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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.