# Ising superconductivity: a first-principles perspective

Darshana Wickramaratne  Center for Computational Materials Science, U.S. Naval
Research Laboratory, Washington, DC 20375, USA  I.I. Mazin  Department of
Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA
Quantum Science and Engineering Center, George Mason University, Fairfax, VA
22030, USA

###### Abstract

The recent discovery of Ising superconductivity has garnered a lot of interest
due in part to the resilience of these superconductors to large in-plane
magnetic fields. In this Perspective we explain the basic concepts that define
the behavior of Ising superconductors, provide an overview of the electronic
structure and magnetic properties with a focus on NbSe2, summarize key
experimental observations that have been made in this class of
superconductors, highlight the role that defects and proximity-induced effects
at interfaces have on Ising superconductivity and finally discuss the
prospects for observing Ising superconductivity in bulk materials.

##  I Introduction

A fundamental concept in the theory of superconductors is the paramagnetic
limiting magnetic field (Clogston-Chandrasekar-Pauli limiting field) [1, 2],
henceforth referred to as the Pauli-limiting field. This is the approximate
magnetic field where superconductivity is suppressed when the Zeeman splitting
of the spin degenerate states at the Fermi level exceeds the magnitude of the
superconducting gap. Recently, superconductivity in a number of two-
dimensional materials was shown to be surprisingly resilient to an in-plane
magnetic field that greatly exceeded the Pauli limiting field. The first
report of this large in-plane critical field was in gated single monolayer
MoS2, a two-dimensional transition metal dichalcogenide (TMD) [3]. Shortly
thereafter large in-plane critical fields that exceeded the Pauli limit were
found in a number of other two-dimensional materials; including monolayer
NbSe2 [4, 5], TaS2 [5], and gated WS2 [6]. Superconductors with thermodynamic
critical fields, $H_{c2}$, that exceed the Pauli limiting field are often
associated with spin triplet superconductors (since the Cooper pairs with
parallel spins having the ability to screen the applied magnetic field), or
due to extrinsic effects such as spin-orbit scattering. The large in-plane
critical magnetic fields in these two-dimensional materials, now referred to
as Ising superconductors has a very different intrinsic origin.

The large in-plane $H_{c2}$ in the initial experiments on the monolayer TMDs
was hypothesized to originate from the large spin-orbit splitting due to the
transition metal in these TMDs. This was based on a general understanding that
the Fermi surface of these hexagonal materials is comprised of two concentric
Fermi rings around the K and K′ points of the Brillouin zone. The large SOC of
the transition metal splits the two Fermi contours within a given K valley and
the pseudospin direction is pinned along $\hat{z}$. This description of the
Fermi surface has formed the basis for a number of model theories that have
been put forth to analyze the experiments that have been reported on Ising
superconductors [7, 8, 9, 10]. We note several authors have also provided an
overview on the different materials where evidence for Ising superconductivity
has been uncovered [11, 12, 13]. While these model theories have been used to
both attempt to explain the wide range of experiments and also predict new
phenomena, they are unable to establish materials specific insight into
different phenomena.

In this Perspective we offer a personal viewpoint on how first-principles
calculations can and have been used to elucidate the properties of Ising
superconductors. Our aim is not to provide a comprehensive overview of the
field of Ising superconductivity given that this remains an active area of
research. Rather, our goal is to show how first-principles calculations enable
insight into materials-specific issues that are otherwise unattainable with
model Hamiltonian descriptions of the electronic structure of the TMD
monolayers. While Ising superconductivity has been reported in a range of
materials, our discussion will primarily focus on the example of the monolayer
NbSe2, the most widely studied Ising superconductor.

Our discussion will highlight the hierarchy of energy scales that provides a
natural explanation for the large $H_{c2}$ in these materials, the presence of
magnetic fluctuations [14, 15], which we show are present in the TMDs that
exhibit Ising superconductivity, and a detailed description of the fermiology
including the role of the third pocket at the zone-center. No material is
ideal, including the TMD Ising superconductors. We will show that non-magnetic
and magnetic point defects which are likely to be present in these materials
can manifest themselves in interesting phenomena such as apparent disorder-
induced enhancement in superconductivity [16, 17], hysteresis in the
superconducting phase [18], oscillations in the superconducting current and
apparent breaking of rotational symmetry [19, 20].

This article is organized as follows - in Section II we provide a general
discussion of the physics of Ising superconductors, III we provide an overview
of the electronic and magnetic properties of monolayer NbSe2, in Section IV we
discuss the impact that defects, doping and alloying have on the
superconducting properties of Ising superconductors, in Section V we show how
experiments involving heterostructures with Ising superconductors lead to a
number of puzzling results, and in Section VI we show how Ising
superconductivity a phenomenon often associated with single monolayers can be
found in bulk materials.

##  II General considerations

We first discuss at a general level the physics that leads to Ising
superconductivity. We start from the simplest description of the Fermi surface
of monolayer TMDs where Ising superconductivity is observed - two quasi-
circular Fermi contours around the K and K′ valleys which are related by time
reversal symmetry.

