7. Superconductivity

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```					                                                                    Topics to cover:

1. Resistance, magnetic ﬁeld and heat capacity observations.
Statistical and Low Temperature Physics (PHYS393)

2. Explanation using macroscopic wavefunction.

7. Superconductivity                                  3. Quantised vortices.

4. Cooper pairs.
Kai Hock
2010 - 2011                                5. Applications: trains, accelerators, ...
University of Liverpool

Superconductivity                   1

Zero resistance.                                                 Examples of superconductors.

Metals conduct electricity. Normally, there is always some
resistance, however small.

In some materials, this resistance suddenly falls to zero below a
certain temperature. In 1911, Kamerlingh Onnes discovered
that this happened with mercury below 4.2 K

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/scond.html

Notice that these are metals, and that the transition
temperatures are close to liquid helium temperature.

Superconductivity               2                                   Superconductivity                   3
Meissner eﬀect.                                                   Measuring magnetic ﬁeld.

When the resistance drops to zero, the superconductor all               This graph shows the magnetisation of lead in liquid helium,
expels all magnetic ﬁeld from its body.                                 plotted against the applied ﬁeld.

Livingston, Physical Review, vol. 129 (1963), p. 1943
http://www.materia.coppe.ufrj.br/sarra/artigos/artigo10114/index.html

Below a certain critical ﬁeld, the magnetisation is equal and
The ﬁeld inside the body of a superconductor can be obtained
opposite to the applied ﬁeld. So the resultant ﬁeld inside the
by inserting it in a coil and measuring the induced voltage.
superconductor is zero.

Superconductivity                   4                                   Superconductivity                    5

Levitation.                                                           Heat capacity.

The expulsion of magnetic ﬁeld from a superconductor is called          Recall the heat capacity of a normal metal:
is Meissner eﬀect.
Cv = γT + AT 3.

A striking demonstration is the levitation of a superconductor          Measurements show that for supercondctors, this changes
above a magnet.                                                         completely below the transition temperature. This graph is the
result of measurement for the niobium metal.

Brown, et al, Physical Review, vol. 92 (1953), p. 52

Superconductivity                   6                                   Superconductivity                    7
If we select the normal conducting state of niobium by applying
Niobium become superconducting below 9.5 K. It is possible to      a strong magnetic ﬁeld, we would measure the curve labelled
prevent it from becoming superconducting by applying a             “normal.”
suﬃciently large enough magnetic ﬁeld.

We know from the Meissner eﬀect that a niobium expels all
magnetic ﬁeld. However, if the ﬁeld is strong enough, it can
“force” its way into the superconductor. This destroys the
superconductivity and returns the niobium to a normal
conducting state - even if temperature is below 9.5 K.

Using this property, it is possible to select between the normal
and the superconducting state.                                     This follows the “normal” behaviour of

Cv = γT + AT 3.

Superconductivity               8                                  Superconductivity                 9

The Macroscopic Wavefunction.

If we do not apply any magnetic ﬁeld, we get the
superconducting state. Then we would get the curve labelled
In 1937, Fritz London suggested that if the electrons in a
“superconducting.”
superconductor somehow forms a macroscopic wavefunction.

Using this assumption, London was able to explain it expels all
magnetic ﬁeld. To understand this, we ﬁrst need to appreciate
why expulsion of the magnetic ﬁeld is strange.

Suppose the resistance going to zero is the only change in a
metal. Consider what happens if we now bring a magnetic to
the metal.
If we subtract the phonon contribution of AT 3, we would ﬁnd
that the curve is closer to the exponential form:                  The change in magnetic ﬂux through the metal induces an
electric current, according to Faraday’s law.
C = a exp(−b/T )
for some connstants a and b. This looks like the Boltzmann
distribution.
Superconductivity              10                                  Superconductivity                 11
Lenz’s law.

