# The Superconducting Ferromagnet UCoGe

**ABSTRACT** The correlated metal UCoGe is a weak itinerant ferromagnet with a Curie temperature T

C=3K and a superconductor with a transition temperature T

s=0.6K. We review its basic thermal, magnetic—onthe 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 T

s=0.6K 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 fluctuations.

KeywordsSuperconducting ferromagnet-

p-wave superconductor-Magneticquantum critical point-Critical spin fluctuations

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- Y. Tada, S. Fujimoto, N. Kawakami, T. Hattori, Y. Ihara, K. Ishida, K. Deguchi, N. K. Sato, I. Satoh[Show abstract] [Hide abstract]

**ABSTRACT:**Identification of pairing mechanisms leading to the unconventional superconductivity realized in copper-oxide, heavy-fermions, and organic compounds is one of the most challenging issues in condensed-matter physics. Clear evidence for an electron-phonon mechanism in conventional superconductors is seen by the isotope effect on the superconducting transition temperatures $T_{\rm SC}$, since isotopic substitution varies the phonon frequency without affecting the electronic states. In unconventional superconductors, magnetic fluctuations have been proposed to mediate superconductivity, and considerable efforts have been made to unravel relationships between normal-state magnetic fluctuations and superconductivity. Here, we show that characteristic experimental results on the ferromagnetic (FM) superconductor UCoGe ($T_{\rm Curie} \sim 2.5 $ K and $T_{\rm SC} \sim 0.6$ K) can be understood consistently within a scenario of the spin-triplet superconductivity induced by FM spin fluctuations. Temperature and angle dependencies of the upper critical magnetic field of the superconductivity ($H_{c2}$) are calculated on the basis of the above scenario by solving the Eliashberg equation. Calculated $H_{c2}$ well agrees with the characteristic experimental results observed in UCoGe. This is a first example that FM fluctuations are shown to be a pairing glue of superconductivity.Journal of Physics Conference Series 02/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**Magnetic fluctuations near to quantum criticality can have profound effects. They lead to characteristic scaling at high temperature which may ultimately give way to a reconstruction of the phase diagram and the formation of new phases at low temperatures. The ferromagnet UGe2 is unstable to p-wave superconducting order—an effect presaged by the superfluidity in 3He—whereas in CeFePO fluctuations drive the formation of spiral magnetic order. Here we develop a general quantum order-by-disorder description of these systems that encompasses both of these instabilities within a unified framework. This allows us to demonstrate that in fact these instabilities intertwine toform a pair density wave.Physical review. B, Condensed matter 03/2013; 87(12). · 3.77 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present an overview of unconventional phenomena arising close to ferromagnetic and ferroelectric quantum phase transitions. The applicability and potential breakdown of traditional field theories of quantum criticality and the emergence of a multiplicity of critical fields in particular will be discussed.Journal of Physics Conference Series 12/2012; 400(3):2048-.

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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

fluctuations.

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.

Corresponding author:

Dr. A. de Visser

Van der Waals-Zeeman Institute, University of Amsterdam

Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

E-mail: a.devisser@uva.nl

* Current address: Hanoi Advanced School of Science and Technology, Hanoi, University of

Technology, 1 Dai Co Viet, Hanoi, Vietnam.

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1. Introduction

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) [3], URhGe [4], UIr (under pressure) [5],

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 [6]. 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

ferromagnets [3,9,10,11].

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 [12]. 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 [1]. Till then UCoGe was thought to be a

paramagnet down to a temperature of 1.2 K [13]. 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 [14], 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 [15]. Muon spin relaxation measurements [16]

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) [15] shows an unusual upward

curvature and is not Pauli limited for B || a and B || b, which provides solid

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evidence for spin-triplet Cooper pairing. High-pressure susceptibility and

transport experiments [17] 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

spin fluctuations.

2. Metallurgical aspects and sample preparation

UCoGe crystallizes in the orthorhombic TiNiSi structure (space group Pnma)

[18], with room-temperature lattice parameters a = 6.845 Å, b = 4.206 Å and

c = 7.222 Å [1]. 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 [1]. 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

875 °C.

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 [19]. 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 [20] 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 [21].

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

0.06 K.

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

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= 3 K as deduced from Arrott plots [1]. 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 [22].

Magnetization measurements on a single-crystalline sample show UCoGe

is a uniaxial ferromagnet [15]. 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 [23], predicted for weak itinerant

ferromagnets [24], 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 [24], 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 [24]. 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.

4. Superconductivity

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

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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

reduced.

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 [1]. 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) [1] 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 [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

Fig. 5).

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 [16]. The experiments were carried out in

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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) [16]. Such an increase is expected when a spontaneous

vortex lattice is formed [25], 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 [15] 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

Bc2

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

Bc2

b

>> Bc2

c, of

a,b(T=0) largely exceeds the

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paramagnetic limiting field, Bc2,Pauli(0) = 1.83Ts, for a weak coupling spin-

singlet superconductor [26]. 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 [28], 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 [29]. 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 ([30], 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 [17]. 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 [27]. For our single crystal it was

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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 [31].

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 [7], 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

calculations [32].

The evolution of the upper critical field with pressure was investigated by

resistivity measurements for B || a and B || c [17]. 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(T0) as

a(0) persist in the paramagnetic phase, from which it has

8. Discussion

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 [1].

Moreover, the amplitude of the muon depolarization signal in the magnetic

phase confirms bulk magnetism, which persists below Ts = 0.6 K [16]. 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 [33]. 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

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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

[34]. This is at variance with the zero-field SR [16] and NQR [33] 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 [25], 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 [35]. 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 [36]. 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 [16] and NQR [33]

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 [21] 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 [29].

Under the constraint of a large spin-orbit coupling and a sufficiently large

U and spin S

U magnetic

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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 [38]. 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 [39]. 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 [39] 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 [40], which demonstrates the

pressure response reported in Fig. 8 is a robust property.

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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 [30] 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 [30]. 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

T [42].

9. Conclusions

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

b(T)

a(T) shows an unusual upward

a,b values reported in Ref.15 (see Fig. 7). The field-reinforced

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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.

Acknowledgement

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

Research (NWO), and of the EC 6th Framework Programme COST Action

P16 ECOM.

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