Analytic expressions for chemical potentials without any approximations are derived for all types of extrinsic (doped) gapped Dirac-cone materials including gapped graphene, silicene, germanene, and single-layer transition-metal dichalcogenides. In setting up our derivations, a reliable piecewise-linear model has been established for calculating the density of states in molybdenum disulfide, showing good agreement with previously obtained numerical results. For spin- and valley-resolved band structures, a decrease of chemical potential with increasing temperature is found as a result of enhanced thermal populations of an upper subband. Due to the broken symmetry with respect to electron and hole states in ${\mathrm{MoS}}_2$, the chemical potential is shown to cross a zero-energy point at sufficiently high temperatures. It is important to mention that the chemical potential at a fixed temperature can still be tuned by varying the doping density and band structure of a system with an external electric or strain field. Since a thermal-convolution path (or a chemical-potential-dependent response function for the thermal convolution of fermions) starting from zero temperature must be selected in advance before obtaining finite-temperature properties of any collective quantities, e.g., polarizability, plasmon modes, and damping, a control of their thermal dependence within a certain temperature range is expected for field-tunable extrinsic gapped Dirac-cone materials.

• Received 19 October 2017
• Revised 20 November 2017

DOI:https://doi.org/10.1103/PhysRevB.96.245403