It looks like we have just “explained” the Meissner eﬀect.
However, let us now look at what happens if the magnet is
According to Lenz’s law, the current would ﬂow in such a way        already there before cooling.
as to produce a magnetic ﬁeld of its own that opposes the
incoming ﬁeld.                                                      We start with a normal metal with a magnetic ﬁeld going
through the body. Then we cool this down and the resistance
In a metal with resistance, this induced current would quickly      falls to zero.
slow down to zero. The induced ﬁeld becomes zero, and only
the incoming ﬁeld remains in the body of the metal.                 According to Faraday’s law, since there is no change in
magnetic ﬂux, no current is induced. So the original ﬁeld from
If the metal has no resistance, the induced current continues to    the magnet remains in the body.
ﬂow. The induced ﬂux has to be opposite to the incoming ﬂux.
Therefore they cancel, and the ﬁeld in the body becomes zero.       In a real superconductor, we know from the Meissner eﬀect
that, even in this case, the magnetic ﬁeld must be expelled.
In this way, the ﬁeld is “expelled.”

This shows that there is something diﬀerent about a
superconductor that the familiar laws of electromagnetism
cannot explain.
Superconductivity                  12                               Superconductivity                  13

Macroscopic wavefunction.                                                  Vector potential

We shall now see how a macroscopic wavefunction, ψ, can
explain the Meissner eﬀect.                                         In order to use the vector potential, lets review its meaning. It
is deﬁned by
Recall the operator in quantum mechanics for momentum:                                             ∇ × A = B,
dψ                                        where B is the magnetic ﬁeld. This a bit similar to relation
−i    = pxψ
dx                                        between the electric ﬁeld and electric potential.
where p is the momentum mv.
http://en.wikipedia.org/wiki/Magnetic_potential
In the presence of an electromagnetic ﬁeld, this is changed to
dψ                                         For a qualitative understanding, the integral form of this
−i = (mv − qA)ψ                            equation is suﬃcient:
dx
where A is the vector potential and q the charge of the particle.                                  A.dl =       B.dS,
C            S
http://quantummechanics.ucsd.edu/ph130a/130_notes/node29.html       where the left integral is along any loop C, and and the right
integral is over any surface S enclosed by the loop.
Both equations are quantum mechanical postulates that have
been shown to give correct results in physics.
Superconductivity                  14                               Superconductivity                  15
Ampere’s law.                                                             Phase.

The right side of this equation is the magnetic ﬂux Φ,               Let us now return to the quantum mechanical equation:
dψ
A.dl =           B.dS                                          −i  = (mv − qA)ψ.
C                S                                                  dx
and the left side is the line integral for magnetic potential.       Recall the wavefunction we used for superﬂuids:

ψ = e−iφ(x)
If we make the following replacements: A → B and B → J,
where J is the current density, we get Ampere’s law.                 where φ(x) is the phase. Substituting into the equation, we get
dφ
= mv − qA.
In the more familiar Ampere’s law, the electric current is related                             dx
to the integral of magnetic ﬁeld over a loop round the current.      This relation along a straight line in x can be extended in a
simple way to any path or loop in 3D.
In the same way, the equation
Consider a loop in a superconductor of length L enclosing an
A.dl = Φ
C                                       area S. Integrating along this loop, we get
tells us that magnetic ﬂux is equal to the integral of vector
∆φ = m        v.dl − q       A.dl.
potential over a loop round the ﬂux.                                                                L              L

Superconductivity                    16                              Superconductivity                  17

Let us now see how this equation
m
∆φ =       J.dl − qΦ.
∆φ = m           v.dl − q        A.dl.                                      ρq L
L              L
can help us understand Meissner’s eﬀect.
The phase change ∆φ is zero or a multiple of 2π, because the
wavefunction returns to the same value after one loop.
For a simple lump of metal, the wavefunction would be
continuous through the whole volume, so the phase change
The integral over A gives the magnetic ﬂux Φ.
would be zero. The equation then simpliﬁes to
m
The velocity v is related to the current density J by                                             J.dl = qΦ.
ρq L
J = ρq v,                              This means that:
where ρ is the number density of the electrons. The above
equation then becomes                                                     if there is a magnetic ﬁeld in the macroscopic
m                                             wavefunction, then is a there is an electric current.
∆φ =        J.dl − qΦ.
ρq L

To see why this is special, consider Faraday’s law again.
Superconductivity                    18                              Superconductivity                  19
Meissner eﬀect.                                               London’s penetration depth.

According to Faraday’s law, a change in magnetic ﬂux is          Flux from the wavefunction, or superconducting, current would
required before a current can be induced.                        cancel some of the incoming ﬂux.

