Lattice-corrected strain-induced vector potentials in graphene

The electronic implications of strain in graphene can be captured at low energies by means of pseudovector potentials which can give rise to pseudomagnetic fields. These strain-induced vector potentials arise from the local perturbation to the electronic hopping amplitudes in a tight-binding framework. Here we complete the standard description of the strain-induced vector potential, which accounts only for the hopping perturbation, with the explicit inclusion of the lattice deformations or, equivalently, the deformation of the Brillouin zone. These corrections are linear in strain and are different at each of the strained, inequivalent Dirac points, and hence are equally necessary to identify the precise magnitude of the vector potential. This effect can be relevant in scenarios of inhomogeneous strain profiles, where electronic motion depends on the amount of overlap among the local Fermi surfaces. In particular, it affects the pseudomagnetic field distribution induced by inhomogeneous strain configurations, and can lead to new opportunities in tailoring the optimal strain fields for certain desired functionalities.

Figure 1

(a) Unstrained graphene's real space lattice with labeled nearest-neighbor vectors ($\overrightarrow{{\delta }_i}$), labeled lattice translation vectors, ${\vec{R}}_i=n{\vec{a}}_{+}+m{\vec{a}}_{-}$ with $m$ and $n$ integers, and black and gray dots representing the A and B sublattices, respectively. (b) The Brillouin zone of unstrained graphene with labeled high symmetry points. (c) The positions of the unstrained (black, solid) and strained (brown, dashed) nearest neighbors, and the corresponding ${\vec{\delta }}_i$ and ${\vec{\delta }}_i^{\prime }$ for 20$\%$ uniaxial strain along the armchair direction. (d) The unstrained (black, solid) and strained (brown, dashed) Brillouin zone, also for 20$\%$ armchair uniaxial strain, with labels for the now inequivalent Dirac points.

Figure 2

Contours of the band structure of graphene under tensile isotropic strain, shear strain, uniaxial armchair strain, and zigzag strain (rows), for $\epsilon =1\%$ near the three $\mathit{K}$ points (columns). The contours are overlaid with the Brillouin zone of unstrained (solid, black) and strained (dashed, brown) graphene. Vectors mark the displacement of the Dirac points predicted by the traditional (${\vec{A}}_p$, dashed/red arrow) and the corrected (${\vec{A}}_{{\mathit{K}}_i}$, solid/orange arrow) form of the pseudovector potential [Eqs. (4)], with the green dots marking the positions of the Dirac points for strained graphene. The red vectors appears as a dot for isotropic strain, because the traditional form of the vector potential does not predict a shift in the Dirac points. Each plot is square with an area of $0.{12}^2$.

Figure 3

The spatial distribution of the pseudomagnetic fields generated when an equilateral triangle with 50-nm sides, 2-nm-radius fillets, and the base oriented 30 degrees counterclockwise from the armchair direction is pressurized to 14 MPa. In (a) the calculation is done using the traditional form of the pseudovector potential, while in (b) it is performed including the differences derived in this work. Here, the three different colored curves correspond to the pseudomagnetic fields felt by the electrons at the three different $\mathit{K}$ points with the magnetic field for the electrons at ${\mathit{K}}_{\mathbf{1}}$ isolated in (c).