REFERENCES

[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. Vukmirovic, V. M. Stojanovic, and M. Vanevic,

Phys. Rev. B 81, 041408(R) (2010).

[5] V. M. Stojanovic, N. Vukmirovic, and C. Bruder,

Phys. Rev. B 82, 165410 (2010).

[6] J. Bai et al., Nat. Nanotechnol. 5, 190 (2010).

CONCLUSIONS & OUTLOOK

• 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 SiO2.

MASS ENHANCEMENT

The phonon-induced renormalization is character-

ized by the quasiparticle weight at the conduction

band-bottom Zc(k = 0), which we evaluate using

Rayleigh-Schrodinger perturbation theory.

8 10 12 14 16 18 203.0

3.2

3.4

3.6

3.8

4.0

4.2

R = 5

(a)

4NNFC VFF

-1 c

L

12 14 16 18 202.6

2.7

2.8

2.9

3.0

3.1

-1 c

R = 7

(b)

L

4NNFC VFF

FIG. 6: The inverse quasiparticle weights Z−1c (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(meff

m∗e

)α=

Z−1c (k = 0)

1 +∂

∂εc(kα)ReΣc(kα, ω)

∣∣kα=0,ω=Ec(0)

.

is rather large. Its anisotropy is determined by that

of the bare-band mass rather than by phonon-related

effects.

ELECTRON-PHONON COUPLING

Dominant mechanism of electron-phonon interac-

tion in all sp2-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(∆ucc) = t + α∆ucc, 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

Hep =α

2

∑R,m,δ,λ

(a†R+dm+δ

aR+dm+ H.c.

)×

[uλ,R+dm+δ − uλ,R+dm

]· δ .

δ ≡ δ/‖δ‖ is the unit vector in the direction of δ,

uλ,R+dmis the phonon (branch λ) normal coordi-

nate of an atom at R + dm, and α = 5.27 eV/A is

the coupling constant. In momentum space

Hep =1√N

∑k,q,λ,n

γλnn(k,q)a

†n,k+qan,k(b

†−q,λ+ bq,λ),

where an,k annihilates an electron with quasimo-

mentum k in the n-th Bloch band and bq,λ a phonon

of branch λ with quasimomentum q. The function

γλnn(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 |γλcc(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.

PHONON 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).

0.00 0.05 0.10 0.15 0.200

20

40

60

80

100

0

20

40

60

80

100

! (eV)

4NNFCVFF

Dph(!

) (

meV

-1)

FIG. 3: The phonon density-of-states for the {L =

17, R = 5} graphene antidot lattice, obtained using

the 4NNFC and VFF models.

ELECTRONIC 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.

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5K M

K M

L=9R=3

E/t

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. Stojanovic1, Nenad Vukmirovic2, and C. Bruder1

1Department of Physics, University of Basel2Lawrence Berkeley National Laboratory, USA

POLARONIC SIGNATURES AND SPECTRAL PROPERTIESOF GRAPHENE ANTIDOT LATTICES