Excitonic Superfluid Phase in Double Bilayer Graphene

LETTERS PUBLISHED ONLINE: 22 MAY 2017 | DOI: 10.1038/NPHYS4140

Excitonic superﬂuid phase in double bilayer graphene

J. I. A. Li1*, T. Taniguchi2, K. Watanabe2, J. Hone3 and C. R. Dean1*

A spatially indirect exciton is created when an electron and For spatially indirect excitons in a magnetic field B, the energy a hole, confined to separate layers of a double quantum well scale of the condensate is conveniently characterized√ by the effective 1,2 ~ system, bind to form a composite boson . Such excitons are interlayer separation, d/`B, where `B = /eB is the magnetic long-lived, and in the limit of strong interactions are predicted length, which describes the carrier spacing within a layer, and to undergo a Bose–Einstein condensate-like phase transition d is the thickness of the tunnelling barrier separating the layers. into a superfluid ground state1–3. Here, we report evidence Reducing d increases the interlayer Coulomb interaction, e2/d of an exciton condensate in the quantum Hall eect regime (and therefore the exciton binding energy), whereas reducing the 2 of double-layer structures of bilayer graphene. Interlayer magnetic length increases the intralayer Coulomb energy, e /`B correlation is identified by quantized Hall drag at matched (increasing interaction energy between the excitons). Here ~ is the layer densities, and the dissipationless nature of the phase is reduced Planck constant, e is the elementary charge and is the confirmed in the counterflow geometry4,5. A selection rule for dielectric constant. For GaAs double layers, a minimum interlayer the condensate phase is observed involving both the orbital separation of d ∼20 nm is required to prevent interlayer tunnelling and valley indices of bilayer graphene. Our results establish and maintain sufficiently high mobility, placing a stringent limit on double bilayer graphene as an ideal system for studying the rich the achievable d/`B. Nonetheless, the EC phase in electron-doped . phase diagram of strongly interacting bosonic particles in the GaAs layers is observed to emerge for d/`B 2, with a characteristic solid state. energy scale of 800 mK (ref. 15). In bulk semiconductors, an optically excited electron–hole pair Graphene double layers possess several advantages for realizing interacts through Coulomb attraction to form a bound quasi- the EC phase, including wide tunability of carrier density across particle, referred to as a spatially direct exciton (Fig. 1a). Such electrons and holes by field effect gating, single atomic layer thick- excitons are easily generated but recombine on the nanosecond ness allowing interlayer spacing down to a few nanometres without timescale. By confining the electrons and holes to separate, significant tunnelling16, and the possibility of the EC phase transi- but closely spaced, two-dimensional (2D) quantum wells, strong tion exceeding cryogenic temperatures17. However, while Coulomb attraction is maintained but recombination is blocked, leading to drag measurements of double monolayer graphene (MLG) het- long-lived excitons. These so-called spatially indirect excitons are erostructures have successfully probed the regime of strong inter- predicted to exhibit a rich phase diagram of correlated behaviours, actions (small d/`B), no evidence of the EC phase has been rep- including a type of superfluid BEC ground state, at temperatures orted16,18. Here, we report measurement of double bilayer graphene much higher than for similar phenomena in atomic gases1,2,6. (BLG) structures in the QHE regime for interlayer separations span- Realizing the exciton condensate (EC) phase in electron–hole ning d =2.5 to 5 nm, where d is the thickness of the hexagonal boron quantum wells (QW) has proved difficult owing to the requ- nitride (hBN) tunnel barrier. In addition to the potentially more irement of fabricating matched electron- and hole-doped layers favourable electronic dispersion in comparison with MLG19–21, the that are strongly interacting but electrically isolated, while zeroth Landau level (ZLL) of BLG is eightfold degenerate, with the maintaining high mobility7,8. On the other hand, an equivalent spin and valley isospin degeneracy supplemented by an accidental condensate state is possible for identically doped (electron–electron orbital degeneracy22. This multitude of broken symmetry states or hole–hole) coupled quantum wells under application of a further expands the phase diagram of possible superfluid states. strong magnetic field. In the quantum Hall effect (QHE) regime, Correlation between the layers in the QHE regime is probed tuning both layers to half filling of the lowest Landau level can by a combination of Coulomb drag and magnetoresistance mea- be viewed as populating the lowest band in each layer with an surements in both counterflow and parallel flow geometries (see equal number of electrons and holes, which then couple across Methods). At B = 9 T and T = 20 K, the longitudinal drag shows the layers, forming an equivalent system of indirect excitons9 conventional behaviour (Fig. 1d), namely a finite response at partial (Fig. 1a). Indeed, with this approach, several studies have revealed Landau level (LL) filling that drops to zero when either layer is tuned the existence of the EC in GaAs double layers4,10–14. The EC to a QHE gap. As T decreases and B increases, we observe complete phase appears at total filling νT = 1 for the balanced case (each LL symmetry breaking with fully developed QHE gaps for all integer layer tuned to ν = 1/2), and remains stabilized when the layer filling fractions. The overall drag signature diminishes at B = 15 T densities are imbalanced as long as νT = 1, since this condition and T = 0.3 K, but apparently remains robust at certain filling maintains an equal number of electron- and hole-like carriers across fractions, as shown in Fig. 1e. Labelling regions of the plot by the 15 the barrier . coordinates of the bottom and top layer filling fraction, (νbot, νtop),

