Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

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October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

Cooper Pair Breaking 241

belonging to group 2 it depends on the special case. The most impor-

tant case is, of course, that of type II superconductors (Shubnikov phase)

with a large κ value. Here the order of transition depends not only on the

ratio T

but also on the Maki parameter and the spin-orbital scattering

parameter b.

5. Scattering Centers with Low-Energy Excitations

As previously discussed, the ﬁeld of Cooper pair breaking started with a

study of the eﬀects of paramagnetic impurities. Originally the scattering of

conduction electrons oﬀ those ions was studied by treating the interaction

(4) in Born approximation. A few important generalizations took place since

then. One consists in including interactions between impurities.

Another

is going beyond the Born approximation. This was done by treating the

impurity spin classically

as well as quantum mechanically, i.e. by including

the Kondo eﬀect in the latter case. For a review see, e.g. Ref. 56. A third

generalization takes into account that magnetic impurities, here especially

rare earth impurities have often intra-atomic excitations with energies in the

meV regime. Since these energies are of the same order as k

one expects

a strong interplay between the two.

In generalizing the Born approximation, it is advantageous to work with

the one-particle T

(ω) and two-particle T

(ω, ω

) scattering matrices.

7,8

terms of those functions the transition temperature is determined by solving

the coupled equations

˜ω

= ω

− n

Im T

(˜ω

)

∆

= ∆ + n

πN(0)T

(˜ω

, ˜ω

)

∆

|˜ω

(42)

where ω

= 2πT

(n + 1/2) with n being an integer. The ﬁrst of the two

equations describes the scattering of a conduction electron in the normal

state giving rise to a self-energy while the second one accounts for a modiﬁ-

cation of the pairing interactions due to inelastic impurity scattering. From

the self-consistency equation (11) one can derive in the limit n

→ 0 the

following lowering of the transition temperature due to pair breaking

−



= −T

+∞

n=−∞

Im T

(ω

)

sgn ω

πN(0)

|ω

+∞

m=−∞

(ω

, ω

)

|ω

(43)

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

242 P. Fulde

We want to include the Kondo eﬀect, which shows up to order J

but

requires going beyond perturbation expansion. For this it is advantageous to

start instead of (4) from the more basis Anderson Hamiltonian. It is written

down for a single ion with a ν

degenerate f orbital of energy 

which

is hybridizing with matrix element V

mσ

(k) with the conduction electrons.

Electrons in the f orbital repel each other with energy U. Therefore

H =

kσ

(k)c

kσ

+

m6=m

kmσ



mσ

(k)f

kσ

+h.c.



(44)

All energies are measured from the Fermi energy E

. As common n

. It is supposed that U → ∞, which implies that the f-site is either

occupied with one electron or empty. This corresponds to Ce

, a well

studied Kondo ion. There are three diﬀerent regimes between which one may

distinguish. They depend on the ratio 

/Γ where Γ = πν

N(0)[V

mσ

)]

(a) 

/Γ  1: the f orbital remains empty and only ordinary, nonmagnetic

scattering takes place;

(b) |

/Γ| ' 1: the f -site is in the intermediate-valence or valence-

ﬂuctuating regime;

/Γ  1: this is the regime in which magnetic scattering and the

Kondo eﬀect take place and on which we shall concentrate.

The essential feature of the Kondo eﬀect is the formation of a singlet state,

in this case of the 4f electron with the conduction electrons. The 4f electron

of Ce

with J = 5/2 can hop via V

mν

(k) from orbital ν into the conduction

band while a conduction electron returns to the site in orbital ν

. The

singlet formation energy is characterized by a temperature T

, the Kondo

temperature. It is often of order meV. For temperatures T  T

the Kondo

ions have a Curie-Weiss susceptibility and an eﬀective magnetic moment

eﬀ

(T ) = 3T χ(T ) , (45)

while for T  T

the susceptibility for the singlet state is Pauli-like. The

aim is to understand the pair-breaking eﬀect of Kondo ions. The conduction

electron scattering rate is

Im T

(ω, T ) = −

(Γ/N (0)) ρ

(ω, T ) (46)

and therefore proportional to the many-body spectral density ρ

(ω, T ) for

addition or removal of an f -electron. The latter consists of a Lorentzian

centered at 

(removal of an f electron) of width ν

Γ and height (πν

Γ)

−1

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

Cooper Pair Breaking 243

and a small resonance-like feature at the Fermi energy (Abrikosov-Suhl res-

onance). There is also a broad peak at 

+ U corresponding to the addition

of a second f-electron. But since U is large it is of no interest here. It is

observed in inverse photoemission, i.e. Bremsstrahlung experiments.

For Ce ions 

' −2 eV, while the narrow resonance at E

has a small

spectral weight of (1 − n

) = πk

/(ν

Γ) and is due to spin ﬂuctuations.