Figure 1: (a) Schematic illustration of the Fermi contours around K and K′ in
a monolayer TMD Ising superconductor in the presence of spin-orbit
interaction. Blue represents $\hat{z}$ pseudospin states and yellow represents
$-\hat{z}$ pseudospin states. (b) Net pseudospin enclosed around each Fermi
contour denoted by the area enclosed by the blue and yellow shaded regions.
(c) Fermi contours and net pseudospin in the presence of a magnetic field that
is parallel to the $c$-axis of the monolayer, i.e parallel to the direction of
the spins. (d) Fermi contours and the net pseudospin in the presence of an in-
plane magnetic field, i.e field is perpendicular to the direction of the
spins. Note that the area bounded by the two Fermi contours within a given
valley is similar in (b) and (d) but different in (c) due to the Zeeman
splitting. (e) Possible in-plane spin textures (Rashba or Dresselhaus) in
addition to the Ising spin texture in monolayer or few-layer TMDs due to
symmetry lowering effects, eg. substrate-induced effects as illustrated in the
top right.

With spin-orbit interaction the spin-degenerate states within a given valley
are split by a spin-orbit coupling (SOC) parameter, $\Delta_{\rm SOC}$,
resulting in two concentric Fermi contours as illustrated in Figure 1(a). The
pseudospin along $\hat{z}$ flips direction between the two Fermi contours such
that within a given valley there is a finite polarization of pseudospins as
denoted by the colored area in Figure 1(b). Moving from one valley (K) to its
time-reversal partner (K′) the direction of the pseudospin flips as
illustrated by the different colored areas in Figure 1(b) so that the overall
net magnetic moment is zero.

If we apply a magnetic field, $H$, parallel to the $c$-axis of the monolayer
TMD, the Fermi contours within a given valley are Zeeman split by an amount
that is linearly proportional to the magnitude of the magnetic field. Assuming
the magnetic field is applied along $+\hat{z}$, the Fermi contours derived
from pseudospin $+\hat{z}$ will increase in area while those derived from
$-\hat{z}$ will decrease. Hence, based on our notation in Figure 1(a), the
area between the two concentric Fermi contours will now be different between K
and K′ as illustrated in Figure 1(c). This manifests itself in a finite net
spin polarization as expected by an amount that is proportional to the
magnetic field and the single-spin density of states of the material
(neglecting the Stoner enhancement).

When the magnetic field is perpendicular to the $c$-axis, the spins that are
$\lesssim\Delta_{\rm SOC}$ away from the Fermi level tilt away from $\hat{z}$.
Since $\Delta_{\rm SOC}$ is significantly larger than the magnitude of the
Zeeman splitting and is also larger than the superconducting gap, $\Delta$,
the tilting of the spins away from $\hat{z}$ means the spin-susceptibility is
defined by states that are approximately an energy $\Delta_{\rm SOC}$ away
from the Fermi level. To first order in the magnitude of the in-plane magnetic
field there is no Zeeman splitting of the Fermi contours. As a result the area
between the two Fermi contours within a single valley does not change.
However, there is a net spin polarization due to the fact that the pseudospins
are tilted away from $\hat{z}$ to be in plane in the two valleys.

With this heuristic understanding of the response of the electronic structure
to out-of-plane and in-plane magnetic fields it is now easy to address why
Ising superconductors exhibit such a large anisotropy in their $H_{c2}$. The
$H_{c2}$ of a superconductor is determined by the difference in the free
energies of the normal state and the superconducting state of a material.
Within BCS theory and at 0 K the thermodynamic critial field, $H_{c0}$ is
defined as $F_{n}-F_{s}\sim\Delta^{2}N(0)/2=(\chi_{n}-\chi_{s})H_{0}^{2}/2$
where $N(0$) is the density of states at the Fermi level. Since the spin
susceptibility for magnetic fields parallel to the $c$-axis is determined by
the Zeeman splitting of the Fermi contours, superconductivity is suppressed
once the magnitude of the Zeeman splitting exceeds $\Delta$, $i.e$, the
superconductivity is Pauli limited. In contrast, when the magnetic field is
perpendicular to the $c$-axis the spin susceptibility is determined by states
approximately $\Delta_{\rm SOC}$ from the Fermi level. Since $\Delta_{\rm
SOC}$ is significantly larger than the magnitude of the Zeeman splitting
induced by the applied magnetic field, $H$, and $\Delta$, the $H_{c2}$ is
formally infinite in an Ising superconductor.

From this discussion it is apparent that the hierarchy of energy scales that
dictates this large in-plane $H_{c2}$ is $H\leq\Delta\ll\Delta_{\rm SOC}$. The
combination of time-reversal symmetry and broken inversion symmetry guarantees
that the pseudospin has pure $\pm\hat{z}$ character around the K valleys.
Moving from a monolayer to a bilayer leads to four bands that cross the Fermi
level, forming two pairs degenerate in energy and comprised of equal
contributions from both monolayers. Within a single K valley the sign of the
pseudospin flips from $+\hat{z}$ to $-\hat{z}$ between the pair of degenerate
states. The second pair of degenerate states is an energy $\Delta_{\rm SOC}$
away. This layer degeneracy of the states within a K valley is due to the
centrosymmetric stacking of the two monolayers. The vertical stacking between
the two monolayers leads to a finite interlayer hopping, $t$, between the
monolayers. We will show in Section III that this interlayer hopping is
$k$-dependent.