For a macroscopic wavefunction, the very presence of the ﬂux     The amount cancelled depends on the density of the electrons
produces the current. No change in ﬂux is needed!                in the wavefunction. The higher the density, the larger the
superconducting current, and more of the incoming ﬂux would
Let us look at the case of transition to the superconducting     be cancelled.
state again. Previously, we have not been able to explain the
expulsion of the ﬁeld using Faraday’s law.                       For a uniform external ﬁeld, this superconducting current would
typically be circulating the metal. So it produces the greatest
We can now explain this assuming that a macroscopic              ﬁeld at the centre, where more cancellation takes place.
wavefunction appears when the metal becomes
superconducting, If there is a magnetic ﬁeld in the metal, it    For larger electron density, the region of cancellation is also
would produce a current. This current would in turn produce a    larger. In a typical superconductor, there is suﬃcent density to
ﬂux.                                                             expel the incoming ﬁeld from most of the volume.

A more detailed reasoning would show that this wavefunction      In practice, some ﬁeld would penetrate to a depth of about 100
ﬂux is in the opposite direction to the incoming ﬂux.            nm on the surface.
Superconductivity             20                                 Superconductivity                 21

Vortices.

The reason for the penetration depth is that a current is        In the lectures on superﬂuid helium, we have seen that a
needed to keep the ﬁeld expelled.                                macroscopic wavefunction can give rise to vortices that
quantised.
Recall that a ﬁeld must be present in the macroscopic
wavefunction in order to produce the current. As the ﬁeld gets   If the electrons in a superconductor also forms a macroscopic
expelled from the center of the superconductor, the current at   wavefunction, quantised vortices should also be possible in the
the center would also stop.                                      electrons. This is indeed observed:

If the ﬁeld is completely expelled from the metal, there would
be no current at all in the metal. Then there would be no
opposing ﬂux to cancel the incoming ﬂux. The external ﬂux
would come in again and start producing current.

For this reason, a balance would to be reached. The ﬁeld
would penetrate until a depth when there is suﬃcient current
to keep the rest of the volume ﬁeld free.
Essmann and Trauble, Physics Letters 24A, 526 (1967)

Superconductivity             22                                 Superconductivity                 23
Observing vortices.

Likewise, Essmann and Trauble sprinkled some cobalt powder
The method used to observe vortices is similar to the method       on a Lead-Indium alloy. This is what they saw under an
for observing magnetic ﬁeld lines in school.                       electron microscope:

Sprinkle some iron ﬁlings on a piece of paper, place a magnet
underneath, tap the paper gently, and this is what you would
see:

The cobalt powder collected at the centres of the vortices,
where magnetic ﬁelds are strongest.

A nice gallery of superconducting vortices can be found here:
http://www.fys.uio.no/super/vortex/index.html

Superconductivity              24                                  Superconductivity                  25

Type II superconductors.                                            Type II superconductors.

The existence of vortices is in fact not consistent with           This shows the magnetisation of Lead alloy with diﬀerent
Meissner’s eﬀect.                                                  amount of Indium:

We have learnt that when a metal becomes superconducting, it
expels all magnetic ﬁeld (except for some near its surface).

A vortex in a superconductor is a circulating current. This
must produce a magnetic ﬁeld in the superconductor. This

Below a certain critical ﬁeld, the magnetisation (e.g. OB) is
It turns out that the Meissner’s eﬀect is only true for some
strong enough to cancel the applied ﬁeld.
metals - mainly pure metals. These are called Type I
superconductors.
For higher ﬁeld, the magnetisation decreases (e.g. curve to the
right of B). It is not enough to cancel the applied ﬁeld, which
For alloys and other materials, it is possible for magnetic ﬁeld
then penetrates the superconductor. These are called type II
to penetrate the body of the superconductor to some extent.
superconductors.
Superconductivity              26                                  Superconductivity                  27
Flux quantisation.                                               Flux measurement.

We have seen that vortices of electrons do exist in a               Deaver and Fairbank measured the ﬂux through a long, thin
superconductor. Lets now look at whether they are quantised.        tube made of Tin:

If the current around a vortex is quantised, so is the magnetic
1. Apply a magnetic ﬁeld to the tube.
ﬂux produced. This can be measured, the has indeed been
found to be quantised.
2. Cool below the 3.7 K transition temperature.

