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 641

The emphasis of this review is on experiment. In

view of the enormous scope of the subject,this article

is not comprehensive and the choice of examples dis-

cussed is very selective. The chapter is divided into

the following sections:

1. Introduction

2. Conventional superconductors containing local-

ized magnetic moments

3. f -electron heavy fermion superconductors

4. Organic superconductors

5. Layered cuprate and ruthenate superconductors

6. Comparison of the propertiesof different classes

of novel superconductors

It should be noted that the magnetic field is denoted

by the symbol H throughout the text, and by both H

and B in the figures.

13.2 Conventional Superconductors

Containing Localized Magnetic

Moments

13.2.1 Introduction

The focus of this article is on the unconventional

superconductivity displayed by several classes of

novel superconductors, including heavy fermion f -

electron compounds, organic compounds, and lay-

ered cuprates and ruthenates. As noted above, there

is a preponderance of evidence for pairing of super-

conducting electronswithangularmomentum L > 0

and a strong perception that magnetic interactions

play a dominant role in mediating the electron pair-

ing in these materials. However, in order to put these

developments in perspective, we first describe the

extraordinary effects that occur when magnetic mo-

ments are embedded in s-wave superconductors. We

consider two situations: one in which magnetic ions

dissolvedin a superconducting hosthave a smallcon-

centration and do not exhibit spin glass or long-range

magnetic order (paramagnetic impurities), and an-

other in which a large concentration of magnetic

ions occupy a sublattice in a superconducting inter-

metallic compound and exhibit long range magnetic

order (magnetically ordered superconductors). Our

discussion is brief; the interested reader is referred

to several more extensive articles on the subject for

further reference [5–13].

The interrelation between superconductivity and

magnetism has a longhistory that spans nearly a half

century. The first theoretical inquiry into the coexis-

tenceof superconductivity and magnetism was made

by Ginzburg in1957 [14] andexperimentalinvestiga-

tions by Matthias,Suhl,and Corenzwit soon followed

in 1958 [15]. The early experiments were carried out

on binary and pseudobinary systems in which a rare

earth rare earth (R) impurity with a partially-filled

4f electron shell and corresponding magnetic mo-

ment was dissolved in a superconducting element

or binary compound; e.g., La

1−x

and Y

1−x

(Note that throughout this Chapter, rare earth R in-

cludes Y and the lanthanides La through Lu, upon

successive filling of the f -electron shell.) These ex-

periments were very provocative, but largely incon-

clusive with regard to questions concerning the co-

existence of superconductivity and long-range mag-

netic order, primarily due to complications associ-

ated with chemical clustering and/or short-range or

“glassy” types of magnetic order. However, the ex-

periments stimulated some imaginative theoretical

developments which, although largely inapplicable

to the binary and pseudobinary systems then being

investigated, anticipated several important phenom-

ena that were later discovered in the 1970s.A signif-

icant “spin-off” of the early experimental and work

on the coexistence problem was the achievement of

a rather good understanding of superconductivity

in the presence of paramagnetic impurities, includ-

ing the effects of the crystalline electric field, Kondo

scattering, localized spin ﬂuctuations, etc. [5,6].

13.2.2 Superconducting-Magnetic Interactions

In a conventional superconductor,the superconduc-

tivity involves electron pairs (Cooper pairs) in which

the two electrons have opposite momentum k and

spin s -(k ↑, −k ↓). An applied magnetic field H or

the magnetic moment  ofanioninasuperconduc-

tor can interact with the superconducting electrons

in two ways: via the Zeeman interaction of H or the

exchange field generated by  with the spin s of a

642 M.B. Maple et al.

conduction electron (s·H), and via the electromag-

netic interaction of the vector potential associated

with H or  with the momentum k of a conduction

electron (the k·A term in the one electron Hamilto-

nian). Both of these interactions raise the energy of

onememberofaCooperpairandlowertheenergyof

the other. Such “pair breaking” interactions are very

destructive for superconductivity and generally lead

to strong depressions of T

.In a type II superconduc-

tor that does not contain ions carrying magnetic mo-

ments,superconducting electron pairs can be broken

by both the Zeeman interaction of the applied mag-

netic field H with the spin of a superconducting elec-

tron, and the electromagnetic interaction of H with

the momentum of a superconducting electron. The

Zeeman interaction leads to the paramagnetic limit-

ing fieldH

=[N(E

)/(

- 

)]

