# Pressure-induced exciton formation and superconductivity in platinum-based
mineral Sperrylite

Limin Wang  Rongwei Hu  Yash Anand  Shanta R. Saha  Maryland Quantum Materials
Center, Department of Physics, University of Maryland, College Park, MD 20742
Jason R. Jeffries  Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, California 94550, USA  Johnpierre Paglione<EMAIL_ADDRESS>Maryland Quantum Materials Center, Department of Physics, University of
Maryland, College Park, MD 20742  Canadian Institute for Advanced Research,
Toronto, Ontario M5G 1Z8, Canada

###### Abstract

We report a comprehensive study of Sperrylite (PtAs2), the main platinum
source in natural minerals, as a function of applied pressures up to 150 GPa.
While no structural phase transition was detected from pressure-dependent
X-ray measurements, the unit cell volume shrinks monotonically with pressure
following the third-order Birch-Murnaghan equation of state. The mildly
semiconducting behavior found in pure synthesized crystals at ambient
pressures becomes more insulating upon increasing applied pressure before
metalizing at higher pressures, giving way to the appearance of an abrupt
decrease in resistance near 3 K at pressures above 92 GPa consistent with the
onset of a superconducing phase. The pressure evolution of the calculated
electronic band structure reveals the same physical trend as our transport
measurements, with a non-monotonic evolution explained by a hole band that is
pushed below the Fermi energy and an electron band that approaches it as a
function of pressure, both reaching a touching point suggestive of an
excitonic state. A topological Lifshitz transition of the electronic structure
and an increase in the density of states may naturally explain the onset of
superconductivity in this material.

##  I Introduction

The enhancement of superconductivity as a function of applied pressures, known
as the method to achieve the record-high transition temperature $T_{c}$ of 164
K at $\sim$31 GPa in the mercury-based cuprate superconductors [1], has
recently come into strong focus due to the discoveries of near-room
temperature superconductivity in hydride materials [2]. It has also long been
a standard method of finding enhanced superconductivity in a wide range of
compounds including elements [3], heavy-fermions [4], topological insulators
[5], and oxides [6]. However, despite efforts to search for superconductivity
in natural minerals [7] there are very few reports [8].

Sperrylite with formula PtAs2 , named after Francis Louis Sperry, an American
chemist in the late 1890s, forms in the cubic Pa-3 crystal structure shown in
the inset of Fig. 1, with Pt and As atoms occuping 4a and 8c _Wyckoff_
positions of the unit cell. Natural Sperrylite is tin-white crystallized and
is known to be metallic, with indistinct cleavage on 001 planes. Recent
studies of PtAs2 and the related material PtSb2 have focused mainly on their
thermoelectric properties. Attributed to the presence of a corrugated flat
electronic band [9], PtSb2 reaches a maximum power factor of 43 $\mu$W/cmK2 at
400 K [10] with Ir partially substituted. A higher value of 65 $\mu$W/cmK2 at
440 K [11] was obtained by Rh substitution into PtAs2 . Recently, the related
pyrite material PtBi2, which was predicted to be a three-dimensional Dirac
semimetal [12], displayed superconductivity under very large applied pressures
[13] that appear to alter the materials electronic structure to be nearly
compensated. Overall, the Pt-based pyrite system hosts several intriguing
properties and presents sensitivity to unit cell density that warrants further
investigation.

Here we have conducted high-pressure measurements on synthesized PtAs2 single
crystals up to 150 GPa in order to investigate whether superconductivity can
also be induced by increase of the unit cell density, as well as to understand
how electronic structure evolves. We find that the very small gap
semiconducting behavior that occurs at ambient pressure is non-monotonically
modified by applied pressures, giving way to a superconducting transition that
emerges above 90 GPa and onsets as high as 3.5 K at the highest explored
pressure of 150 GPa. We consider the evolution of the crystal structure with
pressure and study the electronic structure to reveal a possible emergence of
excitonic insulator behavior as well as a transformation to a new electronic
structure that is supportive of superconductivity.

Figure 1: X-ray diffraction data from powdered PtAs2 single crystals, with a
small amount of lead identified as originating from the growth flux. Black
dots, red lines, and green lines represent the observed, fitting curve and the
difference between the two, respectively. The insert shows the unit cell of
PtAs2 , with Pa-3 cubic crystal structure (grey and pink spheres represent Pt
and As atoms, respectively.

