of 20
1
Type
-
II
Ising Pairing in
Few
-
Layer Stanene
J
oseph Falson
1
, Yong Xu
2,3
, Menghan Liao
2
, Yunyi Zang
2
, Kejing Zhu
2
, Chong Wang
2
,
Zetao
Zhang
2
,
Hongchao Liu
4
, Wenhui Duan
2
,
5
, Ke He
2,
6
, Haiwen Liu
7
*
, Jurgen H. Smet
1
*
,
Ding Zhang
2,
6
*
, and Qi
-
Kun Xue
2,
6
1
Max Planck Institute for Solid State Research, Stuttgart, 70569, Germany
2
State Key Laboratory of Low
-
Dimensional Quantum Physics, Department of Physics,
Tsinghua
University, Beijing, 100084, China
3
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351
-
0198, Japan
4
International Center for Quantum Materials, Peking University, Beijing 100871, China
5
Institute for Advanced Study, Tsinghua Uni
versity, Beijing 100084, China
6
Beijing Academy of Quantum Information Sciences, Beijing 100193, China
7
Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University,
Beijing 100875, China
*Correspondence to:
haiwen.liu@bnu.edu.cn
,
j.smet@fkf.mpg.de
,
dingzhang@mail.tsinghua.edu.cn
Abstract
:
S
pin
-
orbit coupling has proven indispensable in rea
lizing topological materials and
more recently Ising pairing in two
-
dimensional su
perconductors. This pairing mechanism relies
on inversion symmetry breaking
and
sustains anomalously large in
-
plane polarizing magnetic
fields whose upper limit is expected
to
diverge at low temperatures, although experimental
demonstration of this has remained elusive due to the required fields. In this work, the recently
discovered superconductor f
ew
-
layer
stanene, i.e.
epitaxially strained
-
Sn, is shown to exhibit a
new
type of Ising pairing between
carriers
residing in bands with different orbital indices
near the
Γ
-
point
. The bands are split as a result of spin
-
orbit locking
without the participation
of inversion
symmetry breaking. The
in
-
plane upper critical field
is
strongly enhanced at ultra
-
low temperature
and reveals the sought
for
upturn
.
2
Realizing superconducting materials resilient to strong external magnetic fields remains a
particularly important pursuit for both applied and fundamental research alike
(1
-
7)
. The viability
of exotic pairing mechanisms supporting this ambition is under co
nstant scrutiny. One recent
breakthrough has been the identification of “Ising pairing” in two
-
dimensional (2D) crystalline
superconductors
(2).
This pairing mechanism can apparently boost the in
-
plane upper critical field,
B
c2,//
, compared to the Chandras
ekhar
-
Clogston or Pauli limit
(8
-
9)
. For instance, MoS
2
(3
-
4)
,
populated with charge carriers through ionic liquid gating, exhibits a
B
c2,//
exceeding the value
expected from the Pauli limit by a factor of 5
-
6. In atomically thin NbSe
2
,
B
c2,//
was reported to be
as high as 31.5 Tesla
the highest applied field
at a temperature
T
equal to 50% of the
superconducting transition temperature
T
c
(5),
even though the Pauli limit would only yield a field
of up to 5.5 T. In amorphous superconducting
films (Fig. 1
A
) spin
-
flip scattering, as illustrated in
the cartoon in panel A of Fig. 1, has been attributed a key role in enhancing
B
c2,//
. However, in
these high mobility crystalline samples spin
-
flip scattering
(10)
can be safely discarded as the
origi
n of the enhancement as it would imply unphysically short scattering times
(3
-
5)
. Theory has
therefore pointed to properties inherent to the band structure of these 2D materials to account for
the anomalous robustness. As a result of broken inversion symme
try, opposing valleys in
-
space
host states of opposite spin orientation (Fig. 1
B)
. By now it has been established that strong spin
-
orbit coupling (SOC) induces significant spin splitting among these valleys. Consequently, Cooper
pairs formed from carrie
rs in opposing valleys possess locked opposite spins and become resilient
to an in
-
plane pair
-
breaking field. This physical framework inaugurated the search for ever
increasing
B
c2,//
almost exclusively in transition metal dichalcogenides as their crystal
structure
may naturally break in
-
plane inversion symmetry. Single layers of WS
2
and TaS
2
recently unveiled
in plane critical fields that easily exceed the Pauli limit by a factor of 10
(6
-
7)
.
