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 661

serves as an example to illustrate how various spec-

troscopic techniques have been used to elucidate the

nature of heavy-fermion superconductivity.

13.3.2 Normal-State Properties

Speciﬁc Heat

For most metals, the normal-state specific heat C at

low temperatures can be described by the sum of an

electronic contributionthat varies linearly with tem-

perature T, and a phonon term that varies as T

, i.e.,

C(T)= T + ˇT

. (13.16)

This equation is validfor temperatures well belowthe

Fermi and Debye temperatures. Due to significant

correlations between electrons at the Fermi-surface,

the physics of heavy-fermion materials is most ef-

fectively described in terms of quasiparticle excita-

tions. The effective mass of these quasiparticles can

be much larger than that of free electrons,and is pro-

portional to the electronic specific heat coefficient

 = C/T.

For an ordinary metal,  is close to the value

for a free-electron gas 

=(

/3k

)N(E

), where

N(E

) is the density of states at the Fermi Energy;

e.g., for sodium,  /

≈ 1.3. In heavy-fermion com-

pounds,  /

can range from 10–1000. In these ma-

terials, strong electronic or magnetic correlations

between the electrons at the Fermi surface lead to

high effective masses up to several hundred times

the free-electron mass. The effective mass is propor-

tional to the density of states at the Fermi energy,

and therefore to the specific heat coefficient  .Val-

ues of  (T = 0) for various heavy-fermion materials

are shown in Table 13.2.

For many heavy-fermion compounds, C(T)de-

viates from Eq. (13.16) at low temperatures, such

that  (T)=C(T)/T increases with decreasing tem-

perature, sometimes reaching a maximum. The spe-

cific heat of the heavy-fermion compounds UBe

and CeCu

, plotted as C/T vs T

are shown in

Fig.13.22.Thevalueof  of both compoundsdiverges

at temperatures below ∼ 4K,reachingavalueof1

J/mol K

as T approaches 0 K.This may be an indica-

tion of NFL behavior,to be discussed in Sect. 13.3.6,

0.2

0.4

0.6

0.8

1.2

0 10203040506070

CeCu

UBe

K lom/J( T/C

)

UPt

0.3

0.35

0.4

0.45

0 100 200 300

Fig. 13.22. Normal-state specific heat C(T) plotted as C/T

vs T

for CeCu

(circles) [108] and UBe

(triangles)

[136]. The inset shows C/T vs T

for the compound UPt

[137], where the solid line is a fit to Eq. (13.17)

or a low effective Fermi temperature, below which

the density of states at the Fermi energy increases.

For the heavy-fermion systems UPt

and UAl

,the

specific heat at low temperatures canbe describedby

C(T)= T + ıT

ln T + ˇT

(13.17)

for T < 20 K. The additional ıT

ln T term was first

observed in

He [171], and is predicted [172,173]

for long range spin ﬂuctuations. The specific heat of

UPt

is shown in the inset to Fig. 13.22, where the

solid line is a fit to Eq. (13.17).

Magnetization

The magnetic susceptibility in heavy-fermion sys-

tems tends to follow a Curie–Weiss law at high tem-

peratures

(T)=



eff

(T − 

)

, (13.18)

where N

is Avogadro’s number, k

is Boltzmann’s

constant, 

eff

is the effective magnetic moment, and



is the Curie–Weiss temperature. The effective

moment is often reduced somewhat with respect to

the free ion value of the rare earth or actinide ion,

which can be attributed to valence ﬂuctuations, the

Kondo effect, or crystalline electric field effects. The

662 M.B. Maple et al.

Table 13.2. Normal-state properties of heavy-fermion superconductors at ambient pressure, unless noted otherwise. 

is the electronic specific-heat coefficient at low temperatures, taken at T

. % (300 K) and %

are the room temperature

and low temperature resistivities, respectively. T

is the magnetic ordering temperature at ambient pressure. The order

type refers to antiferromagnetism (AFM), ferromagnetism (FM) or spin density wave (SDW). T

coh

is the temperature at

which the screening of local moments by the Kondo effect reduces magnetic scattering, and is determined from a gradual

shoulder or maximum in the electrical resistivity.

