R EFERENCES [1] T. G. Pedersen et al., Phys. Rev. Lett. 100, 136804 (2008). [2]J. Zimmermann, P. Pavone, and G. Cuniberti, Phys. Rev. B 78, 045410 (2008). [3]V. Perebeinos and J. Tersoff, Phys. Rev. B 79, 241409(R) (2009). [4] N. Vukmirovi´ c, V. M. Stojanovi´ c, and M. Vanevi´ c, Phys. Rev. B 81, 041408(R) (2010). [5] V. M. Stojanovi´ c, N. Vukmirovi´ c, and C. Bruder, Phys. Rev. B 82, 165410 (2010). [6] J. Bai et al., Nat. Nanotechnol. 5, 190 (2010). C ONCLUSIONS &O UTLOOK • In graphene antidot lattices, optical phonons play an important role; large mass enhancement ob- tained is a signature of polaronic behavior. • Future study of transport in graphene antidot lat- tices should include inelastic degrees of freedom. • To understand charge transport in a field-effect transistor geometry [6], one should study the in- terplay of Peierls-type coupling and long-range coupling at the interface between graphene anti- dot lattices and polar substrates such as SiO 2 . M ASS ENHANCEMENT The phonon-induced renormalization is character- ized by the quasiparticle weight at the conduction band-bottom Z c (k = 0), which we evaluate using Rayleigh-Schr¨ odinger perturbation theory. FIG. 6: The inverse quasiparticle weights Z -1 c (k = 0) for the {L, R =5} [(a)] and {L, R =7} [(b)] graphene antidot lattices. The e-ph mass enhancement in direction α = x, y m eff m * e α = Z -1 c (k = 0) 1+ ∂ ∂ε c (k α ) Re Σ c (k α ,ω ) k α =0,ω =E c (0) . is rather large. Its anisotropy is determined by that of the bare-band mass rather than by phonon-related effects. E LECTRON - PHONON COUPLING Dominant mechanism of electron-phonon interac- tion in all sp 2 -bonded carbon-based systems is the modulation of π -electron hopping integrals due to lattice distortions (Peierls-type coupling). Optical phonons modulate (elongate or contract) the in- plane C-C bond and thus alter the overlap between the out-of-plane π orbitals. This renders π -electron hopping integrals dynamically bondlength-dependent t(Δu cc )= t + αΔu cc , as illustrated in Fig. 4. FIG. 4: Illustration of Peierls-type coupling. In the tight-binding electron basis, the real space electron-phonon coupling Hamiltonian reads ˆ H ep = α 2 X R,m,δ ,λ ( ˆ a † R+d m +δ ˆ a R+d m +H.c. ) × ˆ u λ,R+d m +δ - ˆ u λ,R+d m · ¯ δ . ¯ δ ≡ δ /kδ k is the unit vector in the direction of δ , ˆ u λ,R+d m is the phonon (branch λ) normal coordi- nate of an atom at R + d m , and α =5.27 eV/ ˚ A is the coupling constant. In momentum space ˆ H ep = 1 √ N X k,q,λ,n γ λ nn (k, q)ˆ a † n,k+q ˆ a n,k ( ˆ b † -q,λ + ˆ b q,λ ) , where ˆ a n,k annihilates an electron with quasimo- mentum k in the n-th Bloch band and ˆ b q,λ a phonon of branch λ with quasimomentum q. The function γ λ nn (k, q) strongly depends on both k and q; for electrons at the bottom of the conduction band, it is largest for small phonon momenta [see Fig. 5(a)]. (a) (b) (c) (d) FIG. 5: The q-dependence of the moduli |γ λ cc (k = 0, q)| of the electron-phonon vertex functions for a conduction-band electron at k =0 and high- energy phonon branches in the Brillouin zones of the {L = 13,R =5} [(a),(b)] and {L = 15,R = 7} [(c),(d)] graphene antidot lattices. P HONON SPECTRA The phonon spectra of the {L, R =5} and {L, R = 7} families of lattices are computed using two mod- els that yield accurate results for graphene itself: the fourth-nearest-neighbor force-constant (4NNFC) model [2] and the valence force-field (VFF) model [3]. The highest optical-phonon energy is essentially inherited from graphene and only weakly depen- dent on L and R; this energy is 195.3 meV in the 4NNFC approach (197.5 meV in the VFF approach). FIG. 3: The phonon density-of-states for the {L = 17,R =5} graphene antidot lattice, obtained using the 4NNFC and VFF models. E LECTRONIC STRUCTURE We study band structure of antidot lattices with 300 - 1600 atoms per unit cell, using a nearest- neighbor tight-binding model for π electrons. FIG. 2: Typical band structure of graphene antidot lattices with circular antidots. Because of particle- hole symmetry inherent to the model, only bands above the Fermi level (E =0) are displayed. Graphene antidot lattices, superlattices of holes (an- tidots) in a graphene sheet, display a direct band gap whose magnitude can be controlled via the an- tidot size and density. For more details, see Ref. 1. FIG. 1: (a) Finite segment of a graphene antidot lattice; (b) hexagonal unit cell of the antidot lattice {L, R} with circular antidots. L and R are dimen- sionless numbers, lengths expressed in units of the graphene lattice constant a ≈ 2.46 ˚ A. Vladimir M. Stojanovi´ c 1 , Nenad Vukmirovi´ c 2 , and C. Bruder 1 1 Department of Physics, University of Basel 2 Lawrence Berkeley National Laboratory, USA P OLARONIC SIGNATURES AND SPECTRAL PROPERTIES OF GRAPHENE ANTIDOT LATTICES