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 671

directional nature of the tunneling measurements

provides informationabout the symmetry of the su-

perconducting order parameter.Order parametersof

odd parity can be ruled out due to the strong Pauli

limitingthatis observedin UPd

,andthereis only

one even d-wave point group that does not show a

gap node along the c-axis. The authors concluded

therefore that only the fully symmetric A

d-wave

representation is consistent with the measurements.

Nuclear Magnetic Resonance

Two types of NMR and NQR measurements have

been performed on UPd

Al measurements by

Kyogaku and coworkers in the early 1990s [219],and

more recently

105

Pd NMR measurements by Matsuda

et al. [206]. Both types of measurements agree as

to the nature of the superconductivity in UPd

The spin-lattice relaxation rate T

derived from

105

NQR results is plotted as 1/T

vs temperature T in

Fig. 13.32. Between T

and T

,1/T

varies as T in

accordance with the Curie–Weiss law. Below T

,1/T

followsa T

temperature dependence,andthe Hebel–

Slichter peak usually seen in BCS superconductors

is absent. These data are consistent with the pres-

ence of unconventional superconductivity with an

anisotropic energy gap that has line nodes on the

Fermi surface.The best fit to the 1/T

vs T data gives

a superconducting energy gap of 2(0) = 5.5 k

This energy gap is larger than that measured by neu-

tron diffraction, perhaps due to anisotropy of the

energy gap.In contrast to the NMR and NQR results,

the T

temperature dependence of the specific heat

implies point nodes on the Fermi surface.Additional

NQR measurements at lower T are necessary to re-

solve this discrepancy. It is also possible that both

point and line nodes are present and only one domi-

nates in each measurement.

The magnetic properties of UPd

can only

be discerned from

105

Pd NMR and not from

NMR measurements. This is because the Al atoms

are located interstitially between Pd-U sheets, so

transferred hyperfine fields from neighboring anti-

aligned U moments are canceled almost perfectly,

and T

reﬂects only the susceptibility of the conduc-

tion electrons. On the other hand, the Pd atoms are

Fig. 13.32. Spin-lattice relaxation rate 1/T

of UPd

a function of temperature T determined from

105

Pd NQR

measurements in the vicinity of the N´eel temperature T

andsuperconducting transition temperatureT

,after[206]

surrounded ferromagnetically by U moments, which

affect the Pd T

relaxation via the hyperfine inter-

action. The results of a recent

105

Pd NMR and NQR

study are described below.

105

Pd NQR measurements show [206] a resonance

line at ≈ 5 MHz which is split into two lines in the

AFM state.The splitting can be explained by the pres-

ence of a 2.9 kOe internal field. The temperature de-

pendence of this internal field, which extrapolates

to zero at T

, closely matches that of the sublat-

tice magnetization measured by neutron diffraction

measurements[183,209].The presence of a local field

throughout the sample below T

confirms the micro-

scopic coexistence of antiferromagnetism and un-

conventional superconductivity.

The hyperfine coupling constant, deduced from

the Knight shift, is very small (0.3kOe/

), indicat-

ing that there is no significant polarization density

of Pd 4d electrons.Furthermore,the large value of T

above T

confirms that the Pd ions are nonmagnetic

and only very weakly coupled to the U ions.

The NMR and NQR measurements of UPd

confirm unconventional superconductivity with an

energy gap that goes to zero at lines on the Fermi sur-

face. The

105

Pd NMR and NQR measurements show

that the antiferromagnetism is due to U local mo-

ments with an internal field H =2.9kOe.Further-

more,the results unambiguously demonstrate the co-

existence of antiferromagnetism and superconduc-

tivity.

672 M.B. Maple et al.

13.3.5 Quantum Critical Points

Introduction

Quantum phase transitions (QPT) are observed in

many f -electron heavy-fermion systems,in which an

antiferromagnetic or ferromagnetic transition tem-

perature, spin glass freezing temperature, or spin

density wave (SDW) is suppressed to T =0Kby

pressure or chemical substitution.When the QPT is

a second-order phase transition, it is referred to as

a quantum critical point (QCP). A wide variety of

f -electron materials display QPTs or QCPs including

CeIn

andCePd

(AFM),UGe

(FM),and CeCu

(SDW).In the vicinityof the criticalpressure or con-

centration ı

at which the QCP occurs, so-called

non-Fermi-liquid (NFL) behavior is often observed;

itmanifestsasunusualpower-lawtemperaturede-

pendencies of the physical properties at low temper-

atures. NFL behavior is discussed in more detail in

Sect. 13.3.6. In some compounds, superconductivity

also occurs in a narrow region around the critical

pressure P

at the QCP. The presence of supercon-

ductivity in the vicinity of the QCP suggests that the

superconducting electron pairing mechanism may

be mediated by magnetic ﬂuctuations. QCPs are also

observed in the high-T

cuprate superconductors as

discussed in Sect. 13.5.

