P461

Semiconductors
1
Superconductivity
•
Resistance goes to 0 below a critical temperature T
c
element T
c
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
T
c
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
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
M
K
E
Sn
t
cons
T
M
vibrations
c
)
(
tan
119
,
117
,
115
5
.
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)
electron
electron
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 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
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)
0
2
1
p
p
P
pair
r
p
i
e
2
2
1
2

....



n
total
different times
different pairs
P461

Semiconductors
7
Energy levels in S.C.
•
electrons in Cooper pair have energy as part of the
Fermi sea (E
1
and E
2
=E
F
D)
plus from their
binding energy into a Cooper pair (V
12
)
•
E
1
and E
2
are just above E
F
(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)
12
2
1
2
1
V
E
E
E
F
E
E
2
2
1
normal
s.c.
12
2
E
E
F
C
T
Temperature
E
gap
C
gap
kT
E
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
c
< 0. Currents are induced which
counter applied field. Usually .00001.
Superconducting
c
=

1 (“perfect” diamagnetic)
vector
scalar
H
T
be
can
lity
susceptibi
magnetic
H
M
M
H
B
,
),
(
),
(
:
)
(
0
c
c
c
c
P461

Semiconductors
9
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
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…)
c
= 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
Fe
most
5
4
10
,
10
c
c
)
(
)
(
parallel
B
el
antiparall
B
B
E
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)
)
(
/
/
n
n
M
n
Ce
n
n
Ce
n
kT
B
kT
B
kT
B
kT
B
kT
B
kT
B
kT
B
kt
B
if
e
e
e
e
n
M
average
kT
B
kT
B
kT
B
kT
B
2
/
/
/
/
)
/
1
(
)
/
1
(
)
/
1
(
)
/
1
(
kT
n
kTH
B
n
H
n
H
M
small
H
M
H
B
2
0
2
0
0
0
)
(
c
c
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,
c
smaller
n
e
C
n
n
e
C
n
kT
E
B
kT
E
B
F
F
1
1
1
1
/
)
(
/
)
(
antiparallel
parallel
E
F
F
E
kT
B
0
turn on B field.
shifts by
B
B
2
antiparallel states drop to
lower energy parallel
P461

Semiconductors
13
•
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
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
F
E
F
E
F
E
P A
P A
vs
R
E(unmagnetized)

E(magnetized)
Mn
Fe Co Ni
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