When (46) is set into (43) the reduction of T

is found to be

−



N(0)J

, (47)

where here J

= −V

hyb

/|

|. This can be compared with the corresponding

expression of the Abrikosov-Gorkov theory which for ν

= 2 and g = 2 is

−



3π

J(J + 1)N (0)J

. (48)

Starting from the Anderson model shows that charge ﬂuctuations from f

to f

are the ones which cause the pair-breaking in the Abrikosov-Gorkov

theory. The strength of the Abrikosov-Suhl resonance is a measure of the

probability of a f

→ f

transition when an f electron is added to the

ion. Remember that the ground state of a Ce

impurity is a coherent

superposition of a f

and f

electron state. At low T and low ω the resonance

dominates the scattering rate of the conduction electron. It depends on the

scaled quantities T/T

and ω/T

only.

The eﬀect of T

(ω

) is supplemented by the eﬀective interaction between

electrons of opposite spin T

(ω

, ω

). It results from a virtual polarization

of the impurity. Here we concentrate on elastic scattering where ω

= ω

It gives the dominant contribution to dT

/dn

and counteracts the eﬀect

of Im T

(ω

, T

) because it reﬂects the increasingly nonmagnetic character

of a Kondo ion for large ratios of T

. When it is added in (43) to the

contribution of Im T

(ω

) we obtain the dependence of −(dT

/dn

)

shown

in Fig. 4. Pair-breaking is strongest for a ratio T

' 5. In the limit

of large T

, pair-breaking reduces to zero because of a strongly bound

singlet state. The superconducting state is a pair condensate of quasiparti-

cles formed by a coherent superposition of conduction and f electrons. The

transition temperature T

of that system will be generally lower than that

of the pure one. We speak here of pair weakening because the system be-

haves like a BCS superconductor except for the change in T

. This is easily

understood. A large T

implies a large ν

Γ and a value of (1−n

) which is in

the intermediate valence or valence-ﬂuctuating regime. In this case Friedel’s

model of a nonmagnetic virtual bound state applies.

The impurity level

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

244 P. Fulde

Fig. 4. Initial depression of T

with impurity concentration n

as function of

), i.e. the ratio of the Kondo temperature (T

= T

) and the supercon-

ducting transition temperature. The pair-breaking eﬀect of an impurity is largest

for T

' 5. (From Ref. 8)

at 

is broadened and partially occupied but U is not large enough to

generate a local moment. At the impurity site two electrons of opposite

spins repel each other and therefore the averaged attractive interaction of

paired electrons is diminished.

8,35,66,78

Therefore we deal here with pair

weakening rather than pair breaking.

The nonmonotonous behavior of dT

/dn

has an interesting consequence,

i.e. a possible reentrant behavior of superconductivity.

For a range of

impurity concentrations for which pair breaking changes strongly with

, one ﬁnds as function of temperature two and even three transition

temperatures T

. This eﬀect has indeed been observed and an example is

shown in Fig. 5.

Rare-earth ions with well localized 4f electrons in an incomplete f -shell

have a (2J + 1)-fold degenerate ground state where J denotes the lowest

Hund’s rule multiplet. The latter is split in the crystalline electric ﬁeld

(CEF) of the surrounding of an ion. Conduction electrons cause transitions

between the CEF energy levels. Of particular interest are non-Kramers ions

which have an even number of 4f electrons and may have a nondegenerate

singlet CEF ground state. An example is Pr

which is in a 4f

conﬁg-

uration. Two interactions with the conduction electron are of particular

importance. One is the isotropic exchange interaction (4) while the second

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

Cooper Pair Breaking 245

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(La, Ce) Al

T / T

c c0

n(a/0)

Fig. 5. Reduced transition temperature versus impurity concentration for (La, Ce)

with reentrant behavior due to the Kondo eﬀect. AG denotes the result of the

Abrikosov-Gorkov theory. (From Ref. 47)

is the aspherical Coulomb charge scattering



4π



1/2

m=−2



s; kd



(J)c

sσ

kdmσ

+ h.c.] . (49)

Here Q

is the quadrupole moment of, e.g. the Pr

ion and the Coulomb

integrals I

s; kd) can be found in Ref. 23. The c

kdmσ

destroy a con-

duction electron with momentum k = |k|, in a ` = 2 state with az-

imuthal quantum number m and spin σ, while c

sσ

creates an electron

with momentum k

in a ` = 0 state. The operators Y

(J) are spheri-

cal harmonics written in terms of J

operators. For example, Y

±1

(J) =

±(J

+ J

) /



(2/3)

1/2

(2J

− J)



. The Hamiltonian transfers angular

momentum ` = 2 between conduction electrons and the 4f

shell.

