# Electron-beam Floating-zone Refined UCoGe

K. E. Avers  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Department of Physics and Astronomy, Northwestern University, Evanston, IL,
USA  Center for Applied Physics & Superconducting Technologies, Northwestern
University, Evanston, IL, USA  M. D. Nguyen  Department of Physics and
Astronomy, Northwestern University, Evanston, IL, USA  J. W. Scott  Department
of Physics and Astronomy, Northwestern University, Evanston, IL, USA  A. M.
Zimmerman  Department of Physics and Astronomy, Northwestern University,
Evanston, IL, USA  S. M. Thomas  Los Alamos National Laboratory, Los Alamos,
NM 87545, USA  P. F. S. Rosa  Los Alamos National Laboratory, Los Alamos, NM
87545, USA  E. D. Bauer  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  J. D. Thompson  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
W. P. Halperin  Department of Physics and Astronomy, Northwestern University,
Evanston, IL, USA

###### Abstract

The interplay between unconventional superconductivity and quantum critical
ferromagnetism in the U-Ge compounds represents an open problem in strongly
correlated electron systems. Sample quality can have a strong influence on
both of these ordered states in the compound UCoGe, as is true for most
unconventional superconductors. We report results of a new approach at UCoGe
crystal growth using a floating-zone method with potential for improvements of
sample quality and size as compared with traditional means such as Czochralski
growth. Single crystals of the ferromagnetic superconductor UCoGe were
produced using an ultra-high vacuum electron-beam floating-zone refining
technique. Annealed single crystals show well-defined signatures of bulk
ferromagnetism and superconductivity at $T_{c}\sim$2.6 K and $T_{s}\sim$0.55
K, respectively, in the resistivity and heat capacity. Scanning electron
microscopy of samples with different surface treatments shows evidence of an
off-stoichiometric uranium rich phase of UCoGe collected in cracks and voids
that might be limiting sample quality.

##  I Introduction

The phenomenon of superconductivity, the abrupt disappearance of electrical
resistance below a material specific temperature, was discovered over 100
years ago in elemental mercury at 4.16 K [1, 2]. Fifty-six years later, the
BCS theory provided a microscopic understanding in which conduction electrons
with opposite momentum form bound states called Cooper pairs mediated by
lattice vibrations [3]. It was thought that superconductivity was a solved
problem until advances in sample growth techniques, such as flux growth,
Czochralski, and solid state reaction techniques, lead to the discovery of
high-temperature cuprate superconductors [4, 5] and heavy fermion
superconductors [6] in which the microscopic origin of the pairing mechanism
is not driven by phonons. In particular, uranium containing heavy fermion
superconductors exhibit exotic behavior including time-reversal symmetry
breaking in the superconducting state as reported for UPt3 [7, 8] and a hidden
order state coexisting with time-reversal symmetry breaking superconductivity
in URu2Si2 [9, 10]. Another exciting feature is that they offer some of the
clearest examples of systems with multiple superconducting phases, with both
the aforementioned UPt3 [11] and Th doped UBe13 [12] demonstrating this
unambiguously. Although many types of superconducting compounds are known,
almost all of them share a common trait: antagonism to external magnetic field
[13], magnetic impurities [14, 15], and ferromagnetic order [16]. The curious
exception to this rule at present is limited to the relatively new compounds
UGe2 [17], UCoGe [18], URhGe [19], and possibly UTe2 [20, 21] in which
ferromagnetic order, or rather ferromagnetic critical fluctuations [22], are
responsible for superconductivity. We emphasize that the ferromagnetic
fluctuation issue is an open question in UTe2 [23, 24] due to recent neutron
scattering experiments showing that antiferromagnetic ones are dominant [25],
which is also supported by recent hydrostatic pressure work [26]. Despite all
these materials having orthorhombic crystal structures [27], each have their
own unique behaviors: UGe2 exhibits crossover behavior between two
ferromagnetic phases with a different ordered moment [28], URhGe has a
magnetic field induced superconducting phase disconnected from its zero field
one [29], and UCoGe shows evidence of a first order ferromagnetic phase
transition [30].

The U-Co-Ge ternary system includes a number of compounds such as the room-
temperature ferromagnet U2Co17-yGey [31], the Pauli paramagnet U3Co12-xGe4
[32], and the heavy fermion low temperature ferromagnet U3Co2Ge7 [33]. Its
metallurgy is largely uninvestigated, which is a potential hindrance to the
improvement of UCoGe single crystals. UCoGe may have a small peritectic window
above 1300 ∘C and up to its melting point at $\sim$ 1360 ∘C [34], making it
feasible to obtain large single crystals from a high temperature
stoichiometric melt. The majority of reported single crystal samples have been
grown by Czochralski [18, 35, 36], with the exception of one growth attempt
with optical floating-zone refining [37]. This motivated our attempt to grow
UCoGe with an ultra-high vacuum (UHV), electron-beam floating-zone technique.
In previous work with this method, some of the highest quality bulk crystals
of UPt3 were reported [38, 39, 40]. However, maintenance of stoichiometry in
vacuum with volatile components, such as Ge and Co, was a potential drawback.
We report here that this drawback is not insurmountable. UCoGe ideally
crystallizes in the orthorhombic TiNiSi structure as shown in Fig. 1 with the
ferromagnetic Ising magnetic moments pointing along the c-axis below $T_{c}$=
2.6 K [27]. UCoGe may also crystallize in the related disordered CeCu2
structure which is characterized by random site mixing between the Co and Ge
sites [41].

Figure 1: Crystal structure of UCoGe. Uranium atoms: large blue balls; cobalt
atoms: light blue balls; germanium atoms: red balls. The lines between U atoms
are drawn to emphasize the zig-zag arrangement. The green arrows show the
Ising ferromagnetic moments on the uranium sites.

In this work, we detail the growth and characterization of UHV electron-beam
floating-zone refined UCoGe single crystals. The crystals were examined under
a variety of conditions (unannealed, annealed, acid-etched, polished, etc)
using scanning electron microscopy (SEM), low-temperature electrical
resistivity, heat capacity and magnetization. We find our crystals are of high
quality and display evidence of bulk superconductivity and ferromagnetism,
which we will show are consistent with previously reported high quality UCoGe
single crystals. By comparing results of these measurements under different
sample conditions we have found evidence of a yet undiscovered/unreported off-
stoichiometric uranium rich phase of UCoGe, referred to as the low temperature
phase (LTP), that coexists within cracks and voids inside the UCoGe single
crystal. This LTP may be a key reason for the difficulty in improving the
quality of UCoGe single crystals. Our work may guide future efforts in
improving the quality of UCoGe single crystals, and hence enabling a better
understanding of the exotic phenomena of ferromagnetic critical fluctuation
mediated superconductivity.

