The superconducting ferromagnet UCoGe
A. Gasparini1, Y.K. Huang1, N.T. Huy1,*, J.C.P. Klaasse1,
T. Naka2, E. Slooten1 and A. de Visser1
1Van der Waals-Zeeman Institute, University of Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
2National Research Institute for Materials Science, Sengen 1-2-1,
Tsukuba, Ibaraki 305-0047, Japan
The correlated metal UCoGe is a weak itinerant ferromagnet with a Curie
temperature TC = 3 K and a superconductor with a transition temperature
Ts = 0.6 K. We review its basic thermal, magnetic on the macro and
microscopic scale and transport properties, as well as the response to high
pressure. The data unambiguously show that superconductivity and
ferromagnetism coexist below Ts = 0.6 K and are carried by the same 5f
electrons. We present evidence that UCoGe is a p-wave superconductor and
argue that superconductivity is mediated by critical ferromagnetic spin
PACS: 74.25.Dw, 74.70.Tx, 75.30.Kz
Manuscript submitted to the Special issue on Quantum Phase Transitions
2010 Journal of Low Temperature Physics.
Dr. A. de Visser
Van der Waals-Zeeman Institute, University of Amsterdam
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
* Current address: Hanoi Advanced School of Science and Technology, Hanoi, University of
Technology, 1 Dai Co Viet, Hanoi, Vietnam.
The intermetallic compound UCoGe belongs to the intriguing family of
superconducting ferromagnets [1,2]. In superconducting ferromagnets, a
superconducting transition takes place at a temperature Ts deep in the
ferromagnetic state, i.e. well below the Curie temperature TC, without
expelling magnetic order. The superconducting ferromagnets discovered
hitherto are UGe2 (under pressure) , URhGe , UIr (under pressure) ,
and UCoGe. In these uranium intermetallics magnetism has a strong itinerant
character and both ordering phenomena are carried by the same 5f electrons.
The coexistence of superconductivity and ferromagnetism is at odds with the
standard BCS theory for phonon-mediated s-wave superconductivity,
because the ferromagnetic exchange field is expected to inhibit spin-singlet
Cooper pairing . The band nature of the ferromagnetic order, however,
allows for an alternative explanation, in which critical spin fluctuations
provide the mechanism for pairing up the spin-split band electrons in spin-
triplet Cooper states [7,8]. In recent years ample evidence has been presented
that such an unusual pairing mechanism is at work in superconducting
With the discovery of superconducting ferromagnets a new research
theme in the field of magnetism and superconductivity has been disclosed.
Research into ferromagnetic superconductors will help to unravel how
magnetic fluctuations can stimulate superconductivity, which is a central
theme running through materials families as diverse as the heavy-fermion
superconductors, high-Ts cuprates and the recently-discovered FeAs-based
superconductors . This novel insight might turn out to be crucial in the
design of new superconducting materials.
The coexistence of superconductivity and weak itinerant ferromagnetism
in UCoGe was reported in 2007 . Till then UCoGe was thought to be a
paramagnet down to a temperature of 1.2 K . However, in a search for a
ferromagnetic quantum critical point induced in the superconducting
ferromagnet URhGe (Ts = 0.25 K, TC = 9.5 K) by alloying with Co , it
was discovered that UCoGe is actually a weak itinerant ferromagnet below
TC = 3 K and, moreover, a superconductor below Ts = 0.8 K.
