Generalized Wigner crystallization in moiré materials

Recent experiments on the twisted transition metal dichalcogenide (TMD) material ${\mathrm{WSe}}_2/{\mathrm{WS}}_2$ have observed insulating states at fractional occupancy of the moiré bands. Such states were conceived as generalized Wigner crystals (GWCs). In this paper, we investigate the problem of Wigner crystallization in the presence of an underlying (moiré) lattice. Based on the best estimates of the system parameters, we find a variety of homobilayer and heterobilayer TMDs to be excellent candidates for realizing GWCs. In particular, our analysis based on $r_s$ indicates that ${\mathrm{MoSe}}_2$ (among the homobilayers) and ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ or ${\mathrm{MoS}}_2/{\mathrm{WS}}_2$ (among the heterobilayers) are the best candidates for realizing GWCs. We also establish that due to larger effective mass of the valence bands, in general, hole crystals are easier to realize that electron crystals as seen experimentally. For completeness, we show that satisfying the Mott criterion $n_{\mathrm{Mott}}^{1/2}{a}_{*}=1$ requires densities nearly three orders of magnitude larger than the maximal density for GWC formation. This indicates that for the typical density of operation, bilayer moiré systems are far from the Mott insulating regime. These crystals realized on a moiré lattice, unlike the conventional Wigner crystals, are incompressible due to the gap arising from pinning with the lattice. Finally, we capture this many-body gap by variationally renormalizing the dispersion of the vibration modes. We show these low-energy modes, arising from the coupling of the WC with the moiré lattice, can be effectively modeled as a Sine-Gordon theory of fluctuations.

Figure 1

Schematic of AA stacked TMD bilayer: (a) the side view shows the $M{X}_2$ layer, with trigonal prismatic (H) coordination, stacked on top of the $M^{\prime }{X}_2^{\prime }$ layer. The large (yellow or brown) balls represent the metal ions $M$ or $M^{\prime }$ and the small (blue or green) balls represent the chalcogens $X$ or $X^{\prime }$. The distance between the metal ions and the chalcogens are, respectively, denoted by $d_M$ and $d_X$. (b) The top view is a honeycomb lattice of lattice of lattice constant $a$.

Figure 2

A cartoon rendition of a WC and the relevant length scales: A Wigner lattice is realized on a moiré superlattice (gray background) at filling fraction $2/3$. The distance between two nearest dark (or bright) spots is the moiré periodicity ${\lambda }_{\mathrm{m}}$. The distance between the two nearest localized particles (red dots) is the Wigner lattice periodicity ${\lambda }_{\mathrm{w}}$. The bell-shaped curve, representing the wave function of a localized particle, has a width of $2\xi$. Our discussion in this paper is confined to a crystal where the moiré electrons are highly localized, $\xi \ll {\lambda }_{\mathrm{w}}$.

Figure 3

Schematic of a MBZ (blue) and a WBZ (red). The BZ vectors are, $|{\mathit{g}}_n|=\frac{4\pi }{\sqrt{3}{\lambda }_{\mathrm{m}}}$ and $|{\mathit{\kappa }}_n|=\frac{4\pi }{\sqrt{3}{\lambda }_{\mathrm{w}}}$. In general, since ${\lambda }_{\mathrm{w}}>{\lambda }_{\mathrm{m}}$, the WBZ is smaller than the MBZ. For the particular case drawn above, ${\lambda }_{\mathrm{w}}=3{\lambda }_{\mathrm{m}}$. In other words, the third WBZ is the same as the first MBZ ($|{\mathit{\kappa }}_n|=3|{\mathit{g}}_n|$).

Figure 4

Dispersion of the longitudinal (solid) and transverse (dashed) modes of a 2D WC (independent of its coupling to a moiré potential). Here, we have set $c_a=1=d_{\parallel }$. $\gamma =1$ corresponds to the long-range Coulomb interaction. With increasing $\gamma$, the interaction becomes increasingly short-range. For $\gamma =2$, as can be seen from Eq. (16), a gap of size $d_{\parallel }$ appears in the longitudinal mode. With further increase in $\gamma$, this gap diverges.