Wigner lattice between two dielectric slabs: Image potential and Casimir effect

We analyze a model of electron lattice between two metallic or polar dielectric slabs. Typically, the electrons are confined on a thin helium layer above a metallic slab and form, under given circumstances, the Wigner lattice. We assume that another slab is placed above the electron layer at a distance away to have negligible influence on basic properties of the Wigner lattice. Here we want to analyze the total force acting on the lattice electrons as well as the total force acting between the metallic (or dielectric) slabs. From the classical electrodynamics one can determine the image potential and calculate corresponding forces. In the quantum electrodynamics, fluctuations of the electromagnetic field will add in attraction between the slabs due to the Casimir effect. We calculate and compare those two forces on the slabs within the same model, since they are an explicit manifestation of the same field. We found out that the resulting forces can be of comparable strengths in the appropriate setup.

Figure 1

Geometry of the system.

Figure 2

The forces on the Wigner lattice as a function of the slab separation $d$: ${\mathcal{F}}_{\mathrm{tot}}^{z_0}$ (full), ${\mathcal{F}}_1^{z_0}$ (dash-dotted), and ${\mathcal{F}}_{\mathrm{ind}}^{z_0}$ (dashed) line. At $z_0=3$ nm, ${\mathcal{F}}_1^{z_0}$ dominates over ${\mathcal{F}}_{\mathrm{ind}}^{z_0}$, but at $z_0=20$ nm, we find ${\mathcal{F}}_{\mathrm{ind}}^{z_0}\approx -{\mathcal{F}}_1^{z_0}$; so the curves are enlarged in order to be presented on the same scale. The lattice parameter is $r_0=30$ nm.

Figure 3

The forces ${\mathcal{F}}_{\mathrm{tot}}^{z_0}$ and ${\mathcal{F}}_1^{z_0}$ on the Wigner lattice as a function of lattice parameter $r_0$. Results are shown for several values of $d$. The position of the lattice is $z_0=20$ nm. ${\mathcal{F}}_1^{z_0}$ is independent of $r_0$ and appears as a straight line for any value of $d$, while ${\mathcal{F}}_{\mathrm{tot}}^{z_0}$ reduces to ${\mathcal{F}}_1^{z_0}$ at low electron densities ($r_0\gg z_0$).

Figure 4

The forces ${\mathcal{F}}_{\mathrm{tot}}^{z_0}$ (full line) and ${\mathcal{F}}_1^{z_0}$ (dashed line) on the Wigner lattice surrounded by the well-separated metallic or dielectric slabs ($d\to \infty$). The $\mathbf{G}=0$ component (dotted line) contributes only if there is no upper plate ($d=\infty$). The lattice position is $z_0=20$ nm. For a dielectric slab we took ${\omega }_T=0.02$ eV, ${\omega }_L=0.03$ eV, i.e., $\Omega =0.38$, appropriate for NaCl.

Figure 5

Image force ${\mathcal{F}}_{\mathrm{im}}^d$ (in units ${\text{N/m}}^2$) as a function of metallic slab separation $d$. The force is shown for three different lattice parameters $r_0$, together with various lattice positions $z_0$: 1 nm (full), 3 nm (dashed), 20 nm (dash-dotted), and 100 nm (dotted) line. The Casimir forces for real (${\mathcal{F}}_C$) and ideal (${\mathcal{F}}_C^0$) metal are shown for comparison.

Figure 6

Total force ${\mathcal{F}}_{\mathrm{tot}}^d$ (full line) on metallic slabs for different lattice parameters $r_0$. The contributions ${\mathcal{F}}_{\mathrm{im}}^d$ (dash-dotted line) and ${\mathcal{F}}_C$ (dashed line) to ${\mathcal{F}}_{\mathrm{tot}}^d$ are also shown, as well as ${\mathcal{F}}_C^0$ (dotted line).

Figure 7

Image force ${\mathcal{F}}_{\mathrm{im}}^d$ on metallic slabs as a function of lattice parameter $r_0$. Full line: ($d=20$ nm, $z_0=6$ nm) and ($d=100$ nm, $z_0=20$ nm). Dotted line describes the same $d$ values, with $z_0=3$ nm. The averaged image potential $\langle {\mathcal{F}}_{\mathrm{im}}^d\rangle$ (dashed line) depends only on $r_0$. The Casimir forces for real (${\mathcal{F}}_C$, full line) and for ideal (${\mathcal{F}}_C^0$, dashed line) metallic slabs are shown for comparison.