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7. Superconductivity


									                                                                    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


                                                                    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

                                                                        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
                                                                   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
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.
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:
where A is the vector potential and q the charge of the particle.                                  A.dl =       B.dS,
                                                                                               C            S       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:
                             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
                                                                                                  = 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
                                                                                          ∆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
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


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
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

                                                                   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:

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
contradicts the Meissner’s effect.

                                                                   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
                                                                   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.
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

                                                                    Recall the relation between flux and current in a macroscopic
                                                                                           ∆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:
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
                                                                                         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
                                                                                                Φ=   .
                                                                    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
                                                                 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

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.
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

                                                                 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.                    4. Detecting weak magnetic field (SQUIDS)

The Cooper pairs would flow like a superfluid, unless there is
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

                                                                   (and Wikipedia)
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.

The Large Hadron Collider:

                                                                  (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

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

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

Superconductivity               52

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