BCS theory

The BCS theory is a many-body theory to explain superconductivity in metals , which was developed in 1957 by John Bardeen , Leon Neil Cooper and John Robert Schrieffer . For this they received the 1972 Nobel Prize in Physics .

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The basis of the BCS theory was the experimental observation that the superconductivity of many metals shows a relatively strong dependence of the transition temperature on the mass of the metal isotope under investigation : ${\displaystyle T_{C}}$ ${\displaystyle M}$

${\displaystyle T_{C}\propto {\frac {1}{\sqrt {M}}}}$

This suggested that a mechanism of superconductivity must be the interaction with the mass-dependent, quantized lattice vibrations (whose quanta are called phonons ).

This can be imagined as follows: A first electron changes the lattice (or a lattice oscillation) by releasing energy in such a way that a second electron (e.g. by changing its path or recording a phonon) achieves an equally large energy gain. This is only possible if the lattice building blocks and the electrons move slowly enough (therefore only below a critical current density ).

The idea of ​​the BCS creators is to postulate the formation of Cooper pairs from two electrons each through a weak attractive interaction . Because of their spin (s _e  = 1/2), electrons are fermions and as such cannot occupy the same state ( Pauli principle ). In contrast, the Cooper pairs with integer spin ( singlet state s = 0 (antiparallel arrangement of the electron spins) or triplet state s = 1 (parallel arrangement of the electron spins)) are bosons and can therefore simultaneously have the same state, and thus also all assume the basic state . This is not only energetically more favorable, but also expresses itself in a Bose-Einstein wave function that spans the entire solid .

This wave function can no longer be influenced by local obstacles ( atomic nuclei and defects in the lattice in general) and thus guarantees a resistance-free charge transport. This prevents interaction with the rest of the metal and establishes the typical properties of a superconductor, such as the vanishing electrical resistance .

Breakdown of superconductivity

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When a Cooper pair is formed, the amount of energy is released. ${\displaystyle \Delta }$

If the external energy is too great, be it through the supply of heat, a current density that is too great, radiation or the like, the pairs are broken up again, and the electrons re-enter their normal interaction with the rest of the metal. This explains why superconductivity can only occur at low temperatures, small currents and low magnetic fields .

This has to be seen in relative terms: Current research results on MgB ₂ superconductors show that current densities of 85 kA / cm² have already been measured when the magnetic field is switched off.

Limits of the BCS theory

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can, contrary to claims to the contrary, also be explained by the BCS theory

The BCS theory originally only explains conventional superconductivity at temperatures close to absolute zero . These type I superconductors, which are also known as soft or ideal , show a complete Meissner-Ochsenfeld effect and a good agreement between theory and experiment.

The high-temperature superconductivity discovered by Bednorz and Müller in 1986 , as it occurs in some ceramics , for example , can also be explained by the BCS theory, contrary to claims to the contrary: it has been proven that Cooper pairs also take over charge transport in high-temperature superconductors. However, the mechanism of pair formation is still unclear; it is out of the question via the direct electron-phonon interaction.

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On the other hand, it has been proven that Cooper pairs take over charge transport even with high-temperature superconductors.

Solid-state physical details

Deformation track as compression of the positively charged hulls ( network levels )

The property of superconductivity assumes that there is a new phase of the electron gas in the metal. The ground state ( T = 0) of an electron gas collapses if an attractive interaction, no matter how small, is allowed between two electrons. In his theory, Cooper used the approach that an electron leaves a deformation trail on the ion trunk on its way through the solid due to its negative charge . The accumulation of positively charged ion cores has an attractive effect on a second electron. The two electrons thus attract each other via the lattice deformation - similar to two spheres in a funnel.

At the moment when an electron flies past, the ions receive a force impulse which only leads to a movement of the ions and thus to a polarization of the lattice after they have passed the electron (see picture).

Compared to the high electron speed, the lattice follows very slowly, it reaches its maximum deformation with a distance ${\displaystyle v_{e}\sim 10^{6}\,{\frac {\mathrm {m} }{\mathrm {s} }}}$

${\displaystyle s={\frac {v_{e}2\pi }{\omega _{\mathrm {D} }}}}$

behind the electron, with the Debye frequency of the phonons of the crystal lattice. ${\displaystyle \omega _{\mathrm {D} }}$

Because of this , the two electrons experience a coupling over a distance of more than 100  nm . That implies u. a. that the Coulomb repulsion is largely shielded. ${\displaystyle {\frac {2\pi }{\omega _{\mathrm {D} }}}\sim 10^{-13}\,\mathrm {s} }$

Quantum mechanical interpretation

Feynman diagram of the interaction of two electrons with the wave vectors k1 and k2 or via a phonon of the wave vector q for the formation of Cooper pairs within the framework of the BCS theory

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

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 one-electron states with wave vectors are filled up to the Fermi edge ( T = 0) and all states remain unoccupied. We now add two electrons to this system with the wave vectors , and the corresponding energies and on states above and assume that the two electrons are coupled via the attractive interaction just described. All other electrons in the Fermi lake should not interact with each other and, because of the Pauli principle, prevent further occupation of the states . During phonon exchange, the two electrons change their wave number vectors, whereby the conservation law must apply: ${\displaystyle {\vec {k}}}$ ${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}k_{\mathrm {F} }^{2}}{2m}}}$ ${\displaystyle E>E_{F}}$ ${\displaystyle {\vec {k}}_{1}}$ ${\displaystyle {\vec {k}}_{2}}$ ${\displaystyle E({\vec {k}}_{1})}$ ${\displaystyle E({\vec {k}}_{2})}$ ${\displaystyle E_{\mathrm {F} }}$ ${\displaystyle |{\vec {k}}|<k_{\mathrm {F} }}$