Several theoretically prognosticated features of Ising superco
nductivity however remain to be
verified experimentally. For example, the
B
c2,//
is predicted to diverge and deviate from the 2D
Ginzburg
-
Landau
(G
-
L)
formula at low temperatures, even if a moderate amount of disorder is
present
(11
-
13)
. Such behavior is r
eminiscent of the Fulde
-
Ferrell
-
Larkin
-
Ovchinnikov (FFLO)
state
(14
-
17)
(Fig. 1
C
), an epitome of robust pairing against spin polarizing fields in clean
superconductors. There, macroscopic coherence gets replaced by a spatially ordered phase in the
presence
of a partial spin polarization at low temperatures, i.e.
<
0
.
5
. The experimental
3
observation of a rapidly increasing
B
c2,//
at low temperature provides strong support to the existence
of the FFLO state in organic superconductors
(17)
. In Ising super
conductors, however, it is the
spin split band structure that imposes a similar renormalization to the
G
-
L
formula at
.
Unfortunately, the relevant magnetic field regime in the phase diagram as
0
is difficult to
access for established Ising super
conductors due to technical limitations in the attainable magnetic
fields.
Here, we address this potential divergence
of
B
c2,//
at low temperature and breakdown of the G
-
L
formula, that should be characteristic for Ising superconductivity, in epitaxial t
hin films of
-
Sn(111), also referred to as few layer stanene
(18, 19)
. This material has recently emerged as a 2D
superconductor. We first establish on theoretical grounds that it too is an Ising 2D superconductor
although of a different type than those
previously reported which derive their Ising nature from
broken inversion symmetry and the spin
-
orbit driven splitting of valleys located at different points
of the Brillouin zone. We assert that in few layer stanene, which possesses both centrosymmetry
and spin
-
degenerate Fermi pockets at the
Γ
-
point, spin
-
orbit
al
locking and reversed effective
Zeeman fields for bands with different orbital indices produce the out
-
of
-
plane spin orientation
and spin splitting required for Ising pairing. We will refer to
this pairing mechanism as type
-
II
Ising superconductivity.
Incidentally
, in this system the in
-
plane magnetic fields needed to
demonstrate the failure of the G
-
L
formula and the divergence of the in
-
plane critical field, that so
far remained elusive in exp
eriment, are within reach.
Figure 2
A
illustrates the atomic structure of trilayer stanene grown on PbTe/Bi
2
Te
3
/Si(111)
substrates with low
-
temperature molecular beam epitaxy
(18)
. Few
-
layer stanene itself has no M
z
mirror symmetry and is centrosymmetri
c
(20,21)
, although surface decoration and the substrate
weakly break inversion symmetry of the films under study (Fig. 2
A
). Figure 2
B
sketches the band
structure of the trilayer. This 3D rendering is based on angle
-
resolved photoemission spectroscopy
(ARP
ES) data as well as first
-
principles calculations
(19)
. In the vicinity of the Fermi level, a
linearly dispersing hole band surrounds a small electron pocket at the
Γ
-
point, giving rise to two
-
band superconductivity
(19)
. Both the atomic and electronic str
ucture are remarkably distinct from
the widely studied transition metal dichalcogenides, for which the existence of M
z
mirror
4
symmetry,
inversion symmetry
breaking, and Fermi pockets at momenta that are not time
-
reversal
invariant (near K and K’) are essen
tial for realizing Ising superconductivity. The absence of this
mirror symmetry as well as broken inversion symmetry necessitates an alternative mechanism in
stanene to produce the out
-
of
-
plane spin
orientations required for Ising pairing. It should not re
ly
on inversion symmetry breaking and in addition be applicable for spin
-
degenerate Fermi pockets
near time reversal invariant momenta. Hence, we have termed this mechanism type
-
II Ising
superconductivity
(22)
in order to distinguish it from previous insta
nces of Ising superconductivity.
