Note for Mechanics - MECH By abhijith anirudhan

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1 An Introduction to Superconductors: Theory and Application Jonathan Cory, Engineering 302 Electromagnetics, Professor Paulo F. Ribeiro, Fellow, IEEE  Abstract— Radical developments have recently taken place in the field of superconductors. More specifically, high temperature superconductors have been a focal point, and therefore many modifications are being made in the types and varieties of products which employ such devices. In order to better appreciate and understand this field and its implications, it is necessary to attain knowledge of the history and basic principles of superconductivity. With this in mind, the exploration of current applications is ensued in light of recent discoveries. With these topics in focus, it is the goal of this paper to obtain a better understanding and appreciation for research in the field of superconductors. The conclusions drawn are that while much advancement has been made in the field of high temperature superconductors, there is much more room for improvement. The effect seen by such improvements are then widespread throughout medical technologies as well as countless other applications. I. INTRODUCTION uperconductors have often been called the bridge between theoretical physics and practical application. As the knowledge of superconducting materials is ever increasing, the possibility of a high temperature superconductor is nearly realized. The applications of this phenomenon are relevant in nearly every technological field and the medical implications relating to magnetic resonance imaging and brain mapping are astounding. This paper presents an overview of some AC and DC applications of superconductors (specifically high temperature). However, In order to better understand and appreciate these applications of superconductors, it is first necessary to gain an understanding of the history and basic theory of superconductors. temperature could trail off to a set value, or thirdly, the resistance could achieve some minimum point and then approach infinity exponentially. The graph of this hypothesis can be seen in figure 1. [2] Tests were then performed by Onnes with a variety of materials leading to the testing of mercury. Previous tests had found that the resistances fell exponentially as the temperature approached zero. However, the testing of mercury broke the mold. Onnes found that at 4.2 degrees Kelvin, the resistance of mercury fell sharply. “At this point (somewhat below 4.2 K) within some hundredths of a degree came a sudden fall not foreseen by the vibrator theory of resistance, that had framed, bringing the resistance at once less than a millionth of its original value at the melting point… Mercury had passed into a new state, which on account of its extraordinary electrical properties may be called the superconductive state.” [2] S II. HISTORY OF THE SUPERCONDUCTOR Fig. 1 Reaction of resistance to extreme low temperatures [21] A. Original Discovery B. Development In 1908, a discovery made by Heike Kamerling Onnes initially opened the door to an entire realm of possibility involving low temperature testing (Superconductivity-Buckel). This was the first successful attempt at liquefying of helium which provided a new temperature range approaching absolute zero. Soon after this discovery, Onnes was able to perform experiments at these temperature for which the focus was the change in resistance at such temperatures. Three possible responses to these low temperature tests were anticipated. First, the resistance could fall linearly toward zero. Secondly, the In 1933 a second important discovery was made by Meissner and Ochsenfeld that, in addition to being a perfect conductor, a superconductor is also a perfect diamagnet where a thin gap existed between the surface of the superconductor and the magnetic field on the order of 500 Angstroms. This means that “Induced currents in it would meet no resistance, so they would persist in whatever magnitude necessary to perfectly cancel the external field change.” [7] This in the vortex state was termed the “Meissner effect.” The superconductor then excludes any magnetic field that would normally flow through by inducing current loops to exactly cancel the magnetic field. An example which can be seen in figure 2 depicts this

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2 phenomenon. temperature decreases, the number of collisions decreases and the resistance follows. A basic equation for this interaction between resistance and temperature can be seen below. R R0 R0 Fig. 2 The Meissner effect [18] Not long after this discovery Fritz London proposed that superconductivity is a quantum phenomenon. This proposal, although at the time was inconclusive, however, is now substantiated by current microscopic theory. Another important breakthrough was the discovery of the Isotope effect which points at a connection between electron and lattice vibrations in relation to superconductivity (Superconductivity in Science and Technology –Cohen). Other Important additions to the current knowledge of superconductors include the tunnel effect (which points to the likelihood of energy gap in the superconducting state), flux quantization and supercurrent flow through tunnel barrier. C. A Quest for High Temperature Until 1986, the highest critical temperature for a superconducting transition was 23.2 degrees Kelvin. The material was Nb3Ge and speculation was that there was not much more room for improvement. The focus for two European scientists then became to make an alloy in which they could enhance the electron-phonon coupling parameter. This led to a critical temperature of slightly more than 30 K. Variations and further experimentation over the past two decades have led to a current critical temperature record of 125 degrees Kelvin (significantly more than the boiling point of liquid nitrogen -77K). These recent discoveries have opened up the door for countless applications, from high powered MRIs and brain mapping (SQUIDs) to magnetic friction free trains. [3] III.  T T0 Einstein proposed a theory that the electrical resistance of the metals would fall sharply at very low (at that time unattainable) temperatures although popular opinion was that the resistances would only reach a resistance of zero at absolute zero temperature. On a microscopic level, superconductivity implies that electron pairs are forming with a spacing of hundreds of nanometers which is three times larger than the lattice spacing. These widely spaced electron pairs have been termed Cooper pairs after the scientist (Leon Cooper) involved in their discovery. Current and past understanding of “normal” electron behavior has been governed by the Pauli Exclusion Principle. This law states that no two electrons (each with half integer spin) can have the same quantum number or in other words, be in the same state simultaneously. [8] Other existing particles known as bosons carry a full integer charge. Because of this full integer spin, bosons are not governed by the Pauli Exclusion Principle and can therefore simultaneously exist in the same state. Einstein called this condensation as the bosons could move in unlimited numbers to a single ground state. [8] The first noticeable proof of Einstein’s observation was the superfluidity of Helium. This trend was found when Helium was cooled to below 2.17 degrees Kelvin, all viscosity of the liquid vanished. Viscosity, which can be termed the resistance to flow of a liquid, is yet another parallel between the waterelectricity analogies for electrical resistance. Yet another parallel between is the comparison between laminar flow and the energy band gap with tunneling effect. A diagram portraying the laminar flow of a fluid through a pipe can be seen in figure 3. MICROSCOPIC THEORY Originally, the basis for investigating the reaction of electrical resistance in relation to temperature was founded on an observable trend in the characteristics of metals. It was noted that with increasing temperatures the resistance of metals increased to a point while with decreasing temperatures, resistances decreased. The understanding of the principle of resistance varying with temperature is easily understood as resistance itself is merely the effect of collisions within a wire. Thus, as the temperature increases, atomic movement and vibration increases causing more collisions. Conversely, as Fig. 3 Laminar Flow Through a Pipe [8]