##  II Methods

Naturally occurring Sperrylite mineral samples were obtained from the
Department of Mineral Sciences at the Smithsonian Museum of Natural History as
part of a broad search for superconductivity in natural minerals [7]. However,
since no superconductivity was detected, we proceed with focus on lab-
synthesized single-crystal samples of PtAs2 in order to characterize the
material’s intrinsic properties and minimize impurity effects. Crystals were
grown using a standard molten flux technique from lead flux with ratio
PtAs2:Pb = 1:20. This mixture was heated to 1100oC in an alumina crucible over
5 hours, then cooled to 600oC over 60 hours, subsequently removing crystals
from the melt by centrifuging the lead flux. Typical dimensions of obtained
single crystals varied from several millimeters to over 1 cm, and samples were
polished to $\sim(2\times 1\times 0.1)~{}$mm3 for transport measurements.

Electrical transport and x-ray diffraction experiments under high pressures
utilized diamond anvil cells (DACs) to generate pressures in excess of 100
GPa. The electrical transport DAC was constructed with a non-magnetic (BeCu),
screw-driven cell body, and the high-pressure chamber comprised a 300 $\mu$m
standard anvil, a non-magnetic MP35N gasket with a 90 $\mu$m hole, and a 260
$\mu$m, 8-probe designer anvil. The mismatched anvil sizes resulted in
significant damage to the exterior of the culet of the designer anvil after
the experiments; the CVD-grown diamond exhibited a ring-shape gouge
approximately 300 $\mu$m in diameter, the same size as the culet of the
opposing anvil in the experiment. Pressure in the chamber was measured via
ruby fluorescence. Electrical transport data was acquired with a 4-probe
technique using an AC resistance bridge to determine resistivity as a function
of temperature using a commercial cryostat.

The DAC used for X-ray diffraction consisted of an asymmetric, steel piston-
cylinder cell body combined with two opposed anvils with 250 $\mu$m diameter
culets compressing a rhenium gasket pre-indented to approximately 35 $\mu$m in
thickness. A 65 $\mu$m hole was drilled into the rhenium gasket using an
electric discharge machine. Copper powder, a few ruby spheres, and the
powdered PtAs2 were loaded into the pressure chamber before gas-loading the
pressure chamber with neon as the pressure-transmitting medium. X-ray
diffraction data was collected at the APS/HPCAT 16 BM-D beamline using a
transmission geometry with a 29.2 keV beam aligned along the compression axis
of the anvils. The x-ray beam was micro-focused to a spot size of 6$\times$17
$\mu$m, and the detector was calibrated with a CeO2 standard. Pressure was
increased using a gas membrane that was included in the DAC. Pressure was
calibrated from the equation of state of copper [14]. Analysis of the x-ray
diffraction data was carried out using Fit2D [15] and EXPGUI/GSAS-I [16].

Electronic structure calculations were obtained via first-principles density
functional theory calculation using the WIEN2K [17] implementation of the full
potential linearized augmented plane wave method within the PBE generalized
gradient approximation. We used the lattice parameters from Ref. 18. The
$k$-point mesh was taken to be 14$\times$14$\times$14\. To simulate the
pressure effects on the electronic structure of PtAs2 , the band structure and
density of states calculations were then performed utilizing the same
structure, but adjusting the ambient pressure cubic lattice constant $a_{0}$
by factors of 0.9, 0.873, and 0.87 to mimic applied pressures of 123, 186 and
194 GPa, respectively, following the measured compression as described below.

Figure 2: Cubic unit cell volume of PtAs2 obtained by X-ray diffraction under
applied pressures. Data from this study (purple diamonds) is fit (solid line)
to the Birch-Murnaghan isothermal equation of state as described in the text,
and compared to data measured in a previous study (blue squares) by Tschauner
et. al. [19]. Figure 3: (a) Temperature-dependent magnetic susceptibility
measured at 1 T of a lab-synthesized PtAs2 single crystal. (b) Electrical
resistivity temperature dependence of lab-synthesized PtAs2 single crystal at
ambient pressure. Inset shows Arrhenius plot of data indicating semiconducting
behavior at low temperatures.

