  We present an analytical theory for the gate electrostatics and the classical
and quantum capacitance of the graphene nanoribbons (GNRs) and compare it with
the exact self-consistent numerical calculations based on the tight-binding
p-orbital Hamiltonian within the Hartree approximation. We demonstrate that the
analytical theory is in a good qualitative (and in some aspects quantitative)
agreement with the exact calculations. There are however some important
discrepancies. In order to understand the origin of these discrepancies we
investigate the self-consistent electronic structure and charge density
distribution in the nanoribbons and relate the above discrepancy to the
inability of the simple electrostatic model to capture the classical gate
electrostatics of the GNRs. In turn, the failure of the classical
electrostatics is traced to the quantum mechanical effects leading to the
significant modification of the self-consistent charge distribution in
comparison to the non-interacting electron description. The role of
electron-electron interaction in the electronic structure and the capacitance
of the GNRs is discussed. Our exact numerical calculations show that the
density distribution and the potential profile in the GNRs are qualitatively
different from those in conventional split-gate quantum wires; at the same
time, the electron distribution and the potential profile in the GNRs show
qualitatively similar features to those in the cleaved-edge overgrown quantum
wires. Finally, we discuss an experimental extraction of the quantum
capacitance from experimental data.
