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Superconductivity and Superconductivity and
Quantum CoherenceQuantum CoherenceLent Term 2007
Credits: Dr Christoph Bergemann
Quantum Matter Group
Also: David Khmelnitskii, John Waldram, …
• 12 Lectures: Mon, Wed 10-11am Mott Seminar Room
• 3 Supervisions, each with one examples sheet
• This is a new course – feedback is welcome!
• Printed lecture notes have intentional gaps for you to fill in the algebra during lectures
Complete versions on course web site:
http://www-qm.phy.cam.ac.uk/teaching.php
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Literature:Literature:J. F. Annett: Superconductivity, Superfluids and Condensatesunified treatment of all three phenomena
J. R. Waldram: Superconductivity of Metals and Cupratesmodern textbook with deep discussions,
including copper oxide superconductors
M. Tinkham: Introduction to Superconductivitytraditional textbook
V. V. Schmidt: The Physics of Superconductorshelpful insights
C. J. Pethick/H. Smith: Bose-EinsteinCondensation in Dilute GasesBEC and superfluidity; recent developments
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Outline:Outline:
• Phenomenology of the SuperconductingState (4 lectures)
• Applications of Superconductivity (1)
• Bose-Einstein Condensates (1)
• Superfluidity in 4He (1)
• Quantum Coherence and BCS Theory (3)
• Unconventional Superconductivity in
Exotic Materials & Superfluidity in 3He (2)
Macroscopic “Ginzburg-Landau” Treatment
Microscopic Theory
New Developments
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Lecture 1:Lecture 1:
• Historical overview
• Macroscopic manifestation of superconductivity: ρ, χ, C/T
• Meissner effect and levitation
• Type-I and type-II superconductivity
• Superconductivity as an ordered state – Landau theory as a precursor to Ginzburg-Landau theory
• Literature: Waldram ch. 4 (or equivalent chapters in Annett, Schmidt, or Tinkham)
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Timeline:Timeline:
Unconventional superconductivityUnconventional superconductivity
(e.g. p(e.g. p--wave)wave)1980s1980s--nownow
Josephson effect and SQUIDsJosephson effect and SQUIDs1962/641962/64
Superfluidity in Superfluidity in 33HeHe19711971
GinzburgGinzburg--Landau theory of superconductivityLandau theory of superconductivity19501950
Prediction of BosePrediction of Bose--Einstein condensation (BEC)Einstein condensation (BEC)19251925
Superfluidity in Superfluidity in 44HeHe1927/381927/38
BEC and BCS in atomic gasesBEC and BCS in atomic gases1990s1990s--nownow
HighHigh--temperature superconductorstemperature superconductors19861986
BCS theory of superconductivityBCS theory of superconductivity19571957
Superconductivity in mercurySuperconductivity in mercury19111911
Liquefaction of Liquefaction of 44HeHe19081908
?
KamerlinghOnnes
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Examples of SuperconductorsExamples of Superconductors
0.3 K
1.5 K
164 K
92 K
~35K
24.5 K
39 K
~0.8 K
10 K
9.3 K
4.1 K
first ferromagnetic superconductorUGe2
cuprate superconductor with Tc above liquid nitrogen temperatures
YBa2Cu3O7-δ
highest Tc superconductor to dateHgBa2Ca2Cu3O8+δ
used in superconducting magnetsup to ~ 9 T
NbTi
used in superconducting magnetsup to ~ 20 T
highest Tc amongst “conventional” superconductors
first of the heavy-fermion superconductors
Nb3Sn
MgB2
CeCu2Si2
p-wave superconductorSr2RuO4
first of the cuprate superconductorsLa2-xBaxCuO4
highest Tc amongst the elementsNb
first superconductor ever discoveredHg
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Superconducting elements:Superconducting elements:
(from www.webelements.com - see also examples sheet)
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Basic experimental facts:Basic experimental facts:
• The resistivity of a superconductor drops to zero below some transition temperature Tc
• The temptation is to explain this merely as an absence of scattering – but we will soon see that there is much more to the story
• Immediate corollary: can’t change the magnetic field inside a superconductor
B = 0 B
Switch on external B:
zero field cooled
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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
Switch off external B:B B
Experimentally, the first step does not work – even when field-cooled, the super-conductor expels the field!
B
field cooled
This is known as the Meissner effect and is the first indication that the superconducting transition is a true thermodynamic phase transition.
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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).
Obviously, flux expulsion carries a pro-gressive energy penalty and will eventually break down at high enough fields. This leads to the destruction of the superconducting state above a critical field Hc
Ideal magnetisation curve…
Hc
…and so-called type-II superconductivity(which we’ll discuss later)
Hc1 Hc2H
M
NB: Subtleties arise from geometry-dependent demagnetisation effects; these curves are strictly valid only for a long rod configuration
B
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• anomaly at Tc confirmssecond order phasetransition
• exponential low-Tbehaviour indicative ofenergy gap (only reallyexplained by BCS)
• powerlaw low-T behaviour →→→→ unconventional superconductivity (to be discussed later)
• areas match to conserve entropy
So, if we are really faced with a phase transition, we should have a look at the specific heat:
Picture credits: A. J. Schofield
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From the form of C/T we find that the entropy vs temperature looks as follows:
T
S
TcThe superconducting state has lower entropy and is therefore themore ordered state. From what we know so far, the nature of the order parameter is totally unclear. However, a general theory based on just a few reasonable assumptions about the hypothetical order parameter is remarkably powerful. It describes not just BCS superconductors but also the high-Tcs, superfluids, and BECs. This is known as Ginzburg-Landau theory.
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Landau Theory:Landau Theory:Near the phase transition, the order parameter – let’s call it ψ –will be small. We can therefore write down a Taylor expansion ofthe free energy density:
if F(-ψ) = F(ψ)
Where is the free energy minimum?
• for α > 0, the minimum is at ψ = 0 →→→→ disordered state
• for α < 0, the minimum is at ψ = ±±±±ψ0 →→→→ order
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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 = 0. Thus, a power series expansion 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).
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So far this description is good, e.g., for magnetic phase transitions (where ψ = magnetisation). The Ginzburg-Landau theory makes a few further assumptions that treat ψ in effect as a macroscopic wave function. This was originally motivated by experimental results, but we will (in a few weeks) see how the wavefunction nature of the order parameter arises from a microscopic theory.
The extra assumptions are:
• ψ can be complex-valued
• ψ can vary in space – but this carries an energy penalty
• ψ couples to the electromagnetic field in the same way as an ordinary wavefunction
Here, A is the magnetic vector potential and q is the relevant charge, which experimentally turns out to be q = –2e.
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This provides the first clue that superconductivity has got something to do with electron pairs. This plays a crucial part in the microscopic theory.
A final part in the free energy that must not be forgotten is the energy cost of expelling the magnetic field:
So finally we arrive at the Ginzburg-Landau free energy density:
We have written the gradient term “QM-style” and thereby introduced an effective mass m, which turns out to be m = 2me . This is consistent with q = –2e.