Gate-Defined Topological Josephson Junctions in Bernal Bilayer Graphene
Ying-Ming Xie ,
1,2,3
Étienne Lantagne-Hurtubise ,
2,3
Andrea F. Young,
4
Stevan Nadj-Perge,
5,3
and Jason Alicea
2,3
1
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
2
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
3
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
4
Department of Physics, University of California at Santa Barbara, Santa Barbara, California 93106, USA
5
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
1200 East California Boulevard, Pasadena, California 91125, USA
(Received 28 April 2023; accepted 7 September 2023; published 5 October 2023)
Recent experiments on Bernal bilayer graphene (BLG) deposited on monolayer WSe
2
revealed robust,
ultraclean superconductivity coexisting with sizable induced spin-orbit coupling. Here, we propose
BLG
=
WSe
2
as a platform to engineer
gate-defined
planar topological Josephson junctions, where the
normal and superconducting regions descend from a common material. More precisely, we show that if
superconductivity in BLG
=
WSe
2
is gapped and emerges from a parent state with intervalley coherence,
then Majorana zero-energy modes can form in the barrier region upon applying weak in-plane magnetic
fields. Our results spotlight a potential pathway for
“
internally engineered
”
topological superconductivity
that minimizes detrimental disorder and orbital-magnetic-field effects.
DOI:
10.1103/PhysRevLett.131.146601
Experimental searches for non-Abelian anyons have to
date largely followed two complementary paths. The first
seeks intrinsic realizations of strongly correlated topological
phases of matter, most notably non-Abelian fractional
quantum Hall states
[1]
and quantum spin liquids
[2]
.
The second endeavors to engineer topological supercon-
ductors by interfacing well-understood building blocks
—
e.g., conventional superconductors and semiconductors
—
that originate from disparate materials
[3
–
18]
. One can,
however, contemplate a middle ground between these
strategies, wherein phases of matter intrinsic to a
single
medium
are leveraged to
“
internally engineer
”
topological
superconductivity. Graphenemultilayerscompriseanattrac-
tive platform for the latter approach given their extra-
ordinarily rich and tunable phase diagrams. As proof of
concept, Ref.
[19]
proposed that gate-defined wires judi-
ciously immersed between gapped phases of twisted bilayer
graphene
[20
–
24]
, or its multilayer generalizations
[25
–
29]
,
could realize topological superconductivity without invok-
ing
“
external
”
proximity effects (see also Refs.
[30
–
33]
for
related architectures).
Untwisted
(i.e., crystalline) graphene multilayers exhibit
phase diagrams whose richness and tunability rival that of
their twisted counterparts. Here, applying a perpendicular
displacement field
D
opens a gap at charge neutrality and
locally flattens the bands near the Brillouin zone corners
—
providing a knob to continuously tune the strength of
electronic correlations. Experiments have reported a series
of correlation-driven symmetry-broken metallic states
together with superconductivity in both Bernal bilayer
graphene (BLG)
[34
–
39]
and rhombohedral trilayer
graphene
[40,41]
, inspiring various theory proposals for
the underlying pairing mechanism
[42
–
58]
. In BLG, super-
conductivity was first observed over a narrow density
window in the presence of in-plane magnetic fields
B
k
≳
150
mT, with a low critical temperature
T
c
≈
30
mK
[34]
.
More recent experiments
[37,38]
found that placing BLG
adjacent to monolayer WSe
2
both generates appreciable
spin-orbit coupling (SOC)
and
promotes Cooper pairing:
Superconductivity appears over a broader density window
within a symmetry-broken parent metallic phase, with
T
c
up
to hundreds of mK, and without any applied magnetic field.
Similar trends have now been observed in several graphene-
based systems
[23,37,59]
, suggesting a deep connection
between SOC and enhanced pairing
[50
–
52,60]
.
Here, we propose BLG
=
WSe
2
as a new platform for
internally engineered topological superconductivity. Our
proposal is inspired by seminal theory works
[61,62]
which showed that spin-orbit-coupled planar Josephson
junctions at a phase difference of
π
can, in principle, host
topological superconductivity at arbitrarily weak magnetic
fields. Numerous experiments have since pursued this
approach in junctions fashioned from proximitized hetero-
structures
[63
–
67]
. We show that Josephson junctions
generated solely by electrostatic gating
in BLG
=
WSe
2
[Fig.
1(a)
] can similarly host a topological regime at weak
magnetic fields, provided two requirements are satisfied:
Superconductivity native to BLG
=
WSe
2
must exhibit a
bulk gap [to ensure well-defined Andreev bound states
(ABSs) in the junction] and descend from a symmetry-
broken normal state with intervalley coherence (to lift
valley degeneracy while maintaining the resonance
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condition for intervalley Cooper pairing). At present, the
superconducting order parameter and normal-state sym-
metry in BLG
=
WSe
2
remain unknown. Hartree-Fock treat-
ments do, however, predict that various types of IVC
states are energetically competitive in BLG
[37,60]
, rhom-
bohedral trilayer graphene
[43,68
–
70]
, and twisted
bilayer graphene
[71
–
74]
(see also a recent tensor-network
study
[75]
). Moreover, IVC order was recently imaged
using STM in monolayer graphene in its zeroth Landau
level
[76,77]
as well as in twisted graphene bilayers
[78]
and trilayers
[79]
. Turning the problem on its head, one can
view our proposal as a probe of IVC order in BLG
=
WSe
2
.
Aside from eliminating the need for heterostructure
engineering, our proposal entails two other key advantages:
(i) Superconductivity in BLG
=
WSe
2
occurs deep in the
clean limit
[34,37]
, thereby mitigating adversarial disorder
effects that remain a major obstruction in the pursuit of
topological superconductivity, and (ii) the atomically thin
nature of the setup reduces detrimental orbital effects of
in-plane magnetic fields
[67,80,81]
.
Symmetry-based low-energy description.
—
First, we
derive a minimal effective Hamiltonian for the valence
band of BLG
=
WSe
2
at large displacement fields
D
—
where
superconductivity emerges. As a baseline, Fig.
2(a)
sketches the large-
D
valence band in the absence of SOC.
The low-energy degrees of freedom carry spin and valley
quantum numbers associated with Pauli matrices
s
x;y;z
and
τ
x;y;z
, respectively. The system preserves time-reversal
symmetry
T
¼
i
τ
x
s
y
K
(
K
denotes complex conjugation)
and threefold rotations
C
3
¼
e
−
i
ð
π
=
3
Þ
s
z
. We also impose the
approximate
x
→
−
x
mirror symmetry
M
x
¼
i
τ
x
s
x
,even
though it is weakly broken by the WSe
2
substrate, and (for
now) enforce valley conservation.
