# BCS Theory of Superconductivity

Updated on July 16, 2015

## Introduction

The BCS theory is the first successful theoretical understanding of superconductivity that was widely accepted in the scientific community. John Bardeen, Leon Cooper, and Robert Schriefferproposed their mathematically complex theory in 1957, and were awarded for their work in 1972 by winning the Nobel Prize. It will be presented how this theory came about from previous discoveries, and how there is still a need for further theoretical understanding. This paper will also discuss the different superconductors and what has been discovered since this Nobel Prize winning theory was proposed. It will end by talking about the applications for superconductivity in a wide variety of areas, and how it is leading to some amazing technological feats.

## Background and History

A superconductor is an element, inter-metallic alloy, or compound that will conduct electricity without resistance below a certain temperature [1]. The certain temperature mentioned is different for each element or compound and is referred to as the temperature of conversion [1] or transition temperature [2]. This temperature is denoted with the variable , and it is distinguished by seeing a sudden drop in resistance once the compound reaches it. With zero resistance it is possible that a current will flow foreverin a material that forms a closed loop and has been super cooled below its transition temperature. This phenomenon, along with superfluidity, proves that dissipation free movement is possible, and is the closest thing to perpetual motion in nature. Its inability to be understood by using classical physics causes scientists to refer to it as a macroscopic quantum phenomenon.

Superconductivity was first observed by a Dutch physicist named Heike KamerlinghOnnes in 1911 when he cooled mercury to 4 degrees Kelvin and noticed its resistance suddenly vanished. He won the Nobel Prize in physics two years later for his discovery and research. The next important discovery was made in 1933 when German researchers Walter Meissner and RoberOchenfeld recognized that superconducting materials will strongly repel and tends to expel a magnetic field. They called this strong diamagnetism the Meissner effect, and it led to the prediction of the existence ofan electromagnetic penetration depth. These discoveries and theories were able to describe the relationships between the observed phenomena, but there weren’t any theories that explained how these phenomena came about based off of the fundamental laws of physics. That was until John Bardeen, Leon Cooper, and Robert Schrieffer proposed their theory in 1957. It was the first widely accepted theory that explained superconductivity, and it was able to explain how superconductivity happened in elements and simple alloys that were cooled to temperatures close to absolute zero.

## Understanding the Theory

The theory was named the BCS theory after the men who created it and would later receive a Nobel Prize for it in 1972. Their theory was based off of the discovery made by Leon Cooper that electrons in a superconductor become grouped together in pairs near the Fermi level. The Fermi level is defined as the potential energy level of an electron in a crystalline solid. This grouping of electrons is now called Cooper pair, and it was found that a system of these pairs would function as a single entity in a superconductor [3]. Figure 1 and 2 make it possible to get a better visual of what is happening.This pairing action is very interesting because up to this point classical physics had always defined electrons as purely repulsive to one another. These Cooper pairs behave like a particle with integral spin which do not obey the Pauli Exclusion Principle.This principle states that no two particles with half-integer spin may occupy the same quantum state simultaneously [4]. Since the Cooper pairs aren’t constrained by this principle it is possible for multiple electrons to be in the lowest quantum state. This leads to the fact that the entire collection of Cooper pairs in the metal can be described by a single wave function [5].

The way these electrons are able to form pairs together is through their interaction with the underlying lattice’s vibrations. One of the electrons in the Cooper pair creates a region of excess positive charge by polarizing the lattice when it attracts the nuclei towards it. This positive charge or potential well then attracts a second electron. So the attraction between the two negatively charged electrons is created through the positively charged nuclei. A good analogy is imaging a bowling ball rolling down a bed. This motion would create an indention in the bed which would make it easier for a second bowling ball to follow in its path [1]. So as a voltage is introduced to the superconductor, the Cooper pairs start to all start to move and continue to move even after the voltage is removed because they don’t experience any opposition to their motion. To stop or reverse the direction of the Cooper pairs’ travel would have to be done by separating a Cooper pair and a number of the Cooper pairs with itat the same time. This would cause the current to stop, but this doesn’t happen on its own with time alone.

One way to cause a material to lose its superconductivity is by warming up the superconductor,which causes the Cooper pairs to separate into individual electrons again. Heike KamerlinghOnnesalso found out shortly after his discovery of this phenomenon that superconductivity could be reversed by applying a large current or a strong magnetic field to the superconductor, and the BCS theory supplied a way to experimentally measure the energy needed to separate the Cooper pairs into their individual electrons.Their theory was also successful in explaining the isotope effect, which what happens when the transition temperature is reduced because a heavier atom of the element making up the superconductor is added.If the conduction was purely electronic, then it wouldn’t matter what the nuclear masses were. Since there is a relation though, it points to the fact that there is an interaction between and the lattice is and the electrons. The mechanical properties of the lattice are what causes this drop in transition temperature.