##  II Methods

Near-stoichiometric amounts of U (electromigration purified to 99.99 %
purity), Co (99.99 %), and Ge (99.9999%) elements were arc-melted into 4
individual buttons of UCoGe on a water cooled copper hearth under $\sim$20
torr of Ar (99.999 %). The average stoichiometry of the buttons determined by
the initial mass of the constituent elements is 33.48 % U, 33.16 % Co, and
33.36 % Ge. All percentages in this work are atomic percent (at. %), not mass
percent. Before each arc-melting operation, a getter made of uranium was
heated into the liquid state in order to absorb oxygen. The buttons were arc-
melted into two final ingots in a long narrow trough formed in the hearth. The
ingots had diameters of $\sim$ 6 mm and lengths of $\sim$ 150 mm and $\sim$ 65
mm with a combined mass of $\sim$ 57 g. The ingots were quite brittle and
tended to crack and eventually break if mishandled, requiring a few attempts
to optimize their casting. During arc-melting it was observed that the ingots
would not remain straight, but rather would bend upward away from the hearth
suggesting there exists a high temperature structural transition in the solid
phase as it cools. Effects of this high-temperature phase have been reported,
with prior samples becoming distorted when annealed close to the melting
temperature of UCoGe [42]. It was also observed that small flakes of a high
temperature solid phase, suspected to be a uranium oxide phase based on later
SEM observations, floated on top of the molten parts of the ingot both during
arc-melting and zone refining.

The final ingots were mounted vertically in a UHV electron-beam floating-zone
refining apparatus, clamped at their ends between molybdenum fixtures and set
screws, with the longer secured ingot above and the shorter ingot below. The
specifics of the zone refining apparatus has been previously described in Ref.
38. The vacuum chamber was evacuated to $\sim$8 nTorr via a high temperature
bake out procedure using a diffusion pump with a liquid nitrogen baffle to
prevent oil vapor contamination. The ingots were then joined in-situ using the
electron gun, and a $\sim$75 mm length of the ingot was refined 3 times by
translating the electron gun upwards with a rate of $\sim$15 mm/hr as shown in
Fig. 2 (a) and (b).

During zone refining a liquid nitrogen Meissner trap was used to capture
volatile impurities. In addition to any undesired volatile impurities, the
viewing ports became darkened due to the relatively high vapor pressure of the
Co and Ge compared to U. It was later shown via energy-dispersive x-ray
spectroscopy (EDS) that the stoichiometry may have changed. The resulting
ingot, after being removed from the chamber, is shown in Fig. 2 (c). The
variation of diameter in the lower parts of the ingot is due to adjustments of
electron-beam power as well as vertical adjustments of the lower ingot
necessary to maintain a stable floating-zone. In contrast, the upper part of
the ingot required fewer adjustments resulting in a more uniform cross section
and smoother as-grown surface. The discolorations are impurities that floated
to the surface with the greater amount of discoloration near the top
indicating that the zone refining process was able to segregate impurities to
the top of the ingot. The impurities and surface structure hindered Laue x-ray
diffraction on the as-cast surface resulting in patterns with smeared Bragg
peaks, or even no observable Bragg peaks at all. This necessitated spark
cutting, polishing, and orienting for the Laue pattern in Fig. 2 (d) aligned
along the [010] (b-axis) direction. Subsequent Laue surveys suggested that the
ingot did not become a large single crystal, but rather multiple macroscopic
crystal grains. This was determined by taking multiple Laue exposures at
different locations along the length of the ingot where we found a uniform
Laue pattern, and hence a crystal grain, that persisted over at least $\sim$
10 mm. Once an appropriate region belonging to a single crystal was found, an
oriented wafer was spark cut from that region of the ingot, polished, and
further Laue surveys conducted to determine if the interior region was single
crystalline. It was observed that neither the a, b, nor c-axes coincided with
the growth direction during zone refining which required cuts to be made at
reasonably large inclination angles ($\sim$30∘) relative to the cylinder axis
of the ingot.

Once the wafer was determined to be a suitable single crystal grain, it was
further oriented and cut into needle shaped samples for further
characterization. Two needles, labeled S1 and S2, were oriented and spark cut
to 3.10 mm $\times$ 0.96 mm $\times$ 1.10 mm, and 4.30 mm $\times$ 1.10 mm
$\times$ 1.30 mm, respectively, along the a, b, and c-axes. S1 was taken from
the lower part of the ingot while S2 was taken closer to the top, with the
approximate location of both needles indicated in Fig. 2 (c). Note that
substantial diameter variation in the region of S1, while the diameter of S2
is relatively smooth. The b-axis normal surface of these needles was polished
to a mirror shine using 1200 grit SiC sandpaper for the x-ray Laue diffraction
measurements performed on the parent wafers.

In the following text, we will discuss the samples in three preparation
conditions, hereafter denoted X, Y, and Z. The initial sample conditions after
zone refinings will be referred to as X or “unannealed”. Samples annealed
under UHV conditions for 2 weeks at 900 ∘C while resting on polycrystalline
UCoGe wafers, but with no further treatment, will be referred to as Y or
simply “annealed”. After annealing, it was observed that a liquified phase
collected on the polished (ac-plane) surfaces, but not the spark cut (ab-
plane) surfaces. This phase will be referred to as the ‘low temperature phase’
(LTP), due to its relatively low melting point. Powder X-ray diffraction (not
shown) did not indicate existence of any phases other than UCoGe in either
arc-melted or zone refined material, so the identity of this phase remains
unknown, although it is clearly visible optically and in the SEM data. Due to
the LTP, the needles were further polished to a final thickness along the
b-axis direction of 0.40 mm and 0.19 mm for S1 and S2, respectively. This
final sample condition will be referred to as Z or “annealed/polished”. In
addition, when discussing results pertaining to the outer diameter of the
ingot it will be referred to as the “as-grown” surface in contrast to
discussing surfaces from the interior that were spark cut and polished.

Figure 2: (a): Photograph of the molten floating-zone during growth of UCoGe.
The green arrow indicates the fully molten floating-zone that was being swept
upward. (b): View of the ingot being zone refined showing the circular
electron gun filament housed in a water cooled copper translation stage, and
the molybdenum grounding plate used to focus the electron-beam held below by
two threaded shafts. (c): Picture of UCoGe sample after three zone refining
passes over a $\approx 75$ mm (3 inches on the ruler) long segment of the
sample with the approximate location of S1 and S2 shown by the green and
purple stars, respectively. (d): Laue x-ray diffraction image of an aligned
piece of the UCoGe zone refined sample. The diffraction pattern shows the
sample is aligned close to the [010] (b-axis) direction demonstrating the
single crystalline nature of the zone refined sample.

