and remains stabilized when the layer densities are imbalanced as long as ν_{T} = 1, since this condition maintains an equal number of electron and holelike carriers across the barrier^{15}.
For spatially indirect excitons in a magnetic field B, the energy scale of the condensate is conveniently characterized by the effective interlayer separation, d/ℓ_{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, e^{2}/εd (and therefore the exciton binding energy), whereas reducing the magnetic length increases the intralayer Coulomb energy, e^{2}/εℓ_{B} (increasing interaction energy between the excitons). Here ℏ 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/ℓ_{B}. Nonetheless, the EC phase in electrondoped GaAs layers is observed to emerge for d/ℓ_{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 tunnelling^{16}, and the possibility of the EC phase transition exceeding cryogenic temperatures^{17}. However, while Coulomb drag measurements of double monolayer graphene (MLG) heterostructures have successfully probed the regime of strong interactions (small d/ℓ_{B}), no evidence of the EC phase has been reported^{16,18}. 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 MLG^{19,20,21}, the zeroth Landau level (ZLL) of BLG is eightfold degenerate, with the spin and valley isospin degeneracy supplemented by an accidental orbital degeneracy^{22}. 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, (ν_{bot}, ν_{top}), 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 coupling^{19,25,26} is expected to vanish in the zerotemperature 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 layers^{13}. 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 (R_{xx}^{drag}), Hall drag (R_{xy}^{drag}), and drive layer Hall conductance (σ_{xy}^{drive}) 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 R_{xx}^{drag} R_{xy}^{drag} and R_{xy}^{drive} following a diagonal line corresponding to total filling fraction ν_{T} = 1, 3 and 5 (ν_{T} = ν_{top} + ν_{bot}). Figure 2d shows R_{xy}^{drag} and R_{xy}^{drive} for varying magnetic field measured along a line of varying drive layer density. R_{xy}^{drive} shows conventional behaviour with welldefined QHE plateaux observed at ν_{drive} = 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 ν_{T} = 1, R_{xy}^{drive} deviates strongly from its singlelayer value and exhibits reentrant behaviour with quantized magnitude h/e^{2}. At the same total filling, R_{xy}^{drag} takes on this same quantized value. The amplitude of R_{xx}^{drag} first rises dramatically in the vicinity of ν_{T} = 1 and then dips rapidly to zero at exact filling. Quantization of both R_{xy}^{drive}and R_{xy}^{drag} at integer total filling, concomitant with a local zerovalued R_{xx}^{drag}, provides strong evidence of the formation of an EC phase^{11}.
Confirmation that a superfluid phase of charge carriers has truly formed is provided by magnetotransport in the counterflow geometry^{9}, in which charge current is carried through the doublewell system by excitons generated (and then annihilated) at the contacts (inset Fig. 2e). Chargeneutral excitons feel no Lorentz force even under very large B, and zero Hall resistance is expected^{4,15}. Indeed, Fig. 2e shows vanishing counterflow Hall resistance when ν_{T} = 1. The dissipationless nature of the EC is revealed by simultaneous observation of zero longitudinal resistance R_{xx}^{CF}. 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 R_{xy}^{∥} shows a prominent peak at ν_{T} = 1, approaching the quantized value of 2h/e^{2}. (This doubling of the quantization is due to the fact that current flows through the double BLG system twice, and R_{xy}^{∥} is defined as V_{xy}/I instead of V_{xy}/2I.) The stark difference between R_{xy}^{CF} and R_{xy}^{∥} provides further evidence and confirmation the origin of the ν_{T} = 1 state lies in the strong correlation and interlayer phase coherence between the two BLG layers.
In Fig. 3a we plot the magnitude of R_{xy}^{drag}, R_{xy}^{drive}, R_{xx}^{drag} and R_{xy}^{CF} versus d/ℓ_{B}. For device 37 (d = 3.6 nm), quantized R_{xy}^{drive} and R_{xy}^{drag} together with zerovalued R_{xy}^{CF} persist only over a narrow range, effectively establishing both an upper and lower critical value for d/ℓ_{B}. The upper bound is understood by the requirement to be in the socalled strongly interacting regime (that is, achieve a minimum effective interlayer interaction). We note that the critical value d/ℓ_{B} ∼ 0.6 is approximately 30% that was reported for GaAs^{4,14}. Reducing the interlayer spacing from 3.6 nm to 2.5 nm results in a decrease of the lower critical d/ℓ_{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/ℓ_{B} tends towards zero.
Figure 3b shows the counterflow Hall resistance R_{xy}^{CF} plotted as a function of filling fractions ν_{top} and ν_{bot}. The EC state, as evidenced by a zerovalued R_{xy}^{CF}, again follows a diagonal line corresponding to ν_{T} = 1. Along this diagonal the state is described by an interlayer density imbalance, which we parametrize as Δν = ν_{bot} − ν_{top} (Δν = 0 only for ν_{top} = ν_{bot} = 1/2). To understand the effect of this layer imbalance, we examine the temperature dependence of the ν_{T} = 1 state over a large range of Δν.
The minimum value of the R_{xy}^{CF} shows activated behaviour with varying temperature (Fig. 3c), allowing us to deduce an associated gap^{4} 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 dependence^{27} 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 ν_{T} = 1 phase in GaAs double quantum wells^{28} and may have the same origin. Activated behaviour is observed also for the EC states at ν_{T} = 3 and 5; however, they exhibit much smaller energy gaps, and are therefore in general less developed compared to the ν_{T} = 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 ν_{T} = 1 state against perpendicular electric field. A voltage bias, V_{bias}, 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 bilayers^{23,24}. 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 level^{29}, the valley (layer) dependence of the exciton condensate could suggest that interlayer coherence occurs mainly between the two adjacent singlelayer graphenes^{30}; however, further work will be necessary to understand the precise role of the valley ordering.
This work marks the beginning of a systematic study of excitonic superfluidity in graphene doublelayer 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 technique^{31}. 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 10^{9} Ω. Correlation between the layers in the QHE regime can be probed by a combination of Coulomb drag^{33} and magnetoresistance measurements in both counterflow and parallel flow geometries^{4,9,15}. In the drag measurement, a current I_{drive} is sent through the drive BLG layer, while the longitudinal and Hall voltage (V_{xx} and V_{xy}) of the drive and drag layers are measured simultaneously. We define the magneto and Hall drag resistance as R_{xx}^{drag} = V_{xx}^{drag}/I_{drive} and R_{xy}^{drag} = V_{xy}^{drag}/I_{drive}. 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 layer^{15} (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 (DMR1507788). 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. DMR1157490 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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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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DOI: https://doi.org/10.1038/nphys4140
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