Technical University of München

Tutorial: The Physics of Superconductivity

H. Kinder

Onnes Landau Ginzburg Abrikosov

Bardeen Cooper Schrieffer Bednorz Müller

Meissner

Outline

• basics

• normal state

• superconducting state, overview

• pair attraction

• interplay of pairs

• BCS Theory

• zero resistance

• Meissner state

• mixed state

• flux flow

Quantum Mechanics for Engineers

• Newton 1704: light consists of particles

• Huygens 1691: light is a wave

• Planck 1900: light-quanta E = h

• Heisenberg 1925: there are no particles nor waves:

both are manifestations of the same thing

a square? a circle? a cylinder!

basics

QM Survival-Kit

• De Broglie relation:

• Heisenberg's uncertainty principle:

• Planck's formula:

• Pauli principle:

E h

x p h

p h / k

Energy Frequency h/2

position momentum

wave length wave vector

they differ in position, momentum, or spin

2 Electrons (Fermions) have never the same state

basics

The Normal State

• electrons in metals can move almost freely

• 1 cm³ of Sn contains 5x1022 electrons:

momentum is fixed position is uncertain: x Lh h

px L

sample dimension

Pauli principle: all of their momenta must differ by p in 3 dimensions:3

3 3 3max

h Np N p N h

L V

3 22 3 34 243maxp N / V h 5 10 /1cm 2 10 Js 2 10 kg m / s

24 -30 6max max ev p / m 2 10 /10 m / s 2 10 m / s high speed!

"Fermi velocity"

normal state

Momentum Space

• states in momentum space

pz

pxp

p

py

• ground state at T=0:

states inside a sphere are occupiedto minimize total momentum

pz

px

"Fermi sphere"

average distance pL

normal state

Normal State at T > 0

• cross section of Fermi sphere:

Fermi momentum

occupation probability

f

pxpFermi

T1

T=0

T2

normal state

Normal State Carrying a Current

Fermi sphere is displaced by applied voltage:

rigid displacement

f

pxpFermi

I = 0I > 0

pz

px

more electrons going right than left

cross section:

normal state

The Superconducting State, Overview

• electrons in superconductors are bound to pairs "Cooper pairs"

• all pairs have the same momentum P = p1+p2

• bound state orbitals depend on materials:

no current: P = 0 or p1= -p2

opposite spins: • the total spin of each pair is zero

L=0: s-wave metallic + MgB2

L=2: d-waveHigh Tc

L angular momentum

• the binding energy of each pair is binding B cE 2 3.52 k T (BCS theory)

SC state

The Superfluid Condensate

• all pairs together form a classical wave

• the wave has amplitude and phase ie complex representation:

• the amplitude squared is the pair density2 Pairs

Pairs

Nn

V

(r, t)

• frequently used terms:macroscopic wave function

pair field

pair amplitude

superfluid

order parameter

gap parameter

the condensate

SC state

Waves on a Ring

n=1 n=2 n=3

. . .

• wave length must fit to the perimeter: n 2 r

• wave length momentum current magnetic flux

• i. e. the magnetic flux is quantized: = n0

0n

15 20

h2 10 Tesla m

e2

pairs

SC state

Josephson-Effect

• dual beam-interference with electron pairs

weak links B

current

magnetic field B (10-5T)cu

rren

t

• Superconducting QUantum Interference Device, SQUID

SC state

• isotope effect:

How can two electrons form a pair?

c

atomic

1T

m

atomic mass

Sn

log(matomic)

log(

Tc)

vibrations (=phonons) must play a role

pair attraction

Electron-Phonon Interaction

• principle: Fröhlich 1950

e-

electron at rest: moving electron:

screening overscreening

supersonic electron:

anti-screening

e-e-

net charge; positivenegative negative

isotope effect OK

c

atomic

1T range of attraction speed of sound

m

pair attraction

Remarks on the Matress picture

• demonstrates indirect interaction via another medium

• however: suggests static attraction

matress should vibrate!

