both layers filled to half filling of the first hole
LL (ntop=nbot=–½). We report results over
the range 0.3 <d/lB< 0.8, where well-defined
exciton superfluid states exist at the lowest ex-
perimental temperature.
To probe the dynamics of the interlayer ex-
citon, we used the Coulomb drag and coun-
terflow geometries ( 35 – 38 )(Fig.1D,inset)( 26 ).
In the Coulomb drag geometry, the exciton
condensate is identified by the emergence of
a quantized Hall resistance plateau equal to
h/e^2 , as measured in both the drive and drag
layers, concomitant with zero longitudinal
resistance in both layers (Fig. 1D). In contrast,
when the two layers are decoupled, the drive
layer exhibits a density-dependent Hall resist-
ance, whereas the Hall resistance of the drag
layer is close to zero ( 39 ). Thus, the Hall drag
resistanceRdragxy provides an experimental mea-
sure of interlayer pairing ( 23 – 25 ). In the coun-
terflow geometry, charge-neutral excitons can
be induced to flow by configuring the current
to move in opposite directions in the two layers
( 40 ). In this geometry, the neutral exciton cur-
rent gives a zero-valued Hall resistance in both
layers, and the dissipationless nature of the
superfluid condensate is revealed by a vanish-
ing longitudinal resistance (Fig. 1D).
Figure 1, E and F, shows the temperature
dependence of the counterflow longitudinal
resistanceRCFxxand the Hall drag resistance
Rdragxy of ad= 3.7 nm device, for a range of
values ofd/lBobtained by varying the mag-
netic fieldB(see also fig. S2). At low temperatures,
the exciton superfluid phase was observed over
the full range of effective layer separation that
we studied, 0.3 <d/lB<0.8,asevidencedby
the vanishingRCFxxin counterflow and quan-
tizedRdragxy ( 23 , 36 – 38 ).
The temperature evolution of these quan-
tities across different values ofd/lBallowed
us to experimentally map key features of the
condensate phase diagram. First, we identified
the critical temperature of the condensate as
the value below which the longitudinal trans-
port becomes dissipationless. We defined this
point as the temperature whereRCFxxdrops to
less than 5% of the high-temperature satura-
tionvalue.IndicatedbyawhitelineinFig.1E,
this boundary identifies a dome below which
thecondensateiswellformed.Thedomeshape
of the critical temperature is consistent with
theoretical expectation ( 29 ). In the strong
coupling limit (smalld/lB), the primary conse-
quence of increasingBis a corresponding
increase of the exciton density (ºB), which
in turn drives upTc. Conversely, in the weak
coupling limit (larged/lB), increasingd/lB
further reduces the interlayer coupling, result-
ing in a diminishing of the pairing between the
two Fermi liquids and causingTcto decrease.
Second, we interpretRdragxy as a measure of
the pair fraction. In the limit of strong coupling,
where electrons and holes occur in tightly
bound pairs, excitons may persist at temperatures
well above the counterflow-superconductivity
critical temperature. In this temperature range,
we would still expect to observe a largeRdragxy
response. On the other hand, at temperatures
sufficiently high that electrons and holes are
dissociated, the value ofRdragxy will be close to
zero. We can therefore identify a temperature
scale for the pair-breaking by the temperature
whereRdragxy deviates from the quantized value
h/e^2. Phenomenologically, we define the pair-
breaking temperatureTpairas the temperature
whereRdragxy drops to half its quantized value,
that is,h/2e^2 (Fig. 1F, black line).
In Fig. 1G, we summarize the experimental
phase diagram by plotting the temperature
derivative of the counterflow resistance,
dRCFxx=dT, versusd/lB. Plotting this way em-
phasizes the three distinct regimes of the
magneto-exciton phase diagram: the low-
temperature superfluidic condensate (phase
I,T<Tc); the intermediate phase, where there
is a dissipative channel (i.e.,RCFxx>0)butthe
two layers remain coupled through exciton
formation (phase II,Tc<T<Tpair); and the
high-temperature normal phase, where the
layers are decoupled and most excitons are
unbound (phase III,T>Tpair). We note that
the temperature range over whichdRCFxx=dT
is finite-valued tracks reasonably well theTc
andTpairphase boundaries identified from
Fig. 1E and Fig. 1F, respectively; this indicates
thatRCFxxandRdragxy are correlated in this phase
diagram and that dissipation continuously in-
creases with temperature in phase II.
