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# Temperature - Pressure phase diagram of the cubic Laves phase Au2Pb

K. W. Chen1,2, D. Graf1, T. Besara1, A. Gallagher1,2, N. Kikugawa1,4, L.
Balicas1, T. Siegrist1,3, A. Shekhter,1 R. E. Baumbach1 1National High
Magnetic Field Laboratory, Florida State University  2Department of Physics,
Florida State University  3Department of Chemical and Biomedical Engineering,
Florida State University  4National Institute for Materials Science, 3-13
Sakura, Tsukuba, Japan

###### Abstract

The temperature ($T$) as a function of pressure ($P$) phase diagram is
reported for the cubic Laves phase compound Au2Pb, which was recently proposed
to support linearly dispersing “topological” bands, together with conventional
quadratic bands. At ambient pressure, Au2Pb exhibits several structural phase
transitions at $T_{1}$ $=$ 97 K, $T_{2}$ $=$ 51 K, and $T_{3}$ $=$ 40 K with
superconductivity below $T_{\rm{c}}$ $=$ 1.2 K. Applied pressure results in a
rich phase diagram where $T_{1}$, $T_{2}$, and $T_{3}$ evolve strongly with
$P$ and a new phase is stabilized for $P$ $>$ 0.64 GPa that also supports
superconductivity below 1.1 K. These observations suggest that Au2Pb is an
ideal system in which to investigate the relationship between structural
degrees of freedom, band topology, and resulting anomalous behaviors.

##  I Introduction

The cubic Laves phase compounds $M_{2}$$X$ can be considered as being
relatives of the pyrochlore structure, where the tetrahedrally arranged $M$
atoms are distributed inside a face centered cubic arrangement of $X$ atoms.
Pearson  Prior work has shown that this structure accommodates a multitude of
binary chemical combinations, stein2004  making it a deep reservoir for
electronic and magnetic states including superconductivity,Finlayson1978 ;
Huxley1993 ; Saini2004  electronic valence changes,Barberis1980 ; Felner1986
itinerant electron magnetism,Matthias1958  geometrically frustrated
magnetism,Shiga1994  and other effects. Recent work addressing Au2Pb indicates
that this family of materials is now expanded to include topologically
protected electronic states. The calculated electronic band structure for
Au2Pb includes a crystallographically protected Dirac point at room
temperature, in addition to other conventional bands at the Fermi energy
($E_{\rm{F}}$). Schoop2015  With decreasing temperature, Au2Pb undergoes
several structural phase transitions at $T_{1}$ $=$ 97 K, $T_{2}$ $=$ 51 K,
and $T_{3}$ $=$ 40 K. Below $T_{3}$, Au2Pb enters a primitive orthorhombic
structure, where electronic structure calculations predict that the Dirac
point is gapped but retains topological protection. Other band crossings at
$E_{F}$ are also present at low $T$, providing conduction electrons that
produce metallic behavior and conventional BCS superconductivity ($T_{c}$
$\approx$ 1.2 K).

While the presence of conventional bands at $E_{F}$ makes Au2Pb distinct from
simple three dimensional Dirac semimetals (at high $T$) Young2012  or
topological insulators (at low $T$) Zhang2009 , it is of interest because it
offers the opportunity to study the relationship between conventional
electrons and quasiparticles associated with linearly dispersing bands. This
includes the question of how conventional BCS superconductivity might interact
with gapped topological bands. It is also possible that the simultaneous
presence of linearly dispersing and quadratic bands could result in unusual
transport and thermodynamic properties. The complex structural evolution
additionally allows for study of how a Dirac point is influenced by changes in
the crystal symmetry.

Hydrostatic pressure is a useful tool to systematically probe lattice
structures and concomitant electronic states. For this reason, we undertook to
study electrical transport under applied pressure $P$ $<$ 1.8 GPa in single
crystals of Au2Pb. Our measurements reveal an unusually rich $T-P$ phase
diagram that includes several structural phases and superconducting regions.
In particular, we find that while $T_{1}$increases monotonically for pressures
up to 1.83 GPa, $T_{2}$ and $T_{3}$ are initially suppressed and meet near
$P_{1}$ $\approx$ 0.64 GPa. For $P$ $\geq$ $P_{1}$ $\approx$ 0.64 GPa, another
phase boundary that likely is structural in nature is seen at $T_{4}$, which
increases with $P$. Superconductivity is seen at low temperatures throughout
the entire measured $P$ range. This multitude of structural phases and the
persistence of superconductivity will be useful for uncovering anomalous
electronic behavior and disentangling topological behaviors from those that
are due to material specific details. We additionally compare to the Laves
phase analogue, Au2Bi, which is a conventional metal that does not exhibit any
structural phase transitions but does support superconductivity.

