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

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1120 P.S. Riseborough,G.M. Schmiedeshoff,and J.L.Smith

Fig. 19.65. The differential conductivity of a UBe

-Au point

contact at T = 330 mK, as a function of voltage. The height

of the zero-bias anomaly is roughly five times greater than

is expected for an s-wave superconductor. The solid line

shows an estimate of the Kondo contribution to G(V).[Af-

ter W¨alti et al. [444]]

In contrast to the very small magnitude of the zero

bias conductance peak found in UPt

and URu

the magnitude of the conductance peak in UBe

[444] is roughly a factor of 5 times larger than the

theoretical value for an s-wave superconductor.The

enhancement of thezero bias anomaly was attributed

to the formation of an Andreev surface state, which

can occur if there is a non-trivial dependence on the

angle subtended by the nodes of the order parameter

to the normal to the interface [445]. As the bias volt-

age was increased, the zero bias conductance peak

of UBe

was followed by a minimum which was re-

duced from the normal state value and then showed

a sharp increase as it recovered to the normal state

value. The variation of the differential conductivity

withvoltageis shownin Fig.19.65.Thesharpincrease

occurred at a temperature-dependent voltage which

was of the order of magnitude 0.25 mV at 330 mK.

Such sharp changes of the conductance havenot been

observed for s-wave superconductors but have been

predicted to occur at the gap voltage eV = 

for

junctions with anisotropic superconductors [445].If

the temperature dependent characteristic voltage is

interpreted as being the magnitude of the order pa-

rameter 

(T), then UBe

wouldcorrespondtoa

universal ratio 2

(0)/k

of about 6.7 which is far

greater than the BCS value of 3.5, and is also greater

than the ratio predicted for most p-wave phases

(ABM 4.6,polar 4.9) butis more comparable with the

value of 6.2 predicted for the m =0d-wave phase.

Josephson Tunneling

Josephson tunneling involves the tunneling of

Cooper pairs between two superconductors [446].A

d.c.tunnel current may appear even in the absence of

an applied voltage but also appears as an a.c. current

if a voltage is applied. The a.c. current is expressed

in terms of the phase difference between the wave

functions of the two superconductors

I(t)=I

sin



2eVt



+ '



. (19.239)

The phase difference not only has contributions from

the intrinsic phases of the superconductors but also

has contributions from any magnetic vector poten-

tial that may be present at the junction.When inte-

gratedover the finitesizeof the contact,this term can

give rise to a Frauenhofer diffraction pattern [447].

The leading order contribution to the Josephson cou-

pling can be calculated using second order perturba-

tion theory [446]. To lowest order there should be

no Josephson current between a singlet and triplet

superconductor [448], since spin is conserved in the

tunneling process.However,even in this case,spinor-

bit interaction and higher order processes may pro-

duce a non-zero Josephson current. If the tunneling

is due to a fourth order process, the Josephson fre-

quency is expected to be given by 4eV/h.Sincethe

presence of the surface locally violates certain rota-

tional symmetries contained in the point group, it

is possible for relatively large Josephson currents to

ﬂow between superconductors with different orbital

symmetries. In fact, the magnitude of the order pa-

rameter at the surface may even vary with the relative

orientation of the boundary and the node directions.

Due to this, it is possible to extract the symmetry of

the order parameter from Josephson tunneling mea-

surements [449]. However, the Josephson effect is a

very weak effect and sensitive to impurities and in-

homogeneities, since the total coupling energy for a

conventional Josephson junction is estimated to be

only of the order of 1 eV. For the heavy-fermion sys-

tems, the preparation and characterization of good

19 Heavy-Fermion Superconductivity 1121

surfaces and junctions represent formidable experi-

mental problems.Therefore, there has been only lim-

ited progress in the investigation of the Josephson

effect in heavy-fermion superconductors.

