Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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13 Unconventional Superconductivity in Novel Materials 681

 ∝



1−a







where |a|≈1andn ≈ 1−1.5;

(13.24)

∝ −









or T

−1+

( ≈ 0.7−0.8);

(13.25)

 ∝



1−







, with n ≈ 0.5 ,

 ∝ −ln





 ∝ T

−1+

( ≈ 0.7−0.8) ;

(13.26)





(!, T) scales as

. (13.27)

In several of these systems, the scaling temperature

can be identified with the Kondo temperature T

The characteristic NFL temperature dependences

were first established from studies of chemically sub-

stituted f -electron materials as Y

1−x

[278,285],

UCu

5−x

[286, 287], and CeCu

6−x

[288, 289],

but more recently have been found in stoichiomet-

ric compounds at ambient pressure (e.g., YbRh

[290]) and under applied pressure (e.g., CeIn

[150],

CePd

[291]). These experiments suggest that dif-

ferent mechanisms may be responsible for the NFL

behavior observed in these f -electron materials.

Several NFL models with distinct origins have been

proposed and can be grouped into three main cate-

gories: (i) single-ion models based on a multichannel

Kondo effect, (ii) disorder-driven NFL models, and

(iii) models involving quantum ﬂuctuations of an

order parameter at a second-order T =0K phase

transition (QCP) Various theoretical models have

been proposed based onsingle-ion physics involving

an unconventional Kondo effect, of either magnetic

or electric (quadrupolar) origin [276, 292–295]. In

these multichannel Kondo models, the magnetic

(quadrupolar) moments are over-compensated by

a number of channels of conduction electrons due

to additional orbital (spin) degrees of freedom, giv-

ing rise to power-law or logarithmic temperature

dependences of the physical properties at low tem-

peratures. Theoretical models which incorporate

the effects of disorder include the Kondo disorder

model [296–298], in which a distribution of Kondo

temperatures T

due to chemical disorder with finite

weight at T

= 0 K leads to NFL properties, and

an inhomogeneous Griffiths–McCoy phase [299],

which consists of magnetic clusters in a paramag-

netic phase formed as a result of the competition

between the Kondo effect and the Ruderman–Kittel–

Kasuya–Yosida (RKKY) interaction in the presence

of magnetic anisotropy and disorder.

The critical behavior near a quantum critical

point has been considered by a number of investi-

gators [220–222,300–304]. Hertz [300] and later Mil-

lis [223,301] studied the ﬂuctuations of an order pa-

rameternearaT = 0 K second order phase transition

using renormalization techniques. They found that

the critical behavior is determined by both spatial

and temporal ﬂuctuations in an effective dimension

eff

= d + z,whered is the spatial dimension and

z is the dynamical exponent. The physical proper-

ties predicted by the Hertz/Millis theory including

(T), (T), C(T)/T and the dependence of the or-

dering temperature on the control parameter ı are

listed in Table 13.5. Similar results were obtained by

Moriya [220,221] with a self consistent renormaliza-

tion group model of spin ﬂuctuations in weak itiner-

ant magnetic systems and also by a spin ﬂuctuation

theory by Lonzarich and coworkers [222]. A recent

paper by Si examines the effect of non-Gaussian ﬂuc-

tuations near aquantumcriticalpoint,andsuggestsa

model in whichtheKondo temperature is suppressed

to zero at the same point as the second order phase

transition, but from the opposite side of the phase

diagram [305].

