P461 - Semiconductors 1

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

Pb 7.2 2.2

Nb 9.2 1.3

• many compounds (Nb-Ti, Cu-O-Y mixtures) have Tc up to 90 K. Some are ceramics at room temp

Res.

T

P461 - Semiconductors 2

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 3

Superconductors observations

• For different isotopes, the critical temperature depends on mass. ISOTOPE EFFECT

• 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

M

KE

SntconsTM

vibrations

c

)(tan 119,117,1155.0

P461 - Semiconductors 4

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)

electronelectron

atoms

P461 - Semiconductors 5

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 6

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)

• 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)

021 ppPpair

rpie

221

2 |....||| ntotal

different times

different pairs

P461 - Semiconductors 7

Energy levels in S.C.• electrons in Cooper pair have energy as part of the Fermi sea (E1 and

E2=EF plus from their binding energy into a Cooper pair (V12)

• E1 and E2 are just above EF (where the action is). If the condition is met then have transition to the lower energy superconducting state

• can only happen for T less than critical temperature. Lower T gives larger energy gap. At T=0 (from BCS theory)

122121 VEEE

FEE 221

normal

s.c.12

2

E

EF

CT

Temperature

Egap

Cgap kTE 3

P461 - Semiconductors 8

Magnetic Properties of Materials

• H = magnetic field strength from macroscopic currents• M = field due to charge movement and spin in atoms -

microscopic

• can have residual magnetism: M not equal 0 when H=0• diamagnetic < 0. Currents are induced which counter

applied field. Usually .00001. Superconducting = -1 (“perfect” diamagnetic)

vectorscalarHTbecan

litysusceptibimagneticHM

MHB

,),(),(:

)(0

P461 - Semiconductors 9

Magnetics - Practical• in many applications one is given the magnetic properties of a

material (essentially its ) and go from there to calculate B field for given geometry

D0 Iron Toroid

beamline sweeping magnet

spectrometer air-gap analysis

magnet

P461 - Semiconductors 10

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…) = 0

• 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

Femost 54 10,10

)(

)(

parallelB

elantiparallB

BE

P461 - Semiconductors 11

Paramagnetism II• Use Boltzman distribution to get number parallel and

antiparallel

• where M = net magnetic dipole moment per unit volume

• can use this to calculate susceptibility(Curie Law)

)(

/

/

nnM

nCen

nCenkTB

kTB

kT

B

kTBkTB

kTBkTB

ktBifee

ee

n

Maverage

kTBkTB

kTBkTB

2

//

//

)/1()/1(

)/1()/1(

kT

n

kTH

Bn

H

n

H

M

smallHMHB2

02

000 )(

P461 - Semiconductors 12

Paramagnetism III• if electrons are in a Fermi Gas (like in a metal) then

need to use Fermi-Dirac statistics

• reduces number of electrons which can flip, reduces induced magnetism, smaller

ne

Cn

ne

Cn

kTEB

kTEB

F

F

1

11

1

/)(

/)(

antiparallel

parallel

EF

FEkTB 0

turn on B field. shifts by B

B2

antiparallel states drop to

lower energy parallel

P461 - Semiconductors 13

• Certain materials have very large (1000) and a non-zero B when H=0 (permanent magnet). 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

Ferromagnetism

P461 - Semiconductors 14

• R small. lots of overlap broad band, many possible energy states and magnetic effects diluted

• R large. not much overlap, energy difference small

• R medium. broadening of energy band similar to magnetic shift almost all in state

Ferromagnetism II

vs

FE

FE

FE

P A

P A

vs

R

E(unmagnetized)-

E(magnetized)

Mn

Fe Co Ni