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Tutorial: The Physics of Superconductivity
H. Kinder
Technical University of München
Onnes Meissner Landau Ginzburg Abrikosov
Bardeen Cooper Schrieffer Bednorz Müller
Outline
• basics
• normal state
• superconducting state, overview
• pair attraction
• interplay of pairs
• BCS Theory
• zero resistance
• Meissner state
• mixed state
• flux flow
basics
Quantum Mechanics for Engineers
• Huygens 1691: light is a wave
• Newton 1704: light consists of particles
• 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
• Planck's formula: E h
Energy Frequency h/2
• Heisenberg's uncertainty principle: x p h
position momentum
• De Broglie relation: p h / k
wave length wave vector
• Pauli principle:
2 Electrons (Fermions) have never the same state
they differ in position, momentum, or spin
normal state
The Normal State
• electrons in metals can move almost freely
h h
momentum is fixed position is uncertain: x L p
x L
sample dimension
• 1 cm³ of Sn contains 5x1022 electrons:
Pauli principle: all of their momenta must differ by p in 3 dimensions:
3
h N
p3
max N p3 N h 3
L V
pmax 3 N / V h 3 5 1022 /1cm3 21034 Js 2 1024 kg m / s
v max p max / m e 2 1024 /10-30 m / s 2 106 m / s high speed!
"Fermi velocity"
normal state
Momentum Space
pz
• states in momentum space py
p
average distance p p px
L
pz
• ground state at T=0:
states inside a sphere are occupied px
to minimize total momentum
"Fermi sphere"
normal state
Normal State at T > 0
• cross section of Fermi sphere:
occupation
probability
f
T=0
T1
T2
px
pFermi
Fermi momentum
normal state
Normal State Carrying a Current
Fermi sphere is displaced by applied voltage:
pz
rigid displacement
px
I=0
f I>0
cross section:
more electrons going right than left
px
pFermi
SC state
The Superconducting State, Overview
• electrons in superconductors are bound to pairs "Cooper pairs"
• all pairs have the same momentum P = p1+p2 no current: P = 0 or p1= -p2
• the total spin of each pair is zero opposite spins:
• the binding energy of each pair is E binding 2 3.52 k BTc (BCS theory)
• bound state orbitals depend on materials: L angular momentum
L=0: s-wave L=2: d-wave
metallic + MgB2 High Tc
SC state
The Superfluid Condensate
• all pairs together form a classical wave (r, t)
• the wave has amplitude and phase complex representation: ei
N Pairs
n Pairs
2
• the amplitude squared is the pair density V
• 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
h
0 2 1015 Tesla m2
pairs 2e n 0
SC state
Josephson-Effect
• dual beam-interference with electron pairs
current
current
weak links B
magnetic field B (10-5T)
• Superconducting QUantum Interference Device, SQUID
pair attraction
How can two electrons form a pair?
• isotope effect:
1
Tc
Sn matomic
atomic mass
log(Tc)
vibrations (=phonons) must play a role
log(matomic)
pair attraction
Electron-Phonon Interaction
• principle: Fröhlich 1950
electron at rest: moving electron: supersonic electron:
e- e- e-
screening overscreening anti-screening
net charge; negative positive negative
1
Tc range of attraction speed of sound
matomic
isotope effect OK
pair attraction
Remarks on the Matress picture
• demonstrates indirect interaction via another medium
• however: suggests static attraction
• isotope effect depends on mass: dynamic attraction
matress should vibrate!
pair attraction
Effective Attraction in Cuprates
• almost no isotope effect antiferromagnetism (AF)
300K and superconductivity (SC)
closely related
• phase diagram
on hole doping:
• neutron scattering: AF fluctuations persist in SC region
• are these the matress??
no generally accepted understanding available yet
interplay of pairs
Pair Size
h
estimate from uncertainty principle: p x h x
p
p 2
2 2
1 (mv) p
momentum p requires kinetic energy: E kin mv
2
2 2m 2m 2m
available Energy: E binding 3.5 k B Tc
for Ekin Ebinding : p 7 m k BTc
h
diameter: x
7 m k BTc
with Tc 20 K: x 50 nm "coherence length 0"
in reality: 0 1...10 nm
HTS LTS
interplay of pairs
Overlap of Pairs
N
electron density in Sn was: 2 1022 cm 3
V
in a volume of 0 :
3
103...105 pairs
HTS LTS
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
this one breaks the ranks
let all pairs go with same speed except for one maverick:
• the maverick is crossing all other's ways not allowed by Pauli principle
• maverick must evade to empty states with higher energy
• this costs too much energy pair is broken up
• conclusion:
to mimimize energy, all pairs must march in lockstep!