Table 13.5.Spin-ﬂuctuation theory predictions for the tem-

perature dependences of the resistivity  and specific heat

divided by temperature C/T, and the dependence of the

magnetic ordering temperature T

on the control parame-

ter ı (pressure or carrier concentration) for 3D and 2D fer-

romagnetic (FM) and antiferromagnetic (AFM) spin ﬂuc-

tuations

3D FM 3D AFM 2D FM 2D AFM

 T

5/3

3/2

4/3

C/T −ln(T/T

) 

− ˇT

1/2

−1/3

−ln(T/T

)

(ı − ı

)

3/4

(ı − ı

)

2/3

ı − ı

682 M.B. Maple et al.

In recent years, non-Fermi liquid behavior has

also been observed in three systems that are based

on UPd

(discussed in Sect.13.3.4):U

1−x

for M=Th, La, and Y. The magnetic phase diagrams

of these three systems are shown in Fig. 13.43. It

is remarkable that although these three systems

have the same parent compound, the phase dia-

gram of U

1−x

is quite different from those

of U

1−x

and U

1−x

.BoththeAFM

Fig. 13.43. Temperature–M concentration (T–x) phase di-

agrams of the U

1−x

(M =Th,La,Y) systems.Re-

gions of antiferromagnetic order (AFM), spin glass freez-

ing (SG), superconductivity (SC) and non-Fermi liquid be-

havior (NFL) determined from measurements of the spe-

cific heat C, magnetic susceptibility , and electrical resis-

tivity as a function of temperature T are indicated in the

T − x plane. The temperature dependences of  and C/T

observed in the NFL regions are indicated in the figure,

after [306]

order and superconductivity are only slightly sup-

pressed with Th substitution (Fig.13.43a) at x =0.2,

and the system crosses over to an NFL regime at

x ≈ 0.6. The NFL properties of U

1−x

low temperatures scale with U concentration indi-

cating the NFL behavior is associated with single-ion

physics such as an unconventional Kondo effect.

In the U

1−x

system, the antiferromag-

netismand superconductivity are rapidly suppressed

with Y substitution (Fig. 13.43c). Similar behavior

is found for U

1−x

(Fig. 13.43b). The N´eel

temperature decreases from T

≈ 14 K in UPd

to T

≈ 7Kbyx =0.2, after which spin glass behav-

ior is observed at T

≈ 1 K. The spin glass transi-

tion is no longer observed at a critical concentra-

tion x

≈ 0.7 near the concentration where the NFL

behavior occurs [307]. It appears that T

vanishes

nearly linearly with x as x → x

. These results sug-

gest that the NFL behavior in this system is associ-

ated with ﬂuctuationsof a magnetic order parameter

above the spin glass freezing transition at T =0K.

An example of the NFL behavior observed in these

systems is shown in Fig. 13.44 for a U

1−x

sample with a composition (x =0.8) close to that

of the QCP at x

≈ 0.7. The physical properties of

0.2

0.8

at low temperatures have the follow-

ing T-dependences:  ∝ T, C/T ∝ (or T

with

n =0.2), and  ∝ [1 − (T/T

)]

1/2

Although the magnetic phase diagram of

1−x

is similar to that of U

1−x

the non-Fermi liquid behavior is close to that of

1−x

.Thissuggeststhatdisorderfromlo-

cal expansion of the lattice by the larger Th and

La ions could be driving the NFL behavior in these

compounds.SubstitutionwithY leaves the lattice un-

changed [306].

Non-Fermi liquid behavior has also been found

in the vicinity of a ferromagnetic quantum criti-

cal point in systems such as Th

1−x

[311]

and URu

2−x

[310]. The substitution of Re

or Tc for Ru in the heavy-fermion superconductor

URu

has been found to suppress both hidden or-

der (HO) [312] and superconductivity and induce

ferromagnetism (FM) [308,309].The superconduct-

ing transition temperature T

of URu

2−x

de-

creases rapidly from T

=1.5Kat x =0to below

13 Unconventional Superconductivity in Novel Materials 683

Fig. 13.44. Electrical resistivity  vs T, specific heat divided

by temperature, C/T,vslogT(upper inset), and magnetic

susceptibility  vs T

1/2

(lower inset)forU

0.2

0.8

,af-

ter [307]

