Dynamical Signatures of Edge-State Magnetism on Graphene Nanoribbons

We investigate the edge-state magnetism of graphene nanoribbons using projective quantum Monte Carlo simulations and a self-consistent mean-field approximation of the Hubbard model. The static magnetic correlations are found to be short ranged. Nevertheless, the correlation length increases with the width of the ribbon such that already for ribbons of moderate widths we observe a strong trend towards mean-field-type ferromagnetic correlations at a zigzag edge. These correlations are accompanied by a dominant low-energy peak in the local spectral function and we propose that this can be used to detect edge-state magnetism by scanning tunneling microscopy. The dynamic spin structure factor at the edge of a ribbon exhibits an approximately linearly dispersing collective magnonlike mode at low energies that decays into Stoner modes beyond the energy scale where it merges into the particle-hole continuum.

Figure 1

Real-space spin-spin correlation function along the zigzag edge of ribbons. Points are the QMC and MFT data and lines are the fits with Eq. (2). Inset: Geometry of the graphene nanoribbon with zigzag edge, $L=6$, $W=4$, and lattice size $N=2\times L\times W=48$. The differently shaded circles represent the positive and negative values of the local magnetization $⟨S_i^z⟩=\frac{1}{2}⟨n_{i,\uparrow }-n_{i,\downarrow }⟩$, calculated from MFT at $U/t=2$. Radii are proportional to the magnitude.

Figure 2

Spectral functions of a width $W=4$ ribbon at $U/t=1$. QMC data are for a ribbon of length $L=30$ whereas MFT corresponds to the thermodynamic limit. The inset shows QMC results for the momentum-resolved spectral function $A(q,\omega )$ along the zigzag edge, while the main panel shows the local spectral function $A(\omega )$ subject to a Lorentzian broadening $\Delta \omega =0.02t$. Note the prominent low-energy peak in $A(\omega {)}_{\mathrm{edge}}$ at $\omega /t\approx 0.1$ which is absent in $A(\omega {)}_{\mathrm{bulk}}$.

Figure 3

(a) Single-particle spectral function $A(\omega {)}_{\mathrm{edge}}$ for several values of $U/t$ and a ribbon of width $W=48$. (b) Comparison of the position ${\omega }_{\max }$ of the maximum of $A(\omega {)}_{\mathrm{edge}}$ between MFT for $W=4$, 48 and QMC for $W=4$ and $L=30$. (c) $A(q,\omega {)}_{\mathrm{edge}}$ for a ribbon with $W=48$ at $U/t=2$. Unless stated otherwise, results are based on MFT for long ribbons and a Lorentzian broadening $\Delta \omega =0.02t$.

Figure 4

QMC results for the spin spectral function $S(q,\omega )$ along the edge of a zigzag ribbon with $L=30$, $W=4$, and $U/t=2$. The inset shows the linewidth (FWHM) of the dominant mode in $S(q,\omega )$ as a function of momentum $q$.