# Giant Valley-Polarized Spin Splittings in Magnetized Janus Pt
Dichalcogenides

Shahid Sattar<EMAIL_ADDRESS>Department of Physics and Electrical
Engineering, Linnaeus University, Kalmar SE-39231, Sweden  J. Andreas Larsson
Applied Physics, Department of Engineering Sciences and Mathematics, Luleå
University of Technology, Luleå SE-97187, Sweden  C. M. Canali  Department of
Physics and Electrical Engineering, Linnaeus University, Kalmar SE-39231,
Sweden  Stephan Roche  Catalan Institute of Nanoscience and Nanotechnology
(ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  ICREA,
08101 Barcelona, Spain  Jose H. Garcia<EMAIL_ADDRESS>Catalan
Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB,
Bellaterra, 08193 Barcelona, Spain

###### Abstract

We reveal giant proximity-induced magnetism and valley-polarization effects in
Janus Pt dichalcogenides (such as SPtSe), when bound to the Europium oxide
(EuO) substrate. Using first-principles simulations, it is surprisingly found
that the charge redistribution, resulting from proximity with EuO, leads to
the formation of two K and K${}^{{}^{\prime}}$valleys in the conduction bands.
Each of these valleys displays its own spin polarization and a specific spin-
texture dictated by broken inversion and time-reversal symmetries, and valley-
exchange and Rashba splittings as large as hundreds of meV. This provides a
platform for exploring novel spin-valley physics in low-dimensional
semiconductors, with potential spin transport mechanisms such as spin-orbit
torques much more resilient to disorder and temperature effects.

Platinum Chalcogenides, Europium Oxide, EuO, Heterostructures, Spin-Splitting,
Valley Polarization, Janus

††preprint: APS/123-QED

One of the key aspects of van der Waals (vdW) heterostructures is the
proximity effect, where interlayer hybridization combines with the electronic
properties of their individual constituents to produce synergistic behaviors
[1, 2, 3, 4]. A salient illustration of such emerging properties occurs in
graphene-based heterostructures, since graphene has exceptional transport
properties due to its linear dispersion at two nonequivalent valleys in the
Brillouin zone, but lacks spin functionality owing to its very small intrinsic
spin-orbit coupling (SOC). However, it displays gate-tunable spin-dependent
phenomena at room temperature [5, 6, 7, 8] and spin-valley coupling [9, 10,
11] when combined with high SOC materials. Graphene also displays proximity-
induced magnetism which is evidenced by a plethora of interesting gate-
dependent magnetoresitive phenomena [12, 13, 14, 15, 16, 17, 18]. The
incorporation of recently discovered materials with exotic individual
properties into vdW heterostructures is a unique way to discover and engineer
disruptive material functionalities, and to efficiently realize complex
effects, such as, valley polarization, SOC-exchange swapping [19, 20, 21, 22],
and symmetry-enhanced spin-orbit torques (SOT) [23, 24, 25, 26], where the
combination of magnetism, SOC and valley control are required.

The recent synthesis of low-symmetry materials such as Fe3GeTe2 [27], 1T’-WTe2
[28] and Janus SMoSe [29], has shown new ways to control the spin degree of
freedom by removing symmetry constraints and led to the observation of unusual
phenomenon such as canted quantum spin Hall effect [30], persistent spin-
textures [31], multi-directional spin Hall effect [32, 33, 34], and interface-
free Rashba effect [35]. However, these systems are typically metastable [36,
37], a challenge that can be overcome by encapsulation or via the substrates,
and that also open a door for novel collective effects. Transition metal
dichalclogenides (TMDs) in their Janus configuration (where one of the
dichalcogenides of the TMD is replaced by a different one) [35] are
interesting low-symmetry materials since they possess an intrinsically broken
inversion symmetry that yields to significant Rashba splittings and spin-
momentum locking, and then offering SOT mechanisms [35, 38]. These two
features are essential elements for realizing spin-orbit torques (SOTs) [39,
40], a phenomenon where the magnetization direction of a magnetic system is
electrically-controlled via the SOC, and that is very promising for efficient
nonvolatile memories with nanosecond dynamics [41, 42].

Ultrathin Pt dichalcogenides (PtX2 (X=S, Se, Te)) constitute a particularly
interesting class of materials with a very strong and highly tunable SOC that
bestow it with exotic electronic and spin properties [43, 44, 45, 46, 47].
Their most stable geometrical phase is the 1T-phase (P$\bar{3}$m$1$) which
displays a semiconducting behavior [43]. Monolayer (1L) SPtSe was recently
synthesized in a Janus form [48], enabling strong spin-momentum locking due to
their low-symmetry nature [45]. To fruitfully exploit such spin-momentum
locking for SOT applications, it is imperative to determine the spin physics
of the Janus-TMD coupled to a magnetic material. Such a coupling should be
sufficiently strong to allow for the magnetization control via the electronic
spins propagating inside the TMD, but weak enough to at least partially
preserve the spin-momentum locking [40]. In Pt-based TMDs, some studies
suggest the formation of magnetism [49, 50], but to date the consequences of
proximity effects between platinum-based TMDs and magnetic substrates have not
yet been explored.

In this letter, we predict that low-symmetry Janus SPtSe on EuO is a suitable
material combination that generates large exchange coupling and valley-
polarization, while preserving spin-momentum locking. Specifically, the EuO
substrate induces the formation of two time-reversal related valleys with C3v
symmetry at low symmetry points along the $\Gamma$-$K$/$\Gamma$-$K^{\prime}$
paths. These valleys display a large spin-splitting of several hundreds meV,
which are traced back to the joint contributions of Zeeman-like and Rashba
interactions. We also discuss how the combination of these effects could lead
to a giant spin-orbit torque and a current-driven magnetic anisotropy,
originating from the valley’s special point group symmetry. Since these
materials possess a modest lattice mismatch amenable to epitaxial growth and a
low growth temperature similar to other recently synthesize systems [51, 52],
we propose them as the first feasible 2D material with all ingredients
required for efficient spin-orbit torques and magnetic-field-free switching of
the magnetization.

