nitride gate dielectrics in which the charge
carrier densityneand electrical displacement
fieldDarecontrolledbysingle-crystalgraphite
top and bottom gates ( 23 ) (fig. S1). We report
data from two devices that show nearly iden-
tical behavior. Data shown in the main text
are from device A, and data from device B are
shown in fig. S11 ( 24 ).
Figure 1E shows inverse electronic com-
pressibilityk¼@m=@ne( 23 , 25 ) measured for
small-hole doping. A series of transitions are
visible as dips in the inverse compressibility,
accompanied by concomitant sharp changes
in the electrical resistivity (see fig. S2 for ad-
ditional data). High-resolution quantum oscil-
lation data as a function of the perpendicular
magnetic fieldB⊥show that these features are
associated with changes in the Fermi surface
topology linked to breaking of the spin and
valley symmetries. Figure 1F shows the Fourier
transform of the magnetoresistance (see fig. S3
for additional data),R(1/B⊥) (whereB⊥is the
out-of-plane magnetic field), measured at dif-
ferent (ne,D) points indicated in Fig. 1E. Fourier
transforms are plotted as a function of the
oscillation frequency normalized to the total
carrier density, which we denote asfn.fncor-
responds to the fraction of the Luttinger vol-
ume encircled by the phase coherent orbit that
generates a given oscillation peak. To deter-
mine Luttinger volume, we use the geometric
capacitance per unit area of the top and bot-
tom gates (ctandcb) and the spectroscopically
determined ( 16 ) bandgap of bilayer graphene
(D) to calculate the carrier densityne. Account-
ing for the finite bandgap, the system will be
doped with holes whenctvcttþþccbbvb< 4 De(where
vbandvtare the bottom and top gate volt-
ages andeis the elementary charge). Then,
ne¼ctvtþcbvbþ
ðÞctþcbD
4 e. At large negative
ne, a prominent peak is visible atfn= 0.25,
along with associated harmonics. This is con-
sistent with four identical Fermi surfaces, each
enclosing one quarter of the Luttinger vol-
ume, as expected for a state that preserves the
four-fold combined spin and valley degeneracy
of the honeycomb lattice (see bottom right
of Fig. 1F for a schematic depiction, with spin
and valley flavors rendered in different colors).
We refer to this symmetric phase as Sym 4. At
low densities and highD, by contrast, the
strongest peak occurs atfn¼^1 = 12. This is again
consistent with intact isospin symmetry but in
the regime of density where trigonal warping
produces three Fermi pockets within a single
isospin flavor. This phase is referred to as Sym 12.
In the single-particle picture, Sym 12 and Sym 4
are the phases that straddle the van Hove sin-
gularity (Fig. 1D).
We also identify regions with lowered de-
generacy. The first, which appears at low
density, is characterized by a broad peak at
fn= 1 and corresponds to a quarter metal with
a single, fully isospin-polarized Fermi sur-
face(wedenotethisisospinferromagnetIF 1 ).
Adjoining IF 1 is a phase with a strong peak
atfn=f 1 that is slightly smaller than 1, and
another peak atfn=f 2 that is close to 0. We
associate this signature with a partially isospin-
polarized phase featuring a large Fermi sur-
face of one isospin flavor and smaller Fermi
surfaces in a second, and we refer to it as PIP 1.For this phase,f 1 þf 2 ≠1, which suggests that
there may be multiple small Fermi pockets per
isospin. An additional phase appears as a sash
at densities intermediate between Sym 4 and
Sym 12. This phase shows two prominent peaks
atfn=f 1 and atfn=f 2 , such thatf 1 +f 2 = 0.5.
We call this phase PIP 2 and associate it with
theexistenceofasingleFermisurfaceineachof
two majority and two minority isospin flavors.
Possible Fermi surface topologies for the ob-
served phases are depicted in Fig. 1F for the
case where spin and valley remain good quan-
tum numbers. Notably, these schematic de-
pictions do not allow for the possibility of
intervalley coherence, which is theoretically
possible but cannot be unambiguously deter-
mined from the quantum oscillation data alone.
The symmetry-breaking transitions move to
higherjjne with increasingjjD, as expected
from a Stoner picture given that increasing
jjDenhancesthesizeofthevanHovesingu-
larity, favoring isospin symmetry-breaking
states at higher carrier concentration ( 26 ). This
behavior resembles that observed in rhombo-
hedral trilayer graphene (RTG) ( 27 ), but the
observed phases differ between the two sys-
tems. First, the critical temperature (TC)ofthe
isospin symmetry-breaking phases is found
to be ~1 K (fig. S10), lower than that in RTG.
Additionally, in BBG we find no signatures
of the annular Fermi sea that hosts super-
conductivity in RTG. BBG also does not appear
to host the spin-polarized, half-metal state
foundinRTG.Finally,thequartermetalstate
IF 1 occupies only a very small domain in the
parameter space. These differences may be
tied to subtle differences in the underlying
band wave functions in these two systems.
Further differentiating BBG and RTG, electri-
cal resistivity remains finite at all densities at
B= 0 in BBG (Fig. 2A).
Most unusually, superconductivity emerges
with the application of a finite magnetic field.
Figure 2B shows the resistivity, measured at
a nominal temperature of 10 mK andB‖=
165 mT applied in the plane of the sample. A
zero-resistance state appears at largeDat the
apparent transition between the PIP 2 and
Sym 12 states (similar data are obtained for
D<0;fig.S4).Figure2,CandD,showsthe
temperature-dependent linear and nonlinear
resistivity within this zero-resistance state. We
defineTBKT=26mK(whereBKTindicatesthe
Berezinskii-Kosterlitz-Thouless theory) from
the temperature where the voltageV¼I^3. The
critical temperature is not found to decrease
appreciably with appliedB‖over the accessible
range (see fig. S5 for additional data). For
the observedTC, the maximum critical field
for a paramagnetic superconductor is ex-
pected atB∥≈ 1 : 23 kBmBTC≈40 mT (wherekBis
the Boltzmann constant andmBis the Bohr
magneton). The entire range of the observed
superconductivity, from its onset until the776 18 FEBRUARY 2022•VOL 375 ISSUE 6582 science.orgSCIENCE
Fig. 3. Fermiology of the
superconducting state.
(A)B⊥dependence ofRxxat
fixedD= 1.02 V nm−^1 and
B‖=165 mT in thenerange
near the superconducting
phase. (B) Fourier transform
ofRxx(1/B), calculated from
the data in (A). Only data
within 0.1 T <B< 0.4 T
are used to calculate the
result. Data are plotted as a
function offn, as in Fig. 1.
In the color bar, cyan and
yellow represent small and
large FFT amplitudes,
respectively. (C)Rxxversus
neatB⊥= 0 andB‖= 165 mT.
RESEARCH | REPORTS