Characterization of Magnetic Field Phase Transitions
in Type II Supercon ductors
Timothy McIntosh, Faculty Mentor: Richard Scalettar,
Department of Physics, University of California, Davis,
The discovery of superconductivity
by Onnes, in 1911, did not bring a wave of expectation as the development
of near room temperature type II superconductors did, in 1986. In both
Type I and Type II superconductors, electrical resistance is zero, yet
the transition from a normal to superconducting state is different in both.
Type II superconductors are special in that they pass through several phases
on their way from a normal to a superconducting metal. The Phases of the
magnetic field change as temperature and field strength vary. Finding characteristics
such as structure and field limits can be very useful in understanding
the process of vortex phase change. These findings allow for more precise
applications, and aid in the development of new materials. I am developing
a Monte Carlo program which simulates Type II superconductors to investigate
the peak effect and phase transitions of dirty Type II superconductors.
The development of a Simulation in FORTRAN and further understanding of
the peak effect may unveil the workings of the field induced phase transitionsi
and lead to higher critical currents.
Superconductivity was discovered in
1911 by a Dutch physicist, H. Kamerlingh-Onn es. The effects of superconductivity
were first seen in an experiment by a studen t of Onnes. This student was
using liquid helium, which was only first manufactur ed by Onnes a few
months before, to measure the resistivity of Lead as it cooled to the record
low temperature of liquid Helium, 4 Kelvin. To the great surprise o f the
student and to Onnes, the resistivity of Lead became absolutely zero near
4 Kelvin. This shocking news was not predicted. It contradicted the thought
that r esistance only would be zero when the material lost all thermal
energy, called ab solute zero. Thinking that a mistake was made in the
test, Onnes himself repeated the experiment and found the same result.
Investigating this phenomena, Onnes pl aced a current in a superconducting
ring and tested the current over time. He fou nd that indeed, the current
was not being degraded by resistance. He later conclu ded, materials superconduct
at low temperatures up until a critical temperature.
The mechanism of superconductivity
wasn't understood until over 50 years after i t was discovered. Three scientists,
J. Bardeen, L. Cooper, and J. R. Schrieffer, d eveloped a theory, named
for their initials ( BCS ), which describes the nature of superconductivity.
This theory is based on Cooper pairing.
Cooper pairing is the pairing of electro ns in a superconducting atomic
lattice. Since electron are so much less massive t han the nuclei of atoms,
they have higher momenta. When an electron moves through the atomic lattice
of a superconductor, the negative electrical charge, creates an attractive
force on the atoms around it. This force distorts the atomic lattic e and
pulls the atoms together toward the position of the electron, but because
t he momenta of the electrons is vastly larger than those of the atoms,
the electro n will quickly leaves this position of atomic attraction. At
this point in time t he electron has left the local area, but the atoms
still remain in the distorted lattice causing an area of overall positive
electrical charge. This positively ch arged area in turn will attract another
electron. It is found that the best attra ction occurs between electrons
which have the same momenta and wave function, bu t opposite spins. In
this way, superconductors are able to pass the energy from o ne electron
to another with out losing it to the distortion of the atomic lattice .
All the energy remains within the electrons which results in a complete
lack of electrical resistance.
For most of the century the highest
temperature known superconductor existed at 25 K. In 1986, superconductors
were found at 40 K and higher, spawning the term h igh temperature superconductors.
Today, superconductor temperatures reach an amaz ing 135 K, noting that
cheaply made liquid nitrogen is 77 K.
In 1933 two German scientists, W. Meissner
and R. Ochsenfeld, observed and repo rted a superconductor's effective
repulsion of a magnetic field. This is now know n as the Meissner effect.
By two basic laws of electrodynamics
we can explain the Meissner effect. By Fara day's and Ampere's Laws, an
applied magnetic field induces a current, and an appl ied current produces
a magnetic field. During the application of a magnetic field , a current
is produced in the surface of the material, and consequentially this current
will induce a magnetic field directly opposing the applied field. In a
su perconductor the produced current will remain indefinitely, so the induced
magnet ic field will oppose the applied field for as long as the superconducting
state i s maintained or the applied field becomes too great.