##  II Experimental Methods

Single crystals of Au2Pb were grown using a molten Pb flux as described
previously Schoop2015 . Elemental Au and Pb were mixed in the ratio 2:3 and
sealed under vacuum in a quartz tube. The mixture was heated at a rate of 50
oC/hr to 600 oC, held at this temperature for 24hr, and slowly cooled at a
rate of 3oC/hr to 300oC. The crystals were separated from the flux by
centrifuging at this temperature. Single crystals of Au2Bi were made using a
similar approach, where the molar ratio was 3:7.

Single crystals of Au2Pb and Au2Bi were structurally characterized at room
temperature by single crystal x-ray diffraction using an Oxford-Diffraction
Xcalibur2 CCD system with graphite monochromated Mo $K\alpha$ radiation. Data
was collected using $\omega$ scans with 1o frame widths to a resolution of 0.5
Å, equivalent to $2\theta$ = 90o. The data collection, indexation, and
absorption correction were performed using the Agilent CrysAlisPro software
CrysAlisPro . Subsequent structure refinement was performed by using CRYSTALS
Crystals . A crystallographic information file (CIF) for Au2Pb has been
deposited with ICSD (CSD No. 430187) ICSD . For subsequent measurements,
single crystals were selected and aligned using a four-axis Enraf Nonius CAD-4
Single Crystal X-Ray Diffractometer. The obtained orientation matrix allowed
for an unambiguous determination of the crystalline axes to within a degree.

Specimens were prepared for electrical resistivity $\rho$ measurements by
attaching Pt wires using a home built micro-spot welder. Samples of Au2Pb were
loaded into a piston-cylinder pressure cell. The pressure $P$ was determined
by measuring the superconducting transition of 6N polycrystalline Pb. Daphne
7474 oil was used as the pressure-transmitting medium Murata1997 ; Murata2008
. The electrical resistivity $\rho$ was measured with currents applied along
the [100] direction (at room temperature) for temperatures 1.8 K $<$ $T$ $<$
300 K and $P$ $\leq$ 1.8 GPa using a Quantum Design Physical Property
Measurement System and an SR830 lock-in amplifier with a Keithley 6221 as the
current source. Additional resistivity measurements under applied pressure
were performed at the National High Magnetic Field Laboratory, Tallahassee,
for temperatures between 500 mK $<$ $T$ $<$ 2 K using a standard He-3
cryostat. Electrical resistivity and heat capacity measurements were performed
at ambient pressure on an aligned single crystal of Au2Bi using the He-3
option in a Quantum Design Physical Properties Measurement System for
temperatures 400 mK $<$ $T$ $<$ 20 K.

##  III Results

Our measurements verify that Au2Pb and Au2Bi crystallize in the space group
$Fd\bar{3}m$ (#227) where Au2Pb has unit cell parameter $a=7.9141(2)$ Å (Fig.
1(a)); the structural parameters resulting from the x-ray refinement for Au2Pb
are summarized in Table 1. Au2Pb and Au2Bi are cubic Laves phases with a
pyrochlore lattice, Pearson  where the gold is in the 16c position at
$(0,0,0)$ and lead/bismuth are in the 8b position at $(3/8,3/8,3/8)$ (Table
2). The single crystal x-ray diffraction for Au2Pb displayed a site vacancy
defect on the Pb-site, yielding a formal stoichiometry of Au2Pb0.95. For
simplicity, the compound will be referred to as Au2Pb throughout the rest of
the manuscript.

Figure 1: Crystal structure of Au2Pb, showing (a) the unit cell, (b) local
coordination of Au along [111], and (c) local coordination of Pb along [111].

The gold atoms form a network of corner-shared tetrahedra connected along the
$\langle 111\rangle$-directions, with a Au–Au distance of 2.798 Å. The gold
network in turn forms voids in which the lead atoms reside as trigonal chains
along $\langle 110\rangle$, with a Pb–Pb distance of 3.427 Å. Au is 12-fold
coordinated, surrounded by six Au-atoms and six Pb-atoms (Fig. 1(b)), forming
a icosahedron. Pb is 16-fold coordinated with twelve Au-atoms and four Pb-
atoms surrounding it (Fig. 1(c)), forming a 16-vertex Frank-Kasper polyhedron
Pearson .

