11

Superconductivity andSuperconductivity andQuantum CoherenceQuantum 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

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”,

Oxford University Press, 2009

33

Literature:Literature:

JF Annett: Superconductivity, Superfluids andCondensates

JR Waldram: Superconductivity of Metalsand 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 Condensation in Dilute Gases

M Tinkham: Introduction to Superconductivity

VV Schmidt: The Physics of Superconductors

GE Volovik: The Universe in a Helium Droplet

44

Outline:Outline:

• Ginzburg-Landau (GL) Theory of the Superconducting State

• Applications of Superconductivity• Bose-Einstein Condensates• Superfluidity in 4He

• Quantum Coherence and Bardeen-Cooper-Schrieffer (BCS) Theory

• Unconventional Superconductivityin Advanced Materials

PhenomenologicalGL Theory

MicroscopicBCS Theory

NewDevelopments

55

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 –

introduction to the Ginzburg-Landau theory

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

66

Timeline:Timeline:

Unconventional superconductors, Unconventional superconductors, includingincludinghigh temperature superconductorshigh temperature superconductors

1970s-now1970s-now

Josephson effect and SQUIDsJosephson effect and SQUIDs1962-641962-64

Superfluidity in Superfluidity in 33HeHe19711971

Ginzburg-Landau theoryGinzburg-Landau theorySuperconducting vorticesSuperconducting vortices

195019501952-571952-57

Prediction of Bose-Einstein condensationPrediction of Bose-Einstein condensation19251925

Superfluidity in Superfluidity in 44HeHeMeissner Meissner effecteffect

1927-381927-3819331933

Cold atomic gases, TopologicalCold atomic gases, Topological phases, phases, ……1990s-now1990s-now

BCS & BCS & Bogoliubov Bogoliubov theorytheory19571957

Superconductivity in mercurySuperconductivity in mercury19111911

Liquefaction of HLiquefaction of H22 & & 44He - the crucial stepsHe - the crucial stepsDewar & Dewar & OnnesOnnes

1898-19081898-1908

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Examples of SuperconductorsExamples of Superconductors

24.5 K

39 K

0.5-1.5 K

10 K

9.3 K

4.1 K

UGe2

CeCu2Si2 0.5 KCs3C60 up to ~40 KIron Pnictides up to ~60 KCopper Oxides up to ~160 K

superconducting magnets to ~ 9 TNbTi

superconducting magnets to ~ 20 T

high-Tc s-wave superconductor

Nb3Sn

MgB2

spin-triplet superconductors onthe border of ferromagnetism

Sr2RuO4

highest Tc amongst the elementsNb

first superconductor Tc =Hg

many additional important examples exist

Unconventional spin-singletsuperconductors on border ofMott transitions and/orantiferromagnetism

88

Superconducting elements:Superconducting elements:

ww

w.w

ebel

emen

ts.c

om

- e

xam

ple

s sh

eet

99

Basic experimental facts:Basic experimental facts:

• 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

Ideal magnetisation curve…

Hc

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

Hc1 Hc2

H

M

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

B

1212

• 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

1313

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

1515

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.

!

f ="#2

+$

2#

4+

1

2m(%ih& + 2eA)# 2+

1

2µo( B % BE )2