To start with, we aim at clarifying if the strong SOC and broken inversion
symmetry in SPtSe can lead to a large spin-splitting and spin-momentum
locking. Besides, one key question is whether a magnetic insulating substrate
such as EuO can induce a strong exchange coupling in SPtSe which will be
essential for SOT applications. To answer those questions, we performed fully-
relativistic first-principles calculations using the Vienna Ab-initio
Simulation Package (VASP) [53] to determine the geometrical, electronic, and
spin properties of the Pt-dichalcogenides in the proximity to the magnetic
substrate EuO. Technical details of the simulations are presented in the
Supplementary Information [54]. We found that all freestanding TMDs are stable
in the configuration shown in Fig. 1.a. Both PtSe2 and PtS2 have similar
electronic structures displaying a semiconducting behavior with an indirect
band-gap and no band-splitting [54]. The lack of spin-splitting, despite the
presence of large SOC, is the result of time-reversal-symmetry Kramer’s
degeneracy combined with inversion symmetry (one of the symmetries of the
lattice $\bar{3}$m point group), which leads to two-fold degenerate bands over
the whole BZ. It is pertinent to mention that the calculated band-gaps values
are underestimated; for example, 1L PtSe2 displays an indirect band-gap of
1.20 eV which is significantly smaller than the experimentally reported value
of 2.2$\pm$0.1 eV [46]. Therefore, an energy shift of $\sim$0.9 eV in the
conduction band is needed to compensate and match the experimental values
[55]. We also highlight the lack of valleys at the $K$ and $K^{\prime}$ points
in the BZ defined in Fig. 1.b, which is a significant difference between these
structures and the more commonly found TMDs in the 2H-phase.

Figure 1: (a) Top and side views of 1L PtSe2 (black rectangle shows unit
cell). (b) Hexagonal Brillouin zone with high-symmetry points. Minimum energy
configurations of (c) PtS2, (d) Janus SPtSe, and (e) 2L PtSe2 on EuO
substrate.

In the Janus TMD SePtS, the presence of dissimilar chalcogen atoms breaks
inversion symmetry by removing a vertical mirror plane reducing the point
group to 3m; this lifts the band degeneracy with a significant splitting in
the range of 10-100 meV [54]. An analysis of different kinds of stacking
showed that the rock-salt crystal structure of EuO oriented along the (111)
hexagonal face, sketched in Fig. 1.c, is the ideal orientation for epitaxial
growth, since in this case the Janus SPtSe presents a very small amount of
strain of 0.3$\%$, while all the other symmetric structures have strains of
about $3\%$ [54]. After the relaxation procedure, we determined that Pt-atoms
prefer to sit on top of the Eu-atoms, resembling the MoTe2/EuO structure [19].
We found that EuO has a ferromagnetic ground state, and a lattice constant of
3.65 Å on its (111) hexagonal face. To identify the proximity-induce exchange
interaction, we have performed calculations in the absence of SOC and found
the typical exchange interactions to be very large, $\sim 400$ meV [54]. We
also observed the unexpected formation of valleys in the vicinity of high-
symmetry $K$ and $K$’ point of the BZ. To our knowledge, this is the first
example of proximity-induced valley formation in 2D materials. Since Eu and Pt
are very heavy elements, we expect strong possible SOC effects at these
induced $K$-points that could be used for purely electrical manipulation of
the magnetic moments in EuO. Systems displaying valleys at the $K$-points with
3m point group symmetries are very important for SOT since they allow for
optimal Rashba-Edelstein effect and efficient SOT [56, 26] due to the
possibility of vertical spin-splitting and symmetric spin-momentum locking in
the vicinity of two time-reversal symmetric $K$-points, which also enables a
novel kind of anisotropy-like SOT [24] hitherto only observed in Fe3GeTe2
[57].  However, unlike Fe3GeTe2, Pt-based TMDs are semiconductors, which
allows for the possibility of gate-controlled SOTs, while the large Zeeman-
exchange and SOC open the door for very large efficiency even with low-current
densities.

The origin of the large proximity-induced magnetism is actually understood by
the observed significant charge redistribution between the proximitized
materials and the surface of the EuO substrate. Fig. 2.a shows the charge
density difference which evidences a substantial charge accumulation on the
bottom chalcogen atom compared to the top one with charge-density extending
inside the van der Waals gap. By further performing a Bader charge analysis,
one demonstrates that bottom S atom acquires 0.52 $|e|$ charge from the
surrounding atoms (Eu loses 0.49 $|e|$ and Pt loses 0.06 $|e|$) resulting in
$n$-doping of the system. Fig. 2.b clarifies that such a charge redistribution
is accompanied by a transfer of spin-polarized electrons from Eu to Pt atoms,
which leads to the formation of a magnetic moment of magnitude $+0.2\mu_{B}$
located on the Pt $d$-orbitals. We also observe S atoms to be anti-
ferromagnetically coupled to it by having magnetic moments of $+0.1\mu_{B}$
and $-0.02\mu_{B}$ located on $p$-orbitals for the top and bottom S atoms,
respectively. In other words, non-magnetic few-layer PtX2 (X=S, Se) becomes
magnetic on the EuO substrate due to the unique anti-ferromagnetic coupling of
the S atoms. Looking at the EuO substrate itself, Eu and O atoms are also
anti-ferromagnetically coupled to each other having a total magnetic moment of
$+7\mu_{B}$ and $-0.1\mu_{B}$ sitting on $f$\- and $p$-orbitals, respectively.
The system also become $n$-doped due the charge transfer between the chalcogen
and surrounding atoms. These results are further validated by the atom-
projected density of states that shows a strong orbital hybridization between
Pt(d), S(p) and Eu(d) orbitals [54]. Since it is quite difficult to grow
monolayer systems in general, and the electronic properties of PtSe2 are
remarkably dependent on the crystalline structure [43, 44, 45, 46, 47], we
also evaluated the bilayer case in the AA-stacking.

