Superconductivity : An Overview

Discovered by accident in 1911 by H. Kamerlingh Onnes, the man who first liquefied Helium thereby making it possible to observe the phenomenon, superconductivity has brought with it the prospects of marvelous applications some of which take advantage of the lack of head dissipation or electrical loss when running current through a superconducting material (Tinkham, 1). Yet, despite such lofty prospects for the use of superconductivity towards commercial applications, the extreme conditions and limits necessary to keep a material in the superconducting state have prevented such wide-spread use of superconductivity as was first supposed.

In order for a material to attain the superconductive state, the temperature and magnetic field must fall below the curve of the equation: Hc(T) = H(0) [ 1 - (T / Tc)2 ] where Hc is the critical magnetic field, T is the temperature, and Tc is the critical temperature (Solymar, 429). This equation limits the temperature of the superconducting state to Tc, the critical temperature without an applied magnetic field, and Hc, the critical magnetic field at 0 K as shown in Figure 1. Below the T vs. H curve, the material will be in the superconducting state, and above the T vs. H curve, the material will revert to its normal state. Since the applied magnetic field is also a factor in keeping a material in the superconducting state, the aspirations of running exceedingly high current through a superconductor and suffering only negligible losses turns out not to be as readily possible with superconductors as was first supposed because if a current is run through a superconducting material that produces a magnetic field higher than that material’s critical field, then the material will cease to be a superconductor and will transport the current with the usual resistance and heat dissipation.

In pure materials, like lead, mercury, and tin, any applied magnetic field will be completely expelled from the inside of the material in the superconducting state; these superconductors are perfect diamagnets (Dull, 9). Eddy currents form on the surface of the superconducting material in the presence of an applied magnetic field in such a way that they completely cancel the field in the interior of the material (Ashcroft, 727). Since these surface currents exist in three dimensions, they must contain a certain penetration depth into the superconductor material (Solymar, 438). This penetration depth is usually on the order of 102 - 105 Angstroms (Ashcroft, 739).

Referring back to Figure 1, no matter which path is taken in the T vs. H diagram, superconductivity will always be achieved if the temperature and magnetic field coordinates fall under the limiting curve. So, when the magnetic field is zero and the temperature falls below the critical temp, the resistance in the material falls to zero. When a magnetic field is applied, but is remains under the T vs. H curve in Figure 1, the increase in the magnetic flux is opposed by eddy currents formed on the superconductor. This course of events concurs with the classical theory of a material that expells magnetic flux. However, if a modest magnetic field is applied to the material above the critical temperature (normal state), and then the temperature is brought down until within the T vs. H curve, the material now in the superconducting state still provides the currents that exactly cancel the applied field even though the applied magnetic flux did not change during the transition from the normal to the superconducting state (Solymar, 431). This change in the flux of the material without the change of flux of the applied field, called the Meissner effect, is yet another instance where superconductors behave differently then classical theory or intuition would predict (431).

Likewise, the physical explanation of how and why the superconducting state forms are difficult to fully explain using classical theory. Perhaps the principle explanation as to how the superconductor forms involves the pairing of two electrons in the material, which are defined as Cooper pairs. These two electrons indirectly form a bond with each other due to an interaction with the lattice atoms of the material in which they are located (Solymar, 426-428). In free space, these two electrons would repel one another due to Coulombic forces; however, taking in to account the forces that all of the other electrons in the material as well as the atomic movements of the lattice of the material, produce a force between two electrons of opposite spin and momenta that is stronger than the Coulombic force repelling them (Ashcroft, 739-740). The electrons in the Cooper pairs no longer obey the Fermi-Dirac statistics, nor do they obey the Pauli-exclusion principle for that matter (Solymar, 428, 432). Only one wave equation is used to describe all of the superconducting electrons making all of the Cooper pairs theoretically identical (428). While Cooper predicted the theory of paired electrons, it was the BCS theory by John Bardeen, Leon Cooper, and Robert Schrieffer in 1957 that officially extended Cooper’s theories to all superconducting electrons (Ashcroft, 741).