A d.c. Josephson current was found to exist be-

tween CeCu

and Al [438]. The Josephson current

vanished at the T

of CeCu

with applied mag-

netic fields greater than the upper critical field of Al

but smaller than the critical field of CeCu

.Hence,

the tunneling current involved the superconducting

phases of both materials. The application of mag-

netic fields produced irregular Frauenhofer diffrac-

tion patterns which were smeared out. The smearing

was attributed to the irregular geometry of the junc-

tions. As the observed value of the critical current

was quite large, having a maximum magnitude that

is 80% of the BCS value, it appears as if the Josephson

coupling is of second order [450] and not of fourth

order. Furthermore, sinceAl is known to be a singlet

superconductor,thelarge coupling strength indicates

that CeCu

is also a singlet superconductor.

No Josephson currents were found to ﬂow across

weak links between UPt

surfaces [438]. However,

a Josephson current, with an irregular Frauenhofer

pattern and well defined Shapiro steps was observed

between UPt

and Nb in a superconductor–normal

metal–superconductor junction [451]. The critical

current measured for junctions where the current

ﬂow is primarily along the c -axis of the UPt

single

crystal is significantly largerthan for junctionswhere

the current ﬂows along the basal plane. The critical

current temperature relations show clear kinks at the

lower critical temperature T

. For temperatures be-

low T

, the slope of the critical current-temperature

relation is large for current ﬂow along the c direc-

tion, and smaller for current ﬂow in the basal plane.

For temperatures above T

, the slope of the criti-

cal current temperature relation for ﬂow in the basal

plane changes to a larger value but the curve rep-

resenting ﬂow along the c-axisisratherﬂat.The

change in the anisotropy at T

seems to indicate

that the anisotropy in the current ﬂow is related to

the unconventional nature of the superconductivity

in UPt

. It should be noted that, since the anisotropy

measurements were made on junctions with different

surfaces, the apparent anisotropy possibly could re-

ﬂect a difference in the properties of the interfaces.A

much larger anisotropy was inferred from measure-

ments of Josephson currents through point contacts

between URu

and Nb [452] where no current was

observed for contacts aligned parallel to the c-axis

but finite currents were found for contacts aligned

parallel to the a–b directions. The absence of Joseph-

son currents ﬂowing along the c-axis could, however,

have many other possible causes.

No Josephson tunneling currents were observed

to ﬂow across a weak link between two surfaces of

UBe

. In a junction between Al and UBe

,super-

conductivitywas introducedinUBe

by the proxim-

ity effect for temperatures below the superconduct-

ing transition of Al but above the superconducting

transition for UBe

. In this temperature regime, a

weak Josephson current was observed between these

materials. A proximity induced Josephson current

was also found to occur across a point contact be-

tween UBe

and Ta at temperatures below T

for

Ta and ab ove T

for UBe

. This Josephson current

was destroyed as the temperature was lowered be-

low the T

of UBe

[453, 454]. This was taken as

evidence that the superconducting order parameter

of UBe

competes with the superconductivity of Ta

and hence, are of different symmetry. The analysis

presented in [454] suggested that the Cooper pairs

were spin-triplet, however, spin-singlet d-wave pair-

ing would be more consistent with the experimental

observations. In particular, when illuminated with

microwave radiation of frequency !,thecurrentd.c.

voltagerelationexhibitedShapiro stepsof magnitude

V = !/2e,insteadofV = !/4e as would be

expected for a fourth order Josephson coupling be-

tween singlet and triplet superconductors [448,455].

The interpretation of these experiments is enigmatic

since a conventional Josephson tunneling current,

with an irregular Frauenhofer pattern (see Figs.19.66

and 19.67) and conventional Shapiro steps (shown

in Fig. 19.68), was observed at temperatures below

0.94 K between UBe

and Nb in a fairly well defined

superconductor–normal-metal–superconductor ge-

ometry [456]. The Shapiro steps were observed to

have a magnitude of V = !/2e, as expected for

a second order Josephson coupling between two sin-

glet superconductors. On the other hand,the magni-

1122 P.S. Riseborough,G.M. Schmiedeshoff,and J.L.Smith

Fig. 19.66. The field dependence of the Josephson current

through a UBe

-Cu-Nb junction.The irregular Frauenhof-

fer pattern suggests that the junction is not uniform. The

inset shows the I − V curve showing the critical Josephson

current. [After Shibata et al. [456]]

Fig. 19.67. The temperature dependence of the critical

Josephson current J

and the junction resistance R,near

for a UBe

-Nb junction.After Shibata et al. [456]

tudes of the critical currents were remarkably small,

being two orders ofmagnitude smallerthan thecriti-

cal currentsobservedin UPt

.However,as the critical

currents were limited by heating, it was not possible

to determine whether the magnitude of the Joseph-

son coupling in UBe

was consistent with the tun-

neling being due to a fourthorderprocessor a second

order process.

