; Vorlesung Supraleitung im Wintersemester 200102
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Vorlesung Supraleitung im Wintersemester 200102


  • pg 1
									Tutorial: The Physics of Superconductivity
                              H. Kinder
      Technical University of München

    Onnes            Meissner Landau         Ginzburg    Abrikosov

       Bardeen   Cooper        Schrieffer   Bednorz     Müller

•   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

    • 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!
                         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:
                                       h  N
                 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

   • states in momentum space                          py

       average distance p                       p        px


   • 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:


                                            Fermi momentum
normal state
                Normal State Carrying a Current

   Fermi sphere is displaced by applied voltage:

      rigid displacement

                                                   f              I>0
      cross section:

      more electrons going right than left
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
   • the amplitude squared is the pair density                 V
   • frequently used terms:
                                    macroscopic wave function
                                    pair field
                                    pair amplitude
                                    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
                                0       2 1015 Tesla  m2
                         pairs        2e                         n  0
SC state

    • dual beam-interference with electron pairs


           weak links             B

                                                   magnetic field B (10-5T)

    • Superconducting QUantum Interference Device, SQUID
pair attraction
                   How can two electrons form a pair?

     • isotope effect:

                                               Tc 
                                     Sn               matomic
                                                           atomic mass

                                          vibrations (=phonons) must play a role
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
                                Tc  range of attraction  speed of sound 

                                                                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
     estimate from uncertainty principle:   p  x  h  x 
                                                                       p 2
                                                                     2     2
                                                     1     (mv)    p
     momentum p requires kinetic energy:   E kin    mv 
                                                                    
                                                     2      2m    2m   2m

     available Energy:                      E binding  3.5 k B Tc

     for Ekin  Ebinding :                  p  7 m k BTc
      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

     electron density in Sn was:      2 1022 cm 3

      in a volume of  0 :
                               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?


        scattering would change the velocity, break the pair and cost energy
                                elastic scattering is forbidden
BCS theory
                                    BCS Theory
    • Bardeen, Cooper, and Schrieffer 1957
                                                microscopic theory of superconductivity

    • BCS ground state (T = 0) in momentum space:
                                                        p          looks similar to NC at Tc
               -pF                                 pF

    • big difference: Pair correlation:
                       state p

                       state p

                     if p occupied, then also -p               if p empty, then also -p
BCS theory

    • 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
                                                       critical Temperature Tc

zero resistance
                       Supercurrents at T > 0

     • normal state resistance:
                                            inelastic scattering
                                 defects    elastic scattering

     • 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
                                                     ns qp
     • "2nd London equation":         jsuper             Btotal

              the current is perpendicular to the field
Meissner state

     • 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
                                                          caracteristic length:
                Bext                                     "magnetic penetration depth" 
             e                jsuper
                                                                    100 nm

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


                    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:

                                                •  > : more loss than gain
                                                          positive interface energy
                        B(x)       nsuper
                                                •  > : more gain than loss
                                                          negative interface energy
                   0                                    spontanous creation of
                                                          internal interfaces
                       Eshielding saved
                                                 material parameter
                                                    /  controls the behavior
                       Ebinding lost
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

                              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


                                               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
                                                   low defect density
                                                   higher defect density

                                      "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


            jc tech

     • decreases in magnetic field   more vortices/pinning center
     • decreases with temperature    thermal activation of vortices
flux flow
               Pinning in external magnetic field
                                                            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:

                                     B                              Ba  Bc2

                                                                    Ba grows

                                                                    Ba < Bc1 (Meissner)
                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
     • here:           0 jy
     • 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

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