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Ising
pairing in atomically thin superconductors
Ding Zhang
1, 2, 3, 4,
∗
and Joseph Falson
5, 6,
†
1
State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics,
Tsinghua University, Beijing 100084, China.
2
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan.
3
Beijing Academy of Quantum Information Sciences, Beijing 100193, China.
4
Frontier Science Center for Quantum Information, Beijing 100084, China.
5
Department of Applied Physics and Materials Science,
California Institute of Technology, Pasadena, CA, USA.
6
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA, 91125, USA
(Dated: August 28, 2021)
Abstract
Ising-type pairing in atomically thin superconducting materials has emerged as a novel means of gen-
erating devices with resilience to a magnetic field applied parallel to the two-dimensional plane. In this
mini-review, we canvas the state of the field by giving a historical account of two-dimensional supercon-
ductors with strongly enhanced in-plane upper critical fields, together with the type-I and type-II Ising
pairing mechanisms. We highlight the vital role of spin-orbit coupling in these superconductors and discuss
other e
ff
ects such as symmetry breaking, atomic thicknesses, etc. Finally, we summarize the recent theo-
retical proposals and highlight the open questions, such as exploring topological superconductivity in these
systems and looking for more materials with Ising pairing.
∗
dingzhang@mail.tsinghua.edu.cn
†
f
alson@caltech.edu
1
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I.
INTRODUCTION
Two years after his discovery of superconductivity, Kamerlingh Onnes envisioned that super-
conductors could be used for generating strong magnetic fields on the order of tens of Tesla [1].
It soon turned out that superconductors discovered in the earlier times, so-called type-I super-
conductors, easily lost their dissipationless property in a magnetic field with a field strength as
low as tens of millitesla. It was until the discovery of type-II superconductors that Onnes’ initial
dream became practical. Today, scientists around the world use such superconductors to generate
high magnetic fields on a daily basis in medical and research settings. The field strengths of su-
perconducting magnets stay on the order of tens of Tesla, which are governed by a fundamental
parameter of the superconductor–the upper critical field (
B
c
2
). For type-I superconductors, the
magnetic field either gets fully screened or penetrates through the whole compound and destroys
superconductivity. A first order transition occurs at a critical field of
B
c
. For type-II superconduc-
tors, however, the magnetic field lines can be first trapped in a bundle of pinholes–vortices with
quantized fluxes, leaving the rest of the bulk still superconducting. The onset of such a mixed state
is marked by the lower critical field–
B
c
1
. It is until the density of the vortices, which increases with
the magnetic field, reaches a certain threshold that the superconducting state and the normal state
become energetically indistinguishable. At this density with a corresponding magnetic field–
B
c
2
,
the superconductor returns to the normal state.
One could lift the upper bound imposed by the vortex e
ff
ect by drastically reducing the cross-
section of a superconductor in the magnetic field. In other words, a thin film superconductor can
be much robust against an in-plane magnetic field. The Ginzburg-Landau theory predicts that such
an in-plane upper critical field–
B
c
2
;
∥
increases rapidly with the reduction of film thicknesses [2].
In ultrathin superconductors, however, the orbital e
ff
ect brought by vortices becomes negligible.
Instead, the spin paramagnetism emerges as the main mechanism that brings the superconductor
with singlet pairing back to the normal state. A su
ffi
ciently strong magnetic field can polarize
all electronic spins such that Cooper pairs with anti-aligned spins break up [3, 4]. This magnetic
field is the so-called Pauli limit:
B
p
. By equating the superconducting binding energy with the
paramagnetic energy of the normal state, one obtains that
B
BCS
p
=
1
:
86
T
c
0
(T), where
T
c
0
is the
transition temperature. We add BCS in our notation because the formula above is obtained by
assuming a standard BCS ratio of 2
∆=
3
:
53
k
B
T
c
0
for weak superconductors and a
g
-factor of 2.
These assumptions are not necessarily followed in realistic situations. For example, the gap to
T
c
0
2
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ratio
may be much larger than the BCS ratio [5]. On the other hand, the
g
-factor can di
ff
er from
2, as pointed out in literatures [6, 7]. The aforementioned e
ff
ects may result in a value of
B
p
that
deviates from
B
BCS
p
. Here, however, we employ
B
BCS
p
as a conventional tool to normalize
B
c
2
such
that
B
c
2
=
B
BCS
p
serves as the figure-of-merit when di
ff
erent superconductors are compared.