200 mK at x =0.01, while the N`eel temperature T

associated with the formation of the small-moment

(∼ 0.03 

/U atom) hidden order phase (often as-

sociated with antiferromagnetism or a spin den-

sity wave) at 17.5 K decreases less rapidly with x

and appears to vanish by x =0.15 as shown in the

T − x phase diagram in Fig.13.45. Ferromagnetic or-

der occurs in the concentration range 0.3 ≤ x ≤ 1.0

where the Curie temperature 

, saturation moment



sat

, and electronic specific heat coefficient  ,ex-

hibit maxima at x =0.8of38K,0.44 

/U atom,

and 160 mJ/mol K

[308], respectively, as displayed

in Fig. 13.46. Magnetic measurements indicate that

the suppression of FM order occurs at x

=0.3.For

samples close to where features associated with the

hidden order phase can no longer be resolved and

the FM critical point (x

=0.3) with Re concen-

trations 0.15 < x ≤ 0.6,the non-Fermi liquidbehav-

ior is characterized by C/T ∝ −lnT and (T) ∝ T

with 1 ≤ n < 2 at low temperatures [310].Therefore,

it appears that the NFL behavior persists well into

the ferromagnetic state.

NFL behavior is observed in a wide range of

CeMIn

(M = Co, Rh, Ir) compounds as discussed

previously. In addition to the NFL behavior found

in stoichiometric CeCoIn

, studies of the chem-

ically substituted systems Ce

1−x

(La,Y)

RhIn

and

1−x

CoIn

have revealed logarithmic or power-

law divergences of the specific heat and magnetic

Fig. 13.45. Magnetic phase diagram of the

URu

2−x

system showing the hidden or-

der (HO, filled circles), superconducting (SC,

open circles), and ferromagnetic (FM, squares)

regions. The power-law exponent of the electri-

cal resistivity,n,isshowninthetop part.Some

data are taken from [308,309],after [310]

684 M.B. Maple et al.

Fig. 13.46. Physical properties of URu

2−x

vs Re concentration x. (a) Saturation moment



sat

(at 5 K); (b) effective moment 

eff

; (c) Curie

temperature T

and Curie–Weiss temperature



;and(d) electronic specific heat coefficient

 (at 1 K). The filled circles are from [308].Data

for x = 0 from [163].The lines are guides to the

eye, after [310]

susceptibility once the superconductivity or antifer-

romagnetism in the parent compounds is suppressed

[313–315]. In Ce

1−x

(La,Y)

RhIn

compounds, addi-

tional humps in the specific heat and peaks in the

magnetic susceptibility are observed that coexist

with NFL behavior and decrease with increasing x.

These features can described by the Ce

J =5/2

Hund’s rule multiplet being split by a tetragonal

CEF [313]. The humps in the specific heat have also

been attributed to short-range 2D antiferromagnetic

correlations. [314,316].

An interesting new theory for NFL behavior has

arisen out of studies of the CeMIn

compounds.

In this model, the specific heat and magnetization

are decomposed into two components: a Kondo-gas

part which obeys the single-ion Kondo model, as-

sociated with the coupling between Ce local mo-

ments and conduction electrons,and a Kondo-liquid

portion, resulting from intersite coupling between

the Ce moments. This model was proposed for the

1−x

CoIn

system [317], and can also be applied

to the Ce

1−x

RhIn

system [318]. For these two sys-

tems, the fraction f

of electrons participating in the

single-ion Kondo effect increases roughly linearly as

afunctionofx until it saturates at x =0.9, at which

point the Kondo-liquid portion goes to zero and the

system entirely obeys the single-ion Kondo model.

The fraction f

also varies as a function of temper-

ature. For pure CeCoIn

, the Kondo gas condenses

into a Kondo liquid below T

∗

∼ 45 K, resulting in a

sudden downturn in the electrical resistivity or “co-

herence shoulder”. This model provides an intrigu-

13 Unconventional Superconductivity in Novel Materials 685

ing new possibility for the origin of NFL behavior,at

least in the CeMIn

compounds, and its applicability

to other systems needs to be investigated.

