Sensory-organ-like response determines the magnetism of zigzag-edged honeycomb nanoribbons

We present an analytical effective theory for the magnetic phase diagram for zigzag-edge terminated honeycomb nanoribbons described by a Hubbard model with an interaction parameter $U$. We show that the edge magnetic moment varies as $\ln U$ and uncover its dependence on the width $W$ of the ribbon. The physics of this owes its origin to the sensory-organ-like response of the nanoribbons, demonstrating that considerations beyond the usual Stoner-Landau theory are necessary to understand the magnetism of these systems. A first-order magnetic transition from an antiparallel orientation of the moments on opposite edges to a parallel orientation occurs upon doping with holes or electrons. The critical doping for this transition is shown to depend inversely on the width of the ribbon. Using variational Monte Carlo calculations, we show that magnetism is robust to fluctuations. Additionally, we show that the magnetic phase diagram is generic to zigzag-edge terminated nanostructures such as nanodots. Furthermore, we perform first-principles modeling to show how such magnetic transitions can be realized in substituted graphene nanoribbons.

Figure 1

Two possible magnetic states of zigzag-edged nanoribbon, (a) antiferromagnetic (AF) and (b) ferromagnetic (FM). In both (a) and (b) the size of the arrow at a site is proportional to the magnetic moment at the site; in both cases the moments are concentrated at the edge sites. (c) Magnetic phase diagram of zigzag nanoribbons for $U=1.5$ obtained from numerical mean-field calculations. Inset: The critical doping required for AF to FM transition is inversely proportional to the ribbon width.

Figure 2

(a) Width dependence of magnetic moment of the AF configuration for different values of $U$ obtained from the numerical mean-field calculations compared with the analytical result Eq. (11). Inset: The logarithmic dependence of the slope of $M_0^{\text{AF}}$ vs $a/W$ on $U$. Points are obtained from the numerical simulations, solid line is from Eq. (11). Results shown are for undoped ribbons.

Figure 3

Dependence of $C\left(U\right)$ on $U$. Critical doping ${\delta }_c$ for transition from AF to FM state is given by ${\delta }_c=C\left(U\right)\frac{a}{W}$ [see Eq. (18)].

Figure 4

(a) Mean-field AF configuration in a zigzag-edge terminated hexgonal nanodot $(U=1.5)$. (b) FM configuration obtained upon doping ($\delta =0.1$).

Figure 5

FM structure of the boron substituted graphene nanoribbon obtained from ab initio calculations. Dark atoms are borons and light ones are carbon. Hydrogen passivation of the edge atoms is also shown.