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-effecttransistor
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 ENHANCEMENTThe 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 Z1c (k = 0)
for the {L,R = 5} [(a)] and {L,R = 7} [(b)]graphene antidot
lattices.
The e-ph mass enhancement in direction = x, y(meffme
)=
Z1c (k = 0)
1 +
c(k)Rec(k, )
k=0,=Ec(0)
.
is rather large. Its anisotropy is determined by that
of the bare-band mass rather than by phonon-related
effects.
ELECTRON-PHONON COUPLINGDominant mechanism of electron-phonon
interac-
tion in all sp2-bonded carbon-based systems is the
modulation of -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 orbitals. This renders -electron
hopping integrals dynamically bondlength-dependent
t(ucc) = t + 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 =
2
R,m,,
(aR+dm+aR+dm + H.c.
)
[u,R+dm+ u,R+dm
] .
/ is the unit vector in the direction of ,u,R+dm is the phonon
(branch ) normal coordi-
nate of an atom at R + dm, and = 5.27 eV/A is
the coupling constant. In momentum space
Hep =1N
k,q,,n
nn(k,q)an,k+qan,k(b
q,+ bq,),
where an,k annihilates an electron with quasimo-
mentum k in the n-th Bloch band and bq, a phonon
of branch with quasimomentum q. The function
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 |cc(k =0,q)| of the
electron-phonon vertex functions fora 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 SPECTRAThe 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 usingthe 4NNFC and VFF
models.
ELECTRONIC STRUCTUREWe study band structure of antidot lattices
with
300 1600 atoms per unit cell, using a nearest-neighbor
tight-binding model for 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