1/2

,whereN(E

)isthe

density of states at the Fermi level E

,and

and 

are the normal and superconducting state Pauli sus-

ceptibilities, respectively [16,17], while the electro-

magnetic interaction yields the orbital critical field



= ¥

/2

,where¥

is the ﬂux quantum and 

is the superconducting coherence length [18,19].Ac-

cording to the BCS theory, 

(0) = 0, and H

= H

=1.84T

[Tesla]. However, spin orbit scattering can

increase 

and, in turn, H

How does a superconductor overcome these mag-

netic pair breaking interactions? One way that has

been experimentally established is to set up a mod-

ulated spin structure with a period (lattice constant)

a < . This condition is satisfied in the known an-

tiferromagnetic superconductors in which   10

Å. In ferromagnetic superconductors, this condition

canbesatisfied by settingup anewsinusoidallymod-

ulated state that screens the exchange interaction

at long wavelengths. Another way involves the pres-

ence of a negative (antiparallel) exchange field that

compensates the applied magnetic field,allowing su-

perconductivity to persist in magnetic fields much

higher than the upper critical field. Yet another pos-

sibility is to utilize magnetic interactions to produce

an unconventional superconducting state involving

triplet spin pairing of electrons. These and other sit-

uations in which superconductivity and magnetism

coexist are surveyed in this Chapter.

13.2.3 Paramagnetic Impurities in Superconductors

The effect of magnetic impurities on superconduc-

tivity has received a great deal of attention ever

since the discovery that they produce a precipitous

drop in the superconducting transition temperature

[15,20]. Herring [21] and Suhl and Matthias [22]

noted that the conduction electron-impurity spin ex-

change interaction could account for this strong de-

pression of T

. Assuming that exchange scattering

of conduction electrons by impurity spins may be

adequately described within the first Born approx-

imation (to second order in the exchange interac-

tionparameterJ ),Abrikosovand Gor’kov(hereafter

AG) [23] developed in 1960 what is now a classic the-

ory for superconductors containing paramagnetic

impurities. Their theory successfully explained the

basic features of these early experiments and further

predicted the striking phenomenon of gapless su-

perconductivity. However, studies of local moments

in metals in the normal state have shown that the

assumptions upon which the AG theory is founded

are not applicable to many matrix-impurity systems

that nonetheless exhibit a strong depression of T

In these systems, the hybridization of the localized

d or f -electron states of the solute ion with conduc-

tion electron states results in a continuous demag-

netization of the solute ion below a characteristic

temperature T

. The characteristic temperature T

can be associated with ﬂuctuations of localized mo-

ments with a frequency 

−1

or compensation of the

localized moments by the conduction electron spins

(Kondo effect). Large depressions of T

are found

even when the solute ions are weakly magnetic at

superconducting temperatures (i.e., T

 T

Exchange Interaction

For transition metal impurity ions where the orbital

angular momentum L is quenched, the exchange in-

teraction Hamiltonian is given by

=−2J S · s , (13.1)

where J is the exchange interaction parameter, S is

the paramagnetic impurity spin, and s is the conduc-

tion electron spin density at the impurity site.For R

13 Unconventional Superconductivity in Novel Materials 643

impurities, where L is not quenched,S is replaced by

its projection on the totalangular momentum vector

J = L + S of the Hund’s rule ground state. Then the

exchange interaction Hamiltonian becomes

=−2J (g

−1)J · s , (13.2)

where g

is the Land´e g-factor for the Hund’s rule

ground state of the R ion.

