Large and stable band gaps in spin-polarized graphene antidot lattices

Introducing a periodic array of holes, i.e., an antidot lattice, in a graphene sheet has been suggested as a route towards the tantalizing objective of “opening the gap” in this otherwise zero-gap semiconductor. Combining density-functional and mean-field Hubbard tight-binding methods, we study the effect of spin polarization on graphene antidot lattices (GALs). Focusing on GALs with extended zigzag edges, we systematically investigate the geometry dependence of spin polarization, electronic structure, and band gaps. A scaling law for the band gap is established, demonstrating marked deviations from that of circular holes without spin polarization. Furthermore, we estimate the robustness of the magnetic ordering against raised temperature and doping and, finally, consider how the optical properties are modified by spin polarization. Our results demonstrate that large, stable band gaps are expected for a range of geometries.

Figure 1

Excerpt of a triangular GAL with hexagonal holes. $W$, $S$, and $L$ represent the subribbon width, edge length, and unit-cell size, respectively, whereas ${\vec{a}}_1$ and ${\vec{a}}_2$ are GAL lattice vectors parallel to carbon-carbon bonds. The edges are zigzags of alternating carbon atoms from the $A$ and $B$ sublattices illustrated by blue and red atoms, respectively. The dashed line indicates the unit cell.

Figure 2

Spin polarization of $\{W=2,S=8\}$ triangular GAL, comparing the $z$-integrated DFT density with the on-site HTB density. Note that the DFT density is plotted as a contour plot, with contour lines at values shown in the left bar, while the HTB model is a scatter plot, with dot sizes and colors indicating the on-site Hubbard spin-density difference.

Figure 3

Density of states calculated using our HTB model for a few triangular GALs, with subribbon width $W=2$ and varying edge length $S$, projected onto the edge atoms of the $A$ sublattice (red and black curves) in addition to the total density of states (green curve).

Figure 4

Band gap (top) and maximum spin polarization (bottom) of triangular GALs of varying dimensions calculated using the Hubbard model. The band gap is seen to vanish with increasing hole size for fixed $W$ if spin is neglected (dash-dotted lines). However, the Hubbard interaction term introduces a finite band gap resembling those of ZGNRs, shown in the shaded region, for large hole sizes. A band-gap increase is seen at edge lengths $S$ larger than 5 for all structures, matching the onset of spin polarization. DFT results are shown in the insets, where the onset of spin polarization is in full agreement with the HTB model. Additionally, the DFT and HTB band gaps are in reasonable agreement, with the position of the band-gap minimum near $S=5$ reproduced by both methods.

Figure 5

Comparison of the tight-binding models including (HTB) and neglecting (TB) Hubbard interaction with scaling rule of Ref. 9 (full black line).

Figure 6

Band gaps of a few rotated triangular GALs calculated using the HTB model. Structures which are metallic in the absence of Hubbard interaction are denoted by circles as indicated by the “M” in the legend, whereas squares indicate semiconducting structures, denoted by “S” in the legend. The full and dashed lines indicate results including and excluding Hubbard interaction, respectively. The inset illustrates the unit cell (full black line) of the rotated triangular structure, together with the side length parameters $S$ and $L$, in addition to a width parameter $W=L-2S$.

Figure 7

Band gap and maximum spin polarization of several triangular GALs at varying temperatures calculated using the HTB model. All species tend to depolarize at temperatures larger than approximately 1000 K, regardless of band gap.

Figure 8

Band gap and maximum spin polarization of triangular GALs calculated for various doping levels and structure parameters using the HTB model.

Figure 9

Optical response of various triangular $W=2$ GALs from the HTB model in units of the static conductivity of graphene, ${\sigma }_0=e^2/\left(4\hslash \right)$.