Compound 

300K



Order T

coh

(mJ/mol K

)(‹§ cm) (‹§ cm) (K) type (K)

CeCu

700 [108] 90 [138] 4.8 [139] 0.8 [140] SDW 20 [138]

CePd

47 2 [141] 10 [142] AFM

CeRh

80 17 6

35 [143] AFM

CeNi

350 [112] 14/16 2.6/0.9 [144] 50 [145]

CeCu

225

120

[146] 4.15 [147,148] AFM

CeIn

0.7 [149] 10.1 AFM 50 [150]

CeRhIn

200

[151] 33

< 1 3.8 AFM 25

[115]

CeIrIn

720 40 1.25 50 [116]

CeCoIn

290 17 3 30 [117]

RhIn

∼400

[152] 70 65 2.8 AFM 10

[118]

CePt

Si 390 [119] 90 5 2.2 AFM 90

CeRhSi

120 [120] 75 [153] 0.4 1.6 AFM

CeIrSi

120 [120] 140 [154] 1.4 5.0 AFM ∼ 200

CeNiGe

40 [121] 250 [155] 1.5 5.5 AFM ∼ 120

90 [156] 180 [157] 1.9 4.8/4.2 [156] AFM ∼ 100

PrOs

350

[123,158] 200 6

UBe

900 [159] 140 200 [136] 10

UPt

420 [160] 225/100 [161] 0.5 6 [162] AFM 10

URu

65 [126,163] 330/170 30/30 [164] 17.5 AFM 50

UPd

150 [127] 140 10 14 AFM 75

UNi

120 [128] 1000 25 4.5 SDW 100

UGe

30 [165,166] 200 0.2 53 FM

UIr 40 [167] 110 [168] 0.44 47 FM

URhGe 160 [169] 2 [109] 10 FM

Fe 280 [170] 50

PuCoGa

[133] 250 20

PuRhGa

[134] 260 25

7.3kbar

160 kbar

16 kbar

determined from normal and superconducting-state properties.

13 Unconventional Superconductivity in Novel Materials 663

Curie–Weiss temperature is generally large and neg-

ative due to valence ﬂuctuations or Kondo interac-

tions if the temperature range of the Curie–Weiss

fits is much larger than the CEF splitting of the lan-

thanide or uranium f -electron energy levels. 

found to be a rough measure of the Kondo tempera-

ture, i.e., T

∼ (3 − 4)

At low temperatures (e.g. ∼ 50 K for CeMIn

= Co, Rh, Ir), 300 K for UPt

) the magnetic suscep-

tibility (T) typically deviates from a Curie–Weiss

law. The direction of this deviation varies from com-

pound to compound. For example, UPt

has a larger

susceptibility at low temperatures than is predicted

by the Curie–Weiss law,whereas UBe

has a smaller

susceptibility. There are various mechanisms which

could be responsible for these deviations, including

CEF effects, magnetic ordering, quenching of the lo-

cal moments by the Kondo effect (which would only

explain a suppression of the magnetization),and va-

lence ﬂuctuations.

The Pauli susceptibility of heavy-fermion materi-

als at the lowest temperatures, (0), is much larger

than that of ordinary metals, typically by a factor of

10–100.It is empirically found[174] to scale with the

electronic specific heat coefficient  (0). In Fig.13.23,

 (0)isplottedasafunctionof(0)for variousheavy-

fermion materials. This scaling of the magnetic sus-

ceptibility and the electronic specific heat coeffi-

cient is predicted by Wilson for Kondo systems [175]

where the Wilson–Sommerfeld ratio

R =





eff

(0)

 (0)

(13.19)

is on the order unity for most heavy-fermion com-

pounds. The predictions give R = 1 for the nonin-

teracting electron model, and R = 2 for the Kondo

model [175]. In a more general context the Wilson

ratio is also known as the Sommerfeld ratio.