Generally, J is approximated as

J ∼ J

+ J

, (13.3)

where J

(> 0) is the Heisenberg exchange term,

while J

(< 0) arises from covalent mixing between

the localized d or f -electron states and conduc-

tion electron states. In terms of the Schrieffer–Wolff

transformation [24], J

is given by

∼

V

k

U





(



+ U)

< 0 , (13.4)

when |



|. Here, V

k

isamatrixelement

mixing localized states with itinerant states,  =

V

k

N(E

) is the Hartree–Fock (HF) half-width

of the resonant state, 



= E



− E

, E



is the energy

of the centroid of the localized d or f -state, E

is the

Fermi energy, and U is the intra-atomic Coulomb

repulsion which splits spin-up and spin-down states

in the Friedel–Anderson model [25,26]. In the limit

that U |



|, this expression takes the form

J ∼

V

k







< 0. (13.5)

Forpartially-filledf -electron shellsof rareearth ions

with stable valence such as Pr,Nd,Gd,Tb,Dy,Ho,and

Er, the hybridization between the 4f -electron and

conduction electron states is weak and J is domi-

nated by J

(ferromagnetic exchange). In contrast,

for partially-filled d or f -electron shells of transition

metal ions, rare earth ions with unstable f -valence

such as Ce, occasionally Pr, Sm, Eu, Tm, and Yb,

and actinide ions, the hybridization is appreciable

and J is dominated by J

(antiferromagnetic ex-

change). The superconducting properties of matrix-

paramagnetic impurity systems for the cases of fer-

romagnetic and antiferromagnetic exchange are dra-

matically different and are described below.

The exchange interaction also produces spin glass

or long-range magnetic ordering in metals contain-

ing localized magnetic moments via the Ruderman–

Kittel–Kasuya–Yosida (RKKY) mechanism. This is

the primary source of long range magnetic ordering

in ternary rare earth compounds, discussed later in

this section, although dipolar contributions may be

important in certain cases. The polarized rare earth

magnetic moments can then interact with the con-

duction electrons in two ways: by means of the Zee-

maninteractionoftheexchangefieldandthecon-

duction electron spins, and via the electromagnetic

interaction of the magnetization and the persistent

current.

Ferromagnetic Exchange Interaction

The effect of paramagnetic impurities on supercon-

ductivity for ferromagnetic exchange is best illus-

trated for Gd impurity ions in situations where the

interactions between Gd impurities are sufficiently

weak that they may be neglected over an appreciable

temperature range below T

, the superconducting

critical temperature of the superconducting matrix

into which they have been dissolved. Since Gd is an

S-state ion, this also avoids complications associated

with the lifting of the degeneracy of the Hund’s rule

ground state by the crystalline electric field (CEF).

Experimentally, it has been found that the curves of

vs n/n

is the superconducting transition

temperature of the alloy with paramagnetic impurity

concentration n,andn

is the critical concentration

above which superconductivitydoes not occur at any

temperature) and C/C

vs T

(C and C

are

the specific heat jumps at T

and T

, respectively)

behave in characteristic manners, both of which can

be represented well by universal functions that have

been calculated on the basis of the theory of AG [6].

Shown in Fig. 13.1 is a plot of T

vs n/n

for

the La

1−x

system; T

=3.3KistheT

of the

LaAl

host compound (x =0)andn

=0.59 at. %

is the critical concentration where T

vanishes [27].

The solid line, which has been fitted to the T

n/n

data, is the AG theoretical curve.Also shown in

Fig.13.1 are theCurie–Weiss temperatures 

,plotted

as 

vs n/n

,for the La

1−x

system where

644 M.B. Maple et al.

Fig. 13.1. Reduced superconducting critical

temperature T

vs reduced Gd impurity

concentration n/n

for the La

1−x

sys-

tem. The value of the superconducting critical

temperature T

of the LaAl

host compound

is 3.3 K, while the value of the critical con-

centration n

where T

=0is0.59 at. % Gd.

The line denoted AG is the theoretical curve

of Abrikosov and Gor’kov [23], which has been

fitted to the data. Inset: Curie–Weiss tempera-

tures 

determined from magnetic suscepti-

bility measurements, plotted as 

vs n/n

Thelinethroughthedataisaguidetotheeye,

after [5,27]

the solid line is a guide to the eye [5,27]. Apparently,

interactions between the Gd magnetic moments are

sufficiently weak in this system to allow the AG the-

ory to be tested to concentrations in the vicinity of

. Shown in Fig. 13.2 is the behavior of C/C

vs T

for the La

1−x

system [28] and the

1−x

system [29], compared to calculations by

Skalski et al. [30] based on the AG theory.