In this paper we review the basic thermal, magnetic and transport
properties of UCoGe. Magnetization measurements show that UCoGe is a
uniaxial ferromagnet, and that the ordered moment m0 = 0.07 B is directed
along the orthorhombic c axis . Muon spin relaxation measurements 
provide unambiguous proof that magnetism is a bulk property, which
coexists with superconductivity on the microscopic scale. The temperature
variation of the upper critical field Bc2(T)  shows an unusual upward
curvature and is not Pauli limited for B || a and B || b, which provides solid
evidence for spin-triplet Cooper pairing. High-pressure susceptibility and
transport experiments  reveal that ferromagnetic order is smoothly
depressed and vanishes at a critical pressure pc 1.4 GPa. Near the
ferromagnetic critical point superconductivity is enhanced, which yields
strong support for superconductivity stimulated by critical ferromagnetic
2. Metallurgical aspects and sample preparation
UCoGe crystallizes in the orthorhombic TiNiSi structure (space group Pnma)
, with room-temperature lattice parameters a = 6.845 Å, b = 4.206 Å and
c = 7.222 Å . Superconductivity and magnetic order were first observed
on annealed polycrystalline samples with nominal compositions U1.02CoGe
(sample #2, RRR 10) and U1.02Co1.02U (sample #3, RRR 30) prepared by
arc melting . Here RRR= R(300K)/R(1K) is the residual resistance ratio.
The coexistence of superconductivity and ferromagnetism is a robust
property of all polycrystalline samples subjected to an appropriate heat
treatment procedure, typically a period of ten days at a temperature of
Single-crystalline samples were pulled from the melt with nominal
composition U1.01CoGe using a modified Czochralski technique in a tri-arc
furnace under a high-purity argon atmosphere . To improve the sample
quality, pieces of the single crystals, cut by spark-erosion, were annealed in
evacuated quartz tubes for one day at 1250 ºC and 21 days at 880 ºC. This
heat-treatment procedure is similar to the one applied to URhGe  and led
to a significant increase of the RRR value from 5 to ~30. The still relatively
low RRR value is possibly caused by remaining disorder due to Co and Ge
site inversion. Notice, the TiNiSi structure is an ordered variant of the CeCu2
structure, in which Co and Ge atoms randomly occupy the 4c positions .
The temperature dependence of the resistivity of annealed single-
crystalline UCoGe for a current along the b axis is shown in Fig. 1. The RRR
value amounts to 40. Proper superconducting and ferromagnetic phase
transitions are observed. The magnetic transition is represented by a sharp
kink at TC = 2.8 K, and superconductivity appears at temperatures below Ts =
0.6 K. However, the superconducting transition is still relatively broad, Ts
3. Weak itinerant ferromagnetic order
Magnetization data taken on polycrystalline samples provide solid evidence
that UCoGe is a weak itinerant ferromagnet, with a Curie temperature TC
= 3 K as deduced from Arrott plots . A hysteresis loop with a coercive
field of ~0.3 mT at T = 2 K corroborates ferromagnetic order. The
polycrystalline averaged ordered moment amounts to m0 = 0.03 μB for T0.
Consequently, the ratio, peff/MS, of the Curie-Weiss effective moment peff =
1.7 B over the saturation moment MS is small, which classifies UCoGe as a
weak itinerant ferromagnet .
Magnetization measurements on a single-crystalline sample show UCoGe
is a uniaxial ferromagnet . The field dependence of the magnetization
M(B) measured in fields up to 5 T applied along the a, b and c axis at a
temperature of 2 K is shown in Fig. 2. In the inset we show the temperature
variation M(T) measured in a field B || c of 0.01 T. The Curie temperature TC
= 2.8 K is determined by the inflection point in M(T). M(T) is well expressed
by the relation M(T) = m0(1 – (T/T*)2)1/2 , predicted for weak itinerant
ferromagnets , with T* ~ TC and the ordered moment m0 0.07 μB/f.u.
Resistivity measurements on a single-crystalline sample show the magnetic
transition is presented by a sharp kink at TC = 2.8 K (see Fig. 1). In the
temperature ranges below and above TC the resistivity follows the
typicalFermi-liquid ~ T 2 and ~ T 5/3 laws , respectively. The
temperature exponent n = 5/3 is characteristic for scattering at critical
ferromagnetic spin fluctuations. Transport measurements in a magnetic field
B || c reveal the ferromagnetic transition is rapidly smeared out (see Fig. 1).
