Theory of magnetic edge states in chiral graphene nanoribbons

Using a model Hamiltonian approach including electron-electron interactions, we systematically investigate the electronic structure and magnetic properties of chiral graphene nanoribbons. We show that the presence of magnetic edge states is an intrinsic feature of smooth graphene nanoribbons with chiral edges, and discover a number of structure-property relations. Specifically, we study the dependence of magnetic moments and edge-state energy splittings on the nanoribbon width and chiral angle as well as the role of environmental screening effects. Our results address a recent experimental observation of signatures of magnetic ordering in chiral graphene nanoribbons and provide an avenue toward tuning their properties via the structural and environmental degrees of freedom.

Figure 1

(a) Atomic structure of a model of chiral graphene nanoribbon. The structure shown corresponds to $\theta =10.9^{\circ }$ chiral GNR characterized by edge repeat vector (4,1) and width $w=6$. Zigzag and armchair units of the edge are indicated. (b) Chirality angles $\theta$ of the considered ($n+1,1$) and ($n+1,n$) series of chiral GNRs.

Figure 2

Evolution of the tight-binding band structures (no $e$-$e$ interactions) of graphene nanoribbons ($w=12$) upon the change of chirality from zigzag ($\theta =0^{\circ }$) to armchair ($\theta ={30}^{\circ }$) via the series of intermediate chiral configurations. The scales of the plots account for the varying Brillouin-zone dimensions. The insets schematically illustrate the projection of points $K$ and $K^{\prime }$ of the 2D BZ of ideal graphene onto the 1D band structures of zigzag and armchair GNRs.

Figure 3

(a) Effects of the electron-electron interactions on the band structure (left panel) and the density of states (right panel) of a zigzag GNR ($w=12$). Dashed and solid curves correspond to the tight-binding model ($U/t=0$) and the mean-field Hubbard model ($U/t=1$), respectively. (b) and (c) Respective band structure and density-of-states plots for (2,1) and (6,1) chiral GNRs. (d) Spin-density distribution in (6,1) chiral GNR ($w=4$). Circle areas correspond to the local magnetic moments from the mean-field Hubbard model solution obtained at $U/t=1$.

Figure 4

(a) Magnetic moment per edge unit length $M$ as a function of chirality angle $\theta$ for three different nanoribbon widths $w$ from the mean-field Hubbard model ($U/t=1$) calculations. The dotted line shows magnetic moments in the limit of infinite width [see Eq. (3)]. (b) Electronic band gap ${\Delta }^0$ and maximum energy splitting ${\Delta }^1$ as a function of $\theta$ for different values of $w$ (mean-field Hubbard model, $U/t=1$). Crosses indicate the tight-binding band gaps ${\Delta }_{\mathrm{g}}$. The values of (c) $M$ and (d) ${\Delta }^0$ and ${\Delta }^1$ obtained using the mean-field Hubbard model at different values of $U/t$ and $w=6$.

Figure 5

Variation of the local density of states (LDOS) (a) across the $(6,1)$ chiral GNR ($x$ axis is oriented along the edge, $y=0$ corresponds to the outermost edge atom) and (b) along its edge obtained using the mean-field Hubbard model ($U/t=1$). (c) and (d) Log-linear plots of LDOS at the energies indicated in panel (a) across the $(6,1)$ chiral GNR and along its edge, respectively. The dashed lines correspond to the tight-binding LDOS at $E=0$. The inset in panel (d) superimposes the edge structure with the plot.