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 231

that implies a non-s wave pairing with an order parameter which changes sign

on translation through Q

= π/c. Here c is the lattice constant perpendicular

to the layers of U . An order parameter with a nodeline was indeed conﬁrmed

experimentally.

32,36,48

A review of pair-breaking in superconductivity which summarizes the

situation of the theory in the late ’60s was given by Maki,

a pioneer in the

ﬁeld. Many experimental results are also found in a review by Alloul et al.

In the following we want to discuss some of the most important concepts.

2. Time-reversal Symmetry Breaking

The Hamiltonian of an interacting electron system is time-reversal invariant,

i.e.

[H, T

]

−

= 0 . (1)

Here T

= −iσ

C is the time-reversal operator with C changing a function

into its complex conjugate and σ

denoting a Pauli matrix. This implies that

for each one-particle eigenstate Hψ

= 

there exists a time reversed one

= T

with the same energy (Kramers’ theorem), i.e. Hψ

= 

These two states can be paired to a ground state

|Φ

BCS

i =



+ v



|0i, (2)

where c

creates an electron with wavefunction ψ

. When momentum and

spin are good quantum numbers it is n = (k, σ) and

n = (−k, −σ).

Let us assume that we add a perturbation to the system which violates

time-reversal symmetry. An example is

int

A + Ap

) − µ

H (3)

where p

,σ

are momentum and spin of the i-th electron and A and H are

the vector potential and magnetic ﬁeld, respectively. Another one is

int

= −(g − 1)J

k,k

;αβ

αβ

kβ

J (4)

where g is the Land´e factor and J is the total angular momentum of a

magnetic impurity. In those cases the time evolution of T

(t) does not

vanish any more and instead follows from

= i [H

int

, T

]

−

. (5)

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

232 P. Fulde

As shown by de Gennes

the linearized Ginzburg-Landau equation from

which the transition temperature T

is determined can be written in the

weak coupling limit in the form

1 = N(0)V

dd

1 − f() − f(

)

 + 

g(

− ) . (6)

Here N(0)V is the BCS parameter f () is the Fermi function and most

importantly

g() =

2π



(0)T

(t)



−it

(7)

is the Fourier transformed correlation function of the time-reversal operator.

One notices that when H

int

= 0 or when H

int

is time-reversal invariant, we

obtain g( − 

) = δ( − 

) and (6) reduces to the well known BCS expres-

sion. Here we are interested in [H

int

, T

] 6= 0. In this case we must distin-

guish between two diﬀerent long-time behaviors of the correlation function

(0)T

(t)i. They are

(a) lim

t→∞



(0)T

(t)



= η , 0 < η < 1

(b) lim

t→∞



(0)T

(t)



= e

−2t/τ

. (8)

In the ﬁrst case one speaks of nonergodic processes while the second one

implies ergodic behavior. The eﬀect on T

and on Cooper pairing is quite

diﬀerent in the two cases. Nonergodic behavior leads to pair weakening,

while ergodic behavior results in pair breaking.

When pair weakening takes place the transition temperature is

reduced to

= 1.13 ω

exp



−

ηN(0)V



, (9)

where ω

denotes a cut-oﬀ in the frequency of the bosons which provide

for the electron attractive interactions. The eﬀect of the non-time-reversal

invariant interaction is just a reduction of the eﬀective electron–electron

interaction. If, however ergodic behavior prevails we may set

g(z) =

2π

1 + z

(10)

and after integrating (6) the equation for T

becomes

1 = N(0)V

n'0

n + 1/2 + 1/ (2πT

)

. (11)

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

Cooper Pair Breaking 233

Note that here a cut oﬀ at ω

is still missing so that the sum is divergent.

After the cut oﬀ is introduced (11) becomes

ln(T

) + ψ



2πT



− ψ(1/2) = 0 (12)

where ψ(x) is the digamma function.

The transition temperature T

obtained when H

int

= 0. The last relation is the same which Abrikosov and

Gorkov obtained in their original work. One ﬁnds from (12) that T

drops continuously with increasing pair-breaking parameter 1/τ

and van-

ishes at a critical value of





crit

πT

2γ

; γ = 1.78 . (13)

3. Nonergodic versus Ergodic Behavior

As pointed out above, nonergodic behavior of time-reversal symmetry break-

ing leads to pair weakening instead of pair breaking. Two examples are given

for illustration. Consider a thin superconducting ﬁlm of thickness d with a

rough surface but no scattering centers inside the ﬁlm, when a magnetic ﬁeld

is applied parallel to the ﬁlm.

