Evaluation of the Green’s function of disordered graphene
W. Zhu,
1,2
Q. W. Shi,
1,2,
* X. R. Wang,
2,3,†
X. P. Wang,
1
J. L. Yang,
1
J. Chen,
4,5
and J. G. Hou
1
1
Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China,
Hefei 230026, China
2
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
3
School of Physics, Shandong University, Jinan, People’s Republic of China
4
Electrical and Computer Engineering, University of Alberta, Alberta, Canada T6G 2V4
5
National Research Council/National Institute for Nanotechnology, Alberta, Canada T6G 2M9

Received 4 August 2010; revised manuscript received 22 September 2010; published 11 October 2010
Accurate simulations of Green’s function and the self-energy function for noninteracting electrons in disor-
dered graphenes are performed. The fundamental physical quantities such as the elastic relaxation time

e
, the
phase velocity v
p
, and the group velocity v
g
are evaluated. New features around the Dirac point are revealed,
indicating that multiscattering-induced hybridization of Bloch states plays an important role in the vicinity of
the Dirac point.
DOI: 10.1103/PhysRevB.82.153405 PACS number
s: 81.05.ue, 71.23.k, 71.55.i
I. INTRODUCTION
Graphene, a single layer of graphite, has been intensively
studied in recent years for its many intriguing transport
properties.
1–3
Examples include minimum conductivity and
linear carrier density dependence of conductivity.
4,5
Elec-
trons in an ideal graphene are governed by the relativistic
massless Dirac equation and exhibit a linear-dispersion rela-
tion in the vicinity of the Dirac point, and zero density of
state at the Dirac point.
6
Among many relativistic effects,
Klein paradox
3
is arguably one of the most important effects
that makes Dirac electrons different from the Schrödinger
electrons in disordered systems.
7
An in-depth understanding of Dirac electrons in disor-
dered graphenes requires a better knowledge of the self-
energy function to extract such fundamental physical quanti-
ties as the phase velocity v
p
, the group velocity v
g
, and the
elastic relaxation time

e
. However, it is known
8
that the
accurate and reliable calculation is quite difficult and non-
trivial, although various approximations have been employed
in different theoretical studies. So far, almost all
calculations
9,10
on electron properties in disordered
graphenes were performed without fully considering the dis-
order effects. The wave nature of Dirac electrons is more
pronounced near the Dirac point because of very large elec-
tron wavelength there. It is known that interference of mul-
tiscattering leads to the weak localization and the Anderson
localization in conventional disordered electron systems.
11
Quantum interference also plays important roles in coherent
wave propagation through quasirandom
12
and random
media.
13,14
As it will be shown below, hybridization of Bloch
states near the Dirac point in realistic disordered graphenes is
essential in understanding the diffusion properties although
previous calculations
9,10
can capture the essential physics far
away from the Dirac point.
In this Brief Report, we present a systematic method for
the precise computation of Green’s function and the self-
energy of large size disordered graphenes. We extract accu-
rate

e
, v
p
, and v
g
values from the spectral function A
k ,E
derived from the self-energy function. Compared the results
from the self-consistent Born approximation
SCBA,itis
found that the

e
is overestimated by the SCBA in the strong
disorder regime. We show that both v
p
and v
g
deviate sig-
nificantly from their unrenormalized values and exhibit sub-
stantial energy dependence, especially near the Dirac point.
The effective group velocity is larger than the effective phase
velocity but substantially lower than its unrenormalized
value when the mixing of the different Bloch states is domi-
nant. Moreover, we generalize the Einstein relation to calcu-
late the conductivity of the disordered graphene.
II. MODEL AND METHOD
 electrons of undoped graphene can be modeled by a
tight-binding Hamiltonian on a honeycomb lattice of two
sites per unit cell, H
0
= t
ij
ij+H.c., where t=−2.7 eV is
the hopping energy. The corresponding eigenvalues and
eigenstates of H
0
near the Dirac point are E
k,
= v
F
0
k and
k=
kAe
i
k
kB /2, respectively.
4,15
Here, v
F
0
is the
unrenormalized Fermi velocity and  is the Planck’s con-
stant. A and B stand for A and B sublattices. 
k is the polar
angle of the momentum k and kA
B
=
1