We express the single-particle Hamiltonian respecting
these symmetries as
H
0
ð
k
Þ¼
h
0
ð
k
Þþ
h
so
ð
k
Þ
;
ð
1
Þ
where
k
denotes momentum measured with respect
to the
K
and
K
0
points. The first term captures the
SOC-free band dispersion and can be decomposed as
h
0
ð
k
Þ¼
ξ
0
ð
k
Þþ
ξ
1
ð
k
Þ
τ
z
. We take
ξ
0
ð
k
Þ
≈−
μ
þ
t
a
k
2
þ
t
c
k
4
with
μ
the chemical potential; the valley-dependent
contribution encodes trigonal warping and, to leading order
in momentum, takes the form
ξ
1
ð
k
Þ
≈
t
b
ð
k
3
x
−
3
k
x
k
2
y
Þ
.By
fitting to the full dispersion
[82]
, we estimate
t
a
¼
4
eV ·
a
2
,
t
b
¼
−
60
eV ·
a
3
, and
t
c
¼
−
1500
eV ·
a
4
, with
a
¼
0
.
246
nm the lattice constant. The second term in
Eq.
(1)
captures SOC in BLG inherited via virtual tunneling
to WSe
2
:
h
so
ð
k
Þ¼
β
I
2
τ
z
s
z
þ
α
R
ð
k
x
s
y
−
k
y
s
x
Þ
:
ð
2
Þ
Here,
β
I
and
α
R
, respectively, denote Ising and Rashba
SOC couplings, whose magnitudes depend on the
BLG
=
WSe
2
interface quality and twist angle
[48,95
–
97]
.
For example,
β
I
∼
0
.
7
–
2
meV was extracted using quan-
tum Hall measurements in different devices
[37,38,98]
. The
Rashba scale is harder to directly measure but can be
conservatively estimated
[82]
as
α
R
∼
1
–
3
meV ·
a
.
Intervalley coherence.
—
Topological superconductivity
emerges when an odd number of Fermi surfaces acquire a
pairing gap. To this end, SOC in coordination with a
Zeeman field facilitates the removal of spin degeneracy,
though valley degeneracy in BLG provides an added
obstruction. Coulomb interactions can lift the latter degen-
eracy by promoting symmetry breaking within the spin-
valley subspace, akin to Stoner ferromagnetism
[34
–
41]
.
Valley polarized states
—
in which electrons preferentially
populate either the
K
or
K
0
valley
—
are antagonistic to
Cooper pairing and, thus, likely irrelevant for the parent
state of superconductivity. We instead focus on IVC orders,
wherein valley degeneracy is lifted via spontaneous coher-
ent tunneling between
K
and
K
0
; such states more naturally
host superconductivity since the resonance condition
for intervalley Cooper pairing can persist. Table I in
Supplemental Material
[82]
classifies possible IVC orders.
While we expect that our proposal holds for generic IVC
(a)
(b)
x
y
00.511.52
-1
-0.5
0
0.5
1
Topological
FIG. 1. (a) Gate-defined Josephson junction in BLG
=
WSe
2
predicted to host Majorana zero-energy modes (MZMs) near a
phase difference
φ
¼
π
with small in-plane magnetic fields
B
.
(b) Andreev bound states spectrum from Eq.
(8)
with Zeeman
energy
̃
h
¼
0
(gray lines) and
̃
h
¼
0
.
5
Δ
(red lines), the latter
opening a topological regime.
FIG. 2. Valence bands of BLG
=
WSe
2
at large displacement
fields, calculated from Eq.
(4)
with
k
y
¼
0
and
λ
1
¼
0
. Para-
meters are (a)
β
I
¼
α
R
¼
λ
0
¼
0
and (b),(c)
β
I
¼
1
.
4
meV,
α
R
¼
2
meV ·
a
, and
λ
0
¼
3
meV; (c) further includes a Zeeman
field
h
¼
0
.
3
meV (along the
x
direction). Insets in (c) show the
Fermi pockets arising with chemical potentials shown by the red
and black dashed lines. Energy bands are colored according to
their valley projection.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
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states, so long as they support gapped zero-momentum
Cooper pairing, we focus for concreteness on the subset
that preserves
T
,
C
3
, and
M
x
. To linear order in momen-
tum, the corresponding IVC order parameter can be
expressed as
Δ
IVC
ð
k
Þ¼
λ
0
τ
x
þ
λ
1
τ
x
ð
k
x
s
y
−
k
y
s
x
Þ
;
ð
3
Þ
where
λ
0
describes a spin- and momentum-independent
contribution and
λ
1
encodes a spin-valley-orbit coupling.
Supplementing Eq.
(1)
with both the above IVC order
parameter and an in-plane Zeeman field
h
yields the
putative normal-state Hamiltonian
H
ð
k
Þ¼
H
0
ð
k
Þþ
Δ
IVC
ð
k
Þþ
h
·
s
ð
4
Þ
that can exhibit fully lifted spin and valley degeneracies.
Figures
2(b)
and
2(c)
sketch the band structure evolution
upon (b) turning on SOC and IVC order and then (c) further
adding a Zeeman field.
When the chemical potential intersects only the upper
pair of bands [red dashed line in Fig.
2(c)
], one can further
distill the model by projecting out the lower, inert bands
—
yielding an effective Hamiltonian
̃
H
ð
k
Þ¼
ξ
0
ð
k
Þþ
̃
β
I
ð
k
3
x
−
3
k
x
k
2
y
Þ
σ
z
þ
̃
α
R
ð
k
x
σ
y
−
k
y
σ
x
Þþ
̃
h
·
σ
ð
5
Þ
valid in the regime
h
≪
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ
2
0
þ
β
2
I
=
4
p
. Here,
σ
x;y;z
are Pauli
matrices acting in the low-energy subspace and
̃
β
I
¼
t
b
β
I
2
ffiffiffiffiffiffiffiffiffiffiffiffiffi
λ
2
0
þ
β
2
I
4
q
;
̃
α
R
¼
α
R
λ
0
ffiffiffiffiffiffiffiffiffiffiffiffiffi
λ
2
0
þ
β
2
I
4
q
þ
λ
1
;
̃
h
¼
h
λ
0
ffiffiffiffiffiffiffiffiffiffiffiffiffi
λ
2
0
þ
β
2
I
4
q
:
ð
6
Þ
Equation
(5)
represents a two-band model for fermions
with cubic-in-momentum Ising SOC and linear-in-momen-
tum Rashba SOC. Note that IVC order suppresses
̃
β
I
but
enhances
̃
α
R
and
̃
h
through a linear coupling to the bare
Zeeman field
h
and Rashba SOC
α
R
, which only would be
quadratic when
λ
0
¼
0
. The spin-valley-orbit term
λ
1
simply contributes to the effective Rashba coupling; we
thus set
λ
1
¼
0
hereafter. Parameter renormalizations in
Eq.
(6)
may be relevant for the unconventional Pauli-limit-
violation trends observed in Ref.
[37]
.
Topological Josephson junctions.
—
We now incorporate
superconductivity, assuming for simplicity an
s
-wave order
parameter
—
though we stress that our scheme readily
extends to more exotic pairings provided they generate a
bulk gap. The corresponding Bogoliubov
–
de Gennes
(BdG) Hamiltonian reads
̃
H
BdG
ð
k
Þ¼
̃
H
ð
k
Þ
i
Δ
σ
y
−
i
Δ
σ
y
−
̃
H
ð
−
k
Þ
ð
7
Þ
with
Δ
the pairing amplitude. To set the stage, we first
consider a simple case where
ξ
0
ð
k
Þ
≈−
k
2
=
2
m
−
μ
and
̃
β
I
k
3
F
≪
̃
α
R
k
F
(
m
is an effective mass and
k
F
is the Fermi
momentum). Equation
(7)
then maps to the Hamiltonian for
a proximitized Rashba-coupled 2D electron gas under
in-plane magnetic fields
—
exactly the ingredients required
to create topological superconductivity in planar Josephson
junctions
[61,62]
.