Note that the conduction-electron part of H

is time-reversal invari-

ant. Therefore transitions between CEF levels caused by H

contribute

to pair formation rather than to pair breaking. Their eﬀect resembles that

of localized phonons. As it turns out, the exchange matrix elements are

usually dominant and for that reason CEF split non-Kramers ions usually

break Cooper pairs. A notable exception is pointed out below. However, the

depression of T

with impurity concentration n

deviates strongly from the

Abrikosov-Gorkov theory. Consider a CEF splitting of the ions as shown in

Fig. 6 and the eﬀect of H

int

given by (4).

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

246 P. Fulde

Fig. 6. Schematic view of CEF eigenstates |ii and energies δ

of an ion some of

which may be degenerate.

The CEF eigenstates |ii and energies δ

are assumed to be known,

e.g. from inelastic neutron scattering experiments. The matrices T

and

can be expressed in terms of the magnetic susceptibility

χ (ω

, ω

) =

|hi|J|ji|

− n

)

(ω

− ω

)

+ δ

. (50)

The thermal population of the diﬀerent states |ii is denoted by n

, and

= δ

− δ

. In the simplest case of two singlets separated by an energy δ

and all other CEF levels neglected, we obtain

χ (ω

, ω

) =

2δ tanh(δ/2T )

(ω

−ω

)

+ δ

|h0|J|1i|

. (51)

The T matrices are shown in Fig. 7.

Fig. 7. Scattering matrices T

(ω

) and T

(ω

, ω

) due to impurities with internal

degrees of freedom χ(ω

) is the spin susceptibility and γ = (g − 1)N(0)J

. Solid

lines refer to electronic Green’s functions.

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

Cooper Pair Breaking 247

Fig. 8. T

as function of impurity concentration for La

1−x

. The dashed

line is the theoretical curve for δ/T

= 4.4. The point denoted by an arrow has

been ﬁtted to give an initial slope of one. AG denotes the Abrikosov-Gorkov results.

(From Ref. 9)

From (42) and (51) we obtain

+ ψ



y (δ/2T

)

y (δ/2T

)



− ψ





= 0 , (52)

where τ

is deﬁned so that the slope d(T

)/d(τ

)

−1

= −1. This

has the advantage that we can compare the result with those of the

Abrikosov-Gorkov theory. The function y(x) has the value y(0) = 2 and is

monotonously decreasing with x. For x = 10 it has dropped to y(10) ' 0.6.

Thus with increasing δ/T

the pair-breaking eﬀect of a single ion decreases

continuously because virtual CEF excitations are reduced. Therefore with

higher concentration n

the pair-breaking eﬀect of a single impurity de-

creases (for more details see Ref. 23). With increasing ratio (δ/T

), i.e. with

decreasing T

a virtual excitation to the upper CEF becomes less favorable.

Experiments by Bucher et al.

on La

1−x

have conﬁrmed the above

results (see Fig. 8 where the results of the Abrikosov-Gorkov theory are

shown too). The diﬀerences to the standard pair-breaking theory are less

pronounced for Kramers ions, which always have a degenerate ground state.

6. Breaking of Anisotropic Pair States

In the high-T

superconducting cuprates pairing takes place in the form

of a strongly anisotropic pair state. An isotropic s-wave order parameter

cannot explain many experimental data. In particular the low-temperature

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

248 P. Fulde

thermodynamic properties such as the speciﬁc heat require instead strongly

anisotropic pairing. Phase sensitive measurements favor d-wave pairing.

Also in other superconductors with strong electron correlations, strongly

anisotropic pair states prevail. Examples are the Fe-pnictides or f-electron

systems with heavy quasiparticles.

12,70

A generalization of the theory of pair

breaking and of pair weakening to strongly anisotropic pairing is therefore

mandatory. Equation (42) is generalized for a momentum dependent order

parameter

˜ω

(

k) = ω

− n

Im T

(

k, ω

)

∆

(

k) = ∆(

k) + n

N(0)

(

, ω

)

∆

(

)

|˜ω

(

. (53)

The two-dimensional unit vector

k runs over the Fermi surface. The function

ˆn(

k) is normalized to one, i.e.

kn(

k) = 1. Within the Born approximation

the scattering matrices of a nonmagnetic impurity with scattering potential

) are

(

, ω

) = −iπsgn ω

N(0)

)|u(k, k

(

, ω

) = |u(

. (54)

Similarly (43) is generalized to

−



= π

N(0)

dkn(

)n(

|∆(

k)|

|u(k, k

−∆(

k)|u(

∆(

)

. (55)

Hereby we used a normalization of the order parameter of the form

kn(

k)|∆(

k)|

= 1 . (56)

When the order parameter ∆(

k) is constant, the right-hand side of (55)

vanishes. This is quite diﬀerent for an order parameter with

kn(

k)∆(

k) = 0 , (57)

a situation encountered, e.g. in d-wave superconductors. Here we have to

account for anisotropies in the scattering matrix elements. The reduction in

is largest for isotropic scattering, i.e. when u(

) = const. The eﬀect can

be reduced to a large extent when the scattering is strongly anisotropic. An

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

Cooper Pair Breaking 249

anisotropic order parameter can be nearly insensitive to impurity scattering

if the latter is also strongly anisotropic. For example, it may happen that a

superconducting, nearly constant order parameter does exist only on parts

of the Fermi surface. The latter may consist of several disconnected pieces.

