Melting temperature of two-dimensional Wigner crystals: Anharmonic effects

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 $\Gamma =103\mathrm{}$, where $\Gamma =\frac{e^2}{\left(r_0{T}_M\right)}$, $r_0$ 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 $r_s\left(\equiv \frac{r_0}{a_B}\right)<\mathrm{}5.57\mathrm{}$, where $a_B$ is the Bohr radius.