In aluminum films, Tedrow and Meservey [2] demonstrated that the spin paramagnetic e
ff
ect
was largely obeyed, although the additional e
ff
ect of spin-orbit scattering should have been taken
into account. Two 5-nm thick aluminum samples, for example, possessed slightly enhanced
T
c
0
of 2.04 and 2.15 K, in comparison to the bulk value of 1.2 K. Their corresponding
B
c
2
;
∥
were 4.01
and 4.05 T, close to the calculated
B
BCS
p
of 3.79 and 4.00 T (One exemplary data set digitized from
the original paper is shown in Fig. 2a). Furthermore, the aluminum thin films hosted the predicted
first-order transition to the normal state at low temperature, due to the spin-paramagetic e
ff
ect.
Since it was derived based on a simple model regarding the electronic spin, the Pauli limit
can be surpassed in situations where the spin orientation departs from the ideal isotropic case. In
superconducting thin films incorporated with heavy elements, spin-orbit scattering can randomize
the spins and substantially weaken the polarizing e
ff
ect of the magnetic field [8]. Thin films may
also experience the breaking of inversion symmetry along the normal direction from the substrate
to the top surface. It leads to the Rashba e
ff
ect that tends to align the spins in di
ff
erent directions in
the plane, which can essentially enhance
B
c
2
up to
√
2
B
B
CS
p
[9]. In clean superconductors, above
B
BCS
p
, the uniformly superconducting state can get replaced by a spatially ordered state –Fulde-
Ferrell-Larkin-Ovchinnikov (FFLO) state with partial spin polarizations [10, 11]. Another exotic
scenario is spin triplet pairing where the system would not be a
ff
ected by the Pauli paramagnetic
e
ff
ect at all.
By the end of 2015, it became clear that
B
c
2
;
∥
in the atomically thin highly crystalline supercon-
ductors [12–14] can greatly surpass
B
BCS
p
. Yet none of the aforementioned mechanisms were able
to explain this enhancement. It marked the advent of so-called Ising superconductors. To date,
the Ising superconductivity had been discovered in many transition metal dichalcogenides [6, 15–
18], as shown in Fig. 1. Their robustness of superconductivity against the in-plane magnetic field
originates from the strong spin-orbit coupling together with the breaking of in-plane inversion
symmetry, as was considered earlier for non-centrosymmetric superconductors of bulk crystal-
s [19, 20]. Lately, enhanced
B
c
2
in atomically thin superconductors without breaking in-plane
inversion symmetry has also been reported [21, 22] (Fig. 1). It suggests a distinct type of Ising
pairing in materials with strong spin-orbit couplings hosting multi-degeneracies at high symmetry
3
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points
[23, 24]. Further theoretical work not only outlines a broader material pool of potential
Ising superconductors but also suggests a possible realization of topological superconductivity in
these systems [25–39].
This mini-review aims at providing a progress report of atomically thin superconductors with
enhanced
B
c
2
that are mostly attributed to Ising pairing (Fig. 1). Reviews on a broader scope or
some other specific aspects of two-dimensional (2D) superconductors can be found in Refs [40–
44]. Here we collect data points from the reported Ising superconductors, hoping to construct a
unique perspective at the enhancement of
B
c
2
. We discuss the influence of spin-orbit coupling
strengths, sample thicknesses, etc. while ending with some open questions in the field.
II. DISCOVERY OF ISING SUPERCONDUCTORS
Thanks to technical advancements, there are now a number of viable pathways to creating ultra-
thin crystals that can show superconductivity [40, 42]. One direction is to directly synthesize thin
films of a material, for example through molecular beam epitaxy (MBE). Another is through me-
chanical exfoliation where sheets of van der Waals materials can be isolated down to a few atomic
layers. Alternatively, it is possible to study the physics of thin layers by selectively accumulating
carriers on the surface of bulk crystals, for example with ionic liquid (IL) gating. The enhanced
B
c
2
;
∥
has become one intriguing property of these 2D superconductors. Here we give a historical
account of selected 2D superconductors with enhanced
B
c
2
;
∥
. Figure 1 acts to guide our discussion
by plotting the ratio of
B
c
2
;
∥
/
B
BCS
p
from reports in the literature according to their publication date.