SEM images were collected using a Hitachi S-3400N-II with an attached Oxford
INCAx-act SDD EDS to determine relative elemental composition via the AZtec
software package. All elemental percentages have an error of $\sim$1 at.% as
reported by the software, although we comment that since _in situ_ elemental
references were not used the actual error may be larger. Electrical
resistivity measurements were performed using a standard four-point method
down to 425 mK for unannealed (X) S1 and for annealed (Y) S1 and S2. The
temperature was measured by a germanium resistance thermometer that was
calibrated below 45 K. Heat capacity measurements were performed using a
Quantum Design PPMS. After the samples were polished along the b-axis
direction to their final thickness as annealed/polished (Z) the resistivity
was remeasured without the surface influence from the LTP. This second series
of resistivity measurements were made with a Quantum Design PPMS. For the
resistivity data that will be presented the excitation current was 0.6 mA for
unannealed (X) S1 X, 0.1 mA for annealed (Y) S1, 0.5 mA for annealed (Y) S2,
and 0.1 mA for annealed/polished (Z) S1 and S2. Magnetization measurements
were done using on a Quantum Design MPMS.

##  III Results

SEM images from UCoGe under various treatment conditions are presented in Fig.
3. These images were collected from wafers near S1 (Figs. 3 (a), (b)) or in
the case of Fig. 3 (c) and Fig. 3 (d) directly from S1. In general, SEM was
not performed on pieces oriented on a crystallographic axis, but rather
circular wafers cut from the main ingot with the cut surface normal along the
growth direction. Fig. 3 (a) shows an interior surface that was polished, and
then etched in a warm mixture of nitric and hydrochloric acid where the
etching produced a large crack in the surface. This suggests that a
metallurgical phase other than UCoGe is present that is more susceptible to
the etching and is likely to be the aforementioned LTP, even though such a
phase could not be detected in powder x-ray diffraction. We acknowledge that
this crack could instead have started as a dislocation or grain boundary. This
would also provide an energetically favorable location for the chemical
reaction, but without detailed transmission electron microscopy results it is
difficult to comment on the presence of larger scale structural defects in
UCoGe. The interior region had EDS spectra consistent with a U:Co:Ge molar
ratio of $\sim$ 1:1:1 ($\pm$0.03, or 1%). The as-grown surface is shown in
Fig. 3 (b) showing a micro-structure that formed at the surface during growth.
Unlike SEM taken from the etched surface of interior regions, the as-grown
surface showed regions with an oxygen signal in the EDS spectra, suggesting
that a uranium oxide segregated to the outside diameter of the ingot during
growth. This is consistent with the aforementioned solid flakes we observed on
the molten surface during arc-melting and zone refining. This slight oxide
contamination is likely due to natural oxidation when the stock material was
exposed to air before and after arc-melting attempts, as well as the $\sim$ 3
months the ingots were left exposed to air between the final arc-melting and
zone refining. The as-grown surface also had measurable regions of $\sim$
1:1:1 $\pm$ 0.03 free of oxygen, indicating that the oxidation is not
completely coating the surface. The surface structure and oxide made Laue
difficult when taken directly from the as-grown surface. Both arc-melted and
zone refined polished surfaces (not shown) are completely featureless other
than scratch marks from polishing, and also had EDS spectra consistent with
UCoGe. Phase boundaries between the UCoGe and any secondary phases could not
be resolved. In contrast Fig. 3 (c) and (d) show the aftermath of annealing on
polished surfaces. The LTP liquified and flowed through micro-cracks (c), and
holes in the material (d). Upon solidification the LTP formed small particles,
with a larger concentration of particles near the cracks and holes.

Figure 3: SEM images taken from various zone refined UCoGe samples near the
bottom of the ingot at different stages of treatment. (a) SEM image of warm
HCl+nitric acid etch displaying a large crack. (b) SEM image of the as-grown
surface displaying micro-structure of the unknown high temperature phase
observed during the growth process, suspected to be a uranium oxide due to the
oxygen detected in EDS on this surface. (c), (d) SEM image of a surface that
was polished to a mirror finish, and then annealed at 900 ∘C for 2 weeks
(condition Y). The LTP liquified and collected on the polished surface via
either large cracks (c), or from small holes (d).

The average elemental composition for each of the different sample treatments
is reported in table 1. The arc-melted results are very similar to the
percentages calculated from the initial mass of the constituent elements used,
suggesting that there is minimal evaporation during arc-melting, and that the
internal calibration used by the AZtec software is reasonably accurate for the
UCoGe system. After zone refining, the amount of uranium detected increased to
35 % consistent with the minor, but noticeable, evaporation of Co and Ge
observed during zone refining. After acid etching, the surface uranium rich
regions were removed and compositions of U, and Co were selectively reduced
such that the surface stoichiometry was close to ideal. Nonetheless, there
were some spectra collected that had % Ge as high as $\sim$43 % consistent
with Ge being chemically inert compared to U and Co. In contrast, the after
zone refined and arc-melting SEM observations had minimal variance from site
to site of a few percent, within error. Most of the measurements are within a
reasonable error consistent with on-stoichiometric UCoGe, but the unannealed
as-grown surface condition X and zone refined annealed condition Y surface are
uranium rich. The as-grown surface became rich in uranium due to the
segregation of uranium oxide(s) to the outer diameter of the ingot during
growth, although it is also possible that the as-grown surface was depleted of
Co and Ge since that is where the evaporation occurs. It is harder to confirm
the latter as we also observed relatively oxygen free regions that had
percentages consistent with UCoGe. However, the annealed surface also had a
noticeable increase in U, with no region that could be identified with
percentages consistent with ideal stoichiometry, unlike all the other EDS
measurements. The highest recorded uranium concentration was $\sim$ 70 %
directly from one of the large particles of LTP that formed. This indicates
that this LTP is richer in U compared to UCoGe. In contrast to our
observations, there have been previous secondary phases resolved in SEM on
UCoGe such as elemental U, U3O8 and U3Co4Ge7 [37]. However, all three of these
previous phases are unlikely to be the LTP, as both U and U3O8 melt at higher
temperatures than the 900 ∘C annealing temperature, whereas U3Co4Ge7 has the
incorrect stoichiometry to be consistent with the increase of U concentration
observed in the condition Y results in table 1.

Condition | % U | % Co | % Ge
---|---|---|---
arc-melted | 34$\pm$1 | 33$\pm$1 | 33$\pm$1
zone refined polished (X) | 35$\pm$1 | 32$\pm$1 | 33$\pm$1
zone refined acid etched (X) | 33$\pm$1 | 32$\pm$1 | 35$\pm$1
zone refined as-grown surface (X) | 63$\pm$1 | 17$\pm$1 | 20$\pm$1
zone refined annealed (Y) | 52$\pm$1 | 26$\pm$1 | 22$\pm$1
Table 1: Average atomic percentages (at. %) of the major elements at various
stages of treatment. With the exception of the arc-melted row all entries are
from material near S1. All entries were from interior regions of the crystal
that were spark cut and polished except the zone refined as-grown surface
entry. Most surfaces had ratios consistent with UCoGe, except the zone refined
as-grown surface due to uranium oxide segregating to the outer surface of the
ingot during growth, and the zone refined annealed surface due to the
liquefaction of the LTP that collected on the polished surface. The 1 % error
is reported by the AZtec software, but this should be interpreted as a lower
bound given the lack of reference sources used for calibration.

