Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene

Analytical expressions are presented for the intraband conductivity tensor of graphene that includes spatial dispersion for arbitrarily wave-vector values and the presence of a nonzero Fermi energy. The conductivity tensor elements are derived from the semiclassical Boltzmann transport equation under both the relaxation-time approximation and the Bhatnagar-Gross-Krook model (which allows for an extra degree of freedom to enforce number conservation). The derived expressions are based on linear electron dispersion near the Dirac points, and extend previous results that assumed small wave-vector values; these are shown to be inadequate for the very slow waves expected on graphene nanoribbons. The new expressions are also compared to results obtained by numerical integration over the first Brillouin zone using the exact (tight-binding) electron dispersion relation. Very good agreement is found between the new analytical expressions and the exact numerical results. Furthermore, a comparison with the longitudinal random-phase conductivity is also made. It is shown analytically that these new expressions lead to the correct value of the quantum capacitance of a graphene sheet and that ignoring spatial dispersion leads to serious errors in the propagation properties of fundamental modes on graphene nanoribbons. DOI: 10.1103/PhysRevB.87.115429