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
• a nonzero momentum of the pairs corresponds to a transport current
• why is the current frictionless?
defect
scattering would change the velocity, break the pair and cost energy
elastic scattering is forbidden
demonstration
BCS theory
BCS Theory
• Bardeen, Cooper, and Schrieffer 1957
microscopic theory of superconductivity
• BCS ground state (T = 0) in momentum space:
occupation
probability
1
2
p looks similar to NC at Tc
0
p
-pF pF
• big difference: Pair correlation:
state p
state p
if p occupied, then also -p if p empty, then also -p
BCS theory
Quasiparticles
• excited sates of the superconductor single electrons, broken pairs
• anti-pair correlation:
state p
state p
if p occupied, then -p empty if p empty, then -p occupied
• minimum excitation energy = binding energy of pairs 2 3.52 k B Tc
energy gap of the superconductor
• quasiparticles exist only at finite temperatures
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
broken pairs yield new quasiparticles
• catastrophe occurs at some finite temperature: all pairs are broken up
(T)
critical Temperature Tc
0
T
Tc
zero resistance
Supercurrents at T > 0
• normal state resistance:
inelastic scattering
0
defects elastic scattering
T
• superconducting state:
pair breaking phonons of Energy h 2 are abundant at TTc/2
inelastic scattering is not forbidden!
dynamic equilibrium:
pair breaking recombination
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
Meissner state
Superconductor in Weak Magnetic Fields
• in magnetic fields, the pairs don't fit together correctly
binding energy will decrease
• but they dont feel the field when they move!
for physicists:
the "kinetic momentum" can compensate the "field momentum"
• consequence: a magnetic field sets the pairs in motion
spontaneous supercurrents occur when sample is cooled in field
2
ns qp
• "2nd London equation": jsuper Btotal
mp
the current is perpendicular to the field
Meissner state
Shielding
• the supercurrents have a magnetic field of their own
• Ampère's law: Bshielding 0 jsuper
• one finds that the field is opposite to the external field
• the total field falls off rapidly into the superconductor
B
SC
Bext
caracteristic length:
Btotal
1
Bext "magnetic penetration depth"
e jsuper
x
100 nm
surface
Bshielding
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
mixed state
Superconductor in strong Magnetic Fields
• in stronger fields, the condensate is no longer rigid
supercurrents cost too much energy
• how to reduce the currents?
let the field come in!
• simple behaviour. SC breaks down totally Type I superconductors
• intelligent behavior: vortices Type II superconductors
mixed
state
NbSe2 MgB2 LuNi2B2C
mixed state
Critical fields
Type I: Type II:
Bint Bint
Bext Bext
0 Bc 0 Bc1 Bcth Bc2
fields up to 0.2 Tesla only can sustain much higher fields
historically discovered first all technical SC are of Type II
mixed state
What makes the difference?
interface energy between NC and SC in magnetic field:
SC
• > : more loss than gain
NC
positive interface energy
Bext
B(x) nsuper
• > : more gain than loss
negative interface energy
x
0 spontanous creation of
internal interfaces
Eshielding saved
material parameter
/ controls the behavior
Ebinding lost
"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
SC SC vortex
flux line
Shielding current
• div B 0 : vortices go throug from surface to surface, or they form rings
flux flow
Vortex motion
magnetic field and transport current simultaneously:
e. g. magnets, motors, transformers
vortices
Lorentz force
vortices move at right angles with field and current; why?
flux flow
microscopic picture
eddy field
F Lorentz Force
sample boundary
force on a pair: F 2e vsuper B wants to push the pairs to the side
Hall voltage forces the pairs to go straight
but: counter force on vortices! motion of the vortex to the side!
flux flow
Resistance due to Flux Motion:
• power consuption of one vortex: P1 F v vortex
• N vortices: PN N F v vortex
• energy conservation: V I transport
V N F v vortex / I transport
voltage drop!
• conclusion: superconductor has resistance
flux flow resistance
flux flow
Experimental Result
ideal
V
low defect density
higher defect density
I
Ic
"technical" critical current
• Ic depends on defect density
• inhomogeneities are locking the vortices: "flux pinning"
• v = 0 is enforced
no work
no voltage drop, no resisitance
flux flow
Pinning Mechanisms:
segregation
with small
(or even NL)
condensation energy
is lost
• segregation: vortex core can stay without cost in binding energy
• to go on will cost again energy
• i.e. segregation has a binding force "pinning - force"
• particularly effective: defect sizee
flux flow
Jc tech as Function of Temperature and Field
E
j
jc tech
• decreases in magnetic field more vortices/pinning center
• decreases with temperature thermal activation of vortices
flux flow
Pinning in external magnetic field
Bi
Hysterese!
frozen-in
flux
ideal type II SC
-Bc2 -Bc1 Bc1 Bc2 Ba
virgin curve
• Bc1 and Bc2 unchanged
• pinning impedes entrance and exit of vortices
• Bi is inhomogenous within the sample
flux flow
Field Distribution in the Sample:
Bi
B Ba Bc2
Ba grows
Ba < Bc1 (Meissner)
x
Bc1 Bc2 Ba SL
• surface: jump ideal magnetisation curve
• inside: field gradient gradient of vortex density
• vortices move only if their repulsion force is greater than the pinning force
• gradient decreases with increasing field strengtn
flux flow
Bean Model:
• density gradient shielding current y
• macroscopic average over vortices:
Bi 0 jAbschirm Ampère
Bi
• here: 0 jy
x
x
• if B/ x small: j < jc vortices pinned
• if B/ x larger: j > jc vortices are ripped away
• move until everywhere j = jc "critical state"
• remark: B
measurement of dBi(x)/dx jc
x