CePd

and CeIn

Lonzarich and coworkers [1, 2] reported measure-

ments of the temperature dependence of the electri-

cal resistivity of single crystal specimens of the AFM

compounds CeIn

and CePd

at various pressures

up to ∼ 30 kbar. The N´eel temperature T

, deter-

mined from a discontinuity in the slope d/dT at

, was found to decrease monotonically with in-

creasing P for both compounds and to extrapolate to

zero at a critical pressure P

of ≈ 26 kbar for CeIn

and ≈ 28 kbar for CePd

. At temperatures below

1 K and pressures in a narrow region near P

, (T)

was found to drop abruptly, indicative of supercon-

ductivity for both compounds.The dependence of T

on P has an inverted parabolic shape,reminiscent of

the shape of the T

vs charge carrier concentration

curvefoundforthehighT

cuprate superconductors.

Fig. 13.33. Temperature-pressure T − P phase diagram of

CeIn

showing theN´eel temperature T

and superconduct-

ing transition temperature T

(plotted as 10T

)Theinset

shows the superconducting transition in the resistivity 

vs temperature T at P =24kbar,after[1].

The N´eel temperature and superconducting transi-

tion temperature are plotted vs pressure in Fig. 13.33

for CeIn

. The inset to this figure shows the super-

conducting transition in (T)at24kbar.AtP = P



the pressure at which the maximum in T

occurs,

the superconductivity and AFM appear to coexist.

However, at pressures above P



, T

can no longer be

defined from the electrical resistivity measurements.

The transition from the AFM to the paramagnetic

state appears to become first order or to be broad-

ened in temperature due to pressure inhomogeneity

such that it can no longer be detected in (T)mea-

surements.

The observation of superconductivity near P

the pressure at which T

extrapolates to zero, sug-

gests that AFM spin ﬂuctuations may be responsi-

ble for the pairing of superconducting electrons in

these two compounds. Spin-ﬂuctuation theories pre-

dict [220–223] that the resistivity in the vicinity of

the QCP should vary as T

with n =3/2for3DAFM

ﬂuctuations and n = 1 for 2D AFM ﬂuctuations (Ta-

ble 13.5). In CeIn

and CePd

, it was observed that

near P

and above T

, (T)variesasT

with n =1.6

for CeIn

and n =1.2 for CePd

. These results

13 Unconventional Superconductivity in Novel Materials 673

suggest that the spin ﬂuctuations have 3D charac-

ter in cubic CeIn

and 2D character in tetragonal

CePd

. Spin-ﬂuctuation theory also predicts that

the magnetic ordering temperature near P

should

vary as (P

− P)

where y =2/3 for 3D and 1 for 2D

ﬂuctuations.While a linear dependence of T

on P

was observed in CePd

, consistent with 2D ﬂuc-

tuations,a power-law has not yet been distinguished

for CeIn

CeMIn

(M = Co, Ir, Rh)

The CeMIn

(M = Co,Rh,Ir)family ofheavy-fermion

compounds was discovered only a few years ago, yet

has attracted an enormous amount of attention and

display a variety of interesting phenomena including

unconventional superconductivity, possible Ferrel–

Larkin Fulde–Ovchinnikov (FFLO) superconductiv-

ity,antiferromagnetism,quantum critical points,and

non-Fermi liquid behavior [115–117, 152, 178, 224,

225]. CeRhIn

displays AFM order at T

=3.8K,but

becomes superconducting under pressure with a T

of about2.2 K at 21 kbar [115].CeIrIn

is a supercon-

ductor at ambient pressure with a bulk T

=0.4K,

but a resistive superconducting transition at a higher

temperature of T

=1.2 K [116].Finally,CeCoIn

be-

comes superconducting at 2.3 K, which is the high-

est T

of any Ce-based heavy-fermion superconduc-

tor yet discovered [117]. Measurements of power-

law T-dependencies in the specific heat [116,117],

NMR relaxation rate [226, 227], and thermal con-

ductivity [228] show that CeCoIn

and CeIrIn

are

unconventional superconductorswith nodal gaps.In

particular, Izawa and coworkers measured [229] the

directional thermal conductivity of CeCoIn

single

crystals in an external magnetic field rotated in the

ab plane.The thermal conductivity data show a four-

fold symmetry with peaks (corresponding to a nodal

direction) when the magnetic field is aligned in the

[110] direction.These data are consistent with d

−y

symmetry of the superconducting energy gap.