13.3.7 Magnetically Ordered H eavy-Fermion

Superconductors

In addition to the antiferromagnetic heavy-fermion

superconductors such as CeIn

and CePd

dis-

cussed in Sect. 13.3.5, several ferromagnetic heavy-

fermion superconductors have been discovered re-

cently and are described in the following.

UGe

Superconductivity under pressure was observed by

Saxena et al. in single crystal specimens of the fer-

romagnetic (FM) compound UGe

within the ferro-

magnetic phase up to the critical pressure P

≈ 16

kbar where the ferromagnetism vanishes [3]. Shown

in Fig. 13.47 is the T − P phase diagram for single

crystals of UGe

[3]. The Curie temperature T

de-

creases from 53 K at ambient pressure to ≈ 12 K at

P = 15 kbar, above which the FM transition can no

longer be determined from the (T) curves,suggest-

ing a first-order-like transition. A small feature in

the resistivity curves is observed at applied pres-

sures P < 13 kbar, perhaps related to a spin den-

sity wave (SDW) or charge density wave (CDW), and

is denoted in Fig. 13.47 as T

[165,201]. Supercon-

ductivity is confined to a narrow pressure region

10 kbar ≤ P < 16 kbar reaching a maximum value

of T

=0.7KatP

= 13 kbar where features associ-

ated with the possible SDW/CDW can no longer be

observed [3] (the phase diagram of polycrystalline

UGe

samples, discussed below, is similar [201]). No

superconductivity is observed at or above P

,indi-

cating that the ferromagnetism and superconductiv-

ity areintimately related.Recent magnetizationmea-

surementsunder pressure confirm the first-order FM

phase transition at P

and reveal that the transition

at T

is also first-order [319].

It has been suggested that UGe

is a p-wave su-

perconductor, in which the superconducting elec-

trons do not suffer from the pair-breaking inter-

action caused by the large internal magnetic field,

Fig. 13.47. Temperature-pressure T − P phase diagram of

UGe

. T

denotes the Curie temperature; T

denotes the

superconducting transition (the T

values are scaled by a

factorof 10 forclarity; T

denotes the feature in the (P, T)

curves possibly due to a SDW/CDW transition. The solid

and dashed lines are guides to the eye, after [3,165]

since the mean free path l ∼ 1500 Å is much larger

than the superconducting coherence length 

∼

130 − 200 Å [3,202]. Specific heat experiments un-

der applied pressure reveal a finite value of C/T as

T → 0 K,which is consistent with the superconduct-

ing gap vanishing at lines or points on the Fermi

surface [166,202]. In addition, neutron diffraction

measurements yield no evidence of a change in the

magnetic moment upon entering the superconduct-

ing state, consistent with p-wave superconductiv-

ity [165].Spin-tripletsuperconductivitymediated by

coupled SDW/CDW spin ﬂuctuations has been put

forth in recent theoretical work by Watanabe and

Miyake [320]. This theory can explain the anoma-

lous reentrant behavior observed in upper critical

field measurements along the a-axis of UGe

and the

anisotropy of H

[165]; however,neutrondiffraction

measurements do not reveal any static order asso-

ciated with a SDW/CDW. While these results can be

explained within a spin-triplet scenario of supercon-

ductivity,an investigation on polycrystalline samples

of UGe

with electron mean free paths of the order

of the superconducting coherence length, discussed

686 M.B. Maple et al.

Fig. 13.48. Electrical resistivity  vs T

for UGe

at various applied pressures P.Thelines are fits

to the data of the form  = 

+ AT

Inset (a):

Coefficient of the T

term of (T), A,vsP.Inset

(b): Residual resistivity 

vs P, after [201]

below, indicates that the superconductivity in UGe

may have s-wave character [201].

It is widely believed that p-wave superconductors

arequitesensitive tononmagnetic impurities,result-

ing in the suppression of superconductivity when

l ≈ 

,wherel is the electron mean free path and



is the superconducting coherence length. Two of

the leading candidates for p-wave superconductiv-

ity are Sr

RuO

[321,322] and UPt

[323,324].When

the ratio l/

∼ 1, T

is completely suppressed for

RuO

[325] and decreases by 15 % for UPt

[324].

