Combined effect of doping and temperature on the anisotropy of low-energy plasmons in monolayer graphene

We compare the two-dimensional (2D) plasmon dispersion relations for monolayer graphene when the sample is doped with carriers in the conduction band and the temperature $T$ is zero with the case when the temperature is finite and there is no doping. Additionally, we have obtained the plasmon excitations when there is doping at finite temperature. The results were obtained in the random-phase approximation which employs energy electronic bands calculated using ab initio density functional theory. We found that in the undoped case the finite temperature results in appearance in the low-energy region of a 2D plasmon which is absent for the $T=0$ case. Its energy is gradually increased with increasing $T$. It is accompanied by expansion in the momentum range where this mode is observed as well. The 2D plasmon dispersion in the $\Gamma$M direction may differ in substantial ways from that along the $\Gamma$K direction at sufficiently high temperature and doping concentrations. Moreover, at temperatures exceeding $\approx300$ meV a second mode emerges along the $\Gamma$K direction at lower energies like it occurs at a doping level exceeding $\approx 300$ meV. Once the temperature exceeds $\approx 0.75$ eV this mode ceases to exit whereas the 2D plasmon exists as a well-defined collective excitation up to $T=1.5$ eV, a maximal temperature investigated in this work.