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science
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content/
full/
science.
aa
x
3873
/DC1
Supp
lementary
Material
for
Type
-
II Ising pairing in few
-
layer stanene
Joseph
Falson
,
Yong
Xu
,
Menghan
Liao
,
Yunyi
Zang
,
Kejing
Zhu
,
Chong
Wang
,
Zetao
Zhang
,
Hongchao
Liu
,
Wenhui
Duan
,
Ke
He
,
Haiwen
Liu
*
,
Jurgen H.
Smet
*
,
Ding
Zhang
*
,
Qi
-
Kun
Xue
*
Corresponding author. Email: haiwen.liu@bnu.edu.cn (H.L.); j.smet@fkf.mpg.de (J.H.S.);
dingzhang@mail.tsinghua.edu.cn (D.Z.)
Published
12 M
arch
2020
as
Science
First Release
DOI:
10.1126/science.
aax3873
This PDF file
includes:
Materials and Methods
Supplementary Text
Figs. S1 to S8
Table S1
References
Materials
Heterostructures were fabricated by molecular beam epitaxy
(18)
. A five quintuple layer
Bi
2
Te
3
(111) buffer film was initially deposited on a Si(111) substrate, followed by an
n
-
layer
(n=6,12,15) PbTe
(111) film. On this substrate,
n
(n=2,3,5) atomic layers of Sn were deposited at
low temperature (
T
≈ 150 K) and subsequently annealed at
T
≤ 400 K to improve the surface
morphology. The reflected high energy electron diffraction (RHEED) pattern displayed
a streak
-
pattern with no detectable shift during the growth, suggesting epitaxial locking of the stanene layer
with the PbTe layer. The estimated lattice constant is
a
= 4.52Å.
Methods
All resistance data were obtained by standard low
-
frequency lock
-
in t
echniques. The data
presented in Figure 2
C
were collected in a
3
He cryostat
operated
down to
T
= 250 mK. All other
low temperature data
were gathered in a Top
-
Loading
-
into
-
Mixture (TLM)
dilution refrigerator
with a base temperature
T
≈ 20 mK. The samples
were immersed in the
3
He
-
4
He mixture
and
mounted on a low
-
friction rotation stage to obtain both
B
and
B
//
-
data sets
.
The temperature of the
3
He
-
4
He mixture was read by a calibrated ruthenium oxide thermometer placed on the mixing
chamber.
The data shown
in
Fig. 2 and
Fig. S
3
B
were recorded by sweeping the magnetic field
from high field to zero. We estimate an error in the angle of 0.3
o
from the perfectly parallel field
orientation. The upper critical fields
of 3
-
Sn/12
-
PbTe
presented in
Fig. 3
A
are obtained from the
position where the resistance drops to half of its normal state resistance.
The
temperature
dependent
upper critical field
data
of 3
-
Sn/6
-
PbTe
in Fig. 3
A
and
those in Fig. 3
B
are obtained
from the resistance data presented
in Fig. S
4
B
, S5
B
, S6
B
. These
se
ts of data were
collected by
sweeping the rotation angle in steps of 0.05
o
through the parallel configuration at a fixe
d total
magnetic field and temperature while recording the resistance
.
Fig. S2
illustrates how this
measurement was done.
The right panel in Fig. S2 shows
the resistance r
ecorded as a function of
the
angle
θ
푟푎푤
(not calibrated, i.e. corrected for an off
set)
.
The
resistance minimum
in
each
trace
at a different angle
forms one data point in the (
B
//
-
T
)
-
plane
on the left
, as i
ndicated
by the
horizontal dotted lines for some of the data points
.
By carrying
out
the measurement
in this fashion
,
the ang
ular
precision is improved to
0.1
o
.
T
he data
is interpolated in t
he left panel. The in
-
plane
upper critical field is then obtained from the position where the interpolated resista
nce drops to
half (or 1%) of the
normal state resistance. The vertical error bars
in Fig. 3, Fig. S6
D
and Fig. S7
arise f
rom ha
lf of
the magnetic fie
ld step size (0.0
5 T). In othe
r figure
s, the error ba
rs are smaller
than the
size
of the
sym
bol.
Supplementary Text
I.
