The theory BCS is a complete theory of the Supraconductivité which was proposed in 1957 by John Bardeen, Leon NR. Cooper, and John Robert Schrieffer. She explains the Supraconductivité by the formation of pairs of electrons ( even of Cooper ) under the effect of a gravitational interaction between electrons resulting from the exchange of Phonon S. For their work, these authors obtained the Nobel Prize of physics in 1972.

Origin of attraction enters the electrons

It is possible to include/understand the origin of attraction between the electrons thanks to a simple qualitative argument. In a metal, the negatively charged electrons exert an attraction on the positive ions which are in their vicinity. These ions being much heavier than the electrons, they have a greater inertia. For this reason, when an electron passed close to a whole of positive ions, these ions do not return immediately to their position of balance of origin. It results an excess from them from positive loads at the place where this electron passed. A second electron will thus feel a gravitational attraction resulting from this excess of positive loads. Obviously, the electrons and the ions must be described by quantum mechanics, by taking account of the indiscernibility of the electrons, and this qualitative argument is justified by more rigorous calculations. The complete theoretical treatment uses the methods of the Second quantification, and is based on the Hamiltonien of Frohlich:

$H=\; \backslash \; sum\_\; \{K,\; \backslash \; sigma\}\; \backslash \; epsilon\; (K)\; c^\; \backslash \; dagger\_\; \{K,\; \backslash \; sigma\}\; c\_\; \{K,\; \backslash \; sigma\}\; +\; \backslash \; sum\_q\; \backslash \; hbar\; \backslash \; omega\_q\; b^\; \backslash \; dagger\_q\; b\_q\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{\backslash \; Omega\}\}\; \backslash \; sum\_\; \{K,\; Q,\; \backslash \; sigma\}\; G\; (K,\; Q)\; b\_q\; c\_\; \{K,\; \backslash \; sigma\}\; +\; c^\; \backslash \; dagger\_\; \{k+q,\; \backslash \; sigma\}\; b^\; \backslash \; dagger\_\; \{-\; Q\}\; c\_\; \{K,\; \backslash \; sigma\}$

where $c\_\; \{K,\; \backslash \; sigma\}$ is a Opérateur of annihilation for an electron of Spin $\backslash \; sigma$, and of quasi-impulse $k$, $b\_q$ is the operator of annihilation of a phonon of quasi-impulse $q$, $c^\; \backslash \; dagger\_\; \{K,\; \backslash \; sigma\}$ and $b^\; \backslash \; dagger\_q$ is the operators of corresponding creation, and $g\; (K,\; Q)$ is the element of matrix of the electron-phonon coupling. This term describes the emission or the absorption of phonons by the electrons. It is noted that in these processes, the quasi-impulse is preserved.

By means of a canonical transformation, one can eliminate the electron-phonon interaction from the Hamiltonian of Frohlich to obtain an effective interaction between the electrons. An alternative approach consists in using the perturbation theory to the second order in the coupling electron phonon. In this approach, an electron emits a virtual phonon which is absorbed at once by another electron. This process is the quantum version of the semi-traditional qualitative argument of the beginning of the paragraph. One finds an element of matrix for the interaction between the electrons of the form:

$\backslash \; langle\; k-q,\; k\text{'}+q\; \backslash \; mid\; V\_\; \{EFF.\}\backslash \; mid\; K,\; k\text{'}\; \backslash \; rangle\; =\; \backslash \; frac\; \{2\; \backslash \; mid\; G\; (K,\; Q)\; \backslash \; mid^2\; \backslash \; hbar\; \backslash \; omega\_q\}\; \{(\backslash \; epsilon\; (K)\; -\; \backslash \; epsilon\; (k+q))^2-\; (\backslash \; hbar\; \backslash \; omega\_q)\; ^2\}$

This element of matrix is in general positive, which corresponds to a repulsive interaction, but for this term becomes negative what corresponds to a gravitational interaction. These gravitational interactions created by exchange of virtual Bosons are not limited to the physics of the condensed matter. Another example is the gravitational interaction between nucleons in the atomic nuclei by exchange of Méson S predicted by Hideki Yukawa.

