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Predicting and Explaining T

and Other Properties of BCS Superc o nductors 381

Fig. 4. Some of the formulas discussed in the text for calculating the supercon-

ducting transition temperature.

Coulomb interaction is modiﬁed (hence the asterisk superscript) because of

the diﬀerence in energy scales between the two pairing potentials.

To include the eﬀects of renormalization and of stronger coupling, McMil-

lan

extended the coupling range to be applicable when the electron-phonon

parameter λ became larger and renormalization of this parameter is included.

The McMillan equation (Fig. 4) can be used for values of λ up to about

1.25. The extensions of the McMillan equation beyond the above limit led

to incorrect estimates of the maximum transition temperature, T

max

.This

gave a pessimistic view since the resulting expression for T

(λ) saturates.

Later, Allen and Dynes

solved the Eliashberg

equations for several model

phonon spectra and found that, for large λ, T

was proportional to

√

λ with

no saturation.

Modern calculations of T

are usually done numerically where the mod-

iﬁed BCS theory, as embodied in the Eliashberg equations, gives precise

estimates when precise normal state properties are used as input. There

are model solutions that result in simple formulas (for µ = 0) such as the

Kresin–Barbee–Cohen

27,28

model based on a ﬂat phonon spectrum

=0.25



ω

(e

2/λ

− 1)

−1/2

(2)

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

382 M. L. Cohen

where ω

 is an appropriately averaged square of the phonon frequency.

This expression gives the correct

√

λ limit at high λ and the exponential

BCS model solution for small λ.

The shift in searching for superconductors and higher T

superconductors,

in particular, by replacing the search for large N(0) materials to looking for

large λ’s was next. The separation suggested by the NV parameter is not

appropriate for most materials. Most systems are not modeled well by the

BCS model solution. A possible exception is the C

superconductors

where it can be argued that N and V are for the most part independent

of each other. The phonons responsible for the pairing are associated with

the intramolecular vibrations of C

.TheN factor comes in through the

electrons that originate from the alkali metals used for doping. Experimen-

tally, increasing the lattice constant to increase N resultsinanincreasein

in an expected way. Increasing N by exploring metal components with

d-electrons has also been suggested.

Another aspect of the fullerene systems is the proposal to increase V

by considering carbon molecules with higher curvature such as the case for

. Higher curvature is expected to yield stronger coupling between the

intramolecular vibrations and the itinerant electrons. A consequence of this

argument is that it suggests a trend in which this coupling would be weakest

in graphite, stronger for carbon nanotubes, and even stronger for C

and

This is consistent with the limited measurements obtained up to

this point. The T

of graphite is about 0.5 K, carbon nanotubes exhibit

superconducting ﬂuctuations around 15 K,

30,31

(A represents an alkali

metal) shows superconductivity above 30 K. No data is available yet for C

systems.

The measurements on carbon nanotubes are currently being reﬁned. At

this point, it appears that the theoretical prediction

of an inverse rela-

tionship between T

and the tube diameter is borne out. The data also

show some evidence that correlations between tubes are important. This

area could yield new insights into the nature of superconductivity for lower

dimensional systems.

To appreciate the thinking behind many of the proposals based on BCS

theory for achieving higher T

’s, it can be argued that for this theory, most

expressions for the transition temperature have the form, T

∼ ω

f(λ), where

is a frequency related to the energy scale of the boson causing electron

pairing and f(λ) is a function that increases with λ. This suggests two

paths to higher T

’s, increasing ω

or increasing λ. For the ﬁrst path, one

wants high frequency phonons which points to exploring compounds with

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Predicting and Explaining T

and Other Properties of BCS Superc o nductors 383

atoms having light masses. Electronic collective excitations have also been

suggested because of their high energy. For the second path, the search

has been for large λ’s. Hence, when considering the shift from looking for

large N(0) to large λ’s, an important caveat to add is that strong electron-

phonon interactions often lead to structural instabilities as described earlier.

Therefore, the desired material may not be stable at atmospheric pressure.

Calculations for Real Material

In the last section, an overview of the approaches used by theorists to explain

and predict properties of class 1 superconductors was given. The assump-

tion was that the BCS theory with phonon mediated electron pairing was

operative, and the focus was on T

calculations using approximate solutions

of the BCS gap equation in its original form. In early work, sometimes

the T

formulas described in Fig. 4 and Eqs. (1) and (2) were used,

and

sometimes multiple square well solutions to the BCS gap equation were em-

ployed.