Filter
ing of the
me
asurement system
Ultra-
low
temperature measuremen
ts were perfor
med in a
TLM diluti
on refrigera
tor de
signed to
enable low electron temperatures by effectively filtering high frequency signals and i
mmersing
the sample directly i
n
to the mixture. Recent reports
(
29
)
studying 2D superconductors have
shown that these samples are extremely sensitive to external noise, necessitating thorough high
frequency filtering of lines in order to accurately characterize the superconducting state. In the
experimental setup, each measurement wire initially passes a damped LC filter stage within the
breakout box at room temperature. Figure S1
A
displays the attenuation characteristics of this
first filter. After entering the sample rod, the signal is carried from room temperature down to the
1K pot using individual thermocoax lines, each with a length of approximately 2.5 m. These
lines consist of a resistive stainless steel inner conductor (
R
cable
200
Ω
) that is isolated from the
outer conductor by a MgO dielectric. High frequency signals are strongly attenuated along these
lines owing to the skin effect of the MgO nanoparticles. These cables essentially act as efficient
distributed RC filters and are therefore widely employed
in cryogenics for suppressing stray
radiation induced thermal fluctuations
(
30
)(
31
)
. Figure
S1
B
presents the attenuation data
measured through such a thermocoax line within the same sample rod used for measuring
stanene. A network analyzer with a frequency range of 300 kHz to 8.5 G
Hz was employed. The
signal is strongly attenuated in the MHz/GHz range where the photon energy approximately
corresponds to the measurement
T
through the relationship
hf
=
k
B
T
. A flexible loom of
superconducting NbTi wires is employed to connect the thermocoax cables, thermalized near the
1K stage, with the sample stage located in the mixture. The measurement electronics i
s powered
by a dedicated phase of a three-phase power supply, operated on individual isolating
transformers, and are isolated from the data gathering measurement computer by an optical
isolator. Significant efforts, such as electrically isolating the cryostat from pumps and diagnostics
electronics, have been taken to minimize ground-loops within the measurement circuitr
y. This
setup has proven extremely effective in
the study
of delicate fractional quantum Hall phases
(3
2
-3
5
)
whose activation gaps are on the order of 100
mK,
and were observed
to continue
to
develop down to base
T
of the cryostat, suggesting good coupling
between the two
-
dimensional electron gas electron temperature and that indicated by the mixing
chamber thermometer.
During the measurements, the stanene devices were excited with a low
frequency AC current of
I
100 nA (
I
I
c
, where
I
c
is the critical superconducting current), with
the local voltage drop being measured in a four
-
point configuration using a SR830 lock
-
in
amplifier.
II.
Estimation of the mean free path and the coherence length
The band structure o
f
a
stanene trilayer as measured
by ARPES, c
onsists of a
hole band
with a
linear dispersion
and a small electron pocket around the
Γ
-
point. For 3
-
Sn/6
-
PbTe, the linear band
plays a dominant role in tra
nsport, as has been shown in
previous work
(21)
. We the
refore estimate
the mean free path
assuming that only this linear band contributes to transport. We start with the
formula for
the conductivity in the Drude picture,
σ
=
2
τ
,
where
σ
=
1
/
푒푒푡
.
푒푒푡
is the
sheet resistance of the sample
,
τ
the scattering time,
n
the
charge carrier density and
m
the effective mass. The carrier density can be calculated from the
Fermi momentum
by
=
2
2
π
(assuming a degeneracy of two)
and
m
from
, where
is the
Fermi velocity
(
3
4
)(
3
5
)
. By using the relation
=
τ
,
we obtain:
=
2
푒푒푡
1
.
A
normal state sheet resistance of 1 kΩ
is used
for the evaluation of
. From the ARPES
data in
our previous work
(21)
, t
he Fermi momentum was
determined to be 0.25
1
.
The
calculated mean
free path then equals
10 n
m. We note that this estimate yields
an upper bound of
because
additional bands with
linear
dispersion cross
the Fermi level. The evaluated mean free path is
consistent with that
calculated by directly multiplying the Fermi velocity obtained from ARPES
(
5
×
10
4
m/s
(21)
) and the
scattering time estimated from a fit of
the temperature dependent
in
-
plane upper critical field with the
theoretical model (
τ
0
0
.05 ps in Table S1):
τ
0
~
2
.
5
nm.
The superconducting coherence
length ξ
is
obtained with the help of the two
-
dimensional
Ginzburg
-
Landau formula
ξ
=
Φ
0
/
2
2
/
푑푇
,
0
.
Here,
2
/
푑푇
is the slope of the temperature dependence of the out
-
of
-
plane
2
data close to
T
c
,
0
(dotted lines in Fig. S3 to S6).