However, in UGe

superconductivity was observed

in some polycrystalline samples with residual resis-

tivities 

as high as 3 ‹§cm, corresponding to a

mean free path l ∼ 100 Å, without any apparent

diminution of T

. Since the superconducting coher-

ence length 

is ∼ 130 − 200 Å [3, 202], this rep-

resents a ratio l/

≈ 1, considerably reduced from

the value l/

≈ 10 observed for the single crystal

specimens of UGe

. This suggests that the supercon-

ductivity observed in UGe

under pressure may have

s-wave character. It is noteworthy that several the-

ories have recently been proposed in which s-wave,

rather than p-wave,superconductivityispredicted to

occur in the ferromagnetic state [326–330].

In the low temperature FL-like phase, the residual

resistivity 

and the coefficient A of the T

con-

tribution to the resistivity, (T)=

+ AT

,are

both strongly dependent on pressure as shown in

Fig. 13.48 for polycrystalline samples of UGe

and

exhibit peaks at 15 kbar and 11 kbar, respectively

(inset of Fig. 13.48). It is interesting that the maxi-

mum in 

correlates with the broad maximum in

(P) that is centered near the same pressure. Spe-

cific heat measurements on polycrystalline samples

of UGe

reveal that  increases with pressure up to

15 kbar [166].The increase of  is consistent with the

behavior of A(P) according to the Kadowaki–Woods

relation A =1.0 × 10

−5



‹§ cm (molK/mJ)

[188].

Since T

of a conventional s-wave superconductorin-

creases with N(E

), which is proportional to  ,and

is not strongly affected by potential scattering, the

correlationbetween T

, ,and

is qualitativelycon-

sistent with s-wave superconductivity.

A precedent for the macroscopic coexistence of

ferromagnetism and s-wave superconductivity is

found in the compound ErRh

, discussed in Sect.

13.2.3, which exhibits localized moment ferromag-

netism, in contrast to UGe

, which is regarded as

an itinerant ferromagnet. The compound ErRh

13 Unconventional Superconductivity in Novel Materials 687

becomes superconducting at a critical temperature

≈ 8.7 K, FM at a Curie temperature T

≈ 1.5

K, and normal again at a second critical temperature

≈ 0.9 K [80]. There is evidence, based on neu-

tron scattering and specific heat measurements [84],

for the macroscopic coexistence of FM and super-

conducting regions in the temperature interval be-

tween T

and 

. Within the superconducting re-

gions between T

and 

, a sinusoidally modulated

magnetic state with a wavelength of ∼ 100 Å that co-

exists with superconductivity has been detected by

means of small angle neutron scattering measure-

ments [84]. The sinusoidally modulated magnetic

state is believed to be produced by screening of the

magnetic field associated with the ferromagnetically

aligned Er magnetic moments by the supercurrents

at long wavelengths [331].Similar behavior has been

found for the localized moment FM superconduc-

tor HoMo

[332]. It seems reasonable that such

an inhomogeneous FM s-wavesuperconducting state

could also exist in UGe

, in spite of the fact that it is

an itinerant ferromagnet.

Irrespective of whether the superconductivity in

UGe

is of s-wave (spin-singlet) or p-wave (spin-

triplet) variety, it appears to be strongly connected

to the ferromagnetism. For s-wave superconductiv-

ity, it seems possible that the onset of superconduc-

tivity occurs when the Curie temperature drops to

a value such that the superconductivity is no longer

suppressed by the internal field in the ferromagnet,

while near P

where the Curie temperature vanishes,

the superconductivity is suppressed by the ferro-

magnetic spin ﬂuctuations. Alternatively, the longi-

tudinal spin ﬂuctuations may actually produce pair-

ing of electrons in spin-singlet s-wave states as pro-

posed recently by Suhl [329] and developed further

by Abrikosov [330]. The competition between these

two effects could then lead to the maximum in T

a pressure below P

.For triplet-spin superconductiv-

ity, the superconductivity would seem to require the

presence of the ferromagnetism and be optimized

at a pressure below P

. Further experiments will be

required to develop an understanding of the origin

and nature of superconductivity in this remarkable

material.

