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

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III. New Superconductors

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August 30, 2010 11:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch15

PREDICTING AND EXPLAINING T

AND OTHER

PROPERTIES OF BCS SUPERCONDUCTORS

Marvin L. Cohen

Department of Physics, University of California,

and

Materials Sciences Division,

Lawrence Berkeley National Laboratory,

Berkeley, CA 94720, USA

mlcohen@berkeley.edu

After providing some history and background material regarding the evo-

lution of research on superconductivity, a description of the use of the BCS

theory for the development of approaches for calculations and predictions

of superconducting properties will be given. The emphasis is on estimates

of T

raising T

, and predicting new superconductors. The basic analysis

will be based on the BCS formalism with modern extensions. The focus

here is on phonon mediated pairing of electrons as described in the origi-

nal BCS paper augmented by current modiﬁcations and the use of modern

calculational approaches.

1. Introduction

The famous photograph (Fig. 1) of the attendees at the 1911 Solvay Confer-

ence was taken at the Metropole Hotel in Brussels in the fall of that year.

Heike Kamerlingh Onnes is shown standing next to Albert Einstein. One

can speculate that the smile on Onnes’ face reﬂects a very happy period

in his research career since superconductivity was discovered

in his labo-

ratory in the spring of 1911, and one can also wonder about what went on

during the conversations between Onnes and Einstein on superconductivity

at that time. It has been reported that Einstein expressed doubt that a

general theory explaining Onnes’ observations could be developed. It took

46 years. The BCS paper

was published in 1957, which was unfortunately

after Einstein’s death in 1955.

The theory has been extremely successful. It explained all the properties

of conventional superconductors and provided concepts and formalisms that

375

August 30, 2010 11:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch15

376 M. L. Cohen

Fig. 1. Photograph of the attendees of the 1911 Solvay Conference taken in the

Metropole Hotel. Kamerlingh Onnes (third from right) stands next to Einstein

(second from right).

eventually found their way into many areas of physics such as nuclear struc-

ture and the physics of Bose–Einstein condensates. The concept of pairing

introduced by Leon Cooper

was an essential ingredient of the BCS theory as

was the mechanism for the use of lattice vibrations to provide the attractive

pairing interaction. Measurements of the isotope eﬀects on T

provided the

stimulus for considering lattice vibrations, and theoretical studies such as the

one by John Bardeen and David Pines

explicitly demonstrated the attrac-

tive phonon-induced pairing interaction and its form. The many-electron

features of the BCS theory evolved from the many-body wave function sug-

gested by J. Robert Schrieﬀer.

The details have been well documented

elsewhere, and the BCS theory has joined the ranks of great theories and

conceptual breakthroughs of twentieth century physics.

Here the focus will be on the electron pairing interaction, the so-called

mechanism of superconductivity, and on the use of BCS theory to solve for

the material properties of superconductivity and, in particular, T

2. Materials and T

After Onnes’ discovery of superconductivity in Hg, many materials were

tested. Empirical rules were established to help ﬁnd more superconductors.

It was soon found that only metals exhibited superconductivity. Another

August 30, 2010 11:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch15

Predicting and Explaining T

and Other Properties of BCS Superc o nductors 377

example of a rule of thumb was that bad conductors were better candidates

for superconductivity than good conductors. After BCS, this latter trend

was attributed to the argument that the stronger electron-phonon scattering

that gave rise to higher resistivities would be helpful for inducing the pairing

interaction. Some rules have been dropped, like the one that suggested

avoiding oxides. The earliest exception for the oxide rule was the discovery

of superconductivity in SrTiO

This rule was later found to be particularly

misguided since copper oxide superconductors

have been found to yield the

highest T

materials found thus far. Another rule was to avoid magnetism

related elements such as Fe. Again, this rule has been broken with the new

results found for Fe compounds.

Despite such shortcomings of the empirical approaches of experimentalists

searching for new superconductors (sometimes referred to as superconductiv-

ity alchemists), they were the ones who did ﬁnd the new superconductors.

The highest T

’s went up at a steady average rate of 1 K every three years

during the 75-year time period from 1911 to 1986. Despite the theoretical

advances by the Londons

in 1935, Ginzburg and Landau

in 1950, and

BCS in 1957, no changes in the time dependence of the maximum T

the frequency of discoveries of new superconductors occurred at those times.

Theory did not lead in this area.

To discuss the role that theory did have, it is convenient to divide su-

perconducting materials into classes 1 and 2, as shown in Fig. 2. Class 1

contains conventional BCS superconductors like Al, Sn, Pb, Hg, etc. For

these materials, there are many experimental tests showing in detail that

the BCS theory works. Similar tests and arguments can be made for the

other members of this class, such as C

-based

superconductors, some or-

ganic superconductors, degenerate semiconductors,

and MgB

which is

the highest transition temperature member of class 1 known at this time. A

theoretical description of their superconducting properties based on electron

pairing induced by electron-phonon coupling works extremely well for class

1, and it is possible to use the theory to predict new superconductors.

For class 2, the high T

copper oxides are the most studied. At this time,

the highest T

’s achieved for these systems

14,15

are approximately 135 K at

atmospheric pressure and 165 K at high pressures. Heavy fermion metals

with electronic masses comparable to that of muons have unusual supercon-

ducting properties but relatively low T

’s, some organics appear to be in this

class, and the interesting, recently discovered Fe-based materials with T

’s

up to around 57 K at this time are usually included in class 2. Although

some features of class 1 superconductors, such as electron pairing of some

August 30, 2010 11:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch15

378 M. L. Cohen

CLASSES OF

SUPERCONDUCTORS

1 BCS: conventional metals, C60,

some organics, doped

semiconductors, MgB2, …

--------------------

2 EXOTIC: copper oxides, heavy

fermion metals, some organics,

(Fe,Ni) oxypnictides , FeSe, …

Fig. 2. Approximate division of superconducting materials into two classes.

kind, is believed to be responsible for superconductivity in this class of ma-

terials, the general view is that augmentations or fundamental changes in

the BCS theory may be necessary to explain class 2 superconductivity. The

search for these modiﬁcations has been going on for more than 20 years,

however at this time, there is not yet consensus among researchers in this

ﬁeld.