As can be seen in Figs. 13.1 and 13.2, the experi-

mental T

vs n/n

and C/C

vs T

curves

are well represented by the AG theory. The basic as-

sumptions embodied in the AG theory are that (1)

the superconducting order parameter does not vary

with position, (2) the spins of the impurities are

fixed and randomly oriented in space, and (3) scat-

tering of conduction electrons by the paramagnetic

impurities can be calculated within the first Born ap-

proximation. The first assumption requires that the

impurities be randomly distributed throughout the

superconducting matrix, while the second precludes

any correlations between the spins of the impurities

such as those that would arise from magnetic order.

Ifoneregardstheimpurityspindirectionasfixed,

the exchange interaction is not time reversal invari-

ant. As a result, the lifetime 

of the time-reversed

single particle paired states of which the supercon-

ducting wave function is comprised is no longer in-

finite. In the AG theory, the superconducting proper-

ties in the presence of solute spins are characterized

Fig. 13.2. Reduced specific heat jump C/C

vs reduced

critical temperature T

for the matrix-impurity sys-

tems La

1−x

and Th

1−x

.TheLa

1−x

data

were derived from C(T) measurements by Luengo and

Maple [28] and the Th

1−x

data were derived from

(T) measurements by Decker and Finnemore [29].The

dashed line represents the BCS law of corresponding states,

whereas the solid line is the result of the AG theory as cal-

culated by Skalski et al. [30],after [6]

13 Unconventional Superconductivity in Novel Materials 645

by a pair breaking parameter ˛ = 

−1

.Thetheory

predicts a second order transition to the supercon-

ducting state and a rapid decrease of the supercon-

ducting transition temperature with ˛ given by the

universal relation









−



+0.14

˛T



, (13.6)

where T

corresponds to ˛ =0,˛

= k

/4

(ln is Euler’s constant) corresponds to T

= 0 (com-

plete destruction of superconductivity) and is the

digamma function.

This expression has the linear asymptotic form

=1−0.691





, (13.7)

as ˛ → 0.

It is apparently sufficient to calculate the pair

breaking parameter ˛ within the first Born approxi-

mation (to second order in J ), which for supercon-

ductors containing paramagnetic R ions gives the

result

˛ ≡







N(E

)



−1)J(J +1), (13.8)

where n is the paramagnetic impurity concentration

and N(E

) is the density of states at the Fermi level

for the matrix.

It is important to note that ˛ is proportional to

n and independent of temperature. Hence, ˛/˛

can

bereplaced by n/n

in Eqs.13.6 and 13.7 where n

the critical concentration for the complete suppres-

sion of superconductivity. Thus, the theory predicts

that the reduced critical temperature T

will be

a universal function of reduced concentration n/n









−



+0.14



, (13.9)

= U





, (13.10)

The specific heat jump C at T

as a functionof T

has also been calculated on the basis of the AG the-

ory by Skalski et al. [30]. They find that the reduced

specific heat jump C/C

is a universal function V

of T

C

= V





, (13.11)

which deviates markedly from the linear BCS law of

corresponding states

C

. (13.12)

One of the most striking predictions of theAG the-

ory is the phenomenon of gapless superconductivity.

This results fromthefact thatthe finitelifetime

cor-

responds to an energy broadening  ∼ /

which

introduces states into the gap and spreads out the

BCS peak in the density of states. As a consequence,

the energy gap §

(˛) no longer corresponds to the

order parameter (˛) and with increasing ˛ (or n),

(˛)goestozerofasterthatT

(˛). For all concen-

trations, the superconductor is gapless at tempera-

tures sufficiently near T

; while for n > 0.91 n

,the

superconductor is gapless at all temperatures.For ex-

ample,the theory predicts an attenuated specific heat

jump at T

(compared to the BCS law of correspond-

ing states; see Fig. 13.2), and a linearly temperature-

dependent term in linearly temperature-dependent

term in the specific heat below 0.91 n

. Other pre-

dictions of the AG theory have been verified exper-

imentally as described in other reviews [5,6] (Note,

the recent study by Kim and Overhauser [31].)