Next, we consider a Josephson junction, with phase
difference
φ
, formed by gate tuning a barrier region of
length
L
into a normal phase with
Δ
¼
0
. The magnetic
field is oriented parallel to the junction (along the
x
direction), which is optimal for accessing the topological
regime
[61,62]
. Within the barrier, the relevant Fermi
velocity
v
F
arises from the large pockets in Fig.
2(c)
;
band-structure estimates give
v
F
∼
5
×
10
5
m
=
s. For rea-
sonable junction lengths
L
∼
50
–
200
nm, the correspond-
ing Thouless energy
E
T
¼ð
π
v
F
=
2
L
Þ
∼
0
.
8
–
3
meV greatly
exceeds both the pairing energy
Δ
and renormalized
Zeeman energy
̃
h
. We therefore assume the short-junction
limit
E
T
≫
Δ
;
̃
h
below.
Topological phase transitions are determined by comput-
ing the ABS spectrum at
k
x
¼
0
using the standard
scattering matrix method (for details, see Supplemental
Material
[82]
). In the absence of normal reflections at the
interfaces, the
k
x
¼
0
ABS energies take the simple form
ε
;
η
ð
φ
Þ¼
η
̃
h
Δ
cos
ð
φ
=
2
Þ
;
ð
8
Þ
where
η
¼
is a pseudospin label. Figure
1(b)
plots these
energies at
̃
h
¼
0
and
̃
h
¼
0
.
5
Δ
. In the zero-field limit,
both
η
¼
branches become gapless at a phase difference
φ
¼
π
. Turning on the in-plane Zeeman field shifts the gap
closing points to
φ
≠
π
, thereby opening up a topological
regime. The above features are consistent with the results in
Refs.
[61,62]
, except that we consider a uniform Zeeman
energy in the barrier and superconducting regions. The
̃
h
contribution in
ε
;
η
descends from the Zeeman energy
in the superconductors
—
which produces nondegenerate
k
x
¼
0
energies at
φ
¼
0
.
Numerical phase diagram.
—
We now verify the above
physical picture via a numerical calculation of the ABSs in
a more realistic BLG
=
WSe
2
model. We rewrite the four-
band Hamiltonian capturing the low-energy bands, Eqs.
(4)
and
(7)
, as a tight-binding model in the
y
direction for a
given
k
x
(see Supplemental Material, Sec. V
[82]
). To
emulate experiments, where quantum oscillations reveal
that superconductivity occurs in a regime with both large
and small pockets
[37]
, we fix the chemical potential in the
superconducting regions to
μ
2
¼
−
1
meV [black dashed
line in Fig.
2(c)
]. The chemical potential
μ
1
in the barrier
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
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can be tuned independently, including to a regime with only
large pockets (where the effective two-band model applies).
For concreteness, unless specified otherwise we set
μ
1
¼
2
meV,
β
I
¼
1
.
4
meV,
α
R
¼
2
meV ·
a
,
λ
0
¼
3
meV,
λ
1
¼
0
, and
Δ
¼
0
.
1
meV; under BCS weak-coupling
assumptions, the latter corresponds to
T
c
∼
600
mK.
Note that this choice of
λ
0
and
β
I
gives
̃
h
≈
h
.
The ABS spectrum follows by diagonalizing the 1D
tight-binding model for each
k
x
. Figures
3(a)
and
3(b)
present the
k
x
¼
0
energies versus
φ
for (a)
h
¼
0
and
(b)
h
¼
0
.
5
Δ
. In Fig.
3(a)
, lifting of valley degeneracy by
IVC order leaves two nearly degenerate ABSs with a small
splitting (away from
φ
¼
0
;
π
). Figure
3(b)
shows that a
Zeeman field nucleates a topological region centered
around
φ
¼
π
, separated from the trivial regions by a
k
x
¼
0
gap closure. These observations agree well with
simpler two-band-model results from Fig.
1(b)
; indeed, the
bound states continue to be well captured by Eq.
(8)
[see
dashed lines in Fig.
3(b)
]. Figure
3(c)
shows the broader
phase diagram, obtained from the
k
x
¼
0
gap (
E
g
) illus-
trated by the color map, versus
h=
Δ
and
φ
. In the
topological phase, the minimal gap
E
g;m
typically appears
at finite
k
x
and is maximized when SOC effects are
dominated by the effective Rashba contribution (see
Supplemental Material, Sec. VI
[82]
). The topological
phase transition lines closely emulate the curves
̃
h
¼
Δ
j
cos
ð
φ
=
2
Þj
predicted by Eq.
(8)
. For the
π
junction,
topological superconductivity sets in at arbitrarily weak
Zeeman fields
—
though more generally normal reflections
at the interfaces push the required Zeeman field to a finite
value
[61]
(see Supplemental Material, Sec. III
[82]
). As the
in-plane field increases, the topological phase persists until
the superconducting regions become gapless, roughly when
the renormalized Zeeman energy
̃
h
∼
Δ
.
Figure
3(d)
reveals the dependence of
E
g
on
h=
Δ
and the
barrier chemical potential
μ
1
at
φ
¼
π
. A robust topological
region occurs for
μ
1
∼
0
–
3
meV, where the barrier hosts
only two large hole pockets. Outside of this range, addi-
tional small pockets arise
—
see Fig.
2(c)
—
that engender
frequent topological phase transitions. Quantum oscilla-
tions, however, indicate that the number of small pockets
above
T
c
is smaller than the six naively predicted by band
theory
[37]
, which was suggested to arise from electronic
nematicity
[99,100]
. Including nematicity in our calcula-
tion (see Supplemental Material, Sec. IV
[82]
), we find that
the overall features of the phase diagram persist, while the
extent of the robust topological region increases due to
fewer
“
polluting
”
low-energy states.
Atomically resolved MZM wave functions.
—
Topological
superconductivity acquires a novel fingerprint in our setup:
The essential ingredient of IVC order generates atomic-
scale translation symmetry breaking with an enlarged
ffiffiffi
3
p
×
ffiffiffi
3
p
Kekul ́
e supercell, which manifests directly in
the MZM wave functions. Figure
4(a)
illustrates the
atomically resolved MZM density of states in the barrier
(see Supplemental Material
[82]
, Sec. VIII); the usual
exponential localization and Friedel-like oscillations are
evident on the scale shown. Fourier transforming the data
reveals characteristic momentum-space peaks [Fig.
4(b)
]
associated with the Kekul ́
e supercell, while Fig.