Other Ferromagnetic Heavy-Fermion Superconductors

The coexistence of superconductivity and ferromag-

netism was recently reported in URhGe [109], UIr

[168] and ZrZn

[333]. While bulk superconductiv-

ity appears to be well established in URhGe given the

specific heat jump at T

[109], the situation is less

clear for UIr and ZrZn

. In UIr, ferromagnetism in

this material at T

= 46 K is rapidly suppressed

with pressure [167] and evolves into another small-

moment ferromagnetic phase that is destroyed at

a critical pressure P

= 26 kbar [168]. Near this

critical pressure, the electrical resistivity follows a

power-law T-dependence ( ∼ T

)withanexpo-

nent n  5/3 as expected for critical spin ﬂuctua-

tions of a 3D ferromagnet [220] (see Table 13.5). In

addition,superconductivityis observed at T

=0.14

KinthevicinityofP

suggesting that ferromag-

netic ﬂuctuations may mediate the pairing in this

material. As discussed in Sect. 13.3.5, UIr is a can-

didate for spin-triplet superconductivity,despite the

lack of inversion symmetry. The large initial slope

of the upper critical field and the large value of the

coefficient of (T) indicate an enhancement of

the effective mass [168].However, the identical value

of T

and similar H

of Ir [334] raise the possibil-

ity of filamentary superconductivity associated with

Ir in UIr; additional measurements are needed to

Fig. 13.49. Pressure-temperature phase diagram of ZrZn

: Curie temperature; T

: Superconducting critical tem-

perature. The values of T

have been multiplied by a factor

of 10 for clarity,after [333]

688 M.B. Maple et al.

clarify the nature of superconductivity in this inter-

esting material. For ZrZn

, a superconducting tran-

sition at T

≈ 0.3 K was observed to decrease with

increasing pressure and vanish near the critical pres-

sure P

at which T

vanishes as shown in Fig.13.49.

The absence of a superconducting anomaly in spe-

cific heat, and the occurrence of superconductivity

within the ferromagnetic state only in very pure sam-

ples suggest the possibility of a spin-triplet pairing

mechanism, although trace amounts of a supercon-

ducting impurity phase cannot be ruled out at this

point.

13.3.8 Multiple SuperconductingPhases

UPt

and U

1−x

The first direct evidence for unconventional super-

conductivity was found for the compound UPt

which displays at least three distinct supercon-

ducting phases. Multiple superconducting phases

were also found in the heavy-fermion compound

1−x

when substituted with 2 − 4 % Th, and

morerecentlyin PrOs

(discussed in Sect.13.3.9).

The appearance of multiple superconducting phases,

whichcannot beexplainedintermsof a conventional

s-wave order parameter, has prompted a ﬂurry of in-

vestigations into these exciting compounds.

UPt

exhibits antiferromagnetic (AFM) order be-

low T

= 5 K with a very small ordered moment of

0.02 

/U atomaligned in thebasal plane [182].Early

measurements on as-grown samples showed a single

broadsuperconducting transition in the specific heat

at 0.5 K; however, later measurements of annealed

samples revealed two peaks in the specific heat near

at temperatures T

and T

−

[335].The specific heat

vs temperature of a single crystal of UPt

is plotted

in Fig. 13.50. The double kink feature has been veri-

fied in many different crystals, and while the values

of the T

’s vary between samples, the kinks are al-

ways separated by about 50 mK. These data, along

with anomalies in ultrasonic attenuation [336], and

akinkinH

vs T point to multiple superconducting

phases.

