Linear response function in the presence of elastic scattering: plasmons in graphene and the two-dimensional electron gas

Within linear-response theory we derive a response function that thoroughly takes into account the influence of elastic scattering and is valid beyond the long-wavelength limit. We apply the theo-ry to plasmons in graphene and the two-dimensional electron gas (2DEG), in the random-phase ap-proximation, and for the former take into account intraband and interband excitations. The scatte-ring-modified dispersion relation shows that below a critical scattering strength $\gamma_c$, simply related to the plasmon frequency $\omega$, no plasmons are allowed. The critical strengths $\gamma_c$ and the allowed $(\omega, q)$ plasmon spectra for intraband and interband transitions in graphene are different. In both graphene and the 2DEG the strength $\gamma_c$ falls rapidly for small $\omega$ but much more slowly for large $\omega$.