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

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Developing BCS Ideas in the Former Soviet Union 121

and for superconducting alloys, correspondingly. Without entering into the

actual calculations, I only list below the main results.

Close to the temperature of transition, |T − T

|  1, Eq. (25) can be

simpliﬁed. Indeed, near T

∆(r) is small and Eqs. (24) for

G and

F ,

can

be solved perturbatively. When calculating

F (ω

; r, r) in Eq. (25), one can

restrict oneself by a few nonzero terms in ∆(r), ∆

∗

(r) and A(r).

For clean

superconductors such expansion results in the equation of the form:





∂ − i

∗

A(r)





− T

−

|Ψ(r)|



Ψ(r) = 0 (27)

Equation (27) has the familiar form of the GL-equation

for the “gap”

parameter:

Ψ(r) = ∆(r)

7ς(3)N /4πT

Note e

∗

= 2e that stands for the charge of the Cooper pair! (N is the density

of electrons, λ = 7ς(3)E

/12(πT

)

The expression for the electrical current is derived in the same way:

j(r) = −

∗



∗

∂Ψ

∂r

− Ψ

∂Ψ

∗

∂r



−

∗2

A|Ψ|

(28)

The GL-theory, as it is known,

contains one important dimensionless

parameter, κ. In terms of the observable characteristics of a material, Gins-

burg and Landau

expressed it in the following way:

κ = (

√

∗

/~c)H

(29)

(Here H

is the thermodynamic critical ﬁeld and δ

is the penetration

depth.) Now κ can be written through the microscopic parameters of a

metal:

κ = 3T

(π/v

)

3/2

(c/ep

)(2/7ς(3))

1/2

(30)

The parameter κ determines the behavior of a superconductor in strong

magnetic ﬁelds. Abrikosov showed

that depending on whether κ is small

or large, a superconductor would belong to the one of two classes: Type

I or Type II, correspondingly. From Eq. (30) one may conclude that the

microscopic theory imposes no limitations on the κ-value, so that even a

pure metal can be a Type II superconductor. I predicted

that this could

be the case for the elemental niobium, Nb.

Type II superconductivity is more common in alloys. The GL equations

for an alloy, derived in Ref. 23, look quite similar to (27, 28) but with the

September 17, 2010 8:58 World Scientiﬁc Review Volume - 9.75in x 6.5in ch7

122 L. P. Gor’kov

coeﬃcients now depending on the transport meantime, τ





∂ − i

∗

A(r)





− T

−

Nχ(ρ)

|Ψ(r)|



Ψ(r) = 0

j(r) = −

∗



∗

∂Ψ

∂r

− Ψ

∂Ψ

∗

∂r



−

∗2

A|Ψ|

(31)

The GL wave function is now connected with the “gap” as:

Ψ(r) = [χ(ρ)7ς(3)N/16π

]

1/2

∆(r) (32)

and:

χ(ρ) =

7ς(3)ρ



2ρ







− ψ



+ ρ



(33)

In Eq. (33) ρ =

2πT

, and ψ(z) is the logarithmic derivative of the Γ-

function. (In Eq. (31) λ

= λχ(ρ)).

The penetration depth and the value of the GL parameter κ also suﬀers

the change:

= δ

χ(ρ); κ = κ

/χ(ρ) (34)

In the short mean free path limit (so-called “dirty” alloys) κ is expressible

in terms of the observable quantities for the normal phase only:

κ = 0.065ecγ

1/2

/σk

(35)

(Here γ and σ stand for the coeﬃcient in the linear electronic speciﬁc heat

and for conductivity, respectively.)

Together with the GL and Abrikosov phenomenology

the above results

are known as the GLAG-theory (after Ginsburg, Landau, Abrikosov and

Gor’kov).

7.5. Paramagnetic impurities

Alloying by ordinary impurities or defects does not change either the

energy gap, or T

, as we have shown in Sec. 7.2. In 1960 Abrikosov and

Gor’kov

studied alloys containing paramagnetic centers, i.e. atoms or ions

with the nonzero spin. It turned out that the presence of such centers is

detrimental to superconductivity. The transition temperature, T

(x) de-

creases with increase of the concentration x of the paramagnetic impurities

and vanishes at some critical concentration, x

. This behavior might have

been expected, for the potential for scattering on such a defect does now

include, in addition, the interaction with the external spin on a center, s,

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Developing BCS Ideas in the Former Soviet Union 123

V (p −p

) ⇒ V (p −p

) +

V (p −p

)(σ ·s), that tends to misalign spins of the

Cooper pair.