Fig. 19.68. The I(V)relationforaUBe

-Nb junction,show-

ing conventional Shapiro steps when an ac current of fre-

quency 100 kHz is superimposed on the dc current. [After

Shibata et al. [456]]

19.4.3 Dynamic Magnetic P roperties

Nuclear Magnetic Resonance

As previously mentioned, nuclear magnetic reso-

nance experiments yield several important quanti-

ties: the Knight shift and the nuclear spin relaxation

rates which are 1/T

the longitudinal relaxation rate,

and the transverse relaxation rate 1/T

.Thechanges

in behaviorof these quantitiesonentering the super-

conducting phase does give information about the

nature of the density of magnetic excitations in the

superconductor.

The Knight Shift

The Knight shift has many contributions, one im-

portant contribution provides a measure of the lo-

cal part of the static susceptibility of the itinerant

electrons which produces a magnetic polarization

at the nuclear site. For heavy-fermion superconduc-

tors, the normal state Knight shift is expected to be

dominated by the enhanced Pauli spin susceptibil-

ity, though spin-orbit coupling will introduce an or-

bital part. However, in the superconducting state,the

Knight shift is expected to be dominated by the dia-

19 Heavy-Fermion Superconductivity 1123

magnetic susceptibility produced by the supercur-

rent shielding the external field. The Pauli spin sus-

ceptibilitywill be modified by the superconductivity,

and provide informationabout the pairing.The zero

field susceptibility is defined as a derivative of the

magnetization (T)=(

∂M

∂H

). The magnetization that

is produced by the electronic spins aligning with a

magnetic field applied along the z-axis is given by



g







f (E

↑,k

)−f (E

↓,k

)



, (19.240)

whichisexpressedin termsof the Fermi-distribution

for quasi-particles with spin  and quasi-particle en-

ergy E

,k

. The field dependence of the quasi-particle

energies depends on the type of spin pairing, so we

shall discuss the different types of spin pairings sep-

arately.

Singlet Pairing

For singlet pairing, the magnetic field couples to the

spins of the quasi-particles via the Zeeman ener-

gies. As can be seen from inspection of the matrix

in Eq. (19.58), only the time reversal partners pair

when

−→

d (k

) ≡ 0. The quasi-particles consist of bro-

ken pairs, i.e., electrons of spin  and holes of spin

−. Since a down-spin hole has the same Zeeman

energy as an up-spin electron, the quasi-particle en-

ergies depend on the magnetic field through

,k

= E

H=0,k

−



g

 H



(19.241)

and so the spin susceptibility takes the usual form



S=0

(T)=4



g



∞



dE 

S=0

(E)



−

∂f

∂E



(19.242)

The BCS density of states should be used in the

above expression, for an s-wave superconductor. In

this case, the susceptibility tends to zero as T →

0 in an exponentially activated way 

BCS

(T) ∼

exp[−

T].The exponential vanishing of the spin

susceptibility occurs as the electrons form singlet

Fig. 19.69. The calculated spin susceptibilities of singlet

superconducting phases. The susceptibilities are normal-

ized to the normals state susceptibility. The susceptibil-

ities in the singlet superconducting phases all vanish as

T → 0. The susceptibility of the BCS phase is given by the

Yosida function, which vanishes exponentially at low tem-

peratures. The susceptibilities of the isotropic and m =1

d-wavephasesvaryaspowersofthetemperature,atlow

temperatures

pairs in the ground state,and the finite spin moment

is caused by thermal population of quasi-particles

[457]. For a singlet superconductor that has point

nodes, the density of states varies as E

at low ener-

gies and, therefore, one expects that 

S=0

(T) ∝ T

for T  T

. Whereas for a singlet superconduc-

tor which has line nodes, the density of states is

given by 

S=0

(E) ∝ E at low energies, hence one ob-

tains 

S=0

(T) ∝ T. The temperature dependence of

the calculated spin susceptibility expected for vari-

ous singlet spin superconducting phases is shown in

Fig. 19.69.