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3 In a superconductor, the Cooper pairs, mentioned earlier, act as bosons. This means that the pairs can exist simultaneously in ultra-low energy states. These pairs form closely to the top of the collection of energy levels (also called the Fermi level) by interacting with the crystal lattice. The slight lattice vibrations attract these Cooper pairs, leaving an energy gap. Because the pairs are not subject to Pauli Exclusion Principle due to their combined integer spin, they can simultaneously occupy the same energy state. This attraction is termed the phonon interaction. The energy gap is then the sort of “tunnel” through which resistance free traveling can occur in the case that the thermal energy is less than the band gap. Where in normal circumstances, collisions would occur causing normal resistivity, at low temperatures, the thermal energy drops to a value less than the band gap, leaving the band gap open for tunneling. This theory of functionality for superconducting principles is known as the Bardeen-CoorperSchrieffer Theory (or BCS theory). [7] [5] The BCS theory not only substantiates the microscopic theory of superconductivity, but also predicts a bandgap based on the critical temperature. The equation for this prediction can be seen below where Eg is the predicted bandgap and Tc is the critical temperature. The graph in figure 4 displays some superconductors and their respective bandgaps and critical temperatures. Fig. 5 Superconductors Energy Gap Reduction. [8] The energy gap reduction then provides the means through which a superconducting state is produced in that when the thermal energy is less than the energy gap the resistance of the material drops to 0. IV. SUPERCONDUCTIVE MATERIALS Currently two basic types of superconductors are recognized. These types are aptly termed type I and type II superconductors. In order to understand applications of superconductors it is first necessary to understand these two types as they relate specific characteristics of materials to certain attainable results. A. Type I Superconductors Contrary to intuition, the best normal conductors (i.e. gold, silver, and copper) are not superconductors at all due to their small lattice vibrations. Metals listed in figure 2 are all type 1 superconductors as their lattice vibrations are an attractive force to the Cooper pairs. These types of superconductors fall under the BCS theory mentioned earlier. Typical metals in this category posses “softer” characteristics. They do not maintain their superconductivity at higher temperatures and exhibit lower temperature magnetic fields than type II. Fig. 4 Superconductors and Relational Bangaps. [8] As the superconductor approaches its critical temperature, the energy gap decreases exponentially. A plot of this relationship can be seen in figure 5.

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4 Mat. Tc Mat. Tc Be 0 Gd* 1.1 Rh 0 Al 1.2 W 0.015 Pa 1.4 Ir 0.1 Th 1.4 Lu 0.1 Re 1.4 Hf 0.1 Tl 2.39 Ru 0.5 In 3.408 Os 0.7 Sn 3.722 Mo 0.92 Hg 4.153 Zr 0.546 Ta 4.47 Cd 0.56 V 5.38 U 0.2 La 6.00 Ti 0.39 Pb 7.193 Zn 0.85 Tc 7.77 Ga 1.083 Nb 9.46 flexibility reasons. A table of type II superconductors can be seen in figure 8. Fig. 8 Table of Type II Superconductors [8] Type II superconductors possess not only harder characteristics, but also retain a higher critical temperature and the ability to produce high powered magnetic fields. An alloy of niobium and titanium is currently used in the making of the MRI and other supermagnets utilized by Fermilab. In addition to harder characteristics and a higher critical temperature, type II superconductors possess a higher threshold tolerance to magnetic fields so that they retain their superconducting properties even when in contact with a stronger magnetic field. Overall, some of the differences between type I and type II superconductors can be seen in Figure 9. Fig. 6 Table of Type I Superconductors [8] In addition to having a lower critical temperature than type II superconductors, type I superconductors also have a lower threshold for magnetic field tolerance. This means that when a magnetic field greater than the threshold is applied to a type I superconductor, its superconducting state ceases. Fig. 7 Reaction of Superconductivity to Magnetic Fields [8] A. Type II Superconductors Type II superconductors are often referred to as the “hard” superconductors. They are composed of alloys of ceramics and metal oxides. An example of a newer type II superconductor is a material known as BSCCO which is an oxide composed of bismuth, strontium, calcium and copper. [4] In addition to these materials, silver was added for Fig. 9 Differences in Trend between Type I and Type II [18]