INTERNATIONAL MATERIALS RESEARCH CONGRES 2008

Cancún, Qro. México, august 17-21

Does of ? tell us something about the magnitude

of Tc in HTSC?

R. Baquero Departamento de Física,

Cinvestav www.fis.cinvestav.mx/rbaquero

SUPERCONDUCTIVITY

SUPERCONDUCTIVITY OCCURS IN SYSTEMS WITH A

METALLIC CHARACTER (FERMI SURFACES, PHONONS, E-PH

INTERACTION, MAY BE ALSO OTHER INTS.) BELOW A

CERTAIN CRITICAL TEMPERATURE, Tc.

IT IS THE PHYSICS OF THE COOPER PAIRS (BCS,

1957)

Cooper pairs are a special kind of bound state of

two electrons. They are labeled with the free

electron quantum numbers but they built up a

different Hilbert space. The binding is supplied

by a boson. In conventional superconductivity the

boson is a phonon. We say that the mechanism is

the e-ph interaction. The binding energy per

electron is called the gap. In general, the gap

is a function of the vector k and of the energy

ek. Ounce we know the gap function, we can

calculate the free energy and from there the

thermodynamic functions of interest.

Whenever a free conduction electron interacts

with a phonon its mass renormalizes so that m

m (1?). The renormalization (averaged) is

called the electron-phonon interaction parameter.

As a consequence, the renormalized energy can

be written as Ek ek/ (1?)

THERE ARE TWO KINDS CONVENTIONAL AND HIGH-Tc

(HTSC)

WE UNDERSTAND WELL CONVENTIONAL SUPERCONDUCTIVITY

(E-PH) THEORIES ASSOCIATED ARE BCS AND

ELIASHBERG

HTSC ARE NOT AT ALL UNDERSTOOD AT PRESENT (

MECHANISM? ).

None of them includes details of the electronic

states

THE PROBLEM

EXCELENT STATE-OF-THE ART VERY RECENT

CALCULATIONS OF THE ELECTRON-PHONON INTERACTION

IN SOME HTSC GIVE VERY SMALL VALUES FOR THE E-PH

INTERACTION PARAMETER, ? (S3-6, Nature,April).

THE MAIN RESULTS IN THESE WORKS ARE IN HTSC, 1-

THE ELECTRON-PHONON INTERACTION PARAMETER, ?, IS

VERY SMALL. 2- THE CONTRIBUTION OF THE E-PH INT.

TO THE KINK IS VERY SMALL

THESE RESULTS SEEM FINAL

Ek

THE KINK HAS BEEN MEASURED AT T 5 Tc

ek

IN THIS TALK I WANT TO DEAL WITH THE FOLLOWING

PROBLEM

I WILL REFER TO THE ELECTRON-PHONON INTERACTION

AND TO ITS ROLE IN DETERMINING THE MAGNITUDE OF

Tc.

A WIDELY USED CRITERIUM IS

THE HIGHER

THE ?, THE HIGHER THE Tc

AND CONSEQUENTLY A SMALL ? MEANS THAT

THE E-PH MECHANISM IS DISCARDED.

I WILL ANALYZE IN DETAIL THIS CRITERIUM.

THESE RESULTS CONTRADICT THE EXISTING CRITERIA

AND BELIEFS ON THE SUBJECT

I WILL SHOW 1- THAT IT CONTRADICTS ELIASHBERG

THEORY IN SOME SENSE. 2- THAT IT IS HARDLY VALID

FOR HTSC. 3- THAT A LOW ? VALUE IS NOT ENOUGH TO

DISCARD THE E-PH MECHANISM. 4- AND, FINALLY, THAT

CONTRARY TO WHAT HAS BEEN CURRENTLY ARGUED, A LOW

VALUE OF ?, MIGHT EVEN BE GOOD NEWS FOR THE

E-PH MECHANISM ALTHOUGH IT IS NOT AT ALL A PROOF

OF IT IN ITSELF.

ELIASHBERG THEORY IS THE MANY-BODY SOLUTION

(NON-RELATIVISTIC FIELD THEORY) OF THE SAME BCS

THEORY IDEA SUPERCONDUCTIVITY IS THE PHYSICS

OF THE COOPER PAIRS

TO SOLVE THE ELIASHBERG EQUATIONS, DETAILS OF THE

SYSTEM ARE REQUIRED. THESE DATA (ELECTRONS,

PHONONS, E-PH INTERACTION) ENTER THE THEORY

THROUGH THE SO-CALLED ELIASHBERG FUNCTION.

ELIASHBERG THEORY GIVES A HIGHLY ACCURATE ACCOUNT

OF THE EXPERIMENTAL RESULTS OF CONVENTIONAL

SUPERCONDUCTORS.

ELIASHBERG THEORY Tc-EQUATION

The Eliashberg function

Electron-electron repulsion parameter

The cutt-off frequency necessary to end the

infinite sum over the Matsubara frequencies

HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?