Table 1: Selected single crystal x-ray diffraction data, along with collection and refinement parameters. (Lattice parameters are for Au2Pb0.951(11))  Compound | Au2Pb
---|---
Composition | Au2Pb0.951(11)
Formula weight | 590.90 g/mol
Space group |  $Fd\bar{3}m$ (#227)
Unit cell parameter |  $a=7.9141(2)$ Å
Volume | 495.69(1) Å3
$Z$ | 8
Data collection range | $4.46^{o}\leq\theta\leq 45.93^{o}$
Reflections collected | 6344
Independent reflections | 130
Parameters refined | 6
$R_{1}$, $wR_{2}$ | 0.0431, 0.0687
Goodness-of-fit on $F^{2}$ | 0.9942
Table 2: Atomic coordinates and equivalent thermal displacement parameters, along with interatomic distances. Atom | Site | Occ. | $x$ | $y$ | $z$ |  $U_{\textrm{eq}}$ (Å2)
---|---|---|---|---|---|---
Au | 16c | 1 | 0 | 0 | 0 | 0.0320(5)
Pb | 8b | 0.951(11) | 3/8 | 3/8 | 3/8 | 0.0191(5)
|  |  |  |  |  |
Bond | Distance (Å) |
Au—Au | 2.798(1) |
Pb—Pb | 3.427(1) |
Au—Pb | 3.281(1) |

The ambient pressure electrical resistivities for Au2Pb and Au2Bi are shown in
Fig. 2. The residual resistivity ratios $RRR$ $=$ $\rho_{300K}$/$\rho_{0}$ for
Au2Pb and Au2Bi are 9 and 20, respectively. For Au2Pb, the structural
transitions at $T_{2}$ $=$ 51 K and $T_{3}$ $=$ 40 K are easily observed and
superconductivity appears below $T_{\rm{c}}$ $=$ 1.2 K (Fig. 2b), as
previously reported. Schoop2015  In comparison, Au2Bi shows typical metallic
behavior with no evidence for structural phase transitions, suggesting that it
is useful as a conventional metallic Laves phase analogue to Au2Pb. We
additionally find that Au2Bi exhibits superconductivity near 1.7 K (Fig. 2b),
as previously reported. Khan1975  Heat capacity measurements (Fig. 2c) show
that the Au2Bi superconductivity occurs in the bulk, where the ratio
$\Delta$$C$/$\gamma$$T_{c}$ $=$ 1.9, is comparable to what is seen for Au2Pb.
Schoop2015  This value is larger than what is expected for a BCS
superconductor ($\Delta$$C$/$\gamma$$T_{c}$ $=$ 1.43), but is smaller than
that of elemental lead. Neighbor1967  A fit to the data using the expression
$C/T$ $=$ $\gamma$ $+$ $\beta$$T^{2}$ gives an electronic coefficient $\gamma$
$=$ 2.3 mJ/mol-K2 and a Debye temperature $\Theta_{\rm{D}}$ $=$ 153 K ($\beta$
$=$ 1.6 mJ/mol-K4). For Au2Pb, we find $\gamma$ $=$ 1.9 mJ/mol-K2 and
$\Theta_{\rm{D}}$ $=$ 133 K, as previously reported. Schoop2015

Figure 2: (a) Cooling curves of the room temperature normalized electrical
resistivity $\rho$/$\rho_{\rm{300K}}$ versus temperature $T$ for Au2Pb and
Au2Bi, where the structural transitions $T_{1}$, $T_{2}$, and $T_{3}$ in Au2Pb
are indicated by arrows. (b) Zoom of the normalized resistivity
$\rho$/$\rho_{\rm{2K}}$ for $T$ $\leq$ 2.2 K, where superconductivity appears
as sharp drops to zero resistivity. (c) Heat capacity divided by temperature
$C/T$ vs. $T^{2}$ for Au2Pb and Au2Bi, where the superconducting transitions
appear as sharp lambda-type transitions. The dotted lines are fits to the data
as described in the text.