4(c)
enlarges the red rectangular region from (a) and clearly
shows the corresponding atomic-scale ordering. Contrary
0
0.5
1
1.5
2
-1
-0.5
0
0.5
1
(b)
E/Δ
Topological
h
/Δ
h
/Δ
h
/Δ
(with nematicity)
E
g
/
Δ
E
g
/
Δ
Topological
0
0.5
1
1.5
2
-1
-0.5
0
0.5
1
h
/Δ=0.5
E/Δ
Topological
Topological
h
/Δ=0
(a)
(c)
(d)
(e)
(b)
FIG. 3. (a),(b) ABS energies
E
at
k
x
¼
0
as a function of
φ
,
obtained from a realistic tight-binding model with Zeeman
energy (a)
h
¼
0
and (b)
h
¼
0
.
5
Δ
. Dashed lines in (b) trace
the analytical result Eq.
(8)
obtained in the short junction limit.
(c)
–
(e) Phase diagrams as a function of (c)
h;
φ
and (d),(e)
h;
μ
1
.
In (c), the dashed white line indicates
h
¼
Δ
j
cos
ð
φ
=
2
Þj
, which
roughly captures the topological phase boundary. (e) is the same
as (d) but with nematicity phenomenologically incorporated.
Data correspond to
L
¼
158
nm and
φ
¼
π
.
Kekulé peaks
Max
Min
k
y
a
y/a
x/a
Min
Max
(a)
(b)
(c)
(e)
Min
(d)
FT
y/a
x/a
y/a
x/a
y/a
x/a
k
x
a
Min
Max
Max
Min
Min
Max
FIG. 4. (a) Atomically resolved density of states for the MZM
wave function in the junction, assuming Kekul ́
e angle
θ
¼
0
.
(b) Fourier transform (FT) of the data in (a) revealing Kekul ́
e
peaks resulting from IVC-induced atomic-scale reconstruction.
(c)
–
(e) Enlargement of the red rectangular region in (a) showing
the evolution of the symmetry-breaking pattern with different
Kekul ́
e angles. Data correspond to
L
¼
160
a
,
h=
Δ
¼
0
.
8
, and
φ
¼
π
.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-4
to the familiar
“
bond-centered
”
Kekul ́
e patterns observed in
monolayer graphene
[76,77]
, here symmetry breaking
manifests primarily on sites due to sublattice and layer
polarization generated at large
D
fields. Figures
4(d)
and
4(e)
explore different Kekul ́
e angles
θ
, obtained by
replacing the
τ
x
order parameter considered thus far with
cos
ð
θ
Þ
τ
x
þ
sin
ð
θ
Þ
τ
y
. The resulting Kekul ́
e pattern inti-
mately relates to
θ
, allowing experimental characterization
of IVC order via the MZMs.
STM measurements that resolve zero-bias peaks
without the atomic-scale structure predicted here could
arise from trivial ABSs originating from disorder
[101
–
103]
or inhomogeneities near the barrier ends
[104
–
106]
(see Refs.
[18,107]
for reviews). Conversely, and more
definitively, observing localized zero modes at both junc-
tion ends that appear near
φ
¼
π
and
exhibit Kekul ́
e order
would strongly support topological superconductivity aris-
ing from an IVC normal state.
Discussion and outlook.
—
We proposed a route to one-
dimensional topological superconductivity where all
required ingredients
—
SOC, Cooper pairing, and the ability
to fabricate planar Josephson junctions
—
appear natively in
a
single
BLG
=
WSe
2
platform. Our proposal relies on two
reasonable but so far untested assumptions: a gapped
superconducting phase and a normal parent state with
intervalley coherence. The ultraclean nature of supercon-
ductivity in BLG
[34,37]
, with an electronic mean-free path
far exceeding the coherence length, presents an enormous
virtue that potentially circumvents disorder effects that
plague proximitized nanowires
[101
–
103]
and Josephson
junctions
[63
–
67]
. Weak disorder arising, e.g., from
imperfections in the geometry of gates defining the
junction, can even enhance the robustness of the topo-
logical phase by decreasing the Majorana localization
length
[108]
. A more controllable route to the same goal
consists of gate defining the junction in a zigzag geometry
designed to enhance Andreev reflections
[109]
.
Our proposal readily generalizes to more exotic order
parameters (see Supplemental Material, Table I
[82]
). If
either the normal or superconducting state spontaneously
breaks time-reversal symmetry, topological superconduc-
tivity could arise without an applied magnetic field
[19]
.
For example, a spin-nematic IVC state described by a
τ
x
s
x
order parameter generates an effective Zeeman field when
projected to the low-energy subspace of Eq.
(5)
. We also
expect our proposal to be relevant for a broader family of
graphene-based structures. Substantial efforts have been
devoted to gate-defined wires and Josephson junctions in
twisted bilayer graphene
[19,30
–
33,110
–
112]
, which also
enjoys exquisite gate tunability but suffers from more
prevalent disorder
[113,114]
. Rhombohedral trilayer gra-
phene offers a cleaner platform for gate-tunable correlated
states and superconductivity
[40,41]
—
thus presenting
another interesting medium for future exploration along
these lines.
We are grateful to Andrey Antipov, Cory Dean, Cyprian
Lewandowski, Alex Thomson, and Yiran Zhang for
enlightening discussions. Y.-M. X. acknowledges the
support of Hong Kong Research Grant Council through
PDFS2223-6S01. É. L.-H. was supported by the Gordon
and Betty Moore Foundation
’
s EPiQS Initiative, Grant
No. GBMF8682. J. A. was supported by the Army
Research Office under Grant No. W911NF-17-1-0323;
the Caltech Institute for Quantum Information and
Matter, an NSF Physics Frontiers Center with support of
the Gordon and Betty Moore Foundation through Grant
No. GBMF1250; and the Walter Burke Institute for
Theoretical Physics at Caltech. The U.S. Department of
Energy, Office of Science, National Quantum Information
Science Research Centers, Quantum Science Center sup-
ported the symmetry-based analysis of this work. S. N.-P.
acknowledges support of Office of Naval Research (Grant
No. N142112635) and NSF-CAREER (DMR-1753306)
programs. Work at UCSB was supported by the U.S.
Department of Energy (Grant No. DE-SC0020305).
[1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das
Sarma, Non-Abelian anyons and topological quantum
computation,
Rev. Mod. Phys.
80
, 1083 (2008)
.
[2] A. Kitaev, Anyons in an exactly solved model and beyond,
Ann. Phys. (Amsterdam)
321
, 2 (2006)
.
[3] L. Fu and C. L. Kane, Superconducting Proximity Effect
and Majorana Fermions at the Surface of a Topological
Insulator,
Phys. Rev. Lett.
100
, 096407 (2008)
.
[4] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,
Generic New Platform for Topological Quantum Compu-
tation using Semiconductor Heterostructures,
Phys. Rev.
Lett.
104
, 040502 (2010)
.
[5] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Majorana
Fermions and a Topological Phase Transition in Semi-
conductor-Superconductor Heterostructures,
Phys. Rev.
Lett.
105
, 077001 (2010)
.
[6] Y. Oreg, G. Refael, and F. von Oppen, Helical Liquids and
Majorana Bound States in Quantum Wires,
Phys. Rev.
Lett.
105
, 177002 (2010)
.