The input for the normal state parameters such as eﬀective masses

and phonon couplings were taken from experiment. The result was several

successful predictions

6,12

of new superconductors such as some degenerate

semiconductors, GeTe, SnTe, and SrTiO

, where the normal state data used

were available because of transport and optical experiments. These success-

ful predictions were surprising because the carrier densities in these systems

are so low

6,12,32

relative to standard metals. The fact that the carrier densi-

ties could be varied and the availability of reliable normal state data supplied

important information made these estimates of T

possible. Another inter-

esting case

is the transition metal compound Nb

NbwhichisNbinthe

A15 structure. Here again the availability of data and normal state calcula-

tions for Nb and Nb compounds in the A15 structure provided input for the

semi-empirical calculation of T

The successes using ab initio calculations came with the development

of the Eliashberg theory

and the advances made in calculating normal

state properties.

If we restrict ourselves to phonon mediated pairing and

Eliashberg theory, then an expression for calculating λ is given in Fig. 4 in

terms of the Eliashberg function,

F (k, k



,ω) ≡ N (ε

)





δ(ω − ω

) , (3)

where N(ε

) is the density of states per spin at the Fermi level, g



is the

electron-phonon matrix element, and ω

is the frequency of the phonon

in jth branch with q = k − k



. The expression in Fig. 4 for the Eliashberg

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

384 M. L. Cohen

function has been averaged over wavevectors k and k



and hence is a function

of frequency only, whereas the expression given in Eq. (3) includes the initial

and ﬁnal electron wavevectors.

It is often convenient to solve the Eliashberg equations in the Matsubara

representation

where

λ(k, k



,n) ≡



∞

dωα

F (k, k



,ω)

2ω

+(2nπT )

. (4)

Here, the conventional λ is represented by the average

λ = λ(k, k



, 0). (5)

The ﬁrst ab initio calculations dealt with the averaged values for the

input to the Eliashberg equations, and it was based on a method that was

developed for calculating electron-phonon couplings.

The application to

high pressure Si was a dramatic example of the power of this approach.

Using only the atomic number and atomic mass for Si the total energy

for

candidate crystal structures were compared at various pressures. Several new

forms of Si were successfully predicted.

Two of these, simple (or primitive)

hexagonal, sh, and hcp Si were studied in detail. The existence of these

phases, their stability, lattice constants, mechanical properties, electronic

structures, vibrational properties and electron-phonon coupling were then

determined from ﬁrst-principles calculations. Using the densities of states

at the Fermi energy as a scaling parameter, the Coulomb potential µ

∗

could

be estimated. The result was a prediction

of superconductivity for both

phases and an estimate of T

and its pressure dependence. Considering the

limited input and successful predictions, it could be said that high pressure

Si became the best-understood superconductor.

As was pointed out earlier, MgB

was not predicted to be superconduct-

ing even though the theoretical tools were available. However, this material

proved to be an interesting prototype system to study. For metals with

fairly isotropic Fermi surfaces, λ can be related to the normal state mass

enhancement which in turn can be extracted from heat capacity data. In

contrast, for MgB

the anisotropic properties of the Fermi surface have to

be considered. This is possible using the theory associated with Eqs. (3)

and (4), and the result was an almost ab initio calculation of the properties

of MgB

20,21,39,40

including the transition temperature, isotope eﬀects, tun-

neling gaps, speciﬁc heat, temperature dependence of the gaps, and other

features. The agreement with experiment was excellent, and the calcula-

tions demonstrated that speciﬁc parts of the Fermi surface played a special

role which could be demonstrated by the existence of two superconducting

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 385

energy gaps. It was shown that a particular phonon mode strongly couples

to speciﬁc electronic states resulting in a relatively high T

5. Increasing T

The quest for higher superconducting transition temperatures began in 1911,

and although our theoretical understanding of superconductivity has ad-

vanced considerably and successful predictions of new superconductors have

been made, especially for class 1 superconductors, we are still far from having

a well-deﬁned path for increasing T

. For strong electron-phonon coupling,

the Allen–Dynes equation is more optimistic than the McMillan equation

which saturates as the coupling gets stronger. As discussed earlier, strong

coupling can lead to lattice instabilities, and it was also shown

that for

some cases, other issues arise such as the role of the frequency diﬀerences

between the Coulomb and phonon interactions.