Before discussing quantum critical points in the

CeMIn

(M = Co, Rh, Ir) systems, a description

of the unusual nature of the superconductivity in

CeCoIn

will be presented. Recent measurements

[197,229–232] of specific heat, magnetization, ther-

mal conductivity, and thermal expansion show that

the superconducting transition of CeCoIn

becomes

first-order for applied magnetic fields above 47 kOe

in the [001] direction, and 100 kOe in the [100]

direction. Evidence for the first order transition is

also found in quasi-adiabatic magnetic field sweeps

where the temperature of a thermally isolated sam-

ple is measured as a function of applied magnetic

field [230]. For fields in the [001] direction, the tem-

perature vs field data show a kink at temperatures

above 0.72 K, corresponding to a second-order tran-

sition, and a step at temperatures below 0.72 K, cor-

responding to the latent heat released by a first-order

transition.

First-order superconducting transitions were pre-

dicted in 1964 by Maki and Tsuneto [19,233] for su-

perconductors in the clean limit where spin-orbit

scattering can be neglected. However, CeCoIn

the first superconductor to date to unambiguously

demonstrate a first-order transition in specific heat

measurements. The application of a magnetic field

to a spin-singlet superconductor can destroy the su-

perconductivity via two effects: orbital pair break-

ing, in which the field-induced kinetic energy of

the Cooper pair exceeds the superconducting con-

densate energy, and Pauli paramagnetic limiting, in

which it is energetically favorable for the electron

spins to align with the magnetic field, thus break-

ing the Cooper pairs (see Sect. 13.2.). The critical

field determined by orbital pair breaking is given

by Eq. (13.22), whereas the critical field from Pauli

paramagnetic limiting is H

= 

√

2

.TheMaki

parameter ˛ =

√

(0)/H

is a measure of the rel-

ative strengths of these two effects.When Pauli para-

magnetism is dominant and ˛ ≥ 1.8, the super-

conducting transition is predicted to become first

order at temperatures below a reduced temperature

= T

[234]. Several groups have calculated the

Maki parameter for CeCoIn

[230,235,236] with re-

sults ranging from 3.2 to 13. While there are vary-

ing techniques and experimentaldifficulties in deter-

mining the precise value of ˛,allthemeasurements

agree that ˛ > 1.8 indicating that CeCoIn

is para-

magnetically limited. Furthermore, thermal conduc-

tivity measurements indicate that CeCoIn

is in the

clean limit with l/

= 14 [228].

674 M.B. Maple et al.

Fig. 13.34. (a) Electronic specific heat of CeCoIn

plotted

as C/T vs temperature T and field H for H || [110]. (b)

Contour plot of the data in (a) with the same color scale.

Plots are from Bianchi et al. [232]