Figure 2: (a,c) Charge density difference (isovalue: $2.5\times 10^{-3}$
electrons/bohr3), and (b,d) Spin density (isovalue: $3.0\times 10^{-3}$
electrons/bohr3 ) of 1L PtS2 and Janus SPtSe on EuO substrate, respectively.
In (a,c), blue and yellow colors represent charge accumulation and depletion,
wheres in (b,d), blue and yellow colors represent spin-up and spin-down
densities, respectively.

We now discuss the results of the calculations when both magnetism and SOC are
included, for the monolayer and bilayer cases of PtSe2 and SPtSe on the EuO
substrate. The binding energy and interlayer distance for these calculations
are given in Table 1. The binding energy is calculated using the definition:
E${}_{b}=E({\rm PtSe_{2}/\rm EuO})-E({\rm PtSe_{2}})-E({\rm EuO}),$ where
$E({\rm PtSe_{2}/\rm EuO})$ is the total energy of the combined system,
$E({\rm PtSe_{2}})$ is the total energy of detached PtSe2, and $E({\rm EuO})$
is the total energy of the EuO substrate. The large value of Eb= -1.70 eV
(-1.84 eV for 2L PtSe2 on EuO) confirms the presence of strong interlayer
interactions. We note that the binding energy is even larger for the lower
atomic-size chalcogen PtS2 relatives of PtSe2 with lesser interlayer distance,
providing a stronger attachment of the former to EuO substrate albeit
perturbing the electronic dispersion as discussed below. These results are
consistent with experimental data on Pt (111) substrate where an interlayer
distance of 2.28 Å is observed [43]. In Fig. 3.(a-d) the band structures for
all the three cases are reported, and unveil a novel effect not present in
previous calculations, namely a valley-dependent band splittings highlighted
in Fig. 3.b for 1L Janus SPtSe. Such an effect provides convincing evidence of
the simultaneous action of exchange and SOC interactions. We estimate the
(Zeeman) exchange strength by computing the average of the splitting at each
valley $\Delta_{\rm ex}\approx(\delta_{\Gamma\rightarrow
K}+\delta_{\Gamma\rightarrow K^{\prime}})/2$, which leads to values between
190 meV to 360 meV as reported in Table 1. Valley polarizations are actually
quite typical of magnetized TMDs in their 1H structural phase, being a
consequence of the hexagonal symmetry (D3h point group) and broken inversion
symmetry that create a specific SOC term named valley-Zeeman, which behaves as
an effective Zeeman field opposite sign at each valley. Therefore, the
combined effect of this valley-Zeeman interaction with a true Zeeman-field
yields a valley-dependent Zeeman splitting with a splitting strength piloted
by the SOC. We note that in principle, Pt-based TMDs are most stable in the
tetragonal phase (D3d point group), a phase preserving inversion symmetry,
which forbids such a type of valley-Zeeman SOC term. However, the presence of
the substrate breaks such a symmetry whereas the preserved three-fold
rotational symmetry induces the formation valley-Zeeman SOC terms at the $K$
points. Consequently, the valley-polarization predicted by our calculations
demonstrate large proximity-induced magnetism and lifting of the inversion
symmetry in these Pt-based TMD supported by a EuO substrate.

Table 1: Binding energy (E${}_{\text{b}}$), interlayer distance (d), conduction band spin-splittings along the $\Gamma$$\rightarrow$K ($\delta_{\text{ $\Gamma$$\rightarrow$K}}$) and $\Gamma$$\rightarrow$K’ ($\delta_{\text{ $\Gamma$$\rightarrow$K'}}$) paths, Zeeman exchange strength ($\Delta_{\rm ex}$) and average difference in splittings ($\lambda_{vz}$).  |  |  | E${}_{\text{b}}$
---
(eV)
| d
---
(Å)
| $\delta_{\text{ $\Gamma$$\rightarrow$K}}$
---
(meV)
| $\delta_{\text{ $\Gamma$$\rightarrow$K'}}$
---
(meV)
| $\Delta_{\rm ex}$
---
(meV)
| $\lambda_{vz}$
---
(meV)
PtS2 | 1L | -1.94 | 2.05 | 362 | 355 | 358.5 | 3.5
2L | -2.05 | 2.04 | 331 | 270 | 300.5 | 30.5
SPtSe | 1L | -2.04 | 2.07 | 313 | 372 | 342.5 | 29.5
2L | -1.87 | 2.05 | 185 | 345 | 265 | 80
PtSe2 | 1L | -1.70 | 2.20 | 215 | 253 | 234 | 19
2L | -1.84 | 2.18 | 171 | 214 | 192.5 | 21.5
Figure 3: (a,c) Electronic band structure of 1L and 2L PtSe2 and (b,d) Janus
SPtSe on EuO substrate, respectively. The SOC effects are included for each
case.

The strength of the valley-Zeeman interaction is estimated by computing the
average difference in the splitting at each valley $\lambda_{\rm
vz}\approx\mathinner{\\!\left\lvert\delta_{\Gamma\rightarrow
K}-\delta_{\Gamma\rightarrow K^{\prime}}\right\rvert}/2$. The values in Table
1 show Zeeman interactions ranging from 3 meV to 80 meV in the symmetric
structures. These values are extraordinarily large compared to other
conventional TMDs, as a consequence of the strong SOC introduced by the heavy
metals. We however note that such estimation does not include possible
additional contribution from Rashba SOC stemming from the interaction with the
substrate, which should slightly renormalize the band splitting.

Figure 4: 2D spin-textures of Janus 2L SPtSe on EuO substrate. Refer to brown
lines in Fig. 3.d, inner contours corresponds to valence bands passing through
an energy $E=-0.85$ eV whereas the outer contours (corners) belongs to
conduction bands passing through an energy $E=-0.20$ eV, respectively. The
red/blue colors indicate spin up/down.

