Spin waves in graphene nanoribbon devices

We investigate spin excitations and electronic properties of graphene nanoribbon devices with zigzag edges. The magnetic region of the device is coupled to nonmagnetic metallic leads. The ground state of the magnetic region is described self-consistently within a mean-field scheme. The magnetic moment is site dependent, in contrast with the case of an infinite nanoribbon. Spin excitations are extracted from the transverse dynamic spin susceptibility. Several standing-wave modes can be identified. We study the behavior of these modes as the coupling between the magnetic region and the leads is varied. A central point found is that for a finite zigzag nanoribbon, spin excitations are damped at all finite energies. The signature of antiferromagnetic correlations is still present in the predominantly linear relationship between the standing-mode energy and the mode wave vector. The effect of an external doping is also considered and, as in the infinite case, it is found that ferromagnetic order along the ribbon's edges becomes unstable at modest doping levels.

Figure 1

Schematic depiction of a zigzag nanoribbon with $M$ atoms in the vertical direction and $N$ atoms along the longitudinal direction. Semi-infinite doped graphene ribbons are considered as right and left metallic leads coupled to the finite ribbon.

Figure 2

Magnetic moment of the edge atoms of a finite zigzag nanoribbon with $M=8$ and (a) $N=10$ and (b) $N=30$. Different gate voltages are considered: $V_g=0.1t$, $0.2t$, and $0.3t$.

Figure 3

Spectral densities associated with spin waves projected at the up edge for nanoribbons with $M=8$, (a) $N=\left(\text{a}\right)10$, (b) 15, (c) 20, (d) 25, and (e) 30 atoms, under a gate voltage of $V_g=0.1t$ applied to the leads.

Figure 4

(a) Excitation energy as a function of the spin-wave mode index (normal mode) for nanoribbons with different lengths and (b) linewidth as a function of the nanoribbon length (given in terms of numbers of atoms) for the first mode. Inset in (b) shows the linewidths for different modes.

Figure 5

Spectral density for a nanoribbon with $M=8$ and $N=30$ atoms and different gate voltages: (a) $V_g=0.1t$, (b) $V_g=0.2t$, and (c) $V_g=0.3t$. The vertical dashed lines mark the shifts of the spin excitation energies due to the gate voltage.

Figure 6

(a) Linewidth and (b) Hartree-Fock spectral density dependences on the coupling intensity between the finite graphene ribbon and the leads for a zigzag nanoribbon with $M=8$ and $N=30$ atoms and gate voltage $V_g=0.1t$. Inset of both panels: dependence of the first spin-wave excitation mode.

Figure 7

(a) Eigenvectors of the zero-frequency mean-field susceptibility for zigzag nanoribbons with $M=8$ and $N=30$ with zero doping. (b) Corresponding eigenvalues as a function of the spin-wave mode.

Figure 8

Top to bottom: first eigenvector (full symbol) of the zero-frequency mean-field susceptibility for zigzag nanoribbons with $M=8$ and $N=30$ with zero doping (top panel) and $0.001,0.002,$ and $0.003$ extra electrons/atom. Open symbols are the magnetic moment results for the corresponding doped nanoribbons.