Melting temperature of two-dimensional Wigner crystals: Anharmonic effects

Abstract

Making use of the Green's-function technique, we obtain the lowest-order corrections to the shear modulus due to anharmonicity. In the classical limit our result agrees with a recent result obtained by Fisher, resulting in $\ensuremath{\Gamma}=103$, where $\ensuremath{\Gamma}=\frac{{e}^{2}}{({r}_{0}{T}_{M})}$, ${r}_{0}$ is the interelectron distance, and ${T}_{M}$ is the melting temperature. More generally we obtain the shear modulus, which includes the quantum corrections as well. Assuming that even in the high-density limit the melting of the Wigner lattice is due to dissociation of bound dislocation pairs, the phase diagram is constructed. We find that the Wigner lattice becomes unstable for ${r}_{s}(\ensuremath{\equiv}\frac{{r}_{0}}{{a}_{B}})<5.57$, where ${a}_{B}$ is the Bohr radius.