Many heavy-fermion superconductors order an-

tiferromagnetically (e.g., UPd

, CeIn

)orexhibit

spin glass freezing at low temperatures, and a very

small number also order ferromagnetically (e.g.,

UGe

). The magnetic ordering temperatures of all

the heavy-fermion superconductors are listed in Ta-

ble 13.3. The heavy-fermion compounds that or-

der antiferromagnetically fall into two classes. Com-

Fig. 13.23. Electronic specific heat coefficient  plotted as a

function of magnetic susceptibility  at low temperatures

ona log–logscale forvarious heavy-fermion materials.The

solid line is the free-electron prediction for the Wilson ratio

(R = 1),after [174]

pounds in the first class (e.g., UPd

,UNi

CeIn

and CeRhIn

) have ordered magnetic mo-

ments in the range (0.2−1)

, and appear to have

standard dipolar order parameters. Compounds in

the second class (URu

and UPt

)haveverysmall

ordered moments (∼ 0.01 

), although the N´eel

temperatures of these compounds are comparable

to those of the first class. One possible explanation

for these small moments is quadrupolar ordering, al-

though only the dipolar moment has been detected

by means of neutron scattering experiments. In UPt

and U

1−x

, an additional tiny (0.001 

)mag-

netic moment is associated with the onset of a second

superconducting phase.

Electrical Resist ivity

The electrical resistivity  of heavy-fermion mate-

rials is remarkably different than that of conven-

tional metals. For a typical metal, (e.g., Cu), the re-

sistivity decreases with decreasing temperature due

to electron–phonon scattering for T > 

,where



is the Debye temperature. At low temperatures,

Fermi liquid theory predicts that (T) saturates as

(T) ∝ AT

∝ (T/T

)

, with a coefficient A of the

664 M.B. Maple et al.

Table 13.3. Magnetic properties of some heavy-fermion superconductors. 

and 

eff

are Curie–Weiss fit parameters to

the susceptibility at high temperatures, and 

ord

is the ordered moment determined from neutron diffraction.

(H parallel/perpendicular to c-axis)

Compound (0) 



eff



ord

(cm

/mol) (K) (

)(

)

CeCu

0.082 140 2.68 [176]

CeNi

> 0.009 170 2.6 [145]

CeCu

0.3 [113]

CeRhIn

0.022/0.005 16/−79 2.38 [115] 0.75 [177]

CeIrIn

0.018/0.009 12.5/−67 2.28 [116]

CeCoIn

0.014/0.008 −83/−54 2.59 [117]

RhIn

0.013/0.016 12/−40 2.38/2.32 0.55 [178]

CePt

Si 0.013/0.012 [179] −45/−75 2.67/2.75 0.16 [180]

CeRhSi

0.095 [120] –128 2.65

CeIrSi

0.032/0.062 [154] –186/−109 2.57/2.64

CeNiGe

0.08 –12 2.58 0.8 [121]

0.044 [156] −52 2.5 0.4 [122]

PrOs

0.06 [123] −16 3.0

UBe

0.015 [174] −53 3.1

UPt

0.19/0.10 [181] −200 3.0 0.01 [182]

URu

0.01/0.06 −65 3.5 0.03 [126]

UPd

0.015 [127] −47 3.2 0.85 [183]

UNi

0.023/0.055 [128,184] 0.2 [185]

UGe

55 [3] 2.7 1.5 [165]

UIr –31 2.91 [167] 0.5 [168]

URhGe 10 [109] 1.8 0.42 [131]

Fe 0.003 [186]

PuCoGa

0.031 −2 0.68 [133]

PuRhGa

0.0015/0.0013 −132/−160 0.85 [134,187]

order of 10

−6

‹§ cm/K

.ThisAT

term is so small

in simple metals that it is often obscured by other

electron scattering processes. In heavy-fermion ma-

terials on the other hand, the AT

term can be up

to five orders of magnitude larger than in ordinary

metals, reﬂecting a low effective Fermi temperature

∗

∼ 10–100 K, and is often observed at low tem-

peratures. A universal relation between the large

magnitude of A and the enhanced electronic spe-

cific heat coefficient  in heavy-fermion materials

was found by Kadowaki and Woods [188]. They ob-

served that most HF compounds obeyed the relation

A/

=1× 10

−5

‹§ cm (mol K/mJ)

,asillustrated

in Fig. 13.24, provided the materials are not too close

to a magnetic instability.Displayed in Fig. 13.25 are

the (T) curves for two typical heavy-fermion com-

pounds UBe

and CeAl

.The resistivity of these and

many other heavy-fermion materials is of the order

of 100‹§cm near room temperature, some 10−100

times that of normal metals, and reﬂects the large

exchange (Kondo) interaction between the local mo-

ments and the conduction electrons.A weak temper-

ature dependence of (T) is most often observed in

these materials at higher temperatures, followed by

13 Unconventional Superconductivity in Novel Materials 665

Fig. 13.24. Kadowaki–Woods plot: coefficient of the T

term in the electrical resistivity A vs electronic specific

heat coefficient  . The line has a slope A/

=1×

−5

‹§ cm (mol K/mJ)

