Superconductivity Resistance goes to 0 below a critical temperature T c element T c resistivity (T300) 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 Pb 7.2 2.2 Nb 9.2 1.3 Res. many compounds (Nb-Ti, Cu-O-Y mixtures) have T c up to 90 K. Some are ceramics at room temp T 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 E 0.5 T c vibrations cons t ( Sn ) K M tan 115,117, 119 again shows superconductivity due to interactions with the lattice. If M infinity, no vibrations, and T c 0 spike in specific heat at T c 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 E F 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) r r ip e r r P pair p p 1 + 2 0 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) 2 ψ total ψ 1+ ψ 2+... ψn 2 P461 - Semiconductors 6
Energy levels in S.C. electrons in Cooper pair have energy as part of the Fermi sea (E 1 and E 2 E F + ) plus from their binding energy into a Cooper pair (V 12 ) E 1+ 2 E1+ E2 V 12 E 1 and E 2 are just above E F (where the action is). If the condition E1 + 2< 2E F is met then have transition to the lower energy superconducting state 2 E F E 12 E gap s.c. normal T C Temperature can only happen for T less than critical temperature. Lower T gives larger energy gap. At T0 (from BCS theory) E 3 gap kt C 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 v B ( r r µ 0 H+ M ) r r M χh χ magnetic susceptibility canbe : χ( T ), χ ( H ), scalar, vector can have residual magnetism: M not equal 0 when H0 diamagnetic χ< 0. Currents are induced which counter applied field. Usually.00001. Superconducting χ -1 ( perfect diamagnetic) P461 - Semiconductors 8
Magnetics - Practical in many applications one is given the magnetic properties of a material (essentially its χ) 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 J0 (Helium, filled shells, molecular solids with covalent S0 bonds ) χ 0 4 χ 10 most, χ 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 r r E B µ + µ B( antiparall µ B( parallel ) el) P461 - Semiconductors 10
Paramagnetism II Use Boltzman distribution to get number parallel and antiparallel n n + M µ ( n µ B / kt µ B / kt where M net magnetic dipole moment per unit volume µ B / kt µ B / kt M e e µ average µ µ B / kt µ B / kt n e + e if µ B<< kt µ Ce Ce can use this to calculate susceptibility(curie Law) n n n 2 ( 1+ µ B / kt ) (1 µ B / kt ) µ B µ (1+ µ B / kt) + (1 µ B / kt) kt B µ 0 H+ µ 0M µ 0H ( χsmall) χ M H nµ H 2 nµ B kth P461 - Semiconductors 11 + ) 2 µ 0nµ kt
Paramagnetism III if electrons are in a Fermi Gas (like in a metal) then need to use Fermi-Dirac statistics 1 n C n ( µ B EF ) / kt e + 1 1 n+ C n ( µ B EF ) / kt e + 1 reduces number of electrons which can flip, reduces induced magnetism, χ smaller antiparallel parallel B 0 kt << E F 2µB E F turn on B field. shifts by µb antiparallel states drop to lower energy parallel P461 - Semiconductors 12
Ferromagnetism Certain materials have very large χ (1000) and a non-zero B when H0 (permanent magnet). χ 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 L2. Use Hund s rules maximize S (symmetric spin) spatial is antisymmetric and electrons further apart. So S2 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 energy states and magnetic effects diluted E F R large. not much overlap, energy difference vs small P P A A E F R medium. broadening of energy band similar to magnetic shift almost all in state E F E(unmagnetized)- Fe Co Ni vs E(magnetized) Mn R P461 - Semiconductors 14