Quantum Nature of Edge Magnetism in Graphene

It is argued that the subtle crossover from decoherence-dominated classical magnetism to fluctuation-dominated quantum magnetism is experimentally accessible in graphene nanoribbons. We show that the width of a nanoribbon determines whether the edge magnetism is on the classical side, on the quantum side, or in between. In the classical regime, decoherence is dominant and leads to static spin polarizations at the ribbon edges, which are well described by mean-field theories. The quantum Zeno effect is identified as the basic mechanism which is responsible for the spin polarization and thereby enables the application of graphene in spintronics. On the quantum side, however, the spin polarization is destroyed by dynamical processes. The great tunability of graphene magnetism thus offers a viable route for the study of the quantum-classical crossover.

Figure 1

Special ribbon geometry which allows for a controlled mapping to a spin-$\frac{1}{2}$ quantum Heisenberg model with a single spin located on each zigzag segment. The effective spin-spin interactions are ferromagnetic (antiferromagnetic) along (across) the edges and sketched here for one reference spin.

Figure 2

(a) Spin-spin correlation length $\xi$ and spin gap $\mathrm{\Delta }$ for different widths $W$, calculated for $12\mu \mathrm{m}$ long ribbons at temperature $T=10\mathrm{mK}$ (errors are smaller than symbol size). Open circles show results for twice as long ribbons (i.e., $24\mu \mathrm{m}$). (b) Spin-spin correlations $⟨s_{xA}^z{s}_{x^{\prime }A}^z⟩$ along the edge for different widths $W$. For comparison, the black line shows the constant polarization as expected from mean-field theory.

Figure 3

Lifetime ${\tau }_{\mathrm{qd}}$ of the Néel state in the Lieb-Mattis approximation for different ribbon dimensions. $L$ is the number of effective edge spins along the edge. For convenience, the width $W$ of the ribbon is given in nm in the upper abscissa scale. On the right side an example ribbon with aspect ratio 2 ($L=4\mathrm{unit}\mathrm{cells}$; $W=12\mathrm{nm}$) is shown. For different $L$ the ribbon’s aspect ratio is invariant. Periodic boundary conditions in the $x$ direction are assumed. The edge spins are indicated as blue ellipses. The inset shows the exact time evolution $D_{\mathrm{st}}(t)$ for $L=2,4,6,\dots ,150$. The time axis is rescaled by the revival time $t_r=2\pi /J^{\mathrm{AF}}$.