Scattering by linear defects in graphene: A continuum approach

We study the low-energy electronic transport across periodic extended defects in graphene. In the continuum low-energy limit, such defects act as infinitessimally thin stripes separating two regions where the Dirac Hamiltonian governs the low-energy phenomena. The behavior of these systems is defined by the boundary condition imposed by the defect on the massless Dirac fermions. We demonstrate how this low-energy boundary condition can be computed from the tight-binding model of the defect line. For simplicity we consider defect lines oriented along the zigzag direction, which requires the consideration of only one copy of the Dirac equation. Three defect lines of this kind are studied and shown to be mappable between them: the pentagon-only, the $zz\left(558\right)$, and the $zz\left(5757\right)$ defect lines. In addition, in this same limit, we calculate the conductance across such defect lines with size $L$ and find it to be proportional to $k_{F}L$ at low temperatures.

Figure 1

Scheme of a graphene sheet with a pentagon-only defect line along the zigzag direction. The primitive vectors are ${\mathbf{u}}_1=a(1,0)$ and ${\mathbf{u}}_2=a(-1/2,\sqrt{3}/2)$, where $a$ stands for the lattice parameter.

Figure 2

(a) Graphene FBZ with the incident vectors used in the TB and CA formalism: $\mathbf{q}=\mathbf{k}-{\mathbf{K}}_{\nu }$. (b) Scheme of the electron scattering through the barrier (in the low-energy limit). When there is no potential difference between the two regions separated by the defect line, incident and transmitted wave vectors are equal, $\mathbf{q}=\tilde{\mathbf{q}}$.

Figure 3

Scheme of the one-dimensional chain obtained by Fourier transformation on the $x$ direction of the TB Hamiltonian of a graphene layer with a pentagon-only defect line along the zigzag direction. The complex hopping amplitude $t^{\prime }$ has the value $t^{\prime }=t(1+e^{i{k}_{x}a})$.

Figure 4

Plot of the transmission probability, $T$, in terms of the angle of incidence on the pentagon-only defect line, ${\theta }_{\mathbf{q}}^{+}$ (for the low-energy limit, around ${\mathbf{K}}_{\nu }$, with $\nu =+1$). We have used the value $\xi =0.3$ to obtain these curves. In each of the panels, we compare the TB result for $\epsilon >0$ (full violet curves) and for $\epsilon <0$ (dashed violet curves) with that obtained from the CA (in green), with $\Delta V=0$. (a–f) Energies $|\epsilon /t|=0.01$, 0.04, 0.08, 0.16, 0.32, and 0.64, respectively.

Figure 5

Scheme of a $zz\left(558\right)$ defect line.

Figure 6

Scheme of a $zz\left(5757\right)$ defect line.

Figure 7

The transmission probabilities for a $zz\left(558\right)$ defect, at $\epsilon /t=.03$, with ${\xi }_1=1,{\xi }_2=0.5$ (dashed red), ${\xi }_1=1.5,$ ${\xi }_2=1.125$ (dash-dotted blue), obtained in a full TB calculation, and the corresponding low-energy approximation (continuous black), given by Eq. (27) with $\xi =1.0$.

Figure 8

Comparison between the transmission probabilities, $T$, across the defect for the pentagon-only [or $zz\left(55\right)$ ] defect line (full orange line), the $zz\left(558\right)$ defect line (dashed red line), and the $zz\left(5757\right)$ defect line (dash-dotted green line). The transmission probabilities were calculated in the low-energy limit from Eq. (4), with all the hopping parameters chosen equal to 1, $\xi ={\xi }_1={\xi }_2={\xi }_a={\xi }_b={\xi }_c=1$. $T$ is plotted in terms of the angle of incidence, ${\theta }_{\mathbf{q}}^{+}$, of $\mathbf{q}=\mathbf{k}-{\mathbf{K}}_{+}$, the momentum around the Dirac point ${\mathbf{K}}_{+}$. The transmission probabilities valid for the vicinity of the other Dirac point, ${\mathbf{K}}_{-}$, are obtained from the ones plotted above, by a reflection along the vertical line $\theta =\pi /2$.

Figure 9

Dependence of the total angular integrated transmission probability associated with one Dirac point, $\mathbb{T}(E,\Delta V)={\mathbb{T}}^{+}(E,\Delta V)={\mathbb{T}}^{-}(E,\Delta V)$, when $\Delta V=0$, in terms of the hopping, $\xi t$, between the atoms $D$ at the defect line (see Fig. 1). The vertical dashed line indicates the value of $\xi$ used to obtain the curves of Fig. 4: $\xi =0.3$.