Metals, Magnetic Fields, and Superconductors

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Metals, Magnetic Fields, and Superconductors
or
Can we really say anything intelligent about
stuff made of 1,000,000,000,000,000,000,000,000 at
oms?
N. Bonesteel, Supernet, NHMFL, Jan. 30, 2007.
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Exclusion Principle the Periodic Table
The Atom
2p
2s
1s
Exclusion Principle
Electron shells
No two electrons can occupy the same quantum
state.
Nucleus (Protons Neutrons)
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Copper Metal
conduction electrons
Crystal Lattice
e-
e-
e-
e-
Lattice Ion 28 core electrons (charge -28e)
nucleus (charge 29e). Net positive charge of e
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The Fermi Sea of Electrons
Because of the exclusion principle, no two
conduction electrons in a metal can move with the
same velocity.
of electrons
Fermi Sea
velocity
-vF
0
vF
Fermi velocity
Typically vF is 1/100 the speed of light !
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The surface of the Fermi sea is where the
action is!
No electrical current
Finite electrical current
of electrons
of electrons
-vF
-vF
0
0
vF
vF
When electrical current flows in a metal, the
Fermi sea is shifted so there are more right
movers than left movers.
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The surface of the Fermi sea is where the
action is!
No electrical current
Finite electrical current
of electrons
of electrons
-vF
-vF
0
0
vF
vF
When electrical current flows in a metal, the
Fermi sea is shifted so there are more right
movers than left movers.
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For a three dimensional gas of electrons, there
is a Fermi sphere.
vz
Fermi surface
We have made many approximations, but it can be
shown that under very general circumstances the
Fermi surface exists in real metals. (Very
deep result)
vF
vy
vx
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Some Fermi Surfaces
Sr2RuO4
http//www-qm.phy.cam.ac.uk/FSgallery.php
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Some Fermi Surfaces
UPt3
http//www-qm.phy.cam.ac.uk/FSgallery.php
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Electrical Resistance
I (Current)
V (Voltage)
V
Resistance R
I
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Resistance vs. Temperature for a Typical Metal
Resistance
0
0
100
200
300
Temperature (K)
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Superconductivity
In 1911 H. Kamerlingh-Onnes measured the
resistance of mercury at very low temperatures
and found that it vanished below a certain
critical temperature.
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Superconductivity
Resistance
Tc (K)
Material
Al Hg Pb Sn YBa2Cu3O7
1.2 4.2 1.4 3.7 90 !
0
0
Tc
Temperature
Critical Temperature
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Many elemental metals are superconducting at low
temperatures, with Tc 1 - 5 K.
superconducting element
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And, for the extreme skeptics....
superconducting ring
persistent current
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In superconducting rings electrical currents
have been observed to persist for years !
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Enter the magnetic field B
Magnetic Field lines of force (made fresh daily
here at the NHMFL!)
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The Meissner Effect
T lt TC Superconducting State
T gt TC Normal State
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The Meissner Effect
Magnet
Superconductor
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The Meissner Effect
Magnet
N
S
Superconductor
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Superconductivity was a great mystery from the
moment of its discovery (1911).
The unlocking of its secrets had to wait for the
discovery of quantum mechanics (1928).
Even so, it took 30 more years to come to the
final answer
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The BCS Theory
John Bardeen
Robert Schrieffer
Leon Cooper
The Nobel Prize in Physics 1972 "for their
jointly developed theory of superconductivity,
usually called the BCS-theory"
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1. Electrons bind into pairs due to the
positively charged lattice ions.
Positively Charged Lattice Ions
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
Accumulation of positive charge
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
Accumulation of positive charge
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
e-
Accumulation of positive charge
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1. Electrons bind into pairs due to the
positively charged lattice ions.
e-
e-
Cooper Pair
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2. Pairs of electrons do not need to satisfy the
exclusion principle. Cooper pairs can then
condense into a single quantum state.
This condensed state is described by a
macroscopic wave function which we will
represent using a red arrow.
phase
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Normal Metal
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
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Superconductor
A superconductor is a metal full of arrows!
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The arrows are governed by rules.
Rule 1 Electrical current flows when the arrows
twist.
Zero current
I (current)
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The arrows are governed by rules
Rule 2 The more the arrows twist, the bigger the
current.
I (current)
I (current)
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The arrows are governed by rules
Rule 3 Arrows must twist continuously.
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The arrows are governed by rules
Rule 3 Arrows must twist continuously.
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The arrows are governed by rules
Rule 3 Arrows must twist continuously.
OK
Not allowed
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Thats all we need to understand persistent
supercurrents!
superconducting ring
persistent current
In superconducting rings electrical currents
have been observed to persist for years !
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Superconducting Ring
Arrows arent twisting, so no current.
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Superconducting Ring
I
Arrows are twisting, so now there is a current.
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Superconducting Ring
I
This current cant dissipate because the arrows
cant go continuously from this state.
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Superconducting Ring
to this state.
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Superconducting Ring
I
So this current will flow forever! (Or at least
as long as the ring is superconducting)
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A Conserved Quantity Twist Number N
N 0, No net twist around the ring
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A Conserved Quantity Twist Number N
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N 1, Arrows do one full rotation going around
the ring
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A Conserved Quantity Twist Number N
N ?
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A Conserved Quantity Twist Number N
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N 2, Arrows do two full rotations going around
the ring
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Magnetic Flux
Magnetic Field induced by supercurrent B
Area of hole in ring A
N 2
Magnetic flux
I
N0,1,2,
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Magnetic Flux (A beautiful formula!)
Plancks constant
Speed of light
charge of the electron
N0,1,2,
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Another rule for the arrows
Magnetic field lines coming out of the screen
Rule 4 Arrows must twist when going around
magnetic field lines.
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And now we can explain the Meissner Effect!
Magnet
N
S
Superconductor
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The Meissner Effect
Arrows cannot twist
Magnetic Field is expelled
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For some superconductors (type II), field lines
can penetrate, if field is strong enough.
What happens here? (Looks like trouble with Rule
4 !)
Vortex
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For some superconductors (type II), field lines
can penetrate, if field is strong enough.
Normal core (i.e. not superconducting)
Vortex
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For some superconductors (type II), field lines
can penetrate, if field is strong enough.
Normal core (i.e. not superconducting)
Vortex
(Flux )
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The Vortex Lattice
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Josephson Junction
Insulator
Superconductor
Superconductor
Twist discontinuity
I (Current)
Current can flow even if voltage drop is zero due
to quantum tunneling of Cooper pairs across the
junction.
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SQUID
(Superconducting QUantum Interference Device)
Magnetic Flux
I
I
I
-hc/2e
hc/2e
hc/e
-hc/e
3hc/2e
0
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A bit more history
Remarkably, the arrow theory came before BCS
(i.e. before anyone knew what the arrows might
be!)
Heinz and Fritz London
Lev Landau
Vitaly Ginzburg
Physicists call the arrow theory
Ginzburg-Landau theory.
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A bit more history
The existence of quantized vortices and the
vortex lattice was predicted by Alexei Abrikosov.
The fact that the arrow theory follows from the
BCS theory was shown by Lev Gorkov.
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Normal Metal
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
e-
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Superconductor
A superconductor is a metal full of arrows!
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Superconductor
More is Different! P.W. Anderson