Abstract. The electronic properties of graphene may be changed from semimetallic to semiconducting by introducing perforations (antidots) in a periodic pattern. The properties of such graphene antidot lattices (GALs) have previously been studied using atomistic models, which are very time consuming for large structures. We present a continuum model that uses the Dirac equation (DE) to describe the electronic and optical properties of GALs. The advantages of the Dirac model are that the calculation time does not depend on the size of the structures and that the results are scalable. In addition, an approximation of the band gap using the DE is presented. The Dirac model is compared with nearest-neighbour tight-binding (TB) in order to assess its accuracy. Extended zigzag regions give rise to localized edge states, whereas armchair edges do not. We find that the Dirac model is in quantitative agreement with TB for GALs without edge states, but deviates for antidots with large zigzag regions.

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Inspired by recent experimental realizations of monolayer Fe membranes in graphene perforations, we perform ab initio calculations of Fe monolayers and membranes embedded in graphene in order to assess their structural stability and magnetization. We demonstrate that monolayer Fe has a larger spin magnetization per atom than bulk Fe and that Fe membranes embedded in graphene exhibit spin magnetization comparable to monolayer Fe. We find that free-standing monolayer Fe is structurally more stable in a triangular lattice compared to both square and honeycomb lattices. This is contradictory to the experimental observation that the embedded Fe membranes form a square lattice. However, we find that embedded Fe membranes in graphene perforations can be more stable in the square lattice configuration compared to the triangular. In addition, we find that the square lattice has a lower edge formation energy, which means that the square Fe lattice may be favored during formation of the membrane.

Abstract. In order to use graphene for semiconductor applications, such as transistors with high on/off ratios, a band gap must be introduced into this otherwise semimetallic material. A promising method of achieving a band gap is by introducing nanoscale perforations (antidots) in a periodic pattern, known as a graphene antidot lattice (GAL). A graphene antidot barrier (GAB) can be made by introducing a 1D GAL strip in an otherwise pristine sheet of graphene. In this paper, we will use the Dirac equation (DE) with a spatially varying mass term to calculate the electronic transport through such structures. Our approach is much more general than previous attempts to use the Dirac equation to calculate scattering of Dirac electrons on antidots. The advantage of using the DE is that the computational time is scale invariant and our method may therefore be used to calculate properties of arbitrarily large structures. We show that the results of our Dirac model are in quantitative agreement with tight-binding for hexagonal antidots with armchair edges. Furthermore, for a wide range of structures, we verify that a relatively narrow GAB, with only a few antidots in the unit cell, is sufficient to give rise to a transport gap.

Bottom-up fabrication of graphene antidot lattices (GALs) has previously yielded atomically precise structures with sub-nanometer periodicity. Focusing on this type of experimentally realized GAL, we perform density functional theory calculations on the pristine structure as well as GALs with edge carbon atoms substituted with boron or nitrogen. We show that p-and n-type doping levels emerge with activation energies that depend on the level of hydrogenation at the impurity. Furthermore, a tight-binding parameterization together with a Green's function method are used to describe more dilute doping.