We formulate our model based on ARPES results and first
-
principles calculations, and focus on
the bands involving the
-
and
-
orbitals of Sn as they are the most relevant for electronic
conduction. The SOC lifts the
four
-
fold degeneracy at the
Γ
-
point (Fig. 2
B
) and results in two sets
of spin
-
degenerate bands mainly composed of
(
|
+
,
|
)
(solid circles in Fig. 1
D
)
and
(
|
+
,
|
)
(dashed circles in Fig. 1
D
)
, respectively
,
where
+
and
refer to the
+
and
orbitals
(20)
.
Due to spin
-
orbit locking (Fig. 1
D
), bands with different orbital indices
experience an opposite out
-
of
-
plane effective Zeeman field. It is parametrized as
SO
(
k
)
and
is
strongly
-
dependent. It is extraordinarily large at the
Γ
-
point itself, where it produces a splitting
of approximately 0.5 eV in monolayer stanene, which is equivalent to a field of about 10
3
Tesla.
However, it significantly weakens due to inter
-
orbital mixing at larger
, since an in
-
plane
magnetic field contri
butes a perturbation term to the Hamiltonian proportional to
+
,
|
|
+
,
, where
is the Pauli matrix.
T
his term is zero for
=
0
and exerts
increasing influence at larger
. Even though
SO
(
k
)
decreases
moderately
with film thickness
in
few
-
laye
r stanene
as a consequence of reduced band splitting in a quantum well setting, Ising
-
like
pairing between
|
+
and
|
within the Fermi pockets near the
Γ
-
point is expected to persist in
few
-
layer stanene and this pairing is anticipated to be robust against in
-
plane magnetic fields.
Panel
C
in Fig. 2
presents
the temperature dependent sheet resistance down to 250 mK of a sample
consisting of trilayer stan
ene that has been grown on a 12 layer thick PbTe buffer (i.e. 3
-
Sn/12
-
PbTe). Details of the sample preparation and measurement techniques are deferred to the
supplementary materials
. The observed transition temperature equals 1.1 K. Figures 2
D
and
E
displ
ay color renditions of the sample resistance in the parameter space spanned by the temperature
and either the perpendicular (
D
) or the in
-
plane (
E
) magnetic field. They reflect the phase diagram
5
of the superconducting ground state. The white color correspo
nds to approximately half of the
normal state resistance and, hence, demarcates the phase transition to the normal state and also
traces the temperature dependence of the upper critical magnetic fields marked by open circles.
Close to the transition temper
ature
T
c
, both
B
c2,
-
T
and
B
c2,//
-
T
follow the 2D G
-
L formula
(19)
and
deviations only become apparent at lower temperatures. The out
-
of
-
plane upper critical field
B
c2,
-
T
exhibits an upturn which is properly captured by the formula of a two band supercond
uctor
(23)
(solid black curve in Fig. 2
D
) that considers the orbital effect of the perpendicular magnetic field.
However, when the magnetic field is applied parallel to an ultrathin superconductor,
superconductivity is primarily suppressed by the paramagne
tic effect and the two
-
band formula
reduces to a simple square root dependence on
T
c
(23)
, indistinguishable from that of the 2D G
-
L
formula (pink curve in Fig. 2
E
). Clearly, such a two
-
band treatment fails to describe
the
enhancement in the in
-
plane upper
critical field observed in experiment, which amounts to 1 Tesla
by cooling below
T
= 0.2 K. Moreover, the in
-
plane upper critical field is about one order of
magnitude higher than the out
-
of
-
plane field. It exceeds the Pauli limit by a factor of 2
-
4 (Fig.