We finally discuss the results obtained for the asymmetric or Janus structure.
In Table 1 a substantial valley polarization is observed, that is 160 meV and
60 meV for the bilayer and monolayer structures respectively. This is three to
ten times larger than their symmetric counterparts. Since the structures are
formed by the same atoms, such a change can be attributed to the giant Rashba
effect expected in this phase. It is pertinent to mention that Janus SPtSe on
EuO show these effects regardless of the presence of S or Se adjacent to the
substrate [54]. It is complicated to extract the Rashba splitting since its
contribution is mixed with different interactions. Therefore, we assume no
Rashba interaction in the symmetric structure and further subtract the average
spin polarization for the Janus case. Following this procedure, we evaluate a
Rashba coupling ranging between 30 meV and 60 meV, which is again much larger
than any other studied Janus-like system to date.

To determine whether the systems in indeed subject to a strong Rashba SOC, the
spin-textures are computed using a 2D k-mesh ($k_{x}\times
k_{y}\mathrel{\mathop{\mathchar 58\relax}}15\times 15$) centered at the
$\Gamma$-point ($k_{z}=0$) and are presented for the energy contours $E=-0.20$
eV and $E=-0.85$ eV in Fig. 4. The colors indicates opposite spin-polarities
along the projected $x$, $y$, and $z$ directions, represented also in the
labels above. In the valence bands, around the $\Gamma$ point, we see a close
contour which points to a mexican-hat shaped dispersion, although it is tilted
due to the effect of the exchange interaction. This contour displays spin-
momentum locking which is opposite for the two concentric bands and hints at
potentially observing large spin-orbit torque in experiments. However, in the
conduction bands, the contour crosses the $K$/$K^{\prime}$ points showing two
distinct features: (i) strong spin-momentum locking in isolated bands, (ii) an
energy contour with $C_{3v}$ point group symmetry. The first feature allows
optimal spin-to-charge conversion via the Rashba-Edelstein effect which is a
precursor of SOT [56, 26]. The out-of-plane spin component reinforces the idea
of a concomitant existence of exchange and Rashba interactions. The presence
of a Fermi-contour with three-fold rotational invariance is a necessary
condition for a novel kind of anisotropy-like SOT [24] hitherto observed only
in Fe3GeTe2 [57].

In conclusion, we employed first-principles calculations to investigate
proximity effects in ultrathin Pt dichalcogenides and Janus SPtSe on magnetic
EuO substrate. Substantial charge redistribution was found within these
systems, resulting in shifting CBMs in the vicinity of high-symmetry K and
K${}^{{}^{\prime}}$ points, with the formation of multiple valleys. Very
importantly, the broken inversion and time-reversal symmetries, together with
proximity effects from the magnetic substrate, generate huge spin-splittings
(of the order of several hundred meV) in the conduction band of few-layer PtX2
(X=S, Se) and Janus SPtSe. Moreover, these ultrathin systems become magnetic,
hosting magnetic moments at different atomic sites with an anti-ferromagnetic
coupling between opposite S atoms demonstrating spin-valley polarization.
These findings provide a versatile platform to explore spin-valley physics in
Pt dichalcogenides and leads to their potential electronic and spintronics
applications.  Nevertheless, efforts are still necessary to improve the air-
sensititvy of the Janus structure and increase the curie temperature of EuO

Fruitful discussions with Udo Schwingenschlögl and Muhammad Tahir are greatly
acknowledged. We thank Knut och Alice Wallenberg foundation, Kempestiftelserna
and Carl Tryggers Stiftelsen for financial support. The computations were
enabled by resources provided by the Swedish National Infrastructure for
Computing (SNIC) at HPC2N and NSC partially funded by the Swedish Research
Council through grant agreement no. 2018-05973. ICN2 authors were supported by
King Abdullah University of Science and Technology (KAUST) through the award
OSR-2018-CRG7-3717 from the Office of Sponsored Research (OSR) and by the
European Union Horizon 2020 research and innovation program under Grant
Agreement No. 881603 (Graphene Flagship). ICN2 is funded by the CERCA
Programme/Generalitat de Catalunya, and is supported by the Severo Ochoa
program from Spanish MINECO (Grant No. SEV-2017-0706 and MAT2016-75952-R).

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Full superconducting gap in the candidate topological superconductor In_1-xPb_xTe for x = 0.2
M. P. Smylie
Department of Physics and Astronomy, Hofstra University, Hempstead, New York 11549, USA
Materials Science Division, Argonne National Laboratory, 9700 S. Cass Ave., Lemont, Illinois 60439, USA

Kaya Kobayashi
Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan
Department of Physics, Okayama University, Okayama 700-8530, Japan

J. Z. Dans
H. Hebbeker
Department of Physics and Astronomy, Hofstra University, Hempstead, New York 11549, USA

R. Chapai
W.-K. Kwok
U. Welp
Materials Science Division, Argonne National Laboratory, 9700 S. Cass Ave., Lemont, Illinois 60439, USA

High-pressure synthesis techniques have allowed for the growth of samples on the indium-rich side of (Pb,In)Te, which have increased superconducting transition temperatures compared to lead-rich compounds. In this study we present measurements of the temperature dependence of the London penetration depth $\Delta \lambda(T)$ in the compound In_0.8Pb_0.2Te, which shows a bulk $T_{c,onset}$ of $\sim4.75$ K. The results indicate fully gapped BCS-like behavior, ruling out odd-parity, topologically nontrivial $A_{2u}$ pairing; however, odd-parity $A_{1u}$ pairing is still possible. Critical field values measured below 1 K and other superconducting parameters are also presented.

