Cooper couple

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Leon Neil Cooper , first descriptor

When Cooper pairs are pairwise associations of mobile electrons referred to special materials. They occur at very low temperatures and are a prerequisite for the superconducting state of these ( metallic or ceramic ) materials. The phenomenon of Cooper pair formation is named after it was first described in 1956 by Leon Neil Cooper and is of fundamental importance in the BCS theory of superconductivity.

More generally, they are pairs in fermionic many-body systems , in which two paired fermions result in a "composite boson ". The same phenomenon occurs in a different context, namely between two atoms in the superfluid state of  3 He below a temperature of 2.6 m K ; on the other hand, there are no Cooper pairs in the superfluid state of  4 He , since here the atoms are bosons.

Another possibility for the formation of Cooper pairs lies in the condensation of ultra-cold fermionic gases of low density using methods of atomic physics , comparable to the Bose-Einstein condensation of bosonic gases.

Cooper pairs also seem to be involved in the high-temperature superconductivity discovered in 1986 , as experimental evidence shows. However, the binding mechanism of the two electrons leading to pair formation is still unclear in this case, despite years of efforts, in contrast to the conventional superconductors known for over 100 years, as described below, in which phonons cause pair formation.

Explanation

In metals, the conduction electrons can move practically freely between the atoms. This “ electron gas ” consists of fermions and is therefore subject to the Fermi distribution , which predicts a certain velocity distribution from zero to very high values ​​(the characteristic temperature is ). The movement of the atomic nuclei , on the other hand, plays a comparatively minor role (the characteristic temperature here is the Debye temperature of around 150 ... 600 K).

Only at even lower temperatures does the atomic movement attract the electrons in pairs, which can no longer be neglected. The strength of this interaction corresponds to temperatures of about

10 K (−263 ° C);

that corresponds to energies

or lifetimes of size

with the reduced Planck quantum of action .

The result for corresponds to typical phonon frequencies , but this does not prove anything: Experiments that show that the particles involved are actually phonons ( quantized atomic vibrations) and not, for example, different excitation states of the system, are based on the isotope effect what brought Leon Neil Cooper to the ideas presented below.

The movement of the atomic nuclei runs through the whole medium as a wave phenomenon and gives (after quantization) the phonons. Due to its higher mass, it takes place with a considerable delay, which results in a weak polarization of the grating , which overcompensates for the Coulomb repulsion . A second electron can now lower its energy in this “polarization track”, i. i.e., it is weakly bound; a Cooper pair is created through the movement of the grid. The formation of the Cooper pairs is based - like all polarization effects - on a weak indirect interaction: the electrons attract each other because the system is polarized by the interaction.

This interaction can be described by the following diagram :

Feynman diagram of a contribution to the electron-electron bond through “exchange” of a phonon (time axis from bottom to top; the phonon, a vibration quantum of the solid, here designated by γ, runs from left to right or from right at a comparatively low speed Left).

The resulting binding effect can be compared to the formation of a weak depression in an eardrum under the gravity of a first particle: As a result, a second particle, which is also moving on the eardrum, is attracted to the first, so that both are bound to each other.

Since the two electrons involved move in opposite directions, the total momentum of the Cooper pair is i. a. small or zero. The impulses do not have to be exactly, but only "approximately" opposite-equal, so that pairing is possible. In fact, the speed of the supercurrent - and thus its strength - is proportional to the difference considered.

The “space requirement” of each electron in a Cooper pair is described by its wave packet . When these move away from each other, Cooper pairs disintegrate because the wave packets hardly overlap, others form anew.

If one uses the uncertainty relation to estimate the extent of the wave packets, one arrives at values ​​of up to 10 −6  m. A comparison with the mean distance between the electrons in the crystal lattice gives the surprising result that the radius of the Cooper pair can be of the specified order of magnitude, so that there can be at least 10 10 other electrons between the electrons of a Cooper pair . Of these, around a million other electrons have so similar and overlapping wave packets that they too form Cooper pairs. The Cooper pairs are almost as numerous as the electrons themselves.