1Department of Physics, Columbia University, New York, New York 10027, USA. 2National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 3Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA. *e-mail: [email protected]; [email protected]

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a B = 0 QHE regime

dedrag Ω −0.8 0.8 drag Ω −0.8 0.8 Rxx (k ) Rxx (k ) Indirect Indirect 4 4 exciton exciton Direct 3 3 exciton 2 2 1 1 QWTunnel QW QWTunnel QW barrier barrier 0 0 top top ν ν −1 −1 bc −2 −2 d = 5 nm d = 5 nm Cr/Au −3 T = 20 K −3 T = 0.3 K BLG BN −4 B = 9 T −4 B = 15 T BLG BN −4 −3 −2 −101 2 34 −4 −3 −2 −101 2 34 SiO ν ν 2 BN bot bot FLG Top BLG Top gate Bot. BLG Bot. gate

Figure 1 | Double bilayer graphene. 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=5nm, as a function of filling factors. d, Measured at B=9T and T =20K. e, Measured at B=15T and T =0.3K. In e diverging response near zero density for each layer has been removed for clarity. an electron–hole asymmetry is apparent. In the electron–electron double-well system by excitons generated (and then annihilated) (e–e) quadrant, magnetodrag is observed whenever there is partial at the contacts (inset Fig. 2e). Charge-neutral excitons feel no filling of both the ν = 1 and 3 LLs [(1, 1), (1, 3)(3, 1) and (3, 3)], Lorentz force even under very large B, and zero Hall resistance whereas in the hole–hole (h–h) quadrant it is partially filled ν = 2 is expected4,15. Indeed, Fig. 2e shows vanishing counterflow Hall and 4 LLs [(−2,−2), (−2,−4), (−4,−2) and (−4,−4)]. resistance when νT = 1. The dissipationless nature of the EC Based on recent understanding of how the eightfold degeneracy is revealed by simultaneous observation of zero longitudinal CF of the ZLL in BLG lifts at large B (refs 23,24), we can assign resistance Rxx . Figure 2e also plots Hall resistance in the parallel a spin, valley, and orbital index to the symmetry broken states flow configuration, which is a linear combination of drag and of each layer (see Supplementary Information), and observe that counterflow measurements. The Hall resistance in the parallel flow k = strong magnetodrag in Fig. 1e appears only where both layers are geometry Rxy shows a prominent peak at νT 1, approaching the in a zero orbital state. Magnetodrag due to momentum or energy quantized value of 2h/e2. (This doubling of the quantization is due coupling19,25,26 is expected to vanish in the zero-temperature limit, to the fact that current flows through the double BLG system twice, k whereas the 0.3K response in Fig. 1e exceeds 1 k in some regions, and Rxy is defined as Vxy /I instead of Vxy /2I.) The stark difference CF k suggesting a different origin. One possibility is the formation of between Rxy and Rxy provides further evidence and confirmation the indirect excitons between the layers that are not yet phase coherent, origin of the νT =1 state lies in the strong correlation and interlayer resembling the EC precursor reported in GaAs double layers13. This phase coherence between the two BLG layers. drag drive drag CF interpretation would suggest a selection rule where EC formation is In Fig. 3a we plot the magnitude of Rxy , Rxy , Rxx and Rxy versus = drive drag limited to the zero orbital ground states only. d/`B. For device 37 (d 3.6nm), quantized Rxy and Rxy together drag CF Figure 2a–c shows the longitudinal magnetodrag (Rxx ), Hall with zero-valued Rxy persist only over a narrow range, effectively drag drive drag (Rxy ), and drive layer Hall conductance (σxy ) for a device establishing both an upper and lower critical value for d/`B. The with interlayer separation d = 3.6 nm, measured at B = 18 T and upper bound is understood by the requirement to be in the so-called T = 20 mK (for simplicity we focus our discussion on the e–e strongly interacting regime (that is, achieve a minimum effective quadrant only, but a complete mapping of the ZLL can be found interlayer interaction). We note that the critical value d/`B ∼ 0.6 in the Supplementary Information). A large response is observed is approximately 30% that was reported for GaAs4,14. Reducing drag drag drive in Rxx Rxy and Rxy following a diagonal line corresponding to the interlayer spacing from 3.6 nm to 2.5 nm results in a decrease total filling fraction νT =1, 3 and 5 (νT =νtop +νbot). Figure 2d shows of the lower critical d/`B (Fig. 3a). However, we note that this drag drive Rxy and Rxy for varying magnetic field measured along a line of boundary corresponds to approximately the same absolute magnetic drive varying drive layer density. Rxy shows conventional behaviour with field value of approximately 18 T. This may relate to the minimum well-defined QHE plateaux observed at νdrive = 1 and 2, while the magnetic field required to fully lift the ZLL degeneracy (set by magnetodrag is near zero at this sample temperature over most of sample disorder, which is approximately the same between these two the density range. However, when the drive and drag layer densities devices). Alternatively this could be signal of a transition to a new, = drive sum to νT 1, Rxy deviates strongly from its single-layer value as yet unidentified, phase as d/`B tends towards zero. 2 CF and exhibits re-entrant behaviour with quantized magnitude h/e . Figure 3b shows the counterflow Hall resistance Rxy plotted as a drag At the same total filling, Rxy takes on this same quantized value. function of filling fractions νtop and νbot. The EC state, as evidenced drag CF The amplitude of Rxx first rises dramatically in the vicinity of by a zero-valued Rxy , again follows a diagonal line corresponding νT =1 and then dips rapidly to zero at exact filling. Quantization of to νT =1. Along this diagonal the state is described by an interlayer drive drag = − = both Rxy and Rxy at integer total filling, concomitant with a local density imbalance, which we parametrize as 1ν νbot νtop (1ν 0 drag = = zero-valued Rxx , provides strong evidence of the formation of an only for νtop νbot 1/2). To understand the effect of this layer 11 EC phase . imbalance, we examine the temperature dependence of the νT = 1 Confirmation that a superfluid phase of charge carriers has state over a large range of 1ν. CF truly formed is provided by magnetotransport in the counterflow The minimum value of the Rxy shows activated behaviour with geometry9, in which charge current is carried through the varying temperature (Fig. 3c), allowing us to deduce an associated