The temperature dependence of the resistivity, $\rho(T)$, taken from the ac-
plane surface is shown in Fig. 4 with electrical current applied parallel to
the a-axis up to 300 K (a), 10 K (b), and 1 K (c). As observed in Fig. 4(a),
all data exhibit the feature of a local maximum in resistivity at roughly 65
K, known to be characteristic of the Kondo coherence effect [43]. It is
apparent that unannealed (X) and annealed/polished (Z) resistivities in Fig. 4
(a) match each other quite well, except at the lowest temperatures in Fig. 4
(c). The annealed (Y) behavior is an anomaly worth examining as the
discrepancy is greater than that which can be explained by measurement error
of the dimensions of the samples. The annealed (Y) S1 resistivity (blue
squares) is substantially greater than unannealed (X) S1 resistivity (yellow
circles) in Fig. 4 (a) suggesting that the cracks and inhomogeneous
distribution of the LTP shown in Fig. 3 (c) and (d) affects electrical
transport. It is evident that the annealed (Y) S1 resistivity is nearly a
factor of 3 larger in magnitude compared to the S2 data (red triangles). After
the samples were polished (Z) the S1 (turquoise diamonds) and S2 (purple
triangles) magnitudes of $\rho(T)$ decreased and increased, respectively, as
represented by the light blue arrow and red arrow. In addition, the
annealed/polished (Z) $\rho(T)$ for both S1 and S2 match the unannealed (X) S1
data at high temperatures demonstrating that most of the damage from the LTP
is confined to the surface and is able to be removed.

This limit of the influence of the LTP to the surface is supported in Fig. 4
(b) by annealed (Y) S1 having a smaller RRR = $\rho(300K)/\rho(T\rightarrow
0K)$ of 20 compared to the unannealed (X) case with a RRR of 30, where the
extrapolation is from the non-superconducting state. The influence of the
cracks and LTP on annealed (Y) S2 results in a much smaller $\rho(T\rightarrow
0K)$, despite having the same RRR. This suggests that the LTP is less
resistive than the bulk UCoGe, and that electrical transport can be partially
shorted depending on the exact location of the electrical contacts relative to
the puddles of LTP. The $\rho(T\rightarrow 0K)$ for both annealed (Y) S1 and
S2 are $\sim$ 18 $\mu\Omega$-cm and $\sim$ 7 $\mu\Omega$-cm respectively,
while for S1 when unannealed (X), it is $\sim$ 13 $\mu\Omega$-cm. The onset of
superconductivity ($T_{s}$) for S1 is at $T_{s}\sim$ 0.61 K determined by
linearly interpolating resistivity between the normal state and
superconducting transition regions.

As shown in Fig. 4 (b), all three data sets have a $\sim T^{\frac{5}{3}}$
power law behavior above a Curie temperature ($T_{c}$) of $T_{c}\sim$2.6 K.
Only when annealed (Y) do both samples show a well-defined power law of $\sim
T^{2}$ in the ferromagnetic state, while unannealed (X) S1 has a very
broadened ferromagnetic transition with no indication of superconductivity.
Both power law behaviors are consistent with a clean metallic ferromagnetic
system dominated by electron-magnon scattering [44], and has been observed
previously in other high quality UCoGe samples [45]. We note that $\sim T^{2}$
is also consistent with electron-electron scattering in a Fermi liquid, as has
been observed in a large number of non-magnetic heavy fermion compounds [46].
It is likely that both electron-magnon and electron-electron scattering are
present in UCoGe, but that the former is dominant. That is indicated from the
Kadowaki-Woods ratio, a comparison between the $\sim T^{2}$ resistivity
coefficient ($\sim$ 2.5 $\mu\Omega$-cm/K2 for our annealed/polished (Z)
samples) and the heat capacity squared ($\sim$ (56)2 mJ2/mol2-K4 as estimated
from heat capacity later), for UCoGe is $\sim$ 800 $\mu\Omega$-cm mol2 K2/J2.
This ratio is much larger than in heavy fermion superconductors ($\sim$ 10
$\mu\Omega$-cm mol2 K2/J2) [46] placing the two systems in distinctly
different categories where the reasons for the enhanced resistivity
coefficient and heat capacity are different. For further insight into this
relation between resistivity and heat capacity in the ferromagnetic
superconductors see Refs. [47, 48].

As can be seen in Fig. 4 (c), the annealed (Y) samples have zero resistance
superconducting temperatures of $T_{s}$ $\sim$ 0.52 K, and $\sim$ 0.44 K for
S1 and S2, respectively. There is no resolvable change in the superconducting
transition temperatures after polishing (Z); both S1 and S2 maintain the same
zero resistance superconducting temperatures, and there is likewise no
apparent change in the widths of the transitions indicating that polishing
after annealing does not introduce significant strain. The most drastic change
of polishing is that S1 now has a RRR of 30, rather than its annealed (Y)
value of 20 reflecting the removal of the undue influence of the LTP. The RRR
of S2 did not change due to polishing and remained at 20. The values of
$\rho(T\rightarrow 0K)$, however, did change for S1 to $\sim$ 8 $\mu\Omega$-cm
and for S2 to $\sim$ 11 $\mu\Omega$-cm, reflecting the overall change in
$\rho(T)$ indicated by the red and light blue arrows as was noted at higher
temperatures in Fig. 4 (a). There was no qualitative change in the shape, and
hence power law behavior, in the ferromagnetic state nor the paramagnetic
state above $T_{c}$ (not shown). We emphasize that the RRR of 30 for the
unannealed (X) S1 sample was obtained by extrapolating to zero excitation
current and that the annealed (Y) samples displayed minimal change in RRR with
excitation current. If this extrapolation was not done then the unannealed (X)
data shown would also exhibit RRR$\sim$20.

Figure 4: Electrical resistivity $\rho$ vs. temperature $T$ of zone refined
UCoGe for current parallel to the a-axis (I$\parallel$a-axis) for the two
samples S1 while unannealed (X), annealed (Y), and annealed/polished (Z) and
S2 while annealed (Y), and annealed/polished (Z) from the bottom and top of
ingot, respectively, from 0 K to 300 K (a), below 10 K (b), and below 1 K (b).
Only annealed (Y) $\rho$ show deviations that can be ascribed to the LTP. (a)
shows the difference that polishing the LTP off the electrical contact surface
made in achieving consistent behavior between S1 and S2. The blue arrow shows
that $\rho$ for S1 decreased at all temperatures, while the red arrow points
out the opposite behavior for S2. Once polished (Z) both samples match
quantitatively and qualitatively with the unannealed (X) S1 indicating that
the damage from the the LTP is limited to the surface of the sample. (b) shows
the ferromagnetic ($T_{c}$) and superconducting transitions ($T_{s}$) for the
annealed (Y) S1 and S2, and the lack of superconductivity and broadened
ferromagnetic transition in unannealed (X) S1. All solid curves are power law
fits to $\rho=\rho_{0}+AT^{\frac{5}{3}}$ above $T_{c}$ or to
$\rho=\rho_{0}+AT^{2}$ in the ferromagnetic state. (c) shows that the
superconducting transition temperature for both samples remains the same
between annealed (Y) and annealed/polished (Z), except afterward the sample
with the higher zero resistance superconducting transition temperature of 0.52
K (S1) now has a lower $\rho(T)$ than the sample with the lower
superconducting transition temperature of 0.44 K (S2).