The thermodynamic signature of the ferromagnetic transition in the
specific heat measured on a polycrystalline sample (labelled sample #2) is
shown in Fig. 3. Here TC = 3 K is identified by the inflection point in c/T at
the high T side of the peak. The linear term in the electronic specific heat
amounts to 0.057 J/molK2, which indicates UCoGe is a correlated metal, but
the electron interactions are relatively weak. The magnetic entropy Smag
involved in the magnetic transition, obtained by integrating cmag/T versus T,
is 0.3 % of Rln2 (i.e. the value for a local moment S = ½ system). Such a
small value is expected for a weak itinerant ferromagnet . In small
applied magnetic fields (B 0.3 T) the magnetic entropy is preserved, but
the ferromagnetic transition broadens significantly, as shown in Fig. 3.
Typical ac-susceptibility, ac, traces taken on polycrystalline UCoGe
samples are shown in Fig. 4. These data were taken at a low frequency (f =
16 Hz) and an amplitude of the driving field of ~10-5 T. The weak peak
observed at 3 K signals the ferromagnetic transition. Below 1 K, ac rapidly
decreases to a large diamagnetic value, which reflects the superconducting
transition. The onset transition temperatures, Ts
onset, are determined at 0.38 K
and 0.61 K for samples #2 (RRR ≈ 10) and #3 (RRR ≈ 30), respectively.
Clearly, superconductivity depends sensitively on the quality of the samples.
These results are in good agreement with resistivity data taken on the same
samples [1,23]. The ac-susceptibility ac starts to drop when the resistive
transition is complete. At the lowest temperature ac reaches a value of 60
70 % of the ideal screening value S = -1/(1 - N) (here N ≈ 0.08 is the
demagnetizing factor of our samples). This indicates UCoGe is a Type II
superconductor which is always in the mixed phase. Because of the intrinsic
ferromagnetic moment, the local field is nonzero and the magnitude of ac is
The temperature dependence of the ac-susceptibility of sample #3 in
magnetic field is shown in the inset of Fig. 4. In an applied field, the peak
associated with the ferromagnetic order broadens and shifts to higher
temperatures, while the onset temperature for superconductivity shifts to
lower temperatures. In a field of 0.1 T, the magnetic transition in ac(T) is
almost washed out.
Specific-heat and thermal-expansion measurements provide solid
evidence for bulk superconductivity. Specific-heat data taken on a
polycrystalline sample show a broad superconducting transition with an
onset temperature of 0.66 K . A rough estimate for the step size of the
idealized transition in the specific heat, using an equal entropy construction
(with a bulk Ts 0.45 K), yields (c/Ts)/ 1.0, which is considerably
smaller than the BCS value 1.43.
The linear coefficient of thermal expansion, = L-1dL/dT, measured on a
polycrystalline sample (#3)  is shown in Fig. 5. Upon entering the
superconducting state, (T) shows a steady increase. Assuming an ideal
sharp transition at a superconducting temperature Ts = 0.45 K, the estimated
step-size is 3.810-7 K-1 , which reflects bulk superconductivity.
