Negative Compressibility and Charge Partitioning Between Graphene and MoS$_{2}$ Two-Dimensional Electron Gases

Electron-electron interactions often have opposite influences on thermodynamic properties of electrons in graphene compared to conventional two-dimensional electron gases (2DEGs), for example by lowering charge and spin-susceptibilities in the graphene case and enhancing them in the ordinary 2DEG case [1]. In ordinary 2DEGs the charge susceptibility diverges at a finite carrier density, below which the compressibility becomes negative. We theoretically explore the influence of this qualitative difference on how charge is partitioned between a MoS$_{2}$ and a graphene sheet 2DEG when they act as a compound capacitor electrode. Our theory is based on a random phase approximation for charge fluctuations in the 2DEGS and the coupling constant formulation for the ground state energy. We find that in the ideal case the MoS$_{2}$ 2DEG carrier density jumps immediately to a finite value when it is initially populated and discuss how this effect is moderated by disorder. \\[4pt] [1] Yafis Barlas, T. Pereg-Barnea, Marco Polini, Reza Asgari, and A.H. MacDonald, \textit{PRL} \textbf{98, }236601 (2007).