The search for larger electron-phonon couplings also depended on the

development of methods capable of calculating the couplings with high pre-

cision. The early approach

was developed to work with a plane-wave basis

set, and calculations at many points in reciprocal space were needed to ob-

tain a good estimate of λ. A recent development

which has had, and

likely will have, a big impact in this area is the use of Wannier functions to

evaluate electron phonon couplings. This real space approach is illustrated

schematically in Fig. 5 where an electron in one cell is coupled to an electron

in another cell via a vibrating atom in a third cell. This approach allows a

detailed calculation of the electron and phonon self-energies illustrated by

the diagrams in Fig. 6. The calculations using this approach are equivalent

to plane wave calculations at millions of points in reciprocal space. At this

point, applications have been made to explore superconductivity in doped

diamond

and SiC,

the relationship between phonon softening and su-

perconductivity,

and to conduct tests of the strength of electron-phonon

couplings in cuprates

and Fe based superconductors.

The latter two are

important tests since there is still considerable interest in the role of phonons

for pairing in these materials. In particular, the kink observed

in angular

resolved photoemission experiments led to speculations that the size of the

λ extracted assuming the kink was caused by electron-phonon eﬀects could

lead to the observed high T

. The calculations, however, did not support this

view when assuming standard theory for the electron-phonon interactions.

As described earlier, the quest for higher T

materials has become a search

for conditions that will lead to large λ’s, either via large densities of states or

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386 M. L. Cohen

Wannier Representation

Fig. 5. Schematic diagram illustrating the interaction of two electrons in diﬀerent

crystal cells coupled by a vibrating atom in another cell.

Electron and Phonon Self-energy

Fig. 6. Feynman diagrams for electron and phonon self-energies that can be cal-

culated using the Wannier function approach.

large electron-phonon pairing potentials. An interesting approach is to look

at λ as a dimensionless ratio of electronic and vibrational spring constants

λ ∼

N(E

)I



Mω



, (6)

where I

 is the square of the displacement induced electron scattering

matrix element averaged over the Fermi surface, ω

 is an average of the

square of the phonon frequencies, and N(E

) is the electronic density of

states at the Fermi energy.

For harmonic systems, the Mω

 and λ are

roughly independent of M, however perhaps a better view is to look at T

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 387

for large coupling, where

∼ (λω

)

1/2

∼





1/2

. (7)

Equation (7) implies that what is needed for higher T

’s is stronger elec-

tronic spring constants η and light masses. This can be extended to explore

compounds with multiple masses in a unit cell, and estimates can be made

using information from molecular fragments.

Roughly speaking, strong

electronic spring constants are expected for strong bonds such as the cova-

lent or partially covalent bonds found in diamond and SiC. The problem

here is getting enough mobile carriers,

but progress is being made.

6. Conclusions

It is clear that for class 1 materials the BCS theory with modern ﬁeld the-

oretic extensions is extremely robust and predictive. There are no counter

experiments this author is aware of. The theory explains the superconducting

properties and gives reliable predictions. Although extensions using bosons

other than phonons appear promising, there are no experimental demonstra-

tions that superconductivity has been achieved in this manner even for class

2 superconductors. Tests of the phonon mechanism for class 2 materials us-

ing standard electron-phonon theory are not encouraging. If phonon induced

pairings are important, modiﬁcations of the current theory are needed. At

this point, it appears that only the electron pairing aspects of the original

BCS theory has received the blessing of the superconductivity community

for this class. No consensus exists at this time for a correct underlying

theory (or combination of theories) appropriate for the members of class 2

superconductors despite the interesting candidates available.

Regarding the quest for higher T

superconductors, it is not clear whether

class 1 or class 2 type materials are more promising. The known materials

of class 2 have the higher T

’s, but without a descriptive theory, it is diﬃcult

to make quantitative predictions. For class 1, at present MgB

and mate-

rials based on C

have the highest T

’s. Modiﬁcations of MgB

suggest

modest increases in T

.ForC

systems, recent veriﬁcations

of theoretical

predictions

are encouraging. There appear to be no barriers to rule out

the notion that class 1 materials can rival class 2 materials for the highest

’s.

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

388 M. L. Cohen

Acknowledgments

This work was supported by the NSF grant DMR07-05941 and Director,

Oﬃce of Science, Basic Energy Sciences, Materials Sciences and Engineering

Division of the U.S. Department of Energy under contract No. DE-AC02-

05CH11231.

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