Moreover, the thermal expansion data reveal that magnetism and
superconductivity coexist. The relative length change in the superconducting
state L/L = 0.110-6 is small compared to the length change L/L =
1.910-6 due to magnetic ordering. Thus magnetism is not expelled below Ts
and coexists with superconductivity. Thermal expansion measurements on a
single-crystalline sample for a dilatation direction along the b axis show
pronounced phase transition anomalies at TC = 2.8 K and Ts = 0.5 K (see
5. Muon spin relaxation measurements
In order to investigate whether the weak magnetic order is a bulk property of
our samples, muon spin relaxation (SR) experiments were carried out at the
Paul Scherrer Institute in Villigen . The experiments were carried out in
zero applied magnetic field on well characterized polycrystalline samples
with RRR ≈ 30 in the temperature range 0.02 – 10 K. In the paramagnetic
phase the SR spectra are best described by a Kubo-Toyabe function GKT(t),
with a Kubo-Toyabe relaxation rate KT = 0.300.01 s-1. In this temperature
range the depolarization of the muon ensemble is attributed to static nuclear
moments on the 59Co atoms. In the ferromagnetic phase a clear spontaneous
muon precession frequency, , appears and the response of the muon is
described by the depolarization function for an isotropic polycrystalline
magnet GM(t) (see Ref.16). At the lowest temperature (T = 0.02 K) =
1.9720.004 MHz, which corresponds to a local field Bloc ~ 0.015 T at the
muon localization site. The temperature variation (T) tracks the
magnetization M(T) (see Fig. 6). The (T) data are well fitted by a
phenomenological order parameter function (T) = 0 [1 - (T/T*)], with T*
= 3.02 K TC, 0 = 1.98 MHz and critical exponents = 2.3 and = 0.4.
The amplitude of the magnetic signal in zero field below T ~ 1.5 K
corresponds to the amplitude measured in a small transverse field of 50 G in
the paramagnetic phase. This confirms magnetic order is present in the
whole sample volume.
Most interestingly, (T) shows a small decrease of about 2% below Ts.
This effect is observed by the whole muon ensemble, which confirms
magnetism and superconductivity coexist on the microscopic scale. The
decrease of (T) is accompanied by a small increase of the corresponding
damping rate 2(T) . Such an increase is expected when a spontaneous
vortex lattice is formed , i.e. when Bloc is larger than the lower critical
field Bc1. This indicates that even in zero magnetic field UCoGe does not
enter the Meissner state, but is always in the mixed state.
6. Upper critical field
The upper critical field, Bc2, as measured on a single-crystalline UCoGe
sample with RRR 30  is reported in Fig.7. The data were extracted
from resistance, R(T), measurements taken for a current along the
orthorhombic a axis in fixed magnetic fields applied along the a axis
(longitudinal configuration), and b and c axis (transverse configuration). The
superconducting transition temperatures, Ts(B), were determined by the mid-
points of the transitions to the zero resistance state. In zero field Ts = 0.6 K.
At least three remarkable features appear in the data: (i) the large value of
a factor ~ 10, and (iii) a pronounced upturn in Bc2(T) when lowering the
temperature for all field directions. Clearly, this behaviour is at odds with
standard BCS spin-singlet pairing. Bc2
a,b(T=0) ≈ 5 T for B || a, b, (ii) the large anisotropy, Bc2
a,b(T=0) largely exceeds the
paramagnetic limiting field, Bc2,Pauli(0) = 1.83Ts, for a weak coupling spin-
singlet superconductor . Taking into account the 3D nature of the
normal-state electronic properties of UCoGe, this provides solid evidence for
an equal-spin pairing triplet superconducting state. A prerequisite for triplet
pairing is a sufficiently clean sample, such that the mean free path ℓ is larger
than the coherence length. An estimate for ℓ and can be extracted from
the large initial slope Bc2
calculated ℓ ≈ 900 Å and ≈ 120 Å and consequently the clean-limit
condition is satisfied. The large anisotropy in Bc2 has been analyzed in terms
of an anisotropic p-wave interaction , which supports a superconducting
gap function of axial symmetry with point nodes along the c axis, i.e. along
the direction of the ordered moment m0. As shown in Fig. 7, Bc2(T) is at
variance with model calculations for polar and axial states of spin-triplet
superconductors with a single superconducting gap function (see Ref. 15).
Superconducting order parameter calculations for orthorhombic itinerant
ferromagnetic superconductors with strong spin-orbit coupling show that
UCoGe is essentially a two-band superconductor . Within this scenario,
the unusual upward curvature at low temperatures can be attributed to a
crossover between two equal-spin pairing states with different
superconducting transition temperatures. Another appealing explanation for
the unusual Bc2 behaviour is the presence of a field-induced quantum critical
point (, see section 8).