The current situation is summarized in Fig. 3. For class 1, all experimen-

tal data have been explained. In some areas of research, such as calculations

of T

, the precision needed has not been achieved. This shortcoming is often

attributed to our inability to calculate the needed input to the BCS theory,

i.e. the normal state properties, with suﬃcient precision. However, there is

little concern at present with the applicability of the BCS theory for this

class of materials.

Although there is lack of consensus for an overall theory for class 2, there

has been some very interesting and imaginative theoretical research in this

area. The search for a general theory, particularly for the cuprates, has

led to a deeper understanding of spin related properties and the physics

of non-Fermi liquid systems.

Although some researchers have ruled out

the electron-phonon induced superconducting mechanism for the cuprates,

others have not.

One early claim often heard was that this mechanism

generally could not account for such high T

’s. For the range of T

’s found

thus far, this is likely to be an incorrect argument. However, the current

status is that most researchers in this area believe phonon induced pairing

in the cuprates is not the primary mechanism.

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Predicting and Explaining T

and Other Properties of BCS Superc o nductors 379

CURRENT SITUATION

CLASS 1--- ELECTRON PAIRS

PAIRING INDUCED BY ELECTRON-

PHONON INTERACTIONS.

ALL EXPERIMENTS EXPLAINED.

--------------------

CLASS 2---ELECTRON PAIRS

MANY THEORIES, BUT NO

CONSENSUS ON THE MECHANISM

Fig. 3. Synopsis of the current status of theory for the two classes of

superconductors.

What about the highest T

’s? The current upper limit record of 165 K

is found in a pressurized cuprate. This has been the record for about 15

years. For class 1 materials, the fullerene systems are pushing toward 40 K,

and T

is around 40 K for MgB

. At this point, it is diﬃcult to predict

whether the next record breaking T

will be for a material in class 1 or class

2. An important consideration when posing the question of higher T

’s is

that there are promising possible high T

systems within class 1, and the

theory is well established and to a great extent understood for those materi-

als. So why has it been so diﬃcult for theory to predict new superconductors

within this class and to raise T

? As mentioned above, the answer to this

question is primarily related to the fact that a precise calculation of the pair-

ing interaction requires a precise description of the normal state. Success

in this area began with the use of experimentally determined parameters

as input. The extension to more ab initio approaches came later when the

theory of the normal state became more reliable.

At that point, successful

predictions were made because structural phases and electron-phonon cou-

plings could be evaluated from ﬁrst principles. By the time superconductiv-

ity was discovered in MgB

, many theorists had access to highly developed

methods to calculate normal-state properties, so the microscopic origins of

the superconducting properties could be considered.

This research had

added beneﬁts. Because of the particular nature of the phonon-induced

pairing in MgB

, new perspectives arose and a great deal was learned.

Similar situations

occurred related to the research on fullerene-based

superconductors.

August 30, 2010 11:8 World Scientiﬁc Review Volume - 9.75in x 6.5in ch15

380 M. L. Cohen

3. Calculations

The Yogi Berra statement that “predicting is hard especially about the fu-

ture” is apropos when it comes to new superconductors. Hence calculations

after an experimental discovery are easier since it is very helpful to know

the possible range of parameters involved and to know the answer. When a

new candidate system is chosen, there are often many unknown variables to

consider. This is true even when working with class 1 materials.

If we focus on estimates of T

and the question of whether superconduc-

tivity will exist in potential candidates for new class 1 superconductors, we

are led to examine success stories in the past. Just as the alchemists devel-

oped useful rules and relationships for making useful materials, the so-called

superconductivity alchemists had useful rules related to the electron to atom

ratio and similar material properties. The successes of these approaches were

later examined in the light of the BCS theory. A common conclusion was

that the successful searches for higher T

materials involved ﬁnding systems

with high densities of electron states at the Fermi level, N(0), and mate-

rials which were near structural instabilities. These re-evaluations in the

light of theory were based primarily on the solution for T

using a simpli-

ﬁed model to solve the BCS gap equation. This model was similar to that

used by Cooper in his study of a single electron pair. The important point

is that the net attractive pairing interaction parameter in this model is di-

rectly proportional to N(0). The reason for lattice instability considerations

is because the attractive phonon induced pairing interaction parameter of-

ten results in a softening of phonons, and when phonon frequencies become

negative, crystals have structural instabilities. The formula for T

discussed

above based on the BCS model assumed a weak coupling pairing interaction

N(0)V where V is the net attractive pairing potential. As shown in Fig. 4,

the exponential dependence of T

on N(0) is the reason for the strong focus

on this quantity. The interaction V is composed of an attractive phonon

component and the repulsive Coulomb potential. These compete as seen in

the formula for the transition temperature T

basedontheBCSmodel,

∼ E(phonon) exp[−1/(λ − µ

∗

)] . (1)

Here E(phonon) represents an average energy of the phonons such as the

Debye temperature, and the net interaction in the exponent is the diﬀerence

between the dimensionless attractive phonon pairing interaction λ and the

modiﬁed repulsive electron-electron interaction µ

∗

These are the two components of N(0)V , i.e. N(0)V canbeviewedas

λ − µ

∗

. Superconductivity is expected when this diﬀerence is positive. The