  In electronic devices where a two-dimensional electron gas (2DEG) comprises
one or both sides of a plane capacitor, the resulting capacitance $C$ can be
larger than the "geometric capacitance" $C_g$ determined by the physical
separation $d$ between electrodes. This larger capacitance is known to result
from the Coulomb correlations between individual electrons within the low
density 2DEG, which lead to a negative thermodynamic density of states
(negative compressibility). Experiments on such systems generally operate in
the regime where the average spacing between electrons $n^{-1/2}$ in the 2DEG
is smaller than $d$, and these experiments observe $C > C_g$ by only a few
percent. A recent experiment [1], however, has observed $C$ larger than $C_g$
by almost 40% while operating in the regime $nd^2 << 1$. In this paper we argue
that at $nd^2 << 1$ correlations between the electronic charge of opposite
electrodes become important. We develop a theory of the capacitance for the
full range of $nd^2$. We show that, in the absence of disorder, the capacitance
can be $4d/a$ times larger than the geometric value, where $a << d$ is the
electron Bohr radius. Our results compare favorably with the experiment of Ref.
[1] without the use of adjustable parameters.