The heat capacity, plotted as $C/T$ vs. T, from the annealed (Y) S1 sample and
the annealed/polished (Z) s2 sample are presented in Fig. 5. Both $T_{s}$ and
$T_{c}$ as determined from $\rho$ in Fig. 4 are indicated and show anomalies
in $C/T$. The Sommerfeld coefficient can be estimated to be $\sim$ 56
mJ/mol-K2 based on its magnitude to $\sim$ 3 K. The $C(T)/T$ data show a minor
inflection and broad hump with the onset of the itinerant ferromagnetic phase
at $T_{c}$ = 2.6 K. Unlike a typical ferromagnetic phase transition with well
localized moments, the weak itinerant ferromagnetic transition in UCoGe
exhibits a minimal entropy change due to the strong hybridization of the
conduction electrons with the U local moments. This minimal entropy change at
$T_{c}$ is relevant to the quantum critical behavior and subtle changes to the
ferromagnetic and paramagnetic states observed as a function of continuous
doping of UCo1-xRhxGe [49]. The ferromagnetic transition for annealed/polished
(Z) S2 is sharper and has slightly more entropy change than annealed (Y) S1
sample. This is unlikely to be due to the different sample conditions (Y vs.
Z) as the LTP that collected at the surface is not expected to make up a large
enough mass fraction to manifest in heat capacity. If the LTP did manifest in
heat capacity then the $C/T$ near $T_{s}$ and at temperatures above $T_{c}$
would not match between the two samples, which is not evident from the data.
In contrast, $T_{s}$ shows the start of a noticeable rise in $C/T$ coinciding
with the zero resistance superconducting temperature of 0.52 K indicating that
the superconducting state occupies a significant volume of the sample, instead
of existing only at the surface. The $C/T$ of both samples are
indistinguishable from each other qualitatively and quantitatively in the
vicinity of $T_{s}$. Comparison with the details of the heat capacity in the
superconducting state between our samples and those of Refs. [50, 47] must
await future specific heat measurements at lower temperatures than what was
achieved in this study. There exists no indication of any behavior in $C/T$
that could be ascribed to the LTP.

It is possible that the LTP is present and undetected or unreported in
previous work on UCoGe and may explain the difficulty in improving sample
quality, as well as the lack of consistency in $\rho(T)$ from different
reports. It may also explain the difficulty in Shubnikov$-$de Haas (SdH)
quantum oscillation measurements [51, 52, 53, 54]. The identity of the low
temperature phase responsible for the significant change in $\rho(T)$ observed
in our samples is an important issue for improving sample quality. Due to the
lack of information about the U-Co-Ge ternary system it is possible that UCoGe
is a single phase that is stable over a finite composition range. This might
be reflected in the aforementioned historical disagreement in the crystal
structure of UCoGe [27, 41].

However, depending on the growth technique, it is likely that regions of
$\sim$1:1:1 stoichiometry UCoGe make up the major volume of the crystal which
hosts bulk superconductivity as shown by the observable rise in $C/T$ in Fig.
5, but off-stoichiometric U(CoGe)1-δ ends up stuck in cracks and voids of the
crystal. It appears that the LTP is more susceptible to chemical attack by
acid possibly due to the lack of Ge, which is relatively chemically inert
compared to U, and Co. In turn, the lack of Co and Ge results in a decrease of
the melting temperature of U(CoGe)1-δ below 900 ∘C where the samples were
annealed. There is evidence that the proper Co and Ge stoichiometry is
critical in determining sample quality, with too low of a Co and Ge
concentration resulting in lack of ferromagnetism and superconductivity [37].
We plan future growth attempts with an excess of Co and Ge to compensate for
the evaporation during growth that led to the formation of regions of the low
temperature phase.

Figure 5: Heat capacity, plotted as $C/T$, vs. $T$ of zone refined UCoGe S1
annealed (Y). Anomalies at the ferromagnetic transition $T_{c}$ = 2.6 K and at
the superconducting transition at $T_{s}$ = 0.61 K are observed, as indicated
by the arrows. The upturn in $C/T$ provides evidence for a bulk
superconducting transition.

The magnetization, $M$, vs. field, $H$, is shown in Fig. 6 (a) with the inset
focusing on the low $H$, low $M$ region for both samples in the
annealed/polished (Z) condition. The $\mathbb{H}$ is parallel to the Ising
c-axis direction. The $|M|$ is $\sim$ 0.24 $\mu_{B}$ per uranium atom at the
maximum $|H|$ of 6 T. At large $|H|$ both samples have nearly identical $|M|$
suggesting that both samples have the same magnetic moment per uranium atom
within experimental resolution. At low fields in the inset of Fig. 6 (a) S1
has a slightly smaller $|M|$ compared to S2, but they both have nearly the
same ferromagnetic coercive field of $\sim$ 0.002 T.

The $M$ vs. $T$ is shown in Fig. 6 (b) taken upon cooling in $H$=0.01 T
parallel to the c-axis. At high temperatures there is minimal spontaneous
magnetic moment, but upon cooling there is a gradual onset of a spontaneous
magnetic moment reflected in the increase of $M$ consistent with a
ferromagnetic transition at $T_{c}$ of 2.5 K, as was indicated by the kink at
2.5 K in $\rho(T)$ from Fig. 4 (b). The solid lines are a fit to the
expression $M^{2}(T)=M_{0}^{2}(1-(T/T^{*})^{2})$ below 2.5 K with $M_{0}$ and
$T^{*}$ as free fit parameters, as was done in Refs. [45, 18]. The obtained
$T^{*}$ for both samples is $\sim$ 2.9 K, but the $M_{0}$ for S1 is $\sim$
0.048 $\mu_{B}$ and for S2 is $\sim$ 0.059 $\mu_{B}$. Our $M_{0}$ values from
fitting the $M$ vs. $T$ at low fields are similar to the previously reported
values of 0.06 $\mu_{B}$ [45] and 0.07 $\mu_{B}$ [18]. The mismatch in $M(T)$
between the two samples is likely due to demagnetization effects, misalignment
from the c-axis, or slight difference of magnetic properties between the two
samples. The $M$ behavior shown in Fig. 6 is qualitatively consistent with
previous high quality samples in Refs. [45, 18] and quantitatively consistent
at high fields.