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a b drag c drag Ω −10−5 0 Ω 02613 σdrive 2 0312 d = 3.6 nm Rxx (k ) d = 3.6 nm Rxy (k ) d = 3.6 nm xy (e /h) 3 3 3 ν ν ν ν νT = 3 T = 3 T = 5 T = 3 T = 5

2 2 2 top top top ν ν ν

1 ν = 1 ν = 3 1 νT = 1 νT = 3 ν = 3 1 T T νT = 1 T

123 1 2 3 123 ν ν νbot bot bot

de2.0

νT = 1 Drag Counterﬂow

1.5 drive Rxy 1.0 h/e2 ) )

2 2 CF e e R / 12 T / 1.0 xy h h

( 15 T ( ν = 1 || xy xy T Rxy R R drag 18 T CF Rxy Rxx 0.5 d = 2.5 nm 0.5 4 R CF B = 18 T xx R (k xx drag 0.5 drag T = 20 mK |Rxx | 15 T (k 2 d = 3.6 nm Ω ) Ω 0.0 0.0 ) 0.0 0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.6 1.0 1.4 1.8 νdrive νtop

drag drag Figure 2 | Superfluid exciton condensate. a–c, Magnetodrag (Rxx ), Hall drag (Rxy ) and drive layer Hall conductance (σxy) in the e–e quadrant for device = = = = drag drive drag 37 with a tunnel barrier thickness of d 3.6nm, measured at B 18T, T 20mK and Vbias 0V. d, Line cut of Rxy , Rxy and the amplitude of Rxx near νT =1 at dierent magnetic fields for device 45 with a tunnel barrier thickness of d=2.5nm. Inset shows the schematic of the Coulomb drag measurement. CF CF k = e, Line cut of counterflow Hall resistance Rxy longitudinal resistance Rxx , and parallel flow Hall resistance Rxy near νT 1 measured from the top BLG at B=18T. 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 .