Recent measurements on CeCoIn

have revealed

that when T

is suppressed below 0.3 K by magnetic

fields, another transition occurs below T

that is

second-order. This transition has been observed in

specific heat vs magnetic field [235], specific heat vs

temperature [232], thermal conductivity [237], and

in penetration-depth measurements using a self-

resonant tank circuit [236]. This second transition

is most prominent for H in the ab plane; however,

Bianchi et al. also observe a small effect for H along

the crystallographic c-axis. The shape of the phase

diagram is strongly suggestive of an “FFLO” state

occurring between the second and the first-order

transitions. In this state, which was independently

predicted by Fulde and Ferrell [238] and Larkin and

Ovchinnikov [239] in 1964, the superconductor low-

ers its energy in a magnetic field by forming periodic

regions of superconductivity separated by domains

of aligned spins. The order parameter is spatially

modulated and changes sign, either sinusoidally or

more abruptly. The FFLO state is predicted to occur

at high fields, and at temperatures T

FFLO

< T < T

in samples which are in the clean limit and for which

spin-orbit scattering can be neglected. CeCoIn

an excellent candidate for an FFLO state. It is in

the clean limit with l/

∼ 14 [228], has a cylindri-

cal two-dimensional Fermi-surface [240,241] and a

d-wave superconducting order parameter. Specific

heat data as a function of temperature and magnetic

field by Bianchi et al. [232] exhibiting a first-order

superconducting transition and the possible FFLO

region is shown in Fig. 13.34. Further work such as

STM measurements or neutron-diffraction will be

necessary to settle the question of FFLO supercon-

ductivityin this system.The FFLO state has also been

suggested to occur in CeRu

and UPd

; however,

in CeRu

the candidate FFLO transition was later

found to be associated with vortex ﬂux-line melt-

ing [242–244], and in UPd

, the transition was

not consistent with theory [245].

The unconventional superconducting state in

CeCoIn

appears to evolve out of an equally unusual

non-Fermi-liquid normal state.The resistivity above

varies linearly with temperature, and the specific

heat diverges logarithmically with temperature when

a magnetic field of 5 T (greater than H

)isapplied

along the [001] direction. However, at higher mag-

netic fields and low temperatures, Fermi-liquid be-

havior is recovered in the specific heat and electrical

resistivity [117, 232,246]. The high field phase dia-

gram has been investigated by Paglione et al. [246]

and by Bianchi et al. [232]. The phase diagram by

Paglione et al. based on electrical resistivity and spe-

cific heat measurements is shown in Fig. 13.35. Fur-

thermore, Bianchi et al. find scaling of the Sommer-

feldcoefficient = C/T with(H−H

)/T over a broad

range of temperatures and fields both in the FL and

the NFL regime [232]. It is extremely unusual to find

13 Unconventional Superconductivity in Novel Materials 675

a QCP associated with a superconducting transition,

and, in fact, the scaling behavior is indicative of a

second-order QCP whereas the superconductivity is

first-order near H

.Both groups have therefore sug-

gested that the QCP responsible for the NFL is in fact

a second-order antiferromagnetic one, where the an-

tiferromagnetic order is prevented or“hidden”by the

superconductivity.

This question of whether the QCP is the direct

result of the destruction of superconductivity, or

only accidentally coincident with the upper critical

field in CeCoIn

has been probed by three stud-

ies that seek to tune H

and watch the evolution

of the quantum critical behavior. If the supercon-

ductivity is ‘hiding’ an incipient antiferromagnetic

quantum critical point, then tuning H

should ei-

ther reveal antiferromagnetism or drive the system

towards Fermi-liquid behavior. In one study, the

quantum critical behavior was investigated for both

H||c where H

=4.95 T and H||ab where H

=12T

[247]. In the second study [248], the suppression

of H

and concomitant QCP behavior was stud-

ied in the series CeCoIn

5−x

. Both studies found

that the quantum critical behavior always occurred

at H

, regardless of how H

was varied. The non-

Fermi liquid behavior near the quantum critical

point was evidenced in the specific heat C(H, T)as

/T ∼ −lnT and scaling of the Sommerfeld co-

efficient [( (H)− (H

QCP

]/(H)

∼ H/T

where

˛ =0.7(1) and ˇ =2.5(5) for x = 0, 0.03, 0.06, with

˛ =0.9(1) and ˇ =3.0(5) for x =0.12. The elec-

trical resistivity shows non-Fermi liquid behavior in

the form of (T) ∼ T

where ˛ =1.5. These two

experiments show that the quantum critical behav-

ior is directly related to the suppression of super-

conductivity, rather than being coincidental. In one

interpretation, the quantum ﬂuctuations present at

low temperatures are responsible for both the su-

perconducting pairing mechanism as well as the

quantum critical behavior when the superconduc-

tivity is destroyed with magnetic fields. The third

recent set of experiments of electrical resistivity at

applied pressure and in magnetic field revealed that

the field for criticality H

QCP

was suppressed more

rapidly than the upper critical field H

with applied

pressure [249]. This suggests that an antiferromag-

Fig. 13.35. Magnetic field H vs temperature T phase dia-

gram of CeCoIn

. Solid squares are the boundary between

Fermi-liquid (FL) behavior with (T)=

+AT

and non-

Fermi liquid (NFL) behavior. Solid circles indicate a max-

imum in magnetoresistivity,solid triangles are H

deter-

mined from resistivity, and open triangles and open circles

are H

from resistivity and specific heat data in [117]. The

inset shows the coefficient A in the FL regime as a function

of field, which diverges near the upper critical field H

CeCoIn

, after [246]