Thus, in the spin singlet phases, the spin suscep-

tibility could be expected to vanish as T → 0. How-

ever, spin-orbit coupling will produce a residual sus-

ceptibility that depends on the ratio of the super-

conducting coherence length, 

,tothemeanfree

path due to spinorbit scattering, l

. In the presence

of spin-orbit coupling, the spin is no longer a good

quantum number for the single-particle eigenstates

and the spin-up and spin-down states are mixed. In

the limit that the strength of the spin-orbit coupling

1124 P.S. Riseborough,G.M. Schmiedeshoff,and J.L.Smith



−→

L .

−→

S is so large that   

, the average value

of 

for a single particle state tends to zero. The

spin susceptibility is, therefore, reduced. From the

Kramers–Kronig relations, one observes that a sig-

nificant contribution to the normal state (T)comes

from single-particle states with excitation energies

of the order of the spin-orbit scattering rate /

which is,by assumption, greater than 

.Asanopen-

ing up of a superconducting gap at the Fermi energy

is not expected to change the contribution of these

higher energy states, one finds that the susceptibility

in the superconducting state can remain comparable

in magnitude to the normal state value. According

to Anderson [458,459], the normalized susceptibil-

ity should have the two limits. The limit of strong

spin-orbit scattering is defined by l

 

,where



= v

/

and l

= v



, and the limit of weak

spin-orbit scattering is defined by l

 

.Inthe

limit of strong spinorbit scattering, the normalized

susceptibility is given by [460]



BCS

(0)



≈ 1−

8 l

3

. (19.243)

For weak spin-orbit scattering, the susceptibility ra-

tio is given by



BCS

(0)



≈

3

. (19.244)

Hence, the existence of a partial Meissner effect at

T = 0 does not necessarily imply the presence of

either triplet or gapless superconducting phases.

Triplet Pairing

For triplet pairing,theabove argument indicates that

a finite susceptibility could be expected to arise due

to the S = 1 spin of the Cooper pairs. The way the

susceptibility depends on the orientation of the field

−→

H relative to the direction of

−→

d ,for a direction of

−→

independent of k

, is described below.

First,if

−→

d is parallel to

−→

H for all k

,then,as

−→

H is the

direction of quantization of  ,

−→

d is parallel to ˆz and

so S

= 0. Since d

is the only finite component of

the order parameter, the off-diagonal blocks in the

mean-field Hamiltonian Eq.(19.58) are proportional

to 

.And so,asin the singlet case,only the same time

reversal partner states are coupled. Thus,one obtains

the same Zeeman type of couplingas the singlet case,

,k

= E

H=0,k

−

(g

 H), and the susceptibility is

given by the same expression as Eq.(19.242) but with

the appropriate triplet density of states.



S=1,

(T)=2



g







−

∂f

∂E



(19.245)



g



∞



dE

S=1

(E)



−

∂f

∂E



Hence, the spin susceptibility has a linear T varia-

tion for a density of states with lines of nodes or a T

variation with point nodes.

Second, if

−→

H is perpendicular to

−→

d for all k

,then,

−→

d is perpendicular to ˆz, the pairs are in a linear

combinationof states with S

= ±1.The off-diagonal

blocks of the mean-field Hamiltonian are of diago-

nal form andonly coupleelectrons andholes withthe

same spin directions. Then, the quasi-particle ener-

gies are given by

,k



e(k)− −



g

 H





(k)+i d

(k)



(19.246)

Furthermore,since



∂E

,k

∂H



=−



g





∂E

,k

∂e



(k)



, (19.247)

the expression for 

S=1,⊥

(T) has the form



S=1,⊥

(T)=







g





()

+∞



−∞



∂E



∂e



−

∂f (E



)

∂E







g



() . (19.248)

Hence, in this case, the uniform static spin suscepti-

bility retains the normal state value. In general, for

triplet pairing, the orientational average of 

S=1

(T)

will have a magnitude of 2/3 the normal state suscep-

tibility.

