Collective electron excitations on a two-dimensional lattice

The frequency and wave-vector-dependent dielectric function ɛ(q,ω) of a two-dimensional electron system on a square lattice is calculated within the random-phase approximation. Effects of the periodic potential on the electronic structure and dielectric properties are taken into account within the framework of the tight-binding model. Both the dispersion relation of long-wavelength plasmons and the energy-loss function Im[1/ɛ(q,ω)] are numerically obtained for different values of the electronic concentration ${\mathit{n}}_{\mathit{s}}$. For low values of ${\mathit{n}}_{\mathit{s}}$, results are well described by the effective-mass approximation. As ${\mathit{n}}_{\mathit{s}}$ increases, the Fermi level moves to regions in k space where the band structure strongly deviates from that of free electrons, and the appearance of structures in the energy-loss spectrum can be observed. Regarding plasmon excitations it is found that for all values of ${\mathit{n}}_{\mathit{s}}$ the long-wavelength behavior of the plasma frequency ${\mathrm{\omega }}_{\mathit{p}}$(q) can be described by a two-dimensional free-electron model provided a renormalized value of the electronic effective mass is introduced.