§ INTRODUCTION

Following the theoretical prediction <cit.> and subsequent discovery of bulk topological insulators such as Bi_2Se_3 <cit.> in which time-reversal symmetry and parity lead to a finite $Z_2$ topological index, the first topological crystalline insulator, SnTe <cit.> was discovered, where the gapless surface states are  protected by crystalline mirror symmetry and a finite mirror Chern number.
It was found <cit.> that doping these topologically nontrivial insulating compounds can result in bulk superconductivity.
ARPES measurements on both superconducting M_xBi_2Se_3 (M = Cu, Sr, Nb) and on superconducting Sn_1-xIn_xTe reveal that the doped materials still possess spin-polarized topologically protected surface states near the Fermi energy <cit.>, raising the potential for discovery of topological superconductivity.
Indeed, a unique nematic topological superconducting state <cit.> was discovered in M_xBi_2Se_3 (M = Cu, Sr, Nb) while reported possible odd-parity pairing and zero-bias conductivity peaks in Sn_1-xIn_xTe point to the topological nature of this material <cit.>.
Due to the immense interest in finding a true $bulk$ topological superconductor, as opposed to topological superconductivity achieved via proximity effect <cit.> at the interface between a topological insulator and a conventional superconductor, it is vital to explore materials related to these compounds.
As several binary compounds in the Sn-In-Pb-Te family form, this family is an obvious one to explore.
The end members of (Sn,In,Pb)Te are well known.
SnTe has rocksalt structure ($Fm\bar{3}m$) and was the first material to be recognized as a topological crystalline insulator <cit.>.
Below 100 K, SnTe undergoes a structural phase transition to a ferroelectric rhombohedral state (space group $R\bar{3}m$) <cit.> and becomes superconducting at sub-kelvin temperatures <cit.>.
PbTe is a well-known thermoelectric material <cit.> which also has the $Fm\bar{3}m$ structure and is not believed to be superconducting in its pure form <cit.>.
InTe has a body-centered tetragonal lattice structure ($I4/mcm$) <cit.> and superconducts at 2.18 K <cit.>.

Ternary phase diagram of the (Sn,In,Pb)Te system.
Compositions above the red dashed line are above the solubility limit encountered in conventional synthesis; the green shaded area indicates the known superconducting region and the orange shaded area is non-superconducting.
HPHT synthesis has found superconductivity in Sn_1-xIn_xTe for all $x$ values and in In_1-xPb_xTe samples with $x < 0.8$ (solid green lines).
In this work we report on In_0.8Pb_0.2Te (pink circle).
No (Pb_1-xSn_x)_1-yIn_yTe obtained in HPHT synthesis has been reported.

Upon doping SnTe with In, the topologically protected surface states near the Fermi energy are preserved <cit.> while the ferroelectric state is suppressed to $T = 0$ with 4% In doping <cit.>.
There is a small window of $0 < x < 0.04$ in which Sn_1-xIn_xTe remains ferroelectric and superconducting <cit.> with $T_c$ below 2 K, independent of $x$.
For doping levels above $\sim$4%, $T_c$ increases linearly with doping to 4.5 K until the solubility limit is reached at $x \sim$50% <cit.>.
In contrast, PbTe is a trivial band insulator which undergoes a topological phase transition to a topological crystalline insulator phase upon Sn doping to 25% <cit.>.
Doping PbTe with In through conventional synthesis techniques does not generate superconductivity up to the In solubility limit of $\sim$24% <cit.>.
Co-doping with Sn and In to form (Pb_1-xSn_x)_1-yIn_yTe preserves the topological state for $x > 0.25$ and also generates potentially topological superconductivity <cit.> in a limited In-doping window which depends on Sn concentration.
A ternary phase diagram following Zhong et al. <cit.> is shown in Fig. <ref>, where the region above the dashed red line is inaccessible by conventional synthesis techniques, the green shaded area indicates the known superconducting region, and the Pb-heavy orange shaded portion of the diagram is reported as nonsuperconducting.
Solubility limits can be bypassed by high pressure/high temperature (HPHT) synthesis <cit.>, allowing the entire range from SnTe to InTe and from PbTe to InTe to be explored.
Currently, only materials of the form In_1-xPb_xTe and Sn_1-xIn_xTe have been grown via HPHT synthesis, with superconductivity observed over a wide range of compositions (green lines in Fig. <ref>).
No (Pb_1-xSn_x)_1-yIn_yTe HPHT synthesis has been reported yet.
When made via HPHT synthesis, all In_1-xPb_xTe compounds adopt the $Fm\bar{3}m$ FCC structure.
This phase seems to be metastable, reverting to the tetragonal structure when samples are kept at room temperature over extended time.
The end-line compound PbTe remains non-superconducting, whereas superconductivity in HPHT-synthesized InTe is observed with onset temperatures of 3 to 3.5 K (as compared to 2.18 K in the conventionally grown tetragonal material, see above).
While the FCC phase of InTe is believed to be topologically trivial, recent measurements and calculations <cit.> suggest that doping InTe may in fact result in topologically nontrivial states as was the case with doping PbTe.
Reports show a full superconducting gap in conventionally synthesized (Pb_0.5Sn_0.5)_0.7In_0.3Te <cit.> and also in Sn_1-xIn_xTe <cit.> for low and high In-doping.
No measurements on the low-Pb In_1-xPb_xTe phase have been reported yet.
It is imperative to measure the bulk properties of newly accessible doping regimes of InTe to look for evidence of an unconventional superconducting gap which may indicate topological behavior.

In this work, we report on magnetization and transport measurements at $^4$He temperatures and measurements of the temperature dependence of the London penetration depth $\Delta \lambda(T)$ down to $\sim$450 mK in superconducting In_0.8Pb_0.2Te grown by HPHT synthesis (pink circle in Fig. <ref>).
The observed temperature dependence of $\Delta \lambda$ indicates a full superconducting gap, which eliminates one of two possible candidate topological superconducting states for this material.
We find the onset of superconductivity at $\sim4.75$ K and a lower and upper critical field of 9 mT and 2.12 T, respectively, for the bulk phase, with a corresponding estimate of 12.5 nm and 254 nm for the coherence length and the London penetration depth, respectively.
There is a small secondary phase likely consisting of a different doping level of In_1-xPb_xTe with an elevated onset $T_c$ (5.75 K) that has a substantially higher zero-temperature upper critical field estimated at 7.14 T.