## Discoveries since the BCS Theory

There have been many interesting breakthroughs in superconductivity since Bardeen, Cooper, and Schrieffer released their theory in 1957. A Cambridge University graduate student predicted that electrons would flow from a superconductor to another superconductor even if they were separated by an insulator or non-superconducting material. This student was a British physicist named Brian David Josephson, and he was able to predict a mathematical relationship for the current and voltage across that barrier. For his work the phenomenon was called the Josephson Effect, and he won the Nobel Prize in physics in 1973 [6]. The electrons are able to make it through a barrier that would classically stop them by a phenomenon called quantum tunneling. Tunneling was first witnessed in 1927 by Friedrich Hund and was mainly used in nuclear physics for decay and half-lives. It was then found through the studying of semiconductors, diodes, and transistors that electrons could tunnel in solids [7]. This led to Josephson’s work with tunneling Cooper pairs.

Research on superconductors continues today and now the interest is in high temperature superconductors. The feasibility of using superconductors in certain applications increases with finding compounds that superconduct at higher temperatures. Having all of the advantages of superconductivity without having to use all of the energy to cool a compound and maintain it at that temperature is an alluring idea. This race for finding high temperature superconductors began after an important discovery was made at the IBM Research Laboratory in Switzerland. Alex Muller and Georg Bednorz created a brittle ceramic superconductor that was able work at 30 K, which was the highest known transition temperature at that time. The discovery of a ceramic superconductor that worked at 30 K was important because of how it could not be explained by the BCS Theory. The othertruly amazing part about their discovery is the fact that they created the superconductor out of ceramics which is normally an insulator of electricity. The idea of creating a superconductor out of a material that is classically known as an insulator hadn’t been thought of before by many researchers. Then it was found in 1987 that just by trading one element for another in Muller and Bednorz’s compound the transition temperature rose to 92 K. This marked the first time that a material was created to superconduct at a temperature that was higher than the temperature of liquid nitrogen, which is 77 Kelvin.There have been many different combinations that have been tried to create a superconductor that works at higher and higher temperatures, and as of right now the highest transition temperature known is in the range of 133-138 K. This compound composed of mercury, thallium, barium, calcium, copper, and oxygen was synthesized in 1993 by researchers at the University of Colorado.

The creation of all of these new superconductors and seeing the differences between them led to the need to make two different classes of superconductors. Some superconductors had a sudden drop in resistance while others’ resistance gradually decreased, and some allowed a magnetic field to partially penetrate its body while others didn’t allow it to at all. These differences created Type 1 and Type 2 superconductors.

Type 1 superconductors are mainly metals and metalloids and they also go by the name soft superconductors. Some examples include lead, aluminum, uranium, mercury, zinc, tin, and titanium. These types of materials were the first superconductors to be discovered and they are generally conductive at room temperature as well. It is interesting to note though that not all materials that are good at conducting at room temperatures can become superconductive. Copper, silver, and gold are three of the best conductors yet they aren’t able to become superconductors.The reason for this is that their lattice structures are too tightly packed together so Cooper pairs can’t form. Their lattice type is called face-center cubic lattice (FCC), and their tight lattice structure constrains the lattice vibrations so the essential Cooper pairs aren’t able to form. Having a FCC lattice doesn’t stop all elements from being superconductors though. Some can still form but it’s because of other factors like the elements elastic modulus. The Type 1 metals and metalloids require the coldest temperatures to become superconductors. The BCS theory works for Type 1superconductors but it has trouble with most of the Type 2 superconductors. This is because the BCS theory says that the Cooper pairs form due to lattice vibrations, and this is not the case with high temperature superconductors. The main difference between Type 1 and Type 2 superconductors though is in how they react to magnetic fields. Type 1 superconductors have perfect diamagnetism and will completely repel a magnetic field where Type 2 superconductors allow a magnetic field to partially penetrate its body.

Type 2 superconductors are mainly metallic compounds and alloys and are also called hard superconductors.These superconductors are more complex compounds and some examples are (), , and . These types of superconductors can reach transition temperatures that are much higher than Type 1 superconductors. Their transition to zero resistance is not a sudden drop like in soft superconductors, but is instead more gradual. Below is an example of a generic graph of the resistance drop in a Type 1 superconductor (Fig. 3) and it can be compared to the gradual drop in a Type 2 superconductor by looking at the graph for (Fig. 4) just below Fig 1.