Figure 6: Magnetization, $M$, vs. field, $H$, with an inset focusing on the
ferromagnetic hysteresis (a) taken at 2 K with $\mathbb{H}$ parallel to the
c-axis. $M$ vs. $T$ upon cooling in an $H=0.01$ T parallel to the c-axis.
Solid lines are fits to $M^{2}(T)=M_{0}^{2}(1-(T/T^{*})^{2})$ as described in
the text.

##  IV Discussion

The UCoGe samples we have prepared are of quite respectable quality with RRR
$\sim$ 20-30, showing signatures of bulk superconductivity and ferromagnetism.
However, UCoGe sample quality can not be uniquely identified with RRR as this
implicitly assumes that $\rho(300K)$ is sample quality independent, which is
not self-evident from the present understanding of this material.

The sample with the highest RRR for which $\rho(T)$ was reported is by Aoki et
al. [47] at 110, and it shows a zero resistance temperature of $\sim$ 0.58 K
with $\rho(T\rightarrow 0K)\sim$ 3 $\mu\Omega$-cm. But in this instance the
superconducting transition in the 110 RRR sample is substantially broadened
suggesting poor sample homogeneity. Most unconventional superconductors are
quite sensitive to disorder [55], and exhibit suppressed transition
temperatures as $\rho(T\rightarrow 0K)$ increases such as in the case of UPt3
[39]. Aoki et al. [47] describe the evolution of $T_{s}$, with RRR and
$\rho(T\rightarrow 0K)$ across multiple single crystal UCoGe samples [47], and
postulate that $T_{s}$ may be correlated linearly with 1/RRR. There is some
uncertainty as to what is the best metric for sample quality for UCoGe and we
suggest that $\rho(T\rightarrow 0K)$ is a more reliable measure than 1/RRR
since the high temperature resistivity appears to vary from sample to sample
[36, 47, 56, 37]. However, we note one inconsistent report with a zero
resistance superconducting temperature of $\sim$ 0.5 K with a comparatively
large $\rho(T\rightarrow 0K)$ of 27 $\mu\Omega$-cm [56], albeit with a sample
having conflicting indications of the presence/absence of bulk ferromagnetism.

Another important indication of sample quality is the consistent indication of
$T_{s}$ from $\rho(T)$ and $C/T$ which is not always the case [57]. Our single
crystals, as well as other high quality single crystals [50], show signatures
of superconductivity in both $\rho(T)$ and $C/T$ at roughly the same
temperature. A further indication of high crystal quality is a smaller
$\rho(T\rightarrow 0K)$ value. The value of $\rho(T\rightarrow 0K)$ for S1 is
$\sim$ 8 $\mu\Omega$-cm , which is low compared with the results in [47, 36,
56], and more in line with the results of [18, 45]. Finally we note that with
our growth method, $\rho(T)$ for unannealed (X) and annealed/polished (Z), is
of the same magnitude rather than being a factor of 10 larger as reported by
[45]. This may be due to the lack of large scale structural defects in our
unannealed (X) samples compared to Czochralski grown crystals. This leads to
the question of why our unannealed (X) samples lack superconductivity and have
a very broadened ferromagnetic transition. It is possible that the main effect
of annealing is correcting random site occupancy of the Co and Ge via
diffusion, rather than removing larger defects like stacking faults. This
would be consistent with both the aforementioned historical disagreement of
the true crystal structure, and the observation of Co and Ge stoichiometry
being vital for robust ferromagnetism and superconductivity [37].

##  V Conclusion

We have used the uncommon but effective method of UHV electron-beam zone
refining to grow single crystals of the ferromagnetic superconductor UCoGe.
The samples exhibit robust, bulk superconductivity and ferromagnetism, as
shown by the $\rho$, $C/T$, and $M$ data, placing our samples in good company
with the best UCoGe made to date. There is evidence of a slight loss of Co and
Ge causing structural integrity issues, as well as negatively impacting the
superconductivity and ferromagnetism by forming regions of off-stoichiometric
U(CoGe)1-δ in cracks and voids in the single crystal samples. We have
uncovered new metallurgical properties of UCoGe that may allow future
improvement of UCoGe single crystal quality. Our ability to zone refine
multiple times on a single ingot indicates that we can further improve sample
quality if we properly account for the Co and Ge volatility under UHV
conditions. We have shown surface treatment to have an influence on the
measured resistance of UCoGe, but not superconducting and ferromagnetic
temperatures, and consequently for future work we recommend a procedure for
careful polishing of the crystals to improve the interpretation of electrical
transport data.

##  VI Acknowledgments

Work at Los Alamos National Laboratory was performed under the auspices of the
U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering. This work made use of the Jerome B. Cohen
x-Ray Diffraction Facility supported by the MRSEC program of the National
Science Foundation (DMR-1720139) at the Materials Research Center of
Northwestern University and the Soft and Hybrid Nanotechnology Experimental
(SHyNE) Resource (NSF ECCS-1542205). Support for crystal growth was provided
by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES),
Division of Material Sciences and Engineering under Award No. DE-
FG02-05ER46248 (WPH). This work made use of the EPIC facility of Northwestern
University’s NUANCE Center, which has received support from the SHyNE Resource
(NSF ECCS-2025633), the IIN, and Northwestern’s MRSEC program (NSF
DMR-1720139). Support is acknowledged from the Northwestern-Fermilab Center
for Applied Physics and Superconducting Technologies (CAPST). Support is also
acknowledged by the U.S. Department of Energy, Office of Science, Office of
Workforce Development for Teachers and Scientists, Office of Science Graduate
Student Research (SCGSR) program. The SCGSR program is administered by the Oak
Ridge Institute for Science and Education for the DOE under contract number
DE-SC0014664.