W
e
obtain
a coherence length of
54 nm for 3
-
Sn/6
-
PbTe. The
value
s of
ξ for other samples are included
in the corresponding figure capt
ions.
The above
coherence length has been extracted based on the criterion that the critical field occurs when the
resistance has dropped to 50% of the normal state resistance. If a more stringent criterion is used,
such as for instance 1% of the normal st
ate resistance, the critical field
becomes smaller such that
2
/
푑푇
,
0
is reduced. Consequently, the coherence length is larger and equal to 80 nm.
III.
C
omparison with the FFLO state
Superconductor
s
based on the
FFLO state can exhibit an
up
-
turn
in
2
(
)
at low temperatures
(16
-
19)
.
However, the FFLO state is very sensitive to impurity scattering and requires the
superconductor to be clean
, i.e.
the mean free path
should exceed
the coherence
length
ξ
. Based
on
the estimate
above, few
-
layer stanene is
however
still not a clean superconductor.
W
e
also made
an
attempt to fit the
experimental data with the 2D FFLO
B
c2
formula
(16)
as
shown in Fig. 2
E
and Fig. S3
D
-
S6
D
and obtain
poor agreement
.
Finally
,
we
note that
Rashba spin
-
orbit coupling
(SOC)
has
an
opposite effect on the 2D FFLO state and the type
-
II Ising pairing
proposed here
.
As shown in ref.
(18)
, the
Rashba SOC
can
prominent
ly enhance
B
c2
at low temperatures for
the
superconductor hosting the
FFLO state.
This is against our experimental observation that the up
-
turn
feature gets smeared out in
bilayer stanene (Fig. 3
B
) in which the Rashba SOC should be
more pronounced than that
in the stanene trilayer or penta
-
lay
e
r
.
IV.
A
nisotropic g
-
factor
The formula
=
1
.
86
for calculating the Pauli limit assumes a g
-
factor of 2. However, in
quasi
-
2D systems, there can be a strong anisotropy of the g
-
factor
(3
6
)
. An ideal two
-
dimensional
system with no dispersion along the out
-
of
-
plane direction (z
-
direction) has
1
/
=
0
. As a
consequence, the in
-
plane g
-
factor, which is proportional to
1
/
z
,
has to be zero. A quasi
-
2D
system with a finite thickness still has
weak dispersion along the z
-
direction, giving rise to a finite
z
1/푚
and
hence
non-zero
in-plane
g-factor:
. Due
to the
strong anisotropy,
can
be greatly
suppressed
relative
to the out-of-plane
g-factor.
However,
it may still
possess
an
absolute
value
that is comparable with the g-factor of a free electron. For free-standing trilayer stanene, we obtain
from first-principles calculations that the in-plane (out-of-plane) g-factors are 1.4 (33), 1.8 (4.6) at
=
-0.1, -0.2 eV. These theoretically chosen energy positions are close to the Fermi levels
in the experimental
situations
under
different
doping
(21)
. The
large
in-plane
g-factor
s indicate
that trilayer stanene
is in the bulk
limit.
In comparison,
we
estimate
based
on the out-of-plane
upper critical fields in Fig. S4
C
and Fig. 2
D
that the upper bound for the out-of-plane g-factors
are 9.9 and 7.0, respectively. The numerical value of 33 is larger than the estimated value of 9.9
from
experiment.
Apart
from
the
uncertainty
in the numerical
evaluation,
this
difference
may
indicate that our few-layer stanene possesses
a larger gap to
ratio than the standard BCS ratio
of
Δ
/푘
푐,0
=
1.76.
V.
Theoretical model
Based on previous studies
(23, 3
9
,
40
)
, the four-band model with the basis (
푥+푖푦,↑
,
푥−푖푦,↑
,
−푖푦,↓
,
푥+푖푦,↓
) and the external magnetic field
applied along the
x
-axis produces the following
Hamiltonian:
=
2
+
[
+
(
)
(
)
]
.
(S1)
Here
±
(
)
=
(
0
1
2
)
+
푣(±푘
+
)
.
,
and
are
the
Pauli
matrices.
and
are
the momenta
in the plane
of the 2D superconductor.
is the Bohr
magneton.
The
energy dispersion is determined by the mass parameter
0
, the Fermi velocity
and the material
specific constants
1
and
. The zero-field version of this Hamiltonian is the Bernevig-Hughes-
Zhang Hamiltonian for the 2D topological insulator
(3
9
)
. The Zeeman terms were considered in
ref.
