1 1 Superconductivity and Superconductivity and Quantum Coherence Quantum Coherence GGL Lent Term 2010 credits to: Christoph Bergemann, David Khmelnitskii, John Waldram, … • 12 Lectures: Tues & Thrs 11-12am Mott Seminar Room • 3 Supervisions, each with one examples sheet • This is a developing course – feedback is welcome! Complete versions on course web site: www-qm.phy.cam.ac.uk/teaching.php
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Superconductivity and Superconductivity and
Quantum CoherenceQuantum CoherenceGGL Lent Term 2010
Basic experimental facts:Basic experimental facts:• The resistivity of a superconductor drops to zero below some transition temperature Tc
• Immediate corollary: can’t change the magnetic field inside a superconductor
B = 0 B
Switch on external B:
zero field cooled
0 since ,0 curl curl ==−≡−=∂
∂ρρΕ J
t
B
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What if we cool a superconductor in a magnetic field and then switch the field off – do we get something like a permanent magnet?
field cooled
BExperimentally, this does not work – even when field cooled, the superconductor expels the field!
B
field cooled
This is known as the Meissner effect. Superconductivity arises through a thermodynamic phase transition (state depends only on final conditions, e.g., Tand B).
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The Meissner effect leads to the stunning levitation effects that underlie many of the proposed technological applications of superconductivity (see examples sheet).
The superconducting state is destroyed above a critical field Hc
NB: These curves apply for a magnetic field along a long rod.
B
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• exponential low-Tbehaviour indicative ofenergy gap(explained by BCS)
• power-law behaviour at low-T in unconventional superconductors(to be discussed later)
• matching areas means entropy is continuous at Tc consistent with second order phase transition
The electronic specific heat around the superconducting transition temperature Tc:
exponential in simple
superconductors
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From the form of C/T we find that the entropy S vs. temperature has the following form:
T
S
TcThe superconducting state has lower entropy than the normal state and is therefore the more ordered state. A general theory based on just a few reasonable assumptions about the order parameter is remarkably powerful. It describes not just BCS superconductors but also the high-Tc superconductors, superfluids, and Bose-Einstein condensates. This is known as Ginzburg-Landau theory.
normal state
superconducting state
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Landau Theory:Landau Theory:
For a second order phase transition, the order parameter vanishes continuously at Tc. In the Landau theory one assumes that sufficiently close to Tc the free energy density relative to the normal state can be expanded in a Taylor series in the order parameter, ψ
This assumes that the order parameter is real and that the free energy density is an even function of the order parameter.
Where is the free energy minimum?
)0(2
42)( >+= βψβ
αψψf
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Free energy curves:
Picture credits: A. J. Schofield
α > 0 α < 0
ψ ψψ0−ψ0
The phase transition takes place at α(Tc) = 0. Thus, a power series expansion of α(T) around Tc may be expected to have the following leading form:
This is enough to describe a second order phase transition, complete with specific heat jump (examples sheet).
f f
0) ( )( >−= aTTa cα
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This is appropriate for, e.g., ferrromagnetism where ψ in the uniform magnetization along a given axis. In the Ginzburg-Landau (GL) theory, however, ψ is assumed to be complex rather than real as is the case for a macroscopic wave function. We will see in a later lecture how a complex order parameter arises naturally from a microscopic theory. The assumptions in the GL theory are:
• ψ can be complex-valued
• ψ can vary in space – but this carries an energy penaltyproportional to
• ψ couples to the electromagnetic field in the same way as an ordinary wavefunction (Feynman, ch. 21)
Here, A is the magnetic vector potential and q is the relevant charge, which experimentally turns out to be q = –2e.
4422 , ψψψψ →→
2ψ∇
h/iqA−∇→∇
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This provides the first clue that superconductivity has got something to do with electron pairs. The idea of electron pairing is central to the microscopic theory.
A final part in the free energy that must not be forgotten is the relevant magnetic field energy density BM
2/2µ0, where BM=B-BE is due to currents in the superconductor and BE is due to external sources. (Note that when the material is introduced the total field energy density changes from BE
2/2µ0 to B2/2µ0, but
BMBE/µ0 is taken up by 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 term involve an effective mass m = 2me , which is consistent withq = –2e. This represents an effective field theory unifying matter field ψ & gauge field A (recall B=curlA) in the static limit.