In a 2012 systematic study of
-doped SrTiO
3
heterostructures, Kim et al. [45] noted that
B
c
2
;
∥
(
T
→
0) exceeded
B
BCS
p
by a factor of 4. We show one exemplary data set digitized from
the original paper in Fig. 2e. An earlier study on LaAlO
3
/
SrTiO
3
interface superconductors [46]
already found a large
B
c
2
;
∥
=
B
BCS
p
ratio of 3.5. The
-doped films embedded between cap and bu
ff
er
layers of SrTiO
3
has the advantage due to its symmetric structure. It helps exclude the Rashba ef-
fect as the mechanism for the enhancement. For a quantitative analysis, Kim et al. [45] employed
the Werthamer-Helfand-Hohenberg (WHH) formula:
ln
(
T
c
0
T
)
=
(
1
2
−
i
SO
4
)
1
2
+
̄
h
+
1
2
SO
−
i
2
T
=
T
c
0
+
(
1
2
+
i
SO
4
)
1
2
+
̄
h
+
1
2
SO
+
i
2
T
=
T
c
0
−
(
1
2
)
;
(1)
4
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where
is
the digamma function and the rest of the terms are:
̄
h
=
ev
2
F
T
B
3
k
B
T
c
0
;
(2)
SO
=
2
~
3
k
B
T
c
0
SO
;
(3)
=
√
(
B
B
k
B
T
c
0
)
2
−
1
4
2
SO
;
(4)
where
v
F
is
the Fermi velocity,
T
/
SO
the transport
/
spin-orbit scattering time,
B
the Bohr mag-
neton. With the sample getting thinner, they found an unusual decrease of the extracted
SO
while
T
stayed almost unchanged. This dichotomy suggested that the intrinsic spin-orbit coupling may
play a role in giving rise to the large
B
c
2
;
∥
(
T
→
0)
=
B
BCS
p
ratio.
Indications for the intrinsic e
ff
ect of spin-orbit coupling were also found in 2013. Sekihara,
Masutomi and Okamoto studied submonolayer of Pb films grown on GaAs(110) substrate [47]
(Fig. 2e). There, the superconducting transition temperature dropped by as little as 2% in the
in-plane magnetic field up to 14 T. The following equation was employed to fit the data:
ln
(
T
c
0
T
)
=
(
1
2
+
3
SO
2
~
(
B
B
)
2
2
k
B
T
)
−
(
1
2
)
:
(5)
This
equation is usually refereed to as the Klemm-Luther-Beasley (KLB) formula [8]. It can be
derived from Eq. (1) when considering strong spin-orbit scattering for superconductors in the dirty
limit, i.e.
SO
→ ∞
and
T
<<
SO
. The extracted
SO
was found to be comparable with the esti-
mated
T
of the Pb films:
T
∼
0
:
8
SO
. This is unusual because
SO
should be orders of magnitude
larger than
T
. The authors therefore proposed that the intrinsic Rashba spin splitting in their sys-
tem may give rise to an inhomogeneous superconducting state similar to the FFLO state, which
might account for the enhanced
B
c
2
;
∥
. A similar conclusion was arrived by Nam et al. [48] when s-
tudying their five-monolayer Pb films grown on Si(111) substrates. The researchers suggested that
the orbital e
ff
ect, which is often neglected when considering purely a 2D superconductor, may be
still responsible for pair-breaking because the spin degree of freedom is quenched. It highlights
the importance of a close inspection of theoretical expectation of
B
c
2
;
∥
from di
ff
erent mechanisms.
The breakthrough of identifying a novel mechanism di
ff
erent from Rashba spin splitting
or spin-orbit scattering happened in 2015
/
16. Three individual groups observed strongly en-
hanced
B
c
2
;
∥
in their atomically thin superconductors of 2H-MoS
2
(Ref.[12 and 13]) or 2H-NbSe
2
(Ref.[14]) (Fig. 2a). Experimentally, the researchers obtained atomically thin crystalline super-
conductors via two approaches. For thin flakes of 2H-MoS
2
, Lu et al. [12] and Saito et al. [13]
5
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emplo
yed the ionic liquid gating technique to electrostatically inject charge carriers and tune the
top one to two layers into the superconducting state. On the other hand, Xi et al. [14] were able to
mechanically exfoliate 2H-NbSe
2
down to a monolayer and still preserve the superconductivity.