[7] J. Alicea, Majorana fermions in a tunable semiconductor
device,
Phys. Rev. B
81
, 125318 (2010)
.
[8] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M.
Bakkers, and L. P. Kouwenhoven, Signatures of Majorana
fermions in hybrid superconductor-semiconductor nano-
wire devices,
Science
336
, 1003 (2012)
.
[9] T.-P. Choy, J. M. Edge, A. R. Akhmerov, and C. W. J.
Beenakker, Majorana fermions emerging from magnetic
nanoparticles on a superconductor without spin-orbit
coupling,
Phys. Rev. B
84
, 195442 (2011)
.
[10] S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A.
Yazdani, Proposal for realizing Majorana fermions in
chains of magnetic atoms on a superconductor,
Phys. Rev. B
88
, 020407(R) (2013)
.
[11] F. Pientka, L. I. Glazman, and F. von Oppen, Topological
superconducting phase in helical Shiba chains,
Phys. Rev.
B
88
, 155420 (2013)
.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-5
[12] J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Topo-
logical Superconductivity and Majorana Fermions in
RKKY Systems,
Phys. Rev. Lett.
111
, 186805 (2013)
.
[13] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J.
Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani,
Observation of Majorana fermions in ferromagnetic atomic
chains on a superconductor,
Science
346
, 602 (2014)
.
[14] X.-L. Qi and S.-C. Zhang, Topological insulators and
superconductors,
Rev. Mod. Phys.
83
, 1057 (2011)
.
[15] J. Alicea, New directions in the pursuit of Majorana
fermions in solid state systems,
Rep. Prog. Phys.
75
,
076501 (2012)
.
[16] C. Beenakker, Search for Majorana fermions in supercon-
ductors,
Annu. Rev. Condens. Matter Phys.
4
, 113 (2013)
.
[17] R. M. Lutchyn, E. P. A. M. Bakkers, L. P. Kouwenhoven,
P. Krogstrup, C. M. Marcus, and Y. Oreg, Majorana zero
modes in superconductor
–
semiconductor heterostructures,
Nat. Rev. Mater.
3
, 52 (2018)
.
[18] K. Flensberg, F. von Oppen, and A. Stern, Engineered
platforms for topological superconductivity and Majorana
zero modes,
Nat. Rev. Mater.
6
, 944 (2021)
.
[19] A. Thomson, I. M. Sorensen, S. Nadj-Perge, and J. Alicea,
Gate-defined wires in twisted bilayer graphene: From
electrical detection of intervalley coherence to internally
engineered Majorana modes,
Phys. Rev. B
105
, L081405
(2022)
.
[20] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.
Kaxiras, and P. Jarillo-Herrero, Unconventional super-
conductivity in magic-angle graphene superlattices,
Nature
(London)
556
, 43 (2018)
.
[21] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K.
Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R.
Dean, Tuning superconductivity in twisted bilayer gra-
phene,
Science
363
, 1059 (2019)
.
[22] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das,
C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A.
Bachtold, A. H. MacDonald, and D. K. Efetov, Super-
conductors, orbital magnets and correlated states in magic-
angle bilayer graphene,
Nature (London)
574
, 653 (2019)
.
[23] H. S. Arora, R. Polski, Y. Zhang, A. Thomson, Y. Choi, H.
Kim, Z. Lin, I. Z. Wilson, X. Xu, J.-H. Chu, K. Watanabe,
T. Taniguchi, J. Alicea, and S. Nadj-Perge, Superconduc-
tivity in metallic twisted bilayer graphene stabilized by
WSe2,
Nature (London)
583
, 379 (2020)
.
[24] M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K.
Watanabe, T. Taniguchi, and A. Yazdani, Evidence for
unconventional superconductivity in twisted bilayer gra-
phene,
Nature (London)
600
, 240 (2021)
.
[25] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and
P. Jarillo-Herrero, Tunable strongly coupled super-
conductivity in magic-angle twisted trilayer graphene,
Nature (London)
590
, 249 (2021)
.
[26] Z. Hao, A. M. Zimmerman, P. Ledwith, E. Khalaf, D. H.
Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath,
and P. Kim, Electric field tunable superconductivity in
alternating-twist magic-angle trilayer graphene,
Science
371
, 1133 (2021)
.
[27] H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang,
R. Polski, K. Watanabe, T. Taniguchi, J. Alicea, and S.
Nadj-Perge, Evidence for unconventional superconductiv-
ity in twisted trilayer graphene,
Nature (London)
606
, 494
(2022)
.
[28] J. M. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T.
Taniguchi, and P. Jarillo-Herrero, Robust superconductiv-
ity in magic-angle multilayer graphene family,
Nat. Mater.
21
, 877 (2022)
.
[29] Y. Zhang, R. Polski, C. Lewandowski, A. Thomson, Y.
Peng, Y. Choi, H. Kim, K. Watanabe, T. Taniguchi, J.
Alicea, F. von Oppen, G. Refael, and S. Nadj-Perge,
Promotion of superconductivity in magic-angle graphene
multilayers,
Science
377
, 1538 (2022)
.
[30] D. Rodan-Legrain, Y. Cao, J. M. Park, S. C. de la Barrera,
M. T. Randeria, K. Watanabe, T. Taniguchi, and P. Jarillo-
Herrero, Highly tunable junctions and non-local Josephson
effect in magic-angle graphene tunnelling devices,
Nat.
Nanotechnol.
16
, 769 (2021)
.
[31] F. K. de Vries, E. Portol ́
es, G. Zheng, T. Taniguchi, K.
Watanabe, T. Ihn, K. Ensslin, and P. Rickhaus, Gate-
defined Josephson junctions in magic-angle twisted bilayer
graphene,
Nat. Nanotechnol.
16
, 760 (2021)
.
[32] E. Portol ́
es, S. Iwakiri, G. Zheng, P. Rickhaus, T.
Taniguchi, K. Watanabe, T. Ihn, K. Ensslin, and F. K.
de Vries, A tunable monolithic squid in twisted bilayer
graphene,
Nat. Nanotechnol.
17
, 1159 (2022)
.
[33] J. Díez-M ́
erida, A. Díez-Carlón, S. Y. Yang, Y.-M. Xie,
X.-J. Gao, J. Senior, K. Watanabe, T. Taniguchi, X. Lu,
A. P. Higginbotham, K. T. Law, and D. K. Efetov, Sym-
metry-broken Josephson junctions and superconducting
diodes in magic-angle twisted bilayer graphene,
Nat.
Commun.
14
, 2396 (2023)
.
[34] H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L.
Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F.
Young, Isospin magnetism and spin-polarized supercon-
ductivity in Bernal bilayer graphene,
Science
375
, 774
(2022)
.
[35] S. C. de la Barrera, S. Aronson, Z. Zheng, K. Watanabe, T.
Taniguchi, Q. Ma, P. Jarillo-Herrero, and R. Ashoori,
Cascade of isospin phase transitions in Bernal-stacked
bilayer graphene at zero magnetic field,
Nat. Phys.
18
, 771
(2022)
.
[36] A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T.
Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quantum
cascade of correlated phases in trigonally warped bilayer
graphene,
Nature (London)
608
, 298 (2022)
.
