Dirac-Kronig-Penney model for strain-engineered graphene

Motivated by recent proposals on strain engineering of graphene electronic circuits we calculate conductivity, shot noise and the density of states in periodically deformed graphene. We provide the solution to the Dirac-Kronig-Penney model, which describes the phase-coherent transport in clean monolayer samples with an one-dimensional modulation of the strain and the electrostatic potentials. We compare the exact results to a qualitative band-structure analysis. We find that periodic strains induce large pseudogaps and suppress charge transport in the direction of strain modulation. The strain-induced minima in the gate-voltage dependence of the conductivity characterize the quality of graphene superstructures. The effect is especially strong if the variation in interatomic distance exceeds the value $a^2/\ell$, where $a$ is the lattice spacing of free graphene and $\ell$ is the period of the superlattice. A similar effect induced by a periodic electrostatic potential is weakened due to Klein tunnelling.

Figure 1

(Color online) Schematic illustration of the graphene setup with metal leads for $x<0$ and $x>L$. The angle $\theta$ specifies the orientation of the honeycomb lattice with respect to the transport direction $x$.

Figure 2

(Color online) Conductance and Fano factor calculated from Eq. (28) for finite systems comprising $N=10$ and $N=50$ supercells are compared with the averaged expressions to be derived in Eqs. (33, 34), which correspond to the limit $N\to \infty$ and ignore contributions from evanescent modes. The plots are for graphene with strain modulations, $r_{\theta x}=1$. In a small vicinity of the Dirac point, $\epsilon ⪡\hslash v/L$, (shown at the insets) the transport is insensitive to strain due to the extended gauge invariance, Eq. (16), and is dominated by the evanescent modes.

Figure 3

The density plot for the averaged transmission spectra, ${\overline{T}}_q\left(\epsilon \right)$, calculated from Eq. (32) for the Dirac-Kronig-Penney model with strain modulation $r_{\theta x}=0.1$ (top left), $r_{\theta x}=2.0$ (bottom left) and electrostatic field modulation $r_0=0.1$ (top right), $r_0=2.0$ (bottom right).

Figure 4

(Color online) The averaged conductance and Fano factor calculated from Eqs. (33, 34) for the one-dimensional Dirac-Kronig-Penney model with different field strengths. Eq. (30). The effect of the scalar potential is suppressed due to Klein tunnelling through potential barriers.

Figure 5

(Color online) The averaged conductance of Eq. (33) (solid) is compared with an estimate, Eq. (35), based on the mode-counting argument (dashed) for the Dirac-Kronig-Penney model with $r_{\theta x}=0.3,r_0=0$. The upper panel shows the corresponding transmission spectrum.

Figure 6

(Color online) The averaged density of states obtained from Eq. (43) for the Dirac-Kronig-Penney model with different strengths of the periodic potentials.

Figure 7

(Color online) The contour plot of the dispersion relation for the Dirac-Kronig-Penney model, Eqs. (19, 20) with strain modulation $r_{\theta x}=0.3$ (top left), $r_{\theta x}=2.0$ (bottom left) and electrostatic field modulation $r_0=0.3$ (top right), $r_0=2.0$ (bottom right) at energy $\epsilon =3.3\hslash v/2\ell$.

Figure 8

(Color online) Typical band-gap boundaries. In all plots $n,b$ was set to 1.