3. Move the tube up and down rapidly.

4. Place a coil near the end of the tube.

5. Measure the voltage induced in the coil.

6. Obtain the ﬂux from the voltage.
Deaver and Fairbank, Physical Review Letters, vol. 7 (1961) p. 43
Superconductivity                  28                               Superconductivity              29

The measured ﬂux is plotted against applied ﬁeld:                   In the case of the superﬂuid, no magnetic ﬁeld is involved. So
we still need to understand how vortex arise in the
superconductor.

Recall the relation between ﬂux and current in a macroscopic
wavefunction:
m
∆φ =       J.dl − qΦ.
ρq L
The Tin tube is a solid with a hole through it. The
wavefunction is no longer continuous over the whole volume, so
phase change around the tube does not have to be zero:
m
The steps show that the possible ﬂux through the tube is                                 2nπ =        J.dl − qΦ.
ρq L
indeed quantised. The magnitude of each step is                     The equation is true for any loop L in the wavefunction. It is
h                           possible to choose the loop in such that the integral over
Φ=         .
2e                          current J is zero.

Superconductivity                  30                               Superconductivity              31
This ﬁgure shows the cross-section of the Tin tube. We are
interested in the ﬂux through the hollow.                           So if we choose the loop L away from either surfaces, then the
current density along L would be zero. The equation
m
2nπ =       J.dl − qΦ.
ρq L
then becomes
2nπ = qΦ.
The “-” sign can be left out if we are only interested in the
magnitudes. Since q is the charge of an electron, the ﬂux is
nh
Φ=   .
e
This means that one quantum step is h/e.
When this is superconducting, current is only possible very near
the surfaces A and B, within the penetration depth. Further in      We have just found a problem.
the bulk, there is no current because there is no ﬁeld, since all
ﬁeld is expelled.
Superconductivity              32                                   Superconductivity              33

Cooper pair.

Using the macroscopic wavefunction, we have found that the
ﬂux is quantised in steps of h/e.                                   Notice the diﬀerence:

The measurement results tell us that the ﬂux is quantised in        Theory predicts h/e. Measurement gives h/2e.
steps of h/2e.
This means that something must be wrong with the theory. It
seems to suggest that, instead of a charge of e, the particle
should have a charge of 2e.

This is one of the evidence to suggest that the electrons might
somehow be moving in pairs.

Superconductivity              34                                   Superconductivity              35
The Isotope Eﬀect.                                            Lattice Vibration.

If electrons repel each other, how can they form a pair?

The clue: In 1950, the superconducting temperature of
Why do the neutrons change the superconducting temperature?
Mercury was found to be diﬀerent for diﬀerent isotopes of
Mercury.
One possible reason is that the movement of the atoms are
somehow involved in causing the superconductivity.

More neutrons means more mass. This would result in slower
movement of atoms.

This provides an important clue: Lattice vibration is known to
scatter electrons and cause resistance.

Reynolds, et al, Physical Review, vol. 78 (1950) p. 487

The only diﬀerence between isotopes is the number of neutrons
in the nuclei. This should not aﬀect the conduction electrons!
Superconductivity                   36                           Superconductivity              37

How electrons “attract”

When a electron moves in a metal, it can attract the positive
ions and bring them closer.                                      The attractive potential between electrons is much smaller
than the kinetic energy of the two electrons. So it should not
normally be able to bind the electrons together.

However, in this case, the two electrons are not in free space.
They are in a Fermi sea - electrons stacked up to the Fermi
energy.

Another electron may then get attracted to the displaced ions.   In the 1950s, Leon Cooper showed that two electrons near the
Fermi energy is is able to form a bound pair.

Bardeen, Cooper and Schrieﬀer (BCS) then developed a
complete theory to that is able to explain the Meissner’s eﬀect,
the zero resistance, the heat capacity behaviour, and other
phenomena of superconductors.
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/coop.html
Superconductivity                   38                           Superconductivity              39
Cooper pair in real space                                             BCS versus BEC

The wavefunction of an electron in a Cooper pair in real space   The electrons in the pair have opposite spin, so that resultant
is not unlike that of an electron around an atom.                spin of the Cooper pair is zero - it is a boson.

So, like the Bose-Einstein condensate, the Cooper pairs can
condense into the ground state and form a condensate.

