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Transcript of Technical University of München Tutorial: The Physics of Superconductivity H. Kinder Onnes Landau...
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