11

Superconductivity and Superconductivity and

Quantum CoherenceQuantum CoherenceGGL 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

22

Literature:Literature:

JF Annett: Superconductivity, Superfluids and Condensates

JR Waldram: Superconductivity of Metals and Cuprates

AJ Leggett: Quantum Liquids – Bose Condensation & Cooper Pairing in Condensed-Matter Systems

R Feynman: Lectures on Physics Volume III

A Altland & B Simons: Condensed Matter Field Theory

CJ Pethick & H Smith: Bose-Einstein Condensationin Dilute Gases

M Tinkham: Introduction to Superconductivity

VV Schmidt: The Physics of Superconductors

GE Volovik: The Universe in a Helium Droplet

33

Outline:Outline:

• Ginzburg-Landau theory of the Super-

conducting State (4 lectures)

• Applications of Superconductivity (1)

• Bose-Einstein Condensates (1)

• Superfluidity in 4He (1)

• Quantum Coherence and Bardeen-

Cooper-Schrieffer (BCS) Theory (3)

• Superfluidity in 3He and Unconventional

Superconductivity in Advanced Materials (2)

PhenomenologicalGL Theory

Microscopic BCS Theory

New Developments

44

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 the Ginzburg-Landau theory

• Literature: Waldram ch. 4 (or equivalent chapters in Annett, Leggett, Schmidt, or Tinkham)

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Timeline:Timeline:

Unconventional superconductors, Unconventional superconductors, includingincluding

high temperature superconductorshigh temperature superconductors

1970s1970s--nownow

Josephson effect and SQUIDsJosephson effect and SQUIDs1962/641962/64

Superfluidity in Superfluidity in 33HeHe19711971

GinzburgGinzburg--Landau theory of superconductivityLandau theory of superconductivity

AbrikosovAbrikosov vorticesvortices

19501950

1952/571952/57

Prediction of BosePrediction of Bose--Einstein condensation (BEC)Einstein condensation (BEC)19251925

Superfluidity in Superfluidity in 44HeHe

MeissnerMeissner effecteffect

1927/381927/38

19331933

BEC and BCS in atomic gasesBEC and BCS in atomic gases1990s1990s--nownow

BCS theory of superconductivityBCS theory of superconductivity19571957

Superconductivity in mercurySuperconductivity in mercury19111911

Liquefaction of Liquefaction of 44HeHe19081908

?

Kamerlingh Onnes

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Examples of SuperconductorsExamples of Superconductors

24.5 K

39 K

0.5 K

1.5 K

0.5 K to

164 K

92 K

35K

10 K

9.3 K

4.1 K

of ferromagnetism – p-waveUGe2

Tc above boiling point of nitrogenYBa2Cu3O7-δ

highest Tc to dateHgBa2Ca2Cu3O8+δ

superconducting magnets to ~ 9 TNbTi

superconducting magnets to ~ 20 T

high-Tc s-wave superconductivity

superconductivity on border of antiferromagnetism

Nb3Sn

MgB2

CeCu2Si2

superconductivity on the borderSr2RuO4

high-Tc d-wave superconductivityLa2-xBaxCuO4

highest Tc amongst the elementsNb

first superconductorHg

}carbon based compounds, iron arsenides, ….

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Superconducting elements:Superconducting elements:

www.webelements.com-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

• 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

99

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).

1010

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

Ideal magnetisation curve…

Hc

…and so-called type-II superconductivity(which we’ll discuss later)

Hc1 Hc2H

M

NB: These curves apply for a magnetic field along a long rod.

B

1111

• 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

1212

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

1313

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

1414

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

1515

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−∇→∇

1616

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.

2242)(

2

1)2(

2

1

2E

oBBeAi

m−++∇−++=

µψψ

βψα hf