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BCS: The Scientiﬁc “Love of My Life” 131

scattering matrix K obeying the Dyson equation

K(E, E

) = U + U

> ω

G(E, E

)K(E

, E

)

since most E

’s are  ∆, G is just the pair propagator −1/(2E), and this

may be solved as

K =

1 + N (0)U ln[E

/ω

]

. (1)

From these insights and with the help of two other contributions from

Russian workers came the now accepted theory of conventional, “BCS”,

superconductors, appropriately called the “dynamic screening” method.

These two other contributions were the time-dependent version of the en-

ergy gap equations worked out by Eliashberg,

and Migdal’s observation

that higher-order terms in the phonon self-energy were smaller in the ratio

[m(electron)/M(phonon)]

1/2

. Pierre’s attention was called to the Russian

work by Schrieﬀer during a visit to Illinois. (We had been using a cruder

version omitting the imaginary parts.) In essence, in this scheme, even

though the parameters λ and µ quantifying the attraction and the repulsion

(see below) are roughly equal, the wave function for the pair manages to

avoid the repulsive region in time — not in space — and feel a net attrac-

tion. In order to do so it requires the large logarithm in Ref. 1, i.e. a very

broad band.

In the original reference,

we normalized such quantities as N(0)V and

N(0)U to dimensionless coupling constants appropriate for an averaged,

jellium-like metal — it makes no sense to separate out an N (0) since the

coupling constants do not depend any more simply on density of states than

does, for instance, the phonon frequency. The dimensionless quantities

λ =













for the attractive phonon interaction,

and µ =



+ 4k



for the repulsion (2)

were relatively crude but physically sound estimates based on Fermi–Thomas

screening theory. Of course, details of the phonon spectrum can give some

variations on this very coarse-grained picture of the conventional polyelec-

tronic metals, but we at the time were interested primarily in achieving

understanding of the broad picture. Here k

is the screening wave number,

the root mean square phonon momentum, and k

the Fermi momentum.

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132 P. W. Anderson

Inserting these parameters, and solving the gap equation numerically, we

obtained a now familiar equation for the energy gap,



2ω

∆





λ −

1 + µ ln(ε

/ω

)



−1

= [λ − µ

∗

]

−1

, (3)

here is not the conventional upper limit of the phonon spectrum but the

hypothetical single Einstein phonon by which we are approximating it.

The simple estimates in (2) for the parameters give λ and µ very similar

values, each of order a bit less than 1/2, and in general with µ slightly larger.

λ after all simply represents the eﬀect of phonon screening in reducing the

Coulomb repulsion described by µ. For elements, as we showed, this gave

a reasonably good account of the general run of T

’s; and, particularly, of

measured isotope eﬀect coeﬃcients which are all less than the BCS 1/2,

and often much less. A few elements, and even more compounds, show

somewhat enhanced values of λ, which I attribute to local ﬁeld eﬀects. µ

∗

for good metals seems to retain a low value in the range 0.10–0.14. (We

also remarked that eventually almost all simple metals would be found to

be superconducting, a guess which has come true in the long run.)

During the winter of 1961–62, while in Cambridge, I gave a number of

seminars at English universities on this work. Rudolf Peierls attended the

one in Birmingham and asked whether I had thought about experimental

ways of seeing whether there was phonon-like structure in the gap, as the

above implies. After some thought I replied that I supposed it would show

up in tunneling spectroscopy — and it was only two weeks later I heard

from John Rowell at Bell Labs that he had seen just that, features in the

tunneling spectrum of Pb which seemed to correlate with (energy gap) +

prominent peaks in the phonon spectrum (known from inelastic neutron

scattering). There is actually a prior paper by Ivar Giaevar showing such a

feature,

which Schrieﬀer had identiﬁed correctly; John Rowell’s contribu-

tion was primarily to turn vague “features” into a spectroscopic tool using

diﬀerentiation techniques.

