Magnetic Flux in Superconductors
by Jan Kycia
Dept. of Physics, University of Waterloo
In 1911, Kamerlingh-Onnes liquefied helium. This
allowed him to study the properties of materials at lower
temperatures than those ever obtained before. Soon after,
a student in his group unexpectedly discovered
superconductivity in mercury. A superconductor can
conduct electricity with no resistance or heat dissipation.
By measuring the resistance of mercury at low
temperatures, the electrical resistance was found to drop
to zero below 4.15 degrees kelvin (K). Many other
materials were later found to become superconducting,
such as lead, niobium, aluminum and tin. Rules of thumb
came about for predicting which materials would
superconduct. Although nowadays many novel
superconducting materials have been found that do not fit
easily into any of these categories, these rules of thumb
were very useful to researchers in the past. One rule was
that the material had to be metallic. Another rule was that
in general, poor electrical conductors at room temperature
would have the highest superconducting transition
temperature, (Tc). Below the transition temperature, the
material becomes superconducting. Aluminum is a good
conductor, and its Tc is about 1 degree kelvin. Lead is a
poor conductor, so its Tc is about 7 K. Gold, silver and
copper do not superconduct at any attainable temperature.
Another rule of thumb was that magnetic materials
do not become superconducting. For example, iron and
nickel do not superconduct. The reasoning here is that the
superconductivity and magnetism compete with each other
since both involve order to produce a lower energy state.
All superconducting materials have a critical magnetic
field, Hc, above which they stop superconducting. There
are many ways to test the properties of superconducting
materials and the phenomenon of superconductivity itself.
The order in which the experimental parameters are changed
can drastically change the final state of the superconductor,
as the final state is “history-dependent”.
One way to test the superconducting properties of
a material is to make a ring of the material and study its
response to magnetic fields. If you had a superconducting
ring in zero magnetic field, T< Tc, H = 0, and then a
magnetic field was applied perpendicular to the plane of
the ring, by Lenz’ law, with no resistance in the ring
material, a current would be induced, flowing to exactly
cancel the change in magnetic field. So the magnetic field
inside the ring would always be zero. If you keep raising
the field, the counteracting current keeps increasing until
the superconductivity breaks down and the material goes
normal. That is, it becomes resistive and all the magnetic
field lines go rushing into the ring so that the magnetic
field becomes uniform.
Now, you can field-cool this ring, which means you
apply the magnetic field to the ring while the material is at
a temperature above Tc, then cool the ring below Tc. The
magnetic field inside the ring is fixed to be what was
initially applied, even if the externally applied magnetic
field is varied. Even if you lowered the magnetic field to
zero and turned off the magnet, a counteracting current
would be running around the ring to exactly maintain the
original magnetic field. This current can persist in a
superconductor indefinitely, as long as it remains
superconducting. Since there is no resistance, it is
predicted that the electrical current could continue running
for at least 108 years.
The research community has determined that two
classes of superconductors exist: Type I and Type II. They
are differentiated by the way they deal with a magnetic
field when they are field-cooled. Lead, which is a Type I
superconductor, will go normal above a critical magnetic
field of about 1000 gauss (i.e. Hc = 1000 gauss). Imagine
field-cooling a thin foil of lead with a magnetic field
perpendicular to the foil. For very low levels of magnetic
field, all the field lines will be expelled from the lead foil
and the magnetic field inside of it will be zero. This field
expulsion is called the Meissner effect. However, as the
magnetic field is increased, it will require too much
energy to bend the magnetic field lines (this energy is
called the bending energy) such a long distance from the
center of the thin foil all the way to the edge. The thinner
the foil and the larger the foil area, the lower the magnetic
field that can be expelled. Above a certain field, the
superconductor breaks down and allows field lines to
penetrate it at the middle of the foil. This minimizes the
bending energy. The magnetic field inside the normal
region happens to have a field of exactly the value of the
critical field, Hc. Now, as the magnetic field is increased
even more, more of these regions are required to reduce
the bending energy.
A Type II superconducting foil will deal with
lowering the bending energy in a different way. Above the
first critical magnetic field, Hc1, the material allows
magnetic fields to penetrate, while trying to minimize the
total energy of the system. A Type II superconducting
material wants to have as much boundary between normal
regions, with field lines going through them, and the
superconducting regions because the energy is lowered if
it can maximize the length of the superconducting/normal
boundaries. In order to do this and also minimize the
bending energy, the magnetic field lines are broken up into
many tiny regions that are all evenly spaced. Each tiny
region contains a single unit of magnetic field, a flux
quantum, or flux vortex. Just as electric charge is quantized
to the charge of an electron, the magnetic field is quantized
to a flux quantum. As the applied magnetic field is
increased, the density of the flux quanta increases. The
flux quanta interact with one another. As they try to move
as far apart from one another as they can, they form a
triangular lattice pattern.
These patterns can be studied by the Bitter technique.