, after [188]

Fig. 13.25. Electrical resistivity  vs T of the heavy-fermion

compounds UBe

and CeAl

. Inset:  vs T

for CeAl

low temperatures, after [136,189,190]

amaximumorshoulderatT

coh

∼ 50–100 K. The

resistivity drops rapidly just below T

coh

due the for-

mation of coherent Bloch states as the lattice of the

local moments of the f -ion sublattice is screened

by the conduction electrons. At the lowest tempera-

tures, the resistivity exhibits Fermi liquid properties,



+AT

,as shown in the inset of Fig.13.25 for CeAl

A departure from the T

behavior of the electrical

resistivity at low temperatures is observed in some

heavy fermion materials, and is referred to as non-

Fermi liquid behavior. This is discussed in detail in

Sect. 13.3.7.

Thermoelectric Pow er

The thermoelectric power S is a sensitive function

of the shape of the density of states near the Fermi

level; therefore, large values of S are often observed

in heavy-fermion compounds, compared to those of

simple metals, due to the presence of a narrow f or

d-band at the Fermi level.A simple expression for the

thermopower due to electron diffusion, that is valid

for T  

,isgivenby

S =



d(ln (E))



, (13.20)

where (E) is the value of the conductivity of the ma-

terial at an energy E.AmaximuminS is expected at a

temperature of the order of the Kondo temperature.

An anomalously large and positive value of S is often

observed for Ce-based heavy-fermion compounds,

with a maximum centered at T

or at the CEF split-

ting if CEF effects are present. For Yb-based heavy-

fermion materials, S is usually large and negative re-

ﬂecting the fact that the electronic configuration of

Yb is the one-hole counterpart to the one-electron

configuration of Ce.

13.3.3 Superconducting-State Properties

Speciﬁc Heat

The specific heat in the superconducting state of

heavy-fermion materials differs markedly from that

of conventional superconductors. Whereas C(T)in

a conventional superconductor varies as C(T) ∝

exp(−/k

T), in accordance with BCS theory, the

666 M.B. Maple et al.

specific heat of heavy-fermion materials can often

be described by a power-law

C(T)=

T + ıT

, (13.21)

where the power-law exponent n ∼ 2−3. Both the

residual specific heat 

and the power law behav-

ior suggest an unconventional superconducting gap

that vanishes at lines (n =2)orpoints(n =3)onthe

Fermi surface.

In heavy-fermion materials, the magnitude of the

specific heat jump C at T

is a large fraction of the

value of C(T

). Values of C/ T

are listed in Ta-

ble 13.4. The quantity C/ T

is predicted by BCS

theory to be 1.43 (weak coupling) or 1.6 (strong cou-

pling). The fact that C/ T

for heavy-fermion ma-

terialsisoftheorderofunity,where is the value

derived from the normal-state specific heat, implies

that the superconductivity in these materials is a

bulk phenomenon, and that the same electrons re-

sponsible for the heavy-fermion behavior in the nor-

mal state are participating in the superconductivity.

The specific heat jumps for the compounds UBe

and CeCoIn

are shown in Fig. 13.26. For CeCoIn

the specific heat jump is unusually large such that

C/ T

=4.5−5 [116,191]; in addition C(T)below

can be fit by Eq.(13.21) with n =2.3and

= 110

mJ/mol K

Fig. 13.26. Superconducting jump in the specific heat plot-

ted as C/T vs T of UBe

(triangles) [136] and CeCoIn

(circles) [191]. The line below T

is a fit to Eq. (13.21)

C =  T + ıT

Upper Critical Field

All of the known superconducting heavy-fermion

compounds are type-II superconductors with upper

critical fields H

of the order of a few tens of kOe

and lower critical fields H

of the order of tens of

Oe, corresponding to values of  ≡ /

∼ 50 − 100,

where  is the penetration depth and 

is the super-

conducting coherence length. These unusually large

values of H

and small values of H

reﬂect the large

effective masses of the superconducting quasiparti-

cles which can be understood by noting that H

∝

1/

∝ 1/m

∗

and H

∝ 1/

∝ m

∗2

.Mostheavy-

fermion superconductorshave a large initialslope of

the upper critical field near T

,−(dH

/dT)