The AG theory yields an initial depression of T

with n that is linear with a rate, for superconductors

containing R impurities, given by





n=0

=−





N(E

)



−1)

J(J +1),

(13.13)

where the quantity (g

−1)

J(J +1) = D(R) is

called the deGennes factor [33]. The variation of

(−dT

/dn)

n=0

with R for the La

1−x

system scales

with the function D(R) if J

is assumed to decrease

slightly with increasing R atomic number, except for

Ce where (−dT

/dn)

n=0

is anomalously large [15].

Investigations of the effect of substituted R ions on

superconductivity of LaAl

, which has a T

of 3.3K,

revealed a variation of (−dT

/dn)

n=0

with R similar

to thatof La

1−x

,withacorrespondinganomalously

646 M.B. Maple et al.

Fig. 13.3. Initial rate of depression of the superconducting

critical temperature T

with paramagnetic impurity con-

centration n,(−dT

/dn)

n=0

,vs rare earth R impurity forthe

1−x

and La

1−x

systems,after [5,32]

large rate of depression of T

with n for R = Ce [32].

The (−dT

/dn)

n=0

vs R data for both La

1−x

and

1−x

, normalized to the value for Gd, are dis-

played in Fig. 13.3 [5,32].

It should be noted that, according to the AG the-

ory, the depression of T

depends on the magnitude

of J but not on its sign. However, neglected in the

AG theory are contributions to ˛ of higher order

than J

which must be taken into account when J is

negative.These contributionsgive rise to spectacular

temperature dependent pair breaking effects which

are considered below.

Antiferromagnetic Exchange Interaction

The anomalous depression of T

with n for Ce sub-

stituents in La and LaAl

is associated with the hy-

bridization of the localized 4f -electron states of Ce

and the conduction electron states [5]. The large de-

pression of T

with Ce concentration is due to the

large negative contribution to the exchange interac-

tion and the occurrence of the Kondo effect with

∼ T

,whereT

is the Kondo temperature and

is the T

of the La or LaAl

matrix (see below).

The Kondo temperature is a characteristic tem-

perature that separates high temperature magnetic

behavior, where (T) behaves as a Curie–Weiss law,

from low temperature nonmagnetic behavior,where

(T) approaches a constant value. A physical picture

of the nonmagnetic ground state that has emerged

from theories of the Kondo effect involves the grad-

ual formation,as T decreases through T

,ofamany

body singlet ground state in which the spin of each

paramagnetic impurity ion is screened by the an-

tiferromagnetically aligned spins of the conduction

electrons. The Kondo temperature is given by

∼ T

exp



−1

N(E

)|J |



, (13.14)

where T

is the Fermi temperature.

The phenomenon of reentrant superconductiv-

ity due to the Kondo effect was discovered in the

1−x

system.

Here,a sample within a certain range of Ce impu-

rity concentrations becomes superconducting below

a critical temperature T

and then loses its supercon-

ductivity below a second criticaltemperature T

[34,

35].The destruction ofsuperconductivity at T

is as-

sociated with the competition between singlet spin

pairing of electrons in the superconducting state,

with characteristic energy k

, and the formation

of the Kondo many body singlet state involving the

conduction electrons and each Ce impurity ion, with

characteristic energy k

. Shown in Figs. 13.4(a)

and (b), respectively, are ac magnetic susceptibil-

ity vs temperature data and the reentrant T

vs n

curve of the La

1−x

system. The values of T

Fig. 13.4(b) and the transition curves in Fig. 13.4(a)

from which they were inferred are identified with

letters [34]. In addition to the remarkable behavior

exhibited by the T

vs n curve, the C/C

curve also displays some interesting features

which are shown in Fig.13.5.Here,it can be seen that

the curve of C/C

vs T

shows a pronounced

downward deviation from both the BCS law of corre-

sponding states and, as well, the AG theory. The two

specific heat jumps for a specimen at 0.64 at.% Ce are

indicated in Fig. 13.5 by the solid squares.