7. Pressure temperature phase diagram
The response to pressure of the ferromagnetic and superconducting phases of
UCoGe was investigated by ac-susceptibility, ac(T), and resistivity, (T),
measurements on single-crystalline samples using a clamp-cell technique for
pressures up to 2.2 GPa . The resulting pressure-temperature phase
diagram is shown in Fig. 8. The Curie temperatures obtained by both
methods nicely agree. However, the values of Ts determined from the (T)
data systematically exceed those determined by ac, since the diamagnetic
signal is representative for the bulk and appears when the resistive transition
is complete. The Curie temperature, TC, gradually decreases with pressure,
and for p 0.4 GPa a linear depression is observed at a rate 2.4 K/GPa. The
phase line TC(p) extrapolates to the suppression of ferromagnetic order at pc=
1.400.05 GPa. An almost equal critical pressure value is deduced from the
pressure variation of the amplitude of ac at TC (see inset Fig.8).
The susceptibility data reveal the magnetic transition is continuous –
hysteresis in the magnetic signal is absent. This strongly suggests
ferromagnetic order vanishes at a second order quantum critical point at pc.
a,b/dT -8 T/K . For our single crystal it was
However, the phase line TC(p) intersect the superconducting phase boundary
Ts(p) near p 1.1 GPa, above which TC no longer can be detected.
Consequently, we cannot exclude that ferromagnetic order vanishes abruptly
and that the ferro- to paramagnetic phase transition becomes first order when
T 0 .
The superconducting transition temperature first increases with pressure.
Ts goes through a broad maximum near the critical pressure for
ferromagnetic order and persists in the paramagnetic phase. This is clearly at
variance with the p-T phase diagram obtained in the traditional Stoner spin-
fluctuation model for triplet superconductivity , where Ts 0 at pc.
However, when the strong-Ising anisotropy of the magnetization is taken
into account a finite value of Ts at pc can be attained in the model
The evolution of the upper critical field with pressure was investigated by
resistivity measurements for B || a and B || c . Bc2
independent, while Bc2
the critical pressure pc = 1.40 GPa, with extrapolated values Bc2
large as 15 T. This demonstrates superconductivity is enhanced near the
ferromagnetic quantum critical point. Measurements at p = 1.66 GPa show
large values of Bc2
been inferred that p-wave superconductivity occurs at both sides of pc.
c is almost pressure
a shows a remarkable enhancement upon approaching
a(0) persist in the paramagnetic phase, from which it has
An important issue in the field of magnetic superconductors is whether
superconductivity and magnetism are of bulk nature and coexist on the
microscopic scale, or are confined to different parts of the sample because of
phase separation on a macroscopic scale. Especially in the case of
superconducting ferromagnets this is of major concern, as superconductivity
and ferromagnetism form normally competing ground states. As regards
UCoGe, solid evidence has been collected for the intrinsic coexistence of
superconductivity and magnetism [1,16]. The thermodynamic signatures of
the magnetic and superconducting phase transitions in the specific heat and
thermal expansion of UCoGe show values characteristic for the bulk .
Moreover, the amplitude of the muon depolarization signal in the magnetic
phase confirms bulk magnetism, which persists below Ts = 0.6 K . The
same conclusion was reached by 59Co-NQR measurements on poly and
single-crystalline samples: below
superconductivity are found to coexist on the microscopic scale . From
the temperature variation of the NQR spectrum, the authors conclude that the
ferromagnetic phase transition is weakly first order. Notice, recent
TC 2.5 K ferromagnetism and
magnetization measurements on single crystals led to the claim that a
magnetic field of the order of a few mT is needed to stabilize magnetic order
. This is at variance with the zero-field SR  and NQR  results,
and indicates metallurgy is an important issue (see section 2). The nuclear
spin-lattice relaxation rate, 1/T1, in the ferromagnetic phase, extracted from
the NQR experiments, decreases below Ts due to the opening of the
superconducting gap. Interestingly, two contributions to 1/T1 were found, i.e.
