Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas

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'' Cg 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. 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>Cg by only a few percent. A recent experiment [L. Li, C. Richter, S. Paetel, T. Kopp, J. Mannhart, and R. Ashoori, arXiv:1006.2847 (unpublished)], however, has observed C larger than Cg by almost 40% while operating in the regime nd²⪡1. In this paper we argue that at nd²⪡1 correlations between the electronic charge of opposite electrodes become important. We develop a theory of the capacitance for the full range of nd². 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 Li et al. [arXiv:1006.2847 (unpublished)] without the use of adjustable parameters.