Half-metallic graphene nanodots: A comprehensive first-principles theoretical study

A comprehensive first-principles theoretical study of the electronic properties and half-metallic nature of finite rectangular graphene nanoribbons is presented. We identify the bisanthrene isomer of the ${\mathrm{C}}_{28}{\mathrm{H}}_{14}$ molecule to be the smallest graphene derivative to present a spin-polarized ground state. Even at this quantum dot level, the spins are predicted to be aligned antiferromagnetically at the two zigzag edges of the system. As a rule of thumb, we find that zigzag graphene edges that are at least three consecutive units long will present spin polarization if the width of the system is $1\mathrm{nm}$ or wider. Room temperature detectability of the magnetic ordering is predicted for ribbons with zigzag edges $1\mathrm{nm}$ and longer. For the longer systems studied, spin wave structures appear in some high spin multiplicity states. Energy gap oscillations with the length of the zigzag edge are observed. The amplitude of these oscillations is found to be smaller than that predicted for infinite ribbons. The half-metallic nature of the ribbons under an external in-plane electric field is found to be preserved even for finite and extremely short systems.

Figure 1

The three sets of finite GNRs studied. The notation corresponds to the number of hydrogen atom passivating the edges. An $N\times M$ ribbon has $N$ hydrogens on the armchair edge and $M$ hydrogens on the zigzag edge.

Figure 2

(Color online) Isosurface spin densities of the antiferromagnetic ground state of the (a) $04\times 03$, (b) $06\times 03$, (c) $08\times 03$, (d) $06\times 08$, and (e) $08\times 07$ GNRs as obtained using the HSE06 functional and the $6\text{−}31{\mathrm{G}}^{**}$ basis set. The sorting of the atoms into zigzag, armchair, and seam regions is indicated in (d). (f) and (g) represent the diradical and hexaradical Clar structures of bisanthrene.

Figure 3

(Color online) Spin density isosurfaces of the $04\times 16$ GNR high spin multiplicity states as obtained at the $\mathrm{PBE}∕6\text{−}31{\mathrm{G}}^{**}$ level of theory. (a) Ferromagnetic arrangement of the $m_s=3$ state. (b) Mixed configuration of the $m_s=1$ state presenting a short spin wave on one of the zigzag edges.

Figure 4

(Color online) Energy differences between the antiferromagnetic (AF) ground state and the above lying higher spin multiplicity ferromagnetic (FM) or mixed state for the three sets of ribbons studied, as calculated by the local spin density approximation (upper left panel), PBE functional (upper right panel), and the HSE06 functional (lower left panel). The horizontal line represents $k_{B}T$ at room temperature. Notice the different energy scales obtained for the different functional calculations.

Figure 5

(Color online) HOMO-LUMO gap values for the three sets of ribbons studied, as calculated by the local spin density approximation (upper left panel), PBE functional (upper right panel), and the HSE06 functional (lower left panel). Notice the different energy scales obtained for the different functional calculations.

Figure 6

(Color online) Spin-polarized HOMO-LUMO gap dependence on the strength of an external in-plane electric field for three representative finite nanoribbons, as calculated by the local spin density approximation (upper left panel), PBE functional (upper right panel), and the HSE06 functional (lower left panel). Fixed geometries of the relaxed structures in the absence of the external field at each level of theory were used. Notice the different energy scales obtained for the different functional calculations.