Tunable edge magnetism at graphene/graphane interfaces

We study the magnetic properties of graphene edges and graphene/graphane interfaces under the influence of electrostatic gates. For this an effective one-dimensional low-energy theory for the edge states, which is derived from the Hubbard model of the honeycomb lattice, is used. We first study the edge-state model in a mean-field approximation for the Hubbard Hamiltonian and show that it reproduces the results of the two-dimensional lattice theory. Quantum fluctuations around the mean-field theory of the effective one-dimensional model are treated by means of the bosonization technique in order to check the stability of the mean-field solution. We find that edge magnetism at graphene/graphane interfaces can be switched on and off by means of electrostatic gates. We describe a quantum phase transition between an ordinary and a ferromagnetic Luttinger liquid—a realization of itinerant one-dimensional ferromagnetism. This effect may provide means to experimentally discriminate between edge magnetism or disorder as the reason for a transport gap in very clean graphene nanoribbons.

Figure 1

(Color online) Lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ and nearest-neighbor vector $\mathbf{\delta }$ at an $\alpha$ edge. The coordinate $n\equiv n_2$ identifies the position perpendicular to the edge direction. The full (green) ellipses indicate the bulk unit cells $\left(n\ge 1\right)$ while the dashed (red) circles indicate the truncated unit cell at the $\alpha$ edge $\left(n=0\right)$. The sublattice index $s=\text{A}$ and B is also shown. The left part of the figure shows a lattice with an $\alpha$ edge and the right part shows a $\beta$ edge.

Figure 2

(Color online) Noninteracting band structure of a narrow $\alpha \beta$ ribbon (20 nm wide) with graphane terminations. The calculation is based on a tight-binding model which takes into account the $\pi$ and $\sigma$ orbitals of the carbon sites (Ref. 17). The dashed line indicates a typical Fermi energy (see text).

Figure 3

(Color online) Mean-field result for the edge magnetization $M_e$ as a function of the bandwidth $\Delta$ of the edge state. The topmost, thick (red) line represents the result for half filling. The other lines show the results for increasing the filling from 1/2 (topmost line, red) to 0.85 (bottom line, blue). The dashed vertical lines indicate the critical bandwidth obtained from the energy comparison of polarized and unpolarized states at half filling.

Figure 4

(Color online) Mean-field single-particle dispersions of the edge states for the different spin species. The edge-state filling has been chosen $\overline{n}=\frac{1}{2}$. The bandwidth is slightly below the critical value, as indicated by the vertical line in the inset. The horizontal line indicates the Fermi energy.

Figure 5

(Color online) Fermi levels in the nonmagnetic ground state and in the magnetic ground state where the Fermi momenta $k_{Fr\sigma }$ are split by $2\Delta k$ in $k$ direction.

Figure 6

Critical bandwidth ${\left[\Delta /U\right]}_{\text{crit}\text{.}}$ as a function of the edge-state filling. 0 stands for a completely empty edge band ($k_{Fr}=\pi$ in the nonmagnetic configuration) and 1 for a completely filled edge band ($k_{Fr}=\pi +r\pi /3$ in the nonmagnetic configuration).

Figure 7

Mean-field configuration of the nonmagnetic regime. Both spin species are equally occupied (indicated by the bold black lines).

Figure 8

(Color online) The prefactor $u_s/K_s$ of the term ${\left({\partial }_x{\varphi }_s\right)}^2$ in the bosonized Hamiltonian near the Stoner point, as a function of $\Delta /U$ for different fillings $\overline{n}$ close to $\overline{n}=\frac{1}{2}$.

Figure 9

(Color online) Mean-field configuration in the weak ferromagnetic regime. The bold lines represent the occupied states and the thin lines represent the unoccupied states. The red arrows indicate the backscattering process, which is not momentum conserving here.

Figure 10

(Color online) Comparison of the edge magnetizations extracted from the result of the numerical calculation and from the effective model. The solid black line is the edge magnetization from the effective model. The solid gray lines show $M_e^{\text{num}}$ of an $\alpha \alpha$ ribbon with different widths (50, 100, and 200 unit cells in the transverse direction). The lower abscissas of both parts of the figure shows $\Delta /U$ where the bandwidth parameter has been fixed to $\Delta =0.02\text{eV}$ in part (a) and $\Delta =0.2\text{eV}$ in part (b). The corresponding Hubbard interaction strength $U$ is shown in the upper abscissas. The dashed lines show the polarization of the edge states $M_e^{\text{num},\text{es}}$.