§ EXPERIMENTAL METHODS

Polycrystalline mm-scale ingots of In_0.8Pb_0.2Te were grown following the method of Katsuno <cit.>.
Samples were kept in a freezer at approximately -20$^{\circ}$C to avoid a transition from the metastable NaCl-type structure to the tetragonal structure.
X-ray diffraction (XRD) measurement performed in a PANalytical X’pert Pro x-ray diffractometer with Cu $K_\alpha$ radiation on the ingot chosen for SQUID magnetization and Tunnel Diode Oscillator magnetometry (TDO) measurements confirmed the sample has the NaCl-type $Fm\bar{3}m$ structure with a lattice constant of 6.225 Å, matching that of ingots measured immediately following synthesis.

Magnetization measurements were performed on an irregularly shaped ingot with a mass of 7.4 mg in a Quantum Design MPMS DC SQUID magnetometer at temperatures down to 1.8 K.
The sample was mounted inside a gelatin capsule held inside a clear plastic drinking straw.
The TDO technique <cit.> was used on a small piece (approximately 0.5 mm x 0.5 mm x 0.1 mm) cut from this ingot to measure the temperature dependence of the London penetration depth $\Delta \lambda(T) =\lambda(T)-\lambda_0$, where $\lambda_0$ is the zero-temperature value, down to $\sim$450 mK in an Oxford $^3$He cold-finger cryostat with a custom-built resonator <cit.> running at $\sim$14.5 MHz.

Due to the brittleness and small sizes of polycrystalline pieces, polishing a flat surface for 4-point contact geometry with silver paint contacts was not viable.
As a result, magneto-transport measurements were performed in a 2-point contact geometry using silver epoxy (Epotek H20E) and 50 $\mu$m gold wires, with the sample mounted on a custom-built probe in a cryostat capable of 1.6 K and 9 T.
Transport measurements were performed with a DC current of 50 $\mu$A.

§ RESULTS AND DISCUSSION

Figure <ref> shows the room temperature XRD pattern of a small piece of polycrystalline In_0.8Pb_0.2Te following background subtraction.
All peaks belong to the $Fm\bar{3}m$ space group, with no impurity peaks due to the tetragonal phase observed.
The lattice constant is found to be 6.225 Å, in agreement with previous reports for this doping level <cit.> despite the sample having been stored in a freezer for 18 months.
In the $Fm\bar{3}m$ space group, this yields an x-ray density of approximately 7.845 g/cm$^3$.

Low-temperature magneto-transport measurements are shown in Fig. <ref>.
A constant contact resistance stemming from the 2-point geometry has been subtracted.
Resistance measurements reveal a two-step transition with onsets of $T_{c,onset} \sim$5.7 K and 4.5 K, respectively.
Judging from the height of the resistive transitions, we attribute the transition at $\sim$4.5 K to the bulk $x = 0.2$ phase, and the transition at 5.7 K to a higher-$T_c$ impurity phase of In_1-xPb_xTe with a slightly different doping concentration; the exact relation between $T_c$ and $x$ is not fully understood <cit.>. Both transitions are suppressed in increasing magnetic fields.

Room temperature XRD pattern of a small piece of polycrystalline In_0.8Pb_0.2Te following background subtraction.
All observable peaks above the noise level can be indexed with the NaCl-type $Fm\bar{3}m$ structure with a lattice constant 6.225 Å.

Two-point resistance measurements as a function of temperature in various applied fields with a current of 50 $\mu$A after subtracting the residual contact resistance.
A broad transition with $T_{c,onset} \sim$4.5 K is observed, with a small secondary transition at $T_{c,onset}\sim$5.7 K.
Both transitions are suppressed with increasing field.

Zero field-cooled magnetization measurements using a DC SQUID magnetometer in an applied field of 0.2 mT are shown in Fig. <ref>.
The sample is diamagnetic with an onset of $\sim$5.25 K.
The magnetization rapidly decreases below $\sim$4.75 K.

Zero field-cooled magnetic susceptibility measurements in a field of 0.2 mT.
The transition has an onset at $\sim$5.25 K and strongly decreases at $\sim$4.75 K.

We determine the lower and upper critical fields $H_{c1}$ and $H_{c2}$ from field and temperature dependent magnetization measurements, respectively.
We deduce $H_{c1}$ from the value of the penetration field $H_p$ at which the magnetization starts deviating from the linear Meissner behavior at fixed temperature, with field sweeps at different temperatures as shown in Fig 5.
By comparing the slope of the $M$ vs $H$ measurements at low field and at 1.8 K to the ideal value of $-1/4\pi $ for a perfect diamagnet, we determine an effective demagnetization coefficient of approximately $N \approx 0.47$, yielding the $H_{c1} \approx H_p/(1-N)$ values shown in the inset of Fig. <ref>.
This estimate neglects possible effects due to surface barriers, grain boundaries, and the geometric effects of the corners and sharp edges of the irregularly shaped piece of polycrystalline material.
The orange line in the inset of Fig. <ref> represents a conventional parabolic temperature dependence $H_{c1}=H_{c1}(0)(1-(\frac{T}{T_c})^2)$ to the $H_{c1}$ datapoints, from which we extrapolate $\mu_0 H_{c1}$ at $T=0$ to be approximately 9 mT.

Measurements of the temperature dependence of the magnetization in multiple applied fields via SQUID magnetometry are shown in Fig. <ref>.
As the field increases, two effects occur.
The superconducting transition shifts to progressively lower temperature, and the paramagnetic response of the Pb dopant becomes more pronounced; subtracting the normal state background yields the diamagnetic signals shown in the inset.
We define the onset $T_c$ as where each curve crosses the purple dashed line chosen close to zero but outside the noise level.
A different criterion will slightly shift the phase boundary but not affect its shape as discussed further below.