netic quantum critical point is responsible for the

NFL behavior in CeCoIn

Another quantum critical point in CeCoIn

may

occur at negative pressures [225]. Evidence for this

comes from the fact that CeRhIn

is an antiferro-

magnet with a larger unit cell than CeCoIn

.TheN´eel

temperature of CeRhIn

can be suppressedwith pres-

sure leading to superconductivity above P =16kbar,

with a T

= 2 K, remarkably close to that of CeCoIn

At this pressure, the unit cell volume of CeRhIn

is also close to that of CeCoIn

. In fact, there is a

general correlation in the CeMIn

(M = Co, Rh, Ir)

compounds between the superconducting transition

temperature T

and the lattice parameters [250]. It

appears that, with the exception of the compounds

displaying antiferromagnetism, T

varies as a linear

function of the lattice anisotropy c/a where c and

a are the tetragonal lattice parameters. In Fig. 13.36,

676 M.B. Maple et al.

0.5

1.5

2.5

1.605 1.610 1.615 1.620 1.625 1.630 1.635 1.640

)K(

lattice c/a

CeIrIn

CeCo

0.5

CeRh

0.25

0.75

16 kbar

CeRh

0.5

CeCo

0.5

CeCo

0.5

CeRhIn

(~ 16 kbar)

CeCoIn

Fig. 13.36. Superconducting transition temperature T

plot-

ted vs the ratio of the tetragonal lattice constants c/a for

various compounds in the CeMIn

family (M = Co, Rh, Ir),

after [250]

is shown as a function of c/a for members of the

CeMIn

family both under ambient pressure and hy-

drostatic pressures.

Thus, CeRhIn

could be considered equivalent to

CeCoIn

with an expanded lattice parameter,or“neg-

ative pressure”.When pressure is applied to CeCoIn

goes through a maximum at P =15kbarandis

suppressed by P ∼ 37 kbar. When the CeRhIn

and

CeCoIn

phase diagrams are combined, considering

CeCoIn

at ambient pressure equivalent to CeRhIn

at P = 16 kbar, the resulting phase diagram looks

remarkably similar to those of the high-T

cuprates

(Sect.13.5).In fact, several groups suggest the forma-

tion of a pseudogap near the QCP based on electrical

resistivity and

115

In NQR measurements [225,252].

The T − P phase diagrams of CeRhIn

and CeCoIn

are shown in Fig. 13.37. Fermi-liquidbehavior in the

resistivity with a T

temperature-dependence is re-

covered at higher pressures, as shown.

The details of the pressure-induced quantum crit-

ical point in CeRhIn

have been extensively studied

in an effort to understand the quantum criticalpoint

and the attendant superconductivity [151,252–257].

Pressure studies in zero applied magnetic field using

specific heat, resistivity, NMR, and neutron scatter-

ing indicate that the line of antiferromagnetic tran-

Fig. 13.37. Upper figure: Temperature-pressure T − P phase

diagrams of CeRhIn

from electrical resistivity and ac sus-

ceptibility measurements (after [251]). Lower figure: T − P

phase diagram of CeCoIn

based on electrical resistivity

(after [225])

sitions T

(P) vanishes abruptly at the top of the su-

perconducting dome at P

.Some of the studies show

that superconductivity and antiferromagnetism co-

exist for P < P

with T

> T

. Recent de Haas-van

Alphen measurements [258] reveal a divergence of

the effective mass at P

∼ 25 kbar indicating an

AFM QCP at that pressure. The dHvA frequencies

change dramatically from a “small” Fermi surface,

where the 4f electrons of Ce are only weakly hy-

bridized with the conduction electrons, to a “large”

Fermi surface in which there is strong hybridization.