[37] Y. Zhang, R. Polski, A. Thomson, É. Lantagne-Hurtubise,
C. Lewandowski, H. Zhou, K. Watanabe, T. Taniguchi, J.
Alicea, and S. Nadj-Perge, Enhanced superconductivity in
spin
–
orbit proximitized bilayer graphene,
Nature (London)
613
, 268 (2023)
.
[38] L. Holleis, C. L. Patterson, Y. Zhang, H. M. Yoo, H. Zhou,
T. Taniguchi, K. Watanabe, S. Nadj-Perge, and A. F.
Young, Ising superconductivity and nematicity in Bernal
bilayer graphene with strong spin orbit coupling,
arXiv:2303.00742
.
[39] J.-X. Lin, Y. Wang, N. J. Zhang, K. Watanabe, T.
Taniguchi, L. Fu, and J. I. A. Li, Spontaneous momentum
polarization and diodicity in Bernal bilayer graphene,
arXiv:2302.04261
.
[40] H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets,
E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-6
Serbyn, and A. F. Young, Half- and quarter-metals in
rhombohedral trilayer graphene,
Nature (London)
598
,
429 (2021)
.
[41] H. Zhou, T. Xie, T. Taniguchi, K. Watanabe, and A. F.
Young, Superconductivity in rhombohedral trilayer gra-
phene,
Nature (London)
598
, 434 (2021)
.
[42] A. Ghazaryan, T. Holder, M. Serbyn, and E. Berg,
Unconventional Superconductivity in Systems with An-
nular Fermi Surfaces: Application to Rhombohedral Tri-
layer Graphene,
Phys. Rev. Lett.
127
, 247001 (2021)
.
[43] S. Chatterjee, T. Wang, E. Berg, and M. P. Zaletel, Inter-
valley coherent order and isospin fluctuation mediated
superconductivity in rhombohedral trilayer graphene,
Nat.
Commun.
13
, 6013 (2022)
.
[44] T. Cea, P. A. Pantaleón, V. o. T. Phong, and F. Guinea,
Superconductivity from repulsive interactions in rhombo-
hedral trilayer graphene: A Kohn-Luttinger-like mecha-
nism,
Phys. Rev. B
105
, 075432 (2022)
.
[45] A. L. Szabó and B. Roy, Metals, fractional metals, and
superconductivity in rhombohedral trilayer graphene,
Phys. Rev. B
105
, L081407 (2022)
.
[46] Y.-Z. You and A. Vishwanath, Kohn-Luttinger supercon-
ductivity and intervalley coherence in rhombohedral tri-
layer graphene,
Phys. Rev. B
105
, 134524 (2022)
.
[47] Y.-Z. Chou, F. Wu, J. D. Sau, and S. Das Sarma, Acoustic-
phonon-mediated superconductivity in moir ́
eless graphene
multilayers,
Phys. Rev. B
106
, 024507 (2022)
.
[48] Y.-Z. Chou, F. Wu, and S. Das Sarma, Enhanced super-
conductivity through virtual tunneling in Bernal bilayer
graphene coupled to WSe
2
,
Phys. Rev. B
106
, L180502
(2022)
.
[49] A. S. Patri and T. Senthil, Strong correlations in ABC-
stacked trilayer graphene: Moir ́
e is important,
Phys. Rev. B
107
, 165122 (2023)
.
[50] J. B. Curtis, N. R. Poniatowski, Y. Xie, A. Yacoby, E.
Demler, and P. Narang, Stabilizing Fluctuating Spin-
Triplet Superconductivity in Graphene via Induced Spin-
Orbit Coupling,
Phys. Rev. Lett.
130
, 196001 (2023)
.
[51] G. Wagner, Y. H. Kwan, N. Bu
ltinck, S. H. Simon, and S. A.
Parameswaran, Superconductivity from repulsive interactions
in Bernal-stacked bilayer graphene,
arXiv:2302.00682
.
[52] A. Jimeno-Pozo, H. Sainz-Cruz, T. Cea, P. A. Pantaleón,
and F. Guinea, Superconductivity from electronic inter-
actions and spin-orbit enhancement in bilayer and trilayer
graphene,
Phys. Rev. B
107
, L161106 (2023)
.
[53] W. Qin, C. Huang, T. Wolf, N. Wei, I. Blinov, and A. H.
MacDonald, Functional Renormalization Group Study of
Superconductivity in Rhombohedral Trilayer Graphene,
Phys. Rev. Lett.
130
, 146001 (2023)
.
[54] D.-C. Lu, T. Wang, S. Chatterjee, and Y.-Z. You, Corre-
lated metals and unconventional superconductivity in
rhombohedral trilayer graphene: A renormalization group
analysis,
Phys. Rev. B
106
, 155115 (2022)
.
[55] H. Dai, R. Ma, X. Zhang, T. Guo, and T. Ma, Quantum
monte carlo study of superconductivity in rhombohedral
trilayer graphene under an electric field,
Phys. Rev. B
107
,
245106 (2023)
.
[56] Z. Dong, A. V. Chubukov, and L. Levitov, Transformer
spin-triplet superconductivity at the onset of isospin order
in bilayer graphene,
Phys. Rev. B
107
, 174512 (2023)
.
[57] G. Shavit and Y. Oreg, Inducing superconductivity in
bilayer graphene by alleviation of the Stoner blockade,
Phys. Rev. B
108
, 024510 (2023)
.
[58] Z. Dong, P. A. Lee, and L. S. Levitov, Signatures of cooper
pair dynamics and quantum-critical superconductivity in
tunable carrier bands,
arXiv:2304.09812
.
[59] R. Su, M. Kuiri, K. Watanabe, T. Taniguchi, and J. Folk,
Superconductivity in twisted double bilayer graphene
stabilized by WSe
2
,
Nat. Mater. (2023).
[60] M. Xie and S. Das Sarma, Flavor symmetry breaking in
spin-orbit coupled bilayer graphene,
Phys. Rev. B
107
,
L201119 (2023)
.
[61] F. Pientka, A. Keselman, E. Berg, A. Yacoby, A. Stern,
and B. I. Halperin, Topological Superconductivity in a
Planar Josephson Junction,
Phys. Rev. X
7
, 021032
(2017)
.
[62] M. Hell, M. Leijnse, and K. Flensberg, Two-Dimensional
Platform for Networks of Majorana Bound States,
Phys.
Rev. Lett.
118
, 107701 (2017)
.
[63] H. Ren, F. Pientka, S. Hart, A. T. Pierce, M. Kosowsky, L.
Lunczer, R. Schlereth, B. Scharf, E. M. Hankiewicz, L. W.
Molenkamp, B. I. Halperin, and A. Yacoby, Topological
superconductivity in a phase-controlled Josephson junc-
tion,
Nature (London)
569
, 93 (2019)
.
[64] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Portol ́
es,
A. C. C. Drachmann, A. Keselman, S. Gronin, C.
Thomas, T. Wang, R. Kallaher, G. C. Gardner, E. Berg,
M. J. Manfra, A. Stern, C. M. Marcus, and F. Nichele,
Evidence of topological superconductivity in planar Jo-
sephson junctions,
Nature (London)
569
, 89 (2019)
.