Additional terms can arise that impact the energy and spin character of these
states. Indeed Shaffer et al. [8] have shown that the coupling between a
single monolayer and a substrate can introduce an in-plane component to the
spin texture (Rashba and/or Dresselhaus) as illustrated in Figure 1(e) in
addition to the Ising component along $\hat{z}$. These considerations also
apply to the bilayer structures. One additional consideration is changes in
stacking away from the ground state centrosymmetric stacking, which can lead
to changes in the interlayer hopping, as well as to a non-centrosymmetric
bilayer structure. Since the ground state stacking of the bulk NbSe2 unit cell
is centrosymmetric, there is often an implicit assumption that Ising
superconductivity cannot survive in bilayer or bulk structures. However in
Section VI we highlight experimental studies where evidence of Ising
superconductivity in bulk compounds has indeed been observed and point to work
that indicates there is a region of the parameter space that involves
$\Delta_{\rm SOC}$, interlayer hopping and the coupling to a
substrate/superstrate that leads to Ising protection in bilayer structures.

These general considerations provide qualitative insight into two hallmarks of
Ising superconductivity, the large in-plane $H_{c2}$ that greatly exceeds the
Pauli-limiting field and the reduction in $H_{c2}$ for few-layer and bulk
films. In the following sections we will discuss materials-specific aspects of
the electronic properties that provide insight that cannot be attained using
the considerations we have detailed above.

##  III Electronic structure

For the discussion that follows we will focus on monolayer NbSe2, which is the
most widely studied Ising superconductor.

Figure 2: Electronic properties of monolayer NbSe2. (a) Top view and side view
of monolayer NbSe2 where Nb atoms (blue) are in a trigonal prismatic
coordination with the Se atoms (yellow). (b) Band structure with SOC
calculated with DFT of monolayer NbSe2. The colors correspond to the
contribution by the different $d$ orbitals denoted in the legend above. (c)
Renormalized spin-susceptbility as a function of spin spiral q vector across
the Brillouin zone of monolayer NbSe2. (d) The Fermi contour of NbSe2 where
the colors denote the different pseudospin character of the states. Panel (d)
was obtained from Ref. 21

The building block for monolayer NbSe2 is a Nb atom that is in a trigonal
prismatic coordination with the Se atoms. The overall structure is non-
centrosymmetric and belongs to space group P$\bar{6}$m2. The trigonal crystal
field splits the $4d$ states of Nb4+ into the following groups, $d_{z^{2}}$,
[$d_{x^{2}-y^{2}}$, $d_{xy}$] and [$d_{xz}$,$d_{yz}$], which in the absence of
SOC leads to a spin degenerate band that crosses the Fermi level several
times. This band structure, which has been reported in several studies leads
to three Fermi contours, one that encircles $\Gamma$ due to contributions from
the Nb $d_{z^{2}}$ states and a pair of Fermi contours that encircle that K
and K′ points of the Brillouin zone that are contributed by the Nb
[$d_{x^{2}-y^{2}}$, $d_{xy}$] states. The combination of SOC due to Nb and the
lack of an inversion center leads to momentum-dependent spin-orbit splitting
everywhere except along the $\Gamma$-M line. The electronic structure of
monolayer NbSe2 is summarized in Figure 2. Note that there is also a minor
admixture of the Se $p$-states, where at $\Gamma$ the Se $p_{z}$ states
contribute while at K the Se $p_{x,y}$ states contribute as illustrated in
Figure 2(b).

To understand this momentum dependent spin-orbit splitting that pins the
pseudospins to be along $\hat{z}$ we consider that the bands that cross the
Fermi level a state at a given momentum $k$ can be defined as
$\left|{\phi}\right\rangle=\eta\left|d_{x^{2}-y^{2}}\right\rangle+\beta\left|d_{xy}\right\rangle+\gamma\left|d_{z^{2}}\right\rangle$
where $\eta^{2}+\beta^{2}+\gamma^{2}=1$. Here, we ignore the minor
contribution of the chalcogen $p$-states to the bands that cross the Fermi
level. Note that $d_{z^{2}}$ corresponds to
$\left|l,m\right\rangle=\left|2,0\right\rangle,$ $d_{x^{2}-y^{2}}$ to
$(\left|2,2\right\rangle+\left|2,-2\right\rangle)\sqrt{2}$, and $d_{xy}$ to
$(\left|2,2\right\rangle-\left|2,-2\right\rangle)/\mathrm{i}\sqrt{2}$, where
$l$ is the angular momentum quantum number and $m$ is the magnetic quantum
number.