##  III Results

Fig. 1 presents powder X-ray diffraction data measured using the synthetic
PtAs2 crystals at ambient pressure, with fitting performed using the
previously reported pyrite structure [18]. Refinement of the diffraction
spectrum yields a lattice constant of 5.9752(2) Å, which is consistent with
previous results [19]. Powder X-ray diffraction spectra of PtAs2 measured in
the DAC were fit to extract unit cell dimensions as a function of pressure,
plotted as volume (i.e., lattice parameter $a^{3}$) in Fig. 2. PtAs2 does not
undergo any structural phase transition within the measured pressure range,
rather showing a continuous decrease of the unit cell volume as the pressure
increases to 150 GPa. This evolution follows the Birch-Murnaghan isothermal
equation of state (EOS):

$\displaystyle
P(V)=\frac{3B_{0}}{2}[(\frac{V_{0}}{V})^{7/3}-(\frac{V_{0}}{V})^{5/3}]\times$
$\displaystyle[1+\frac{3}{4}(B_{0}^{\prime}-4)((\frac{V_{0}}{V})^{2/3}-1)],$
(1)

where $P$ is the pressure, $V_{0}$ is the reference volume, $V$ is the
deformed volume, $B_{0}$ is the bulk modulus, and $B_{0}^{\prime}$ is the
derivative of the bulk modulus with respect to pressure. Fitting the
experimental data in Fig. 2 to this model yields values $B_{0}$=224 $\pm$ 3
GPa and $B_{0}^{\prime}$=3.62 $\pm$ 0.07 for the bulk modulus and derivative
term, respectively. The resultant fit is displayed in Fig. 2 along with a
comparison to previously measured data [19]. The EOS provides an excellent
fit, while the comparison to previous work shows a slight discrepancy which
could possibly be due to the use of natural mineral specimens by Tschauner et
al., which are likely not as pure as the crystals synthesized for this study.
The previous study also claimed a better fit using the Vinet equation due to
the small pressure derivative of the bulk modulus, however the excellent fit
to Birch-Murnaghan EOS shown in Fig. 2 suggests the value of $B_{0}$ obtained
in the present study is more valid for pure PtAs2 .

A study of the magnetic susceptibility for synthesized PtAs2 , presented in
Fig. 3(a), shows paramagnetic behavior on both warming and cooling curves,
with no evidence of magnetism. The lack of temperature dependence is similar
to previous reports for PtAs2 as well as RhAs2 and IrAs2 [20], although the
historical study reports a diamagnetic response. Fig. 3(b) presents the
electrical resistivity of PtAs2 at ambient pressure, which is also lacking any
significant features. At high temperatures, the transport behavior is that of
a semi-metal where resistivity is nearly flat, exhibiting a slight decrease on
cooling down to a broad minimum centered near 200 K, followed by an increase
with moderately activated behavior. Fitting to a standard Arrhenius activated
behavior using $\rho(T)=\rho(0)e^{\Delta/k_{B}T}$ through the range 50-200 K
as illustrated in the inset, we obtain a rather small value for the thermally
activated energy gap $\Delta$ = 3.96 meV. Below 50 K, $\rho(T)$ deviates from
activated behavior, which indicates that there may be another conduction
channel or scattering mechanism at low temperatures that requires further
investigation.

Fig. 4 presents the resistance temperature dependence of a PtAs2 crystal
mounted in the DAC cell and pressurized up to 150 GPa. The $R(T)$ data exhibit
a non-monotonic evolution as a function of applied pressure, with the lowest
applied pressure data (4.2 GPa) exhibiting a semiconducting behavior similar
to that of ambient pressure, that evolves to a more insulating behavior at mid
range pressures before becoming more metallic at higher pressures. As shown in
the inset in Fig. 4, an abrupt drop in resistivity below about 4 K emerges at
pressures above 77 GPa, which we attribute to the emergence of
superconductivity. Due to the extremely high pressures of this phase, it is
difficult to perform other experiments to confirm the superconducting state.
However, the increasing drop that is evident with increasing pressure is
consistent with such a state. Given the appearance of superconductivity near 2
K in the related pyrite material PtBi2 near $\sim 10$ GPa [13], it is not
surprising that a similar onset occurs in PtAs2, albeit at much higher
pressures. However, we note that the purported electronic structures are quite
different as discussed below, so further work is required to understand the
relationship between the two materials.