• isotope effect depends on mass: dynamic attraction

pair attraction

Effective Attraction in Cuprates

• almost no isotope effect

• neutron scattering: AF fluctuations persist in SC region

• phase diagram

on hole doping:

• are these the matress??no generally accepted understanding available yet

300K

antiferromagnetism (AF)and superconductivity (SC)closely related

pair attraction

2 2 22

kin

1 (mv) p pE mv

2 2m 2m 2m

hp x h x

p

estimate from uncertainty principle:

Pair Size

momentum p requires kinetic energy:

available Energy:

B c

hx

7 m k T diameter:

with Tc 20 K: x 50 nm

in reality: 0 1...10 nm

HTS LTS

binding B cE 3.5 k T

for Ekin Ebinding : B cp 7 m k T

"coherence length 0"

interplay of pairs

22 3N2 10 cm

V electron density in Sn was:

Overlap of Pairs

in a volume of : 103...105 pairs

HTS LTS

30

strong overlap!

e– must fulfil the Pauli principle like in normal conductors

the pairs are Fermions on the atomic scale

interplay of pairs

Synchronized Motion of Many Pairs

let all pairs go with same speed except for one maverick:

this one breaks the ranks

• the maverick is crossing all other's ways

• maverick must evade to empty states with higher energy• this costs too much energy

not allowed by Pauli principle

pair is broken up

to mimimize energy, all pairs must march in lockstep!

• conclusion:

interplay of pairs

Pairs Running in Lockstep

• all pairs have their centers of gravity in the same momentum state

"boson-like behavior", similar to photons in a coherent light wave

• why is the current frictionless?

• a nonzero momentum of the pairs corresponds to a transport current

demonstration

defect

scattering would change the velocity, break the pair and cost energy

elastic scattering is forbidden

interplay of pairs

BCS Theory

• Bardeen, Cooper, and Schrieffer 1957microscopic theory of

superconductivity

• BCS ground state (T = 0) in momentum space:

looks similar to NC at Tc

• big difference: Pair correlation:

if p occupied, then also -p

1

pF

p

occupation probability

0

2p

-pF

if p empty, then also -p

p

p

state

state

BCS theory

Quasiparticles

• excited sates of the superconductor

• anti-pair correlation:

single electrons, broken pairs

• minimum excitation energy = binding energy of pairs B c2 3.52 k T

energy gap of the superconductor

• quasiparticles exist only at finite temperatures

p

p

state

state

if p occupied, then -p empty if p empty, then -p occupied

BCS theory

Superconductor at Finite Temperatures

• a quasiparticle in state p: blocks 2 pair states p and -p:

pair binding energy 2 is weakened

more pairs are broken in thermal equilibrium

• catastrophe occurs at some finite temperature: all pairs are broken up

broken pairs yield new quasiparticles

critical Temperature Tc

T

(T)

0

Tc

BCS theory

Supercurrents at T > 0

pair breaking phonons of Energy are abundant at TTc/2

dynamic equilibrium:

h 2

pair breaking recombination

• normal state resistance:

inelastic scattering

elastic scatteringdefects

T

0

phon

ons

• superconducting state:

inelastic scattering is not forbidden!

can phonons stop the pairs?

zero resistance

Can Phonons Stop the Pairs?

• on pair breaking, two quasiparticles are created:

• the quasiparticles block two pair states

• the blocked pair states move with the same speed as all other pairs

• recombination can only occur with quasiparticles of the same speed

• after recombination, the pair condensate goes on as before

• the total momentum of the condensate is always conserved

zero resistance

Superconductor in Weak Magnetic Fields

• in magnetic fields, the pairs don't fit together correctly

• but they dont feel the field when they move!

binding energy will decrease

for physicists:the "kinetic momentum" can compensate the "field momentum"

• consequence: a magnetic field sets the pairs in motion

2s p

super totalp

n qj B

m

spontaneous supercurrents occur when sample is cooled in field

• "2nd London equation":

the current is perpendicular to the field

Meissner state

B

Btotal

x

Bext

SC

Bshielding

surf

ace

Shielding

• the supercurrents have a magnetic field of their own

shielding 0 superB j

• Ampère's law:

• one finds that the field is opposite to the external field

the total field falls off rapidly into the superconductor

caracteristic length:

"magnetic penetration depth" ext

1B

e

100 nm

jsuper

Meissner state

Meissner Effect

is small, so macroscopic objects are virtually field-free

magnetic fields are expelled from superconductors

even when in-field-cooled

• this holds only in weak fields

when the supercurrents don't cost too much energy

Meissner state

Superconductor in strong Magnetic Fields

• in stronger fields, the condensate is no longer rigid

• how to reduce the currents?

supercurrents cost too much energy

let the field come in!

