Posts General Wigner Crystal in Moiré Superlattice
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General Wigner Crystal in Moiré Superlattice

Ratio of Potential and Kinetic Energy

The dimensionless ratio of the potential energy ($U$) and the kinetic energy $K$, known as $\gamma_s$, can be written as 1

\[\begin{align} \gamma_s &= \frac{U}{K} = \frac{g}{a_0} \, \frac{m^{\ast}}{\epsilon} \, \frac{1}{\sqrt{\pi n_e}} \\[9pt] &= \frac{g}{a_0} \, \frac{m^{\ast}}{\epsilon} \, \frac{\lambda_m}{\sqrt{2\pi\nu/\sqrt{3}}} \\ \end{align}\]

where

  • $g=2$ is the valley degeneracy factor,

  • $m^\ast$ is the effect carrier mass in unit of electron mass $m_0$. From DFT calculations of 1ML CoCl2 (by Mr. Aolei Wang), one can get the effective carrier mass at different band extrema.

    • VBM($\Gamma – M$): $m^\ast = 4.26$
    • VBM($\Gamma – K$): $m^\ast = 2.78$
    • CBM($K – \Gamma$): $m^\ast = 9.61$
    • CBM($K – M$): $m^\ast = 6.81$
  • $a_0 = \hbar^2 / m_0 e^2 = 0.529\,\mathring{A}$ is the Bohr radius,

  • $\epsilon$ is the effective dielectric constant. By treating the bilayer as two dielectrics in series,

    \[\begin{equation} \frac{d_1 + d_2}{\epsilon} = \frac{d_1}{\epsilon_1} + \frac{d_2}{\epsilon_2} \end{equation}\]

    where $\epsilon_i$ and $d_i$ are the dielectric constants and the thickness of the top and bottom layers, respectively. For CoCl2/HOPG bilayer, we choose

    • $d_{\mathrm{CoCl}_2} = 4.8\,\mathring{A}$ from Ref 2
    • $d_\mathrm{HOPG} = 3.4\,\mathring{A}$ from Ref 2
    • $\epsilon_{\mathrm{CoCl}_2} = 4$ (estimated from Ref 3)
    • $\epsilon_{\mathrm{CoCl}_2} = 1.558$ (DFT result from Mr. Aolei Wang)
    • $\epsilon_\mathrm{HOPG} = \infty$ for a metal
  • $\lambda_m$ is the lattice parameter of the moiré lattice. If two hexagonal sublattices with lattice constants $a_1$ and $a_2$ ($a_2 > a_1$) rotate by an angle $\theta$, then the moiré periodicity is given by 4

    \[\begin{equation} \lambda_m \approx \frac{a_2}{\sqrt{\delta^2 + 4\sin^2(\theta / 2)}}, \qquad \delta = 1 - \frac{a_1}{a_2} \end{equation}\]

    For CoCl2/HOPG

    • $a_{\mathrm{CoCl}_2} = 3.528\,\mathring{A}$ from Ref 5

    • $a_\mathrm{HOPG} = 2.468\,\mathring{A}$ from Ref 6

    There is a large lattice mismatch and the moiré periodicity can not be estimated from the above equation. Experimentally, $\lambda_m$ for CoCl2/HOPG falls within $[6, 12]\,\mathring{A}$.

  • $\nu$ is the filling factor,

  • $n_e$ is the charge density,

    \[\begin{equation} n_e = \frac{\nu}{\sqrt{3}\lambda_m^2/2} \end{equation}\]
  • critical value for GWCs is $\gamma_s = 29.5$.

Mott criterion

Mott states and Wigner crystal are both due to the strong Coulomb repulsion between electrons, with the former dominated by intrasite and the latter intersite interaction. Moreover, the Mott states must satisfy the Mott criterion. [1,7]

\[\begin{equation} \sqrt{n_e}\,a_0^\ast \approx 1 \end{equation}\]

where

  • $a_0^\ast$ is the effective Bohr radius

    \[\begin{equation} a_0^\ast = \frac{\hbar^2\epsilon}{m^\ast m_0\,e^2} = \frac{\epsilon}{m^\ast}\,a_0 \end{equation}\]

$\gamma_s$ as function of charge density

Figure 1. Dimensionless parameter $\gamma_s$ as a function of moiré lattice parameter $\lambda_m$ and filling factor $\nu$. The blue dotted line indicates the critical value $\gamma_s = 29.5$ and the red dotted line shows the Mott criterion. The effective mass of the charge $m^\ast$ for the colormap plot was chosen to be 0.5. The figure was generated by this script.
  • As can be seen from the above figure, for $\lambda_m$ in the range of $[6, 12]\,\mathring{A}$ and filling factor $\nu = 1$, $\gamma_s$ is far below the critical value of 29.5, which indicates it is not possible to Wigner crystalize in CoCl2/HOPG based on this simple model.
Figure 2. Filling factor ($\nu$) as a function of moiré lattice parameter at the critical value of $\gamma_s = 29.5$ with effective carrier mass estimated from DFT calculations. The vertical blue shaded area indicates the experimental range of the moiré lattice parameter. The figure was generated by this script.
  • Clearly, with much larger effective mass, the possibility to Wigner crystalization increases.

Reference

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