Effects of disorder range and electronic energy on the perfect transmission in graphene nanoribbons

Numerical calculations based on the recursive Green's function method in the tight-binding approximation are performed to calculate the dimensionless conductance $g$ in disordered graphene nanoribbons with Gaussian scatterers. The influence of the transition from short- to long-ranged disorder on $g$ is studied as well as its effects on the formation of a perfectly conducting channel. We also investigate the dependence of electronic energy on the perfectly conducting channel. We propose and calculate a backscattering estimate in order to establish the connection between the perfectly conducting channel (with $g=1$) and the amount of intervalley scattering.

Figure 1

Electronic band structure of a zigzag GNR with $M=10$ sites across the width (see text). $K$ and $K^{\prime }$ are the two inequivalent points of the first Brillouin zone of graphene. The left (right) arrows indicate positive (negative) group velocity and forward (backward) electronic propagation.

Figure 2

Zigzag graphene ribbon geometry. $n$ and $m$ label the atomic sites on the sublattices $A$ (yellow dots) and $B$ (blue dots). The atoms labeled $n=1,...,N$ belong to the ribbon region and the atoms $n=0$ and $n=N+1$ belong to the left and right contacts, respectively.

Figure 3

Average dimensionless conductance $\langle g\rangle$ as a function of the potential range $d$ for a ribbon of length $L=500a_0$ and width $W=5\sqrt{3}{a}_0$ for the impurity concentrations (a) $n_{\mathrm{imp}}=0.10$ and (b) $n_{\mathrm{imp}}=0.01$.

Figure 4

Disorder range above which the PCC appears, $d_c/a_0$, as a function of the electron energy $E$. Inset: Scaled average conductance $(\langle g\rangle -g_0)/(1-g_0)$ as a function of $d/d_c$ for different energies $E/t$.

Figure 5

Backscattering matrix element $V_{\mathrm{back}}$ calculated as a function of the scattering range $d/a_0$ for different values of the energy $E$ for a single Gaussian impurity center placed (a) at the center of the GNR ($n_0=100,m_0=5$) and (b) at its edge vicinity ($n_0=100,m_0=9$). The GNR has the dimensions $N=200$ and $M=10$. For this value of GNR width, the edge state appears for $E<0.1t$ (Refs. 16 and 19).

Figure 6

Average backscattering matrix element $\langle |V_{\mathrm{back}}{|}^2\rangle$ as a function of the disorder range $d/a_0$ for a number of energies $E$. The average is taken over ${10}^3$ disorder configurations for an impurity concentration of (a) $n_{\text{imp}}=0.01$ and (b) $n_{\text{imp}}=0.10$. The GNR has the dimensions $L=500a_0$ and $W=10(\sqrt{3}/2)a_0$. This ribbon presents an edge state for $E<0.1t$.