Absence of Wigner crystallization in graphene

Graphene, a single sheet of graphite, has attracted tremendous attention due to recent experiments which demonstrate that carriers in it are described by massless fermions with linear dispersion. In this Brief Report, we consider the possibility of Wigner crystallization in graphene in the absence of an external magnetic field. We show that the ratio of potential and kinetic energy is independent of the carrier density, the tuning parameter that usually drives Wigner crystallization, and find that for given material parameters (dielectric constant and Fermi velocity), Wigner crystallization is not possible. We comment on how these results change in the presence of a strong external magnetic field.

Figure 1

Static dielectric constant $ϵ(q\to 0,\omega =0)$ as a function of the spin-orbit interaction energy ${\Delta }_{\mathit{so}}$. We have used $\hslash v_G=5.8\mathrm{eV}\mathrm{\AA }$. We find that, at ${\Delta }_{\mathit{so}}=4\mathrm{meV}$, $ϵ\simeq 1.1$ approaches the minimum value possible, $ϵ=1$.