Wigner crystal

A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in 2D or 3D in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a bcc (body-centered cubic) lattice in 3D, a triangular lattice in 2D and an evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed. A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner-crystallization that has led to observation of insulating behaviour in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization.

There is a single dimensionless parameter characterising the state of a uniform electron gas at zero temperature, the so-called Wigner-Seitz radius r_s = a / a_b, where a is the average inter-particle spacing and a_b is the Bohr radius. Quantum Monte Carlo simulations indicate that the uniform electron gas crystallizes at r_s = 106 in 3D and r_s = 31 in 2D.

For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: G = e² / (k_BTa). The Wigner transition occurs at G = 170 in 3D and G = 125 in 2D. It is believed that ions, such as those of iron, form a Wigner crystal in the interiors of white dwarf stars.

More generally, a Wigner crystal phase can also refer to a crystal phase occurring in non-electronic systems at low density. In contrast, most crystals melt as the density is lowered. Examples seen in the laboratory are charged colloids or charged plastic spheres.

In practice, it is difficult to experimentally realize a Wigner crystal because quantum mechanical fluctuations overpower the Coulomb repulsion and quickly cause disorder. Low electron density is needed. One notable example occurs in quantum dots with low electron densities or high magnetic fields where electrons will spontaneously localize in some situations, forming a so-called rotating "Wigner molecule", a crystalline-like state adapted to the finite size of the quantum dot.

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