The dependence of T

(x) on the concentration is given

by the equation:

ln(T

/T ) = ψ





− ψ





(36)

where 1/τ

is the inverse mean time for scattering with the overturn of the

electronic spin and, as in Sec. 8.4, ψ(x) = Γ

(x)/Γ(x). The value of the

critical concentration is given by:

c,cr

πT

2γ

≡

∆

(37)

The gap in density of states (DOS) also decreases with the x increase.

It was, however, somewhat unexpected to ﬁnd that the energy gap in DOS

closes before the superconductivity is fully destroyed at x

given by Eq. (37):

there exists a narrow range of concentrations below x

in which supercon-

ductivity persists in spite of the zero energy gap in DOS. The result is

another proof that it is the nonzero superconductivity order parameter, ∆,

that controls the capability of a superconductor to carry persistent currents,

not the ﬁnite gap value. The Landau criterion of Eq. (2) is, hence, of the

lesser generality.

8. Applications of Quantum Field Theory Methods

While for many properties (kinetic properties, such as the sound absorption

or thermal conductivity, or the eﬀects, like tunneling, and others) working in

terms of the Bogolyubov qps was more straightforward, it is not so as far as

the electrodynamics of superconductors is concerned, because the new qps

does not possess the ﬁxed electric charge. There were other fundamental

questions for which ﬁnding the solution without the diagrammatic methods

would be next to impossible.

Thus, the extension of BCS theory to the case of strong electron–phonon

coupling had been done by Eliashberg,

who in 1960 extended Migdal’s

paper

on electron–phonon interactions for normal metal by introducing,

instead of the mere gap parameter, ∆, ∆

, in Eqs. (24) for the “local”

Hamiltonian (7), the new self energy parts build up on the anomalous func-

tions F , F

∆(ω

; p) = T

D(ω

− ω

; p −p

)F (ω

; p

)

(2π)

(38)

September 17, 2010 8:58 World Scientiﬁc Review Volume - 9.75in x 6.5in ch7

124 L. P. Gor’kov

In Eq. (38) D(ω

; k) stands for the Green function of phonons multiplied by

the square of the e-ph coupling constant:

D(iω

; p − p

) = g

(p − p

)

(iω

)

− ω

(p − p

)

. (39)

As was ﬁrst demonstrated in Ref. 13, the adiabatic approximation, ω

 E

makes it possible to neglect the so-called “vertex” corrections both in the

normal and the anomalous self-energy parts. In the new equations, replac-

ing Eqs. (24), the “gap” functions ∆(ω

; p) and ∆

∗

(ω

; p) appear together

with the self-consistency condition of Eq. (38). The Bogolyubov principle of

“dangerous diagrams” could not be relevant in the ﬁrst place, because there

are no divergences in (38) due to the frequency dependence of the phonon

Green function.

It is a point to emphasize here that the Quantum Field Theory formu-

lation also makes it easy to generalize to the case of multi-band supercon-

ductors, or to take into account the anisotropy of the electronic spectrum.

The latter allows the rigorous microscopic derivation of the anisotropic GL

equations.

The Gor’kov method, as shown by Eilenberger

in 1968, can be advanced

even further. These considerable simpliﬁcations arise with the explicit use

of the so-called “quasi-classic” approximation: T

, ω

 E

Under this

provision that obviously has a quite general character, one may re-write

the Gor’kov equations into the master equations for the Green functions

integrated over the energy variable:

G(ω

; p, ξ)

dξ

πi

g(ω

; p) f(ω

; p)

(ω

; p) ¯g(−ω

; p)

(40)

(Here momentum p may run on the Fermi surface.)

The emerging set of non-linear equations is local in space and is easier for

the numerical analysis and for the treatment of the inhomogeneous problem.

We shall not dwell here upon these results.

9. Conclusion

The purpose of the sketchy review above was to trace how the development

of powerful new theoretical methods made it possible to extend BCS to a

much broader class of phenomena. Many new notions and results of a fun-

damental character came to light since the early 1960s, both experimentally

and theoretically. The BCS model grew into the mighty theory of super-

conductivity, one of the most accomplished theories in condensed matter

September 17, 2010 8:58 World Scientiﬁc Review Volume - 9.75in x 6.5in ch7

Developing BCS Ideas in the Former Soviet Union 125

physics. I would like to stress again that, to a considerable extent, the main

advances came about through the formulation of the theory in terms of Green

functions.