The experimental phase diagram shown in
Fig. 1G additionally reveals distinct temper-
ature behavior between the smalld/lB(strong
coupling) and larged/lB(weak coupling) re-
gimes. At smalld/lB,Tpairis much larger than
Tc, with a gradual evolution observed between
the condensate phase (phase I) and the high-
temperature layer-decoupled phase (phase III).
This signifies that in the strong coupling limit,
the exciton pairing establishes well above the
condensation temperature, consistent with
the behavior expected for a BEC condensate.
By contrast, at larged/lB,TpairapproachesTc,
reaching toward the BCS limit. The similarity
of these behaviors at small and larged/lBto
the well-known temperature dependence of
the BEC and BCS limits (Fig. 1A) establishes the
graphene double layer as a uniquely tunable
platform where fermion pair condensation can
be studied in both strong- and weak-pairing
regimes ( 1 – 3 , 28 ).
The condensate phase transitions of magneto-
excitons in QHBs can be further examined in the
context of two-dimensional (2D) phase transition
nature. AtT<Tc, the exciton condensate is ex-
pected to be a 2D superfluid described by the
Berezinskii-Kosterlitz-Thouless (BKT) theory
( 41 – 43 ). To produce a counterflow voltageVxxCF,
it is necessary that topological defects, namelyvortices in the condensate order parameter (Fig.
2, A and B), should move across the sample in a
direction perpendicular to the voltage gradi-
ent. Because the energy of an isolated vortex in
a 2D superfluid diverges logarithmically with
the size of the system, vortices can exist at low
temperatures only in bound pairs of opposite
signs (Fig. 2B). Counterflow resistance would
not be produced by the motion of such pairs.
As temperature rises, the vortices unbind at
the critical temperatureTBKT(Fig. 2A). Above
TBKT, the movement of free vortices leads to a
counterflow resistance. BelowTBKT, although
the linear counterflow resistance is predicted
to vanish, there can be a nonlinear response,
giving a nonzero voltage at finite measuring
currents. Specifically, it is predicted that for
small counterflow currentsICF, one should find
a power-law relation:VxxCFºðÞICF
a
, where
the exponent is given bya=1+[prs(T)/T], and
rs(T) is the temperature-dependent phase-
stiffness constant for the order parameter ( 44 ).
According to BKT theory,TBKT=(p/2)rs(TBKT),
soashould be equal to 3 atTBKTand should
increase monotonically with decreasing tem-
perature belowTBKT( 44 ). In principle, the mea-
sured exponent should drop discontinuously
toa=1aboveTBKT, but this decrease should
be gradual for a finite measuring current.
Figure 2C plots experimental current-voltage
(I-V) curves measured in the counterflow ge-
ometry in logarithmic scale. For our smallest
measuring currents, below ~100 nA, we indeed
observed power-law behavior, and we extracted
a measured exponenta(T) by fitting the slope
of theI-Vcurve at low currents. The result is
plotted as a function ofTin the bottom left
inset of Fig. 2D. At larged/lB,aincreases with
decreasingT, allowing us to extractTBKTac-
cording to the criterion ofa=3,forthosea-T
curves that go abovea= 3 at the lowest tem-
perature. Figure 2D shows the experimentally
obtainedTBKTover a large range ofd/lBfor
two graphene double-layer devices; in the
larged/lBlimit,TBKTobtained from theI-V
curves follows the trend of the critical tem-
peratureTcin Fig. 1G.
In the BCS framework,rs(T) collapses at the
mean-field transition temperatureTmthanks
to the proliferation of unpaired quasiparticles,
and thusTBKTis bounded by the mean-field
transition temperatureTm( 44 ). Because in-
creasingd/lBcorresponds to weakening the
interlayer BCS pairing,Tm(and thusTBKT)
should decline asd/lBincreases, in agreement
with the experimental observation shown in
Fig. 2D ford/lB>0.5.Asd/lBdecreases from
the BCS limit, we find thatTBKTfirst increases
and then tends to saturate as thed/lBreaches
~0.5, following the trend ofTc. Eventually the
BKT transition becomes ill defined. Even for
large magnetic fields, the measured value of
adoes not diverge as predicted forT→0, but
instead saturates at a finite value (Fig. 2D,208 14 JANUARY 2022•VOL 375 ISSUE 6577 science.orgSCIENCE
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