Superconductivity by UUzgpZav

VIEWS: 16 PAGES: 14

									         Superconductivity
• Resistance goes to 0 below a critical temperature Tc

  element     Tc resistivity (T=300)
    Ag         --- .16 mOhms/m
    Cu        --     .17 mOhms/m
     Ga     1.1 K   1.7 mO/m
     Al     1.2    .28
     Sn      3.7    1.2       Res.
     Pb      7.2    2.2
    Nb        9.2   1.3
                                              T
• many compounds (Nb-Ti, Cu-O-Y mixtures) have
  Tc up to 90 K. Some are ceramics at room temp




                    P461 - Semiconductors         1
Superconductors observations
• Most superconductors are poor conductors at
  normal temperature. Many good conductors are
  never superconductors
•  superconductivity due to interactions with the
  lattice
• practical applications (making a magnet), often
  interleave S.C. with normal conductor like Cu
• if S.C. (suddenly) becomes non-superconducting
  (quenches), normal conductor able to carry current
  without melting or blowing up
• quenches occur at/near maximum B or E field and
  at maximum current for a given material. Magnets
  can be “trained” to obtain higher values




                   P461 - Semiconductors        2
Superconductors observations
• For different isotopes, the critical temperature
  depends on mass. ISOTOPE EFFECT
              M 0.5Tc  cons tan t ( Sn115,117,119 )
                              K
              Evibrations 
                              M
• again shows superconductivity due to interactions
  with the lattice. If M  infinity, no vibrations, and
  Tc 0
• spike in specific heat at Tc
• indicates phase transition; energy gap between
  conducting and superconducting phases. And what
  the energy difference is
• plasma  gas  liquid  solid  superconductor




                       P461 - Semiconductors           3
            What causes
         superconductivity?
• Bardeen-Cooper-Schrieffer (BCS) model
• paired electrons (cooper pairs) coupled via
  interactions with the lattice
• gives net attractive potential between two electrons
• if electrons interact with each other can move from
  the top of the Fermi sea (where there aren’t
  interactions between electrons) to a slightly lower
  energy level
• Cooper pairs are very far apart (~5,000 atoms) but
  can move coherently through lattice if electric field
   resistivity = 0 (unless kT noise overwhelms 
  breaks lattice coupling)
                         atoms


electron                             electron
                    P461 - Semiconductors         4
            Conditions for
          superconductivity

• Temperature low enough so the number of random
  thermal phonons is small
• interactions between electrons and phonons large
  ( large resistivity at room T)
• number of electrons at E = Fermi energy or just
  below be large. Phonon energy is small (vibrations)
  and so only electrons near EF participate in making
  Cooper pairs (all “action” happens at Fermi energy)
• 2 electrons in Cooper pair have antiparallel spin 
  space wave function is symmetric and so electrons
  are a little closer together. Still 10,000 Angstroms
  apart and only some wavefunctions overlap (low E
   large wavelength)




                    P461 - Semiconductors        5
           Conditions for
         superconductivity 2
•    2 electrons in pair have equal but opposite
    momentum. Maximizes the number of pairs as
    weak bonds constantly breaking and reforming. All
    pairs will then be in phase (other momentum are
    allowed but will be out of phase and also less
    probability to form)                 
                   
                 ip  r
                                            Ppair  p1  p2  0
             e

                      different times
                          different pairs



• if electric field applied, as wave functions of pairs
  are in phase - maximizes probability -- allows
  collective motion unimpeded by lattice (which is
  much smaller than pair size)
             |  total |2 |  1   2  .... n |2

                              P461 - Semiconductors          6
         Energy levels in S.C.
• electrons in Cooper pair have energy as part of the
  Fermi sea (E1 and E2=EFD) plus from their
  binding energy into a Cooper pair (V12)
                  E12  E1  E2  V12
• E1 and E2 are just above EF (where the action is). If
  the condition E12  2 EF is met then have
  transition to the lower energy superconducting state
  2 EF
               Egap
                                          normal
  E12
                      s.c.