Electrical resistivity measurements on the

1−x

system in the normal state revealed the

occurrence of a Kondo effect, as evidenced by a min-

imum in (T) at low temperatures [39]. The Ce im-

purity contribution to (T) was found to diverge as

13 Unconventional Superconductivity in Novel Materials 647

Fig. 13.4. (a) Normalized transition signal,

based on ac magnetic susceptibility measure-

ments, vs temperature for the La

1−x

sys-

tem.(b) Reduced superconducting critical tem-

perature T

vsCe impurity concentrationn

for the La

1−x

system.The value of the su-

perconducting critical temperature T

of the

LaAl

host compound is 3.3K.Thedataidenti-

fied by letters are derived from the correspond-

ing transition curves in (a), after [34]

−lnT with decreasing T down to ∼ 1 K,indicating a

low Kondo temperature T

 1 K.Shown in Fig. 13.6

are specific heat C vs T data for a La

1−x

al-

loy (x =0.64 at.% Ce) in various magnetic fields

up to 38 kOe [38]. Curves a − c, which have been

drawn to fit more accurate data for a La

1−x

al-

loy with x =0.906 at.% Ce, correspond to an entropy

of Rln2 per mol Ce, showing that the Ce

ground

state is a doublet. Curve d is consistent with calcu-

lations of Bloomfield and Hamann [40] for S =1/2

and T

= 0.42 K. The two zero field superconduct-

ing critical temperatures T

and T

are indicated by

arrows in the figure.The Kondo anomalies in the nor-

mal state properties of the La

1−x

system have

also been well characterized and used to test the-

oretical models of the Kondo effect. Subsequently,

reentrant superconductivity was discovered in the

(La

1−y

)

1−x

system in the low Th concentra-

tion range 0 < y ≤ 0.25 [41].

Superconductivity was studied in superconduct-

ing matrix-impurity systems in which the impurity

ions were nonmagnetic at superconducting temper-

atures (i.e., T

 T

,whereT

is a characteris-

tic temperature that separates low temperature non-

magnetic behavior from high temperature magnetic

behavior, such as the Kondo temperature T

). These

648 M.B. Maple et al.

Fig. 13.5. Reduced specific heat jump C/C

vs reduced

critical temperature T

for the La

1−x

system.The

solid triangles, circles and squares represent data from Lu-

engo et al. [36],Armbr¨uster et al.[37], and Bader et al.[38],

respectively. The dashed line represents the BCS law of cor-

responding states,the dot-dashed line indicates the AG re-

sult, and the solid line is a smooth curve drawn through

the data, after [6]

studies revealed that the T

vs n curves have pos-

itive curvature and can be described by a relation

proposed by Kaiser [42], who considered the effect

of nonmagnetic resonant d and f -electron states of

impurity ions on superconductivity. Inthis case T

a modified exponential relation of the form

= T

exp



−

(1 − Dn)



, (13.15)

where A and D are fitting parameters that can be re-

lated to parameters describing the conduction elec-

trons (N(E

)andV ,whereV is the electron–phonon

interaction parameter) and the localized d or f -

electrons (N



), U,andL,whereN



)istheden-

sity of states at E

of the localized d or f -electrons,

U is the intra-atomic Coulomb interaction param-

eter, and L is the d or f -electron orbital angular

momentum). Shown in Fig. 13.7 are plots of T

vs n/n

for three systems [6] in which T

 T

1−x

[43],Th

1−x

[44],and Al

1−x

[45].Dis-

played in Fig. 13.8 are reduced specific heat jump

C/C

vs reduced critical temperature T

data

for the Th

1−x

[46–48], Al

1−x

[49, 50], and

1−x

[51] systems. The nonmagnetic charac-

ter of the f -electron states in these systems at su-

perconducting temperatures where T  T

is re-

ﬂected in the conformance of the C/C

vs T

data with the BCS law of corresponding states. The

1−x

system [43] provides an interesting exam-

ple of the Kondo effect in the electrical resistivity.

Shown in Fig. 13.9(a) are  vs T data for Th

1−x

samples with x =0, 0.01, 0.015, and 0.02, in which

the resistivity minimum phenomenon is evident.

Displayed in Fig. 13.9(b) are  vs T

data, where

(x, T)=(x, T)−(0, T) is the U contribution

to the resistivity, that can be described by the rela-

tion (x, T)=(x, 0)[1 − (T/T

)

]withT

∼ 100

K, consistent with the local Fermi-liquid behavior

expected for the Kondo effect at low temperatures

T  T

[43].