When we account for spin, the Hamiltonian at each $k$-point is a ($2\times 2$)
matrix. Since the single monolayers have $z/-z$ mirror symmetry around the Nb
atom, the Hamiltonian does not include contributions from the $\left|2,\pm
1\right\rangle$ orbitals. Hence, the nondiagonal matrix elements $L_{\pm}$ are
zero. However, the diagonal element can be defined as
$L_{z}=2(\eta\operatorname{Im}\beta-\beta\operatorname{Im}\eta).$ One phase
can always be selected as real, for instance, $\eta,$ which leads to
$L_{z}=2\eta\operatorname{Im}\beta.$ Only along $\Gamma$-M is the value of
$\beta$ real by symmetry which leads to $L_{z}=0$ and zero splitting due to
SOC. Elsewhere along the Brillouin zone, this leads to an orbital moment that
can only be parallel or antiparallel with respect to $\hat{z}$. With the
inclusion of spin-orbit coupling this leads to the pseudospin to point along
$\hat{z}$, which is indeed what we find in electronic structure calculations
of monolayer NbSe2. The spin-orbit induced splitting, which is maximal at K
(and K′), depends largely on the transition metal element of the TMD
monolayer. From DFT calculations, the splitting at K is 150 meV in monolayer
NbSe2, in monolayer NbS2 it is 115 meV, while it is 280 meV in monolayer TaS2.

One of the experimental challenges when studying single monolayer films is
that conventional bulk probes of the electronic structure of materials are not
readily applied to single monolayer films. However, one can still obtain
valuable information on the properties of the single monolayer by comparing
the calculated and experimentally measured properties of the bulk compounds in
addition to comparisons of trends associated with calculated properties of the
single monolayer versus bulk. Hence, it is instructive to briefly discuss the
properties of bulk NbSe2 — a well known superconductor although not an Ising
superconductor.

The bulk NbSe2 unit cell is comprised of two monolayers of NbSe2 that are
vertically stacked leading to a centrosymmetric structure that belongs to
space group $P6_{3}/mmc$. Superconductivity which has been measured below 7.2
K in bulk NbSe2 [22] is widely thought to be driven entirely by electron-
phonon coupling [23, 24]. First-principles calculations of the Fermi surface
of bulk NbSe2 at the level of the generalized gradient approximation are
generally in agreement with ARPES measurements [25, 26].

One experimental fact that had largely been overlooked are spin susceptibility
measurements of bulk NbSe2, which reports a low temperature spin
susceptibility of 3.04$\times$10-4 emu/mole [27]. The Pauli susceptibility of
bulk NbSe2 using the DOS at the Fermi level from DFT calculations yields a
spin susceptibility that is a factor of 3 lower than experiment. This is an
indication of a considerable Stoner renormalization, corresponding to a Stoner
factor $I\approx\frac{2}{3}N(0)$. On the other hand, the calculated Stoner-
renormalized susceptibility is about 40% larger than the experimental
number[14].

This overestimation of the calculated spin susceptibility in comparison to
experiment is a well known consequence of using a mean-field theory such as
the generalized gradient approximation implementation of DFT to calculate the
properties of itinerant metals that are close a magnetic instability. Other
examples of this overestimation in the calculated spin response at the DFT
level in comparison to experiment include studies on bulk Pd [28] and the
iron-based superconductors [29]. The magnetic moments and the tendency towards
long-range magnetic order, which are overestimated at this mean field level in
itinerant metals, are in reality suppressed due to the presence of long range
magnetic fluctuations [30].

This difference in the calculated Pauli susceptibility and the experimental
susceptibility of bulk NbSe2 therefore points to the presence of strong spin
fluctuations in NbSe2, which would renormalize the Pauli paramagnetic
susceptibility. Further indirect evidence for the presence of spin
fluctuations in the bulk structure is that state-of-the-art first-principles
calculations of the superconducting Tc that only account for electron-phonon
coupling largely overestimate $T_{c}$ compared to the experimentally
established value [31]. This discrepancy is likely due to pair-breaking
effects such as spin fluctuations not being accounted for in the calculations.

Returning to the monolayer structure, the calculated ferromagnetic spin
susceptibility of monolayer NbSe2 has been calculated to be a factor of 1.5
larger than the bulk - suggesting spin fluctuations are, not unexpectedly,
stronger in the single monolayer compared to bulk NbSe2 [14]. Das et al. [21]
have also convincingly shown that monolayer NbSe2 exhibits antiferromagnetic
spin fluctuations with a q vector of [0.2,0,0] as illustrated in Figure 2(d).
The role of magnetism in NbSe2 has also been highlighted by a number of
theoretical studies by other groups [15, 32].

There are several ramifications of this combination of the electronic
structure and the presence of spin fluctuations on the Ising
superconductivity. Arguably the most interesting ramifications of the presence
of these spin fluctuations are on the pairing interactions. If we only
consider Cooper pairs formed from states that reside at K and K′ (i.e
neglecting contributions at $\Gamma$), in the simplest approximation, one may
assume that the amplitude of the order parameter across the two Fermi contours
illustrated in Figure 3 is similar.

