# Vorlesung Supraleitung im Wintersemester 200102 by dfhdhdhdhjr

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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  21034 Js  2 1024 kg m / s

 v max  p max / m e  2  1024 /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:  = n0
h
0       2 1015 Tesla  m2
pairs        2e                         n  0
SC state
Josephson-Effect

• dual beam-interference with electron pairs

current

current

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 TTc/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
• 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

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