                              TC
                      Temperature


• can only happen for T less than critical
  temperature. Lower T gives larger energy gap. At
  T=0 (from BCS theory)       E gap  3kTC
                         P461 - Semiconductors     7
     Magnetic Properties of
          Materials
• H = magnetic field strength from macroscopic
  currents
• M = field due to charge movement and spin in
  atoms - microscopic
              
     B  0 (H  M )
           
     M  cH  c  magnetic susceptibility
    can be : c (T ), c ( H ), scalar, vector

• can have residual magnetism: M not equal 0 when
  H=0
• diamagnetic  c < 0. Currents are induced which
  counter applied field. Usually .00001.
  Superconducting c = -1 (“perfect” diamagnetic)


                   P461 - Semiconductors         8
               Magnetics - Practical
• in many applications one is given the magnetic properties of a
  material (essentially its c) and go from there to calculate B
  field for given geometry
                                                  beamline
                                                  sweeping
                                                   magnet



                                                spectrometer air-
                                                  gap analysis
                                                    magnet




                    D0 Iron
                    Toroid
                        P461 - Semiconductors                  9
            Paramagnetism
• Atoms can have permanent magnetic moment
  which tend to line up with external fields
• if J=0 (Helium, filled shells, molecular solids with
  covalent S=0 bonds…)  c = 0

          c  10 4 most , c  10 5 Fe
• assume unfilled levels and J>0
  n = # unpaired magnetic moments/volume
  n+ = number parallel to B
  n- = number antiparallel to B
  n = n+ + n-
• moments want to be parallel as
                   
           E    B
                 B (antiparallel )
                 B ( parallel)


                    P461 - Semiconductors        10
          Paramagnetism II
• Use Boltzman distribution to get number parallel
  and antiparallel            B / kT
                       n  Ce                n
                       n  Ce  B / kT  n
                       M   ( n  n  )
• where M = net magnetic dipole moment per unit
  volume
                     M       e B / kT  e  B / kT
         average        B / kT
                      n      e          e  B / kT
       if B  kt 
              (1  B / kT )  (1  B / kT )  2 B
                                          
              (1  B / kT )  (1  B / kT ) kT

• can use this to calculate susceptibility(Curie Law)
         B  0 H  0 M  0 H             ( c small )
            M n n 2 B  0 n 2
         c           
            H   H   kTH   kT
                    P461 - Semiconductors                 11
          Paramagnetism III
• if electrons are in a Fermi Gas (like in a metal) then
  need to use Fermi-Dirac statistics
                                       1
               n  C        ( B  E F ) / kT
                                                       n
                          e                       1
                                        1
               n  C        (  B  E F ) / kT
                                                         n
                          e                         1
• reduces number of electrons which can flip,
  reduces induced magnetism, c smaller
                        antiparallel
                                                    B  0 kT  EF
                              parallel
                          EF
  2B
                                           turn on B field.
                                             shifts by B


                                        antiparallel states drop to
                                            lower energy parallel



                     P461 - Semiconductors                          12
            Ferromagnetism
• Certain materials have very large c (1000) and a
  non-zero B when H=0 (permanent magnet). c will
  go to 0 at critical temperature of about 1000 K (
  non ferromagnetic)
    4s2: Fe26 3d6        Co27 3d7        Ni28 3d8
  6s2: Gd64 4f8           Dy66 4f10
• All have unfilled “inner” (lower n) shells. BUT lots
  of elements have unfilled shells. Why are a few
  ferromagnetic?
• Single atoms. Fe,Co,Ni              D subshell L=2.
  Use Hund’s rules  maximize S (symmetric spin)
   spatial is antisymmetric and electrons further
  apart. So S=2 for the 4 unpaired electrons in Fe
• Solids. Overlap between electrons  bands
  but less overlap in “inner” shell
  overlapping changes spin coupling (same atom or
  to adjacent atom) and which S has lower energy.
  Adjacent atoms may prefer having spins parallel.
  depends on geometry  internuclear separation R
                    P461 - Semiconductors       13
           Ferromagnetism II
• R small. lots of overlap 
  broad band, many possible                                 EF
  energy states and magnetic
  effects diluted

• R large. not much overlap,                 P A
  energy difference  vs                                  EF
  small
                                             P A

• R medium. broadening of
  energy band similar to
                                                             EF
  magnetic shift  almost all
  in  state
                                              vs 
 E(unmagnetized)-          Fe Co        Ni

  E(magnetized)
                                                        R
                       Mn


                    P461 - Semiconductors                   14

								
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