In that case

(pA + Ap) T

= −i

dφ

(14)

where φ(t) is the phase of the propagating electron. When the electron

moves straight from one surface of the ﬁlm to the other, its phase does not

change (the vector potential has only a component perpendicular to the ﬁlm

which is symmetric with respect to the ﬁlm center). Therefore the only

contribution to dT

/dt comes from the parts of the path before it hits the

surface for the ﬁrst time and after the last hit. Thus hT

(0)T

(t)i remains

ﬁnite even in the limit t → ∞. This results in pair weakening.

A second example concerns a staggered ﬁeld imposed onto the conduction

electrons, i.e.

int

= −I

k,Q



k↑

k+Q↑

−c

k↓

k+Q↓



. (15)

The Q’s are reciprocal lattice vectors of the magnetic lattice and are re-

stricted to the ﬁrst Brillouin zone. It is

= 2iH

int

. (16)

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

234 P. Fulde

While [H

int

, T

] 6= 0 we ﬁnd that the operator Y = T

R commutes with

int

, where R shifts the system by a vector connecting the two sublattices.

Thus [H

int

, Y ]

−

= 0. This is obvious: after application of T

the spins change

sign which increases their energy in the staggered ﬁeld. By shifting the

electrons from one sublattice to the other the spin direction is again in line

with the staggered ﬁeld. This implies that when ψ

kσ

(r) is an eigenfunction in

the staggered ﬁeld so is Y ψ

kσ

(r) with the same eigenvalue. Therefore we may

pair ψ

kσ

(r) with e

iϕ

Y ψ

kσ

(r) where the phase ϕ is chosen by convenience.

Being able to pair electrons properly even when [H

int

, T

] 6= 0 shows that in

the presence of the interaction (15) Cooper pairs are possibly weakened but

not broken.

Next we turn towards ergodic behavior of perturbations. Thereby su-

perconductivity is discussed within the weak coupling limit. Systems with

pair-breaking interactions can be divided into two groups. Group 1 includes

all those cases for which a standard theory can be worked out for all temper-

atures. Group 2 comprises those situations for which a general pair-breaking

theory can be worked out only for the Ginzburg-Landau (GL) regime where

the order parameter is small. The GL equation is of the form



+ ψ



+ ρ



− ψ





|∆(r)|

2(2πT )

(ρ)

|∆(r)|

= 0

(17)

where f

(ρ) varies from case to case. The parameter ρ = (2πT τ

)

−1

a measure of the strength of pair breaking. The coeﬃcient of h|∆(r)|

i is

the same as met before (see (12)) and is generic for all ergodic systems.

It determines T

. At lower temperatures the spatial variation of the order

parameter requires an individual treatment which is the reason for diﬀerent

forms of f

(ρ).

We start with perturbations belonging to group 1. As pointed out before,

for that group a theory is available in closed form for all temperatures.

Dif-

ferent pair-breaking processes lead to equivalent forms of the single-particle

Green’s function where they enter in form of a pair-breaking parameter 1/τ

As a consequence, the thermodynamic properties expressed in terms of this

pair-breaking parameter are the same in all cases. For transport coeﬃcients

this holds true too, but with one exception. We may obtain diﬀerent expres-

sions when the relevant correlation function contains an s-wave spin triplet

vertex and when in addition the momentum transfer is less than (`ξ

)

−1/2

where ξ

is the coherence length.

This is the case, e.g. for the spin suscep-

tibility χ

(q, ω) when q is small. Otherwise diﬀerent pair-breaking mecha-

nisms are completely equivalent.

The equivalence holds true in particular

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

Cooper Pair Breaking 235

for the density of states. In the BCS theory it is given by

(E) = N(0) Im

E/∆

1 − (E/∆)

(18)

where N(0) is the density of states per spin direction in the normal state. In

the presence of a pair-breaking parameter 1/τ

this expression is modiﬁed

(E) = N(0) Im

√

1 − u

(19)

where u(E) is a solution of the equation

∆

= u



1 −

∆

√

1 − u



. (20)

The latter has to be solved numerically and a plot is shown in Fig. 1.

For parameter values 0.91 < [(τ

)

crit

/τ

] < 1 where (τ

)

crit

denotes the

critical pair-breaking parameter at which T

vanishes, one ﬁnds no gap in

the excitation spectrum, i.e. the superconductor is gapless.