[65] C. T. Ke, C. M. Moehle, F. K. de Vries, C. Thomas, S.
Metti, C. R. Guinn, R. Kallaher, M. Lodari, G. Scappucci,
T. Wang, R. E. Diaz, G. C. Gardner, M. J. Manfra, and S.
Goswami, Ballistic superconductivity and tunable
π
–
junc-
tions in InSb quantum wells,
Nat. Commun.
10
, 3764
(2019)
.
[66] M. C. Dartiailh, W. Mayer, J. Yuan, K. S. Wickramasinghe,
A. Matos-Abiague, I.
Ž
uti
ć
, and J. Shabani, Phase Sig-
nature of Topological Transition in Josephson Junctions,
Phys. Rev. Lett.
126
, 036802 (2021)
.
[67] A. Banerjee, O. Lesser, M. A. Rahman, H.-R. Wang, M.-R.
Li, A. Kringhøj, A. M. Whiticar, A. C. C. Drachmann, C.
Thomas, T. Wang, M. J. Manfra, E. Berg, Y. Oreg, A.
Stern, and C. M. Marcus, Signatures of a topological phase
transition in a planar Josephson junction,
Phys. Rev. B
107
,
245304 (2023)
.
[68] T. Wang, M. Vila, M. P. Zaletel, and S. Chatterjee,
Electrical control of magnetism in spin-orbit coupled
graphene multilayers,
arXiv:2303.04855
.
[69] Y. Zhumagulov, D. Kochan, and J. Fabian, Emergent
correlated phases in rhombohedral trilayer graphene in-
duced by proximity spin-orbit and exchange coupling,
arXiv:2305.14277
.
[70] J. M. Koh, J. Alicea, and Étienne Lantagne-Hurtubise,
Correlated phases in spin-orbit-coupled rhombohedral
trilayer graphene,
arXiv:2306.12486
.
[71] N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A.
Vishwanath, and M. P. Zaletel, Ground State and Hidden
Symmetry of Magic-Angle Graphene at Even Integer
Filling,
Phys. Rev. X
10
, 031034 (2020)
.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-7
[72] Y. Zhang, K. Jiang, Z. Wang, and F. Zhang, Correlated
insulating phases of twisted bilayer graphene at commen-
surate filling fractions: A Hartree-Fock study,
Phys. Rev. B
102
, 035136 (2020)
.
[73] B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A.
Yazdani, and B. A. Bernevig, Twisted bilayer graphene.
iv. exact insulator ground states and phase diagram,
Phys.
Rev. B
103
, 205414 (2021)
.
[74] G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and
S. A. Parameswaran, Global Phase Diagram of the Normal
State of Twisted Bilayer Graphene,
Phys. Rev. Lett.
128
,
156401 (2022)
.
[75] T. Wang, D. E. Parker, T. Soejima, J. Hauschild, S. Anand,
N. Bultinck, and M. P. Zaletel, Kekul ́
e spiral order in
magic-angle graphene: A density matrix renormalization
group study,
arXiv:2211.02693
.
[76] X. Liu, G. Farahi, C.-L. Chiu, Z. Papic, K. Watanabe, T.
Taniguchi, M. P. Zaletel, and A. Yazdani, Visualizing
broken symmetry and topological defects in a quantum
Hall ferromagnet,
Science
375
, 321 (2022)
.
[77] A. Coissard, D. Wander, H. Vignaud, A. G. Grushin, C.
Repellin, K. Watanabe, T. Taniguchi, F. Gay, C. B.
Winkelmann, H. Courtois, H. Sellier, and B. Sac ́
ep ́
e,
Imaging tunable quantum Hall broken-symmetry orders
in graphene,
Nature (London)
605
, 51 (2022)
.
[78] K. P. Nuckolls, R. L. Lee, M. Oh, D. Wong, T. Soejima,
J. P. Hong, D. C
ă
lug
ă
ru, J. Herzog-Arbeitman, B. A.
Bernevig, K. Watanabe, T. Taniguchi, N. Regnault,
M. P. Zaletel, and A. Yazdani, Quantum textures of the
many-body wavefunctions in magic-angle graphene,
Nature (London)
620
, 525 (2023)
.
[79] H. Kim, Y. Choi, Étienne Lantagne-Hurtubise, C.
Lewandowski, A. Thomson, L. Kong, H. Zhou, E.
Baum, Y. Zhang, L. Holleis, K. Watanabe, T. Taniguchi,
A. F. Young, J. Alicea, and S. Nadj-Perge, Imaging inter-
valley coherent order in magic-angle twisted trilayer
graphene,
arXiv:2304.10586
.
[80] B. Nijholt and A. R. Akhmerov, Orbital effect of magnetic
field on the Majorana phase diagram,
Phys. Rev. B
93
,
235434 (2016)
.
[81] M. e. Aghaee (Microsoft Quantum Collaboration), InAs-
Al hybrid devices passing the topological gap protocol,
Phys. Rev. B
107
, 245423 (2023)
.
[82] See Supplemental Material at
http://link.aps.org/
supplemental/10.1103/PhysRevLett.131.146601
for de-
tails, including: (1) minimal model for BLG at finite
displacement fields, (2) spin-orbit coupling in the minimal
model, (3) determining the topological region with the
scattering matrix method, (4) incorporating nematicity,
(5) effective tight-binding Hamiltonian for BLG
=
WSe
2
Josephson junction, (6) the minimal topological gap in
various parameter regions, (7) tunneling spectroscopy and
Majorana zero-modes, and (8) computing atomically
resolved Majorana zero mode wavefunctions, which in-
cludes Refs. [83
–
94].
[83] J. Jung and A. H. MacDonald, Accurate tight-binding
models for the
π
bands of bilayer graphene,
Phys. Rev.
B
89
, 035405 (2014)
.
[84] Z. Wang, D.-K. Ki, J. Y. Khoo, D. Mauro, H. Berger, L. S.
Levitov, and A. F. Morpurgo, Origin and Magnitude of
‘
Designer
’
Spin-Orbit Interaction in Graphene on Semi-
conducting Transition Metal Dichalcogenides,
Phys. Rev.
X
6
, 041020 (2016)
.
[85] B. Yang, M. Lohmann, D. Barroso, I. Liao, Z. Lin, Y. Liu,
L. Bartels, K. Watanabe, T. Taniguchi, and J. Shi, Strong
electron-hole symmetric Rashba spin-orbit coupling in
graphene/monolayer transition metal dichalcogenide het-
erostructures,
Phys. Rev. B
96
, 041409(R) (2017)
.
[86] D. Wang, S. Che, G. Cao, R. Lyu, K. Watanabe, T.
Taniguchi, C. N. Lau, and M. Bockrath, Quantum Hall
effect measurement of spin
–
orbit coupling strengths in
ultraclean bilayer graphene
=
WSe
2
heterostructures,
Nano
Lett.
19
, 7028 (2019)
.
[87] J. Amann, T. Völkl, T. Rockinger, D. Kochan, K.