The superconducting phase diagram of UPt

de-

termined fromsound velocity measurements [337] is

shown in Fig. 13.51, where “N” indicates the normal

Fig. 13.50. Specific heat of UPt

in the vicinity of the super-

conductingtransition plottedas C/T vs T.Thedashed lines

represent two idealized jumps,and the solid line represents

a single jump with the same entropy, after [335]

(antiferromagnetic) state, and the three supercon-

ducting phases are labeled “A”,“B”, and “C”. The T

and T

−

transitions observed in the specific heat at

zerofieldcorrespondtotheN-to-AandtheA-to-B

phase transition lines. Measurements of muon spin

relaxation reveal that the transition to the B phase at

−

is accompanied by the onset of a small internal

field at 475 mK. These data might lead to the con-

clusion that the phase transition at T

−

just signifies

the onset of magnetic order.However,the peak in the

specific heat at the first transition is so large that it

occupies most of the Fermi surface, and there is not

enough entropy left for an equally large magnetic

transition at T

−

. The second transition can only be

explained in terms of a transition to a different su-

perconducting phase. Both transitions satisfy the re-

lation C/ T

∼ 1 indicating that heavy electrons

are responsible for the superconductivity (see [288]

for a review).

The heavy-fermion compound U

1−x

also

shows a rich phase diagram with multiple super-

conducting phases. The phase diagram determined

from specific heat,upper criticalfield, and ‹SR stud-

ies is shown in Fig. 13.52 where “A” and “B” indicate

the two superconducting phases [338]. As in UPt

the second superconducting transition is accompa-

nied by a transition to very small moment mag-

netism (0.001 

/U)[339].Recent point contact spec-

troscopy measurements on UPt

[340] are consistent

13 Unconventional Superconductivity in Novel Materials 689

0.5

1.5

2.5

0 100 200 300 400 500

)alseT( H

T (mK)

UPt

0.5

1.5

0 100 200 300 400 500

)alseT( H

T (mK)

H || c

Fig. 13.51. Superconducting phase diagram of UPt

deter-

mined from ultrasonic velocity measurements. [337]. The

data points delineate the second-order phase transition

lines between three superconducting phases “A”,“B”, and

“C”, and the normal state “N”,after [337]

Fig. 13.52. Phase diagram of U

1−x

, after [338]

with line nodes in the ab plane and point nodes along

thec-axis,whileNMR measurements of both systems

show T

temperature dependence below T

,indicat-

ing lines of nodes.

Measurements of T

(P)ontheU

1−x

sys-

tem are displayed in Fig. 13.53 as isobars of T

vs x

for 0 ≤ x ≤ 12 kbar [341]. Two different types of

behavior are present for each pressure, separated by

Fig. 13.53. Isobars of the superconducting critical temper-

ature T

vs x for U

1−x

between 0 and 12 kbar, af-

ter [341]

the minimum in T

(x)atx

min

.Forx < x

min

,amono-

tonic decrease of T

(x) is observed as the pressure

increases, and the magnitude of the slope dT

/dP

also becomes larger with increasing x.Forx > x

min

an abrupt increase of T

occurs with a maximum

and subsequent decrease for higher x. The position

of this maximum is pressure dependent, moving to

higher concentration as P increases. It is tempting

to infer from the T

(x) curves the existence of two

distinct superconducting phases, one in the regions

0 ≤ x ≤ x

min

(A phase) and the other at x > x

min

(B phase), where x

min

is a function of pressure. The

way the T

(x) curves evolve with pressure suggests

that the T

(x) phase boundary for x < x

min

extends

approximately linearly with the same slope to the re-

gion x > x

min

A variety of theories have been advanced to ex-

plain the unconventional superconductivity in UPt

690 M.B. Maple et al.

and U

1−x

. These theories have been exten-

sively reviewed in [342–344]. For UPt

,mostofthe

data points to a spin singlet [345] or spin triplet

[344,346] complex (two component) superconduct-

ing order parameter [342]. Another possibility that

is difficult to rule out is the presence of two differ-

ent superconducting phases with different symme-

tries that happen to be nearly degenerate in tem-

perature [347,348]. The small moment magnetism

in the“B” superconducting states could be explained

by a superconducting order parameter that breaks

time reversal symmetry.On the other hand, the mag-

netism could simply result from the onset of a spin

density wave. The coupling between superconductiv-

ity and magnetism is a key issue that still needs to be

resolved.

