Temperature-dependent compressibility in graphene and two-dimensional systems

We calculate the finite-temperature compressibility for two-dimensional (2D) semiconductor systems, monolayer graphene, and bilayer graphene within the Hartree-Fock approximation. We find that the calculated temperature-dependent compressibility including exchange energy is nonmonotonic. In 2D systems at low temperatures, the inverse compressibility decreases first with increasing temperature, but after reaching a minimum, it increases as temperature is raised further. At high enough temperatures, the negative compressibility of low-density systems induced by the exchange energy becomes positive due to the dominance of the finite-temperature kinetic energy. The inverse compressibility in monolayer graphene is always positive and its temperature dependence appears to be the reverse of the 2D semiconductor systems, i.e., it increases first with temperature and then decreases at high temperatures. The inverse compressibility of bilayer graphene shows the same nonmonotonic behavior as ordinary 2D systems, but at high temperatures, it approaches a constant that is smaller than the value of the noninteracting bilayer graphene. We find the leading-order temperature correction to the compressibility within Hartree-Fock approximation to be T(2) ln T at low temperatures for all three systems.