19 Heavy-Fermion Superconductivity 1125

The sum of the quasi-particleenergiesare lowered

when

−→

H is perpendicular to

−→

d (k

).Foranisotropic

system like

He, and a pairing vector which has a di-

rection independent of k

such as in the ABM phase,

the vector

−→

d will rotate to remain perpendicular to

the field and the susceptibility will remain constant.

On the other hand, heavy-fermion systems are ex-

pected to have strong spin-orbit coupling and crys-

talline anisotropy, therefore, the direction of

−→

d is

expected to remain constant when

−→

H is applied. If

the material has an anisotropy such that spins have

an easy axis or plane, then 

S=1

for that phase will be

reduced if

−→

H tilts away from the axis or plane.In such

cases of strong anisotropy, the susceptibility tensor

is given by the directional average



i,j

S=1

(T)=2



g







−

∂f

∂E



(19.249)



∗

(k)d

(k)

∗

(k).d(k)



∂E

∂e



i,j

−

∗

(k)d

(k)

∗

(k).d(k)



In the BW phase the zero temperature susceptibility

is 2/3 of the normal state value, this occurs as the k

averaged value of

−→

has one component parallel to

−→

H which is suppressed,the other two components re-

tain their normal values. The calculated spin suscep-

tibilities expected for various triplet superconduct-

ing phases is shown in Fig. 19.70. The above discus-

sion indicates that, if the behavior of the hyperfine

field is known (perhaps extrapolated from the nor-

mal state),the temperature dependence of the Knight

shift in the superconducting state can yield informa-

tion about the nature of the pairing.

In pure UBe

, the muon Knight shift [320] ex-

hibits a large reduction below T

with a magnitude

similar to that found in weak coupling s-wave su-

perconductors.The Knight shifts found for Th [320]

and B doped samples [461] show no appreciable

changes on entering the superconducting states

which is consistent with the formation of spin-

singlet Cooper pairs but with strong spin-orbit

scattering or gapless phases caused by impurity

scattering. The anisotropic Knight shift observed

in CeCu

[462, 463] undergoes a significant re-

duction in the superconducting state, similar to the

Fig. 19.70. The calculated spin susceptibilities of triplet su-

perconducting phases. The susceptibility for the BW phase

is isotropic. The susceptibilities of the ABM and polar

phases are anisotropic, if the order parameter is pinned by

the lattice. The susceptibilities vanish at low temperatures

for applied fields which are parallel to the Cooper pair spin

but remain unaffected for fields oriented perpendicular to

the spin

behavior observed in UBe

. The reduction found in

CeCu

is consistent with the vanishing of the spin

susceptibility at zero temperature, as expected for a

spin-singlet superconducting phase.In CeCoIn

[62],

the decrease of the

Co Knight shift in the supercon-

ducting state is large for fields along the c-axis and

is hardly noticeable for fields in the perpendicular

directions. However, the lack of any decrease in the

perpendicular directions was attributed to a large

temperature independent contribution to the sus-

ceptibility from the Co 3d orbitals. The observation

of a smaller but isotropicdecrease in the Knight shift

115

In below T

is supportive of a superconducting

state which involves spin-singlet Cooper pairs.

The Knight shift has been measured in the super-

conducting state of single crystals of UPt

[464,465].

For large fields [464], no appreciable change was ob-

served on entering the superconducting phase for

all directions of the applied field. However, a later

study [465] showed a very small decrease occurred

for fields aligned along either the b-axis or c-axis but

only forfields such that H

< 5kOeandH

< 2kOe.

This was interpreted as showing evidence for spin-

1126 P.S. Riseborough,G.M. Schmiedeshoff,and J.L.Smith

Fig. 19.71. Temperature dependence of the muon Knight

shifts in UPd

, for fields applied parallel and perpen-

dicular to the c-axis. [After Feyerherm et al. [466]]

triplet pairing, in which the superconducting vector

order parameter is pinned to the lattice for small ap-

plied fields but for sufficiently large fields is free to

rotate so as to follow thedirection of theapplied field.