Acknowledgments

The work was supported by NHMFL through the NSF Cooperative agree-

ment No. DMR-0654118 and the State of Florida.

References

1. L. N. Cooper, Phys. Rev. 104, 1189 (1956).

2. J. Bardeen, L. N. Cooper and J. R. Schrieﬀer, Phys. Rev. 106, 162 (1957).

3. J. Bardeen, L. N. Cooper and J. R. Schrieﬀer, Phys. Rev. 108, 1175 (1957).

4. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics: Vol. 9, Statis-

tical Physics, part 2 (1998); ibid, Vol. 6, Fluid Mechanics (1999) (Butterworth-

Heinemann, Oxford).

5. N. N. Bogolyubov, J. Phys. USSR 11, 23 (1947).

6. W. S. Corak, B. B. Goodman, C. B. Satterthwaite and A. Wexler, Phys. Rev.

102, 656 (1956); W. S. Corak and C. B. Satterthwaite, ibid 662 (1956); R. E.

Glover and M. Tinkham, ibid 104, 844 (1956); M. Tinkham, ibid 845 (1956).

7. H. Fr¨ohlich, Proc. Roy. Soc. (London) A215, 291 (1952).

8. J. Bardeen and D. Pines, Phys. Rev. 99, 1140 (1955).

9. L. C. Hebel and C. P. Slichter, Phys. Rev. 107, 901 (1957).

10. N. N. Bogolyubov, Sov. Phys. JETP 7, 41, 51 (1958).

11. L. P. Gor’kov, Sov. Phys. JETP 7, 505 (1958).

12. V. M. Galitski and A. B. Migdal, JETP 7, 95 (1958).

13. A. B. Migdal, Sov. Phys. JETP 7, 996 (1958).

14. L. D. Landau, Sov. Phys. JETP 8, 70 (1959).

15. L. D. Landau, Sov. Phys. JETP 7, 182 (1958).

16. A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 8, 1090 (1959).

17. A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 9, 220 (1959).

18. A. A. Abrikosov, L. P. Gor’kov and I. E. Dzyaloshinskii, Sov. Phys. JETP 9,

636 (1959); E. S. Fradkin, ibid 912 (1959).

19. P. W. Anderson, J. Phys. Chem. Solids 11, 26 (1959).

20. T. Matsubara, Progr. Theor. Phys. 14, 351 (1955).

21. L. P. Gor’kov, Sov. Phys. JETP 9, 1364 (1959).

22. V. L. Ginsburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950)

(in Russian).

23. L. P. Gor’kov, Sov. Phys. JETP 10, 998 (1960).

24. A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957).

25. A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 12, 1243 (1961).

26. G. M. Eliashberg, Sov. Phys. JETP 11, 696 (1960); ibid 12, 1000 (1961).

September 17, 2010 8:58 World Scientiﬁc Review Volume - 9.75in x 6.5in ch7

126 L. P. Gor’kov

27. V. L. Pokrovskii, Sov. Phys. JETP 13, 628 (1961); V. L. Pokrovskii and N. S.

Ryvkin, ibid 16, 67 (1963).

28. L. P. Gor’kov and T. K. Melik-Barkhudarov, Sov. Phys. JETP 18, 1031 (1964).

29. G. Eilenberger, Z. Phys. 214, 195 (1968).

August 30, 2010 11:1 World Scientiﬁc Review Volume - 9.75in x 6.5in ch8

BCS: THE SCIENTIFIC “LOVE OF MY LIFE”

Philip W. Anderson

Department of Physics, Princeton University,

Jadwin Hall, Princeton, NJ 08544, USA

After short comments on my early addenda to BCS — gauge invariance

and the Anderson–Higgs mechanism, the dirty superconductor “theorem,”

and the spinor representation — I focus on the interaction mechanisms

which cause electron–electron pairing. These bifurcate into two almost non-

overlapping classes. In order to cause electrons to pair in spite of the strong,

repulsive, instantaneous Coulomb vertex, the electrons can evade each

others’ propinquity on the same site at the same time either dynamically,

by retaining Γ

(s-wave) relative symmetry, but avoiding each other in time

— called “dynamic screening” — or by assuming a non-symmetric relative

wave function, avoiding each other in space. All simple metals and alloys,

including all the (so far) technically useful superconductors, follow the for-

mer scheme. But starting with the ﬁrst discovery of “heavy-electron” super-

conductors in 1979, and continuing with the “organics” and the magnetic

transition metal compounds such as the cuprates and the iron pnictides,

it appears that the second class may turn out to be numerically superior

and theoretically more fascinating. The basic interaction in many of these

cases appears to be the “kinetic exchange” or superexchange characteristic

of magnetic insulators.

