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 a uniform, inert, neutralizing background (i.e. Jellium Model) 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. ^{[citation needed]} 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.^{[clarification needed]}

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.^{[citation needed]}

A uniform electron gas at zero temperature is characterised by a single dimensionless parameter, 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. The kinetic energy of an electron gas scales as 1/r_s², this can be seen for instance by considering a simple Fermi gas. The potential energy, on the other hand, is proportional to 1/r_s. When r_s becomes larger at low density, the latter becomes dominant and forces the electrons as far apart as possible. As a consequence, they condense into a close-packed lattice. The resulting electron crystal is called the Wigner crystal.

Based on the Lindemann criterion one can find an estimate for the critical r_s. The criterion states that the crystal melts when the root-mean-square displacement of the electrons ${\displaystyle {\sqrt {\langle r^{2}\rangle }}}$ is about a quarter of the lattice spacing a. On the assumption that vibrations of the electrons are approximately harmonic, one can use that for a quantum harmonic oscillator the root mean square displacement in the ground state (in 3D) is given by

with ${\displaystyle \hbar }$ the Planck constant, m_e the electron mass and ω the characteristic frequency of the oscillations. The latter can be estimated by considering the electrostatic potential energy for an electron displaced by r from its lattice point. Say that the Wigner–Seitz cell associated to the lattice point is approximately a sphere of radius a/2. The uniform, neutralizing background then gives rise to a smeared positive charge of density ${\displaystyle 6e/a^{3}\pi }$ with ${\displaystyle e}$ the electron charge. The electric potential felt by the displaced electron as a result of this is given by

with ε₀ the vacuum permittivity. Comparing ${\displaystyle -e\varphi (r)}$ to the energy of a harmonic oscillator, one can read off

or, combining this with the result from the quantum harmonic oscillator for the root-mean-square displacement

The Lindemann criterion than gives us the estimate that r_s > 40 is required to give a stable Wigner crystal. Quantum Monte Carlo simulations indicate that the uniform electron gas actually crystallizes at r_s = 106 in 3D and r_s = 31 in 2D.