## References

  * Onnes [1911] H. K. Onnes, Proceedings 13 II, 1274 (1911).
  * van Delft and Kes [2011] D. van Delft and P. Kes, The Discovery of Superconductivity, Europhysics News 42, 21 (2011).
  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev. 108, 1175 (1957).
  * Bednorz and Muller [1986] J. G. Bednorz and K. A. Muller, Possible High Tc Superconductivity in the Ba-La-Cu-O System, Z. Phys. B 64, 189 (1986).
  * Wu  _et al._ [1987] M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu, Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Compound System at Ambient Pressure, Phys. Rev. Lett. 58, 908 (1987).
  * Steglich  _et al._ [1979] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Schafer, Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu2Si2, Phys. Rev. Lett. 43, 1892 (1979).
  * Schemm  _et al._ [2014] E. R. Schemm, W. J. Gannon, C. M. Wishne, W. P. Halperin, and A. Kapitulnik, Observation of Broken Time-Reversal Symmetry in the Heavy-Fermion Superconductor UPt3, Science 345, 190 (2014).
  * Avers  _et al._ [2020] K. E. Avers, W. J. Gannon, S. J. Kuhn, W. P. Halperin, J. A. Sauls, L. DeBeer-Schmitt, C. D. Dewhurst, J. Gavilano, G. Nagy, U. Gasser, and M. R. Eskildsen, Broken Time-Reversal Symmetry Breaking in the Topological Superconductor UPt3, Nature Phys. https://doi.org/10.1038/s41567-020-0822-z (2020).
  * Wartenbe  _et al._ [2019] M. Wartenbe, R. E. Baumbach, A. Shekhter, G. S. Boebinger, E. D. Bauer, C. C. Moya, N. Harrison, R. D. McDonald, M. B. Salamon, and M. Jamie, Magnetoelastic Coupling in URu2Si2: Probing Multipolar Correlations in the Hidden Order State, Phys. Rev. B 99, 235101 (2019).
  * Schemm  _et al._ [2015] E. R. Schemm, R. E. Baumbach, P. H. Tobash, F. Ronning, E. D. Bauer, and A. Kapitulnik, Evidence for Broken Time-Reversal Symmetry in the Supercondcuting Phase of URu2Si2, Phys. Rev. B 91, 140506(R) (2015).
  * Adenwalla  _et al._ [1990] S. Adenwalla, S. W. Lin, Q. Z. Ran, Z. Zhao, J. B. Ketterson, J. A. Sauls, L. Taillefer, D. G. Hinks, M. Levy, and B. K. Sarma, Phase Diagram of UPt3 from Ultrasonic Velocity Measurements, Phys. Rev. Lett. 65, 2298 (1990).
  * Stewart [2019] G. R. Stewart, UBe13 and U1-xThxBe13:Unconventional Superconductors, J. of Low Temp. Phys. 195, 1 (2019).
  * Werthamer  _et al._ [1966] N. R. Werthamer, E. Helfand, and P. C. Hoenberg, Temperature and Purity Dependence of the Superconducting Critical Field, Hc2. III. Electron Spin and Spin-Orbit Effects, Phys. Rev. 147, 295 (1966).
  * Suhl  _et al._ [1959] H. Suhl, B. T. Matthias, and E. Corenzwit, Some Further Results on Ferromagnetism in Relation to Superconductivity, J. Phys. Chem. Solid 11, 346 (1959).
  * Matthias  _et al._ [1958] B. T. Matthias, H. Suhl, and E. Corenzwit, Ferromagnetic Superconductors, Phys. Rev. Lett. 1, 449 (1958).
  * Liu  _et al._ [2017] Y. Liu, Y.-B. Liu, Y.-L. Yu, Q. Tao, C.-M. Feng, and G.-H. Cao, RbEu(Fe1-xNix)4As4: From a Ferromagnetic Superconductor to a Superconducting Ferromagnet, Phys. Rev. B 96, 224510 (2017).
  * Saxena  _et al._ [2000] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, _et al._ , Superconductivity on the Border of Itinerant-Electron Ferromagnetism in UGe2, Nature 406, 587 (2000).
  * Huy  _et al._ [2008] N. T. Huy, D. E. de Nijs, Y. K. Huang, and A. deVisser, Unusual Upper Critical Field of the Ferromagnetic Superconductor UCoGe, Phys. Rev. Lett. 100, 077002 (2008).
  * Aoki  _et al._ [2001] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Coexistence of Superconductivity and Ferromagnetism in URhGe, Nature 413, 613 (2001).
  * Ran  _et al._ [2019] S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Nearly Ferromagnetic Spin-Triplet Superconductivity, Science 365, 684 (2019).
  * Aoki  _et al._ [2019a] D. Aoki, A. Nakamura, F. Honda, D. Li, Y. Homma, Y. Shimizu, _et al._ , Unconventional Superconductivity in Heavy Fermion UTe2, J. Phys. Soc. Japan 88, 043702 (2019a).
  * Wu  _et al._ [2017] B. Wu, G. Bastien, M. Taupin, C. Paulsen, L. Howald, D. Aoki, and J.-P. Brison, Pairing Mechanism in the Ferromagnetic Superconductor UCoGe, Nature Commun. 8, 14480 (2017).
  * Sundar  _et al._ [2019] S. Sundar, S. Gheidi, K. Akintola, A. M. Cote, S. R. Dunsiger, S. Ran, N. P. Butch, S. R. Saha, J. Paglione, and J. E. Sonier, Coexistence of Ferromagnetic Fluctuations and Superconductivity in the Actinide Superconductor UTe2, Phys. Rev. B 100, 140502(R) (2019).
  * Tokunaga  _et al._ [2019] Y. Tokunaga, H. Sakai, S. Kambe, T. Hattori, N. Higa, G. Nakamine, _et al._ , 125Te-NMR Study on a Single Crystal of Heavy Fermion Superconductor UTe2, J. Phys. Soc. Japan 88, 073701 (2019).
  * Duan  _et al._ [2020] C. Duan, K. Sasmal, M. B. Maple, A. Podlesnyak, J.-X. Zhu, Q. Si, and P. Dai, Incommensurate Spin Fluctuations in the Spin-Triplet Superconductor Candidate UTe2, Phys. Rev. Lett. 125, 237003 (2020).
  * Thomas  _et al._ [2020] S. M. Thomas, F. B. Santos, M. H. Christensen, T. Asaba, F. Ronning, J. D. Thompson, E. D. Bauer, R. M. Fernandes, G. Fabbris, and P. F. S. Rosa, Evidence for a Pressure-Induced Antiferromagnetic Quantum Critical Point in Intermediate-Valence UTe2, Sci. Adv. 6, eabc8709 (2020).
  * Aoki  _et al._ [2019b] D. Aoki, K. Ishida, and J. Flouquet, Review of U-Based Ferromagnetic Superconductors: Comparison Between UGe2, URhGe, and UCoGe, J. Phys. Soc. Japan 88, 022001 (2019b).
  * Pfleiderer and Huxley [2002] C. Pfleiderer and A. D. Huxley, Pressure Dependence of the Magnetization in the Ferromagnetic Superconductor UGe2, Phys. Rev. Lett. 89, 147005 (2002).
  * Levy  _et al._ [2005] F. Levy, I. Sheikin, B. Grenier, and A. D. Huxley, Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe, Science 309, 1343 (2005).
