Anisotropic States of Two-Dimensional Electrons in High Magnetic Fields

We study the collective states formed by two-dimensional electrons in Landau levels of index $n\ge 2$ near half filling. By numerically solving the self-consistent Hartree-Fock (HF) equations for a set of oblique two-dimensional lattices, we find that the stripe state is an anisotropic Wigner crystal (AWC), and determine its precise structure for varying values of the filling factor. Calculating the elastic energy, we find that the shear modulus of the AWC is small but finite (nonzero) within the HF approximation. This implies, in particular, that the long-wavelength magnetophonon mode in the stripe state vanishes like $q^{3/2}$ as in an ordinary Wigner crystal, and not like $q^{5/2}$ as was found in previous studies where the energy of shear deformations was neglected.

Figure 1

Evolution of the WC appearance for various values of the anisotropy parameter $\epsilon$ of Eq. (1). For $\epsilon \ge 0.5$, the lattice shows a pronounced channel-like structure, with each channel consisting of a one-dimensional periodic chain of electron guiding centers.

Figure 2

Upper panel: values of the anisotropy parameter $\epsilon$ that minimize the cohesive energy of the WC vs ${\nu }^{*}$. Lower panel: cohesive energies of the Wigner crystal (circles), $2e$ bubble solid (squares), and $3e$ bubble solid (triangles). For the WC, $\epsilon$ values from the upper panel are used. Both panels are for LL $n=2$.

Figure 3

Shear modulus of the anisotropic Wigner crystal in $n=2$ in units of $e^2/\kappa \ell$. The values of $c_{66}$ obtained for a shear polarized along $\widehat{\mathbf{x}}$ and along $\widehat{\mathbf{y}}$ are denoted by $c_{66,x}$ and $c_{66,y}$, respectively.

Figure 4

Dispersion curve for the stripe state at filling factor $\nu =4.45$. There is a change of behavior from $\omega \sim q^{1.8}$ to $\omega \sim q^{2.5}$ around $q_{\mathrm{cr}}=1/10a$ (in the $y$ axis label, $\omega$ is expressed in units such that $\hslash =1$).