I had also seen a preprint from Bob Schrieﬀer in which he was us-

ing the then completely new technique of online computation to solve the

Eliashberg equations, on which Pierre and I had to use a complicated iter-

ation procedure. It seemed to me therefore, once John and I got the data

sorted out, that the wise way to proceed was to hand over the data to Bob’s

group, together with our suggested phonon spectrum (which, as mentioned

above, consisted of two broadened Einstein modes, one representing longi-

tudinal phonons and the other transverse) and to ask them to try to ﬁt the

rather detailed series of peaks which John saw. The results were beyond

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

BCS: The Scientiﬁc “Love of My Life” 133

expectation, and came out in two simultaneously submitted letters, one by

John, myself and his technical assistant;

and one by Bob and two students,

Doug Scalapino and John Wilkins.

These seemed to the group of us to be

clinching proof that the dynamic screening mechanism was correct for the

classic superconductors. I called it, in a historical talk in 1987, the moment

when “the fat lady sang.”

Bill McMillan took over working with Rowell on this problem at Bell

Labs with great competence, particularly ﬁnding a computer method for

inverting the Eliashberg equation in order to deduce the frequency spectrum

of the interaction α

F (ω), as we named it, in detail from given tunneling

data, thus dispensing with our rough “Einstein Mode” approximation (but

still neglecting propagation). He and Rowell successfully analyzed some half-

dozen further simple superconductors,

and the program was further reﬁned

and utilized by a number of later authors. McMillan,

in a very insightful

paper, expanded on the type of general estimation which Morel and I had

attempted, and created a formula very like our (3) above, with a few frills

and attempting to also estimate the Debye frequency parameter ω

in a

similar way.

In that period there were a number of suggestions extant as to how to

achieve the “Holy Grail” of appreciably higher superconducting transition

temperatures. One that was unaccountably popular was W. A. Little’s “ex-

citonic” mechanism

where he envisaged a metallic polymer chain with

the electron–electron interactions screened by electronic excitations in side-

groups attached to the chain. These excitations could be assumed to have an

arbitrary resonant frequency, and, taking the BCS gap equation literally, T

would scale with this frequency and be as large as one likes. This kind of idea

was taken up by others to justify other types of hypothetical inhomogeneous

systems.

A glance at the formula (3) shows that the Debye frequency appears

in it in two places, as the scale factor outside and as the large logarithm

in the exponent. These two have opposite eﬀects on T

, and for given λ

and µ there is actually an optimum ω

, which tends to be only slightly

above the physical value. (In the case of MgB

, I think the optimum is

very close to being achieved.) Marvin Cohen and I wrote a very short,

simple paper

pointing out these rather obvious conclusions and remarking

that there seemed therefore to be an upper limit to T

achievable by the

dynamic screening mechanism, in the neighborhood of 40

◦

K. So far this

prediction has proved to be rather precisely correct. The iron pnictides

and the cuprates, which strongly violate this rule, clearly use quite diﬀerent

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

134 P. W. Anderson

mechanisms for pairing. The “Bucky-ball” M

superconductors do not

violate it literally but do not satisfy the conditions for dynamic screening,

and may be presumed to require a special mechanism involving their unusual

band structure

and strong correlation eﬀects.

Our paper aroused considerable opposition; it seemed that there was a

strong emotional commitment on the part of Bardeen’s group, as well as

some of the Russian school, to the dream that someday there might appear

a room-temperature superconductor; in particular, our claim that values

of λ much greater than µ would lead to dynamic instability was attacked

strongly as a rigorous statement rather than the practical rule of thumb

that it actually is. Fortunately, we now know that dynamic screening is

not the only mechanism, and await hopefully some further still undiscovered

mechanism that may yet achieve the Holy Grail.