3
A
), assuming the common estimate of
B
p
=1.86
T
c,0
. This indicates that an unusual mechanism
renders the Cooper pairs robust against in
-
plane fields. The spin
-
flip scattering mechanism
(10)
can be readily ruled out, as it fails to agree with the experimental data (light blue curve in Fig. 2
E
marked as KLB). A full theoretical derivation of the temperature dependence of
B
c2,//
based on
atomic orbitals of the system is presented in the supple
mentary materials. The lower curve in Fig
.
3
A
shows the close agreement between model and experiment. They both demonstrate a
prominent upturn feature in the low temperature regime, which deviates from both the 2D G
-
L as
well as the spin scattering formul
a. This upturn behavior as temperature drops is the
key
characteristic of Ising superconductivity that remained elusive in previous experiments. Its
physical origin can be traced back to the peculiar spin
split bands associated with different orbitals
as s
hown in Fig. 1
D,
which are protected by the crystal structure. At
T
close to
T
c
, thermal
activation results in a
partia
lly polarized system, suppressing the contribution of the spin
-
orbit
induced spin
split effect on
B
c2,//
. Data in this regime therefore
overlap with the 2D G
-
L formula.
As
T
approaches zero, however, the spin orientation gets frozen and causes the upturn of
B
c2,//
.
Quantitatively, the dimensionless
parameter
푆푂
,
0
controls the deviation point between the
enhancement behavior charac
teristic for “Ising” superconductivity enhancement and the behavior
6
governed by the G
-
L formula (
T
c,0
denotes the zero field superconducting transition temperature).
Note here
푆푂
is the disorder renormalized
SOC
strength (see supplementary materials).
Typically,
푆푂
/
,
0
4
in our samples (3
-
Sn/12
-
PbTe for example)
and
a clear up
-
turn appears at
/
,
0
0.6.
In Fig
.
3 we also examine the role of the sample design to substantiate our model, in particular the
number of layers of
-
Sn as wel
l as the PbTe buffer layer thickness serve as parameters. Figure 3
A
compares the upper critical field of two trilayer stanene samples with differing PbTe buffer layer
thicknesses. The latter is known to raise the position of the Fermi level as its thickne
ss is increased
due to the donation of carriers from PbTe
(19)
. This results in a lower
T
c,0
for trilayer stanene on
six
-
layer PbTe. Although this sample possesses a higher
B
c2,//
/
B
p
at
0
compared to the
previously examined 3
-
Sn/12
-
PbTe sample, the divergence is missing. We attribute this
smoothening to the variation of the spin locking strength as one moves away from the
Γ
-
point
along the inverted Mexican hat band shape (inset to Fig. 3
A
). Spins of the
|
+
and
|
orbitals
are strongly locked out
-
of
-
plane at the
Γ
-
point. This Ising
-
like orientation, however, becomes less
favorable at larger momenta. Lowering the Fermi level therefore suppresses the spin polarization
of the outer hole band,
which can be
simulated
by an effective Rashba term in the Hamiltonian.
The experimental data can be fitted well by taking into account this effect. The modified formula
also nicely describes the upper critical fields of bilayer stanene (Fig. 3
B
). Here, in
version
symmetry breaking is stronger as the top Sn layer is decorated by hydrogen atoms while the bottom
Sn layer sits on Te atoms of PbTe. In comparison, the middle layer of a trilayer stanene retains
inversion symmetry. Following this argument, a penta
-
layer stanene better preserves the inversion
symmetry and thus experiences weaker Rashba effect, giving rise to an apparent enhancement of
B
c2,//
at low
T
in Fig. 3
C
. These observations highlight that the
-
Sn layer thickness is a key
ingredient to revea
l the delicate features of Ising superconductivity, and that there may exist a
broader range of materials hosting such pairing mechanisms
without the participation of
inversion
symmetry
breaking
.