Kadin, Spatial Structure of the Cooper Pair (2005)

Ketterle and Zwierlein, Making, probing and understanding ultracold Fermi gases (2006)
The size of the Cooper pair is a few hundred times the spacing
between atoms, so there is a lot of overlap between Cooper       However, because of the considerable overlap, it is normally
pairs.                                                           called a BCS condensate instead.
Superconductivity                          40                    Superconductivity                          41

Energy gap.

As an example of a prediction by the BCS theory, recall the      Rearranging the relation gives this ratio:
behaviour of heat capacity in a superconductor,                                                      2∆
C = a exp(−b/T ). This can be written in the form:                                                        = 3.52.
kB Tc
∆          The ratio for measured values are shown here:
Cv = D exp −
kB T
This looks like the Boltzmann factor, in which ∆ is the energy
between two levels. In the BCS theory, ∆ is the energy needed
to excite one electron from the BCS condensate.

This energy is now called the energy gap. It can be obtained
directly from a heat capacity measurement by ﬁtting the above
formula.

Meservey and Schwarz, in Parks (1969) Superconductivity
BCS theory predicts that the energy gap and the transition
temperature are related by:
The ratios are all fairly close to 3.52. This is another evidence
2∆ = 3.52kB Tc.                 that supports the BCS theory.
Superconductivity                          42                    Superconductivity                          43
BCS superﬂuid.                                              Applications of superconductors.

The most obvious property about a superconductor is the zero
resistance. Unfortunately, there does not appear to be a simple    Existing applications of superconductivity include:
way to explain this.

1. Maglev train.
The Cooper pairs can carry electric current, but why does it
not get scattered by phonons and experience resistance?
2. Magnetic Resonance Imaging (MRI)
Victor Weisskopf suggested that the Cooper pairs are packed
like atoms in the helium-4 superﬂuid, and has zero resistance
for similar reasons. So the diﬃculty in scattering a Cooper pair    3. Particle accelerators (e.g. LHC)
is a result of interaction with other Cooper pairs.

http://cdsweb.cern.ch/record/880131/files/p1.pdf                    4. Detecting weak magnetic ﬁeld (SQUIDS)

The Cooper pairs would ﬂow like a superﬂuid, unless there is       http://www.superconductors.org/uses.htm
enough energy to break all of them. This would happen at the
transition temperature: kB Tc ≈ ∆.

Superconductivity                  44                              Superconductivity                  45

Maglev train.                                               Magnetic Resonance Imaging

A train can be levitated above its track using powerful,           MRI requires a very strong magnetic ﬁeld. This is produced
superconducting magnets, so that there is little friction.
using supercondctors.

One, built in Japan in 2005, travelled at half the speed of
sound.

(and Wikipedia)
http://en.wikipedia.org/wiki/Maglev_(transport)
Superconductivity                  46                              Superconductivity                  47
Particle accelerators                                       Detecting weak magnetic ﬁeld

A superconducting device called SQUID can detect very weak
Particle accelerators use superconducting magnets and rf          magnetic ﬁelds.
cavities to accelerate particles to high energies.

(Wikipedia) It is useful for:

-   detecting brainwave,
-   diagnosing problems in various parts of the human body,
-   as an MRI detector,
-   oil prospecting,
-   earthquake prediction,
-   submarine detection, etc.
Superconductivity                  48                             Superconductivity                49

High Temperature Superconductors                                         Characteristics

In 1986, materials that become superconducting above liquid
nitrogen tempratures are discovered. This generated a lot of      The ﬁrst of the high temperature superconductors discovered is
excitement about possible applications, because liquid nitrogen   YBCO (Yttrium-Barium-Copper-Oxide),
is much cheaper than liquid helium.
Being copper oxides, these materials are very poor conductors
of electricity at room temperature.

When they do become superconducting at liquid nitrogen
temperatures, there are fewer Cooper pairs compared to
metallic superconductors.

As a result, they are strongly type II when superconducting -
ﬂux lines can penetrate.

Notice that the examples above 77 K are copper oxides.

Superconductivity                  50                             Superconductivity                51
Applications?

Using these high temperature superconductors for MRI, trains,
LHC and SQUID would remove the need for expensive liquid
helium.

Unfortunately, being copper oxides again, they are brittle and
very diﬃcult to make into electrical wires.

Today, scientists are still trying to solve these engineering
problems.

Superconductivity               52

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