This has been interpreted as a localized/itinerant

crossover of the f electrons at the QCP. It is inter-

esting to note that the dHvA frequencies above P

CeRhIn

closely resemble those of CeCoIn

at ambi-

ent pressure[259] indicating that CeCoIn

isthe itin-

erant version of localized CeRhIn

.A recent study of

13 Unconventional Superconductivity in Novel Materials 677

Fig. 13.38. ac calorimetry data for CeRhIn

in applied hy-

drostatic pressures of 2.1 GPa for various applied mag-

netic fields. Superconducting critical temperature T

and

the onset of antiferromagnetic order T

are indicated (af-

ter [260])

CeRhIn

in magnetic fields under pressure using ac

calorimetry shows that application of magnetic field

allows the line of quantum phase transitions T

(P)

to extend into the superconducting dome such that

AFM and SC coexist with T

> T

[260]. The ac-

calorimetry data is shown in Fig.13.38 and the T − P

phasediagramforvariousmagneticfieldsisshown

in Fig. 13.39.

An interesting question is what happens

when CeRhIn

and CeCoIn

are mixed to form

CeRh

1−x

, and whether tuning the lattice pa-

rameters via concentration is equivalent to tuning

the lattice parameters with pressure. Substitution

studies of both CeRh

1−x

and CeRh

1−x

systems have been carried out and the phase dia-

grams are shown in Fig. 13.40 [191, 261]. Both sys-

tems show a strikingly similar suppression of the

N´eel temperature T

with increasing x,withT

ex-

trapolating to zero at a QCP at x ∼ 0.6−0.8(see

Fig. 13.37). The fact that T

depends only on the

Rh concentration in these two systems, and not on

the lattice parameters, strongly suggests a percola-

tionproblem.AntiferromagnetismisdestroyedasRh

atoms are removed from the system.This percolation

is nontrivial however,since the magnetic moment re-

Fig. 13.39. Temperature-pressure phase diagram of

CeRhIn

for various magnetic fields. In applied fields,

the antiferromagnetism extends into the superconducting

dome, allowing antiferromagnetism and superconductiv-

ity to coexist with T

< T

.Thequantumcriticalpoint

above which antiferromagnetism vanishes for H =0is

indicated (after [260])

sides on the Ce atom and not on the Rh atom. SR

measurements [262] show that a small ordered mo-

ment may reside on the Rh atom; however, a final

understanding of the role of the Rh atom in creating

a three-dimensional Fermi surface and long-range

AFM order in these systems is still lacking. Thus, the

phase diagrams of the substituted systems are pro-

foundly different that those of CeRhIn

and CeCoIn

under pressure (see Fig. 13.37). The superconduc-

tivity coexists with antiferromagnetism over a very

broad range of concentrationsand does not show the

dome-shape typical of quantum critical points.

For the CeRh

1−x

system, the specific heat

plotted as C/T vs T for various Co concentrations x

is shown in Fig. 13.41 [191].(These data arevery sim-

ilar to those of the CeRh

1−x

system in [261].)

Each specific heat curve contains either one or two

peaks. The peaks for T < 2 K are associated with

the superconducting transition, whereas the peaks

for T > 3 K are attributed to AFM transitions. The

C(T)/T data for samples with 0.4 ≤ x ≤ 0.6con-

tain two peaks, indicating that these samples ex-

hibit coexistence of superconductivity and antifer-

romagnetism. The magnitudes of the superconduct-

678 M.B. Maple et al.

Fig. 13.40. Phase diagrams of CeRh

1−x

and

CeRh

1−x

.TheN´eel temperature T

, superconduct-

ing transition temperature T

, and the temperatures at

which the resistivity goes to zero T

=0

are shown vs Ir

or Co concentration x.Thelines are guides to the eye (af-

ter [191,261])

ing peaks are quite large, with C/ T

≈ 1formost

samples, and ∼ 5 for CeCoIn

,indicating that the su-

perconductivity is a bulk phenomenon involving the

heavy electrons.The magnitudeof the superconduct-

ing peaks is suppressed with decreasing x,whereas

the magnitude of the AFM peaks is enhanced with

decreasing x. The entropy at 7 K, which corresponds

Fig. 13.41. Specific heat C(T)shownasaplotofC/T vs

temperatureT for various CeRh

1−x

compounds.The

peaks for T < 2 K are attributed to superconducting tran-

sitions, whereas the peaks for T > 3 K are ascribed to

antiferromagnetic phase transitions, after [191]

to the total area under the superconductingand AFM

peaks, is constant for the samples with x < 1, im-

plying that the same heavy electrons participate in

both the superconductivity and the antiferromag-

netism. The question arises whether the supercon-

ductivity and magnetism coexist on a microscopic

scale or if there are macroscopic concentration vari-

ations of Co and Rh within the sample.This question

has been investigated for the CeRh

1−x

system

by Morris and coworkers [263]. ‹SR measurements

in zero field of the CeRh

0.5

sample show that

the antiferromagnetism exists over at least 85% of

the sample, and ac magnetization measurements in-

dicate at least 85% shielding of the sample. Thus,

magnetism and superconductivity most likely coex-

istmicroscopicallyin thissample.Further discussion

of the CeRh

1−x

system can be found in the mul-

tiple superconducting phases section.