They found that
B
c
2
;
∥
exceeded
B
BCS
p
by a factor of 4 to 6. Notably, the extracted
SO
by using
Eq. (5) to fit the data became shorter than
T
. This is unreasonable because the transport scattering
time should be the shortest. By ruling out the spin-orbit scattering mechanism, the researchers
proposed an intrinsic mechanism. It arises from the strong spin-orbit coupling together with the
lack of in-plane inversion symmetry in the individual atomic layers of MoS
2
and NbSe
2
. Taking a
monolayer of MoS
2
–one S-Mo-S sequence–for an example, its band structure consists of electron
pockets located around the
K
and
K
′
points of the Brillouin zone. Strong spin-orbit coupling
induces spin splitting of the bands. Due to the broken inversion symmetry of its crystal structure,
the spin splitting has opposite signs at the
K
and
K
′
points. In the superconducting state, Cooper
pairs are built up by the electrons from the two valleys. Their spins are locked to the out-of-plane
direction due to the rotational symmetry. This is schematically shown in Fig. 1. An in-plane mag-
netic field therefore has to compete with the Zeeman-like spin splitting built into the bands, thus
the superconducting state becomes more resilient to the paramagnetic e
ff
ect. Such a mechanism
in 2D superconductors is called Ising superconductivity.
In fact, an enhanced
B
c
2
;
∥
over the Pauli limit was proposed earlier in bulk crystals without sym-
metry centers–noncentrosymmetric superconductors by Bulaevskii, Guseinov, and Rusinov [19]
as well as by Frigeri et al. [20]. By employing the specific Hamiltonian suitable for transition
metal dichalcogenide monolayers, Ilic, Meyer and Houzet [49] derived the following formula for
the temperature dependence of the upper critical field for Ising superconductivity:
ln
(
T
c
0
T
)
=
2
B
B
2
∗
2
SO
+
2
B
B
2
Re
1
2
+
i
√
∗
2
SO
+
2
B
B
2
2
k
B
T
−
(
1
2
)
;
(6)
where
∗
SO
is
the e
ff
ective spin-orbit coupling strength, which may be renormalized from the in-
trinsic spin-orbit coupling by the disorder scattering [5].
The concept of Ising superconductivity was further bolstered by the subsequent studies of other
transition metal dichalcogenides monolayers. In a WS
2
monolayer, superconductivity was again
realized by the ionic liquid gating technique [15]. There, for the superconducting state with
T
c
0
=
1
:
54 K, Lu et al. found that the transition temperature only decreased by 5% when a strong in-plane
magnetic field of 35 T was applied [15](Fig. 2b). In a monolayer exfoliated from 2H-TaS
2
crystals,
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de
la Barrera et al. found that
B
c
2
;
∥
exceeded the highest magnetic field of their magnet (34.5 T)
when the temperature was below 60% of
T
c
0
[16] (Fig. 2b). They further carried out a systematic
study by comparing the enhanced
B
c
2
;
∥
in both TaS
2
and NbSe
2
. The larger
B
c
2
;
∥
=
B
BCS
p
ratio in TaS
2
originated from its stronger spin-orbit coupling strength, which was consistent with the density
functional theory calculations. Enhanced
B
c
2
;
∥
was also found in monolayer WTe
2
[50, 51] as well
as few-layers of 1T
d
-MoTe
2
[52] (Fig. 2f). A variant of Ising superconductivity was proposed that
considered the tilted spins in the momentum space. Recently, Rhodes et al. [53] further observed
enhanced
B
c
2
;
∥
in a monolayer of MoTe
2
. These materials suggest a broader scope for locked spins
in superconductors with strong spin-orbit couplings [54].
We remark that there exist two di
ff
erent approaches for estimating the spin-orbit coupling
strength for Ising pairing. Some studies employed Eq. (5) for fitting their data but interpret
3
~
=
2
S O
as
S O
[14, 16]. Others used the formula for Ising superconductivity–similar to Eq. (6)–
to extract
∗
S O
[12, 15]. Due to the completely di
ff
erent origins, the fitted values can greatly
di
ff
er. For example, applying Eq. (5) directly to the data of MoS
2
(orange curve in Fig. 2a) gives
rise to 3
~
=
2
S O
=
18 meV. By contrast, the best fit to the same data set by using Eq. (6) yields
∗
S O
=
3
:
4 meV. This value is slightly smaller than the reported value of 6 meV in literature [12].
This di
ff
erence stems from the fact that Eq. (6) does not take into account the Rashba e
ff
ect. An-
other example is NbSe
2
. Applying Eq. (6) to the data set in Fig. 2a gives rise to
∗
S O
=
4
:
1 meV,
which is much smaller than the reported value: 2
S O
=
76 meV [14] when Eq. (5) was employed.
