1. Does of λ tell us something about the magnitude of Tc in HTSC?
INTERNATIONAL MATERIALS RESEARCH CONGRES 2008
Cancún, Qro. México, august 17-21
Does of λ tell us something about the magnitude of Tc in HTSC?
R. Baquero
Departamento de Física, Cinvestav
06/04/2019
R. Baquero

2. IT IS THE PHYSICS OF THE COOPER PAIRS (BCS, 1957)
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 εk. 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+λ).
The renormalization (averaged) is called the electron-phonon interaction parameter.
As a consequence, the renormalized energy can be written as;
Ek = εk/ (1+λ)
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
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R. Baquero

3. THE MAIN RESULTS IN THESE WORKS ARE: IN HTSC,
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, λ (S3-6, Nature,April).
THE MAIN RESULTS IN THESE WORKS ARE: IN HTSC,
1- THE ELECTRON-PHONON INTERACTION PARAMETER, λ, 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
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R. Baquero

4. I WILL ANALYZE IN DETAIL THIS CRITERIUM.
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 λ, THE HIGHER THE Tc” AND CONSEQUENTLY A SMALL λ 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 λ 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 λ, MIGHT EVEN BE “GOOD NEWS” FOR THE E-PH MECHANISM ALTHOUGH IT IS NOT AT ALL A PROOF OF IT IN ITSELF.
06/04/2019
R. Baquero

5. 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.
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R. Baquero

6. 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
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R. Baquero

7. 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
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R. Baquero

8. BCS THEORY Tc = 1.14 ~ωD exp{ - 1 / N(0)V}
HOW DO WE CALCULATE THE CRITICAL TEMPERATURE?
BCS THEORY
Tc = 1.14 ~ωD 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
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R. Baquero

9. 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~ωDexp{- 1/(λ - 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.
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R. Baquero

10. The higher the λ, the higher the Tc
HOW DO WE SET A LIMIT TO THE CRITICAL TEMPERATURE?
The higher the value of (λ - µ*) > 0, the higher the exponential
For conventional superconductors,
λ is of the order of 0.7 to 1.5 and µ* is of the order of 0.1– 0.13.
The higher the λ, the higher the Tc
06/04/2019
R. Baquero

11. Tc = 1.14~ωDexp{- (1+ λ) /(λ - µ*)}
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~ωDexp{- (1+ λ) /(λ - µ*)}
- 1 / N(0) V
WHICH IS A LITTLE BIT MORE ELABORATED EQUATION
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R. Baquero

12. 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
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R. Baquero

13. 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?
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R. Baquero

14. HOW DO WE SET A LIMIT TO THE CRITICAL TEMPERATURE?
From all these equations the criterium “the higher the λ, the higher the Tc” emerges and a finite limit to Tc can be obtained by taking λ to infinity.
Tc < 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?
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R. Baquero

15. DOES THIS CORRELATION EXISTS? THE HIGHER THE λ, THE HIGHER THE Tc
Tc [meV]
Material
lambda
0.1017 Al
0.43
0.2034 Tl Ta
0.69
0.2931 In Sn
0.72
0.3233
0.8
0.3612 Hg V
0.3862
0.81
0.434 La Mo
0.9
0.4621
0.98
0.5267 Bi
Nb(Rowell)
0.6198 Pb
Nb(Arnold)
1.01
0.7379 Ga
Nb(Butler)
1.22
0.7586
1.55
0.7931 Nb
1.62
2.25
2.45
DOES THIS CORRELATION EXISTS?
THE HIGHER THE λ, THE HIGHER THE Tc
AS WE CAN SEE FROM THE TABLE THE CORRELATION IS RATHER POOR EVEN FOR LOW-Tc SUPERCONDUCTORS.
06/04/2019
R. Baquero

16. 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 λ  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
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R. Baquero

17. THE ELIASHBERG FUNCTION FOR Nb AND FOR Nb-Zr
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
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R. Baquero

18. THIS IS A VERY IMPORTANT RESULT !
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 !
06/04/2019
R. Baquero

19. 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”
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R. Baquero

20. THEY CONTRADICT EACH OTHER!
THE CRITICAL TEMPERATURE IN ELIASHBERG AND BCS THEORY
TO GET A HIGH CRITICAL TEMPERATURE:
According to the λ 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 λ alone. And therefore leaves also without theoretical support the idea that “the higher the λ, 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.
06/04/2019
R. Baquero

21. 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.
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R. Baquero

22. IS THERE A HOPE TO PREDICT Tc?
THE CRITICAL TEMPERATURE IN ELIASHBERG THEORY
What can we learn from the equation
9 K
35 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!!!
06/04/2019
R. Baquero

23. What is the best that you can do with YBa2Cu3O6+X according to E.T.?
A should not be extremely big. A crystal lattice with a big A might not be stable.
06/04/2019
R. Baquero

24. YBa2Cu3O6+X ¡And we get a very reasonable value for A!
A should not be extremely big. A crystal lattice with a big A might not be stable.
YBa2Cu3O6+X
Let us consider a λ of the order of 2.5 (a value that could be guessed if we believe that “the higher the λ , the higher the Tc”. Then:
2.5 = 2A / 80  A = 100 meV
(Nb: A = 8 meV, Tc = 9 K and λ = 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 λ = 0.2 In this case, we get:
λ = 0.2 = 2 A/ 80  A = 8.
¡And we get a very reasonable value for A!
06/04/2019
R. Baquero

25. A should not be extremely big
A should not be extremely big. A crystal lattice with a big A might not be stable.
YBa2Cu3O6+X
λ = 0.2  A = 8
So, very low values for λ, 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.
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R. Baquero

26. CAN WE APPLY ELIASHBERG-MIGDAL THEORY AS IT IS FORMULATED NOW TO HTSC?
YBa2Cu3O6+X
PROBLEMS
1- RESONANCES REQUIRE A MORE DETAILED DESCRIPTION OF THE ELECTRONIC STATES
2- THE VALIDITY OF THE MIGDAL’S THEOREM SHOULD BE EXAMINED IN MORE DETAIL.
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R. Baquero

27. CONCLUSIONS
1- A low value of the electron-phonon interaction parameter, λ, is not an argument solid enough to discard the e-ph mechanism in HTSC.
2- The several assumptions made in Eliashberg theory ( Migdal’s 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)
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R. Baquero

28. Thank you!
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R. Baquero