Density functional study of graphene antidot lattices: Roles of geometrical relaxation and spin

Graphene sheets with regular perforations, dubbed as antidot lattices, have theoretically been predicted to have a number of interesting properties. Their recent experimental realization with lattice constants below 100 nanometers stresses the urgency of a thorough understanding of their electronic properties. In this work, we perform calculations of the band structure for various hydrogen-passivated hole geometries using both spin-polarized density functional theory (DFT) and DFT based tight-binding (DFTB) and address the importance of relaxation of the structures using either method or a combination thereof. We find from DFT that all structures investigated have band gaps ranging from 0.2 to 1.5 eV. Band gap sizes and general trends are well captured by DFTB with band gaps agreeing within about 0.2 eV even for very small structures. A combination of the two methods is found to offer a good trade-off between computational cost and accuracy. Both methods predict nondegenerate midgap states for certain antidot hole symmetries. The inclusion of spin results in a spin-splitting of these states as well as magnetic moments obeying the Lieb theorem. The local-spin texture of both magnetic and nonmagnetic symmetries is addressed.

Figure 1

(Color online) The unit cell of the ${\left\{4,2\right\}}_{○}$(left) and ${\left\{6,5\right\}}_{△}$ (right) system. The hexagonally shaped unit cells are repeated in plane to form a honeycomb lattice of antidots. The carbon atoms (green/gray) are hydrogen terminated (white) along the hole edges.

Figure 2

(Color online) The band gaps for DFTB, DFT on DFTB-relaxed geometry, and DFT on unrelaxed geometry plotted versus pure DFT results. Points above (below) the dotted line are thus overestimated (underestimated) compared to pure DFT. Note, that DFTB calculates the electronic structure without spin and fails to predict a band gap for the ${\left\{6,5\right\}}_{△}$ structure.

Figure 3

(Color online) Band structures for the ${\left\{4,2\right\}}_{○}$ (left column) and ${\left\{6,5\right\}}_{△}$ (right column) systems using DFTB (upper panels) and DFT (lower panels). The dotted blue/dark gray curves in the lower panels are DFT results using structures relaxed with DFTB. The arrow pointing up (down) indicates that bands are filled with majority (minority) electrons only.

Figure 4

(Color online) Amplitudes of certain important states. (a,b): degenerate midgap states, (c): nondegenerate midgap state of ${\left\{6,5\right\}}_{△}$, (d): highest filled bands of ${\left\{6,3.6\right\}}_{○}$. All states are calculated at the $\Gamma$ point.

Figure 5

(Color online) The difference in majority spin and minority spin densities for the ${\left\{6,5\right\}}_{△}$ structure. Blue/dark gray (white) indicates surplus of majority(minority) spin.