I promised to come back to Pierre’s ﬁrst problem. As I said on entering

the subject of electron–electron interactions, the crucial problem is to some-

how evade the hard core screened Coulomb repulsion that a pair of electrons

inevitably incurs. In the dynamic screening mechanism, the evasion is in

time — the electrons are not on the same atom at the same time. There is

only one other possibility — to ensure that the electrons avoid each other

in space, i.e. that the pair wave function, the solution of the BCS gap equa-

tion, is zero at the origin. For two particles in empty space, interacting via

a symmetric potential, it can be shown that the lowest energy bound state

necessarily is the lowest s-wave, which cannot vanish at the origin; but the

case of a BCS pair is quite diﬀerent because the kinetic energy necessary

to create the node at the origin can be provided by the Fermi zero-point

energy. For this reason it is quite reasonable that the best solution may be

asymmetric around the Fermi surface and belong to a degenerate represen-

tation of the crystal point group (or, in an approximately isotropic case, to

a nonzero angular momentum).

This idea occurred independently to Lev Pitaevskii

and to me in 1959,

and we both published in 1960; I had been inﬂuenced by a remark of John

Fisher and some words in the book by David Thouless.

I had suggested

that Pierre look into it, and I also described it to Keith Brueckner during

his visit to Bell in the summer of 1959. Keith, who also knew of the work

at Los Alamos on the Fermi liquid properties of liquid He

, immediately

pricked up his ears and suggested that He

was a good candidate for such a

state, resulting from the interactions of not electrons but He

atoms, which

are also Fermions and below 1–2

◦

Kelvin behave as an isotropic Fermi sea.

(The atoms have an even harder repulsive core than electrons.) I agreed and

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BCS: The Scientiﬁc “Love of My Life” 135

relayed the suggestion to Pierre, who set out to work out the most likely

angular momentum given the He–He interaction, and to ﬁnd the properties

of such a state.

Within a month or two I was a little shocked to receive a preprint from

Keith along with a student, Soda, in which he estimated transition tempera-

tures for L = 1 and L = 2 states for He

as well as several possible higher L

states for nuclear matter. I quite frankly did not have any conﬁdence in such

estimates, using the bare interatomic potential, but, feeling very defensive of

Pierre’s rights in the problem, I argued that we deserved co-authorship for

thinking of the problem, and Keith was very accommodating. It turned out

he was right to jump in, there were at least two other sets of authors with

similar ideas and we only barely achieved priority for the idea of anisotropic

superﬂuid He

, though our choice of state and estimate of T

were both very

wrong (as were the competitors’). The world had to wait 12 years before the

real thing came along.

What I thought was worthwhile in what we were doing was to establish

some matters of principle. Were these anisotropic states real superconduc-

tors? (In spite of the fact that they violated Landau’s criterion for super-

ﬂuidity, that excitations must have a minimum velocity, and that most of

them were gapless in that the gap had zeroes.) They were indeed “super,”

and this was very enlightening as to the true nature of superﬂuidity. I still

ﬁnd Landau’s criterion quoted in the textbooks, but it became meaningless

in 1959. Also, they turned out to exhibit a number of diﬀerent states for any

given L and S, all with the same T

because the equation for T

is linear,

but with diﬀerent free energies; how did they choose among them? What

was the meaning of the apparent possibility of orbital ferromagnetism, and

how would it manifest itself? All of these questions really had to wait for

ﬁnal solution until we had the real stuﬀ at hand, but we could begin to ask

sensible questions. I gave a preliminary talk on these kinds of matters at

Utrecht in summer 1960; and Pierre wrote a long and detailed paper in ’61

on a possible L = 2 state for He

, which fortunately mentioned in passing

the other possibility of L = 1, S = 1.