Consequence of the existence of a gravitational interaction

Leon NR. Cooper predicted by considering two electrons in the presence of an inert sea of Fermi and having a weak gravitational interaction, that whatever the force of this interaction these two particles would form a bound state, called pair of Cooper. This result is not commonplace, because it is known in quantum mechanics that in three dimensions, for two isolated particles, a too weak gravitational interaction does not allow the formation of dependant states (see Landau and Lifchitz t.3). The presence of the sea of Fermi, which prohibits the two particles from occupying the energy levels lower than the energy of Fermi is the element which allows the existence of the dependant state for a weak interaction. The energy of this dependant state is cancelled with the force of attraction with an essential singularity, which indicates that the dependant state cannot be obtained by a perturbation theory in the interaction electron-electron.

The calculation of Cooper is criticizable in the sense that he considers only two electrons and supposes that the other electrons which are under the surface of Fermi do not feel the effect of the interaction. Theory BCS raises this objection by treating all the electrons on an equal footing. The Hamiltonian of theory BCS is written in second quantification:

$H=\; \backslash \; sum\_\; \{K,\; \backslash \; sigma\}\; \backslash \; epsilon\; (K)\; c^\; \backslash \; dagger\_\; \{K,\; \backslash \; sigma\}\; c\_\; \{K,\; \backslash \; sigma\}\; -\; \backslash \; frac\; \{G\}\; \{\backslash \; Omega\}\; \backslash \; sum\_\; \{K,\; k\text{'},\; Q\; \backslash \; atop\; \backslash \; sigma,\; \backslash \; sigma\text{'}\}\; c^\; \backslash \; dagger\_\; \{k+q,\; \backslash \; sigma\}\; c^\; \backslash \; dagger\_\; \{k-q,\; \backslash \; sigma\text{'}\}\; c\_\; \{K,\; \backslash \; sigma\text{'}\}\; c\_\; \{K,\; \backslash \; sigma\}$

Bardeen, Cooper and Schrieffer introduced a function of variational wave for described the fundamental state of this Hamiltonian of the form:

$\backslash \; mid\; \backslash \; Psi\; \backslash \; rangle\; =\; \backslash \; prod\_k\; (u\_k\; +\; v\_k\; c^\; \backslash \; dagger\_\; \{K,\; \backslash \; uparrow\}\; c^\; \backslash \; dagger\_\; \{-\; K,\; \backslash \; downarrow\})\; \backslash \; mid\; 0\; \backslash \; rangle$.

This function of variational wave describes the creation of pairs of Cooper by the operator $c^\; \backslash \; dagger\_\; \{K,\; \backslash \; uparrow\}\; c^\; \backslash \; dagger\_\; \{-\; K,\; \backslash \; downarrow\}$. A pair of Cooper is thus made of two opposite electrons of spin and opposite quasi-impulses. More generally, a pair of Cooper is made of two electrons in states connected one to the other by inversion of time. This property makes it possible to include/understand the existence of the Meissner effect in a superconductor. Indeed, in the presence of a magnetic field, the degeneration between states connected by inversion of time is raised what reduces the binding energy of the pairs of Cooper. To keep the free energy gained by forming the pairs of Cooper, it is advantageous when the magnetic field is sufficiently weak to expel it superconductor. Beyond a certain magnetic field, it is more advantageous to destroy supraconductivity is locally (superconductive of type II) or overall (superconductors of the type II).

