Electronic structure and magnetic properties of graphitic ribbons

First-principles calculations are used to establish that the electronic structure of graphene ribbons with zigzag edges is unstable with respect to magnetic polarization of the edge states. The magnetic interaction between edge states is found to be remarkably long ranged and intimately connected to the electronic structure of the ribbon. Various treatments of electronic exchange and correlation are used to examine the sensitivity of this result to details of the electron-electron interactions, and the qualitative features are found to be independent of the details of the approximation. The possibility of other stablization mechanisms, such as charge ordering and a Peierls distortion, are explicitly considered and found to be unfavorable for ribbons of reasonable width. These results have direct implications for the control of the spin-dependent conductance in graphitic nanoribbons using suitably modulated magnetic fields.

Figure 1

(Color online) A monohydrogenated ribbon of width $N=5$ along $y$. The system is periodic only along $x$ and the dashed lines delimit the periodic unit cell of length $a$.

Figure 2

(Color online) The $N=5$ ribbon with its zigzag edges terminated by an alternation of methylene groups and H atoms. The dashed lines represent the unit cell.

Figure 3

Relevant portion of the electronic band structure of (a) a nonmagnetic, $N=10$, monohydrogenated ribbon, and (b) a graphene sheet representated in the unit cell of the ribbon. Dashed lines represent “edge” bands and the dotted line is at the Fermi level.

Figure 4

(Color online) Electron density of a nonmagnetic, monohydrogenated, $N=10$ ribbon contributed by (a) the states near the Fermi level and (b) the rest of the occupied $\pi$ states. The range of isovalues is $[0:0.028]e∕{\mathrm{\AA }}^3$ in case (a) and $[0:0.12]e∕{\mathrm{\AA }}^3$ in case (b).

Figure 5

(Color online) Wave-function symmetry at the $k=\pi ∕a$ point for a ribbon with $N=5$ (a) and for trans-polyacetylene (b). The circles represent nonbonding orbitals at the Fermi level while the ovals depict those bulk states that are about $3\mathrm{eV}$ below the Fermi level [see Fig. 3a]. The phase of the orbitals is represented by filled $(+)$ and empty $(-)$ symbols.

Figure 6

The energy gap between the highest occupied and lowest unoccupied bulk states of a nonmagnetic, monohydrogenated ribbon as a function of the ribbon’s width.

Figure 7

(Color online) Isovalue surfaces of the spin density for the antiferromagnetic case (a) and ferromagnetic case (b). The dark grey (red online) surface represents spin up density and the black (blue online) surface spin down density. The range of isovalues is $[-0.28:0.28]{\mu }_B∕{\mathrm{\AA }}^3$ in case (a) and $[-0.09:0.28]{\mu }_B∕{\mathrm{\AA }}^3$ in case (b).

Figure 8

The energy difference (per unit cell) between the nonmagnetic and antiferromagnetic case as a function of $N$.

Figure 9

Spin-polarized band structure of a $N=10$ ribbon. The antiferromagnetic case is shown in the upper panels and ferromagnetic case in the lower panels.

Figure 10

The magnetic moment per unit cell as a function of the ribbon width in the ferromagnetic state.

Figure 11

The magnetic interaction strength as a function of ribbon’s width. Empty circles refer to the monohydrogenated ribbon and are fitted using a power-law decay (dotted line) of $1∕N^{1.82}$. The plus and triangle symbols refer to the methylene-group terminated and nonterminated ribbons whose decay laws are $1∕N^{1.89}$ and $1∕N^{1.86}$, respectively.

Figure 12

Magnetic interaction strength as a function of ribbon’s width within PBE (plus) and LSDA (circle). The decay power laws are, respectively, $1∕N^{1.44}$ and $1∕N^{1.39}$.