Ferromagnetism in armchair graphene nanoribbons

Due to the weak spin-orbit interaction and the peculiar relativistic dispersion in graphene, there are exciting proposals to build spin qubits in graphene nanoribbons with armchair boundaries. However, the mutual interactions between electrons are neglected in most studies so far and thus motivate us to investigate the role of electronic correlations in armchair graphene nanoribbon by both analytical and numerical methods. Here we show that the inclusion of mutual repulsions leads to drastic changes and the ground state turns ferromagnetic in a range of carrier concentrations. Our findings highlight the crucial importance of the electron-electron interaction and its subtle interplay with boundary topology in graphene nanoribbons. Furthermore, since the ferromagnetic properties sensitively depend on the carrier concentration, it can be manipulated at ease by electric gates. The resultant ferromagnetic state with metallic conductivity is not only surprising from an academic viewpoint, but also has potential applications in spintronics at nanoscale.

Figure 1

(Color online) (a) Armchair GNR of $L_x=4$ and $L_y=5$. Open edges are present at $y=1$ and $L_y$. The solid circles and squares represent sublattice $A$ and $B$, respectively. The shaded circles show the amplitudes of a local Wannier orbital of $E=t$ at $x=2$, with opposite signs indicated by light/dark colors. Band structures for the infinitely long GNR with (b) $L_y=5$ and (c) $L_y=7$ clearly show the pair of flat bands at $E=\pm t$.

Figure 2

(Color online) Doping dependence of the energy difference $\Delta \left(x_d,\delta S\right)$ between higher-spin and lowest-spin states in the armchair GNR with $L_x=2$ and $L_y=5$ for $U/t=4$. The squares, triangles, and diamonds represent the data for $\delta S=S-S_0=1$, 2, and 3, respectively.

Figure 3

(Color online) Spin polarization profile $⟨S^z\left(\mathbf{r}\right)⟩$ of the ferromagnetic ground state for the armchair GNR with $L_x=2$ and $L_y=5$. It is at the optimal doping $x_d^{\ast }=0.30$ with the interaction strength (a) $U/t=4$. (b) $U/t=0^{+}$. The values of $⟨S^z\left(\mathbf{r}\right)⟩$ are represented by the areas of the shaded circles. It is remarkable that the profiles with intermediate and weak interaction strength are almost identical. (c) The values of $⟨S^z\left(\mathbf{r}\right)⟩$ at each site for $U/t=0^{+},2,4,8$ represented by solid circles, squares, triangles, and diamonds.

Figure 4

(a) Band structure of the undoped, infinitely long armchair GNR with $L_y=5$. (b) Doping dependence of the spin polarization per unit cell. (c) Energy difference per unit cell $\Delta \left(x_h\right)$ between ferromagnetic and paramagnetic states as a function of the hole doping.

Figure 5

(Color online) (a) Spin-decomposed band structure of the infinitely long GNR with $L_y=5$ at the optimal hole doping $x_h^{\ast }\simeq 0.28$. (b) Top and (c) side views of the spin-density distribution (shown in gray color). The yellow (light) and red (dark) spheres denote C and H atoms, respectively.