This explanation rests on the unjustified assumption

of very weak spin-orbit coupling in UPt

and also

does not explain the extremely small magnitude of

the change of Knight shift that occurs when the or-

der parameter is pinned. However, since the crystals

were of very high quality,the absence of a reduction

of the spin susceptibility at low temperatures is not

attributable to spin-orbitor to impurity scattering in

a spin-singlet superconducting phase.

Although the normal state susceptibility and the

SR Knight shift of UPd

are anisotropic (see

Fig. 19.71), the small and anisotropic change in the

Knight shift which occurs below T

has been taken as

evidence that the electrons form spin-singlet Cooper

pairs. An analysis of the normal state data indicates

that part of the anisotropy originates in the cou-

pling. The dipole contribution is estimated to be

three times stronger than the hyperfine field cou-

pling. Although a small anisotropic change in the

Knight shift is observed as the temperature is de-

creased below T

, the analysis attributes the change

to an isotropic reduction of the f spin susceptibil-

Fig. 19.72. The calculated temperature dependence of the

spin-lattice relaxation rate 1/T

for the BCS, BW,ABM and

polar phases. The rates are normalized to the normal state

relaxation rate at T

. A Hebel–Slichter peak occurs below

for the BCS and BW phases, but is absent for the ABM

and polar phases

ity [466]. The reduction has a magnitude of 11%

which is only compatible with spin-singletpairing of

the quasi-particles if one partitions the f suscepti-

bility into separate quasi-particle and local moment

contributions. This is in direct contrast with con-

clusions from Al NMR Knight shift studies of the

isostructural compound UNi

. The Knight shift

measurements of UNi

have been interpreted as

indicating that the quasi-particles form triplet-spin

Cooper pairs [467]. Since the crystals are not axially

symmetric, the crystals contain two inequivalent Al

sites. As a consequence, the experiments on UNi

show two distinct NMR peaks for magnetic fields

applied parallel to the a-axis. In UNi

,theKnight

shift predominantly comes from the hyperfine cou-

pling which has a similar magnitude to that found

in UPd

. However, on reducing T below T

,the

Knight shift observed for one peak did not show a

change comparable to that found in UPd

and

hence, it is argued that these measurements provide

evidence for triplet-spin superconductivity. Due to

the much larger 1/T

contribution to the peak width

for fieldsalongthe c-axis,these experiments were un-

able to determine if the Knight shift exhibits the full

anisotropic Meissner behavior expected from a spin-

19 Heavy-Fermion Superconductivity 1127

Fig. 19.73. The temperature dependence of the

Be nuclear

spin relaxation rate 1/T

in U

1−x

,forx =0and

x =0.033. The solid line is a fit to a T

power law below T

and the dashed line is the normal state Korringa law.[After

MacLaughlin et al [25]]

triplet superconductor. Further results are shown in

Figs. 19.72 and 19.73.

Longitudinal Nuclear Spin Relaxation

The temperature variation of the Spin Lattice relax-

ation rate 1/T

can be drastically changed at the on-

set of superconductivity.Thisoccursasa result of the

change in the quasi-particle excitation spectrum but

alsoisduetoamodificationofthematrixelements

for the coupling of the quasi-particle excitations to

the nuclear spins. As the modification of the ma-

trix elements of the coupling interaction, known as

the coherence factor,involves the gap function,1/T

could provide additional information about the sym-

metry of the superconducting phase. The relaxation

rate for a spin distribution aligned along the z-axis

is driven byspin-ﬂip ﬂuctuations, andis given by the

expression

1/T

= k

T(g

)



A(q)



Im

+,−

(q; !

)

!



(19.250)

where !

is the nuclear Larmor frequency,and A(q)

is the averaged strength of the local hyperfine field.

The factor Im

+,−

(q; !) represents the density of

states for magnetic spin-ﬂip ﬂuctuations. A typical

valueofthenuclearLarmorfrequencyis!

