Diapositiva 1 - PowerPoint PPT Presentation

Loading...

PPT – Diapositiva 1 PowerPoint presentation | free to download - id: 56da0d-OGNhY



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

Diapositiva 1

Description:

Title: Diapositiva 1 Author: Dr. RAFAEL BAQUERO PARRA Last modified by: Dr. RAFAEL BAQUERO PARRA Created Date: 4/18/2008 2:14:19 AM Document presentation format – PowerPoint PPT presentation

Number of Views:72
Avg rating:3.0/5.0
Slides: 29
Provided by: DrRAFAEL4
Learn more at: http://www.fis.cinvestav.mx
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Diapositiva 1


1
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
2
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
3
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
4
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.
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.
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
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
8
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
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?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.
10
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
11
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
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
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?
14
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?
15
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.
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 ? ? 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
17
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
18
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 !
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
20
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.
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.
22
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!!!
23
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.
24
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!
25
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.
26
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.
27
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)
28
Thank you!
About PowerShow.com