Watanabe, T. Taniguchi, J. Fabian, D. Weiss, and
J. Eroms, Counterintuitive gate dependence of weak
antilocalization in bilayer graphene
=
WSe
2
heterostruc-
tures,
Phys. Rev. B
105
, 115425 (2022)
.
[88] M. Gmitra, D. Kochan, P. Högl, and J. Fabian, Trivial and
inverted Dirac bands and the emergence of quantum spin
Hall states in graphene on transition-metal dichalcoge-
nides,
Phys. Rev. B
93
, 155104 (2016)
.
[89] M. Gmitra and J. Fabian, Proximity Effects in Bilayer
Graphene on Monolayer WSe
2
: Field-Effect Spin Valley
Locking, Spin-Orbit Valve, and Spin Transistor,
Phys. Rev.
Lett.
119
, 146401 (2017)
.
[90] M. P. Zaletel and J. Y. Khoo, The gate-tunable strong and
fragile topology of multilayer-graphene on a transition
metal dichalcogenide,
arXiv:1901.01294
.
[91] C. W. J. Beenakker, Universal Limit of Critical-Current
Fluctuations in Mesoscopic Josephson Junctions,
Phys.
Rev. Lett.
67
, 3836 (1991)
.
[92] Y.-M. Xie, K. T. Law, and P. A. Lee, Topological super-
conductivity in EuS/Au/superconductor heterostructures,
Phys. Rev. Res.
3
, 043086 (2021)
.
[93] S. Manna, P. Wei, Y. Xie, K. T. Law, P. A. Lee, and J. S.
Moodera, Signature of a pair of Majorana zero modes in
superconducting gold surface states,
Proc. Natl. Acad. Sci.
U.S.A.
117
, 8775 (2020)
.
[94] J. P. Hong, T. Soejima, and M. P. Zaletel, Detecting
Symmetry Breaking in Magic Angle Graphene using
Scanning Tunneling Microscopy,
Phys. Rev. Lett.
129
,
147001 (2022)
.
[95] Y. Li and M. Koshino, Twist-angle dependence of the
proximity spin-orbit coupling in graphene on transition-
metal dichalcogenides,
Phys. Rev. B
99
, 075438 (2019)
.
[96] A. David, P. Rakyta, A. Kormányos, and G. Burkard,
Induced spin-orbit coupling in twisted graphene
–
transition
metal dichalcogenide heterobilayers: Twistronics meets
spintronics,
Phys. Rev. B
100
, 085412 (2019)
.
[97] T. Naimer, K. Zollner, M. Gmitra, and J. Fabian, Twist-
angle dependent proximity induced spin-orbit coupling in
graphene/transition metal dichalcogenide heterostructures,
Phys. Rev. B
104
, 195156 (2021)
.
[98] J. O. Island, X. Cui, C. Lewandowski, J. Y. Khoo, E. M.
Spanton, H. Zhou, D. Rhodes, J. C. Hone, T. Taniguchi, K.
Watanabe, L. S. Levitov, M. P. Zaletel, and A. F. Young,
Spin
–
orbit-driven band inversion in bilayer graphene by
the van der Waals proximity effect,
Nature (London)
571
,
85 (2019)
.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-8
[99] J. Jung, M. Polini, and A. H. MacDonald, Persistent
current states in bilayer graphene,
Phys. Rev. B
91
,
155423 (2015)
.
[100] Z. Dong, M. Davydova, O. Ogunnaike, and L. Levitov,
Isospin- and momentum-polarized orders in bilayer gra-
phene,
Phys. Rev. B
107
, 075108 (2023)
.
[101] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Zero-Bias
Peaks in the Tunneling Conductance of Spin-Orbit-
Coupled Superconducting Wires with and without Major-
ana End-States,
Phys. Rev. Lett.
109
, 267002 (2012)
.
[102] D. Bagrets and A. Altland, Class
d
Spectral Peak in
Majorana Quantum Wires,
Phys. Rev. Lett.
109
, 227005
(2012)
.
[103] S. Das Sarma and H. Pan, Disorder-induced zero-bias
peaks in Majorana nanowires,
Phys. Rev. B
103
, 195158
(2021)
.
[104] C. Moore, T. D. Stanescu, and S. Tewari, Two-terminal
charge tunneling: Disentangling Majorana zero modes
from partially separated Andreev bound states in semi-
conductor-superconductor heterostructures,
Phys. Rev. B
97
, 165302 (2018)
.
[105] J. Chen, B. D. Woods, P. Yu, M. Hocevar, D. Car, S. R.
Plissard, E. P. A. M. Bakkers, T. D. Stanescu, and S. M.
Frolov, Ubiquitous Non-Majorana Zero-Bias Conductance
Peaks in Nanowire Devices,
Phys. Rev. Lett.
123
, 107703
(2019)
.
[106] T. D. Stanescu and S. Tewari, Robust low-energy Andreev
bound states in semiconductor-superconductor structures:
Importance of partial separation of component Majorana
bound states,
Phys. Rev. B
100
, 155429 (2019)
.
[107] E. Prada, P. San-Jose, M. W. A. de Moor, A. Geresdi,
E. J. H. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado,
and L. P. Kouwenhoven, From Andreev to Majorana
bound states in hybrid superconductor
–
semiconductor
nanowires,
Nat. Rev. Phys.
2
, 575 (2020)
.
[108] A. Haim and A. Stern, Benefits of Weak Disorder in One-
Dimensional Topological Superconductors,
Phys. Rev.
Lett.
122
, 126801 (2019)
.
[109] T. Laeven, B. Nijholt, M. Wimmer, and A. R. Akhmerov,
Enhanced Proximity Effect in Zigzag-Shaped Majorana
Josephson Junctions,
Phys. Rev. Lett.
125
, 086802 (2020)
.
[110] Y.-M. Xie, D. K. Efetov, and K. T. Law,
φ
0
-josephson
junction in twisted bilayer graphene induced by a valley-
polarized state,
Phys. Rev. Res.
5
, 023029 (2023)
.
[111] J.-X. Hu, Z.-T. Sun, Y.-M. Xie, and K. T. Law, Josephson
Diode Effect Induced by Valley Polarization in Twisted
Bilayer Graphene,
Phys. Rev. Lett.
130
, 266003 (2023)
.
[112] H. Sainz-Cruz, P. A. Pantaleón, V. o. T. Phong, A. Jimeno-
Pozo, and F. Guinea, Junctions and Superconducting
Symmetry in Twisted Bilayer Graphene,
Phys. Rev. Lett.
131
, 016003 (2023)
.
[113] A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani, D.
Rodan-Legrain, Y. Myasoedov, K. Watanabe, T. Taniguchi,
P. Moon, M. Koshino, P. Jarillo-Herrero, and E.
Zeldov, Mapping the twist-angle disorder and Landau levels
in magic-angle graphene,
Nature (London)
581
,47
(2020)
.
[114] J. H. Wilson, Y. Fu, S. Das Sarma, and J. H. Pixley,
Disorder in twisted bilayer graphene,
Phys. Rev. Res.
2
,
023325 (2020)
.
PHYSICAL REVIEW LETTERS
131,
146601 (2023)
146601-9