Superconductivity

Discovery of Superconductivity

H. Kammerlingh Onnes discovered superconductivity in 1911 when it was observed that the resistance of Mercury (Hg) dropped to zero at approximately 4.2K. After this was observed, many other metals exhibited the same phenomena when they reached a certain critical temperature T_c. Creating a loop and inducing a current onto a superconductor with zero resistivity will result in that current being sustained almost infinitely, so long as the metal remains at zero resistivity. This is known as a persistent current. Compared to an ordinary metal with an induced current going around in a loop, the current in the ordinary metal ring will decay quickly because of resistance.

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Brief Introduction to Superconductivity

In order to explain superconductivity, knowing a little bit about atomic structure in metals is useful. In most metals, the atoms are arranged in a lattice. Electrons are somewhat free to move around a metal because they are held on loosely to the lattice. This is also what makes metals good conductors. As these electrons move around the metal, they can collide with other atoms and lose energy by way of heat. In a superconductor, these electrons are able to move freely as much as they please without colliding due to certain forces and is explained by BCS Theory.

Critical Temperature of Superconductors

The critical temperature for superconductors T_c is the temperature where the electrical resistivity of a metal drops to zero. This occurs quite suddenly once the temperature is reached and is thought to have undergone some kind of phase transition. This phase of superconductivity is described by BCS theory.

BCS Theory

John Bardeen, Leon Cooper, and Robert Schrieffer modeled the properties of Type 1 superconductors, which eventually became known as BCS Theory. An important part of this theory requires the pairing of the electrons in the superconductor into what are known as Cooper pairs. The pairing of these electrons is a result of attraction from distorted lattice vibrations, which in turn causing a phonon interaction. What happens is one of the electrons comes in contact with the lattice and causes the distortion. This results in a net positive charged localized at that location and then attracts the other electron in the pair. These pairs of electrons behave differently compared to a single electron. The electrons in a Cooper pair have opposite spins and momentum. Because they are also in a superconductor, they have zero resistance as well and do no scatter. These electrons behave somewhat like bosons, which are able to condense into the same energy level. The Pauli Exclusion principle doesn't allow electrons to occupy the same energy level so theoretically there could have been a phase transition to induce this behavior. Compared to normal electrons, the electrons in a Cooper pair have less energy and leave an energy gap. They are known as superelectrons. This energy gap is one of the things that sets these superelectrons apart and prevents collisions with other electrons which could lead to resistivity, as well as suffer no scattering as a result of no collisions. Also, this lack of resistivity allows a persistent current in which can flow for very long durations of time and carry large amounts of electrical current without and dissipation due to resistivity.

To find the concentration of superelectrons:

$n_{s}=n[1-\left({\frac {T}{T_{s}}}\right)^{4}]$

Superelectrons have a lower energy than normal electrons due to the energy gap $\Delta$.

$\Delta E$ ~ $g(0)\Delta ^{2}$

In order to break up a Cooper pair, an energy of $2\Delta$ is required.

Consequences of BCS Theory

Electronic Density of States (DOS) of a superconductor:

$g(E)=g_{n}(E){\frac {|E|}{\sqrt {E^{2}-\Delta ^{2}}}}$

Critical Field

It is possible to destroy superconductivity if a magnetic field is applied. The magnetic field would have to be strong enough to destroy superconductivity, even at temperatures below T_c.

$H_{\text{c}}(T)=H_{\text{c}}(0)[1-\left({\frac {T}{T_{c}}}\right)^{2}]$

Specific Heat

Along with the resistivity of the compound being affected at the critical temperature T_c, the specific heat is affected as well because it is a second order phase transition. What occurs is a discontinuity.

$c_{\text{n}}-c_{s}=-\left\{\mu _{0}TV_{m}\left({\frac {dH_{c}}{dT}}\right)^{2}\right\}_{T=T_{c}}$

The specific heat is also exponentially time dependent.

$c_{s}=ae^{-b\left({\frac {T_{c}}{T}}\right)}$

An energy gap appears at E_f.

$E_{g}=2\Delta$ which is proportional to kT_c (~kT_c)

$H_{\text{c}}(0)\approx {\sqrt {\frac {2nk^{2}}{\mu _{0}E_{f}}}}T_{c}$

Meissner Effect

The transition from a normal metal to a superconductor causes the Meissner Effect, which is a state of perfect diamagnetism. In this state, the superconductor excludes magnetic fields from its interior. In other words, it expels magnetic flux completely. The Meissner effect is shown in MagLev trains, where the trains are held up in suspension due to the magnetic field.

image from http://graneyandthepig.files.wordpress.com/2009/02/maglev-trainjapan.jpg?w=450&h=316

London Equation

The London Equation describes the Meissner Effect. It describes the decay of a magnetic field inside a superconductor in terms of a parameter known as the London penetration depth $\lambda \,\!$.

${\overrightarrow {B}}=-{\frac {m}{n_{s}e^{2}}}\nabla \ \times \ {\overrightarrow {J}}_{s}$

$B_{y}(x)=B_{y}(0)e^{\frac {-x}{\lambda }}$

$\lambda ={\sqrt {\frac {m}{\mu _{0}n_{s}e^{2}}}}$

The London Penetration Depth $\lambda \,\!$ is also temperature dependent. At T_c, it is expected that the magnetic field will penetrate the whole sample.

