  We introduce a method for calculating the dielectric function of
nanostructures with an arbitrary band dispersion and Bloch wave functions. The
linear response of a dissipative electronic system to an external
electromagnetic field is calculated by a self-consistent-field approach within
a Markovian master equation formalism (SCF-MMEF) coupled with full-wave
electromagnetic equations. The SCF-MMEF accurately accounts for several
concurrent scattering mechanisms. The method captures interband
electron-hole-pair generation, as well as the interband and intraband electron
scattering with phonons and impurities. We employ the SCF-MMEF to calculate the
dielectric function, complex conductivity, and loss function for supported
graphene. From the loss-function maximum, we obtain plasmon dispersion and
propagation length for different substrate types [nonpolar diamondlike carbon
(DLC) and polar SiO$_2$ and hBN], impurity densities, carrier densities, and
temperatures. Plasmons on the two polar substrates are suppressed below the
highest surface phonon energy, while the spectrum is broad on the nonpolar DLC.
Plasmon propagation lengths are comparable on polar and nonpolar substrates and
are on the order of tens of nanometers, considerably shorter than previously
reported. They improve with fewer impurities, at lower temperatures, and at
higher carrier densities.