Gaplessness can again be rederived from the behavior of the time-reversal

operator T

Let us see under which circumstances the excitation energy

can be expanded for small order parameter ∆ in the form

= |

| +

|∆(r)|

|hn |T

|mi|



− 

(21)

where |mi, |ni are single-electron states. When [H

int

, T

]

−

= 0 the matrix

elements hn|T

|mi 6= 0 only when |mi and |ni are time reversed. Then

Fig. 1. Tunneling density of states (per spin) as a function of energy E/∆ for

various values of τ

−1

. Curves (a)–(d) correspond to (τ

∆)

−1

= 1.33; 1.0; 0.5; 0.05,

respectively. (From Ref. 68)

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

236 P. Fulde



= 

and the expression diverges. An expansion of that form is therefore

not allowed. If however [H

int

, T

]

−

6= 0 than hn|T

|mi 6= 0 for a range of

|

− 

| ' τ

−1

. In this case we can write (21) in analogy to (6) as

E(p) = |(p)| +

|∆|

d

(p) + 

g(

− (p))

= |(p)| +

|∆(r)|

|(p)|

(p)

+ (τ

)

−2

. (22)

An expansion of the form (21) is therefore possible. Note that there is no

gap in E(p) in the limit of small ∆. From the last relation we obtain

(E) = N (0)







1 +

∆

− (1/τ

)



+ (1/τ

)









(23)

which agrees with the corresponding expression obtained from (19) and (20).

Perturbations belonging to group 1 include:

Paramagnetic impurities. As pointed out in the introduction, supercon-

ductors with paramagnetic impurities were the ﬁrst in which gapless super-

conductivity was observed. The Hamiltonian is the one given by (4). The

spins of the magnetic impurities are treated classically. The scattering of con-

duction electron by the impurities is treated in Born approximation. This

excludes a possible Kondo eﬀect which is discussed later. The pair-breaking

parameter is found to be

= 2πn

N(0)(g − 1)

J(J + 1) . (24)

The dependence of T

) follows from (12) and a plot is found below in

Fig. 8.

Thin ﬁlm in a parallel magnetic ﬁeld. Here we have to distinguish

between local and nonlocal electrodynamics. In the ﬁrst case the mean free

path ` must satisfy `  d where d is the ﬁlm thickness. In addition the

conditions d  λ, (`ξ

)

1/2

have to be fulﬁlled, where λ is the penetration

depth. The following pair-breaking parameter is obtained

39,41

edH)

. (25)

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

Cooper Pair Breaking 237

Note that τ

is the transport mean free time while v

is the Fermi velocity.

In case of nonlocal electrodynamics d . ` the expression (25) is generalized

edH)



π`



, (26)

where g(x) = (3/2x

)[(1 + x

)arctan x − x].

The limit `  d reproduces

(25).

Supercurrent. In case of a uniform current time-reversal symmetry is

broken in the center of mass system because of scattering processes. When

`  ξ

we ﬁnd

(27)

where q denotes the shift of the electron distribution in momentum space in

the presence of a current.

In the Ginzburg-Landau regime (see (17)) the function f

(ρ) applies to

all perturbations belonging to group 1

(ρ) = −

(2)



+ ρ



−

(3)



+ ρ



(28)

where ψ

(2)

(x) and ψ

(3)

(x) are higher derivatives of the digamma function.

Next we turn towards perturbations belonging to group 2 according to

the above classiﬁcation scheme. The pair-breaking parameter τ

−1

applies

here to the Ginzburg-Landau regime (17) only. Two examples are discussed.

Type II superconductors. By restricting oneself to the eﬀect of a mag-

netic ﬁeld on the electron orbits, i.e. by neglecting the Zeeman term we

obtain in the dirty limit (`/ξ

 1),

15,18,42

. (29)

When a magnetic ﬁeld is applied, nucleation starts at the surface. In the

surface regime

= 0.59

. (30)

In the bulk as well as in the surface regime the function f

(ρ) is

(ρ) = −

(2)



+ ρ



. (31)

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

238 P. Fulde

Contact between a superconducting and a magnetic ﬁlm. Boundary

conditions require that, at an interface between a superconductor and a

magnetic metal, the order parameter vanishes.