7
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9
Acknowledgments:
W
e thank Yijin
Zhang for fruitful discussions.
Funding
:
This work is financially supported by the National Natural Science Foundation of China
(grant No. 11790311, 11604176
, 51788104
); the Ministry of Science and Technology of China
(2017YFA0304600,
2017YFA0302902
,
2018YFA0307100,
2018YFA0305603,
2016YFA0301001
); and the Beijing Advanced Innovation Center for Future Chip (ICFC).
Author
contributions
:
J.F. and D.Z. performed the low temperature electrical measurements with
the assistance of M. L. Y. Z., K. Z., and K.
H. grew the samples. Y. X., C. W.,
Z. Z.,
and W. D.
carried out first
-
principles calculations and theoretical analysis. H.
-
W. L. derived the microscopic
model of superconductivity with the assistance of H.
-
C. L. J. F., Y. X., H. L., J. H. S., and D. Z.
an
alyzed the data and wrote the paper with input from Q.
-
K. X. All authors discussed the results
and commented on the manuscript
.
10
Fig. 1
.
Mechanisms for an enhanced in
-
plane upper critical field.
(
A
) spin
-
flip scattering:
electronic spins get randomi
zed via scattering with impurities. (
B
) Type
-
I Ising superconductivity:
pairing of electrons on opposite spin split valleys. Here, only one pair of electron pockets centered
on K and K’ points
are highlighted
.
(
C
)
Fulde
-
Ferrell
-
Larkin
-
Ovchinnikov (
FFLO
)
state: Cooper
pairs form with
a
finite moment
um defined by the magnetic field
.
Only a small section of the Fermi
surface can host pairs (solid curves). Due to this finite momentum
, the order parameter gets
spatially modulated along the same direction,
i.e.
Δ
=
Δ
0
풒풓
.
(
D
) Type
-
II Ising superconductivity:
pairing of charge carriers on orbits around
Γ
point with their spins aligned in the out
-
of
-
plane
orientation. Hole bands are illustrated here as an example. Electron bands or
bands with a
more
complic
ated dispersion are also allowed as long as the spin splitting is caused by the same SOC.
The red and blue circles indicate two energetically degenerate bands with opposite spin
orientations, each of which have spin
split counterparts below the Fermi level
(indicated by the
dashed circles).
11
Fig. 2.
Superconducting properties of trilayer stanene.
(
A
) Atomic structure of hydrogen
decorated trilayer stanene on a PbTe
substrate. Dashed lines mark the three layers of Sn atoms.
Red dotted lines indicate the inversion symmetry. (
B
) Three
-
dimensional schematic of the band
structure of trilayer stanene. Blue and red circles reflect the hole/electron bands intersecting with
the Fermi level. The right panel illustrates the band splitting around the
Γ
point due to SOC. (
C
)
Temperature dependent sheet resistance of trilayer stanene grown on 12 layers of PbTe. (
D
)/(
E
)
Color coded resistance of the trilayer stanene on 12
-
PbTe as a
function of vertical/in
-
plane
magnetic field at a set of temperature points. The white stripe represents the boundary between the
superconducting (SC) and normal state. Circles represent the magnetic fields where the resistance
becomes 50%
of the normal s
tate resistance
R
n
at a fixed
T
. Due to the smooth nature of this
transition,
when determining
B
c2
by using
another definition
such as 1%
R
n
or 10%
R
n
would not
change the general temperature dependent behavior obtained. Solid curves are theoretical fits. Th
e
solid curve in
D
is based on the formula derived for a two
-
band superconductor
(23)
. The blue
curve in
E
is obtained using the formula that takes into account the spin
-
flip scattering as derived
by Klemm, Luther and Beasley (KLB)
(10)
. The pink curve in
E
is based on the 2
-
dimensional
12
Ginzburg
-
Landau formula
(19)
. The white dashed line marks the Pauli limit using the standard
BCS ratio
(3)
as well as a
g
-
factor of 2.