Reentrant superconductivity associated with the

Kondo effect was predicted to occur in the limit

 T

,whereT

is the criticaltemperature of the

superconducting host metal, by M¨uller–Hartmann

and Zittartz (MHZ) [52] and Ludwig and Zucker-

mann (LZ) [53]. MHZ and LZ obtained exponential-

like curves of T

vs n in the limit T

 T

.Cal-

culations of Zuckermann [54] and MHZ [55] for the

initialdepression of T

with n,(−dT

/dn)

n=0

,yielded

a maximum as a function of T

at T

≈ 10.

The magnetic state of an impurity in a supercon-

ducting matrix-impurity system can be continuously

tuned by applying an external pressure P or alloying

the matrix with a suitable element. An example of

the first case is illustrated in Fig. 13.10, which shows

vs pressure for the system La

1−x

with several

concentrations x of Ce [56].With increasing Ce con-

centration,aminimumin T

as a function of pressure

P develops near 15 kbar which becomes more pro-

nounced the higher the Ce concentration. The data

for the La

1−x

sample with x =0.02areparticu-

larly striking; the application of pressure drives T

to zero at about 5 kbar, T

remains zero up to ∼

15 kbar,whereupon superconductivityreappearsand

increases with pressure. The initial depression of

, T

/n, increases markedly with pressure up to a

maximum near 15 kbar and thereafter decreases to a

13 Unconventional Superconductivity in Novel Materials 649

Fig. 13.6. Specific heat of a La

1−x

alloy

(0.64 at.% Ce) vs temperature in various mag-

netic fields up to 38 kOe. Curves a–c, which

have been drawn to fit more accurate data for

aLa

1−x

alloy with 0.906 at.% Ce, corre-

spond to an entropy of Rln2 per mole Ce, show-

ing thatthe Ce

groundstateisadoublet.Curve

d is consistent with calculations of Bloomfield

and Hamann [40] for S =1/2andT

=0.42 K.

The two zero field superconducting transitions

are indicated, after [38]

Fig. 13.7. Reduced superconducting critical

temperature T

vs reduced impurity con-

centration n/n

for the Th

1−x

,Th

1−x

,and

1−x

systems.The solid lines represent the

modified exponential relation given in the fig-

ure that describes the weakening of supercon-

ducting electron pairs by nonmagnetic local-

ized states [42]. The dot-dashed line is the AG

curve, while the dashed line is a straight line to

which the other curves have been scaled in the

initial linear region at low concentrations. The

values of the characteristic (or Kondo) temper-

ature T

, superconducting critical temperature

of the hostT

,and the scalingconcentration n

are indicated in the figure, after [6]

value that, above ∼ 100 kbar, is more than an order

of magnitude smaller than the maximum depression.

Isobars of T

vs n in the inset of Fig. 13.10 evolve

with increasing pressure from nearly linear depres-

sions with slight negative curvature to depressions

with strong positive curvature. These results can be

interpreted in terms of an increase of T

,and,inturn,

the ratio T

, with pressure.As T

increases

under pressure, the initial depression of T

with x is

expected to increase, pass through a maximum,and

650 M.B. Maple et al.

Fig. 13.8. Reduced specific heat jump C/C

vs reduced

critical temperature T

for the Th

1−x

,Th

1−x

,and

1−x

systems. The solid line represents the BCS law of

corresponding states,whereas thedot-dashed lineindicates

the AG result. Open circles: data from Luengo et al. [46,47].

Solid circles: data from Watson et al.[48]. Open square:data

from Martin [49].Solid squar es: data from Smith [50].Solid

triangles: data from Dempesy [51], after [6]

Fig. 13.10. Superconducting critical temperature T

vs pres-

sure for the La

1−x

system. Isobars of T

vs Ce concen-

tration are shown in the inset, after [56]

Fig. 13.9. (a) Electrical resistivity  vs temperature T of Th

1−x

alloys with x =0.5, 1, and 2 at. % U at low temperatures.

(b) Incremental resisitivity (x, T)=(x, T)−(0, T)vsT

of Th

1−x

alloys with x =0.5, 1, and 2 at. % U, after [43]