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winkwellmadeUrban and Civil

Nov 15, 2013 (3 years and 10 months ago)

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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