The essential mechanism for explaining superconductivity (see below), however, is that, in contrast to the electrons, which in a sense "avoid each other" due to Fermi statistics, they can condense to a coherent state , as is the case for superconductivity and in general is characteristic of superfluids. Although the commutation relations of two Cooper pairs do not exactly correspond to those of the Bose particles, they are nevertheless similar to them.

Significance in superconductors

Electrons belong to the particle group of fermions and have spin  1/2 (cf. spin statistics theorem ). The Fermi-Dirac statistics show that in a two-electron system without spin-orbit coupling with a symmetrical position function, the spin function must therefore be antisymmetrical , i.e. approximately

This clearly means that the spin of one electron points "up" (i.e., it is +1/2, in units of the reduced Planck's constant ), while the other spin points "down" (i.e., it is −1 / 2, in the same units), i.e. aligned anti-parallel. The total spin of the Cooper pair is zero in this case. This corresponds to the singlet state.

Another, albeit rarer, case is the parallel alignment of the individual spins of the Cooper pair electrons, with the total spin adding up to one. This is called the triplet state. Such a state can be proven experimentally by tunneling experiments , as these Cooper pairs can tunnel through larger ferromagnetic barriers.

In both cases, the Cooper pairs as composite particles are no fermions, but bosons because of their integral spin . For these, the Fermi-Dirac, but the Bose-Einstein statistics apply . It says - clearly speaking - that the Cooper couples follow a “herd instinct”, so that the above-mentioned. coherent state can result: All pairs move at the same speed in the same direction and are strictly coupled to each other.

The latter addition means u. a. that the situation can basically not be compared with a Bose-Einstein condensate , since the Cooper pairs cannot be regarded as independent particles of a Bose gas .

Nevertheless, the Bose-Einstein statistics explain the properties of metallic superconductors, since all Cooper pairs as effective Bose particles are allowed to occupy the same quantum mechanical state (anti- Pauli principle ). So you are definitely dealing with a macroscopic , collective quantum phenomenon.

Since the expansion of the wave packets of each Cooper pair is almost macroscopically large, they can tunnel through thin insulator layers ( Josephson effect ). Experiments have shown that two electrons always tunnel through the barrier.

Energy gap

Mathematically, the tendency to form Cooper pairs is expressed by the fact that in the Hamilton operator of the system, in addition to the usual bilinear terms (with the electron creation operators and the associated annihilation operators ), quadratic terms of the unusual form and occur:

It is

  • the wave number of the electrons
  • their energy in the normally conducting state
  • a pairing parameter assumed to be real.

The interaction not only changes the basic state and the excited states of the system quantitatively but also qualitatively. The ground state energy is only slightly increased: but - what is more important - an energy gap of the size to the excited states is now formed. That has u. a. As a result, the electrical resistance is zero everywhere at correspondingly low temperatures.

Compound particles in high energy physics

The formation of composite particles is also discussed in high energy physics , e.g. B. in connection with the Higgs boson .

References and footnotes

  1. ^ Leon N. Cooper: Bound electron pairs in a degenerate Fermi gas . In: Physical Review . 104, No. 4, 1956, pp. 1189-1190. doi : 10.1103 / PhysRev.104.1189 .
  2. ^ Osheroff DD, Richardson RC, Lee DM: Evidence for a new phase in solid 3 He . In: Phys. Rev. Lett. . 28, 1972, pp. 885-888.
  3. CA Regal et al. : Observation of Resonance Condensation of Fermionic Atom Pairs . In: Physical Review Letters . 92, 2004, p. 040403. doi : 10.1103 / PhysRevLett.92.040403 .
  4. Cooper pair formation in high temperature superconductors is dealt with in the following article: [1]
  5. ^ Robert Schrieffer: Theory of Superconductivity. Benjamin 1964; see above all the last chapter.
  6. Press release of the Ruhr University Bochum, December 1, 2010: The "pairing behavior" of electrons
  7. The creation operators and annihilation operators act both on the wave vectors and on the spin states.