One should therefore be cautious in drawing comparison between the spin-orbit coupling strengths
based on the listed values in the literatures.
Figure 3 compares attributes of transition metal dichalcogenide superconductors based upon
their layer number and chemical composition. We draw upon two key parameters; the thickness
of samples studied and the cation size. While the data set is incomplete, a trend is visible in both
graphs. The
B
c
2
;
∥
=
B
BCS
p
ratio is enhanced by both thinning the sample (Fig. 3a) and enhancing
the cation size (Fig. 3b). Thinning the sample, preferably to the monolayer limit, is necessary to
realize the inversion symmetry breaking of the lattice. It is nevertheless important to note that the
enhancement in the upper critical field persists even in multi-layered NbSe
2
and TaS
2
. It indicates
that the local inversion symmetry breaking in the plane is su
ffi
cient for spin-orbital locking, al-
though the global inversion symmetry is not broken in multi-layer systems. The e
ff
ect of cation
is associated with the increase in spin-orbit coupling with its increasing
Z
value, which enters the
superconducting behavior in the form of
∗
SO
. The data in Fig. 3 also highlight that the enhanced
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B
c
2
;
∥
is
a reproducible result. Three independent studies of NbSe
2
[6, 14, 16] present similar val-
ues of
B
c
2
;
∥
=
B
BCS
p
at the lowest measured temperatures, along with the value obtained at
T
=
0
through a KLB fit. Notably, one of the studies explored the macroscopic NbSe
2
sample grown by
MBE [6]. The authors employed a pulsed magnet in combination with ultra-low temperature facil-
ity to experimentally determine
B
c
2
;
∥
=
B
BCS
p
down below 0
:
15
T
c
0
. The large enhancement in MBE
grown samples is of technical importance for further investigations of Ising pairing and potential
applications.
Among the transition metal dichalcogenide superconductors, a missing feature of Ising super-
conductivity predicted by theory was the up-turn behavior of
B
c
2
;
∥
at low temperature as described
by Eq. (6). In monolayer cases of NbSe
2
and TaS
2
[14, 16] as well as in the ionic liquid gated
WS
2
[15], the upper critical field quickly exceeded the highest magnetic field available when the
temperature is still close to
T
c
0
, such that the behavior of
B
c
2
;
∥
at
T
→
0 was unattainable. For
bilayer NbSe
2
, trilayer TaS
2
, as well as ionic liquid gated MoS
2
[13], on the other hand,
B
c
2
;
∥
showed a saturating behavior at
T
=
T
c
0
<
0
:
5. It was suggested that the Rashba e
ff
ect [12, 13]
and intervalley scattering [49] in these films can renormalize the temperature behavior so that the
up-turn at low temperature gets smeared out.
The non-saturating behavior was first found in a two-dimensional superconductor outside the
transition metal dichalcogenide family. This time, samples grown by MBE became the protagonist.
Liu et al. measured thin films of Pb grown on a special surface reconstruction layer of Pb–striped
incommensurate phase–on Si(111) substrate. They observed pronounced enhancement of
B
c
2
;
∥
[5].
There, a more realistic
B
p
rather than
B
BCS
p
was estimated based on the known gap to
T
c
ratio of
Pb. Intriguingly, they observed that
B
c
2
;
∥
in the Pb films with a thickness of six monolayers kept
increasing with decreasing temperature down to
T
∼
0
:
25
T
c
0
(Fig. 2b). Although the crystal
structure of Pb itself is centrosymmetric, the authors pointed out that their special substrate hosted
lattice distortion with its symmetry breaking e
ff
ect permeating to thin films grown on top. Together
with the strong spin-orbit coupling, ultrathin Pb films therefore possessed Ising superconductivity.
