In two papers published in the 1930s [1] Wigner considered the effect of Coulomb interactions of electrons in metals (an effect neglected in the free electron model). He found that there can be a ground state (meaning state at T = 0) of the electronic system where the electrons crystallize, i.e. each electron is localized at a lattice site. This crystallization will occur only if the density of the electrons is low enough. Only then the overlap of the electronic wave functions will be small enough that the formation of energy band with delocalized electrons will be overcome by the long-range Coulomb interaction leading to crystallization of the electrons.

Real metals have an electron density much higher than the threshold predicted for the Wigner crystal and therefore no crystallization of electrons as in Wigner's original idea was observed up to now. However, there are several experiments with crystallized electrons that are often called Wigner crystal, although other terms like electron crystal, Wigner solid etc. are also in use.

Electrons on liquid helium
Atop the surface of liquid helium electrons can move freely parallel to the helium surface but perpendicular to the surface the electrons are trapped in a potential well. Thus the electrons form an almost ideal two-dimensional system. Unfortunately the electron density accessible in experiments is rather low, therefore the electrons behave classically and not quantum mechanically. Here the electrons form a crystal due to classical effects if the Coulomb interaction energy is much higher than the kinetic energy. Therefore the electrons crystallize if the density is increased above a critical value or if the temperature is decreased. The first experimental evidence for crystallized electrons on liquid helium was published in 1979 [1].

Electrons in semiconductor heterostructures at high magnetic fields
In 1975 a new path towards crystallization of non-classical electrons was proposed [3]: if a strong magnetic field is applied perpendicular to a two-dimensional system of electrons then the electrons are forced into cyclotron orbits with decreasing radius for increasing field. The spatial extension of the electronic wave-function is limited to the cyclotron orbit and therefore the overlap of the wave-functions can be decreased by increasing the magnetic field leading to a crystallization of quantum-mechanical electrons.
After an extensive search for this magnetically induced Wigner solid (MIWS) in semiconductor heterostructures (with the discovery of the fractional quantum Hall effect as a side effect) evidence accumulated in the 1990s [4].

    References
  1. E. Wigner, Phys. Rev. 46, 1002 (1934), E. Wigner, Trans. Farad. Soc. 34, 678 (1938).
  2. C.C. Grimes and G. Adams, Phys. Rev. Lett. 42, 795 (1979).
  3. Yu. E. Lozovik and V. I. Yudson, JETP Lett. 22, 11 (1975).
  4. Two reviews can be found in "Perspectives in Quantum Hall Effects", edited by Shankar Das Sarma and Aron Pinczuk, New York, 1997: H.A. Fertig, "Properties of the Electron Solid" pp. 71-108 and M. Shayegan, "Case for the Magnetic-Field Induced Two-Dimensional Wigner Crystal" pp. 343-384.

Contact: M. Scheffler, M. Dressel