In the Ginzburg-Landau

regime close to T

the order parameter varies normal to the interface x = 0

like ∆(x) ∼ cos

πx

where d

is the thickness of the superconducting ﬁlm

which extends from 0 < x < d

. In the limit of `  d

and (`ξ

)

1/2

< d

the

pair-breaking parameter is

24,26





, (32)

while f

(ρ) is given by

(ρ) = −

(2)



+ ρ



(3)



+ ρ



. (33)

By comparison with (28) one notices that the term proportional to

(3)



+ ρ



depends sensitively on the spatial variation of ∆(r).

4. Zeeman Splitting of Quasiparticles

When a magnetic ﬁeld is applied parallel to a very thin ﬁlm, e.g. of Al with

A thickness, the eﬀect of the ﬁeld on the spins dominates the one on the

orbits to the extent that the latter may be neglected. A similar argument

holds true for type II superconductor when the κ value, i.e. the ratio of pen-

etration depth to coherent length is very large. In both cases the transition

to the normal state is of ﬁrst order provided that the spin-orbital mean free

path `

= v

is not too small. Generally the density of states splits into

two parts

(E) =

N(0)

sgnE <e







− 1

−

−1







. (34)

The u

(E) are the solutions of the following coupled equations

E ∓ µ

∆

= u

3τ

∆

− u

∓

− 1

. (35)

The root has to be chosen so that (u

− 1)

1/2

→ u

sgn(Re u

) when

E → ±∞. In the special case that τ

∆

 1 (35) reduces to (20) with

the pair-breaking parameter given by

3τ

(µ

, (36)

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

Cooper Pair Breaking 239

Fig. 2. Zeeman split quasiparticle density of states N

(E). Due to the presence

of small spin-orbit scattering the two components are slightly mixed.

i.e. we are back to the standard pair-breaking theory. The Zeeman-split

density of states has been observed by Meservey and Tedrow in tunneling

experiments.

52,72

Since electrons do not change their spin during the tun-

neling process, the method can be applied as a tool to measure magnetic

properties.

An example of N

(E) based on (34) is shown in Fig. 2.

When the eﬀect of the ﬁeld on the orbits cannot be neglected, the pref-

actor α of the h|∆(r)|

i term in the Ginzburg-Landau equation is of the

form

α =





1 +

√

− h





+ ρ

−





1 −

√

− h





+ ρ



− ψ





. (37)

Here the abbreviations b = (3τ

∆

)

−1

and h = µ

H/∆

have been intro-

duced. Furthermore

∆

2πT

a ±

−h

. (38)

The pair-breaking parameter a depends on the particular situation. For a

type II superconductor in the dirty limit it is given by

a =

3∆

+ b (39)

while for a thin ﬁlm in a parallel ﬁeld it is in the same limit

a =

18∆

edH)

+ b . (40)

The two forms of a should be compared with (29) and (25).

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

240 P. Fulde

Eu Sn Mo S Se

0.75 0.25 6 8−y y

y = 0.8

transition temperature T[K]

H[T]

1 2 3 4

Fig. 3. Field induced superconductivity based on the Jaccarino-Peter eﬀect. (From

Ref. 53)

Equation (37) for a type II superconductor in a high ﬁeld was ﬁrst de-

rived by Werthamer, Helfand and Hohenberg.

It had been obtained before

for the case that the Zeeman term µ

σ ·H is caused by spin-polarized mag-

netic impurities, i.e. by a term −n

(g − 1)J

σhJ

Adding both terms,

i.e. working with a total Zeeman interaction of the form

Zeem

= −

(µ

H − n

(g − 1)J

i) (41)

can lead to a Jaccarino-Peter eﬀect.

Here a compensation takes places

between the eﬀect of an external ﬁeld and that of polarized impurities which

couple via exchange to the conduction electrons. As a result, supercon-

ductivity may become magnetic ﬁeld induced. This will be the case when

in the absence of a ﬁeld the (unpolarized) magnetic ions have destroyed

superconductivity and when their eﬀect is reduced by the aforementioned

compensation eﬀect. Clearly that requires a delicate balance between dif-

ferent material properties. Nevertheless, ﬁeld induced superconductivity

has been observed in Eu

0.5

(see Fig. 3) and apparently also in

λ-(BETS)

FeCl

The order of the phase transition from the superconducting to the

normal state in the presence of pair-breaking perturbations is obtained from

the sign of the function f

(ρ) in (17). When f

(ρ) = 0 a system changes

from a second- to a ﬁrst-order phase transition. For perturbations belonging

to group 1 the phase transition is always of second order, while for those