Transcript Diapositiva 1

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
www.fis.cinvestav.mx/~rbaquero
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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
Whenever
pairs
a “free”
are a special
conduction
kind electron
of boundinteracts
state of two
withelectrons.
a phonon They
its mass
are
labeled with the free electron
renormalizes
quantum numbers
so that but they built up a different
THERE ARE TWO “KINDS”:
CONVENTIONAL
AND HIGH-Tc (HTSC)
Hilbert
space.
m* = m (1+λ).
The binding is supplied by a boson. In conventional superconductivity the
The renormalization (averaged) is called the electron-phonon interaction
boson is a phonon. We say that the mechanism is the e-ph interaction.
parameter. SUPERCONDUCTIVITY (E-PH)
WE UNDERSTAND WELL CONVENTIONAL
The binding energy per electron is called “the gap”.
As aTHEORIES
consequence,
the renormalized
energy
be written as;
ASSOCIATED
ARE: BCS
AND can
ELIASHBERG
In general, the gap is a function of the vector k and of the energy εk. Ounce we
Ek = εk/ (1+λ)
know the gap function, we can calculate
the free energy and from there the
functions
interest. ( MECHANISM? ).
HTSC ARE NOT thermodynamic
AT ALL UNDERSTOOD
AT of
PRESENT
None of them includes details of the electronic states
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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
Ek
k
Ek 
(1   )
THESE RESULTS
SEEM FINAL
cal  0.2 needed  1
THE KINK HAS BEEN
MEASURED AT T ~ 5 Tc
ek
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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.
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
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PROOF OF IT IN ITSELF.
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ELIASHBERG THEORY IS THE MANY-BODY SOLUTION (NONRELATIVISTIC 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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ELIASHBERG THEORY Tc-EQUATION
Tc  [ F (), *, c ]
2
The Eliashberg function
Electron-electron repulsion parameter
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The cutt-off frequency necessary to end
the infinite sum over the Matsubara
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frequencies
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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
Tc  [ F (), *, c ]
2
SINCE THIS PARAMETER CANNOT BE
NEITHER CALCULATED NOR MEASURED
WITH ENOUGH ACCURACY TO BE USEFUL
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ELIASHBERG EQUATIONS CANNOT
ACTUALLY PREDICT Tc
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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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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
N(0)
V }
Tc = 1.14~ωDexp{- 1/(λ
- µ*)
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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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.
 1
Tc  1.14 D exp   
 
The higher the λ, the higher the Tc
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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
1 / N(0)
Tc = 1.14~ωDexp{--(1+
λ) /(λV - µ*)}
WHICH IS A LITTLE BIT MORE ELABORATED EQUATION
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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:
ln


1.04(1   )
Tc 
exp  

1.2
    *(1  0.62 ) 
donde

GUESS
 F ( )
ln  exp{  ln( )
d}
0

2
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12
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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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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Tc [meV]
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.
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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
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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:
c
 F ( )
  2
d

0
2
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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 F ()
2
THE ELIASHBERG FUNCTION FOR Nb AND FOR Nb-Zr
Nb
Nb0.75Zr0.25
You can move some weight in
the Eliashberg function by
inserting some impurities into
the sample
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ELIASHBERG THEORY
 Tc
 F ( )
2
THE FUNCTIONAL DERIVATIVE OF Tc WITH THE ELIASHBERG
FUNCTION IS A FUNDAMENTAL RESULT OF THIS THEORY.

 Tc
2
Tc   2
( F ( ))d

F
(

)
0
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.
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THIS IS A VERY IMPORTANT RESULT !
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THE CRITICAL TEMPERATURE IN ELIASHBERG THEORY
HOW DOES THE FUNCTIONAL DERIVATIVE LOOKS LIKE?
opt  7KBTc
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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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”
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.
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DOES THE FUNCTIONAL DERIVATIVE UNIVERSAL RESULT WORK?
Nb3Ge
Tc= 23 K
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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IS THERE A HOPE TO PREDICT Tc?
THE CRITICAL TEMPERATURE IN ELIASHBERG THEORY
What can we learn from the equation
9K
 Tc

0?
2
35 K
  F ( )
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!!!
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What is the best that you can do with
YBa2Cu3O6+X according to E.T.?
 2 F ()  A  (  0 )
c
 F ( )
  2
d

0
0  opt
2

2A
opt

A    2 F ( ) d
0
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A should not be extremely big. A
crystal lattice with a big A might not be
stable.
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
2A
A should not be extremely
big. A crystal lattice with a
big A might not be stable.
opt
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!
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
2A
opt
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
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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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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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Thank you!
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