Figure 3: Schematic illustration of Fermi contours around K and K${\prime}$
illustrating the possible pairing interactions involving phonons. The yellow
and blue contours represent states with pseudospin $m_{z}=1+$ or $m_{z}=1-$,
respectively. The black dotted arrows denote intravalley and intervalley
pairing interactions that can occur due to electron-phonon interaction.

Since the pseudospins flip sign between K and K′, the formation of Cooper
pairs due to phonons can only involve processes that do not require a change
in the sign of the pseudospin. Superconducting states are either classified as
singlet to triplet depending on whether the total spin of the Cooper pair is 0
or 1. One interesting consequence of the broken Kramer’s degeneracy at the K
and K${\prime}$ valleys in the single monolayer TMDs is that the
superconducting state is neither single nor triplet but a combination of
singlet and triplet [14].

To put these qualitative arguments on more firm theoretical footing recent
first-principles calculations [31] have determined the impact of electron-
phonon interaction and spin fluctuations on the pairing interactions in
monolayer NbSe2. The principal findings of this study is that (1) the
electron-phonon interaction is highly anisotropic and the dominant pairing
mechanism is due to intervalley processes between the Fermi contours with
similar pseudospins at the K and K${\prime}$ valleys (solid black arrows in
Figure 3), (2) the $T_{c}$ calculated due to electron-phonon interactions
alone is greatly overestimated compared to the experimental $T_{c}$, and (3)
spin fluctuations weaken the strength of pairing interactions and bring the
calculated $T_{c}$ in closer agreement with experiment. The fact that the
calculated ferromagnetic spin susceptibility of the monolayer structure is
larger than bulk NbSe2 may also explain why the $T_{c}$ of the bulk structure
is larger than that of the monolayer.

Other competing mechanisms have also been proposed to be at play and impact
superconductivity. The role of the charge density wave (CDW) phase in NbSe2,
and its impact on superconductivity continues to be actively debated. Some
theoretical studies have proposed that the CDW phase is responsible for the
reduction in the superconducting $T_{c}$ compared to the $T_{c}$ obtained
entirely due to electron-phonon interaction [33]. However, recent experiments
have shown that the suppression of the CDW in monolayer NbSe2 due to alloying
with Mo also leads to a suppression of the superconducting $T_{c}$ [34], which
is incompatible with the proposal in Ref. 33. Furthermore applying pressure
[35] and introducing defects by irradiation [36] have been shown to suppress
the CDW while leading to a minor change in the superconducting transition
temperature.

Recent tunneling measurements on monolayer NbSe2, which observed a number of
satellite peaks on either side of the primary coherence peaks, were
interpreted as a manifestation of a Leggett mode between a singlet $s$-wave
and a spin-triplet $f$-wave channel [34]. We note however, that this
postulation is incompatible with the findings of first-principles calculations
[31] where it was shown that the $f$-wave pairing interactions within the K
valley are weak. Other tunneling measurements in the presence of an in-plane
magnetic field have interpreted their measurements by pointing to the possible
presence of a subleading spin-triplet order parameter [37]. Transport
measurements in the presence of an in-plane magnetic field found a surprising
two-fold periodicity in the magnetoresistance which was interepreted as
evidence of a competing nematic superconducting instability [20] that coexists
with a conventional singlet superconducting state. In order to confirm the
experimental manifestation of these proposed mechanisms another issue to
contend with is the role of defects and disorder, which we discuss in the
following section.

##  IV Defects, doping and alloying

Experiments on the role of disorder either through alloying, doping or defects
have led to a number of puzzling results. Alloying NbSe2 with sulfur (which is
isovalent to selenium) was found to change $T_{c}$ non-monotonically with
sulfur content — a pronounced increase in $T_{c}$ up to a critical sulfur
content subsequently followed by a monotonic suppression. Qualitatively
similar non-monotonic changes in $T_{c}$ was found in NbSe2 that had been
exposed to silicon [16]. This increase in $T_{c}$ for intermediate
concentrations of the alloying element was postulated to be evidence of
fractal superconductivity — $i.e.,$ a disorder-induced enhancement of
superconductivity. However, this purported enhancement was reported with
respect to NbSe2 samples that had a $T_{c}$ of $\sim$1 K — significantly lower
than the widely reported value of 3-4 K. Finally, doping NbSe2 with Mo was
found to enhance $T_{c}$ slightly up to a critical concentration of doping
after which superconductivity was suppressed [38]. These changes in $T_{c}$
with doping in monolayer NbSe2 is summarized in Figure 4(a).

Figure 4: Effect of alloying and doping in monolayer NbSe2. (a) Summary of
experimental reports on change in $T_{c}$ as a function of alloying monolayer
NbSe2 with sulfur, silicon and molybdenum. (b) Comparison of STM images from
experiment and DFT calculations for substitutional Mo in NbSe2. The isosurface
can be related to the . (c) First-principles calculations of the effect of
alloying Mo in NbSe2 on the density of states at the Fermi level and the shift
in the Fermi level as a function of Mo content. Data points from experiment
are included as blue dots. (d) First-principles calculations of the effect of
alloying on the Se site with S on the density of states (left vertical axis)
and the magnitude of ferromagnetic spin fluctutations normalized with respect
to monolayer NbSe2. Panel (b-d) was obtained from Refs. 39, 38.

