Background Physics

What are superconductors?

Superconductors are materials that are characterized by perfect conductivity and expulsion of the magnetic field when cooled below some critical temperature Tc. Perfect conductivity means that no energy is lost to resistance when a current is flowing. Scientists have set up currents in superconducting rings and found no measurable decay in over a year! Expulsion of the magnetic field means that when a superconductor is in a magnetic field, a current is automatically generated along its surface that exactly cancels out the magnetic field within the superconductor. This is called the Meissner effect.

Kammerlingh-Onnes resistivity curve


Fig. 1(a) Vanishing of electrical resistivity below a critical temperature Tc, discovered in mercury by Kamerlingh-Onnes in 1911.

Meissner effect


Fig. 1(b) Expulsion of magnetic flux below a critical field Hc, discovered by Meissner and Ochsenfeld in 1933.
How they work

The mechanism for superconductivity in metals is very complex- so much so that it was not explained until the BCS theory was made in 1957, 46 years after the initial discovery of superconductivity. The basic principles behind the BCS theory are as follows:

Fermi surface
Fig. 2: Each point in this plane represents an electron state with wavevector k=(kx, ky). Energy increases (in general) with increasing |k|, so the lowest energy configuration will have electrons filling all states inside the circle and none of the states outside. The green line is the Fermi surface that separates empty and filled states, and the Fermi energy is the energy of the states on the Fermi surface.

We can think of any crystalline solid as a lattice of positively charged ion cores, with electrons moving between them. Based on the configuration and masses of the ion cores, there are a number of different quantum mechanical states the electrons can be in, each with its own energy and wavevector k. In the simplest approximation we can assume that the only interactions between the electrons is through the Pauli exclusion principle, which states that two electrons can not be in the same state, so that as we add electrons to the system they will tend to just fill up each state starting with the state of the lowest energy. At low temperatures, where very few electrons are excited by thermal oscillations, essentially all states with energy up to a certain level, called the Fermi energy EF, are filled, and all states with energies above the Fermi energy are empty. This is illustrated in Figure 2.

However, this model is incomplete because it ignores the effects of the interactions between the electrons and between the electrons and lattice. The most obvious interaction of this type is the repulsion between electrons due to Colomb's law, since they are all negatively charged. However, the interactions with the lattice cause an attractive force between the electrons, which can in some circumstances overcome their natural repulsion. In physical terms, this attraction occurs because the electron pulls the surrounding lattice ions towards itself, causing the area around it to become positively charged, which attracts the other electron. On a more technical level, which is important for understanding the zero-resistance behavior of superconductors, the attraction occurs because an electron moving with wavevector k1 can scatter to new wavevector k1', producing a phonon with wavevector k=k1-k1'. This phonon may be subsequently absorbed by another electron with wavevector k2, which is scattered to the state k2+k. This is shown in Figure 3.

Phonon transfer diagram
Fig. 3: Electron lattice interactions allow phonons to transfer momentum between electrons. These phonon-mediated interactions have been shown to be attractive for small values of k.

In superconductors at low temperatures, this attractive force is stronger than the electrons' Colomb repulsion. This is still a weak attraction, which would in general not be strong enough to bind a particle. However, because of the quantum mechanical nature of electrons, Cooper showed that a pair of electrons with opposite spins can experience this attractive force from the other electrons in a way that adds up constructively, so that this pair can have energy lower than the Fermi energy without using any of the states inside the Fermi surface. Therefore, since the system will tend towards the lowest energy state, pairs of electrons inside the Fermi surface will condense into these Cooper pairs. The two electrons that make up each Cooper pair are bound, since perturbing one while leaving the other would increase the energy in the system. Also, because the Cooper pairs are each made up of two electrons, they act as bosons, so many pairs are allowed to condense into the same quantum state.

Inside the material, it is energetically favorable to maximize the effect of the attractive forces between the electrons, which have condensed into Cooper pairs. By conservation of momentum, the phonon interaction shown in figure 3 can only occur between two pairs with wavevectors (k1, k1') and (k2, k2') if k1+k1' = k2+k2'. Therefore, the system will tend towards a configuration in which each Cooper pair has the same average momentum p. Then, scattering a single electron or Cooper pair to a new momentum would require overcoming the attractive force between that pair and every other Cooper pair. Since at low temperatures there are no energy fluctuations large enough to overcome this attraction, once the Cooper pairs as a group have a common momentum p, they will keep this momentum permanently, which explains the absolute zero resistance to electric current seen in superconductors.

Superconductors in magnetic fields

As described above, when a superconductor is placed in a weak magnetic field, it exhibits the Meissner effect and expels the field from its interior. However, expelling a magnetic field requires current flow in the superconductor, which increases the energy requirements of staying in the superconducting state. So, if the magnetic field gets large enough, the superconducting state eventually must begin to break down. Superconductors are classified as type I or type II based on how they respond to increasing magnetic field.

Type I superconductors, which include most pure metals and generally have low Tc, always exhibit the full Meissner effect when in the superconducting state. So, if the magnetic field is increased to a critical level, denoted Hc, type I superconductors are abruptly no longer able to expel the magnetic field and revert to the normal metallic state. Type II superconductors, on the other hand, which include all organic and high temperature superconductors, exhibit the full Meissner effect only up to a magnetic field Hc1. After the field reaches Hc1, type II superconductors begin to allow the magnetic field to penetrate, but still maintain the zero resistance characteristic of the superconducting state. However, for all known type II superconductors, there is a second critical level of the magnetic field denoted Hc2 after which the superconductor reverts to the normal state.

One important reason that all standard type II superconductors must eventually revert to the normal state in a sufficiently high magnetic field is that the Cooper pairs are typically arranged in a singlet spin configuration, so that if one electron has spin in the direction of the magnetic field, the other electron has spin in the opposite direction. However, since the an electron in a magnetic field has the lowest energy with its spin pointing in the same direction as the field, a large enough magnetic field will eventually flip the electron spins to the same direction irregardless of the Meissner effect, and so the Cooper pairs will eventually be disrupted. This effect creates an absolute upper bound on the critical field Hc2 for any superconductor in which Cooper pairs are formed with singlet spin configurations.

Why study organic superconductors?

One aspect of organic superconductors such as the Bechgaard salts that makes them interesting topics of study are that the are strongly anisotropic in structure (for more information see the crystal structure section), so that their conductivity differs along the three axes by multiple orders of magnitude. This inhibits the generation of the circular currents that cause the Meissner effect, potentially significantly increasing the critical field of these materials. Furthermore, the superconducting electrons in the Bechgaard salts are believed to form with triplet spin configurations, so that both electron spins are in the same direction, and the pairs could potentially be unaffected by magnetic fields large enough to disrupt any singlet configuration Cooper pair. Therefore, organic superconductors are good candidates for being very high critical field materials, which is important in many applications of superconductors, in particular the carrying of large amounts of current, since the limiting factor in the current capacity of superconducting wires is that too much current will make a magnetic field strong enough to destroy the superconductivity.

Even if organic superconductors do not exhibit high critical fields, they are still important to study since the standard theory of superconductivity does not apply very well to them. Their unique structure causes the current to move primarily in one or two dimensions, which creates many considerations not present in metallic superconductors, and the probable existence of spin-triplet electron pairings suggest that the mechanism for superconductivity in organic superconductors is significantly different than in regular superconductors.



National Science Foundation

Funding for organic superconductor research in the Hoffman lab was provided by the National Science Foundation under grant number DMR-0508812.

This organic superconductor website was written by former undergraduates Julia Mundy and Sam Cross.