Another important difference between Type 1 and Type 2 superconductors is the Type 2 superconductors’ ability to reach such high transition temperatures. This phenomenon is still not fully understood. The BCS theory is unable to explain how this happens, and 25 years after the discovery of the first anomaly we still don’t have an answer.

## Applications

There are many fascinating applications for superconductors that we have already found, and it will be interesting to see what will be thought of next in this relatively new and mysterious area of study. One of the most well known applications of superconductivity is its use in high speed trains. Otherwise known as Maglev vehicles, they use magnets to suspend, guide, and propel the vehicle rather than through classical mechanical components. This technology is able to do some amazing things by using superconductivity. The highest recorded speed of a maglev train is 361 miles per hour which was set in 2003 by the CJR’s MLX01 superconducting maglev [8]. The magnetic field in the trains can be created by permanent magnets or by superconducting magnets. A positive feature in the superconducting system is that the liquid nitrogen used to cool their magnets is inexpensive and leads to higher speeds and pay loads. On the other side of that though a strong magnetic field is created so magnetic shielding is necessary, and the repulsive force is not strong enough at low speeds so wheels are necessary for those speeds.

Another way that superconductors have improved existing designs is in magnetic resonance imaging (MRI). By using a superconducting magnet rather than a conventional magnet, they are able to reduce the size of the magnet. Superconductors are also helpful in special instruments like a superconducting quantum interference device (SQUID). This tool takes advantage of the Josephson Effect and is able to detect even the weakest magnet field. It can be used for very accurate motion detection systems. Creating generators that use superconducting wires would possibly save energy since the wires wouldn’t have any resistance in them.

The advantages of superconductivity are also getting recognized by the military. They can be used to help reduce the length of low frequency antennas that are used in submarines. They can also help submarines in the making of degaussing cables. American Superconductor has announced that they are making a superconducting degaussing cable, which will lead to less weight, size, and energy. Submarines can emit residual magnetic fields and this could lead to the sub being detected, so the degaussing cables eliminate those electric fields. The military is also using superconductors to create E-bombs. These E-bombs create a large electromagnetic pulse which is capable of taking out the enemy’s electronic equipment.

While superconductors themselves continue to defy our understanding, they are also helping us to learn more about the universe. They are used in high energy particle colliders, and they wouldn’t be possible without the use of superconducting magnets. The Large Hadron Collider (LHC)in Switzerland has 5000 superconducting magnets to accelerate its particles close to the speed of light. These magnets in the LHC have already helped lead to rumors about evidence for the discovery of the Higgs Boson, and I know scientists will continue to find important applications for superconductors.

## Conclusion

The abilities and applications of superconductors are seemingly unbelievable, and I believe it is their potential in technology that makes them so appealing. It has been 100 years since it was observed that an element could be super cooled to the point that it would lose all of its resistance to electricity, and we still don’t have a full answer as to how this is possible. John Bardeen, Leon Cooper, and Robert Schrieffer pioneered a way for us to explain this puzzling phenomenon with their BCS Theory, but it is incomplete. We are already seeing the benefits of its discovery and of Bardeen, Cooper, and Schrieffer’s work in this field, and I know we will continue to learn more amazing applications of superconductivity as research continues. I also know that one day we will have a theoretical answer for how this is all possible, but for now it seems that superconductors will continue to resist our understanding.

## References

[1] Eck, Joe (July 2, 1999). Superconductors. Retrieved February 28, 2012 from

http://www.superconductors.org/INdex.htm

[2] Ginsberg, Donald M. (2012). Encyclopedia Britannica.Superconductivity. Retrieved February 28,

[3] Encyclopedia Britannica. BCS Theory. Retrieved February 28, 2012 from

http://www.britannica.com/EBchecked/topic/57052/BCS-theory

[4] (February 27, 2012).Wikipedia.Pauli Exclusion Principle. Retrieved February 28, 2012 from

http://en.wikipedia.org/wiki/Pauli_exclusion_principle

[5] Serway, Moses, & Moyer (2005). Modern Physics. David Harris

[6](January 21, 2012). Wikipedia.Josephson Effect. Retrieved February 28, 2012 from

http://en.wikipedia.org/wiki/Josephson_effect

[7] (February 9, 2012). Wikipedia.Quantum Tunneling. Retrieved February 28, 2012 from

http://en.wikipedia.org/wiki/Quantum_tunneling

[8] (February 27, 2012). Wikipedia.Maglev. Retrieved February 28, 2012 from

http://en.wikipedia.org/wiki/Maglev

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