A metal foil is cooled with the magnetic field applied in a
small gas filled container and then a small amount of iron
is very carefully vapourized. The iron particles drift like
cigarette smoke and fall onto the foil. The local magnetic
field affects how the iron collects on the surface. The iron
prefers to go to the higher magnetic field areas. By
warming up the foil and either looking at it with an optical
microscope or, for very high resolution, with a scanning
electron microscope, the original field distributions can
be seen. Figure 1. shows several images of the vortex
lattices (Abrikosov lattices) running through a Type II
superconductor, YBa2Cu3O7. Figure 2. is an image of a
Type I superconductor which was field-cooled and
decorated with iron. Here you do not see the individual
flux quanta running through the superconductor but instead
see relatively large domains of either superconductor (no
field penetration, therefore no iron build-up) and normal
conductor (field penetrates and iron collects).
Superconducting wires can be extremely useful for
making electromagnets. Because the wire has no electric
resistance, no power is required to maintain the magnetic
field. To be practical, the wire is desired to have a high
critical temperature, as it will be less expensive to keep
cold. The higher the Tc the better, but in fact certain values
of temperatures are very significant. Liquid helium, which
has a temperature of 4.2K, can be easily stored in efficient
storage dewars (thermos'). Useful superconducting magnet
wires should then have a Tc higher than 4.2K.
Another issue is that the superconductor is influenced
by the size of the magnetic field. As the magnetic
field increases, the superconductor's Tc decreases. The
rate of decrease of Tc in magnetic field depends on the
superconducting material. The best magnet wires are type
II superconductors, and so flux lines penetrate the wire and
produce vortices. As the field is increased the density of
the vortices increases and the superconducting regions
decrease. Superconducting materials have a critical current
density above which they stop superconducting. As
the field is increased, the current density increases for a
given total current because of the increase in normal
conducting regions. A failure mode for superconducting
wires is that although there is no electrical resistance due
to the electrons colliding with defects or the material’s
ion lattice, resistance can arise from flux lines moving
around. To avoid this resistance, the flux lines must be
pinned into energy wells. This is done by purposely
introducing defects in the superconductor in anticipation
for flux lines to get pinned. Not every flux line has to be
pinned because they interact with each other to produce
the vortex lattice. If many of the vortices are pinned, they
can hold down the whole vortex lattice from moving and
dissipating energy and producing a finite resistance.
Another problem with superconducting magnet wire
is that superconductors do not conduct heat very well.
Electrons are usually the most significant contributor to
thermal conductivity The conduction electrons in a normal
metal act as a gas running through the ionic lattice,
bouncing against the lattice and transferring energy. This
electron cooling is similar to air cooling or heating an
object by blowing on it. Due to the nature of the superconducting
state, the superconducting electrons do not collide
with the ionic lattice, do not produce electrical resistance,
and also do not transfer their heat energy and provide
thermal conductivity. This is a problem because if any part
of the magnet wire goes normal, a lot of energy is dumped
into a very small area. If the heat energy is not conducted
away quickly, there is enough energy to melt the wires and
destroy the magnet. To improve the thermal conductivity
of the magnet wires, the strands of superconducting wire
are imbedded in copper, an excellent thermal conductor.
Figure 3 shows a cross sectional view of a superconducting
wire.
You can see small (about 1um diameter) superconducting
fibers, surrounded by a thin diffusion barrier,
which are then imbedded in copper. The diffusion barrier
is extremely important to avoid the Sn reacting with the
copper. Parameters have to be optimized to have the
proper amount of thermal conductivity (copper), protection
from diffusion (diffusion barrier thickness) and current
capacity (superconducting wires).
Magnetic flux can be measured in a very sensitive
way with a Superconducting Quantum Interference Device
(SQUID). In short, the basic element of a SQUID is a
superconducting metal ring into which one or more
Josephson junctions (ie. weak links) are integrated. Current
is applied to two leads attached to the ring and the magnetic
flux that the ring contains induces a voltage change. In this
way, as little as 10-10 gauss of flux can easily be measured
(the earth’s field is ½ gauss and the strongest magnets are
on the order of 105 gauss). The SQUID’s extreme sensitivity
consequently makes it an attractive device for a wide range
of applications in which one desires to quantify small
physical quantities that are associated with magnetic flux.
Specific examples include current, voltage, movement
and of course magnetic fields and their gradients. The
resolution of such measurements is only limited by so
called ‘1/f noise’ (a type of ‘flicker’ noise that varies
reciprocally with frequency) caused by the thermally
activated hopping of the flux vortices between pinning
sites in the superconducting films. A significant research
effort is presently being made to better understand and
reduce 1/f noise in order to make more sensitive amplifiers
and sensors. The noise levels have nearly entered the
quantum limit due to the Heisenberg uncertainty principle.
With superconducting devices running in the quantum
noise limited regime, new ways of implementing these
devices and applications such as quantum computers will
be feasible.