(
40
).
We consider first the situation where only a single band crosses the Fermi level.
The
critical field
푐2
can be solved within the Werthamer-Helfand-Hohenberg (WHH) framework
(
41
)
. We derive the temperature dependence of the in-plane upper critical field with the help of
the
Gor’kov Green function:
푙푛
,
0
+
2
2
푠표
2
+
2
2
푅푒
[
(
1
2
+
푠표
2
+
2
2
2
)
(
1
2
)
]
=
0
,
(S
2
)
where
푠표
=
(
0
1
2
)
2
+
2
2
1
+
2
0
,
0
is the d
isorder renormalized SOC strength.
denotes the Fermi
momentum,
the digamma function and
0
the scattering time.
In the case where Rashba SOC cannot be neglected,
an additional term needs to be considered in
the Hamiltonian:
=
α
(
σ
σ
)
σ
. Here,
α
denotes the Rashba SOC strength, and
σ
and
σ
repre
sent the Pauli matrices for
real
spin
and pseudo
-
spin, respectively. T
he
temperature
dependence
then
becomes:
푙푛
,
0
+
1
2
(
+
+
)
(
1
2
)
=
0
,
(S
3
)
with
±
=
(
1
2
(
)
2
+
푠표
2
2
2
+
2
2
)
푅푒
[
(
1
2
+
±
2
)
]
and
2
±
=
(
+
)
2
+
(
)
2
+
푠표
2
±
(
)
2
+
(
)
2
+
푠표
2
. Here,
denotes the
disorder renormalized Rashba SOC strength.
In the more realistic situation where
0
,
1
,
and
give rise to
the inverted Mexican hat shaped
band of few
-
layer stanene, there are two bands crossing the Fermi level. We can join the single
band formula in virtue of the quasi
-
classical two
-
band Usadel equations
(22)
. The final equation
that governs the temperature de
pendence of the critical field is
2
0
1
2
+
(
1
+
0
)
1
+
(
1
+
0
)
2
=
0
, (S
4
)
with
=
푙푛
,
0
+
1
2
(
+
,
+
,
)
(
1
2
)
,
±
,
=
(
1
2
(
,
)
2
+
푠표
,
2
2
2
+
,
2
,
2
)
푅푒
[
(
1
2
+
±
,
2
)
]
and
=
1
,
2
.
11
,
22
and
12
represent the BCS electron
-
phonon coupling constants,
±
=
11
±
22
,
0
=
2
+
4
12
2
and
=
11
22
12
2
.
In order t
o minimize the number of fit
parameters, we employ equations (S
2
) and (S
3
) to fit our
experimental data. This is valid because equation
(S
4
) reduces to the single band form if the two
bands have similar SOC or one band dominates (neglecting interband scattering).
The
extract
ed
fit
parameters are summarized in Table S1. We estimate
0
by setting the
renormalization factor
1
+
(
2
0
,
0
)
=
50
, which accounts for the reduced SOC strength
푠표
from the pure component originating from the electronic bands
푠표
. The estimated
0
in few
-
layer stanene is comparable to that of MoS
2
in previous studies
(3)
.
For the perpendicular uppe
r critical field, we use the following formula
(2
2
)
to fit the data
ln
,
0
=
[
(
1
)
+
(
2
)
+
0
]
2
+
[
(
(
1
)
(
2
)
)
2
4
+
12
2
2
]
1
2
. (S
5
)
Here,
(
)
=
(
1
2
+
)
(
1
2
)
with
the digamma function.
1
and
2
are the diffusivities of
the two bands.
F
ig. S1.
Transmission characteristics of measurement lines. A
characteristics of the low
-
pass
filter
in the
breakout box
of the measurement system
.
B
a
ttenuation
from the breakout box to the
sample stage along
the
thermocoax line
with
out
(blue)
or with
(gray)
the low
-
pass filter
.
F
ig. S2. Illustration of the data points taken in the in
-
plane situation.
For each curve shown in
panel
B
of Fig. S4
-
S6, we fix the te
mperature and record the resistance as a function of rotation
angle around the in
-
plane situation at a set of magnetic fields. The right panel exemplifies the
obtained data. We take the minimum resistance in each rotation as one data point of the resistanc
e
at fixed
풄ퟐ
,
and
.
The solid curve in the left panel is
obtained by
interpolat
ion
.