Edge magnetization and local density of states in chiral graphene nanoribbons

We study the edge magnetization and the local density of states of chiral graphene nanoribbons using a $\pi \text{-orbital}$ Hubbard model in the mean-field approximation. We show that the inclusion of a realistic next-nearest hopping term in the tight-binding Hamiltonian changes the graphene nanoribbon band structure significantly and affects its magnetic properties. We study the behavior of the edge magnetization upon departing from half-filling as a function of the nanoribbon chirality and width. We find that the edge magnetization depends very weakly on the nanoribbon width, regardless of chirality as long as the ribbon is sufficiently wide. We compare our results to recent scanning tunneling microscopy experiments reporting signatures of magnetic ordering in chiral nanoribbons and provide an interpretation for the observed peaks in the local density of states, that does not depend on the antiferromagnetic interedge interaction.

Figure 1

Representation of a chiral nanoribbon. The chiral vector is ${\mathbf{C}}_h=(5,1)$, and the width is characterized by $\mathbf{W}=(-4,8)$. The sites with dangling bonds at the GNR edges are indicated by circles.

Figure 2

Band structure (left column) and corresponding density of states (right column) of a zigzag graphene nanoribbon of $N=24$ for $t^{\prime }=0$ (upper row) and $t^{\prime }/t=0.1$ (lower row). The solid lines stand for the case of $U/t=1$, whereas the dashed ones stand for $U=0$.

Figure 3

Edge magnetization $M_1$ of zigzag graphene nanoribbons as a function of their width $N$. Inset: Band gaps ${\Delta }_0$ and ${\Delta }_1$ versus $N$. In both cases, $U/t=1.0$ and $t^{\prime }/t=0.0,0.1$, and $0.2$.

Figure 4

Edge magnetization $M_1$ as a function of the chemical potential $\mu$ for zigzag nanoribbons with different GNR widths $N$ for (a) $t^{\prime }/t=0.0$ and (b) $t^{\prime }/t=0.1$. In both cases, $U/t=1$.

Figure 5

Electronic band structure for GNRs of width $w=12$ and $U/t=1$ with chiralities (a) (2,1) and (c) (3,1). Corresponding density of states for the (b) (2,1) and (d) (3,1) chiralities. The solid lines stand for the case of $t^{\prime }/t=0.1$, whereas the dashed ones stand for $t^{\prime }/t=0.0$. The energy gaps ${\Delta }_0$ and ${\Delta }_1$ are only indicated for the $t^{\prime }/t=0.1$ case.

Figure 6

Gap ${\Delta }_1$ as a function of the GNR width $w$ for different next-nearest-neighbor hopping parameters $t^{\prime }/t$. Here, we use $U/t=1.0$.

Figure 7

Edge magnetization $M$ as a function of the chemical potential $\mu$ for chiral nanoribbons of different widths $w$ for (a) $t^{\prime }/t=0.0$ and (b) $t^{\prime }/t=0.1$. In both cases, $U/t=1$.

Figure 8

(a) Band structures of (8,1) chiral graphene nanoribbons of $w=12$ and (b) edge magnetization $M/a_0$ as a function of the chemical potential $\mu /t$. The dashed (red) lines stand for the case of $t^{\prime }/t=0.0$, and the solid (blue) ones stand for $t/t=0.1$.

Figure 9

Local density of states for a (3,1) chiral graphene nanoribbon for (a) $t^{\prime }/t=0$ and (b) $t^{\prime }/t=0.1$. Here, $w=12$ and $U/t=1$.

Figure 10

Local density of states of an (8,1) chiral graphene nanoribbon for (a) $t^{\prime }/t=0$ and (b) $t^{\prime }/t=0.1$. Here, $w=12$ and $U/t=1$.

Figure 11

Local density of states (logarithmic scale) and edge magnetization (linear scale) $M$ of an (8,1) chiral graphene nanoribbon for (a) $t^{\prime }/t=0$ and (b) $t^{\prime }/t=0.1$. Here, $w=12$ and $U/t=1$.