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Characterization of Magnetic Field Phase Transitions in Type II Supercon ductors


Timothy McIntosh, Faculty Mentor: Richard Scalettar, Ph.D
Department of Physics, University of California, Davis, CA 95616
   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.
Introduction

    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 important.

Results

    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.

Discussion

    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 be seen.

    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.

Conclusion

    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.

References

Crabtree, George W., and Nelson, David R. " Vortex Physics in High-Temperature Superconductors. " Physics Today, April, 1997.

Dresner, Lawrence. Stability of Superconductors. New York. Plenum Press, 1995 .

Ginzburg, V. L., and Andryshin, E. A. Superconductivity. NewJersey. World Sc ientific, 1994.

Huse, David A., Fisher, Mathew P. A., and Fisher, Daniel S. " Are Superconducto rs really superconductiong? " vol. 358. Nature, Aug., 1992.

Owens, Frank J. and Poole, Charles P. Jr. The New Superconductors. New York. Plenum Press, 1996.

Vidali, Gianfranco. Superconductivity: The Next Revolution? Cambridge. Cambridge University Press, 1993.


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