I must stop indulging myself in He

nostalgia and get back to the subject

of electron–electron interactions — though actually the issues involved are

not all that dissimilar. But there is one vital aspect to consider — the

response of the superconducting state to disorder and scattering. It is very

hard to make He

impure, nothing dissolves in it — even He

— and the

only “disordered” He

is that contained in porous media, so in spite of its

minute T

of 2.6mk He

’s condensation was ﬁnally observed in 1972 and

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136 P. W. Anderson

conﬁrmed to be in an L = 1, triplet pair state. But standard metals and

alloys all tend to exhibit the “dirty superconductor” immunity to scattering,

which, it was easy to see, was not at all characteristic of anisotropic pair

states, since for these ordinary scattering around the Fermi surface was “pair-

breaking”, equivalent to magnetic scattering of a BCS superconductor. So

even when Balian and Werthamer

pointed out an error in our treatment

of spin in the triplet case, which allowed there to be a fully gapped state

which, however, was not immune to pair-breaking by ordinary scattering, it

was not considered plausible that any of the conventional superconductors

were other than BCS, dynamic-screening states.

There the matter stood until 1979–80, when two unexpected, but utterly

diﬀerent, types of materials were discovered to be superconducting, neither

of which were likely to be caused by dynamic screening. The ﬁrst discovery,

in 1979, was a so-called “heavy electron” mixed valence metal, CeCu

These are compounds of certain rare earth metals, Ce, Yb and the actinide

U, in which the f-shell metal is a magnetic ion at high temperature, but its

electrons form very narrow metallic f-bands at low T . The electrons that

become superconducting are dominated by the high state density in the

F band; although the transition temperatures are low, they are not much

smaller than the eﬀective band width so that the entropy gained at T

, and

hence the speciﬁc heat, is much larger than for conventional superconductors.

Three more such compounds, UPt

, UBe

, and URu

, were discovered

in 1983 by the Los Alamos group around Z. Fisk and H. R. Ott, and the

community ceased to think of Steglich’s compound as a rare anomaly. In

fact, in recent years a dozen or so more such compounds have been found,

many due to the eﬀorts of Lonzarich’s group at Cambridge.

The fact

that the f -shell ion is magnetic at high temperatures proves that the intra-

atomic Coulomb repulsion (which is often designated by U) dominates the

Fermi energy in the f-shell bands and that there can be no pseudopotential

renormalization to µ

∗

as in the elemental metals. Prima facie one must

assume that the pairing is unconventional, involving an asymmetric pair

function of some sort. This conclusion is supported by the fact that T

very structure-sensitive in most of these compounds.

It is somewhat inexplicable why this simple conclusion was resisted for so

long by the community consensus. Somehow, the quite subtle, complicated

dynamic screening mechanism became the old reliable accepted standard,

while the simpler mechanism, in principle, of building a pair function with

a node at the origin became thought of as strange and exotic. For decades

the burden of proof remained on anyone who claimed that the new classes of

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BCS: The Scientiﬁc “Love of My Life” 137

superconductors were not phonon-motivated and involved an unsymmetric

gap function.

In 1980 this new class of superconductors was joined by another, with

the discovery of an “organic superconductor,”

which also turned out to

be the ﬁrst of a large and growing family of compounds unlike either of

the ﬁrst two groups. Following work in the 70’s which led to the “organic

metal” TTF-TCNQ,

Bechgaard’s salts were “charge-compensated” stacks

of moderately large aromatic molecules like TTF. (See ﬁgure for a typical

such molecule):

The chemical structure and electronic properties of such a stack structure

are quite diﬀerent from the covalently-bonded polymers like polyacetylene

envisaged in Ref. 19. The electrons shared along the stack, which make the

substance a metal, are from the highest occupied π molecular orbital

(the HOMO) of the unsaturated part of the molecule. At ambient pressure

the substance is often an antiferromagnetic Mott insulator, and is induced

to become metallic and superconducting only by applying pressure, in many

cases. The ﬁrst discoveries were of approximately one-dimensional sys-

tems, the Bechgaard salts, but these were followed by systems which formed

approximately two-dimensional nets,

and the number of such materials

continues to grow, as does T

, within limits.