The function of wave of BCS presents a certain analogy with the functions of wave of coherent states of the harmonic oscillator and more generally the functions of wave of coherent states bosonic. This analogy states in particular that in the fundamental state of the Hamiltonian of BCS quantity: $\backslash \; langle\; c^\; \backslash \; dagger\_\; \{K,\; \backslash \; uparrow\}\; c^\; \backslash \; dagger\_\; \{-\; K,\; \backslash \; downarrow\}\; \backslash \; rangle\; \backslash \; 0$. This property is the signature of a not-diagonal order with long distance. This not-diagonal order is related to the crack of the symmetry of gauge U (1). Indeed, if one changes the phases of the operators of creation, (what is a symmetry of Hamiltonian BCS), one changes the median value of the parameter of order. The function of wave BCS which has a symmetry lower than Hamiltonian BCS thus describes a spontaneous crack of symmetry of gauge. In the Theory of Ginzburg-Pram, the Paramètre of order $\backslash \; psi$ which describes the superconductive state is proportional to $\backslash \; langle\; c^\; \backslash \; dagger\_\; \{K,\; \backslash \; uparrow\}\; c^\; \backslash \; dagger\_\; \{-\; K,\; \backslash \; downarrow\}\; \backslash \; rangle$ like showed it L.P. Gor' kov by methods of functions of Green.

A simpler method was introduced by Bogoliubov and Valatin to study Hamiltonian BCS. It is based on the introduction of new particles by the transformation of Bogoliubov. P.W. Anderson also introduced a method using of the operators of pseudospins. Lastly, it is possible to reformulate theory BCS using functions of Green and diagrams of Feynman.

Thermodynamics of a superconductor according to theory BCS

To supplement. cf Tinkham

Consequences of theory BCS

• isotopic Effect

• Peak of coherence in the rate of relieving $1/T\_1$ in nuclear magnetic magnetic resonance (Hebel and Slichter).

• Effect Josephson

• Observation of the superconductive gap by tunnel effect (Giaever).

Theory of Bogoliubov-in Genoa

Theory of Eliashberg

In certain materials such as lead, it is not possible to treat the interaction electron-phonon by the theory of the disturbances. A more complete theory of supraconductivity taking account the electron-phonon coupling is necessary. This theory was developed by Eliashberg.

Applications to Helium 3

In helium 3, a superfluid transition was observed in the years 1970 by Douglas Osheroff, Robert C. Richardson and David Mr. Lee. As helium 3 is made of fermions (whereas helium 4 is made of bosons), this transition from phase cannot be a condensation of Bump. However, the superfluidity of helium 3 can be explained in a way similar to the supraconductuvity of metals by the formation of pairs of Cooper between the Helium 3 atoms. There exists in helium 3 two superfluid phases described by the theories of Balian-Werthamer and Anderson-Brinkman-Morel.

references

• L.P. Levy Magnetism and Supraconductivity (EDP Sciences)
• W. Jones and NR. H. March Theoretical Solid State Physics (Dover)
• Mr. Tinkham introduction to superconductivity (Mc Graw-Hill)
• P.G. of Genoa superconductivity off metals and alloys (Addison-Wesley)
• J.R. Schrieffer Theory off Superconductivity (Addison-Wesley)
• G. Rickayzen Theory off Superconductivity (Academic Near)
• J. Mr. Blatt Theory off Superconductivity (Academic Near)
• P.W. Anderson BASIC Concepts off Condensed Matter Physics (Addison-Wesley)
• A.L. Fetter and J.D. Walecka Quantum Theory off Many-particle systems (Dover)
• G.D. Mahan Many-particle physics (Plenum)
• A.A. Abrikosov, L.P. Gor' kov and I.E. Dzialoshinskii Methods off Quantum Field Theory in Statistical Physics (Dover)
• Bardeen, Cooper and Schrieffer '' on the site of the foundation Nobel ''
• B. Douçot course of DEA '' Introduction to supraconductivity ''
• P. Hirschfeld '' Solid State Physics II ''
• P. Hirschfeld '' High-Temperature Superconductivity ''
• NR. B. Kopnin '' Theory off Superconductivity ''