If impurity scattering were to connect only parts of the Fermi surface where

the order parameter is nonzero, then its eﬀect on superconductivity would

be very small.

7. Pair Formation versus Pair Breaking

After the discovery of high-T

superconductivity it was suspected from an

early stage on, that the dominant pairing interaction does not result from

phonons.

54,55,62,69

The same holds true for superconductors with heavy

quasiparticles

although here, because of the low transition temperatures,

the argument is not as compelling. In the following we discuss two cases

where there is strong evidence for non-phononic pairing interactions. Con-

trary to the above, the electronic excitations are associated with localized

electrons which couple to itinerant ones.

The ﬁlled skutterudite PrOs

has a transition temperature T

(= 1.85 K) which is more than twice as large as the corresponding one of

LaOs

= 0.74 K). Both superconductors seem to be of conventional

s-wave type.

38,71,77

Since the only diﬀerence are the 4f

electrons of Pr, they

must be responsible for the large increase in pairing interactions. Inelastic

neutron-scattering experiments have demonstrated

that the CEF ground

state of the J = 4 multiplet is a Γ

singlet with a low-lying excited triplet

of T

symmetry at 8 K. All other levels are so high in energy that their

inﬂuence can be neglected. The triplet is a superposition of two triplets of

symmetry,

|Γ

, mi =

1 − d

|Γ

, m i + d |Γ

, mi , m = 1, 2, 3 . (58)

From experiments one can deduce |d| = 0.26, which implies that |Γ

, mi has

mainly |Γ

, mi character. But transitions |Γ

i → |Γ

, mi can take place only

via H

but not via H

. Therefore these processes are pair forming. The

boson propagator due to intra-atomic CEF excitations is

χ (q, ν

) =

αβm



αβ

(



2δ

+ δ

(59)

with

αβ

(

q) =

√

ˆq

hΓ

|(J

+ J

)|Γ

i (60)

October 4, 2010 10:37 World Scientiﬁc Review Volume - 9.75in x 6.5in ch11

250 P. Fulde

and ν

= 2πT n. The matrix element results from H

which is rewritten

for cubic symmetry of site i in the form

(i) =

√

kqσ

αβcycl.

+ J

)

ˆq

k−qσ

kσ

ikR

. (61)

The Eliashberg equations

allow for the determination of T

when the

dynamic two-particle interactions are known. The latter are mediated by

phonons in the case of LaOs

and by phonons supplemented by χ(q, ν

)

in the case of PrOs

A detailed discussion of Eliashberg’s equations

is found, e.g. in Ref. 63. When the equations are solved, one obtains the

observed enhancement of T

in PrOs

for a coupling constant of g =

0.04 eV. This is a very reasonable value. It also gives the observed eﬀec-

tive electron mass enhancement in the normal state of the system when the

self-energy due to χ(q, ν

) is evaluated.

Another system with pairing interactions based on intra-atomic excita-

tions is UPd

, a superconductor with T

= 1.8 K and an antiferromagnet

with T

= 14.3 K. Due to strong intra-atomic or Hund’s rule correlations

the 5f electrons of U ions remain localized in some of the f orbitals while

they delocalize in others.

Another way of stating the same is by say-

ing that intra-atomic correlations renormalize the hopping matrix-elements

of some of the 5f orbitals to zero. This Dual Model has been successful

in explaining a number of experiments. In UPd

two of the nearly 2.5

5f-electrons with j

= 5/2 and 1/2 remain localized and form a Hund’s rule

ground state. Inelastic neutron scattering shows dispersive CEF excitations

(magnetic excitons) below the Ne´el temperature. They result from coopera-

tive eﬀects of localized electrons. Coupling of conduction electrons to these

excitations explains the strongly anisotropic, heavy-quasiparticle masses in

that system. The same excitations act as bosons which provide for binding

of electrons to Cooper pairs. However, a special form of the order parameter

is required, i.e.

∆(p) = ∆

cos(cp

) or ∆(p) = ∆

sin(cp

) (62)

with c denoting the lattice constant perpendicular to the U planes. Eliash-

berg’s equations are solved with a magnetic exciton propagator given by

χ (q

, ν

) =

+ ω

(q)

. (63)

The experimentally observed exciton dispersion can be ﬁtted with

(q) = ω

[1 + β cos(cq

)] , (64)