In another elemental superconductor–Sn, Liao et al. [55] found that
B
c
2
;
∥
could exceed
B
BCS
p
by a factor of 3. Their superconductors were made of a few atomic layers of strained
-Sn–so-
called stanene–grown by MBE on PbTe
/
Bi
2
Te
3
epitaxial layers upon a silicon (111) substrate. By
measuring few-layer stanene at even lower temperatures down to 0.02
T
c
, Falson et al. [21] were
able to map out almost the complete phase diagram of
B
c
2
;
∥
(
T
). The same sample, as studied in
Ref. [55] actually hosted an ever-growing
B
c
2
;
∥
as
T
decreased such that
B
c
2
;
∥
=
B
BCS
p
eventually
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xceeded 4 (Fig. 2c). Of particular interest, was the up-turn behavior at low temperature, which
echoed the prediction for Ising superconductivity [Eq. (6)]. However, stanene is sharply di
ff
erent
from NbSe
2
or MoS
2
: its crystal structure is centrosymmetric; the electronic bands are around
the
Γ
point, as found by angular resolved photoemission spectroscopy (ARPES). The criteria of
the then established Ising superconductivity were not satisfied. A new theoretical framework was
proposed to explain this distinct type of behavior–type-II Ising superconductivity [23, 24]. Their
theory was applicable to centrosymmetric materials with multiple degenerate orbitals. Taking
Sn for example, bands around the Fermi level stem from
p
x
and
p
y
orbitals. Without spin-orbit
interactions, the orbitals have four-fold degeneracy at the
Γ
point:
|
p
x
⟩
,
|
p
y
⟩
, spin-up (
| ↑⟩
) and
spin-down (
| ↓⟩
). The spin-orbit coupling lifts one layer of degeneracy and gives rise to two sets
of bands: (
|
p
x
+
ip
y
;
↑⟩
,
|
p
x
−
ip
y
;
↓⟩
) and (
|
p
x
−
ip
y
;
↑⟩
,
|
p
x
+
ip
y
;
↓⟩
). Here the e
ff
ective Zeeman
splitting occurs between
|
p
x
±
ip
y
;
↑⟩
and
|
p
x
±
ip
y
;
↓⟩
. This spin-orbit locking e
ff
ect produces
essentially the same spin configurations required by Ising superconductivity, and is schematically
shown in Fig. 1. Based on this argument, Falson et al. [21] demonstrated that their theoretical
model could satisfactorily fit the experimental data. Moreover, the model also explained why the
up-turn was not always prominent even for the same triple layer Sn but on PbTe with di
ff
erent
thicknesses (Fig. 2c). It stemmed from the enhanced spin-tilting of the hole band as the Fermi
momentum increased with thinner PbTe. To further confirm this scenario, it is desirable to carry
out a more systematic study of the enhanced
B
c
2
;
∥
as a function of the Fermi level. We further
note that few-layer stanene constitutes a unique platform as properties close to
T
=
T
c
0
=
0 can be
addressed. Future experiments should be directed to the understanding of the transition from the
purely paramagnetism dominated regime to the orbital e
ff
ect dominated regime.
In fact, there exists not just one type-II Ising superconductor. In ultrathin PdTe
2
grown by
MBE, Liu et al. [22] observed largely enhanced
B
c
2
;
∥
, exceeding
B
p
by a factor of 6-8 (Fig. 2d).
Notably, the PdTe
2
compound also has a centrosymmetric crystal structure and hosts electronic
bands around
Γ
point. Theoretical analysis pointed out that the enhanced
B
c
2
;
∥
was caused by
type-II Ising pairing. Notably, few-layer stanene and PdTe
2
are quite robust against air-exposure.
By contrast, ultrathin NbSe
2
and TaS
2
are quite fragile and require handling in glove-boxes and
capping with protection layers. In retrospect, the enhanced in-plane upper critical field in liquid
gated SnSe
2
as well as 1T
′
-MoS
2
also might be accounted for by type-II Ising paired superconduc-
tivity [23, 56]. Together, these recent findings highlighted a broader scope of 2D superconductors
hosting largely enhanced
B
c
2
;
∥
due to the strong spin-orbit coupling.
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F
ollowing the initial experimental discoveries, further theoretical analysis suggests more exotic
physics lurking in Ising superconductors, with the in-plane magnetic field as a tuning parameter.
