Giant capacitance of a plane capacitor with a two-dimensional electron gas in a magnetic field

If a clean two-dimensional electron gas (2DEG) with a low concentration $n$ comprises one electrode of a plane capacitor, the resulting capacitance $C$ can be higher than the “geometric capacitance” $C_g$ determined by the physical separation $d$ between electrodes. A recent paper [B. Skinner and B. I. Shklovskii, Phys. Rev. B 82, 155111 (2010)] argued that when the effective Bohr radius $a_B$ of the 2DEG satisfies $a_B\ll d$, one can achieve $C\gg C_g$ at a low concentration $n{d}^2\ll 1$. Here we show that even for devices with $a_B>d$, including graphene, for which $a_B$ is effectively infinite, one also arrives at $C\gg C_g$ at low electron concentrations if there is a strong perpendicular magnetic field.

Figure 1

Schematic of a capacitor made with a low-density 2DEG next to a metal electrode in the presence of a transverse magnetic field. Electrons (filled circles) create image charges (open circles) in the metal electrode. The transverse magnetic field confines electrons to their lowest Landau level wave function with characteristic lateral size $l_B$, as shown schematically for the central electron by the shaded (red) area.

Figure 2

Effective capacitor thickness $d^{*}$, in units of $d$, plotted as a function of filling factor $\nu$ in the LLL at different values of $d/l_B$.

Figure 3

Effective capacitor thickness, in units of $d$, as a function of filling factor in the LLL for various values of $d/l_B$. The result of Eq. (7) is plotted as the dashed (red) lines, and the dash-dotted (blue) lines correspond to the coefficients given by Eq. (9). Solid (black) lines are the result of a general numerical fitting procedure that determines the values of the coefficients ${\alpha }_k$ at each $d/l_B$.

Figure 4

Schematic depiction of the effect of disorder on the measured inverse capacitance. The dashed line shows $d^{*}/d$ in the absence of disorder, and the solid line shows $d^{*}/d$ in the presence of a finite but small disorder potential, for which ${\nu }_{\mathrm{m}}\ll 1$.