The introduction of lattice anisotropy causes Dirac cones to shift in response to the applied strain, leaving a pseudogap at the original Dirac points. Here, a group-theory analysis is combined with first-principles calculations to reveal the movement characteristics of Dirac points and band gaps in various graphynes under rotating uniaxial and shear strains. Graphene, where linear effects dominate, is different from α-, β-, and γ-graphynes, which generate strong nonlinear responses due to their bendable acetylenic linkages. However, the linear components of the electronic response, which are essential in determining material performance such as intrinsic carrier mobility due to electron–phonon coupling, can be readily separated, and are well described by a unified theory. The movement of the Dirac points in α-graphyne is circular under a rotating strain, and the pseudogap opening is isotropic with a magnitude of only 2% that in graphene. In comparison, the movement in β-graphyne is elliptical and the center is displaced from the origin. For γ-graphyne, three branches of gaps change with the applied strains with a sine/cosine dependence on the strain angle. The developed methodology is useful in determining the electronic response to various strains of Dirac materials and two-dimensional semiconductors.

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.