In order to experimentally verify many of the predictions, it is important to obtain information
beyond temperature and magnetic field dependent resistances. Attempts to reveal the density of
states in Ising superconductors have provided fruitful results. Both Sohn et al.[57] and Dvir et
al.[58] fabricated planar tunnel junctions on NbSe
2
down to a trilayer. Either a few layers of MoS
2
or a thin film of AlO
x
were used as the barrier. On the other hand, Costanzo et al. [59] utilized
the band bending in the liquid gated MoS
2
on multilayer graphene to realize a superconductor-
insulator-metal junction. All three groups found that the superconducting gap could be protected
by spin locking and stayed almost constant at small in-plane magnetic fields below around 10
T. Intriguingly, as revealed by Sohn et al., the gap gets continuously suppressed by ramping up
the in-plane magnetic field to about 38 T, rather than a first-order phase transition expected for
a BCS superconductor [57]. The continuous transition lends further support to the scenario of
Ising superconductivity. Due to the strong spin-locking of the Ising superconductor, its in-plane
spin susceptibility in the superconducting state stays comparable to that in the normal state. The
free energy of the superconducting state therefore continuously connects to that of the normal
state. Further exploring the potential of planar junctions, Kang et al. [60] fabricated a sandwich
structure consisted of NbSe
2
/
CrBr
3
/
NbSe
2
(CrBr
3
is a ferromagnetic insulator). They observed
nearly 100% tunneling anisotropic magnetoresistance.
III. OUTLOOK
One enticing property of Ising superconductors is the predicted non-trivial topology. Indi-
cations of unconventional pairing were recently found in few-layer NbSe
2
, which manifested as
two-fold anisotropy in the in-plane upper critical field [61]. In fact, possible
p
-wave pairing was
predicted theoretically for a monolayer of MoS
2
before the strongly enhanced
B
c
2
;
∥
was experi-
mentally observed [62]. A spin triplet
s
-wave was also proposed for certain on-site and nearest
neighbor interaction strengths. M
̈
ockli and Khodas predicted that the in-plane magnetic field may
induce nodes in the dispersion with Majorana flatbands connecting them, giving possibility of re-
alizing nodal topological superconductivity [31]. Similar conclusion was arrived by He et al. [29].
By further taking into account the possible Rashba splitting in monolayer NbSe
2
, Sha
ff
er et al. [35]
predicted that the system may enter a topologically trivial phase unless the in-plane magnetic field
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is
applied along one of
Γ
-
K
lines, where a crystalline topological superconducting phase occurs.
It was also proposed that nodal topological phases could be achieved in magnetic islands placed
on Ising superconductors [36].
As already alluded to, triplet pairing is another intriguing property that may accompany Ising
superconductivity. M
̈
ockli and Khodas [31, 32] suggested that the superconducting state in mono-
layer TMD becomes parity mixed with triplet pairing–particularly the
f
-wave. The mixing of
singlet and triplet pairing in monolayer NbSe
2
was also confirmed by Wickramaratne et al. [34]
through density functional theory (DFT) calculations. This triplet pairing may account for the
further enhancement of
B
c
2
;
∥
and can still persist when disorder e
ff
ect is taken into account [33]. It
can also a
ff
ect the density of states [63]. A recent experimental study on the superconducting gap
has indeed unveiled indication of such triplet pairing [64].
However, the triplet pairing may get masked by the dominant singlet pairing. Several theoret-
ical groups thus outlined ways to extract the triplet pairing component. Zhou et al. [25] proposed
that the triplet pairing in Ising superconductors can induce Majorana fermions in a semi-metal
wire via proximity e
ff
ect. The configuration is similar to the case of nano-wires with strong Rash-
ba splittings that are proximitized by conventional
s
-wave superconductors [65]. However, the
spin directions for the proposed semi-metal wire on a Ising superconductor lie in-plane instead
of out-of-plane. The induced superconductivity was further proven to be robust against the in-
plane Zeeman fields [66]. In a similar spirit, spin triplet Andreev reflection was proposed to occur
between a ferromagnet-Ising superconductor junction [30]. An in-plane junction made of Ising
superconductor-ferromagnet-Ising superconductor may further allow the spin triplet Josephson
current to pass [67]. However, constructing such an in-plane junction may require technical inge-
nuity. In general, heterostructures involving Ising superconductors serve as test-beds not only for
further validating Ising pairing but also for achieving exotic properties.
Experiments have shown that the spin-locking e
ff
ect persists even in bilayer or trilayer of NbSe
2
superconductors, for example. It indicates that the local inversion symmetry breaking is su
ffi
cient
for Ising superconductivity, even though the material itself is globally centrosymmetric. By con-
sidering the bilayer case, Liu [27] as well as Nakamura and Yanase [28] predicted that odd-parity
superconductivity may occur. This unconventional superconductivity manifests itself either as an
inhomogeneous FFLO state or a pair-density-wave state under an in-plane magnetic field. How-
ever, it is experimentally quite challenging to detect these exotic states. The current techniques for
measuring the FFLO state are often developed for bulk materials. Detecting the pair-density wave
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further
requires the understanding of the local density of states in these ultrathin superconductors
by using scanning tunneling microscopy (STM).