terms proportional to T and T 3. This has been interpreted to indicate the
superconducting state is inhomogeneous. An appealing explanation for the
inhomogeneous nature is the presence of a spontaneous vortex lattice , in
which case the term linear in T probes the non-superconducting regions of
the sample, while the T 3 term probes the superconducting regions
characterized by a line node in the superconducting gap function. This is in
line with the interpretation of the SR data (see section 5). UCoGe may be
the first material in which a self-induced vortex state is realized. Small angle
neutron scattering experiments and/or scanning squid probe techniques are
needed to put this on firm footing.
Another important issue is the nature of the small ordered moment m0. In
analogy with other superconducting ferromagnets, it is natural to assume that
the moment m0 = 0.07 B, deduced from the magnetization data, is due to U
5f electrons. Electronic structure calculations indeed predict a magnetic
moment on the U site . The calculated moment U ~ 0.1 B is small, due
to an almost complete cancellation of the orbital L
moment. However, the calculations predict the presence of a much larger
moment Co ~ 0.20.5 B on the Co site as well. Recently, a polarized
neutron diffraction study was conducted to solve the nature of the weak
ferromagnetic moment . Experiments carried out on a single-crystalline
sample for B || c reveal that in low magnetic fields the ordered moment is
predominantly located at the U moment. Thus ferromagnetic order is due to
the 5f electrons. This is supported by the zero field SR  and NQR 
data. However, in a magnetic field the situation changes: the ordered
moment grows to U ~ 0.3 B in a field of 12 T (B || c) and, most remarkably,
induces a substantial moment Co ~ 0.2 B on the Co atom, directed
antiparallel to U. Such an anomalous polarizability of the Co 3d orbitals is
unique among uranium intermetallics  and reflects the proximity to a
magnetic instability of UCoGe in zero field.
The enhancement of superconductivity in UCoGe near the ferromagnetic
quantum critical point provides an important clue that critical ferromagnetic
spin fluctuations stimulate p-wave superconductivity. The condensation into
spin-triplet Cooper pairs is in line with symmetry group considerations for
superconducting ferromagnets with orthorhombic crystal symmetry .
Under the constraint of a large spin-orbit coupling and a sufficiently large
U and spin S
exchange splitting, equal-spin pairing results in two-band superconductivity
with gaps and . Only two superconducting gap-structures are
possible. By taking the ordered moment m0 directed along the z axis (as for
an uniaxial ferromagnet), the gap has zeros (nodes) parallel to the magnetic
axis (kx = ky = 0) or a line of zeros on the equator of the Fermi surface (kz =
0). Accurate measurements of the electronic excitation spectrum in the
superconducting state on high-purity single crystals are needed to
discriminate between these two cases.
Since p-wave superconductivity is extremely sensitive to scattering at
non-magnetic impurities and defects [7,37], a necessary condition for triplet
pairing is a ratio of the mean free path over the coherence length ℓ/ > 1. As
mentioned above (section 6), our single crystals are sufficiently clean and we
calculate ℓ/ ≈ 7. The sensitivity of superconductivity to the reduction of the
mean free path has been investigated by doping UCoGe with Si . Ac-
susceptibility and resistivity measurements, carried out on a series of
polycrystalline UCoGe1-xSix samples, show that superconductivity and
ferromagnetism are progressively depressed with increasing Si content and
simultaneously vanishes at a critical concentration xcr 0.12. Since the RRR
value rapidly drops with doping, and concurrently ℓ decreases, it is
surprising triplet superconductivity survives till ~12 at.% Si. This would
require a strong doping-induced reduction of as well. In the case of
UCoGe1-xSix, however, the defect-driven depression of Ts is partly
compensated by its increasing due to chemical pressure. Also, upon
approach of the ferromagnetic quantum critical point, ferromagnetic spin
fluctuations will promote triplet superconductivity even stronger.