Noncentrosymmetric Superconductivity

It was thought for a long time that spin triplet su-

perconductivity can only occur if parity is satisfied

[264].In practice,this means that for heavy-fermion

superconductors the crystal structure must have in-

version symmetry (be centrosymmetric) since the

strong spin-orbit coupling in the HF compounds al-

13 Unconventional Superconductivity in Novel Materials 679

lows the electrons at the Fermi surface to see the

symmetry of the crystal. Most heavy-fermion super-

conductors to date have inversion symmetry. How-

ever, two HF superconductors were recently discov-

ered, CePt

Si [119] and UIr [168], whose crystal

structureslack inversion symmetry,and yetboth are

strong candidates for spin-triplet superconductiv-

ity.Other noncentrosymmetric pressure-induced su-

perconductors CeIrSi

[154], CeIrSi

[153], CeNiGe

[155], and Ce

[157], have also been recently

reported.

In UIr, ambient pressure ferromagnetism with

= 46 K can be suppressed with 2.6 GPa of pres-

sure[167,168],leading to ferromagnetismcoexisting

with superconductivity against the background of

a noncentrosymmetric orthorhombic crystal struc-

ture below T

= 0.14 K [168].

CePt

Si orders antiferromagnetically below T

2.2 K andexhibits superconductivity below T

=0.75

K [119]. The crystalline lattice is shown in Fig.13.42.

In CePt

Si [119], the upper critical field H

=4-5T

significantly exceeds the paramagnetic limit for sin-

glet superconductivity H

∼ 1T.Thus,thereare

strong suggestions for spin triplet superconductivity

in both UIr and CePt

Si despite the lack of inver-

sion symmetry in the lattice structure. This apparent

coexistence of spin-triplet superconductivity with a

noncentrosymmetric symmetry challenges our un-

derstanding.

In general, Cooper pairs form between electrons

with equal and opposite momentum k.Broken inver-

sion symmetry splits the two spin-degenerate bands

forcing the electron with momentum k to pair with

an electron with momentum −k + Q, creating a fi-

nite momentum Cooper pair. It has been predicted

that this should result in a mixing of the spin sin-

glet andspin triplet branches [264].These arguments

were used to explain the lack of superconductivity in

the noncentrosymmetric compound MnSi, which is

a strong candidate for triplet superconductivity due

to the presence of a ferromagnetic quantum criti-

cal point [265]. If only pure triplet superconductiv-

ity cancoexist microscopically with ferromagnetism,

and the noncentrosymmetric symmetry prevents a

pure triplet superconductivity from forming,no su-

perconductivity should be observed at all.

Fig. 13.42. Tetragonal P4mm crystal structure of the heavy-

fermion superconductor CePt

Si, which does not have in-

version symmetry.Figure reproduced from [119].

A recent theoretical paper seeks to resolve the

issue of spin triplet superconductivity in noncen-

trosymmetric crystals. Frigeri et al. show that un-

der certain symmetry conditions that are satisfied

by CePt

Si, the paramagnetic limiting effect is sup-

pressed for both spin-singlet and spin-triplet super-

conductivity,and either type of superconducting or-

der parameter should be possible [265]. Another in-

triguing theory suggests thatthe paramagnetic limit-

ing fieldislifted in CePt

Si for fields along the tetrag-

onal a-axis through the formation of a helical vortex

phase [266].

Two other d-electron systems have been discov-

ered to our knowledge that exhibit superconductiv-

ity in a noncentrosymmetric crystalline lattice. In

the family R

3−y

(R = La,Y), there is no significant

spin-orbit coupling, so the symmetry of the lattice

does not necessarily affect the Cooper pairing wave

function.In the heavy-fermion compoundCd

heavy-fermion superconductivity occurs in a non-

centrosymmetric lattice; however experimental evi-

dence points to nodeless s-wave superconductivity

and no magnetic order, thus there is no a priori

reason to assume spin triplet superconductivity (see

Sergienko et al. [267] and references therein).