,and

sometimes exhibit a linear T-dependence at lower

temperatures. For example, the initial slope of UBe

is enormous, −(dH

/dT)

= 420 kOe/K [136], as

shown in the inset of Fig. 13.27. Below T =0.6K,

the upper critical field of UBe

is linear in tempera-

ture with a slope of −91 kOe/K (Fig. 13.27). The zero

Fig. 13.27. Upper critical field H

vs T of UBe

. Inset:Ex-

panded view of H

(T)ofUBe

near T

. The line is a fit

to the data yielding a slope −(dH

/dT)

= 420 kOe/K,

after [136]

13 Unconventional Superconductivity in Novel Materials 667

temperature value of the orbital critical field can be

estimated from the initial slope of the upper critical

field by the following formula [203]:

∗

(0) = 0.693T

(−dH

/dT)

, (13.22)

from which the superconductingcoherence length 

can be determined from the relation [204]

∗

(0) = ¥

/2

, (13.23)

where ¥

=2.07 × 10

−7

Oe cm

is the ﬂux quan-

tum. The value of 

can be used to estimate the

Fermi velocity from the relation 

∝ 0.18v

/kT

;

thus, the effective mass can be obtained via m

∗

k

within a spherical-Fermi-surface approxi-

mation. The values of the coherence length and

the effective mass estimated in this fashion for the

heavy-fermion superconductors are collected in Ta-

ble 13.4. Many heavy-fermion superconductors have

anisotropic upper critical fields of the order of

⊥

≈ 1.5nearT

(with respect to the c -axis).

Heavy Fermion Superconducting Properties Section

The spin susceptibility 

in the superconducting

state determined from Knight shift measurements is

sensitive to the symmetry of the superconducting or-

der parameter.For spin-singlet (even parity) pairing

(in the absence of spin-orbit coupling), 

decreases

below T

, while for spin-triplet (odd parity) pairing,

the spin susceptibility remains constant (for certain

directions).For nearly all of the heavy fermionsuper-

conductors,a decrease in 

belowT

is observed (the

notable exception is UPt

), implying that the spin

partof the Cooperpairwavefunctionisa spin-singlet

(anti-symmetric under particle exchange). Since the

orbital part of the wavefunction must then be sym-

metric to satisfy the Pauli exclusion principle, the

total angular momentum in these materials is even,

i.e., L =0, 2,...(s-wave,d-wave,...).Spin-latticere-

laxation (1/T

) measurements that probe the shape

of the density of states in the superconducting state

have been used to distinguish between these two sce-

narios. For conventional s-wave superconductors, a

Hebel-Slichter peak at T

and an exponential tem-

perature dependence of 1/T

ispredicted[Tinkham],

while no Hebel-Slichter peak and a power law T-

dependence is expected for an unconventional su-

perconductor (either T

for line nodes or T

for point

nodes). A T

dependence of 1/T

is generally found

in the heavy fermion superconductors indicating a

d-wave superconducting order parameter.

13.3.4 UPd

Introduction

The heavy-fermion compound UPd

belongs to

the ever-growing class of materials which display the

coexistence of antiferromagnetic order and super-

conductivity at low temperatures. This class, which

includes such compounds as UPt

and CeIn

,hasat-

tracted considerable attention during the last decade

due to the unusual nature of the superconductivity.

The detailed nature of the superconductivity in these

compounds has been investigated using an assort-

ment of optical and spectroscopic techniques includ-

ing NMR,Josephson tunneling,andinelastic neutron

scattering,which suggests that the superconductivity

is magnetically mediated and has d-wave order pa-

rameter symmetry. In this section, we use UPd

as an example to illustrate how spectroscopic tech-

niques have been used to elucidate the nature of

heavy-fermion superconductivity.

