Wigner crystal

A Wigner crystal is an ordered arrangement of electrons stabilized by their mutual repulsion when localization costs sufficiently little kinetic or thermal energy. In a degenerate low-density gas, the Coulomb-to-kinetic energy ratio of an electron gas increases with mean spacing. Classical electrons can also order at large Coulomb-to-thermal coupling, as in electron crystallization on liquid helium. A neutralizing background or confining environment is needed to discuss a stable electron assembly. Dimension, screening and disorder affect the transition.

The 1979 experiment of C. C. Grimes and colleagues studied a two-dimensional sheet of electrons above liquid helium. It reported crystallization at densities $3\times 10^8$ to $9\times 10^8{\mathrm{cm}}^{-2}$, with melting temperatures $0.35$ to $0.65\mathrm{K}$. The sheet was classical: Coulomb energy competed with thermal motion, unlike the kinetic-energy comparison used for a zero-temperature degenerate Wigner crystal. It supplies evidence for electron ordering while keeping these two physical regimes distinct.^[ref]

In three dimensions, let $a=(3/4\pi n{)}^{1/3}$ be the mean-spacing radius and $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$ the effective Bohr radius. The Coulomb scale is $E_C\sim e^2/(4\pi \epsilon a)$ and the Fermi energy scale is $E_F\sim {\hslash }^2/(m^{\ast }{a}^2)$, so $E_C/E_F\sim a/a_B^{\ast }=r_s$. Decreasing density favors repulsion relative to delocalization energy, motivating a Wigner crystal. This scaling does not determine the numerical quantum melting value of $r_s$. A classical nondegenerate gas instead compares Coulomb energy with $k_{B}T$.