Even more than the Steglich heavy-electron group, these materials cried

out for treatment as a “strongly-interacting” system where U is greater than

the bandwidth. In fact, at ﬁrst sight they look like simple one-band Hubbard

models with anisotropic hopping energies t

. But in actuality the approx-

imately one-dimensional cases exhibited very complex behavior, especially

at high magnetic ﬁelds, much of which seems to be associated with Landau

levels and with nesting. The only very noteworthy fact about the supercon-

ductivity, which occurs below a large but reasonable Hc

, seems to be that

it is triplet in at least one case.

The strongly two-dimensional materials have certain resemblances to the

cuprates except that they cannot be doped and often are just on the metallic

side of a Mott transition. Hesitantly, we classify them as being what Bernevig

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138 P. W. Anderson

et al.

called “gossamer” superconductors, that is superconductors just on

the smaller U side of a Mott transition.

The next class of superconductors to be discovered were the notorious

“high T

” cuprate superconductors, in 1987.

These are by far the most

exciting, and ostensibly the most controversial, superconductors. Nonethe-

less I argue here that they are, in terms of electron–electron interactions, by

far the simplest and best understood case. Most of the physics excitement

(which still fully occupies those of us still in this ﬁeld) has to do not with

controversy about the causative interactions but with the many fascinating

and unusual phenomena that Nature has chosen to tease out of what is in

essence a rather simple system, so simple that we can use it as a guide to

the others I have just been describing.

I was the ﬁrst to point out

that the crystal and electronic structure

of the cuprates was conducive to being described by the “Hubbard Model”

Hamiltonian in an extremely simple case. This Hamiltonian (which was used

by Hubbard but probably not invented by him) is simply

H =

i,j,σ

∗

i,σ

j,σ

+ U

↑

↓

(4)

where the sites i lie on some simple lattice (in the cuprate case, a layer

structure of simple square planar Cu’s.) When U is small, (4) describes

a simple metal, but when U is large and there is one electron per atom,

we have a “Mott insulator”. As I ﬁrst pointed out,

the Mott insulator

is usually antiferromagnetic because of the “superexchange” eﬀect (I used

the preferable designation “kinetic exchange”), which is the second-order

perturbative eﬀect of the hopping integrals t; the exchange constant J is

proportional to t

/U. But the cuprate has the almost unique ability to

become a doped Mott insulator, that is to embody the Hamiltonian (4)

in the large U case without having exactly one electron per site.

The superexchange eﬀect can be made explicit by means of a canonical

transformation due to Rice and Ueda.

This transformation U

= e

may

be derived by gradually turning on the ﬁrst term of (4) while requiring that

matrix elements between diﬀerent eigenvalues of the second term remain

zero. The result, in the lowest energy subspace, is the “t-J Hamiltonian”

t−J

= U

−1

= P





i,j,σ

∗

i,σ

j,σ





P + J

i,j

· S

; J ∼ t

P = Π

(1 − n

↑

↓

) (5)

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BCS: The Scientiﬁc “Love of My Life” 139

here c

∗

and c are electron destruction and creation operators on sites i and

j, S is the spin on the designated site, and P is the “Gutzwiller” projection

operator which eliminates all doubly-occupied sites.

(Parenthetically, (5) can be re-expressed in terms of the “hat” or pro-

jected operators ˆc

∗

i,σ

= (1−n

i,−σ

∗

i,σ

and ˆc

i,σ

= (1−n

i,−σ

i,σ

that eliminate

the need for explicit P operators surrounding the kinetic energy. This is the

device used in Anderson’s “Hidden Fermi Liquid” theory of the overdoped

cuprates.)