Ever since the day in the spring of 1957 when I fell in love at ﬁrst sight

with the BCS theory, a very large fraction of my scientiﬁc activity has been

related to that theory and its implications. Thus when asked to contribute

to this volume I had to choose among 50 years of work and of memories.

Earliest, as I have described elsewhere,

was my strong feeling of defen-

siveness in the face of the attacks on the theory by Wentzel, Kohn and oth-

ers on the basis of gauge invariance. After an initial rather awkward — but

essentially correct — explication

of the physics of gauge invariance, more or

less in parallel with papers of Nambu and of Bogoliubov–Shirkov,

this led

127

August 30, 2010 11:1 World Scientiﬁc Review Volume - 9.75in x 6.5in ch8

128 P. W. Anderson

to the complete (within RPA) theory of the “Anderson-Higgs” mechanism

which allows superconductivity with a true gap not sullied by “Goldstone

modes”. (As was clearly indicated by experiment already at that time.)

In the course of this work I developed the spinor representation of BCS

pairs, which Nambu elaborated into the scheme that Schrieﬀer used in his

indispensable book. For me, it has always been a perspicuous way of under-

standing the coherence properties of the BCS theory. Personally, it came in

very handy on my one contact with the formidable Lev Landau; intrigued

that there was yet another way to describe the BCS theory, he let me run

on for an hour or more in his dreaded seminar.

A second early interest was in the problem of impurity scattering. HBG

Casimir’s famous remark about “A Mile of Dirty Lead Wire” intrigued me, as

did the contrast between the eﬀects of magnetic impurities and of alloying

or deformation that my friend Bernd Matthias kept pointing out to me.

As I remember, it was during a stroll on the Berkeley campus with Jim

Phillips (I was a summer visitor in 1958, he an assistant professor) that the

solution came to me. I called it the “n-representation”: in the presence of

static disorder, let us consider the actual “scattered” — but extended —

eigenfunctions ϕ

of the one-electron potential, with energies E

. If the

one-electron potential is static and time-reversal invariant — which is as

true of spin-orbit interactions as of ordinary potentials — every one of these

must have a time-reversed partner ϕ

∗

which is distinct from it, by Kramers’

theorem. (In the absence of spin-orbit eﬀects, to get the partner one may just

reverse the spin of a real function.) Then one makes up one’s Cooper pairs

out of these two functions, and proceeds with BCS as before. Of course, there

are two conditions for this representation to give nearly the same result as

the pure perfect crystal. One is that the density of states of the E

be nearly

the same as that of the pure crystal. For relatively weak scattering this will

almost always be the case unless the pure crystal has deep structure near

the Fermi surface — which is indeed the case for Bi, and Bi is indeed very

sensitive to purity — a kind of “exception proves the rule” case. The second

is that the interaction not be enormously anisotropic. The physical reason

for this not being the case will become clear as we discuss the interactions;

at that early date we were all still using BCS’s single parameter N (0)V .

The n-representation turned out to be useful in the hands of de Gennes;

I myself found it valuable in explaining the ﬁnite Knight shifts in small

particles.

But let me ﬁnally get to the topic I was invited to discuss here: “electron–

electron interactions in superconductors.” I will follow a line here that I

August 30, 2010 11:1 World Scientiﬁc Review Volume - 9.75in x 6.5in ch8

BCS: The Scientiﬁc “Love of My Life” 129

adumbrated in 2002 in a general talk

which started out from the incontro-

vertible fact that electrons, if brought close enough to each other, experience

the strong Coulomb repulsion e

/r, so that if one is to bind them together

in pairs one must somehow evade this hard core repulsion.