InthecaseofU

1−x

,there are three second-

order phase transitions thatmeet at a tricritical point,

which is theoretically not allowed [349].This implies

the existence of other phase transition lines.An addi-

tional phase transition has indeed been observed by

assuming that Th substitution is equivalent to pres-

sure,andmeasuring specific heat as a functionof ap-

plied pressure for a sample with x = 0.019.One of the

more popular models for U

1−x

[343,350] as-

sumes multiple degenerate superconducting phases

with different symmetries, as was mentioned for

UPt

. The pure UBe

compound is assumed to have

a d-wave symmetry and an s-wave phase with a lower

.Thed-wave phase is suppressed with Th doping

more rapidly than the s-wave phase, causing the two

transition lines to cross. Thus for intermediate Th

concentrations there is a pure s-wave phase at high

temperatures and a mixed s − d phase at lower tem-

peratures.

PrOs

Recently, superconductivity was observed in the

filled skutterudite compound PrOs

at T

=1.85

K [123, 135]. The superconductivity evidently in-

volves heavy-fermion quasiparticles with an effec-

tive mass m

∗

≈ 50 m

, as inferred from the normal-

state electronic specific heat coefficient  ,thejump

in the specific heat at T

, and the slope of the up-

per critical field near T

. This compound appears

to be the first example of a Pr-based heavy-fermion

superconductor.When this material was first discov-

ered, the quadrupolar Kondo effect, which involves

screening of Pr

electric quadrupole moments of a

nonmagnetic 

doublet ground state in the CEF by

the charges of the conduction electrons [276, 277],

analogous to the screening of the magnetic dipole

moments of paramagnetic ions by conduction elec-

tronspinsthatoccursinthemagneticKondoeffect,

was put forth as a possible scenario for the heavy-

fermion behavior [123,135]. Another scenario could

involve virtual excitations from either a 

singlet or

a 

nonmagnetic doublet ground state to a low lying

(∼10 K) 

triplet state [351].

Evidence for the heavy-fermion behavior in

PrOs

was provided by the magnitude of the

superconducting specific heat jump on a pressed

pellet of single crystals as displayed in the inset of

Fig. 13.54.(TheC(T) data havebeen corrected for ex-

cess Sb derived from the molten Sb ﬂux in which the

crystals were grown.) An equal entropy construction,

in which the entropy is conserved just above and be-

low T

,yieldsC/T

= 632 mJ/mol K

.Thevalueof

the electronicspecific heatcoefficientfromthe weak-

coupling BCS prediction (C/ T

=1.43) is  ≈ 440

mJ/mol K

. In the normal state, the specific heat can

be described by C(T)= T + ˇT

+ C

Sch

(T), where

 T and ˇT

areelectronicand phonon contributions,

respectively, and C

Sch

(T) is a Schottky anomaly for a

two level system consisting of a doublet ground state

and a triplet excited state at an energy  above the

ground state. The best fit of this expression to the

data yields the values  = 607 mJ/mol K

, ˇ =3.95

mJ/mol K

(corresponding to a Debye temperature



= 203 K), and  =7.15 K.

A number of measurements on PrOs

pro-

vide evidence for the formation of a nonmagnetic



ground state due to splitting of the J =4 mul-

tiplet of Pr in a cubic crystalline electric field.

this scenario, hybridization between the 

and ex-

In O

symmetry, the J = 4 states are labeled: 

(singlet), 

(doublet), 

,and

(triplets) [352]. The slightly reduced

symmetry appropriate for the filled skutterudite structure, mixes the magnetic triplets (now labeled 

(1)

and 

(2)

respectively),and the doublet is now labeled 

[353]. We use the O

notation throughout for simplicity.