Polaronic signatures and spectral properties of graphene antidot lattices

We explore the consequences of electron-phonon (e-ph) coupling in graphene antidot lattices (graphene nanomeshes), i.e., triangular superlattices of circular holes (antidots) in a graphene sheet. They display a direct band gap whose magnitude can be controlled via the antidot size and density. The relevant coupling mechanism in these semiconducting counterparts of graphene is the modulation of the nearest-neighbor electronic hopping integrals due to lattice distortions (Peierls-type e-ph coupling). We compute the full momentum dependence of the e-ph vertex functions for a number of representative antidot lattices. Based on the latter, we discuss the origins of the previously found large conduction-band quasiparticle spectral weight due to e-ph coupling. In addition, we study the nonzero-momentum quasiparticle properties with the aid of the self-consistent Born approximation, yielding results that can be compared with future angle-resolved photoemission spectroscopy measurements. Our principal finding is a significant e-ph mass enhancement, an indication of polaronic behavior. This can be ascribed to the peculiar momentum dependence of the e-ph interaction in these narrow-band systems, which favors small phonon momentum scattering. We also discuss implications of our study for recently fabricated large-period graphene antidot lattices.

Figure 1

(Color online) A finite segment of the graphene antidot lattice $\left\{L,R\right\}$ with circular antidots. Here ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the superlattice basis vectors; $L$ and $R$ are, respectively, the side length of the hexagonal unit cell and the antidot radius, both expressed in units of the graphene lattice constant $a\approx 2.46\text{Å}$. The vectors ${\mathbf{\delta }}_1$, ${\mathbf{\delta }}_2$, and ${\mathbf{\delta }}_3$ specify positions of the nearest neighbors of a carbon atom on sublattice $A$.

Figure 2

The phonon density of states for the (a) $\left\{L=9,R=5\right\}$ and (b) $\left\{L=13,R=7\right\}$ graphene antidot lattices, obtained using the 4NNFC (solid line) and VFF (dashed line) methods.

Figure 3

(Color online) The atomic displacement pattern corresponding to the zone-center phonon mode at energy $E_{\lambda }\approx 197.5\text{meV}$ of the $\left\{L=13,R=5\right\}$ graphene antidot lattice (top) and its zoomed version (bottom). The displaced (undisplaced) atomic positions are designated by the squares (dark circles).

Figure 4

(Color online) The atomic displacement pattern corresponding to the zone-center phonon mode at energy $E_{\lambda }\approx 197.5\text{meV}$ of the $\left\{L=15,R=7\right\}$ graphene antidot lattice (top) and its zoomed version (bottom). The displaced (undisplaced) atomic positions are designated by the squares (dark circles).

Figure 5

(Color online) The $\mathbf{q}$ dependence of the moduli $\left|{\gamma }_{cc}^{\lambda }\left(\mathbf{k}=0,\mathbf{q}\right)\right|$ of the bare e-ph vertex functions, for selected high-energy phonon branches $\lambda$, in the hexagonal Brillouin zones of the $\left\{L=13,R=5\right\}$ [(a) and (b)] and $\left\{L=15,R=7\right\}$ [(c) and (d)] graphene antidot lattices. The energies of the corresponding optical phonons at $\mathbf{q}=0$, as obtained using the VFF approach, are $E_{\lambda }\approx 197.5\text{meV}$ [(a), (c), and (d)] and $E_{\lambda }\approx 197.4\text{meV}$ [(b)].

Figure 6

Pictorial representation of the Dyson equation (top). Illustration of the SCBA self-energy, given by a sum over all noncrossing diagrams (bottom).

Figure 7

(Color online) The real and imaginary parts of the SCBA conduction-band electron self-energy in the (a) $\left\{L=13,R=5\right\}$ and (b) $\left\{L=15,R=7\right\}$ graphene antidot lattices for $\mathbf{k}=0.2\mathbf{b}$, where $\mathbf{b}=\left(0,1\right)\ast 4\pi /\left(3La\right)$.

Figure 8

(Color online) The SCBA electron propagators in the (a) $\left\{L=13,R=5\right\}$ and (b) $\left\{L=15,R=7\right\}$ graphene antidot lattices for $\mathbf{k}=0.2\mathbf{b}$, where $\mathbf{b}=\left(0,1\right)\ast 4\pi /\left(3La\right)$.