Inspired by the great development of graphene, more and more research has been conducted to seek new two-dimensional (2D) materials with Dirac cones. Although 2D Dirac materials possess many novel properties and physics, they are rare compared with the numerous 2D materials. To provide explanation for the rarity of 2D Dirac materials as well as clues in searching for new Dirac systems, here we review the recent theoretical aspects of various 2D Dirac materials, including graphene, silicene, germanene, graphynes, several boron and carbon sheets, transition-metal oxides (VO₂)_n/(TiO₂)_m and (CrO₂)_n/(TiO₂)_m, organic and organometallic crystals, so-MoS₂, and artificial lattices (electron gases and ultracold atoms). Their structural and electronic properties are summarized. We also investigate how Dirac points emerge, move, and merge in these systems. The von Neumann–Wigner theorem is used to explain the scarcity of Dirac cones in 2D systems, which leads to rigorous requirements on the symmetry, parameters, Fermi level, and band overlap of materials to achieve Dirac cones. Connections between existence of Dirac cones and the structural features are also discussed.

The carrier mobility of anisotropic two-dimensional semiconductors under longitudinal acoustic phonon scattering was theoretically studied using deformation potential theory. Based on the Boltzmann equation with the relaxation time approximation, an analytic formula of intrinsic anisotropic mobility was derived, showing that the influence of effective mass on mobility anisotropy is larger than those of deformation potential constant or elastic modulus. Parameters were collected for various anisotropic two-dimensional materials (black phosphorus, Hittorf's phosphorus, ${\mathrm{BC}}_2\mathrm{N}$, MXene, ${\mathrm{TiS}}_3$, and ${\mathrm{GeCH}}_3$) to calculate their mobility anisotropy. It was revealed that the anisotropic ratio is overestimated by the previously described method.

Using first-principles $GW$ Bethe-Salpeter equation calculations and the $\mathbf{k}·\mathbf{p}$ theory, we unambiguously show that for two-dimensional (2D) semiconductors, there exists a robust linear scaling law between the quasiparticle band gap ($E_g$) and the exciton binding energy ($E_b$), namely, $E_b\approx E_g/4$, regardless of their lattice configuration, bonding characteristic, as well as the topological property. Such a parameter-free universality is never observed in their three-dimensional counterparts. By deriving a simple expression for the 2D polarizability merely with respect to $E_g$, and adopting the screened hydrogen model for $E_b$, the linear scaling law can be deduced analytically. This work provides an opportunity to better understand the fantastic consequence of the 2D nature for materials, and thus offers valuable guidance for their property modulation and performance control.

A tight-binding analytic framework is combined with first-principles calculations to reveal the mechanism underlying the strain effects on electronic structures of graphene and graphene nanoribbons (GNRs). It provides a unified and precise formulation of the strain effects under various circumstances-including the shift of the Fermi (Dirac) points, the change in band gap of armchair GNRs with uniaxial strain in a zigzag pattern and its insensitivity to shear strain, and the variation of the k-range of edge states in zigzag GNRs under uniaxial and shear strains which determine the gap behavior via the spin polarization interaction.