Finite-temperature Coulomb Excitations in Extrinsic Dirac Structures

We have derived algebraic, analytic expressions for the chemical potential without any restriction on temperature for all types of doped, or extrinsic, gapped Dirac cone materials including gapped graphene, silicene, germanene and single-layer transition metal dichalcogenides. As an important intermediate step of our derivations, we have established a reliable piecewise-linear model for cal- culating the density-of-states in molybdenum disulfide, showing good agreement with previously obtained numerical results. For the spin- and valley-resolved band structure, we obtain an additional decrease of the chemical potential due to thermally induced doping of the upper subband at finite temperature. It has been demonstrated that since the symmetry between the electron and hole states in $MoS_2$ is broken, the chemical potential could cross the zero-energy level at sufficiently high temperature. These results allow us to investigate the collective properties, polarizability, plasmons and their damping. Emphasis is placed on low temperatures, when initial electron doping plays a crucial role. We clearly demonstrated the contribution of the initial doping to the finite-temperature collective properties of the considered materials.