BCS theory

The BCS theory is a many-body theory to explain superconductivity in metals. The BCS theory was developed by John Bardeen, Leon Neil Cooper and John Robert Schrieffer 1957. The name derives from the initials of the surnames of its founders that it received the 1972 Nobel Prize in Physics.

Content

The basis of the BCS theory was the experimental observation that the superconductivity of many metals, a relatively strong dependence of the critical temperature of the isotope of the metal is investigated (TC * (M) 0.5 = const. Wherein M denotes the mass isotope ). This suggested that a mechanism of superconductivity, the interaction with the mass-dependent quantized lattice vibrations (whose quanta are called phonons) should be.

This can be thought of as follows: A first electron changes the lattice (respectively a lattice vibration ) through energy output such that a second electron achieved an equal energy gain (eg by changing its path or recording of a phonon ). This is only possible if the grating elements and the electrons ( therefore below the critical current density) move slowly enough.

The idea of ​​the BCS- creator is to postulate the formation of so-called Cooper pairs of two electrons by a weak attractive interaction. Electrons due to their spin (s = 1/2) fermions and as such can not occupy the same state ( Pauli principle). In contrast, the Cooper pairs are of integral spin ( singlet state s = 0 ( anti-parallel arrangement of the electron spins ) or triplet state S = 1 ( parallel arrangement of the electron spin ) ) bosons and can therefore at the same time the same condition, and thus accept all the default state. This is not only energetically favorable, but also manifests itself in one, spanning the whole solid, Bose -Einstein (BE ) wave function.

This wave function can vary from local obstacles ( atomic nuclei and impurity of the grid in general) are not affected, assuring a resistless charge transport. Characterized an interaction is prevented with the rest of the metal and reasons for the typical properties of a superconductor such as the vanishing electrical resistance.

Collapse of superconductivity

Forms a Cooper pair, the amount of energy is released. Due to excessive exposure to energy from the outside, either by application of heat, a large current density, radiation or the like, however, the couples broken again, and the electrons pass their interaction with the rest of the normal metal again. This explains why superconductivity can only occur at low temperatures, low currents and low magnetic fields, and this is relative: Current research results of MgB2 superconductors show that off, the magnetic field has a current density of 85 kA / cm ² were measured.

Limits of the BCS theory

The BCS theory explains originally only the conventional superconductivity at temperatures close to absolute zero temperature. This also soft or ideal -called type I superconductors show a complete Meissner effect and a good agreement between theory and experiment.

The 1986 Bednorz and Müller discovered high-temperature superconductivity, as occurs for example in some ceramics, contrary to conflicting assertions can also be explained by the BCS theory. However, the mechanism of pair formation is still unclear. A pair formation via the direct electron -phonon interaction is not in question.

It has, however, shown that even when high-temperature superconductors Cooper pairs take charge transport.

Solids Physical details

The property of superconductivity requires that there must be a new phase of the electron gas in the metal. The ground state (T = 0) of an electron gas collapses, albeit ever so small attractive interaction is allowed between two electrons. Cooper used in his theory of the approach is that an electron due to its negative charge leaves a trace deformation of the ion cores on its way through the solid. The accumulation of positively charged ion cores is attractive to a second electron. Thus, the two electrons pull on the lattice deformation - like two balls in a funnel.

At the moment of an electron flying past the ions receive an impulse that leads only after passing the electron to a motion of the ions and thus to a polarization of the grid ( see picture).

With respect to the high electron velocity follows the grid only very slowly, reaching its maximum deformation at a distance behind the electron. is the Debyefrequenz the phonons in the crystal lattice. Ways and learn the two electrons coupling over a distance of more than 100 nm This implies inter alia that the Coulomb repulsion is largely shielded.

Quantum mechanical interpretation

This model can be described by understanding the lattice deformation as the superposition of phonons, which constantly emits the electron due to its interaction with the lattice absorbs and also quantum mechanically.

Let us first consider a non- interacting Fermi gas (see Fermi -Dirac statistics ) of the electrons. The ground state in the potential well is then given by the fact that all electron states with wave vector to the Fermi level (T = 0) are filled and all states remain with unfilled. We now add this system, two electrons with the wave vectors, and the corresponding energies and on conditions above added to and assume that the two electrons are coupled by the attractive interaction just described. All other electrons in the Fermi sea should continue to not interact with each other and prevent further occupation of the states due to the Pauli principle. When Phononenaustausch replace both their electron wave vectors, the conservation law must apply:

We recall that the interaction is limited in space on a shell of energy width, which must be as already mentioned above. In the picture you can see that all pairs for which the above conservation law, end in the blue shaded volume ( rotationally symmetric about the axis given by ).

This volume is directly related to the number of the energy lowering Phononenaustauschprozesse. That is the strength of the attractive interaction is exactly at a maximum when this volume is maximum. This is the case when the two spherical shells overlap, which in turn is only realized. Thus must apply:

Consider the following pairs of electrons with opposite wave vector. The associated Zweiteilchenwellenfunktion must satisfy the Schrödinger equation:

The energy of the electron pair is based on the interaction condition. This gives the following relationship:

Z is half the density of the Debye- off frequency and the attractive potential.

Thus there is a two-electron bound state whose energy is compared to the fully occupied Fermi sea to be lowered. If so turned on even such a small attractive interaction between the electrons, the ground state of the non-interacting free electron gas is unstable. This instability leads in reality to the fact that a high density of such pairs of electrons, which are also called Cooper pairs, forms. This new basic condition is the same as the superconducting phase. It should be mentioned that the Pauli principle is valid for both electrons with respect to the states in the Fermi sphere. Since the approach to Zweiteilchenwellenfunktion to a permutation of the electrons is symmetrical, the total wave function including the spins, must be antisymmetric, the two electrons must have opposite spins.

The actual cause of the supercurrent is that the spin of a Cooper pair is an integer. This means that Cooper pairs no longer Fermi, but by the Bose -Einstein statistics interaction- free particles are described by the particular that they are no longer subject to the Pauli principle. You can use it all take a quantum mechanical state simultaneously.

It is therefore possible that the Cooper pairs in the whole grid by a single wave function to describe. As shown, all the Cooper pairs are together in an energy level deeper. This energy difference is required for cleavage of the Cooper pairs and is larger than any placeable by lattice scattering energy. Thus, in the band model to the Fermi energy is generated an energy gap of width (see picture), which corresponds to the breaking of a Cooper pair. Now exists for potential scattering centers in the grid, instead of individual Cooper pairs or even single electrons, a continuum that could be lifted only with a correspondingly larger amount of energy to a higher level. Since it does not power can be lost by scattering processes, the current flow is lossless.

Note that the binding is a dynamic equilibrium: Cooper pairs decay constant and constantly regenerate. The binding energy of a Cooper pair is about 1 meV, and is therefore against the metallic bond of 1 ... 10 eV is very small. A bond of electrons to form Cooper pairs can only take place in metallic superconductors, if the thermal energy of the lattice is small compared to the binding energy.

At a temperature just below the transition temperature of only a small portion of the conduction electrons is condensed to form Cooper pairs. The lower the temperature decreases, the greater this percentage, until at T = 0 all electrons in the interaction region are connected ( to the Fermi level ) to Cooper pairs.