The superconducting phases of UCoGe under pressure, labelled S1 and S2
in the p-T diagram (Fig. 8), can be discriminated in close analogy to the
familiar superfluid phases of 3He . The state S1 in the ferromagnetic
phase breaks time reversal symmetry and is equivalent to the non-unitary A2
phase of 3He (i.e. the A phase of 3He in magnetic field), which is a linear
combination of the equal-spin pairing states |Sz = 1, m = 1 and |Sz = -1, m =
1 with different population. The large upper-critical field values observed
for state S2 provide solid evidence it is a spin-triplet state as well. Model
calculations  predict it is a unitary triplet state, which does not break
time reversal symmetry. In this sense, S2 is equivalent to the planar state of
3He, which is an equally weighted superposition of the two states |Sz = 1, m =
-1 and |Sz = -1, m = 1. UCoGe is unique as regards its response to pressure,
as it is the only superconducting ferromagnet for which superconductivity
persists above the critical pressure for suppression of ferromagnetism. Very
similar p-T phase diagrams have been obtained by other research groups on
poly- and single-crystalline UCoGe samples , which demonstrates the
pressure response reported in Fig. 8 is a robust property.
explored, namely magnetic field tuning. When carrying out a high-field
magnetotransport study on high quality single crystals (RRR 30) at ambient
pressure, Aoki and co-workers  made a remarkable observation: Bc2
is strongly enhanced and shows an unusual S-shaped curve which
extrapolates to the large value of 20 T when T 0. For B || a even larger
upper critical field values are attained: Bc2
curvature and extrapolates to 30 T for T0. A key ingredient in the
magnetotransport experiment is precise tuning of the magnetic field along
the orthorhombic axes of the crystal. A misalignment of a few degrees
inhibits the observation of these phenomena, which most likely explains the
much lower Bc2
superconducting phase for B || b appears to be connected to the depression of
the Curie temperature in large magnetic fields (with a critical field Bc ~ 16 T
for T 0) [30,41]. Compelling evidence for a close link between critical
spin fluctuations at Bc and superconductivity is obtained by analyzing the
resistivity data in the normal state. The underlying idea is that critical spin
fluctuations, which are the source of the pairing interaction, give rise to an
enhanced quasiparticle mass, m*, which can be probed via the Fermi liquid
term in the resistivity, ~ AT 2. This can be made quantitative by use of a
simple McMillan-like relation between Ts and m* A for ferromagnetic
superconductors, recently proposed in Ref. 11. As expected, for B || b the
transport coefficient A strongly increases with field and shows a pronounced
maximum in the field range of reinforced superconductivity . A similar
relation was recently established for URhGe [10,11], where field re-entrant
superconductivity is due to critical spin fluctuations associated with a spin-
reorientation process which is induced in a large magnetic field (B || b) of 12
In this review we have presented the thermal, magnetic, and transport
properties of the superconducting ferromagnet UCoGe. The data obtained on
high-quality poly- and single-crystalline samples show the unusual
coexistence of ferromagnetism and superconductivity is robust on the
macroscopic and microscopic scale. Notably, the absence of Pauli limiting in
the BT phase diagrams and the enhancement of superconductivity near the
magnetic quantum critical point in the pT diagram provide evidence for
triplet superconductivity mediated by critical ferromagnetic spin
fluctuations. UCoGe offers a unique possibility to further unravel the
intimate link between ferromagnetism and triplet superconductivity.
Especially, the two routes to quantum criticality – pressure tuning and
Recently, a second route to quantum criticality in UCoGe has been
a(T) shows an unusual upward
a,b values reported in Ref.15 (see Fig. 7). The field-reinforced
magnetic field tuning – make it an unrivalled laboratory tool to probe spin-
fluctuation mediated superconductivity. We expect in the near future
measurements of the electronic and magnetic excitation spectra in the
superconducting and magnetic phases will reveal crucial information on the
superconducting gap structure and pairing mechanism. These experiments
inevitably should be performed on high-purity single crystals, which calls for
a strong commitment to further improve the sample preparation process.