$\lambda \,\!=\lambda \,\!(0)(1-{\frac {T^{4}}{T_{c}^{4}}})^{-1/2}$

The spatial distribution of supercurrent is then

$J_{z}(x)=-({\frac {n_{s}e^{2}}{\mu _{0}m}})^{1/2}B_{y}(x)=-J_{s}(0)e^{-x/\lambda }$

Coherence Length

The coherence length $\zeta \,\!$of a superconductor is related to the Fermi Velocity v_F of the material as well as the energy gap. It represents the extent of the superelectron wave function. When magnetic flux penetrates a sample, the magnetic field goes through regions in the form of cylinders. These cylinders are actually filaments that consists of vortex of flux lines. Supercurrent flows around the walls of these vortexes. The core of these vortexes has a diameter of $\zeta \,\!$ and the circulating supercurrent that is present around the normal core of depth $\lambda \,\!$.

$\xi \approx {\frac {\hbar }{\Delta p}}\approx {\frac {\hbar v_{F}}{2\Delta }}\propto {\frac {1}{T_{c}}}$

Flux Quantization

The magnetic flux going through a superconducting ring can be quantified by the following:

$\Phi \,\!=n{\frac {h}{2e}}=n\Phi _{0}$

Types of Superconductors

Type 1

Type 1 superconductors are modeled by BCS theory. Typically, they are composed of metals that are able to show some conductivity at room temperature. In order for a type 1 superconductor to exhibit superconductivity, they must be cooled. They require some of the coolest temperatures for this to occur. In general, cooling things tend to result in processes or reactions slowing down. In this case, what is actually occurring is the molecular vibrations of the metal itself are slowing down. This lowers resistance because it allows electrons to flow freely without collisions. Collisions cause resistance and heat as a result of energy transfer in the collision. BCS theory states that the electrons form Cooper pairs, which come together through an attraction force as a result of distorted lattice vibration and a phonon interaction. This can be described as coupling. Once a type 1 superconductor reaches the critical temperature where superconductivity is shown, a very sharp phase transition occurs. In this state, they exhibit perfect diamagnetism, which means that they repel any magnetic flux that tries to go into the superconductor. In order for there to be current in the superconductor, a very strong magnet will be required. What this does is induce an opposing magnetic field which causes a phenomenon known as the Meissner effect. What is observed is that the magnet will float above the superconductor. An example is the video above.

Type 2

Unlike type 1 superconductors, which are usually pure elements, type 2 superconductors are made from alloys, which are solid solutions that consist of one or more elements, and metallic compounds. What sets type 2 superconductors from type 1 superconductors is their transition from their normal state to the superconducting state. Instead of having a sharp transition state, as seen in type 1 superconductors, it tends to be a gradual process. They do not require the coldest temperatures as needed in type 1 superconductors, which allows them superconduct at higher temperatures. As a result of this, type 2 superconductors are able to conduct more current. They are not perfect diamagnets because they do allow some magnetic flux to penetrate. This occurs because the coherence length is shorter than the London penetration depth, provided that the magnetic field applied is strong enough. This state is known as the vortex state. It is like an intermediary state between the normal state and the Meissner state. The magnetic flux that penetrates in this state can be quantified as a fluxoid. This causes various kinds of phenomena such as the ability to superconduct at higher temperatures.

Superconductivity at High Temperatures

Other than at low temperatures, it has been discovered that superconductivity also occurs at high temperatures. In 1968, Karl Müller and Johannes Bednorz of IBM's Zurich research lab discovered this occurrence of high temperature superconductivity, or HTS for short. They discovered the first high temperature superconductor using a ceramic made of Lanthanum Barium Copper Oxide (LaBaCuO), which experiences a transition at 35K. This led to a family of high temperature superconductors based on cuprate and perovskite ceramic materials. Perovskite materials tend to have a crystal structure in which many superconducting ceramics seem to duplicate. Ceramics are thought to be insulators however this discovery of them exhibiting superconductivity opened new doors in physics. All High Temperature Superconductors are type 2 superconductors.

Cuprate Based Superconductors

Perovskites

Iron Based Superconductors

Discovered by the Chinese and is a whole new class of superconductor. These are interesting objects as the ferromagnetism should destroy the superconducting properties but this is not what is observed. More on this subject in years to come. These are also known as iron pnictides.

Uses of High Temperature Superconductors

High Temperature Superconductors have the possibility to be used in all kinds of technological and mechanical applications. The benefits of High Temperature Superconductors are many. In hospitals, Magnetic Resonance Imaging or MRI for short are basically a giant superconductor. What can be improved with MRIs is the cooling. In order for superconductivity to occur, the magnet must be cooled to its critical temperature, which typically requires the use of liquid-helium. Instead of liquid-helium, liquid-nitrogen would be less costly with the use of a high temperature superconductor.

Ongoing Research

Imagine a room temperature superconductor. That would be amazing!

Applications

With advancements in superconducting material being made at an increasing pace there will surely be an increasing number of applications.

• Superconducting wires can make the transport of electricity much more efficient.

• Using the Meissner Effect, Magnetic Levitation, or MagLev Trains provide faster, quieter, and smoother rides than traditional trains.