Fig. 13.28. Specific heat divided by temperature C/T plot-

ted vs temperature T of polycrystalline UPd

.TheN´eel

temperature T

and superconducting transition tempera-

ture T

are indicated, after [127]

668 M.B. Maple et al.

Table 13.4. Superconducting properties of heavy-fermion superconductors. T

: superconducting transition temperature

at P

; H

: upper critical field at T =0K;−(dH

/dT)

: slope of the upper critical field; m

∗

: effective mass deter-

mined from H

as discussed in Sect. 13.3.3; C/ T

: superconducting specific heat jump C/T

divided by electronic

specific heat coefficient  ; 

: superconducting coherence length determined from Eq. (13.22). Values are listed as H

parallel/perpendicular to the c-axis

Compound T

(0) −(dH

/dT)



C/ T

∗

(K) (kbar) (kOe) (kOe/K) (Å) (m

)

CeCu

0.49 [108] 20/25 230/230 [138] 65 351 [192]

CePd

0.43 28 [142] 13/7 16/12.7 [193] 230/300 97

CeRh

0.4 6 35 13 307 118 [194]

CeNi

0.23 23 > 15 5 [144] 642 75

CeCu

∼ 2 165 [195] 35 11 [146] 147 37

CeIn

0.17 25 [2] 4.5 [196] 32 300 191

CeRhIn

2.17 16 [115] 160 [151] 140 [115] 40 0.36 [151] 104

CeIrIn

0.40 5/9 24/48 223/141 0.76 [116] 100

CeCoIn

2.3 [117] 50/120 7/15 [197] 50/32 4.5 [117] 121

RhIn

2 [118] 23 −/54 −/92 51 94

CePt

Si 0.75 [119] 28/34 [179] 75/57 95/109 0.25 [119] 140

CeRhSi

1.0 [153] 20 70 120 62 140

CeIrSi

1.6 [154] 25 111 114 51 105

CeNiGe

0.48 [155] 65 23 71 118 120

0.26 [157] 36 8 45 200 145

PrOs

1.85 [123] 22 19 116 ∼ 3 [158] 44

UBe

0.85 90 [198] 420 [198] 37 1.23 [136] 351

UPt

0.55 [199] 18 63 117 0.5−1.0 [160] 265

URu

1.5 [126,163] 100 92 59 0.56 184

UPd

1.9 30 43 76 [200] 1.2 [127] 187

UNi

1.0 10 13 191 [200] 0.4 [128] 147

UGe

0.7 [3,201] 10 20 58 [202] 108 [202] 0.25 235

UIr 0.14 [168] 26 0.26 3 1063 137

URhGe 0.3 [109] 7 40 199 0.5 281

Fe 3.7 [170] ∼ 100 34 61 ∼ 2 109

PuCoGa

18.5 [133] ∼ 1270 99 16 ∼147

PuRhGa

8.7 [134] ∼ 270 20/35 52/40 ∼140

UPd

is a heavy-fermion compound with an

electronic specific heat coefficient  of ≈ 100

mJ/mol K

, which corresponds to an effective elec-

tron mass of m

∗

≈ 50 m

. The specific heat of

UPd

reveals antiferromagnetic order at T

=14

K, and superconductivity at T

=2K,asshownin

Fig. 13.28.

13 Unconventional Superconductivity in Novel Materials 669

Magnetic and Crystal Structure

Neutron diffraction measurements [205] show that

UPd

crystalizes in the hexagonal PrNi

struc-

ture with lattice parameters a =5.37 Å and c =

4.19 Å. The antiferromagnetic structure of UPd

consists of ferromagnetic sheets in the easy hexag-

onal plane with a relatively large ordered magnetic

moment of 0.85 

, which alternate along the c-axis.

The magnetism arises from localized uranium mo-

ments with practically no spin transfer from the Pd

ions, as confirmed by studies of neutron diffraction,

NMR, and band-structure calculations [206–208].

The first neutron diffraction measurements [209,

210] on UPd

found that the AFM order remains

almost unchanged in the superconducting state.

However, recent high-resolution diffraction data in-

dicate [211] that as the temperature is lowered be-

low the superconducting critical temperature T

small 1 % reduction occurs in the intensity of the

AFM Bragg peaks.