• simple behaviour. SC breaks down totally Type I superconductors

• intelligent behavior: vortices Type II superconductors

NbSe2 MgB2 LuNi2B2C

mixedstate

mixed state

Critical fields

0Bext

Bint

Bc

Type I: Type II:

Bc1 Bc20Bext

Bint

Bcth

can sustain much higher fields

all technical SC are of Type IIhistorically discovered first

fields up to 0.2 Tesla only

mixed state

What makes the difference?

interface energy between NC and SC in magnetic field:

x

NC

nsuper

SC

Bext

B(x)

0

Ebinding lost

Eshielding saved

> : more loss than gain

> : more gain than loss

spontanous creation of

internal interfaces

positive interface energy

negative interface energy

material parameter

/ controls the behavior

"Ginzburg-Landau-Parameter"

mixed state

Vortices as "Interfaces"

as many interfaces as possible:

• disperse flux as finely as possible:

• smallest possible flux in SC: 1 0 one flux quantum

0

vortexflux line

• vortices go throug from surface to surface, or they form ringsdiv B 0 :

SC SC

Shielding current

mixed state

Vortex motion

e. g. magnets, motors, transformers

magnetic field and transport current simultaneously:

vortices

Lorentz force

flux flow

vortices move at right angles with field and current; why?

microscopic picture

force on a pair: superF 2e v B

Lorentz ForceF

Hall voltage forces the pairs to go straight

but: counter force on vortices! motion of the vortex to the side!

flux flow

eddy field

sample boundary

wants to push the pairs to the side

Resistance due to Flux Motion:

• power consuption of one vortex: 1 vortexP F v

• N vortices:

transportV I

voltage drop!

• conclusion: superconductor has resistance

flux flow resistance

N vortexP N F v

• energy conservation:

flux flow

vortex transportV N F v / I

Experimental Result

• Ic depends on defect density

"technical" critical current

• inhomogeneities are locking the vortices: "flux pinning"

• v = 0 is enforced no work

no voltage drop, no resisitance

flux flow

V ideal

low defect density

higher defect density

IIc

segregationwith small (or even NL)

• segregation: vortex core can stay without cost in binding energy

Pinning Mechanisms:

condensation energyis lost

"pinning - force"

• particularly effective: defect sizee

• to go on will cost again energy

• i.e. segregation has a binding force

flux flow

Jc tech as Function of Temperature and Field

• decreases in magnetic field more vortices/pinning center

• decreases with temperature thermal activation of vortices

E

jjc tech

flux flow

Pinning in external magnetic field

• pinning impedes entrance and exit of vortices

• Bi is inhomogenous within the sample

Hysterese!

• Bc1 and Bc2 unchanged

frozen-influx

Bi

BaBc1 Bc2-Bc1-Bc2

virgin curve

ideal type II SC

flux flow

Field Distribution in the Sample:

• surface: jump ideal magnetisation curve

B

SLx

Ba < Bc1 (Meissner)

Ba Bc2

Ba grows

Bi

BaBc1 Bc2

• inside: field gradient gradient of vortex density

• gradient decreases with increasing field strengtn

• vortices move only if their repulsion force is greater than the pinning force

flux flow

Bean Model:

• density gradient shielding current

i 0 AbschirmB j

Ampère

• macroscopic average over vortices:

• here: i0 y

Bj

x

• if B/ x small: j < jc vortices pinnedx

y

• if B/ x larger: j > jc vortices are ripped away

"critical state"

• remark:

x

B

• move until everywherej = jc

measurement of dBi(x)/dx jc

flux flow