A distinct difference in superconductors
appears when magnetic fields are applie d to different superconducting
materials. This difference divides superconductors into low temperature
type I, and high temperature type II superconductors. Type I superconductors
classically retain their superconducting properties until a cri tical magnetic
field is reached. Most type I superconductors exist in a magnetic field
up to around 0.1 Tesla ( The earth's magnetic field is 0.00005 Tesla ).
Typ e II superconductors experience a transitional state as the magnetic
field rises above the first critical field. In this transitional state,
superconductors funct ion as both a normal and a superconducting metal.
After the superconductor passes the first critical field the magnetic field
begins to penetrate the material, bu t it does so in a quantized state.
As each quantized, or unitary magnetic line en ters the superconductor,
it creates a circulating electric microcurrent perpendic ular to the field.
These microcurrents circulate about each magnetic field line a nd are called
vortices. The vortices create small regions of normal metal leaving the
space in between vortices in a superconducting state. When the field reaches
a second critical value the space between the vortices becomes small and
the mat erial no longer superconducts.
In the transitional phases between
the two critical fields in type II Supercond uctors there is a range of
phases the vortices can take. These states range from lattices, to glasses
and liquids. When the magnetic field being applied to the su perconductor
is near the lower critical field, after the vortices begin forming, the
vortices arrange into a very structured triangular Abrikosov Lattice. As
the field rises closer to the second, higher, critical field, the vortices
get compre ssed into a more disordered state, called a Bose-glass. The
vortices begin to mov e from the force of the applied current as the field
increases and more vortices are crammed into the superconductor. The flux
of the vortices requires energy, an d in this system the sources of energy
are the thermal energy and the current. A t fixed temperatures below the
critical temperature, the thermal energy will not contribute to vortex
flux. Since the vortex flux energy is only drawn from the cu rrent, there
will be a current reduction, which is a form of resistance. As the v ortex
motion increases so does the resistance, and the superconducting regions
di sappear. In this way Superconductors can only be maintained until a
maximum carry ing current, called the critical current.
In type II Superconductors, the atomic
arrangement of the material affects the m otion of the vortices. The atomic
arrangements create areas of attraction to the vortices, called pin-scapes
and pinning sites. Vortices travel through these pin- scapes, getting knocked
about by other vortices and thermal fluctuations, sometim es getting stuck.
About five years ago, it was noticed
in some superconductors that at low magneti c fields near the critical
temperature, a large peak in the critical current occu rs. This peak results
from a unique balance between the thermal energies and inte raction energies.
At these low magnetic fields the vortices form a firm triangula r lattice,
but as the temperature increases, the vortices begin to move around sl
ightly out of their Abrikosov lattice. These slight movements allow each
vortex t o find a deep pin, which restricts the vortices' motion more than
the current can move them. Further experimentation showed that this peak
effect, also called the fish tail effect also occurs with small temperatures
and large fields ( In this case the increased field allows the vortices
to find deep pins, instead of the th ermal energy as described ). These
pinning actions result in a region of greater critical current, which we
see in the peak and fish tail effects.
The explanation of the peak effects
still not entirely complete and assumes that the effect originates in the
copper oxide planes of the Superconductor. It is co nceivable that the
peak effect may occur across planes as well. While searching f or evidence
of a peak across the copper layers, a base study of the critical curr ent
is possible. This study would show the action of vortices without the interac
tion forces, and comparing the two may result in a clearer understanding
of the f ishtail and peak effects.
Materials and Methods
Testing hypothesis in physical matter
is a challenging task, and often there are no means for which measurements
can be made. In these cases it is necessary to e mploy numerical simulations.
In superconductors, vortices act as
a series of flux lines that interact with th e material, the current, and
with each other. Implementing molecular dynamics, a series of particles
connected by springs obeying material dependent magnetic bend ing forces
can represent vortices. In this way it is possible to simulate and stu
dy a vortex motion in a superconducting material.