The shift of the TDO frequency with temperature and/or field is a measurement of the degree of screening of magnetic flux in the sample which is either due to superconductivity or the normal-state skin depth <cit.>.
In a TDO measurement, the transition into or out of the superconducting state is accompanied by a large shift in oscillator frequency, allowing mapping of the temperature dependence of the upper critical field.
TDO measurements on an irregular section cut from the ingot measured in the SQUID are shown in Fig. <ref>(a) in applied magnetic field values of up to 5.5 T in 0.25 T steps.
In all measurements, the sample was field cooled from above $T_c$, then data were collected during a slow warming ramp through the transition and beyond.
As the resonant frequency of the TDO circuit is field-sensitive, the presented curves were all offset in frequency by constant amounts such that the TDO responses in the normal state aligned.
In zero field, there is a large diamagnetic transition with onset temperature of $\sim$4.69 K, which is continuously suppressed in field.
Zooming in near the onset of the transition [Fig. <ref>(b)], there is clear evidence for a second phase with a higher onset temperature of  $\sim$5.75 K, which is suppressed in field at a slower rate than the bulk phase.
The onset temperature found here is comparable to the value observed in higher-doped In_1-xPb_xTe <cit.>, the most likely candidate for this minority phase.
The frequency shifts associated with each transition ($\sim$950 Hz for the bulk phase, $< 5$ Hz for the minority phase) allow for a rough estimation of the volume fraction indicating that this minority phase is a very small part of the bulk.

Zero-field cooled magnetization vs applied field at multiple temperatures.
The penetration field, used to estimate $H_{c1}$, is taken as where the magnetization first deviates from linearity.
The inset shows $H_{c1}$ values after correcting for demagnetization effects and an extrapolation to the zero-temperature value, which we estimate at $\sim$9 mT.

Zero field-cooled magnetic susceptibility vs temperature for polycrystalline In_1-xPb_xTe with $x = 0.2$ as a function of temperature in different applied fields.
The inset shows the diamagnetic transition onsets after background subtraction, with the onset $T_c$ defined as where the transition crosses the dashed purple line.

The TDO, SQUID and magneto-transport measurements can be used to determine a superconducting phase diagram which contains both the bulk phase and the higher-$T_c$ minority phase.
We define the transition temperature of the minority phase as the onset temperature where deviation from the temperature-independent normal-state behavior occurs.
Due to the higher transition temperature phase, the transition temperature in the bulk phase must be estimated as where the signal first deviates from linearity below the onset of the higher temperature phase; an example is shown in Fig. <ref>(b) as the intersection of two dashed red lines superimposed on the $H$ = 1 T dataset.
The phase boundaries for both the bulk phase and the minority phase as derived from TDO measurements (circles), magneto-transport (squares), and SQUID measurements (diamonds) are shown in Fig. <ref>(a), with the minority phase having a higher onset $T_c$ and a steeper $dH_{c2}/dT$.
The minority phase was not clearly distinguishable in SQUID measurements; otherwise, there is very good agreement between all three techniques.
The $T_c$ criteria used here emphasize the onset of superconductivity and may overestimate $H_{c2}$ as compared to other techniques such as the resistive midpoints.
A phenomenological fit to the TDO data following $H_{c2} (T)= H_{c2}(0)(1-T^2)/(1+T^2)$ is shown in red in Fig. <ref>(a) to estimate the zero-temperature values of $H_{c2}$.
The fits extrapolate to an upper critical field at $T=0$ of $\mu_0 H_{c2} (0)=2.12$ T for the bulk phase and $\mu_0 H_{c2} (0)=7.14$ T for the minority phase; both are clearly lower than expected from the Pauli limit of $1.85*T_c$.
From the bulk phase value of $H_{c2} (0)$, we calculate the zero-temperature Ginzburg-Landau coherence length $\xi_0$ to be approximately 12.5 nm via the definition $\mu_{0} H_{c2}=\Phi_0/2\pi\xi^2 (0)$.
Additionally, using the Ginzburg-Landau relation $ H_{c1}=\Phi_0/(4\pi\lambda^2)(ln[\lambda/\xi]+0.5)$ we estimate the zero-temperature London penetration depth $\lambda_0$ to be approximately 254 nm.
This value suggests a Ginzburg-Landau parameter $\kappa=\lambda/\xi$ of 20, identifying In_1-xPb_xTe as extreme type-II.
The two phase boundaries from all three techniques agree with each other.
In Fig. <ref>(b), we present the phase boundary data for both the bulk phase and the minority phase, with each dataset scaled by its own $T_c$ in zero field and by the extrapolated $H_{c2}$ at $T=0$ values.
With this scaling, all datasets fall on a universal line.
We take this as evidence that the minority phase is indeed just a higher doping level of the bulk phase.
There are no other known candidates consisting of Pb, In, or Te in this $T_c$ range or with such high critical fields.

(a) TDO resonator frequency shift vs temperature for a single piece of polycrystalline In_1-xPb_xTe with $x = 0.2$ in zero field (black) and in magnetic fields from 0.25 T to 5.5 T in 0.25 T steps.
The diamagnetic transition onset is shifted to lower temperatures as field is increased.
(b) Expanded view of the onset of the superconducting transitions; there is clearly a small second phase with higher $T_c$ which is suppressed more slowly with field.
The intersection of dashed red lines indicates the criterion used for determining $T_c$ of the lower-temperature phase.

(a) Superconducting phase diagram for polycrystalline In_1-xPb_xTe with $x = 0.2$ measured by multiple techniques.
SQUID measurements did not clearly distinguish the minority phase seen in TDO and magneto-transport data.
The red lines are a $\frac{1-T^2}{1+T^2}$ fit to the TDO data.
Extrapolated zero-temperature values for the upper critical field are 2.12 T and 7.4 T for the bulk phase and the minority phase, respectively.
(b) When transport and TDO curves are normalized to the extrapolated $T_{c0}$ and $H_{c2}(0)$ values, all data approximately falls on a universal curve, suggesting the minority phase is indeed the same material with a different doping level.