Figure 2: (Dull, 6) In physical terms, when an electron moves between a group of lattice atoms, the negatively charged electron slightly attracts the neighboring atoms distorting the lattice as it moves by and so a net positive charge forms directly behind the moving electron. A following electron would feel the effects of the net positive charge of the off-centered lattice atoms and thus be drawn into the wake behind the first electron before the lattice atoms move back to their original positions (Dull, 6). Figure 2 depicts this scenario in which the leading electron creates a phonon from the lattice interaction that is in effect absorbed by the following electron, and this essentially binds the two electrons together (6). Although Cooper pairs are continually forming and breaking within the superconducting material, these Cooper pairs are best thought of as a single entity, not separate electrons (Ashcroft, 470). The wave equation that describes the superconducting electron or Cooper pair, does in fact essentially describe the movement of the center of mass between the two electrons (Tinkham, 10). The Cooper pairs cease to form when vibrations in the lattice atoms, perhaps due to rising temperatures or other such disturbances, provide enough energy to overcome the energy gap that surrounds the Fermi level; otherwise, the Cooper Pairs remain together and the material retains the superconducting state. The energy gap, typically on the order of 1 meV steadily decreases as the temperature decreases and then levels off, approaching approximately 2.5kTc as the temperature approaches 0 K (Tinkham, 8-9). When the Cooper pairs split up back into individual electrons, then the material ceases to be a superconductor and returns to its normal state (Dull, 7). When the Cooper pairs are traveling through a superconducting material, they do not scatter off of defects or vibrations in the lattice unless these defects or vibrations produce enough energy to first break the Cooper pair (Ashcroft, 750-751). It is this lack of scattering that causes the characteristically immeasurable resistance found in superconductors. These Cooper pairs are also theoretically responsible for the super- current that does not observably decay even without the presence of an applied electric field. So far, of the superconductor experiments that have been done have only attempted to measure the decay in supercurrent over a period of a few years (Solymar, 426). However, these cases do show that the current in superconductors does not appreciably decay over time.

As long as it is energetically unfavorable for a Cooper pair to disassociate, the electrons will remain bonded, even while tunneling through a potential barrier (Solymar, 452). Just as single electrons may tunnel trough an insulating material from one metal to another, so too can Cooper pairs tunnel trough a thin insulator from one superconductor to another. Even more remarkable is the notion that just as supercurrent will flow without the presence of an applied Electric field, so too can supercurrent tunnel without any resistance through a barrier without an applied voltage across the barrier as long as the current remains less than the limiting critical current inherent to that junction. This critical current that flows through the barrier is however, extremely sensitive to the magnetic flux within the insulating barrier (453). The higher the applied magnetic field, the lower the supercurrent that is allowed to tunnel. Since the total current passing through the junction remains the same despite any applied magnetic field, when an applied magnetic field does causes the critical current to fall below the actual current, some of the current that passes through is non-superconducting current and thus causes resistance within the junction. This phenomenon is known as the Josephson effect and the junction the Josephson junction (Dull, 12). When a normal wire is run along-side these Josephson Junctions, any current through the wire produces an applied magnetic field in the Josephson Junction which lowers the critical current allowed and produces some resistance thereby causing a voltage across the junction (Solymar, 455). These Josephson Junctions, therefore, are stable at zero and a finite voltage making them excellent switches. Another advantage to the Josephson Junctions is that no phase change is necessary during their operation. The junctions always remain in the superconductive state, and only the type of tunneling, be it strictly Cooper pair tunneling or a split between Cooper pairs and individual electrons, changes (455). When used as a switching device, the Josephson Junction outperforms semiconductor switches by an order of magnitude (Dull, 12).

Josephson Junctions are also extremely useful as highly sensitive magnetometers, called SQUIDS (for Superconducting Quantum Interference Device). Since the supercurrent through a Josephson Junction is very closely dependent upon the magnetic flux through the junction, measuring the supercurrent through the junction provides information about the detection and measurement of potentially minute magnetic fields. The tiniest magnetic field measured by a Josephson Junction has been recorded at approximately 10-7 Wb/m2 (Solymar, 456). The difficulty in producing and maintaining superconducting devices is one of the largest drawbacks to the marketability of these superconducting devices. Other drawbacks such as the low upper limit of the maximum applied magnetic field to a superconductor; however, were to a large extent overcome as a whole new class of superconducting materials were discovered.