Simulating a dynamic system, such as
in superconductors, requires knowledge of t he energies and forces involved,
and requires the use of Newton's equations of mo tion:
a = dv/dt = F(R)/m
vnew = dR/dt = vold + a(t)
Rnew = Rold + vnew(t)
F = force a = acceleration R = position v = velocity
m = mass t = time
Determining and totaling the forces acting on the particles
in the system provide s information to determine acceleration, velocity
and position. Forces in these s ystems, include interaction forces, bending
forces, the Lorentz force, pinning fo rces, thermal forces and a frictional
factor resulting from the material. With va lues for acceleration, velocity,
and position, and including real uncertainties, integrating over time allows
us to simulate particle motion.
Having a simulation of a superconducting
material tells us nothing about its wor king. Measuring values such as
kinetic energy and potential energy gives insight into the interactions
of the vortices. An important characteristic of the flux ph ases is the
organization of the vortices, or structure. Structure is the measure of
Fourier series for the potential energy. If the vortices are very periodic
an d organized the Fourier series will result in a peak in the sinusoidal
coefficien ts, while a disordered arrangement will result in a even distribution
of the coe fficients. Structure is a good measure for determining if the
vortices are in an Abrikosov Lattice, or if they have melted into a glass
or liquid. Another importa nt measurement is the kinetic energy. Determining
where the kinetic energy of the vortices begins to increase reveals when
the current has begun to move the vorti ces and tells us what the critical
current is. In my examination of the peak effe ct this measurement is most
Running the simulation with no interaction
forces and fixed at To/Tc = 0.9 ( To = Temperature, Tc = Critical Temperature
) and the magnetic field at values betwe en 0.36 and 1.20 Tesla for a single
vortex in 50 copper layers revealed no criti cal current peak across planes.
The individual vortex motion across the pin-scape resulted in motion that
had high levels of noise. This resulted in high error ba rs in the calculations.
Averaging over several pin-scapes did not reveal any chan ge.
A second run with the same values for
field and temperature with a 7x7 lattice t hrough 50 layers, again showed
no critical current peak across planes. and simila rly large error bars.
With no interaction forces, the critical current values we re found to
be 5.0 x 10-3 at 0.36 Tesla and 2.0 x 10-3 at 0.94
Tesla. With intera ction forces, the critical current values were found
to be approximately 2.1 x 1 0-2 at 0.36 Tesla and 2.6 x 10-2
at 0.94 Tesla. There was a 10 fold difference be tween simulations with
and without interaction forces.
Runs with only a single vortex revealed
no peak across planes. This is not surpr ising, and the results are sufficient
to determine that there is little to no pea k effect across planes. The
bending force is the only force that acts across plan es, it does not behave
similarly to the interaction forces and as a result no pea k effect should
Finding a 10 fold difference in the
critical currents for a 7 x 7 lattice across 50 planes with and without
interaction forces was very unexpected. It was though t that vortices would
be able to travel into the absolute largest pinning sites w hen there was
no interaction with other vortices. It was thought that this would cause
the vortices to have a greater overall pin and consequently a larger critic
al current. Finding that these ideas are not backed up by numerical results
is in triguing and is reason for a more detailed look. Research is ongoing.
Raising the critical current is one
step to productive superconductors, and the understanding and utilization
of the peak effect proves promising. Current High t emperature superconductors
are far from what commonly is known as high temperatur es, yet the potential
usefulness of superconductivity is a great prize. Having s uperconductors
at room temperature, in every home, driving everyday machines insp ires
scientist to search for practical superconductors. Superconductivity will
ge nerate a new generation of new scientific super tools that will further
open the doors of physics and other fields. The knowledge gained from testing
and resear ching will push materials and low temperature physics into a
new time, where anot her scientist in a lab will happen upon another unpredicted
and similarly bizarre phenomena as superconductivity.
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