Interpreting the experiments on alloying or doping requires information on
where in the lattice the impurity is incorporated since this determines the
electrical properties of the impurity. For example when Mo is incorporated in
NbSe2 it incorporates substitutionally on the Nb site as shown by STM studies
[38] and further corroborated by first-principles calculations (cf. Figure
4(b)). The extra electron from Mo dopes NbSe2. This leads to a narrow range of
doping where $T_{c}$ is initially enhanced beyond which $T_{c}$ gradually
decreases and superconductivity is suppressed at a critical Mo composition.
Doping NbSe2 with Mo leads to a monotonic reduction in the density of states
which leads to two effects. It decreases the magnitude of the electron-phonon
coupling that is responsible for pairing which suppresses $T_{c}$ and it would
also suppress the magnitude of pair-breaking magnetic fluctuations which would
enhance $T_{c}$ as illustrated in Figure 4(c). It is likely that a combination
of these two changes as a function of doping leads to the non-monotonic change
in $T_{c}$ that has been observed in experiments where NbSe2 is doped with Mo.

A number of experiments have also explored the impact of alloying NbSe2 with
isovalent elements across the entire composition range showing a surprising
enhancement in $T_{c}$ as illustrated in Figure 4(a). At this point it is
useful to recall that based on Anderson’s theorem, non-magnetic impurities
will not lead to pair-breaking in a conventional $s$-wave superconductor.
However, magnetic impurities are pair breaking. The tendency towards magnetism
in these materials suggests native point defects may lead to finite magnetic
moments. Indeed, first-principles calculations have shown selenium vacancies
to have low formation energies [39]. These calculations have also uncovered a
large modulation of the spin density that extends several lattice sites away
from the vacancy with a magnetic moment amplitude of $\sim$ 0.5 $\mu_{B}$
[40]. If selenium vacancies are magnetic point defects in NbSe2, this would
provide a natural explanation for a number of experimental puzzles observed
when monolayer NbSe2 has been alloyed with sulfur and silicon.

First, selenium vacancies can act as a source of scattering and lower the
residual resistivity ratio. The magnetic nature of these vacancies would also
render them pair-breaking leading to a lower $T_{c}$. This also coincides with
the fact that in the experiments low values of the residual resistivity ratio
were found in samples where the $T_{c}$ of monolayer NbSe2 was low [36].
Furthermore, in the experiments where sulfur and silicon were alloyed into
NbSe2 it was assumed that the NbSe2 monolayer prior to alloying was
stoichiometric and in the case of silicon that Si was being adsorbed on the
surface [16]. Our first-principles calculations have shown that this
assumption is incorrect and depending on the concentration of selenium
vacancies that are incorporated during growth, sulfur and silicon can occupy
these vacant selenium sites [39]. This can lead to non-monotonic changes in
the electronic and magnetic properties of NbSe2 as a function of alloy content
as shown in Figure 4(d) and it is likely these non-monotonic changes in the
electronic and magnetic properties that lead to non-monotonic change in
$T_{c}$, not fractal superconductivity.

Non-magnetic defects in the prescence of a magnetic field can also have
nontrivial effects in Ising superconductors. Let us first consider an Ising
superconductor with non magnetic impurities in the absence of a magnetic
field. Intervalley scattering between the two outer (or inner) contours is not
permitted since this requires a spin flip. If $H$ is parallel to $\hat{z}$,
there is Zeeman splitting which suppresses superconductivity but the
conditions that prohibit intervalley scattering between the contours where a
spin flip is required remains the same. Hence, no scattering occurs since the
pseudospin remains pinned to $\hat{z}$. When $H$ is in-plane the spins around
the K and K′ valleys tilt in plane in the direction of $H$ as discussed in
Section II and illustrated in Figure 5, acquiring a triplet component.

Figure 5: Schematic illustration of the effect of a magnetic field on the
canting of the spins at the K and K′ points. Figure obtained from Ref. 41.

Since the pseudospins now have a finite in-plane component intervalley
scattering between the outer (and inner) contours due to non-magnetic
impurities is allowed due to the finite overlap of spin states. Since the
degree by which the pseudospins tilt away from $\hat{z}$ to be in-plane is
proportional to the magnitude of the applied in-plane magnetic field, the
magnitude of the scattering is expected to increase as a function of the
magnitude of the magnetic field. This interplay between the Ising spin-orbit
coupling, non magnetic impurities and scattering induced by an in-plane
magnetic field would be expected to lead to a broadening of the coherence
peaks associated with tunneling due to an Ising superconductor. Indeed this
prediction manifests itself in recent tunneling experiments that have been
performed on NbSe2 tunneling heterostructures where CrBr3 was used as a
magnetic insulator tunnel barrier, which we discuss in the following section.