∼ 10

−7

meV and so, for all but in systems with exceptionally

slow spin-ﬂuctuations, one may approximate !

= 0. This approximation could be expected to

fail near a quantum critical point. Just to illustrate

the nature of the coupling to the superconductivity,

we shall replace the full susceptibility by the lowest

order polarization part or quasi-particle susceptibil-

ity.

The first step in evaluating the imaginary part of

the susceptibilityis to express the spin-ﬂip operators

in terms of the four component fields ¦ (k

), so the

spin raising operator ˆ

(q)isgivenby

ˆ

(q)=¦

†

(−k − q)





0−

−



¦ (−k

) , (19.251)

where 

and 

−

are the 2 × 2 Pauli spin matrices.

The spin lowering operator ˆ

−

(q)isgivenby

ˆ

−

(q)=¦

†

(−k − q)





−

0−



¦ (−k

) . (19.252)

In general, the matrix ˜

corresponding to a spin

operator ˆ

is a 2 × 2 block diagonal matrix with

the upper and lower diagonal elements given,respec-

tively,by 

and (i

)

(−i

), where the lower diag-

onal element reﬂects the effect of time reversal on

the spins. The lowest order contribution to the sus-

ceptibility tensor 

˛,ˇ

(q; !) is expressed in terms of

the Fourier transform of the imaginary time single-

particle Green’s functions. These Green’s functions

are defined in terms of theWick’s time ordered prod-

uct of two four-component fields

G(k

; )=−< |

T ¦ (k, )¦

†

(k, 0)| >. (19.253)

1128 P.S. Riseborough,G.M. Schmiedeshoff,and J.L.Smith

The susceptibility tensor can be written as the ana-

lytic continuation onto the real frequency axis of the

trace



˛,ˇ

(q; 

)=k



g





n,k

trace

× G(k

+ q; i(!

+ 

)) ˜

G(k; i!

) ˜

(19.254)

where !

are the fermion Matsubara frequencies,

!

= k

T(2n +1)and

is a boson Matsubara

frequency. For unitary states where

is diagonal,

the quasi-particle contribution to the Fourier trans-

formed Green’s function can be written as

G(k

; i!

)=−Z

−1

(i!

)



+ E

. (19.255)

Hence,on evaluating Eq.(19.254) and continuing the

imaginary frequency i

onto the real ! axis,the en-

ergy difference components of the imaginary part of

the susceptibility tensor are given by the expression,





˛,ˇ

(q; !)



= Z

−2



g





+∞



−∞



f (E)−f (E + !)



trace



(E +

) ˜

(E + ! +

) ˜

4E(E + !)





ı(E − E

)ı(E + ! − E

k+q

)

+ ı(E + E

)ı(E + ! + E

k+q

)



(19.256)

This represents a quasi-elastic process involving

scattering from the thermal population of quasi-

particles.The factorsin front of the spinoperators,as

modified by the matrix nature of ˜ ,producetheco-

herencefactors.Thecoherence factorsdifferbetween

the singlet and triplet pairing cases.

Singlet Pairing

In this case, the spin-ﬂip excitation spectrum is ob-

tained as





+,−

(q; !)



= Z

−2



g





+∞



−∞



f (E)−f (E + !)





(k

)(k + q)+D(k)D(k + q)

k+q





ı(E − E

)ı(E + ! − E

k+q

)

+ ı(E + E

)ı(E + ! + E

k+q

)



. (19.257)

The two summations over k

and k + q in Eq.(19.250)

when combined with Eqn. (19.257) have the effect

that the bi-linear term in (k

)=e(k)−,which

is anti-symmetric about the Fermi energy, vanishes.

Also, since for singlet superconductivity, the order

parameter is an even function of k

these terms are

retained. Then, in the s-wave phase, the relaxation

rate is given by

/T

= k

−2

(g

)

∞





−

∂f

∂E















BCS

(E)

, (19.258)

wherewehaveset!