The electrical resistivity as a function of temperature under several values
of hydrostatic pressure is shown in Fig. 3. Metallic behavior is seen for all
pressures with a room temperature resistivity of 35.9 $\mu$$\Omega$cm for $P$
$=$ 0 which decreases to 33.0 $\mu$$\Omega$cm for $P$ $=$ 1.83 GPa. We find
residual resistivity ratios $RRR$ $=$ $\rho_{295K}$/$\rho_{0}$ in the range
8.1-9.2, slightly increasing with increasing $P$. The previously reported
transition at $T_{1}$ appears as a weak reduction in $\rho$, which becomes
more pronounced and increases with increasing $P$ at a rate of 19 K/GPa (Fig.
3b), indicating that this phase transition would occur at room temperature
near $P$ $\approx$ 9 GPa. $T_{2}$ and $T_{3}$ appear as sharp reductions in
the resistivity, consistent with an earlier report of structural phase
transitions at these temperatures under ambient pressure (Fig.
3a,d,e).Schoop2015  The features at $T_{2}$ and $T_{3}$ decrease linearly at
rates of 21.5 K/GPa and 1.8 K/GPa, respectively, and finally intersect near
$P_{1}$ $\approx$ 0.64 GPa (determined by extrapolation). For pressures larger
than 0.64 GPa, another phase transition appears as a sharp reduction in the
resistivity at $T_{4}$, which produces a more pronounced decrease in
resistivity than $T_{2}$ and $T_{3}$ combined. Unlike $T_{2}$ and $T_{3}$,
$T_{4}$ increases linearly at a rate of 12.6 K/GPa, reaching 52 K at $P$ $=$
1.83 GPa (Fig. 3d). All of the observed phase transitions show hysteresis
under $P$ (summarized in Fig 5). At low temperature and zero pressure, bulk
superconductivity is observed below $T_{\rm{c}}$ $=$ 1.2 K. Superconductivity
persists at 1 GPa, with a slightly reduced transition temperature $T_{c}$ $=$
1.1 K (Fig. 3e).

Figure 3: (a) Cooling curves of the electrical resistivity $\rho$ versus
temperature $T$ at pressures 0 GPa $\leq$ $P$ $\leq$ 1.83 GPa for Au2Pb. Note
that the sharp kink for $P$ $=$ 0 near $T$ $=$ 100 K is an artefact due to a
change in the $T$ sweep rate in this data set only. (b) Zoom of $\rho(T)$ at
several $P$ near the anomaly $T_{1}$, after background subtraction. The arrow
is a guide to the eye showing the transition temperatures and their evolution.
(c) Zoom of $\rho(T)$ for $P$ $\leq$ 0.46 GPa at temperatures near the sharp
anomalies at $T_{2}$ and $T_{3}$. (d) Zoom of $\rho(T)$ for $P$ $\geq$ 0.82
GPa at temperatures near the sharp anomaly at $T_{4}$. (e) $\rho(T)$ for $P$
$=$ 0.03 and 1 GPa showing superconductivity at low $T$. The superconducting
temperature evolves from $T$c $=$ 1.2 K at 0.03 GPa to $T$c $=$ 1.1 K at 1
GPa.

Plots of the resistivity at constant temperature under varying pressure
$\rho(P)$ are shown in Fig. 4. For $T$ $>$ 51 K, $\rho(P)$ is nearly constant
for $P$ $<$ 1.8 GPa. For $T$ $\leq$ 51 K, there is a large reduction of $\rho$
which first appears at large $P$ and $T$ and moves to lower $P$ with
decreasing $T$. For 38 K $<$ $T$ $<$ 51 K, there is a second transition at
lower pressures that evolves with temperature. For $T$ $<$ 38 K, there is a
single transition pressure that remains nearly constant at $P_{1}$ $\approx$
0.64 GPa with decreasing $T$. We infer that these features are associated with
the low temperature structural phase transitions seen in $\rho(T)$.

Figure 4: Electrical resistivity $\rho$ versus pressure $P$ at several
temperatures $T$. The arrows indicate the phase transitions. Several
structural transitions appear as sharp kinks in $\rho(P)$ as described in the
text. The black arrows indicate $T_{2}$, the red arrows indicate $T_{3}$, and
the blue arrows indicate $T_{4}$.

These results are collected in Fig. 5 to build the $T-P$ phase diagram. The
solid circles are the $\rho(T)$ cool down curves (Fig. 3), the hollow circles
are from the $\rho(T)$ of warm up curves and the triangles are from $\rho(P)$
(Fig. 4). While the precise location of the phase boundaries in the phase
diagram need additional measurements to establish the extent of the phase
fields (particularly where $T_{2}$ and $T_{3}$ intersect), there appear to be
five distinct regions including the previously described Dirac semimetal phase
which extends from room temperature to $T$ $=$ 97 K for $P$ $=$ 0 and several
other low $T$ phases labeled 1-4. Region 1 spans the entire pressure range.
Region 2 is a small wedge separating 1 and 3. Regions 3 and 4 are the low
$T$/low $P$ and low $T$/high $P$ phases which are separated by the nearly
vertical phase boundary at $P_{1}$ $\approx$ 0.64 GPa. We further find that
superconductivity is present in both regions 3 and 4, where the transition
temperature is slightly suppressed by $\approx$ 0.1 K in region 4.

