Wigner crystal of a two-dimensional electron gas with a strong spin-orbit interaction

The Wigner-crystal phase of two-dimensional electrons interacting via the Coulomb repulsion and subject to a strong Rashba spin-orbit coupling is investigated. For low enough electronic densities the spin-orbit band splitting can be larger than the zero-point energy of the lattice vibrations. Then the degeneracy of the lower subband results in a spontaneous symmetry breaking of the vibrational ground state. The ${60}^{\circ }$ rotational symmetry of the triangular (spin-orbit coupling free) structure is lost, and the unit cell of the new lattice contains two electrons. Breaking the rotational symmetry also leads to a (slight) squeezing of the underlying triangular lattice.

Figure 1

Schematic visualization of the electronic density in a two-dimensional electron crystal. (a) The ring of minima in momentum space for the lower Rashba subband $E_{p-}={(p-m\lambda )}^2/\left(2m\right)$ ($\lambda$ is the strength of the Rashba interaction). The colored ellipse shows the electronic density [for structure (b)]. The density is centered around a particular minimum, but is strongly elongated along the line of minima, $\Delta p_x\ll \Delta p_y\ll m\lambda$, which helps to lower the interaction energy. (b)–(d) Three possible periodic configurations (see text). The dark ellipses indicate the directions of electrons' oscillations. Structure (c) has the lowest zero-point energy of the lattice vibrations. The arrows in (b) indicate the in-plane spin orientations.

Figure 2

The vibration density of states as a function of the frequency (scaled by ${\omega }_0$) for the Wigner-crystal configuration of Fig. 1 (dashed line) and of Fig. 1 (solid line). Though the spectrum extends all the way down to $\omega =0$ for 1 the average frequency turns out to be smaller for 1. Note the linear vanishing of the density (characteristic of a Dirac cone) in the middle of spectrum for 1 and especially the inverse square-root divergence of the density at the lower edge, $\omega \approx 0.36{\omega }_0$.

Figure 3

Definition of the basis vectors for the original triangular lattice, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, and for the superlattice with three electrons per unit cell, ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$. Electrons from the three different sublattices are shown by different colors.