The evolution of resistivity with applied pressure is quantified in Fig. 5. As
shown in panel a), the non-monotonic evolution is evident when comparing the
evolution of resistance values at 2 K and 300 K, which exhibit different
dependence as a function of pressure. Normalizing the absolute changes by
plotting their ratio in panel b) reveals a striking peak near $\sim$20 GPa.
The temperature dependence of resistivity qualitatively changes with pressure
evolution and exhibits deviations from simple semiconductor-like activated
behavior at low temperatures, suggesting there is a more complex behavior that
is not captured by a simple Arrhenius function. To quantify this evolution, we
apply a power law fit of the form

$\displaystyle R(T)$ $\displaystyle=\frac{1}{G_{0}+aT^{n}},$ (2)

where $G_{0}$ is the residual zero-temperature conductance, $a$ is a scaling
coefficient for the temperature ($T$) dependence and $n$ is the power law
exponent. In Fig. 5(c) we see the pressure effect on the conductance model
described in equation (1), which shows the evolution from a simpler $1/T$
power law at low pressures to a stronger behavior with a peak in the power law
exponent $n$ at a similar pressure as the resistance ratio plot in panel b).
While the power law fits deviate from the measured data, this model is meant
to capture a general phenomenological trend rather than fit a specific model.
Below we discuss possible sources of this non-monotonic behavior tied to the
evolution of the electronic band structure.

Figure 4: Resistance of single crystal PtAs2 as a function of applied
pressures in the range 4.2-150 GPa. Data points represent experimentally
meaured data, and solid lines are results of phenomenological power law fit to
equation 2. The inset shows a zoom of the low-temperature resistance at
pressures (top to bottom) of 77, 92, 110, 115, and 150 GPa, respectively,
highlighting the drop in resistance due to onset of a superconducting state.
Figure 5: Evolution of PtAs2 sample resistance features as a function of
applied pressure, shown (a) at room temperature (orange squares) and 2 K (blue
triangles), and (b) as a ratio (purple triangles). (c) Evolution of the
temperature power law exponent $n$ from fits to conductance model
$R(T)=1/(G_{0}+aT^{n})$ as a function of pressure, as explained in the text.
Figure 6: Low energy electronic band structure (top panels) and electronic
density of states (bottom panels) of PtAs2 as a function of unit cell density,
using lattice parameters of $a=1.0a_{0}$ (a), $a=0.90a_{0}$ (b),$a=0.873a_{0}$
(c),$a=0.87a_{0}$ (d), where $a_{0}$ is the lattice constant for the ambient
pressure unit cell. The corresponding value of pressure P for each calculation
is estimated based on formula (1).  Figure 7: Position of electron (circles)
and hole (squares) band edges with respect to the Fermi energy (E=0) as a
function of calculated pressure from DFT band structure calculations.

In Fig. 6 we present the band structure and density of states for PtAs2, which
appears to be consistent with database calculations [21] that indicate a
trivial topological structure, unlike that of PtBi2 [12] and likely due to the
gapped nature of the band structure and lighter pnictogen anion in PtAs2. We
study the evolution of the PtAs2 band structure as a function of cubic unit
cell densities calculated for lattice constants of $a_{0}$, $0.9a_{0}$,
$0.873a_{0}$ and $0.87a_{0}$ with respect to the ambient pressure lattice
constant $a_{0}$. These densities correspond to estimated applied pressures of
0, 123, 186 and 194 GPa as calculated using the unit cell volume dependence
measured by x-ray diffraction shown in Fig. 2. As is evident, there is a hole
band centered between the $\Gamma$ and M Brillouin zone points that remains
very close to the chemical potential at all densities, an electron pocket at R
that rapidly moves higher in energy (i.e. further from the chemical potential)
while another electron pocket at $\Gamma$ gradually drops toward the chemical
potential. Together, this evolution of the band structure is broadly
consistent with the measured electrical transport behavior, including the non-
monotonic evolution of its temperature dependence as discussed below.

As shown in Fig. 6(a), the ambient pressure hole band (blue) centered between
$\Gamma$ and M points just barely cuts across the Fermi level, reflecting a
non-metallic semimetal behavior consistent with our experimental observation.
By applying pressure, this band is very subtly pushed down below the chemical
potential, opening an indirect semiconducting gap between this band and the
electron band centered at the $\Gamma$ point (red). This change in the hole
band is hard to discern when comparing to the first high pressure panel for
$a=0.9a_{0}$ (123 GPa), but is clear when we plot the position of the band
edges versus pressure with higher point density in Fig. 7. As shown, the
calculated hold band edge crosses the chemical potential at approximately 20
GPa effective pressure, which is precisely where we observe a non-monotonic
change in the evolution of transport features (see Fig. 5(b)). Thus, this
semimetal-semiconductor crossover in the low-pressure regime provides a good
explanation of our transport data in the same pressure range.