abcCF Ω 0 30 35 Rxy (k ) 0.9 4.4 d = 3.6 nm 30 drive 4.0 Rxy B = 18 T 4.2 3.0

25 (K) drag 0.7 Δ 2.0 Rxy 4.0 ) ) 20 νT = 1 CF xy Ω

R 1.0 top (k

ν 3.8 R 15 −0.4 −0.2 0.0 0.2 0.4 log( 0.5 Δν = νbot − νtop 10 3.6 nm 3.6 Δ 2.5 nm Δν = 0 d = 3.6 nm 0.63 K drag RCF 1.42 K Δν = 0.2 5 |Rxx | xy B = 18 T 3.4 3.46 K Δν = 0.4 0.3 T = 20 mK 0 3.2 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 2 3 4 5 ν d/lB bot 1/T

drag drive | drag| CF =− Figure 3 | Tunability of the condensate phase. a, Rxy , Rxy , Rxx and Rxy measured at 1ν 0.3 as a function of eective interlayer separation d/`B for = = drag drive device 37 (d 3.6nm) and 45 (d 2.5nm). Device 45 shows a much smaller lower critical value of d/`B above which full quantization of Rxy and Rxy are CF = = = = observed. b, Rxy measured at B 18T, T 20mK from device 37, as a function of ﬁlling factors for the νT 1 state. Points along νT 1 can be parametrized = − CF = by the interlayer density imbalance 1ν νbot νtop. c, Temperature dependence of Rxy for device 37 measured at B 18T, plotted in an Arrhenius scale, revealing that the energy gap ∆ varies with 1ν. Inset, the activation gap ∆ obtained from c, (open circles) appears symmetric with 1ν, and ﬁts well to a parabola. gap4 as a function of the layer imbalance. In the inset of Fig. 3c, density imbalance. The behaviour of the activation energy sug- we plot the activation gap versus 1ν. The data are fitted well by gests that an interlayer density imbalance strengthens the interlayer a parabolic dependence27 with a minimum of ∆∼0.6K near zero correlation. Similar observations were previously reported for the

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a 20 d = 3.6 nm Received 19 November 2016; accepted 12 April 2017; T = 20 mK published online 22 May 2017

) 15 B = 18 T Ω 10 References (k

drag K+K+ K−K+ K−K− K+K− K+K+ xy

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Spatially indirect exciton condensate phases in Methods double bilayer graphene. Phys. Rev. B 95, 045416 (2017). Methods, including statements of data availability and any Acknowledgements associated accession codes and references, are available in the The authors thank A. Levchenko for helpful discussions. This work was supported by online version of this paper. the National Science Foundation (DMR-1507788). C.R.D. acknowledges partial

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© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS4140 LETTERS support by the David and Lucille Packard Foundation. A portion of this work was Additional information performed at the National High Magnetic Field Laboratory, which is supported by Supplementary information is available in the online version of the paper. Reprints and National Science Foundation Cooperative Agreement No. DMR-1157490 and the permissions information is available online at www.nature.com/reprints. Publisher’s note: State of Florida. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Correspondence and requests for materials should be Author contributions addressed to J.I.A.L. or C.R.D. 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 Competing ﬁnancial interests the manuscript. The authors declare no competing financial interests.

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Methods longitudinal and Hall resistance in each layer15 (see Supplementary Information for Our devices are assembled using the van der Waals transfer technique31. The device schematics of each configuration). 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 Data availability. The data that support the plots within this paper and other are separated by a thin layer of hBN. Even for the thinnest hBN used (2.5 nm) the findings of this study are available from the corresponding author upon interlayer tunnelling resistance is measured to be larger than 109 . Correlation reasonable request. 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 References 4,9,15 geometries . In the drag measurement, a current Idrive is sent through the drive 31. Wang, L. et al. One-dimensional electrical contact to a two-dimensional BLG layer, while the longitudinal and Hall voltage (Vxx and Vxy ) of the drive and material. Science 342, 614–617 (2013). drag layers are measured simultaneously. We define the magneto- and Hall drag 32. Li, J. I. A. et al. Negative Coulomb drag in double bilayer graphene. Phys. Rev. drag = drag drag = drag resistance as Rxx Vxx /Idrive and Rxy Vxy /Idrive. Except where indicated, both Lett. 117, 046802 (2016). BLG layers are grounded with no interlayer bias applied across the hBN tunnelling 33. Solomon, P. M., Price, P. J., Frank, D. J. & La Tulipe, D. C. New phenomena in barrier. In the counterflow (parallel flow) measurement, equal current is sent coupled transport between 2d and 3d electron-gas layers. Phys. Rev. Lett. 63, through both layers, flowing in the opposite (same) direction, while measuring 2508–2511 (1989).

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