ELIASHBERG EQUATIONS REQUIERE THE PREVIOUS

KNOWLEDGE OF THE MECHANISM THIS MEANS ALL THE

DATA (ELECTRON STATES, PHONONS, E-PH INTERACTION)

WHICH ARE INCLUDED IN THE ELIASHBERG FUNCTION

SINCE THIS PARAMETER CANNOT BE NEITHER CALCULATED

NOR MEASURED WITH ENOUGH ACCURACY TO BE USEFUL

ELIASHBERG EQUATIONS CANNOT ACTUALLY PREDICT Tc

HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?

BCS THEORY Tc 1.14 ?D exp - 1 / N(0)V

HERE THE PROBLEM IS THAT V CANNOT BE NEITHER

CALCULATED NOR MEASURED.

TO PREDICT Tc, PHENOMENOLOGICAL EQUATIONS WERE

BUILT UP

HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?

One of the first attempts to obtain some equation

that would allow to predict Tc was to keep the

form of the BCS equation but replacing the

unknown product VN(0) by some parameters

associated to Eliashberg formulation that could

eventually be measured or reasonably guessed.

BCS theory Tc 1.14?Dexp- 1/(? - µ)

N(0) V

THE IDEA HERE IS THAT THE TWO EXPRESSIONS HAVE

THE SAME MEANING, i. e.. BOTH REPRESENT A MEASURE

OF THE ATTRACTION BETWEEN THE TWO ELECTRONS THAT

CONSTITUTE A COOPER PAIR.

HOW DO WE SET A LIMIT TO THE CRITICAL TEMPERATURE?

The higher the value of (? - µ) gt 0, the higher

the exponential For conventional

superconductors, ? is of the order of 0.7 to

1.5 and µ is of the order of 0.1 0.13.

The higher the ?, the higher the Tc

HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?

AGAIN WITH THE SAME IDEA, WE CAN START FROM THE

Tc ELIASHBERG EQUATION TO OBTAIN AN EQUATION OF

THE DESIRED FORM BY MAKING SOME AD HOC

ASSUMPTIONS AND REPLACEMENTS. THIS PROCEDURE

RESULTS IN

BCS Theory Tc 1.14?Dexp- (1 ?) /(? - µ)

- 1 / N(0) V

WHICH IS A LITTLE BIT MORE ELABORATED EQUATION

HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?

THERE IS QUITE AN AMMOUNT OF EQUATIONS OF THIS

TYPE IN THE LITTERATURE THAT TAKE INTO

CONSIDERATION DIFFERENT ASPECTS IN THE HOPE THAT

AT THE END THEY WILL GET A REASONABLY GOOD

REPRODUCTION OF THE EXPERIMENTAL RESULTS. THE

MOST USED IS THE Mc MILLAN EQUATION AS MODIFIED

LATER ON BY ALLEN AND DYNES

GUESS

WE WILL BE DEALING WITH THE ELECTRON-PHONON

MECHANISM

ONE FIRST QUESTION

CAN WE SET A LIMIT TO THE CRITICAL TEMPERATURE

THAT HAS A THEORETICAL JUSTIFICATION?

HOW DO WE SET A LIMIT TO THE CRITICAL TEMPERATURE?

From all these equations the criterium the

higher the ?, the higher the Tc emerges and a

finite limit to Tc can be obtained by taking ? to

infinity.

Tc lt limit

IT IS IMPORTANT TO BE AWARE AT THIS POINT THAT

THESE EQUATIONS ARE RELATED, IN ONE WAY OR

ANOTHER, TO ELIASHBERG THEORY BUT THAT THEY ARE

NOT A PART OR A CONSEQUENCE OF IT

DOES THE CRITERIUM WORK?

Tc meV Material Material lambda

0.1017 Al Al 0.43

0.2034 Tl Ta 0.69

0.2931 In Sn 0.72

0.3233 Sn Tl 0.8

0.3612 Hg V 0.8

0.3862 Ta In 0.81

0.434 La Mo 0.9

0.4621 V La 0.98

0.5267 Bi Nb(Rowell) 0.98

0.6198 Pb Nb(Arnold) 1.01

0.7379 Ga Nb(Butler) 1.22

0.7586 Mo Pb 1.55

0.7931 Nb Hg 1.62

0.7931 Nb Ga 2.25

0.7931 Nb Bi 2.45

DOES THIS CORRELATION EXISTS?

THE HIGHER THE ?, THE HIGHER THE Tc

AS WE CAN SEE FROM THE TABLE THE CORRELATION IS

RATHER POOR EVEN FOR LOW-Tc SUPERCONDUCTORS.

THE ELECTRON-PHONON INTERACTION PARAMETER AND Tc

WHAT IS IT LAMBDA? IT IS THE PARAMETER THAT

CHARACTERIZES THE AVERAGE STRENGHT OF THE

ELECTRON-PHONON INTERACTION. IT IS A PROPERTY OF

THE NORMAL STATE. .