680 M.B. Maple et al.

In UIr, the experimental conditions of high pres-

sure and millikelvin temperatures required to ob-

serve superconductivity have limited the number of

studiestodateonthiscompound.InCePt

Si how-

ever, a number of studies have already been com-

pleted since its recent discovery.We will therefore re-

view some keyexperimental findings here that probe

the question of spin singlet or spin triplet supercon-

ductivity surviving in the absence of parity.

Theexperimentalevidencetodateprovidescon-

ﬂicting evidencefor the natureofthepairingsymme-

try in CePt

Si. The Sommerfeld coefficient  = 390

mJ/mol K

results in a value of C/ T

= 0.25-0.55

that is much lower than the BCS weak coupling pre-

diction of 1.43.A similar depressed value of C/ T

has been observed in the spin-triplet superconduc-

tor SrRu

. However, the low value may also indi-

cate that the Fermi surface is divided between super-

conductivity and antiferromagnetism. Substitution

studies of La on the Ce site show a strong suppres-

sion of superconductivity with only 2% La, which

supports a spin-triplet scenario [268].

Neutron scattering results [269] have shown an

ordered moment on the Ce atom of 0.16 

/Ce atom

in the AFM phase, which is strongly reduced from

the Hund’s rule prediction.This most likely indicates

itinerant antiferromagnetism. On the other hand,

well-defined CEF behavior in magnetization and

NMR measurements indicates at least a partial local-

ization of Ce spins[180,269].The Ce momentsareor-

dered ferromagnetically in the basal plane and stack

antiferromagnetically along the c-axis[269].SR ex-

periments show that magnetism occurs in the whole

sample volume, indicating microscopic coexistence

of superconductivity and antiferromagnetism [270].

NMR measurements [269] indicate that CePt

Si is

the first heavy-fermion superconductor to exhibit a

Hebel-Slichter peak in 1/T

T below T

.CePt

Si does

not show the 1/T

∝ T

behavior that is observed

in most other HF superconductors, and the behav-

ior indicates point-nodesin the superconductinggap

function that would be consistent with a spin-triplet

pairing wave function. On the other hand, thermal

transport and magnetic penetration [271,272] ex-

periments show power-law behavior at low temper-

atures indicating line nodes in the superconduct-

ing gap. Recent theoretical work proposes that line

nodes are possible even in the absence of parity in

CePt

Si [267,273,274].The nature ofthe pairing sym-

metry and its interaction with the crystal lattice sym-

metryremainopenquestionsinCePt

Si and we look

forward to many future studies probing this interest-

ing compound.

13.3.6 Non-Fermi Liquid Behavior

At the time of their discovery, nearly all of

the heavy-fermion superconductors were found to

have Fermi liquid properties, i.e., (T) ∝ T

and

C(T)/T ∝ (T) ∝ constant. However, unusual tem-

perature dependences of the physical properties,

such as a linear T-dependence of the electrical re-

sistivity [275], known as non-Fermi liquid behavior,

have been observed in many heavy-Fermion com-

pound since their first observation in UBe

.Inpar-

ticular, the relatively weak magnetic field depen-

dence of the electrical resistivity and specific heat,

and the absence of a quasi-elastic peak in neutron

scattering measurements in UBe

do not follow the

expected behavior of a conventional Kondo effect,

which is thought to be the origin of the heavy-

fermion characteristics. These experimental data led

Cox to propose a two channel quadrupolar Kondo

model, in which the interaction between the electric

quadrupole moments of a U

nonmagnetic doublet

ground state in a cubic crystalline electric field and

the charges of the conduction electrons give rise to

NFL behavior at low temperatures [276,277].

A fewyearslater,NFLbehavior wasobserved [278]

in the Y

1−x

system over a wide range of U con-

centrations x in the paramagnetic phase below the

critical concentration x

=0.2 where the magnetic

transition temperature (later identified as the spin

glass freezing temperature [279,280])vanished. This

discovery started a period of intense research in f -

electron materials in order to further investigate the

occurrence and origin of non-Fermi liquid behavior

[281–284]. Non-Fermi liquid behavior in f -electron

systems is characterized by weak power-law or log-

arithmic temperature dependences of the physical

properties at low temperatures T  T

for most f -

electron systems and can be summarized as follows: