Magnetism in amorphous carbon

We investigate reported ferromagnetism in a new form of amorphous carbon. We use spin constrained first-principles simulations to obtain amorphous carbon structures with the desired magnetization. We show the existence of $s{p}^2$-like threefold coordinated carbon atoms plays an important role in causing magnetism in amorphous carbon. We predict detailed geometries of threefold carbon atoms that induce the magnetic order in amorphous carbon.

Figure 1

(a) Structure and spin charge density of amorphous carbon with constrained magnetization of $0.4{\mu }_B$/atom and density of $3.4{\mathrm{g}/\mathrm{cm}}^3$. Gray and orange spheres represent fourfold and threefold coordinated carbon atoms, respectively. Here the bonding threshold distance between two carbon atoms is 1.8 Å. A green isosurface illustrates the spin charge density. Schematic illustrations of (b) a threefold carbon atom surrounded by fourfold atoms and (c) two threefold atoms bonded but their $p$ orbitals are ${90}^{\circ }$ rotated relative to each other.

Figure 2

Distribution of local magnetic moments at 216 carbon atomic sites in amorphous carbon ($0.4{\mu }_B$/atom and 3.4 g/${\mathrm{cm}}^3$). Green and red circles represent threefold and fourfold coordinated carbon atoms, respectively. The local magnetic moments are estimated based on the Lowdin charge analysis of spin up and down charge densities.

Figure 3

Structure and spin charge density of amorphous carbon with constrained magnetization of 0.1 ${\mu }_B$/atom. The mass density is $3.4{\mathrm{g}/\mathrm{cm}}^3$.

Figure 4

Radial distribution functions of $3.4{\mathrm{g}/\mathrm{cm}}^3$ amorphous carbon with different constrained magnetization. Structures are relaxed under nonmagnetic (black lines) and constrained magnetization of 0.1 (green), 0.2 (blue), and $0.4{\mu }_B$/atom (red). Two vertical dashed lines indicate the optimized bond lengths of graphene (1.424 Å) and diamond (1.540 Å) using the same computational method. The inset shows the magnified view around the first peak.

Figure 5

Structure and spin charge density of amorphous carbon (mass density of $2.6{\mathrm{g}/\mathrm{cm}}^3$) with constrained magnetization of $0.4{\mu }_B$/atom. Red spheres represent twofold coordinated carbon atoms.

Figure 6

Spin charge density of amorphous carbon obtained with normal spin-polarized calculations without a magnetic constraint (density of $3.4{\mathrm{g}/\mathrm{cm}}^3$). The initial structure is $0.1{\mu }_B$/atom constrained structure and relaxed without a constraint. Red and blue isosurfaces represent majority and minority spin charge densities, respectively. See Fig. 3 for comparison with the spin-constrained structure.

Figure 7

(a) Relative energy as a function of rotation angle of a $p$ orbital in a molecule constructed from amorphous carbon. Green and red lines represent antiferromagnetic and ferromagnetic phases, respectively. The energy is measured from the total energy of the nonmagnetic phase at $0^{\circ }$. Here $0^{\circ }$ indicates the original structure without modification, which appears in the amorphous structure. Structures and spin charge densities at rotation angles of (b) ${60}^{\circ }$ and (c) ${150}^{\circ }$. Gray and white spheres represent carbon and hydrogen atoms. The structures are not relaxed and the only difference is the rotation of the $p$ orbital.