1 Superconductivity and Superconductivity and Quantum Coherence Quantum Coherence Gil Lonzarich Lent Term 2012 Acknowledgements: Christoph Bergemann, John Waldram, David Khmelnitskii, … and, importantly, former students • 12 Lectures: Wednesday & Friday 11-12 am, Mott Seminar Room • Three Supervisions, each with one examples sheet • Questions and suggestions are welcome Complete versions will be made available on the course web site: www-qm.phy.cam.ac.uk/teaching.php
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• Three Supervisions, each with one examples sheet
• Questions and suggestions are welcome
Complete versions will be made available on the course web site:www-qm.phy.cam.ac.uk/teaching.php
22
“Superconductivity, once called one of the best understoodmany-body phenomenon in physics, became again 100 yearsafter its discovery a problem full of questions, mysteries and
challenges.”
X.-G. Wen, MIT“Quantum Field Theory of Many Body Systems”,
• The resistivity of a superconductor drops to zero below sometransition temperature Tc
• Immediate corollary: can’t change the magnetic field inside asuperconductor
B = 0 B
Switch on external B:
zero field cooled
0 since ,0 curl curl ==!"!=#
#$$% J
t
B
1010
What if we cool a superconductor in a magnetic field and thenswitch the field off – do we get something like a permanentmagnet?
field cooled
BExperimentally, this does not work– even when field cooled, thesuperconductor expels the field!
B
field cooled
This is known as the Meissner effect.Superconductivity arises through athermodynamic phase transition (statedepends only on final conditions, e.g., Tand B).
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The Meissner effect leads to the stunninglevitation effects that underlie many of theproposed technological applications ofsuperconductivity (see examples sheet).
The superconducting state is destroyedabove a critical field Hc
NB: These curves apply for amagnetic field along a long rod.
B
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• exponential low-Tbehaviour indicative ofenergy gap(explained by BCS)
• power-law behaviour atlow-T in unconventionalsuperconductors(to be discussed later)
• matching areas means entropy is continuous at Tc: consistentwith second order phase transition
The electronic specific heat around the superconductingtransition temperature Tc:
exponential in simplesuperconductors
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From the form of C/T we find that the entropy S vs. temperaturehas the following form:
T
S
Tc
The superconducting state has lower entropy than the normalstate and is therefore the more ordered state. A general theorybased on just a few reasonable assumptions about the orderparameter is remarkably powerful. It describes not justconventional superconductors but also the high-Tcsuperconductors, superfluids, and Bose-Einstein condensates.This is known as Ginzburg-Landau theory.
normal state
superconducting state
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Free energy near a second orderFree energy near a second orderphase transition:phase transition:
For a second order phase transition, the order parametervanishes continuously at Tc. In the conventional description,known as the Landau model, one assumes that sufficiently closeto Tc the free energy density relative to that of the normal statecan be expanded in a Taylor series in the order parameter, ψ
This assumes that the order parameter is real and that the freeenergy density is an even function of the order parameter.
Where is the free energy minimum?
)0(2
42)( >+= !"!
#""f
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Free energy curves:
Pict
ure
cre
dits:
A.
J. S
chofiel
d
α > 0 α < 0
ψ ψψ0−ψ0
The phase transition takes place at α(Tc) = 0. Thus, a powerseries expansion of α(T ) around Tc may be expected to havethe following leading form:
f f
0) ( )( >!= aTTa c"
This is consistent, in particular, with a specific heat jump thatcharacterizes most superconductors (examples sheet).
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This is appropriate, e.g., for ferromagnetism where ψ in theuniform magnetization along a given axis. In the Ginzburg-Landau(GL) theory, however, ψ is assumed to be complex rather than realas is the case for a macroscopic wave function. We will see how acomplex order parameter arises naturally from a microscopictheory. The assumptions in the GL theory are:
• ψ can be complex-valued
• ψ can vary in space – but this carries an energy penaltyproportional to
• Crucially, ψ couples to the electromagnetic field in the sameway as for an ordinary wavefunction (Feynman III, ch. 21)
Here, A is the magnetic vector potential and q is the relevantcharge, which is found to be q = –2e.
!
"2 # "2
, "4 # "4
2!"
h/iqA!"#"
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This provides the first clue that superconductivity has gotsomething to do with electron pairs.
A final part in the free energy is the relevant magnetic fieldenergy density BM
2/2µ0, where BM=B-BE is due to currents in thesuperconductor and BE is due to external sources. (Note thatwhen the material is introduced the total field energy densitychanges from BE
2/2µ0 to B2/2µ0, but the part BMBE/µ0 is taken upby the external sources (Waldram, Ch.6)).
So finally we arrive at the Ginzburg-Landau free energy density:
We have written the free energy so that the gradient terminvolve an effective mass m = 2me , which is consistent withq = –2e. This represents an effective field theory unifyingmatter field ψ & gauge field A (recall B=curlA) in the static limit.