BUT LAMBDA CAN ALSO BE CALCULATED FROM THE

ELIASHBERG FUNCTION LIKE THIS

IF A HIGH ? ? A HIGH Tc

THEN IT IS IMPLIED THAT

THE WEIGHT THAT THE ELIASHBERG FUNCTION HAS AT

LOW FREQUENCIES IS WHAT DETERMINES THE MAGNITUDE

OF Tc

THE ELIASHBERG FUNCTION FOR Nb AND FOR Nb-Zr

Nb

Nb0.75Zr0.25

How does a change in the Eliashberg function

influences the magnitude of Tc?

You can move some weight in the Eliashberg

function by inserting some impurities into the

sample

ELIASHBERG THEORY

THE FUNCTIONAL DERIVATIVE OF Tc WITH THE

ELIASHBERG FUNCTION IS A FUNDAMENTAL RESULT OF

THIS THEORY.

This equation gives us the answer. It tells us by

how many degrees the critical temperature will

change due to the change that we have introduced

in the spectral (Eliashberg) function. THIS IS A

VERY IMPORTANT RESULT !

THE CRITICAL TEMPERATURE IN ELIASHBERG THEORY

HOW DOES THE FUNCTIONAL DERIVATIVE LOOKS LIKE?

THIS RESULT IS A FUNDAMENTAL PART OF ELIASHBERG

THEORY

IT HAS A MAXIMUM WHICH TURNS OUT TO BE UNIVERSAL

THIS MEANS THAT THERE EXISTS A PARTICULAR

FREQUENCY IN THE ELIASHBERG FUNCTION THAT SETS

THE VALUE OF THE CRITICAL TEMPERATURE. IT IS

CALLED THE OPTIMAL FREQUENCY

THE CRITICAL TEMPERATURE IN ELIASHBERG AND BCS

THEORY

TO GET A HIGH CRITICAL TEMPERATURE

According to the ? criterium the lowest

frequencies are the important ones

THEY CONTRADICT EACH OTHER!

According to Eliashberg theory The higher

frequencies are the important ones

This resul leaves without theoretical foundations

all the Tc approximate equations based on the

knowledge of ? alone. And therefore leaves also

without theoretical support the idea that the

higher the ?, the higher the Tc. This, as we

just saw, is specially important when we are

refering to a HTSC. The Tc approximate

equations and the citerium should be rejected.

DOES THE FUNCTIONAL DERIVATIVE UNIVERSAL RESULT

WORK?

Nb3Ge Tc 23 K

IT WORKS!

The conclusion is therefore that Nb3Ge is an

optimized system. It is only when you know the

functional derivative that you can arrive at this

kind of sharp conclusions.

IS THERE A HOPE TO PREDICT Tc?

THE CRITICAL TEMPERATURE IN ELIASHBERG THEORY

What can we learn from the equation

35 K

9 K

What is exactly involved in shifting the position

of the maximum that occurs in the functional

derivative? The components are i- the phonons

in the system ii- The e-ph interaction Iii- The

wave functions of the free electrons.

THIS EQUATION IS THE KEY AND IT ONLY HOLDS IN

THE SUPERCONDUCTING STATE!!!

What is the best that you can do with YBa2Cu3O6X

according to E.T.?

A should not be extremely big. A crystal lattice

with a big A might not be stable.

YBa2Cu3O6X

A should not be extremely big. A crystal lattice

with a big A might not be stable.

Let us consider a ? of the order of 2.5 (a value

that could be guessed if we believe that the

higher the ? , the higher the Tc. Then 2.5

2A / 80 ? A 100 meV (Nb A 8 meV, Tc 9 K

and ? 1) A lattice with such a big value of A

would most probably be unstable.

On the other hand, a state-of-the art recent

calculation gives for YBCO7 ? 0.2 In this

case, we get ? 0.2 2 A/ 80 ? A 8. And

we get a very reasonable value for A!

YBa2Cu3O6X

A should not be extremely big. A crystal lattice

with a big A might not be stable.

? 0.2 ? A 8

So, very low values for ?, in an e-ph

superconductor with a high critical temperature

for which Eliashberg theory holds (if it exists)

might be necessary to keep the value of A

reasonably low. This might be compulsory for the

lattice to remain stable.

YBa2Cu3O6X

CAN WE APPLY ELIASHBERG-MIGDAL THEORY AS IT IS

FORMULATED NOW TO HTSC?

PROBLEMS

1- RESONANCES REQUIRE A MORE DETAILED

DESCRIPTION OF THE ELECTRONIC STATES

2- THE VALIDITY OF THE MIGDALS THEOREM SHOULD

BE EXAMINED IN MORE DETAIL.

CONCLUSIONS

1- A low value of the electron-phonon interaction

parameter, ?, is not an argument solid enough to

discard the e-ph mechanism in HTSC. 2- The

several assumptions made in Eliashberg theory (

Migdals theorem, description of the electron

states, mean field approximation) should be

examined for their validity in HTSC. 3- According

to the results deduced from Eliashberg theory,

the factors that determine Tc are to be search in

relations that occur only in the superconducting

state (not the konk, not the e-ph int. parameter)

Thank you!