The exchange (J) term in (5) is a product of four fermion operators

and thus can be looked at as an interaction vertex in the particle–particle

channel. Therefore it can be the source of electron pairing, as may ﬁrst have

been pointed out by Scalapino and Hirsch

in the context of heavy-electron

superconductors; if antiferromagnetic, it will tend to favor singlet pairs, and

if one is to assume that a node at the origin is required, that suggests d-

wave pairing, which they indeed mentioned. Rather confusingly, these and

other authors who followed them described this as “pairing caused by spin

ﬂuctuations” which implies to the incautious that the pairing is caused by

propagating or diﬀusing low-frequency modes in a comparable mechanism to

the BCS phonons; but the derivation of J above shows that the J interaction

is an essentially instantaneous vertex (on T

scales) in the case of the doped

Mott insulators, i.e. the cuprates. The spins that interact are created by a

process that resembles the spin ﬂuctuation hole-particle resummation of Berk

and Schrieﬀer,

so there is a sense in which the original claim is true, but

misleading rather than wrong. But common sense

tells us that the large

interaction J must be the key to the pairing, and successful calculations

deriving the observed d(x

− y

) pairing using (5) alone have made that

certain, in my opinion. Nonetheless controversy on the dynamic nature of

the interaction continues, much of it fueled by appeals to the irrelevant

Eliashberg formalism.

In my experience common sense usually trumps

formalism.

The convoluted history of cuprate theory will be treated in more detail by

other authors in this book, so I will not go further than the above expression

of my conviction that the simple Hubbard model can account for almost

all of the fascinating phenomenology which is exhibited by these marvelous

compounds.

Yet another whole class of magnetic-type superconductors has recently

come to light, the iron pnictides such as FeAsLaO, doped by substitution of

F for O.

The T

’s are not up to the cuprates — maximum about 55

◦

K —

but the sheer number of quite diﬀerent chemical structures staggers the mind.

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

140 P. W. Anderson

All, however, have as the working portions a square planar Fe layer bound by

out-of plane pnictide atoms in a very characteristic structure. The undoped

system is usually magnetic but not a true insulator, more like a semimetal.

What is in common with cuprates is this square planar structure and that

it is both dopable and magnetic; but there is no escaping that it cannot be

a one-band Mott system, and in fact multiple Fermi surfaces are observed.

Moderate success has been achieved with rather complex renormalization

schemes based on the assumption that the interaction is purely electronic as

in the cuprates.

Without going in detail into the enormous literature on these four large

classes of unconventional superconductors, I will repeat and expand the

remark I made previously

: It seems very likely that they are all based

on the second method of avoiding the hard Coulomb core of the electron–

electron interaction, and therefore are all cases of unconventional, non-BCS

pairing (although of course they are all, at a deeper level, based on the BCS

coherent state wave-function.) Therefore an unbiased count of superconduc-

tors might well ﬁnd that this is the majority case. In most of these there is

good reason for ignoring the phonon coupling; and the prevalence of nearby

magnetic states strongly suggests that the magnetic interaction exempliﬁed

by the t-J renormalized Hamiltonian is the major culprit in the pairing.

In summary: The electron–electron interaction which is responsible for

turning most metals into superconductors at low enough temperature seems

to belong to one or the other of two almost non-overlapping classes. For the

simple metals and alloys, the dynamic screening mechanism elaborated from

the original BCS phonon scheme is one of the best-attested truths of quan-

tum materials theory. This is responsible for almost all technically useful

superconductors, so far. But there is a numerically far larger, if technolog-

ically unimportant, category of materials that depend on exotic Pitaevskii-

Thouless pairs with complex internal wave functions, vanishing at the origin

in order to evade the Coulomb self-interaction U.

References

1. P. W. Anderson, presented at “BCS50”, 2008.

2. P. W. Anderson, Phys. Rev. 110, 827 (1958).

3. N. N. Bogoliubov, V. Tolmachev and D. V. Shirkov, New Method in the The-

ory of Superconductivity, Academy of Sciences, Moscow (1958), Translation,

Consultants’ Bureau, NY, 1959.

4. P. W. Anderson, Phys. Rev. 112, 1900 (1958).

5. P. G. De Gennes, Superconductivity of Metals and Alloys (Benjamin, NY, 1966).