Inelastic Neutron Scattering

The inelastic neutron scattering profile measured at

the AFM wave vector Q =(0, 0, 0.5) is shown in

Fig. 13.29. In addition to the zero energy divergence

of the Bragg peak, two finite energy (inelastic) fea-

tures are evident in the data: a peak at E =1.5meV

due to the spin density wave, and a sharper peak at

E =0.36 meV which results from an energy gap in

the magnetic spectrum. The spin density wave is un-

usual in that it persists at temperatures well below T

It is localized about the AFM vector Q =(0, 0, 0.5),

since the peak intensity of the spin density wave peak

drops off rapidly away from Q.The orientationof the

spin density wave has been determined to be in the

a-b crystallographic plane, transverse to the sublat-

tice magnetization, from polarized inelastic neutron

scattering measurements [212].

The energy gap in the magnetic spectrum ob-

served in Fig. 13.29 occurs only in the supercon-

ducting state below T

=2KandH

=30kOe.The

temperature dependence of the gap is surprisingly

similar to the BCS theory prediction for a supercon-

ducting energy gap. At the lowest temperatures, the

Fig. 13.29. Inelastic neutron diffraction data for UPd

at temperatures above and below T

=2K.Thelines are

guides to the eye,after [211]

energy gap extrapolates to 0.36meV, which corre-

sponds to 2 =2.2k

. This is comparable to the

weak-coupling BCS theory result of 2 =3.5 T

.The

appearance of this energy gap in the magnetic spec-

trum in the superconducting state suggests magnet-

ically mediated superconductivity.

Superconducting energy gaps at slightly larger

energies (up to 5.5k

) have been observed in

UPd

by NMR [206,213,214]and electron tunnel-

ing [215] measurements. The discrepancy between

the results of the different techniques is probablydue

to anisotropy of the energy gap; the power-law be-

havior of the specific heat and 1/T

relaxation rate

from NMR measurements imply that the energy gap

in UPd

should vanish at lines or points on the

Fermi surface.

The onset of superconductivity has a marked ef-

fect on the spin density wave peak in the neutron

diffraction data.As shown in Fig. 13.30, both the en-

ergy of the spin density wave peak and its line width

increaseabruptly atT

.At the same time,the intensity

of the peak reaches a maximum at T

,andthende-

creases at lower temperatures. These effects are only

apparent at fields below the critical field of UPd

and the normal-state spin density wave reappears

above H

. Thus, there appears to be a coupling be-

tween the superconducting and magnetic order pa-

rameters.

670 M.B. Maple et al.

Fig. 13.30. UPd

inelastic neutron diffraction data [211].

Top pane l : Activation energy E of the spin density wave

peak vs T. Bottom panel:Maximumofthespindensity

wave peak vs temperature T at various wave vectors

A temperature T

, which characterizes the fre-

quency spread (line width) of the spin wave, can

be derived by fitting the energy spectrum to a

Lorentzian function, yielding T

≈ 28 K for UPd

[217].Ithas been suggestedonthebasisof theoretical

work [216] that T

is proportional to T

for spin-wave

mediated superconductors[217].

Electron Tunneling

Further evidence for magnetically-mediated super-

conductivity in UPd

, as well as details of the

symmetry of the order parameter, have been eluci-

dated by recent directional tunneling measurements

on thin films [218]. The UPd

thin films, which

were grown by molecular beam epitaxy in the [001]

orientation, are separated from a thin film of Pb by

a 4 nm layer of aluminum oxide. Applying a mag-

netic field H = 300 Oe, which is above the H

of Pb, but well below the H

of UPd

,results

in a superconductor-insulator-normal tunnel junc-

tion. The resulting tunneling spectrum is shown in

Fig. 13.31, where the thin line is a fit to the Dynes

formula. The most important feature of this tunnel-

ing spectrum is a peak at V =1.22 meV, which is

above the gap energy, and below typical phonon fre-

quencies (

≈ 150 K for UPd

). The energy of

this feature is very close to the energy gap E =1.5

meV of a dispersive spin wave at the magnetic Bragg

point Q =(0, 0, 0.5), which was determined from in-

elastic neutron scattering measurements.Thus,these

measurements strongly suggest a coupling of the su-

perconducting order parameter to a spin-ﬂuctuation

with an energy gap E =1.5 meV. Furthermore, the

Fig. 13.31. Normalized differential conductivity of a

UPd

-AlO

-Pb tunnel junction. The thinner line is

a fit to the Dynes formula with  = 235 ‹eV and

 =35‹eV. The inset shows a feature associated with

strong-coupling spin-ﬂuctuations,after [218]