Carrying out tunneling spectroscopy on Ising superconductors is still under-explored. Many
predicted properties such as a ”mirage” gap at high energies of the order of
S O
[38] remain to be
verified. It is also of particular interest to obtain the density of states in the regime where the tem-
perature dependent
B
c
2
;
∥
shows an up-turn, such as in 6-Pb, few-layer stanene, as well as PdTe
2
.
Theories predicted unconventional pairing and possible topological properties in this regime. Fur-
thermore, it is favorable to probe the local density of states. Theoretically, the interplay between
Ising superconductors and magnetic impurities were considered by Sharma and Tewari [26] as
well as by Zhang et al. [37]. They predicted features that could be identified by STM. Here we
list some of the technical challenges of STM investigation on Ising superconductors and possible
solutions: 1. Many Ising superconductors are mechanically exfoliated flakes with typical sizes
of tens of micrometers. Specially designed markers [68] or optical lens [69] may be necessary
to help locate the flake with the tunneling tip. Alternatively, a macro-scale sample, realized for
example by MBE, may be favorable; 2. the ionic liquid gating employed, for example in inducing
superconductivity in MoS
2
and WS
2
Ising superconductors, is incompatible with STM. One may
employ alternative gating methods with similarly charge carrier tunability, such as ionic solid
/
gel
gating [70, 71]; 3. the exotic phases of Ising superconductors, predicted by theory, often occur
at an in-plane magnetic field beyond the Pauli limit. To clearly probe into these phases, it may
also be necessary to measure down to low temperature such that
T
=
T
c
0
<
0
:
5. The combination
of these requirements make the STM measurement a formidable task. One feasible route may be
to realize Ising superconductors with lower
T
c
0
such that
B
c
2
;
∥
stays in the range o
ff
ered by a com-
mercially available magnet. Reduction of
T
c
0
can be achieved by modulating the carrier density in
the existing Ising superconductors.
In order to expand the experimental tools used for studying Ising superconductivity, one may
also need to search in the material pool for new systems [39]. There exist several reasons for this
proposal. First, many of the existing Ising superconductors require very high magnetic field to in-
duce the predicted exotic states. This is sometimes challenging to be incorporated with other tech-
niques such as STM. Second, nuclear magnetic resonance and thermal capacitance measurements,
which are powerful in identifying the FFLO state, may not be easily applied to those atomical-
ly thin superconductors. Superlattices, where superconducting sheets are protected by insulating
layers, as recently demonstrated by Devarakonda et al. [72] may provide one feasible route. This
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e
xpands the approach to encompass looking for bulk crystals with a stack of Ising superconduc-
tors separated by e
ff
ectively inert layers [73]. Apart from the type-I Ising superconductors that
require inversion symmetry breaking, type-II Ising superconductors rely on spin-orbit coupling
induced band splittings around high-symmetry points. By using this guiding principle, Wang et
al. [23] looked into the database of 2D materials and found a batch of potential candidates for
type-II Ising superconductivity, as shown in Fig. 4. This approach was based upon parameterizing
the strength of spin-orbit coupling vs Fermi momentum (carrier density). It was taken without
significant regard for whether the predicted density could be realized (many materials are in fact
semiconductors) or whether a material is actually superconducting. Nevertheless, a number of
interesting synthesis paradigms are revealed. While chalcogenides have a relatively rich history
of study, the properties of halides are remain relatively poorly explored, especially in the context
of thin films, which is the only viable path to making samples su
ffi
ciently thin if flakes can not
be exfoliated from larger crystals. In general, Ising superconductivity is a vibrant field that has
just shown its tip of an iceberg. Experiments led in the beginning, followed by theoretical under-
standings and predictions for novel physics. Now it is back to the task of experimentalists to probe
deeper and wider into this budding field.
ACKNOWLEDGMENTS
We thank Guangtong Liu, Yong Xu for kindly sharing their data. DZ acknowledges fund-
ing provided by the Ministry of Science and Technology of China (2017YFA0302902, 2017Y-
FA0304600); the National Natural Science Foundation of China (grant No. 11922409, 11790311);
the Beijing Advanced Innovation Center for Future Chips (ICFC). JF acknowledges funding pro-
vided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF
Grant PHY-1733907).
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