Quantum Monte Carlo studies of edge magnetism in chiral graphene nanoribbons

We investigate chiral graphene nanoribbons using projective quantum Monte Carlo simulations within the local Hubbard-model description and study the effects of electron-electron interactions on the electronic and magnetic properties at the ribbons’ edges. Static and dynamical properties are analyzed for nanoribbons of varying width and edge chirality and compared to a self-consistent Hartee-Fock mean-field approximation. Our results show that for chiral ribbons of sufficient width, the spin correlations exhibit exceedingly long correlation lengths, even between zigzag segments that are well separated by periodic armchair regions. Characteristic enhancements in the magnetic correlations for distinct ribbon widths and chiralities are associated with energy gaps in the tight-binding limit of such ribbons. We identify specific signatures in the local density of states and low-energy modes in the local spectral function which directly relate to enhanced electronic correlations along graphene nanoribbons. These signatures in the local density of states might be accessed by scanning tunneling spectroscopy on graphene nanoribbons.

Figure 1

(a) Armchair graphene nanoribbon of width $w=12$, denoted 12-aGNR. (b) Chiral ribbon of chirality $(3,1)$ ($\theta =13.9^{\circ }$) and width $w=6$, denoted $6(3,1)$-cGNR. (c) Zigzag nanoribbon of width $w=6$, denoted 6-zGNR. In all cases, the two sublattices of the bipartite lattice structure are shown using white and black spheres and the ribbons’ unit cells are indicated by dashed lines. Armchair (zigzag) segments along the upper edge within the unit cell in all cases are highlighted in red (blue).

Figure 2

Real-space spin correlations illustrated by disks proportional to $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ across GNRs of various chiralities of width $w=6$ (corresponding to a 12-aGNR in the armchair case) at $U/t=2$. In each case, the reference site is indicated by an arrow. Red (blue) circles correspond to positive (negative) values.

Figure 3

Spin correlation function $C\left(r\right)$ along the edge of GNRs of various chiralities of width $w=6$ (corresponding to a 12-aGNR in the armchair case). For each case, the number of unit cells, $N_{\mathrm{C}}$, used is indicated. Lines show the fitting function $C_{\mathrm{fit}}\left(r\right)$ for each GNR case. Inset: Position of the reference site for the case of a (3,1)-cGNR edge (arrow), as well as the distance to the corresponding sites (triangles) within further unit cells, each separated by a distance $a$.

Figure 4

Spin correlation function $C\left(r\right)$ along the edge of GNRs of various chiralities of width $w=12$ (corresponding to a 24-aGNR in the armchair case). For each case, the employed number of unit cells $N_{\mathrm{C}}$ is indicated. Lines show the fitting function $C_{\mathrm{fit}}\left(r\right)$ for each GNR case.

Figure 5

Spin correlation function $C\left(r\right)$ along the edge of aGNRs of different widths $w$. Lines in this figure are guides for the eye.

Figure 6

Spin correlation function $C\left(r\right)$ along the edge of (a) (2,1)-cGNR and (b) (3,1)-cGNR of different widths. Lines in this figure are guides for the eye.

Figure 7

Map of the single-particle excitation gap for the $U=0$ tight-binding limit for $w(n,1)$-cGNR of varying $n$ (i.e., chirality) and width $w$.

Figure 8

Momentum-resolved single-particle spectral function for a 12-aGNR for edge sites (top) and bulk sites (bottom) for $U=0$ (left) and from MFT (middle) and QMC simulations (right) for $U/t=2$.

Figure 9

Momentum-resolved single-particle spectral function for a 6-zGNR for edge sites (top) and bulk sites (bottom) for $U=0$ (left) and from MFT (middle) and QMC simulations (right) for $U/t=2$.

Figure 10

Momentum-resolved single-particle spectral function for a 6(2,1)-cGNR for edge sites (top) and bulk sites (bottom) for $U=0$ (left) and from MFT (middle) and QMC simulations (right) for $U/t=2$.

Figure 11

Momentum-resolved single-particle spectral function along the edge sites for a 6(3,1)-cGNR (top) and a 12(3,1)-cGNR (bottom) for $U=0$ (left) and from MFT (middle) and QMC simulations (right) for $U/t=2$.

Figure 12

Local single-particle spectral function for a 6-zGNR for edge sites and bulk sites. Inset: QMC data at low energies.

Figure 13

Local single-particle spectral function for a 6(2,1)-cGNR for edge sites and bulk sites. Inset: QMC data at low energies.

Figure 14

Local single-particle spectral function at low energies for a 6(3,1)-cGNR from QMC simulations along the ribbon edge (shown at the bottom), within the majority sublattice, as indicated by the arrows.

Figure 15

Position of the low-energy peak in the edge-site LDOS of chiral graphene nanoribbons of width $w=6$ as a function of the chirality angle $\theta$ within MFT (open symbols) and from QMC simulations (filled symbols), for both $U=t$ and $U=2t$. The crosses indicate the single-particle gap in the tight-binding ($U=0$) limit.

Figure 16

Position of the low-energy peak in the edge-site LDOS of chiral graphene nanoribbons of width $w=12$ as a function of the chirality angle $\theta$ within MFT (open symbols) and from QMC simulations (filled symbols), for both $U=t$ and $U=2t$. The crosses indicate the single-particle gap in the tight-binding ($U=0$) limit.