In pure materials with a minimum of defects, once the applied magnetic field exceed the critical magnetic field, then the material can no longer set up high enough currents to expel the applied field and the magnetic field breaches the interior thereby reverting the material to its normal state. However, these critical fields, typically 102 gauss, are usually too low to be of commercial value since one of the main dreams of superconductor applications is the use of high magnetic field current without decay of heat dissipation (Dull, 9). Therefore, pure materials are not used for such purpose. On the other hand, if the material contains numerous defects, then under superconducting conditions, that material forms a mixed state where normal and superconducting states exist side-by-side, called a mixed state (Tinkham, 12). In this case, the material will form vortices, or ‘flux tubes,’ of superconducting states through the material in a triangular fashion, as shown in Figure 3, surrounded by regions of material in the normal state (12). This structure is known as a fluxon lattice (Dull, 11). In this mixed state, the supercurrent is carried in the superconducting regions alone and so the current still flows with negligible resistance until the Lorenz force of the current causes too much drift in the fluxon lattice (Tinkham, 12- 13). The kinds of material producing this mixed superconducting state are called Type II superconductors. Type I superconductors would refer to the pure material in which the superconducting state would rapidly vanish as the applied magnetic field exceeded the critical field. However, for a Type II superconductor, in the normal areas, the magnetic field may penetrate to a greater extent without disrupting superconductivity entirely. Since part of the applied magnetic field occupies the normal states, the diamagnetic energy needed to keep the field out of the superconducting states decreases allowing for a much higher critical field (Tinkham, 12).

Figure 3: Fluxon Lattice (Ashcroft, 733)

Figure 4: Type I vs. Type II Superconductors (Dull, 10) So, instead of the critical field, Hc, as shown in Figure 4, as the limiting applied magnetic field to the superconductor, Type II superconductors may still remain in the superconducting state until Hc2, typically on the order of 105 gauss (Dull, 9). Even though an applied magnetic field surpassing Hc2 fully penetrates the material thereby forcing the change to a normal state, Hc2 is a significantly higher limit than the limiting Hc for Type I superconductors. This is one of the reasons why Type II superconductors such as YBa2Cu3O7 (YBCO), and Bi2CaSr2Cu2O9 are more commonly used for commercial applications than the Type I superconductors.

Another highly motivating reason why these Type II superconductors are chosen over the pure superconductors is that many of these Type II superconductors, such as the two mentioned above, are of a new class of Oxide Superconductors which have been coined ‘High Temperature’ superconductors. This temperature distinction is in relative terms since up until oxide superconductors, that have critical temperatures as high as 125 K for Tl2Ba2Ca2Cu3O10 (TBCCO), were discovered, the critical temperatures previous to this were below 10 K (Solymar, 458). High-temperature superconductors are generally those whose temperatures are greater than that of liquid nitrogen, or 77 K (Dull, 1). While these high temperature superconductors have eased the difficulty in manufacturing of superconducting devices, they still do not satisfy the dreams or inspire the excitement that room-temperature superconductors would.

Despite these drawbacks, many different applications have evolved to exploit the superconducting materials that are now available. Superconducting devices have been used in Metrology to measure universal constants to a greater degree of accuracy such as Plank’s constant, h, which has recently been modified to 6.626196 x 10-34 Js (Solymar, 456-457). Another useful application of superconductors involves its ability to detect incident radiation. If the temperature for a given superconductor is kept just below the critical temperature, heat provided by incident radiation will radically affect the resistance in the superconductor due to the huge discontinuity in resistance between the superconducting and normal state of the material. The observed change in resistance would in this case reflect the change in incident radiation (457).

Superconducting devices also hold the promise of highly efficient motors and suspension systems. Under the right conditions, a superconductor can actually float over a magnet when the magnetic flux is compressed as the superconductor is pushed towards the magnet due to gravity, the repelling forces increase to match the force of gravity. Once one material is floating above another, rapid rotations of the levitating body are possible at close to 100% efficiency due to the nearly frictionless system (457).

While the initial energy and excitement surrounding the field of superconductivity has waned since its discovery due to the rigorous environmental limitations necessary to keep the known superconducting materials in the superconductive state, the significant progress made in just the last few years in understanding superconductivity, developing more suitable materials, as well as the potential applications of superconductivity are so rewarding that the field of superconductivity will more than likely remain viable long into the future.