Role of direct exchange and Dzyaloshinskii-Moriya interactions in magnetic properties of graphene derivatives: ${\mathrm{C}}_2\mathrm{F}$ and ${\mathrm{C}}_2\mathrm{H}$

According to Lieb's theorem the ferromagnetic interaction in graphene-based materials with bipartite lattice is a result of disbalance between the number of sites available for $p_z$ electrons in different sublattices. Here we report on another mechanism of the ferromagnetism in functionalized graphene that is the direct exchange interaction between spin orbitals. By the example of the single-side semihydrogenated (${\mathrm{C}}_2\mathrm{H}$) and semifluorinated (${\mathrm{C}}_2\mathrm{F}$) graphene we show that such a coupling can partially or even fully compensate antiferromagnetic character of indirect exchange interactions reported earlier [Phys. Rev. B 88, 081405(R) (2013)]. As a result, ${\mathrm{C}}_2\mathrm{H}$ is found to be a two-dimensional material with the isotropic ferromagnetic interaction and negligibly small magnetic anisotropy, which prevents the formation of the long-range magnetic order at finite temperature in accordance with the Mermin-Wagner theorem. This gives a rare example of a system where direct exchange interactions play a crucial role in determining a magnetic structure. In turn, ${\mathrm{C}}_2\mathrm{F}$ is found to be at the threshold of the antiferromagnetic-ferromagnetic instability, which in combination with the Dzyaloshinskii-Moriya interaction can lead to a skyrmion state.

Figure 1

(a) Total densities of states and band structures of ${\mathrm{C}}_2\mathrm{F}$ and ${\mathrm{C}}_2\mathrm{H}$ calculated by using LDA+SO method and their comparison with the one-band model. (b) Band splitting $\mathrm{\Delta }E$ due to spin-orbit coupling calculated for the one-band model.

Figure 2

Wannier functions describing the band at the Fermi level in ${\mathrm{C}}_2\mathrm{F}$ (a) and (c) and ${\mathrm{C}}_2\mathrm{H}$ (b) and (d). Red sphere denotes the center of the Wannier orbital.

Figure 3

Schematic representation of the hopping paths in the triangular model for ${\mathrm{C}}_2\mathrm{F}$ and ${\mathrm{C}}_2\mathrm{H}$. The gray spheres denote the Wannier functions centered at nonbonded carbon atoms.

Figure 4

Schematic representation of the Dzyaloshinskii-Moriya vectors in ${\mathrm{C}}_2\mathrm{F}$. Light and dark red arrows denote the Dzyaloshinskii-Moriya vectors with positive and negative $z$ components, respectively.

Figure 5

The ferromagnetic (left) and ${120}^{\circ }$ Néel (right) solutions obtained within the Hartree-Fock approximation.

Figure 6

Spin configuration (a), static spin structure factors (b), and local chirality (c) of the skyrmion phase realized in ${\mathrm{C}}_2\mathrm{F}$. Spin components in the $xy$ plane are indicated with black arrows.