  * Ohta  _et al._ [2010] T. Ohta, T. Hattori, K. Ishida, Y. Nakai, E. Osaki, K. Deguchi, N. K. Sato, and I. Satoh, Microscopic Coexistence of Ferromagnetism and Superconductivity in Single-Crystal UCoGe, J. Phys. Soc. Japan 79, 023707 (2010).
  * Chevalier  _et al._ [1996] B. Chevalier, P. Graverau, T. Berlureau, L. Fournes, and J. Etourneau, Ferromagnetic Properties of U2M17-yGey, J. Alloys Compounds 233, 174 (1996).
  * Soude  _et al._ [2010] A. Soude, O. Tougait, M. Pasturel, D. Kaczorowski, and H. Noel, Crystal Structure and Electronic Properties of the New Compounds U3Co12-xX4 with X=Si, Ge, J. Solid State Chem. 183, 1180 (2010).
  * Bobev  _et al._ [2007] S. Bobev, E. D. Bauer, F. Ronning, J. D. Thompson, and J. L. Sarrao, Synthesis, Structure and Physical Properties of the New Uranium Ternary Phase U3Co2Ge7, J. Solid State Chem. 180, 2830 (2007).
  * Bauer [2018] E. D. Bauer, Personal Communication (2018).
  * Wu  _et al._ [2018] B. Wu, D. Aoki, and J.-P. Brison, Vortex Liquid Phase in the p-Wave Ferromagnetic Superconductor UCoGe, Phys. Rev. B 98, 024517 (2018).
  * Hattori  _et al._ [2012] T. Hattori, Y. Ihara, Y. Nakai, K. Ishida, Y. Tada, S. Fujimoto, N. Kawakami, E. Osaki, K. Deguchi, N. K. Sato, and I. Satoh, Superconductivity Induced by Longitudinal Ferromagnetic Fluctuations in UCoGe, Phys. Rev Lett. 108, 066403 (2012).
  * Pospisil  _et al._ [2011] J. Pospisil, K. Prokes, M. Reehuis, M. Tovar, J. P. Vejpravova, J. Prokleska, and V. Sechovsky, Influence of Sample Preparation Technology and Treatment on Magnetism and Superconductivity of UCoGe, J. Phys. Soc. Japan 80, 084709 (2011).
  * Kycia [1997] J. B. Kycia,  _Growth and Characterization of High Quality UPt 3 Single Crystals and High Resolution NMR Study of Superfluid 3He-B_, Ph.D. thesis, Northwestern University, 2145 Sheridan Road Evanston, IL 60208 (1997).
  * Kycia  _et al._ [1998] J. B. Kycia, J. I. Hong, M. J. Graf, J. A. Sauls, D. N. Seidman, and W. P. Halperin, Suppression of Superconductivity in UPt3 Single Crystals, Phys. Rev. B 58, R603 (1998).
  * Joynt and Taillefer [2002] R. Joynt and L. Taillefer, The Superconducting Phases of UPt3, Reviews of Modern Physics 74, 235 (2002).
  * Gasparini  _et al._ [2010] A. Gasparini, Y. K. Huang, N. T. Huy, J. C. P. Klaasse, T. Naka, E. Slooten, and A. deVisser, The Superconductor Ferromagnet UCoGe, J. of Low Temp. Phys. 161, 134 (2010).
  * Taupin  _et al._ [2014] M. Taupin, L. Howald, D. Aoki, and J.-P. Brison, Superconducting Gap of UCoGe Probed by Thermal Transport, Phys. Rev. B 90, 180501(R) (2014).
  * Hewson [1993] A. C. Hewson,  _The Kondo Problem to Heavy Fermions_, Cambridge Studies in Magnetism (Cambridge University Press, 1993).
  * Brando  _et al._ [2016] M. Brando, D. Belitz, F. M. Grosche, and T. R. Kirkpatrick, Metallic Quantum Ferromagnets, Rev. Mod. Phys. 88, 025006 (2016).
  * Huy  _et al._ [2009] N. T. Huy, Y. K. Huang, and A. deVisser, Effect of Annealing on the Magnetic and Superconducting Properties of Single-Crystalline UCoGe, J. Magn. Magn. Mater. 321, 2691 (2009).
  * Jacko  _et al._ [2009] A. C. Jacko, J. O. Fjaerestad, and B. J. Powell, A Unified Explanation of the Kadowaki-Woods Ratio in the Strongly Correlated Metals., Nature Phys. 5, 422 (2009).
  * Aoki and Flouquet [2014] D. Aoki and J. Flouquet, Superconductivity and Ferromagentic Quantum Criticality in Uranium Compounds, J. Phys. Soc. Japan 83, 061011 (2014).
  * Hardy  _et al._ [2011] F. Hardy, D. Aoki, C. Meingast, P. Schweiss, P. Burger, H. v. Lohneysen, and J. Flouquet, Transverse and Longitudinal Magnetic-field Response in the Ising Ferromagnets URhGe, UCoGe, and UGe2, Phys. Rev. B 83, 195107 (2011).
  * Pospisil  _et al._ [2020] J. Pospisil, Y. Haga, A. Miyake, S. Kambe, Y. Tokunaga, M. Tokunaga, E. Yamamoto, P. Proschek, J. Volny, and V. Sechovsky, Intriguing Behavior of UCo1-xRhxGe Ferromagnets in Magnetic Field Along the b Axis, Phys. Rev. B 102, 024442 (2020).
  * Aoki and Flouquet [2012] D. Aoki and J. Flouquet, Ferromagnetism and Superconductivity in Uranium Compounds, J. Phys. Soc. Japan 81, 011003 (2012).
  * Aoki  _et al._ [2011] D. Aoki, I. Sheikin, T. D. Matsuda, V. Taufour, G. Knebel, and J. Flouquet, First Observation of Quantum Oscillations in the Ferromagnetic Superconductor UCoGe, J. Phys. Soc. Japan 80, 013705 (2011).
  * Bay  _et al._ [2014] T. V. Bay, A. M. Nikitin, T. Naka, A. McCollam, Y. K. Huang, and A. deVisser, Angular Variation of the Magnetoresistance of the Superconducting Ferromagnet UCoGe, Phys. Rev. B 89, 214512 (2014).
  * Bastien  _et al._ [2016] G. Bastien, A. Gourgort, D. Aoki, A. Pourret, I. Sheikin, G. Seyfarth, J. Flouquet, and G. Knebel, Lifshitz Transitions in the Ferromagnetic Superconductor UCoGe, Phys. Rev. Lett. 117, 206401 (2016).
  * Knafo  _et al._ [2012] W. Knafo, T. D. Matsuda, D. Aoki, F. Hardy, G. W. Scheerer, G. Ballon, M. Nardone, A. Zitouni, C. Meingast, and J. Flouquet, High-field Moment Polarization in the Ferromagnetic Superconductor UCoGe, Phys. Rev. B 86, 184416 (2012).
  * Gorkov [1987] L. P. Gorkov, Superconductivity in Heavy Fermion Systems, Sov. Sci. Rev. A 9, 1 (1987).
  * Troc  _et al._ [2010] R. Troc, R. Wawryk, W. Miiller, H. Misiorek, and M. Samsel-Czekala, Bulk Properties of the UCoGe Kondo-Like System, Philos. Mag. 90, 2249 (2010).
  * Huy  _et al._ [2007] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmulder, A. deVisser, A. Hamann, T. Gorlach, and H. v. Lohneysen, Superconductivity on the Border of Weak Itinerant Ferromagnetism in UCoGe, Phys. Rev. Lett. 99, 067006 (2007).