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Lecture
21:
Superconductors
Superconductivity:
The
electrical
resistance
of
some
solids
disappears
completely
at
sufficiently
low
temperature.
Look
to
the
next
figure:
Resistivity
of
superconductors
versus
temperature
TC:
critical
temperature
<10‐23
Ω ∙ cm
compared
to
𝜌
of
10‐9
Ω ∙cm
in
Cu.
‐Sharp
decrease
in
𝜌 at
TC
(critical
temperature)
1
𝜌
‐Metallic
elements:
TC:
Hg
4.15K
W
0.01
K
Nb
9.3K
‐Metal
oxides
(Ceramics):
High
temperature
semiconductors
YBaCuO7
HgCuo
(contain
CuO2
or
NiO2).
‐Cu,
Ag,
Au,
…
or
Fe,
Ni,
Co
are
not
superconductors
The
Two
Characteristics
of
Superconductivity:
• The
absence
of
resistance
to
current
flow
• Exclusion
of
magnetic
flux
(Meissner
effect)
Meissner
Effect
As
a
superconducting
substance
is
cooled
below
its
critical
temperature
in
the
presence
of
applied
magnetic
field,
it
expels
all
magnetic
flux
from
its
interior.
2
Ill
explain
in
more
detail
This
leads
to
a
very
interesting
phenomena
(magnetic
levitation)
3
A
permanent
magnet
floating
over
a
superconducting
surface
‐magnetic
train
Let’s
go
back
to
the
previous
figure:
4
Now,
if
the
external
magnetic
field
increases
beyond
a
certain
value
(the
critical
field
HC)
the
material
ceases
to
be
superconductor
and
becomes
normal.
‐If
the
temperature
increases
beyond
TC,
the
material
becomes
normal.
Thus,
there
is
a
critical
temperature,
critical
magnetic
field,
and
critical
current.
5
‐Critical
temperature,
critical
magnetic
field,
critical
current
Let’s
look
again
to
(H‐T)
curve:
HC:
critical
magnetic
field
Above
HC,
the
superconductivity
is
destroyed
𝑇!
𝐻! = 𝐻! 1 − ! 𝑇!
A:
normal
state
Magnetic
field=zero
6
Path:
ABCD
Cooling
down:(
A⟶B)
B:
superconductor
H:
still
zero
C:
Turn
in
magnetic
field,
changing
flux
creates
electric
field
that
sets
up
a
current
that
opposes
magnetic
field
(Lenz
law).
Since
there
is
no
resistivity,
eddy
currents
don’t
decay
but
create
magnetic
field
that
cancels
the
applied
field.
Superconductors
act
as
perfect
diamagnet
We
can
summarize:
Now,
let’s
go
through
the
path
ADCB:
i.e.
apply
a
field
to
a
normal
state,
then
cool
down.
What
happened?
7
It
expelled
the
magnetic
field.
We
call
the
two
cases
Meissner
effect:
generated
field
exactly
cancels
out
the
applied
field
in
superconductor
What
is
Behind
Superconductivity
Before
I
give
some
explanation
for
the
phenomena,
let
me
remind
you
what
we
have
so
far:
− Classical
theory
of
metallic
conduction
treated
electrons
as
a
gas
of
independent
particles.
− Quantum
Mechanics:
treated
electrons
as
independent
particles,
and
took
into
account
the
wave
nature
of
electrons
and
exclusion
principle.
∗ Fermi
distribution
∗ Quantization
of
energy
level
and
band
theory
of
solids
Independent
particles
failed
to
explain
superconductivity.
We
need
many‐body
effects
There
was
an
obvious
experimental
evidence
for
the
need
of
many‐body
effects.
Experimental
Evidence:
Isotope
effect:
The
critical
temperature
of
a
superconductor
depends
on
the
total
mass
of
the
nucleus
1
𝑇! ∝
𝑀
if
we
use
the
isotope,
(this
means
we
add
more
neutrons)
the
critical
temperature
decreases.
This
8
suggests
that
the
lattice
vibrations
play
a
role
in
superconductivity.
We
must
take
into
account:
• electron‐phonon
interaction
• electron‐electron
interaction
BCS
Theory
(Bardeen,
Cooper,
Schrieffer,
1957)
takes
that
into
account.
Although
thermal
vibrations
are
behind
the
resistivity,
in
some
materials
and
below
a
certain
temperature,
the
lattice
vibrations
play
a
role
as
an
intermediate
between
two
electrons.
It
results
in
an
attractive
force
between
the
two
electrons.
9
The
first
electron
deforms
the
lattice.
The
second
electron
is
attracted
by
the
deformation
(region
of
increased
positive
charge
density).
The
first
electron
has
emitted
a
phonon.
A
second
electron
passing
by
the
region
will
absorb
the
phonon.
The
two
electrons
have
exchanged
momentum
with
each
other.
Attractive
interaction
exceeds
repulsive
interaction.
The
two
electrons
are
weakly
bound
and
form
Cooper
pair.
Cooper
pairs
are
responsible
for
superconductivity.
10
11
‐What
are
the
conditions
for
superconductivity?
1. 2. 3. 4. 12
• According
to
BCS
theory,
the
binding
energy
of
a
cooper
pair
at
absolute
zero
is
about
3kbTC
,
where
kb
is
Boltzmann
constant,
TC
is
critical
temperature
• Cooper
pairs
do
not
obey
Fermi‐Dirac
statistics.
They
are
bosons.
• BCS
theory
presently
fails
to
explain
superconductivity
of
high
temperature
super
conductors
Summary
of
Superconductor
Properties:
• They
behave
as
(no
DC
resistors)
• They
behave
as
a
perfect
dimagnet
and
experience
“Meissner”
effect
• A
band
gap
was
implied
by
the
very
fact
that
the
resistance
is
precisely
zero.
If
charge
carriers
can
move
through
a
crystal
lattice
without
interacting
at
all,
it
must
be
because
their
energies
are
quantized
such
that
they
do
not
have
any
available
energy
levels
within
reach
of
the
energies
of
interaction
with
the
lattice.
• Energy
gap
13
The
lower
energy
of
superconductivity
electrons
are
separated
from
the
normal
state
by
energy
gap
about
EF
Classification
of
Superconductors:
Type
I
(soft)
Type
II
(hard)
‐Type
I
superconductors:
most
superconductive
elements
except
Nb
14
Superconductivity
is
destroyed
by
applying
a
small
magnetic
field
HC
‐Type
II
superconductors:
The
elimination
of
superconductivity
state
is
gradually
15
mixed
(vortex)
state
Less
than
Hc
1→
superconductor
Above
HC
2
→
normal
It
has
There
are
intermediate
fields,
where
material
has
both
normal
and
superconductor
regions
16
Type
II
includes:
‐ Nb
‐ All
alloys
Note:
Type
II
have
higher
TC,
higher
HC
and
higher
IC
17
!
𝜙! = ℎ = 2.07 𝑥 10!! 𝑔𝑎𝑢𝑠𝑠 − 𝑐𝑚! known
as
a
!!
fluxoid
or
flux
quantum.
18
19
20