Low-temperature penetration depth measurements were carried out via the TDO technique in the temperature range of 0.45 K to 10 K.
In the TDO technique, the frequency shift $\Delta f$ of the resonator is proportional to the change of the penetration depth <cit.>:

\begin{equation}
\Delta f(T) = G\Delta \lambda(T)
\label{eq:F(T)}
\end{equation}

where the geometrical factor G depends on the geometry of the resonator coil as well as the sample volume and shape.
The small amplitude of the AC field in the coil ($\sim$2 $\mu$T) ensures that the sample remains fully in the Meissner state below $T_c$.
The low-temperature variation of the London penetration depth $\Delta\lambda(T)=\lambda(T)-\lambda_0$ can provide information on the superconducting gap structure <cit.>.
In the limit of low temperature, conventional BCS theory for an isotropic s-wave superconductor yields an exponential variation of $\Delta\lambda(T)$:

\begin{equation}
\frac{\Delta\lambda(T)}{\lambda_0} \approx \sqrt{\frac{\pi\Delta_0}{2T}}\exp \left (\frac{-\Delta_0}{T}\right)
\end{equation}

where $\Delta_0$ and $\lambda_0$ are the zero-temperature values of the superconducting energy gap and the penetration depth.
In nodal superconductors, however, a stronger temperature dependence is observed due to the enhanced thermal excitation of quasiparticles near the gap nodes, where the gap amplitude decreases.
This results in a power-law variation in the penetration depth, $\Delta\lambda \sim T^n$, where the exponent n depends on the nature of the nodes and the degree of electron scattering; line nodes in the energy gap will generate a $T$-linear response, whereas point nodes will generate a $T^2$ response.
Behavior with $n\geq3$ is generally considered indistinguishable from exponential and is taken as evidence of a full superconducting gap.

The evolution of the normalized low-temperature TDO frequency shift of In_1-xPb_xTe with $x = 0.2$ is shown in Fig. <ref>.
Instrument noise is on the order of 0.25 Hz, and with averaging slow temperature sweeps back and forth between 0.42 K and 1.6 K, a noise level of $< 0.05$ Hz is achievable, with a full superconducting transition frequency shift of $\sim$950 Hz.
The variation of the frequency shift can be well described (red line) by an exponential dependence with a BCS-like gap value across a wide range of temperature.
The observed gap ratio of $\Delta_0/T_c=1.07$ is smaller than the standard BCS value $\Delta_0/T_c= 1.76$ for an isotropic single-gap s-wave superconductor.
A possible cause could be a sizable anisotropy of the gap <cit.>.
Furthermore, low-energy features found in inelastic neutron scattering measurements <cit.> are not consistent with conventional BCS-like behavior, and calorimetry measurements <cit.> across the (Pb,In)Te family shows an abrupt dip in the Sommerfeld coefficient in the doping window where $T_c$ is approximately constant, suggesting superconductivity in this very low carrier density family of compounds may not be entirely BCS-like.
A $T^2$ fit, characteristic of nodal behavior (dashed, blue), poorly describes the data.
The inset shows the exponent in a fit of the form $AT^n$ with the exponent $n$ a free parameter, while the fit is calculated from the minimum temperature to various cutoff values of reduced temperature $t=T/T_c$.
Over the entire low-temperature range, the fitted exponent $n$ is at least 3, and increases quickly as the upper temperature cutoff for the fit decreases.
Such a fit is essentially indistinguishable from the exponential, BCS-like response shown in red in the main panel.
The presence of the higher-temperature phase does not affect the fit due to its small relative volume fraction as seen in the TDO data.

Normalized low-temperature frequency shift $\Delta f(T)$ for polycrystalline In_1-xPb_xTe with $x = 0.2$.
The BCS-like fit (red) well describes the data; a nodal $T^2$ fit (blue, dashed) poorly describes the data.
(inset) Exponent of a $T^n$ fit to the data vs reduced temperature cutoff for the fit; fits with $n\geq3$ are generally taken as indistinguishable from exponential.

§ CONCLUSION

In summary, we have investigated the superconducting gap symmetry and critical fields of polycrystalline heavily In-doped PbTe, with doping well beyond the previously known In saturation limit.
This doping regime, only available via high-pressure synthesis techniques, shows a higher $T_c$ while maintaining an FCC crystal structure.
A small minority phase appears with higher $T_c$ than the bulk and is most probably associated with higher Pb-doped material due to inhomogenous doping in the growth; both phases have the same normalized magnetic phase boundaries down to sub-K temperatures.
The superconducting gap structure is full and likely anisotropic, and the material is an extreme type-II superconductor with a $\kappa$ of approximately 20.
The zero-temperature upper critical field in the $\sim$5.7 K phase is 7.14 T, and is quite a bit higher compared with another report <cit.> on HPHT-synthesized In_1-xPb_xTe which found $\mu_0 H_{c2}=3.1$ T for $T_c$ of $\sim$5.3 K.
Katsuno reports a nonlinear decrease in the Sommerfeld coefficient $\gamma$ with increasing Pb doping and increasing $T_c$, with $\gamma$ sharply decreasing near max $T_c$.
This is contrary to BCS expectations of increasing $\gamma$ corresponding to increasing density of states and thus to higher $T_c$ values.
Further investigation across the doping diagram is necessary to understand the origin of this behavior.
Regarding any possible topological nature of the superconducting state, extending the symmetry arguments applied to the isostructural superconducting topological insulator Sn_1-xIn_xTe <cit.> suggests the superconducting gap in highly doped In_1-xPb_xTe is either the conventional, topologically trivial $A_{1g}$ state, or the odd-parity, topologically nontrivial $A_{1u}$ state.
We eliminate the odd-parity, topologically nontrivial nodal $A_{2u}$ state as a possibility.
Further synthesis and study of the newly available region of the ternary phase diagram (Fig. <ref>) via HPHT synthesis, especially of the (Pb_1-xSn_x)_1-yIn_yTe formulations, should be pursued to look for superconductivity in topologically nontrivial compounds.

§ ACKNOWLEDGEMENTS

TDO and magnetization measurements at Argonne were supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.
Crystal synthesis was supported by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) under Grants No. 18K03540 and No. 19H01852 and No. 20K05272.