In 1959 David Pines decided to move from his assistant professorship at

Princeton to the University of Illinois, where his ﬁeld of many-body physics

was more appreciated, and he, John Bardeen and Gordon Baym formed

a very strong nucleus in that ﬁeld. At Princeton he left behind the sec-

ond of his brilliant French graduate students, Pierre Morel (the ﬁrst was

Philippe Nozieres). Pierre was employed by the French Consulate in New

York and could not follow him, so David bequeathed him to me — per-

haps the greatest of the many great favors David has done for me in the

course of a long association. Socially Pierre was a little rich for my blood

(one of my most vivid memories of him and his very beautiful wife was

their elegantly seeing us oﬀ on our cut rate Icelandair ﬂight to Cambridge

in Sept 1961 with an enormous bouquet of ﬂowers), but physics wise he was

just what I needed, extremely able mathematically yet not too enamored of

his own ideas.

I had a problem all ready for him, the possibility of exotic BCS-like

states with anisotropic gaps, which I had scribbled down in a notebook a

few months previously. I had in mind a mere theoretical exercise, useful

for the unusual properties it might exhibit in principle, but shortly it was to

become more physical, as we will see presently. But let me take the story out

of chronological order and write ﬁrst about the other problem we worked on,

the interactions responsible for the then-known, true BCS superconductors.

In fact, Pierre had already on his CV a published paper on this subject,

done under David’s direction, and it was a bit of an imposition for me to

make him revisit the subject — but with supreme self-conﬁdence, or blatant

cheek, I did.

Not that the previous work was wrong — it makes the perfectly valid

point that polyelectronic metals tend to have higher transition temperatures

because of the Umklapp processes allowed by the large Fermi surfaces. But

actually the arguments for this eﬀect were not very clearly presented. (It

may be characterized as a “crystal ﬁeld” eﬀect that enhances the electron–

phonon coupling somewhat over that of a pure “jellium” model.) There was

a great deal of physics yet to be done.

I had two physical insights that I wanted to explicate. The ﬁrst was

the idea that the long-wavelength phonons produced only a tiny portion

of the interaction. Presented rather clumsily in our eventual paper,

August 30, 2010 11:1 World Scientiﬁc Review Volume - 9.75in x 6.5in ch8

130 P. W. Anderson

appealed to the understanding that at long wavelengths the electron wave

functions move with the lattice as a neutral whole, so that the Coulomb

energy to couple the electrons and phonons is decreased by a factor q which

is proportional to ω, which makes the coupling, second-order in this en-

ergy, down by ω

. Indeed, later, when we were later able to measure

the spectrum of the interactions the low-frequency part seemed always to

be vanishingly small. (This contradicts recent claims

that a supposed

peak in forward scattering could cause d-wave superconductivity in the

cuprates.)

The second insight was that the phonon interaction is essentially local.

We put it that the phonons could be better approximated (taking into ac-

count that the long-wavelength spectrum is ineﬀective) by an Einstein model

(somewhat broadened) than by a Debye one. (In the end, the successful

approximate model for lead was two Einstein modes rather than one —

one each for longitudinal and transverse modes.) That is, there is no very

relevant structure in momentum space, especially after averaging over the

whole Fermi surface, so that the entire problem is reduced to a local dynam-

ics with the only relevant variation taking place in frequency space. This

gives one a very attractive visualization of the interaction process as the ﬁrst

electron whizzing past an atom site and giving its neighbors an attractive im-

pulse, and then the second electron coming by a little later, encountering the

attractive local change in ion charge density after the original electron has

long since gone but before the displacement of the ions has died out. One

may, that is, to a good ﬁrst approximation, neglect the k-dependence of the

interaction in solving the gap equation.

A third insight was not mine alone but had already been noticed by

Bogoliubov in his early papers,

in the course of solving the BCS gap equa-

tions. This is that the repulsive screened Coulomb interaction could be

replaced by a weaker pseudopotential insofar as its eﬀect on low-energy pro-

cesses is concerned. In fact, we argued that measures of the average strengths

of the Coulomb repulsion and the phonon-mediated attraction were neces-

sarily about equal, and the predominance of the latter stemmed only from

the pseudopotential weakening of the repulsive term.

The screened Coulomb potential is almost instantaneous on the scale of

phonon frequencies — its energy scale is the Fermi energy — and essen-

tially local — basically a constant. If the real action takes place at phonon

frequencies, we would wish to eliminate repeated scatterings up to much

higher energies. Since the early days of many-body theory we have known

to take this into account by replacing the “hard core” like potential U by a