##  V Heterostructures with magnetic and non-magnetic insulators

Since the Ising superconductors are two-dimensional materials they also
represent an ideal platform to explore the role of interfaces and the
interaction of Ising superconductivity with different phenomenon such as
magnetism, charge-density waves, and topological order. One such
heterostructure are atomically thin Josephson junction heterostructures where
the Ising superconductor is used as the top and bottom superconducting
contacts sandwiched between an insulating barrier such as WSe2 [37] or
magnetic insulating barriers such as CrBr3 [20], to Cr2Ge2Te6 [18, 42, 43] as
summarized in Figure 6. In the case of the experiments that use WSe2 as a
tunnel barrier, the tunneling measurements that were performed in the presence
of an in-plane magnetic field find evidence of equal spin-singlet-triplet
pairs in the tunneling conductance.

In the case of the experiments with the magnetic insulating barriers, even
through the barrier materials are different there are a number of unifying
observations. This includes the observation of an apparent pair-breaking of
$\Delta$ but a surprising strengthening of the coherent peak width in
tunneling, hysterersis that sets in below the superconducting $T_{c}$,
evidence of broken rotational symmetry that occurs only in the superconducting
state, and spin-filtering tunneling processes [18, 42, 43, 20]

Figure 6: Tunneling measurements in NbSe2-magnetic insulator heterostructures.
(a) Schematic illustration of the device geometry used in tunneling
experiments along with measurements of the change in the superconducting gap
and coherence peaks as a function of in-plane magnetic field reported in Ref.
44, (b) Hysteresis in the switching current of the NbSe2/Cr2Ge2Te6/NbSe2
junction as a function of applied in-plane magnetic field, and (c) Field
dependence of the magnetoresistance in a NbSe2/CrBr3 heterostructure. Figures
adapted from Ref. 44, 20, 18.

One idea put forth to explain the apparent symmetry breaking is the
possibility of a two-component order parameter where a nematic phase coexists
with the Ising superconducting phase in NbSe2. It was postulated that the
formation of such a heterostructure can couple to both order parameters and
the nematic phase would be responsible for the rotational symmetry breaking
observed in the experiments [19]. Another possible explanation for this
rotational symmety breaking is the role played by nonmagnetic defects which in
the presence of an in-plane magnetic field can behave as pair-breaking defects
and lead to signatures that are consistent with those reported in experiment
[40, 41].

An alternative approach to explore magnetic-Ising superconductor interfaces is
to replace the magnetic layer that is in proximity with the Ising
superconductor with magnetic transition metal ions that are intercalated. Such
approaches have been explored experimentally in few-layer NbSe2 and NbS2
layers where magnetic ions such as Cr and Fe are intercalated in between the
layers [45, 46, 47]. The intercalated ions have been shown to form an ordered
superlattice within the basal plane of the TMD. These superlattices such as
Fe1/3NbS2 exhibit degenerate magnetic states [45] that are tunable by the
concentration of magnetic ions that are intercalated. The impact of these
intercalated ions and magnetic phases on superconductivity remains to be
addressed. One intriguing possibility is the potential to observe signatures
of Ising superconductivity in these intercalated structures given that Ising
superconductivity can occur in bulk strucures as we will discuss in the next
section.

##  VI Ising superconductivity in bulk materials

The principal signature of Ising superconductivity is a large in-plane
$H_{c2}$ which is pronounced in single monolayers of the superconducting TMDs.
Surprisingly, experiments have shown a range of bulk materials that are
comprised of the TMDs exhibit large values of in-plane $H_{c2}$ that are
comparable to those found in single monolayers. This includes experiments
performed on misfit compounds that contain monolayers of NbS2 and NbSe2
sandwiched in between LaS and LaSe layers [48, 49, 50, 51, 52], twisted
monolayers of TaS2 [53], bulk TaS2 intercalated with different organic
molecules [54, 55] and the 4H, 3R, and 6R polymorphs of TaSe2 and TaS2 [56,
57, 58, 59] . The early reports of these large in-plane $H_{c2}$ values was
interepreted as arising from spin-orbit scattering (using short spin-orbit
scattering times) or the presence of a Rashba-like spin texture of the states
at the Fermi surface which would enable in principle finite-momentum pairing
[52]. A majority of these proposed interpretations were made without taking
into account the electronic structure of these misfit compounds and assessing
whether the details of the Fermi surface would in fact favor Ising
superconductivity.

We suggest that in each of these experiments the large in-plane $H_{c2}$ is
likely a manifestation of Ising superconductivity in bulk compounds. The
ground state stacking of the metallic TMDs is 2Ha, which is centrosymmetric
with a center of inversion between the layers. In some misfit compounds, the
misfit layer is intercalated between the pair of TMD monolayers thereby
weakening the interlayer coupling of the bilayer structure. In other misfit
compounds such as (LaSe)1.14(NbSe2)2, which is schematically illustrated in
Figure 7(a), the misfit layer is intercalated between pairs of the bilayer TMD
that are stacking in their equilibrium 2Ha stacking. In both configurations
the misfit layer leads to doping of the TMD layers. First-principles
calculations combined with photoemission data have recently shown that the
states at the Fermi level in these misfit compounds is best described as a
heavily electron-doped TMD layer [51]. Interestingly these bulk misfit
compounds exhibit in-plane $H_{c2}$ values that greatly exceed the Pauli limit
[50].