UCoGe is a unique test-case material for addressing the central issue of how
a ferromagnetic superconductor accommodates an intrinsic internal magnetic
field. It may be the first material to reveal proof for the existence of the long-
searched-for spontaneous vortex phase.
The authors are grateful to T. Gortenmulder, D.E. de Nijs, A. Hamann, T.
Görlach, H. v. Löhneysen, C. Baines, D. Andreica and A. Amato for their
help at various stages of the research. We thank D. Aoki, V.P. Mineev and
A.D. Huxley for helpful discussions. This work is part of the research
programme of the Foundation for Fundamental Research on Matter (FOM),
which is financially supported by the Netherlands Organisation for Scientific
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I || b axis
B || c axis
Fig.1 Temperature variation of the resistivity of annealed single-crystalline UCoGe
for a current I || b in zero magnetic field (upper curve) and in applied fields B || c of
0.02, 0.05, 0.10 and 0.15 T (lowest curve). The residual resistivity 0 = 7 cm and
RRR = 40. Inset: Resistivity in zero field, where the solid lines represent fits to ~
T 2 and ~ T 5/3 in the temperature ranges below and above TC, respectively. Arrows
indicate the Curie temperature TC = 2.8 K and the onset temperature for the
superconducting transition Ts
onset = 0.6 K.
T = 2 K
Fig.2 Magnetization of UCoGe for fields along the a, b and c axis at T= 2 K .
Ferromagnetic order is uniaxial with m0 pointing along the c axis. The inset shows
M(T) for B = 0.01 T along c. In the limit T 0 m0 = 0.07 B.
Fig. 3 The temperature dependence of the specific heat of polycrystalline UCoGe
(sample #2) plotted as c/T versus T in fields of 0, 0.1, 0.2 and 0.3 T .
B = 10-5 T
B= 0 T
Fig. 4 Temperature dependence of the ac-susceptibility ac in polycrystalline UCoGe (samples
#2 and #3) [1,23]. Arrows indicate TC and Ts. Inset: The ac-susceptibility of sample #3
measured in fields of 0, 0.02 and 0.1 T.
Fig. 5 Temperature variation of the coefficient of linear thermal expansion of
UCoGe. Open circles: polycrystal (sample #3) . Closed circles: single-crystal (sc
#1, ann #1) b axis.
R (arb. u.)
Fig. 6 (a) Resistivity versus temperature of polycrystalline UCoGe. (b) Spontaneous
muon precession frequency (T) of UCoGe. The solid line represents a fit to a
phenomenological order parameter function (see text). Notice the 2% decrease of
(T) below the bulk Ts of 0.5 K. Figure adapted from Ref.16.
19 Download full-text
Fig. 7 Temperature dependence of the upper critical field of single-crystalline
UCoGe for B along the principal axes . The solid lines show the calculated
dependence for a superconducting gap function with axial (along c axis) and polar
symmetries (along a and b axis).
FM + S1
Fig. 8 Pressure temperature phase diagram of UCoGe . Ferromagnetism (FM) -
blue area; superconductivity (SC: S1, S2) - yellow area. TC(p) extrapolates to a FM
quantum critical point at pc = 1.400.05 GPa. SC coexists with FM below pc - blue-
yellow hatched area. Symbols: closed blue and red circles TC and Ts from ac(T);
blue and white triangles TC and Ts from (T) (up triangles I || a, down triangles I || c);
closed blue and red squares TC and Ts at p=0 taken from a polycrystal. Inset:
Amplitude of ac(T) at TC as a function of pressure. The data follow a linear p-
dependence and extrapolate to pc = 1.460.10 GPa.