${\displaystyle {\vec {k}}_{1}+{\vec {k}}_{2}={\vec {k}}'_{1}+{\vec {k}}'_{2}={\vec {K}}}$

We remember that the interaction in space is limited to a shell of the energy width, which, as already mentioned , must lie above . In the figure you can see that all pairs, for which the above conservation law applies, end in the blue shaded volume (rotationally symmetrical about the axis given by ). ${\displaystyle {\vec {k}}}$ ${\displaystyle \hbar \omega _{\mathrm {D} }}$ ${\displaystyle E_{\mathrm {F} }}$ ${\displaystyle {\vec {K}}}$

Illustration of electron pair collisions in the reciprocal space of the wave numbers

This volume is directly related to the number of phonon exchange processes that lower the energy. This means that the strength of the attractive interaction becomes maximum precisely when this volume becomes maximum. This is the case when the two spherical shells overlap, which in turn can only be achieved by . So the following must apply: ${\displaystyle {\vec {K}}=0}$

${\displaystyle {\vec {k}}_{1}=-{\vec {k}}_{2}}$

In the following we consider electron pairs with opposite wavenumber vectors. The corresponding two-particle wave function must satisfy the Schrödinger equation:

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\left(\Delta _{1}+\Delta _{2}\right)+V({\vec {r}}_{1},{\vec {r}}_{2})\right)\psi ({\vec {r}}_{1},{\vec {r}}_{2})=(\varepsilon +2E_{\mathrm {F} })\psi ({\vec {r}}_{1},{\vec {r}}_{2})}$

It is the energy of the electron pair in relation to the interaction-free state. The following relationship is obtained: ${\displaystyle \varepsilon }$

${\displaystyle \varepsilon \approx -2\hbar \omega _{\mathrm {D} }e^{-2/V_{0}Z(E_{\mathrm {F} })}}$

Z is half the density of states, the Debye cut-off frequency and the attractive potential. ${\displaystyle \omega _{\mathrm {D} }}$ ${\displaystyle V_{0}}$

So there is a bound two-electron state, the energy of which is lower than that of the fully occupied Fermi Sea . So if even the smallest attractive interaction between the electrons is switched on, the ground state of the non-interacting free electron gas becomes unstable. This instability actually leads to the formation of a high density of such electron pairs, also known as Cooper pairs . This new ground state is identical to the superconducting phase . It should also be mentioned that the Pauli principle applies to both electrons with regard to the states in the Fermi sphere. Since the approach for the two- particle wave function is symmetric with respect to an exchange of the electrons, but the total wave function including the spins must be antisymmetric , the two electrons must have opposite spins . ${\displaystyle \varepsilon =E-2E_{\mathrm {F} }<0}$

The real cause of the supercurrent , however, is that the spin of a Cooper pair is an integer. This means that Cooper pairs are no longer described by the Fermi , but by the Bose-Einstein statistics of interaction-free particles and, in particular, that they are no longer subject to the Pauli principle . You can all assume a quantum mechanical state at the same time .

It is therefore possible to describe the entirety of the Cooper pairs in the lattice by a single wave function. As already shown, all Cooper pairs are together in a lower energy level. This energy difference is required to split the Cooper pairs and is greater than any energy that can be conveyed by lattice scattering. Thus, in the band model around the Fermi energy, there is an energy gap of width (see picture), which corresponds to the breaking up of a Cooper pair. For potential scattering centers in the lattice, instead of individual Cooper pairs or even individual electrons, there is a continuum that can only be raised to a higher level with a correspondingly greater expenditure of energy. Since no energy can be lost through scattering processes, the flow of electricity is lossless . ${\displaystyle E_{\mathrm {F} }}$ ${\displaystyle \Delta }$

Simplified representation of the excitation spectrum of a superconductor

Note that the bond is a dynamic equilibrium: Cooper pairs are constantly falling apart and are constantly being re-formed. The binding energy of a Cooper pair is about 1 meV, so it is very small compared to the metallic bond of 1… 10 eV. A binding of electrons to Cooper pairs can only take place in metallic superconductors if the thermal energy of the lattice is small compared to this binding energy.

At temperatures just below the transition temperature, only a small part of the conduction electrons is condensed into Cooper pairs. The lower the temperature sinks, the greater this proportion becomes, until at T = 0 all electrons in the interaction area (around the Fermi edge) are connected to form Cooper pairs.

Web links

• M. Kathke: Superconductivity, an introduction. (PDF; 365 kB) Aachen, June 7, 1999, accessed on November 29, 2012 (an elaboration on the lecture in the seminar Solid State Physics WS 1997/98). 

literature

• J. Bardeen, LN Cooper and JR Schrieffer, Phys. Rev. 108 , 1175-1204 (1957)
• 'Bardeen-Cooper-Schrieffer theory' on Scholarpedia '
• Ibach, Lüth Solid State Physics , Chapter Superconductivity, Springer Verlag