Enhanced superconductivity in spin–orbit
proximitized bilayer graphene
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authors and unedited
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Supplementary information
https://doi.org/10.1038/s41586-022-05446-x
Supplementary Information:
Enhanced Superconductivity in Spin-Orbit Proximitized Bilayer Graphene
Yiran Zhang, Robert Polski, Alex Thomson,
́
Etienne Lantagne-Hurtubise, Cyprian Lewandowski,
Haoxin Zhou, Kenji Watanabe, Takashi Taniguchi, Jason Alicea, and Stevan Nadj-Perge
1
Supplementary Figures
SI Fig. 1
|
n
-
D
phase diagram of device D4 including electron doping.
R
xx
versus doping
density
n
and displacement field
D
including electron doping for device D4. A node-like resistive
feature appears around
n
= 0
and
D
= 0
(marked by a black arrow; zoom-in data on the right),
showing a small offset in
D
field (
D/ε
0
≈
0
.
025
±
0
.
005
V/nm).
2
SI Fig. 2
|
Alignment between BLG and WSe
2
for D1. a
–
c
, optical image of WSe
2
(
a
), BLG
(
b
) and alignment of BLG-WSe
2
(
c
) for device D1. Coloured dashed lines indicate the possible
crystal edges with zigzag or armchair orientation. Black dashed lines in
b
and
c
indicate one edge
of BLG. The scale bar in each panel corresponds to
20
μ
m.
Theoretical Analysis
1 Continuum model band structure of bilayer graphene
We consider the low-energy continuum model commonly used to describe Bernal-stacked bilayer
graphene (BLG)
S1
, under a perpendicular displacement field
D
which generates a potential differ-
ence
u
=
−
d
⊥
D/ε
BLG
between the top and bottom layers. Here
d
⊥
= 0
.
33
nm is the interlayer
distance and
ε
BLG
∼
4
.
3
is the relative permittivity of bilayer graphene. A continuum approxima-
tion of the band structure returns a Hamiltonian of the form
H
0
=
X
ξ
=
±
X
k
ψ
†
ξ
(
k
)
h
0
,ξ
(
k
)
ψ
ξ
(
k
)
, h
0
,ξ
(
k
) =
u/
2
v
0
Π
†
−
v
4
Π
†
−
v
3
Π
v
0
Π
∆
′
+
u/
2
γ
1
−
v
4
Π
†
−
v
4
Π
γ
1
∆
′
−
u/
2
v
0
Π
†
−
v
3
Π
†
−
v
4
Π
v
0
Π
−
u/
2
(1)
where
Π = (
ξk
x
+
ik
y
)
and
v
i
≡
√
3
a
2
γ
i
. Here,
ξ
=
±
1
indicates the valley that has been expanded
about:
K
,
K
′
= (
ξ
4
π/
3
a,
0)
with
a
= 0
.
246
nm the lattice constant of monolayer graphene. The
4
×
4
matrix
h
ξ
(
k
)
is expressed in the sublattice/layer basis corresponding to creation/annihilation
operators of the form
ψ
ξ
(
k
) = (
ψ
ξ,A
1
(
k
)
,ψ
ξ,B
1
(
k
)
,ψ
ξ,A
2
(
k
)
,ψ
ξ,B
2
(
k
))
T
, where
A
/
B
indicate the
sublattice,
1
,
2
indicate the layer, and the momentum
k
is measured relative to
K
ξ
(indices denot-
ing the spin degrees of freedom have been suppressed). It will sometimes be convenient below to
express the Hamiltonian in terms of the spinors
ψ
(
k
) = (
ψ
+
(
k
)
,ψ
−
(
k
))
T
.
The common values quoted for the five parameters entering the continuum model in Eq. (1) are
3
γ
0
= 2
.
61
eV (intralayer nearest-neighbor tunneling),
γ
1
= 361
meV (leading interlayer tunnel-
ing),
γ
3
= 283
meV (also known as trigonal warping term),
γ
4
= 138
meV, and
∆
′
= 15
meV
(potential difference between dimer and non-dimer sites)
S2
.
A TMD monolayer adjacent to the graphene, such as is the case here with WSe
2
, is known to
induce SOC via virtual tunnelling
S3–S5
:
H
SOC
=
X
ξ
=
±
X
k
ψ
†
ξ
(
k
)
h
SOC
,ξ
ψ
ξ
(
k
)
, h
SOC
,ξ
(
k
) =
P
1
λ
I
2
ξs
z
+
λ
R
2