Wigner crystal

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A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934.[1] 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 triangular lattice in 2D and a b.c.c. (body-centered cubic) lattice in 3D. A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field.

There is a single dimensionless parameter characterizing the state of a uniform electron gas at zero temperature, the so-called Wigner-Seitz radius rs = a / ab, where a is the average inter-particle spacing and ab is the Bohr radius. Quantum Monte Carlo simulations indicate that the uniform electron gas crystallizes at rs = 106 in 3D[2][3] and roughly rs = 35 in 2D.[4][5]

For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: G = e2 / (kB Ta). The Wigner transition occurs at G = 170 in 3D[6] and G = 125 in 2D.[7] 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.

Electrons confined in quantum dots at low densities will spontaneously localize in some situations, forming a so-called Wigner molecule, a crystalline-like state adapted to the finite size of the quantum dot.

  1. ^ E. P. Wigner, Phys. Rev. 46, 1002 (1934).
  2. ^ D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
  3. ^ N. D. Drummond et al., Phys. Rev. B 69, 085116 (2004).
  4. ^ B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).
  5. ^ F. Rapisarda and G. Senatore, Aust. J. Phys. 49, 161 (1996).
  6. ^ D. Dubin, Rev. Mod. Phys. 71, 87 (1999)
  7. ^ W. J. He et al., Phys. Rev. B 68, 195104 2003.