With increasing pressure, e.g. from $a$=$0.9a_{0}$ (123 GPa) to $0.837a_{0}$
(186 GPa), the indirect gap narrows and eventually the conduction and valence
bands touch the Fermi level at the same time, very close to $a=0.873a_{0}$ as
shown in Fig. 6(c). This approach results in a situation favoring the
formation of an exciton insulator state [22], where the formation of electron-
hole pairs occurs with a binding energy typically within tens of meV. In
PtAs2, the pressure-induced shrinking of the energy gap toward the touching
point at $a=0.873a_{0}$ provides ripe conditions for a thermodynamically
stable excitonic state, which should in principle onset once the gap energy
falls below the excitonic binding energy. Without further experiments, it is
unclear what energy range is required to achieve this condition, but the
experimental transport data shown in Fig. 5(b) clearly marks a trend toward
smaller and smaller activation energies above the $\sim$20 GPa resistance
maximum, with a plateau reached above approximately 50 GPa. More interesting,
the calculated pressure of the band touching point is very near the pressure
where we have observed a sudden drop in resistance at low temperatures, which
we attributed in the discussion above to an onset of superconductivity. It is
certainly tempting to consider whether there is a correlation between these
two events, and whether a more exotic type of pairing mechanism may be at
play. Excitonic pairing mechanisms have long been proposed as a possible route
to superconductivity, and mostly focus on a proximitized excitonic medium
[23]. Howeve, more recent proposals have suggested an intrinsic pairing
mechanism is possible that involves a spin-triplet model that avoids strong
Coulomb repulsion issues [24]. What relation the band structure,
superconductivity and exciton formation have requires further study to
elucidate.

In any case, the interesting evolution of band structure at least entails
several notable topological changes. These so-called Lifshitz transitions are
known to give rise to interesting changes in electronic and physical
properties, such as shown in the iron- [25] and nickel-pnictides, for example
in the electronic nematic system BaNi2As2 [26] where a Lifshitz transition
appears to be associated with phase transitions involving superconductivity.

As shown in Figs. 6-7, there are at least two notable Lifshitz transitions
that occur PtAs2 which we have attributed to measured changes in transport
properties. This is also observed in the evolution of the electronic density
of states (DOS) at the Fermi level (see Fig. 6), which first decreases and
then increases with increasing pressure. Therefore, both the evolution of the
band structure and DOS reveal the same physical trend as shown in Fig. 5.
Furthermore, the components of the DOS plotted in Fig. 6 exhibit an
interesting evolution that entails a valence valence band mainly composed of
As 3$p$ orbitals and a conduction band made up of Pt 5$d$ orbitals at ambient
pressure: upon increasing pressure, the Pt-5$d$ bands are pushed away from the
Fermi level and eventually the bands around the Fermi level are dominated by
As-3$p$ orbitals, which no longer have a covalent character but appear to be
metallic. At the highest pressures, the higher DOS at the Fermi level could in
principle also play a role in stabilizing superconductivity, but to understand
how exciton formation also plays a role will require a microscopic model and
further study.

##  IV Conclusion

In conclusion, we present structural and electrical transport properties of
the platinum-based mineral Sperrylite (PtAs2) as a function of applied
pressures up to 150 GPa, and compare our results with electronic structure
calculations as a function of the cubic unit cell lattice density to help
elucidate an unusual non-monotonic evolution of transport from semi-conducting
to metallic at high pressures. As oppposed to the monotonic evolution of the
unit cell density with pressure, which is well described by a third-order
Birch-Murnaghan equation of state, the calculated evolution of electronic
bands – from semi-metallic to insulating to metallic with increasing pressure
– well explains the evolution of electrical transport, and suggest an
excitonic instability may arise near the high pressure band-touching point. At
the highest pressures, the observation of a sudden drop in resistance at 3 K
suggests the onset of superconductivity very close to the calculated band-
touching point, suggesting a strong interplay between the excitonic band
structure evolution and appearance of superconductivity in this compound.

##  V Acknowledgements

The authors thank C. Santelli and J. Post at the Department of Mineral
Sciences, Smithsonian Institution National Museum of Natural History, for
access to the natural mineral collection. Work at the University of Maryland
was supported by the the Air Force Office of Scientific Research under Grant
No. FA9950-22-1-0023, the Gordon and Betty Moore Foundation’s EPiQS Initiative
through Grant No. GBMF9071, and the Maryland Quantum Materials Center.

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