=0,andA is the local hy-

perfine field. The above expression indicates that a

peak in 1/T

should occur just at the superconduct-

ing transition.The peak occurs since the BCS density

of states diverges at E = 

, which yields a diver-

gence in the integrand in the superconducting state.

The logarithmic divergence of 1/T

in the supercon-

ducting state is actually suppressed by small residual

anisotropy in the gap and also by the quasi-particle

lifetimes. As the derivative of the Fermi function



−

∂f

∂E





E=

(19.259)

19 Heavy-Fermion Superconductivity 1129

has its largest value in the superconducting state at

T = T

, the above argument indicate that a peak

in 1/T

occurs just at T

. The resulting experimen-

tally observed peak is known as the Hebel–Slichter

peak [468], and is often seen in weak-coupling s-

wave superconductors. The peak actually occurs at

roughly 0.8 T

. This reduced temperature and the

suppression of the logarithmic divergence is well de-

scribed by interactions between thermally excited

phonons and the quasi-particles [469].Fibich found

that the temperature dependence of the rate could be

expressed as



∝ 2k

Tf (

)





(1 − f (

))

× ln



2

 (T)



4/3



, (19.260)

where  (T) represents the imaginary part of the gap

which is a characteristic energy for quasi-particle

scattering from a thermal distribution of phonons,

 (T) ∼ (k

/!

. However, in strong-coupling

s-wave systems, quasi-particle interactions and life-

time effects can substantially suppress the peak. At

low temperatures, T  T

, the relaxation rate is ex-

pected to show a thermally activated temperature

variation, 1/T

∝ exp[−

T].

Triplet Pairing

Thecoherence factorsin thetripletor p-wave pairing

phases should be replaced by





(k

)(k + q)−(ˆz .

−→

d (k))(

−→

∗

(k + q) . ˆz)



k+q



(19.261)

Since (k

)=e(k)− is an odd function about the

Fermi energy and

−→

d (k

) is an odd function of k,the

bi-linear terms vanish, leaving a coherence factor of

unity. Hence, the rate is given in terms of an inte-

gration over an appropriate quasi-particle density of

states squared for the p-wave phase,

/T

=  k

−2

(g

)

∞





−

∂f

∂E





S=1

(E)

(19.262)

We note that the weak logarithmic divergence of the

density of states in the ABM phase is integrable, and

the Hebel–Slichter peak is also completely absent in

the polar phase.

At low temperatures in the BW phase, the same

type of exponentially activated behavior is recovered

as in the BCS phase. The low temperature form of

1/T

in the ABM phase is given by

/T

=4!

(g

)

−2

()



(19.263)

and in the polar phase

/T

= 



(g

)

−2

()

2

, (19.264)

where the inequality T  T

should hold for both

expressions. Thus, the characteristic energy depen-

denceassociated withthe distributionof nodes could

govern the low temperature quasi-particle contribu-

tion to the spin-latticerelaxationrate.Also,thequasi-

particle contribution to the relaxation rate should

remain unrenormalized.However,it should be noted

that only the lowest order polarization part of the

susceptibility has been considered but the higher or-

der interactions between the quasi-particles and ver-

tex correctionsarealso important.Thevertexcorrec-

tions may be expected to partially cancel the effect

of the self-energy which is responsible for Z.Thein-

teractions are responsible for adding the low-energy

collective spin-ﬂuctuations to the continuum of sin-

gle quasi-particle Stoner excitations.

The spin-lattice relaxation rates of the heavy-

fermion superconductors UBe

[25], UPt

[470],

CeCu

[472], UPd

[471] and CeIrIn

[62,63]

do not show any evidence of a Hebel–Slichter peak at

, but instead show a cross-over between a linear T-

like Korringa variation above T

to an approximate

variation in the superconducting state. In UPt

the T

law is only followed closely in the restricted

temperature range between 0.1 and 0.3K,whichis

far below the superconducting transition tempera-

ture (T

∼ 0.5 K). However, in UBe

,UPd

and

to a lesser extent CeCu

,theT

dependence holds

right up to T

.Sincethegap

(T)isexpectedtobe

rapidlyvaryingwithT in this temperatureregion,the

variation should not be interpreted in terms ofthe