Figure 5: The temperature-pressure $T-P$ phase diagram for Au2Pb. The
filled/hollowed circles are from the cooling/warming curves of $\rho(T)$ (Fig.
3) and the triangles are from the $\rho(P)$ at various $T$ (Fig. 4). The
ambient pressure phase transitions $T_{1}$(=97K) for both cooling and warming
curves are from ref. [Schoop2015, ].

##  IV Discussion

The past several years have witnessed intense interest in materials that
support novel electronic states arising from unusual band structures. Examples
include materials with topologically protected linearly dispersing bands such
as Dirac semimetals (Na3Bi Liu2014  and Cd3As2 Neupane2014 ), Weyl semimentals
(Nb,Ta)(P,As), Huang2015  and topological insulators (Bi2(Se,Te)3), Zhang2009
as well as other unusual metallic phenomena such as the axial anomaly (PdCoO2
kikugawa ). While these materials exhibit remarkable behavior, they are often
limited in terms of opportunities to test microscopic models by transforming
their thermodynamic ground state: e.g., using probes such as applied pressure.
As demonstrated in this study, Au2Pb and potentially the alloy Au2Pb1-xBix
provide the unusual examples where the evolution of topological behavior under
the influence of intrinsic structural phase transitions can be systematically
explored. In order to do this, it will first be important to determine the
structures and electronic characteristics of regions 1, 2, and 4 and to carry
out accompanying band structure calculations.

The presence of superconductivity in both the low and high pressure regions
for Au2Pb presents additional opportunities. Prior work suggests that the
ambient pressure superconductivity in Au2Pb occurs in the bulk and can be
described as being due to weak coupling electron-phonon interactions in the
BCS framework. It is appealing to attribute this superconductivity to the
quadratically dispersing bands that are predicted to cross $E_{\rm{F}}$ along
the $\Gamma$ $-$ $K$ line: i.e., the “conventional bands”. Schoop2015  If this
is the right picture, then Au2Pb is a model system in which to investigate
what happens in a material that hosts both conventional superconductivity and
gapped linearly dispersing bands. Prior work suggests that such a material
might support Majorana fermions, Leijnse2012  although this has yet to be
substantiated. Comparison to Au2Bi will help to independently determine
whether behavior seen in Au2Pb is uniquely associated with topological bands.

Our pressure study further amplifies the novelty of this situation by showing
that there is a distinct high pressure phase which also supports
superconductivity. It will be of interest to determine whether the high
pressure phase includes topological bands and whether they are significantly
different from those at ambient pressure. In the case that the high pressure
bands are all conventional, this would give the opportunity to definitively
separate what superconducting behaviors are uniquely associated with the
presence of topological bands. In particular, further studies of the
magnetoresistance and magnetization will be of use to search for unusual power
law behaviors and to index quantum oscillation Landau levels, which would
unambiguously characterize topological behavior.

##  V Conclusion

We have presented the $T-P$ phase diagram for the cubic Laves phase compound
Au2Pb, which supports several structural phases and superconductivity over a
broad range of pressures. At ambient pressure, we compare it to the
conventional metallic Laves phase analogue Au2Bi. Our observations have
important implications for understanding the interplay between conventional
and linearly dispersing bands, particularly with respect to possible anomalous
transport behavior and the relationship with superconductivity. Owing to its
structural tunability, both with temperature and pressure, it will be possible
to systematically explore these effects under different conditions. Such
information is needed to separate what features should be attributed to
topological behavior and which are merely due to specific details of this
material.

##  VI Acknowledgements

This work was performed at the National High Magnetic Field Laboratory
(NHMFL), which is supported by National Science Foundation Cooperative
Agreement No. DMR-1157490, the State of Florida and the DOE. T.B. and T.S. are
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Materials Sciences and Engineering Division, under Award DE-SC0008832. L.B. is
supported by DOE-BES through award DESC0002613. N.K. acknowledges the support
of overseas researcher dispatch program in NIMS.

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