Figure 7: Ising superconductivity in bulk compounds. (a) Schematic
illustration of the (LaSe)1.14(NbSe2)2 misfit compound where evidence of large
in-plane critical fields that exceed the Pauli limit were meaured. (b)
Experimental reports of the in-plane critical field as a function of
temperature for bulk misfit compounds that contain NbSe2 layers. Figure (b)
was obtained from Reference 50.

A natural question then is what leads to the large values of $H_{c2}$ in these
bulk compounds. In the case of the misfit compounds where the misfit layer is
intercalated in between the bilayer TMD, the increase in the interlayer
separation is sufficient to weaken the interlayer coupling so that the TMDs in
the misfit mimic act as two decoupled monolayers. Even though these monolayers
are doped, the Fermi level shifts to an energy where the Ising protection is
still present. In the misfit compounds where the misfit layer is in between a
pair of bilayers one aspect to consider is that the misfit layer above and
below the bilayer is stacked asymmetrically. This can lead to an asymmetric
on-site potential that effectively breaks the layer degeneracy that is
otherwise present in the electronic structure of the bilayer. Identifying the
competition between the relevant energy scales — $\Delta_{\rm SOC}$,
interlayer coupling between the TMD layers with different stacking
configurations of the misfit layer — and the combined effect of all of this on
the details of the Fermi surface are parameters that can be extracted using
first-principles calculations. Indeed there is a range of parameter space
where indeed bilayer and bulk misfit compounds comprised of the TMDs can
exhibit signatures of Ising superconductivity such as large in-plane $H_{c2}$
[60]. Given the wide range of misfit compounds where large values of in-plane
$H_{c2}$ have been observed [48, 49, 50, 51, 52, 61, 62], we expect these
calculations when analyzed in the context of the experiments will play a key
role in unraveling this signature of Ising superconductivity in these bulk
compounds.

Another route to access Ising superconductivity in bulk materials is
considering the fact that while the ground state stacking of the bulk TMDs is
centrosymmetric there are a large number of polytypes that these materials can
exist in due to the different stacking configurations that are possible. We
list some of these polytypes in Table 1 and schematically illustrate their
stacking configurations in Figure 8.

Polytype |  | Space group
---
(#)
Centrosymmetric? | Symmorphic?
2H |  | P63/mmc
---
(190)
$\checkmark$ | $\times$
3R |  | R3m
---
(160)
$\times$ | $\checkmark$
4Ha |  | P3m1
---
(156)
$\times$ | $\checkmark$
4Hb |  | P63/mmc
---
(194)
$\checkmark$ | $\times$
Table 1: List of some of the possible polytypes of the bulk TMDs, their space
group, whether they are centrosymmetric or not and if they belong to a
symmorphic space group or not.

We note that some of these polytypes such as the 3R and 4Ha configuration are
not centrosymmetric and indeed there have been experimental reports of Ising
superconductivity in these bulk polytypes [62].

Figure 8: Different polytypes of the bulk transition metal dichalcogenides.
The transition metal is denoted with the small yellow circles and the
chalcogens with the large red circles. The labels A, B, and C at the bottom
can be used to determine the stacking sequence of each polytype. For the
centrosymmetric stacking configurations, the center of inversion is denoted
with a black star and a black dotted line. The materials where these polytypes
have been identified are listed in a yellow box. Figure adapted from Ref.63.

##  VII Conclusions

Since the first report of Ising superconductivity in two-dimensional materials
in 2015 the number of materials that exhibit large in-plane magnetic fields
which is a signature of Ising superconductivity has grown to include a wide
range of low-dimensional materials. There is sometimes a myth (which we show
to be unfounded) that Ising superconductivity is a rather esoteric phenomenon.
In this Perspective article we discuss how Ising superconductivity can in fact
be understood at the basic level using our general understanding of the
electronic structure of Ising superconductor materials and thermodynamic
arguments for the susceptibility in the normal and superconducting state.
Beyond providing this heuristic understanding, one of the major themes of this
Perspective article is that first-principles calculations provide materials-
specific microscopic insight into the wide variety of experiments that have
been perfomed thus far on Ising superconductors. We highlighted the important
role played by non-magnetic and magnetic point defects and proximity induced
effects at interfaces in Ising superconductor heterostructures. Finally we
show that Ising superconductivity is not a phenomenon that only occurs in two-
dimensional materials - there are a wide range of bulk materials where Ising
superconductivity is likely to occur as well.

###### Acknowledgements.

D.W was supported by the Office of Naval Research through the Naval Research
Laboratory’s Basic Research Program. I.I.M was supported by the Office of
Naval Research through grant N00014-20-1-2345. References

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