Anderson disorder in graphene nanoribbons: A local distribution approach

Disorder effects strongly influence the transport properties of graphene-based nanodevices even to the point of Anderson localization. Focusing on the local density of states and its distribution function, we analyze the localization properties of actual size graphene nanoribbons. In particular, we determine the time evolution and localization length of the single-particle wave function in dependence on the ribbon extension and edge geometry as well as on the disorder type and strength.

Figure 1

(Color online) Mean (solid red) and typical (dashed blue) DOS for zigzag (left column, $N_z=6$) and armchair (right column, $N_a=10$) GNRs of width $W=1.1\text{nm}$. Top panels: ordered case. Lower panels: in each $\left(2\times 2\right)$ block, we compare for a fixed value of disorder the influence of bulk disorder (${\gamma }_b$, upper rows) to edge disorder (${\gamma }_e$, bottom rows). To illustrate the localization properties, in each panel ${\rho }_{\text{ty}}$ is given for $L=213\left(1064\right)\text{nm}$ by light blue dot-dashed (dark blue dashed) lines. These system sizes correspond to $10000$ $\left(50000\right)$ lattice sites for the armchair and $10392$ $\left(51960\right)$ for the zigzag case. Disorder averaging was performed over ${10}^5$ realizations. In the longitudinal direction, periodic boundary conditions (pbc) are applied.

Figure 2

(Color online) Gap size ${\Delta }_a$ for armchair GNRs as a function of ribbon width $N_a=3n+1$ for edge disorder (main panel) and bulk disorder (inset). Results are based on the averaged DOS for GNRs with a length of 1000 atoms using 4096 realizations of disorder.

Figure 3

(Color online) Normalized LDOS at the band center $\left({\rho }_i/{\rho }_{\text{me}}\right){\mid }_{E=0}$ for particular zigzag GNRs. In addition to contrasting bulk and edge disorder, we compare in the left (right) column the influence of the aspect ratios $L\times W=31.4\times 13.5\left(31.4\times 3.3\right){\text{nm}}^2$, corresponding to $256\times 64\left(256\times 16\right)$ sites. Results obtained by exact diagonalization.

Figure 4

(Color online) Time evolution of the normalized particle density $N{\left|\psi \left({\mathbf{r}}_i\right)\right|}^2$ on disordered GRNs with zigzag and armchair geometries for different values of bulk disorder ${\gamma }_b$. Device dimensions: $\left(1.1\times 213\right){\text{nm}}^2$ corresponding to $6\times 1732$ atoms (zigzag) and $10\times 1000$ atoms (armchair). Times are measured in units of the inverse hopping element $t_0=1/\overline{t}$.

Figure 5

(Color online) Localization length in dependence on bulk $\left({\gamma }_b\right)$ and edge $\left({\gamma }_e\right)$ disorder strength for armchair and zigzag GNRs. The values are sample averages obtained for ten GNRs of $L=213\text{nm}$ when the state has become quasistationary.