N
A
B

r
A
B
e
ik·r
A
B
r
A
B
, where r
A
B
is the position vector
of A
B lattice and N
A
B
is the total A
B-lattice points. The
plus
minus sign denotes the conduction
valence band.
The Green’s function of the clean graphene is, in a diagonal
basis, G
0

k ,E=
1
E+i0
+
−v
F
0
k
k+k++
1
E+i0
+
+v
F
0
k
k−k−.A
weak pointlike disorder is introduced through V=
i
ii
with n
imp
randomly distributed impurity sites where the on-
site energy 
i
of each impurity can take −V
0
or V
0

measured
in the unit of t with equal probability. In this Brief Report,
V
0
is chosen V
0
3 to guarantee no resonant states around
the Dirac point.
16,17
A dimensionless parameter =
n
imp
V
0
2
A
c
2
v
F
0

2
can be used to characterize disorder strength. Here, A
c
is the
area of the unit cell. Our calculation shows that physical
quantities such as self-energy are only determined by the
parameter .
18
The ensemble-averaged Green’s function is defined as
G
k ,E= k
1
E+i
−H
0
−V
k, where the bar means the en-
PHYSICAL REVIEW B 82, 153405
2010
1098-0121/2010/82
15/153405
4 2010 The American Physical Society153405-1

semble average. It can be calculated by using the well-
developed Lanczos recursive method.
16,19,20
In order to ob-
tain an accurate ensemble-averaged Green’s function near
the Dirac point with high-energy resolution,
21
a large sample
containing N=L
x
L
y
6.0 millions carbon atoms
2400
2400 and over 1000 ensembles are used in our simulation,
where L
x
and L
y
is the number of atoms in x and y directions,
respectively. The large lattice samples guarantee that the cal-
culated Green’s function is free of finite-size errors.
22
The
periodic boundary condition has been used and the wave
vectors satisfy k
x
=n
x
4 /3aL
x
and k
y
=n
y
4 /

3aL
y
, where
n
x
y
is an integer and a is the lattice constant.
III. SELF-ENERGY FUNCTION
Figure 1 shows the calculated
a real and
b imaginary
parts of the self-energy function for n
imp
/N=10% and V
0
=0.5
squares and 2.0
circles, respectively. The self-energy
function is defined based on the Dyson’ equation G
k ,E
=G
0

k ,E+G
0

k ,E
k ,EG
k ,E, thus 
k ,E
=G
0
−1

k ,E−G
−1

k ,E.
8
In principle, the self-energy func-
tion depends on energy E and wave vector k. However, our
simulation shows that the self-energy function is not sensi-
tive to wave vector k while the one-particle Green’s function
depends on both E and k. This is not surprising since the
scatterer size is much smaller than the electron wavelength
near the Dirac point so that the inhomogeneous structure of
disordered graphene can be well described by an effective
homogeneous medium. This is also why the self-energy
function is assumed to be k independent in many perturba-
tive calculations. Our finding validates these assumptions.
23
Our calculation should be compared with widely used re-
sults from the SCBA that predicts
15,24

E =

− E/2 − i
0
, E 
0
,
−2
E + i Eln


E
c
E


− i E , E 
0
,

where
0
=E
c
e
−1/2

E
c
is the cut-off energy. As shown in
Fig. 1, SCBA results agree well with our exact self-energy
function for very weak disorder
V
0
=0.5 and
n
imp
/N=10%.
25
When the disorder strength increases sev-
eral times
V
0
=2.0 or 0.07, the perturbative results can-
not capture the main features, especially near the Dirac point.
The discrepancy is obvious for Im 
0 as shown in Fig.
1
b.Im
E at the Dirac point is
0
10
−3
t
E
c
=

3t
Ref.
6 from the SCBA while our exact value is Im 
0
10
−2
t. Therefore, the true broadening of states at or near
the Dirac point is much bigger than what is predicted by
SCBA. This difference can be attributed to the mixture of
Bloch states caused by the impurities.
5,26
Furthermore, the
level-repulsion effect pushes all energy level toward the
Dirac point so that the density of states at the Dirac point
increases more in the presence of impurities
see inset of Fig.
4 below. Therefore, the impurities increase the imaginary
part of the self-energy function
directly associated with the
density of states at the Dirac point. When the wavelength
becomes short, and the quantum interference as well as the
Bloch state mixing are less important, Im 
E are deter-
mined by the disorder scattering. The difference between our
exact simulation and that of the SCBA is small, as shown in
Fig. 1
b.
IV. SPECTRAL FUNCTION
The single-particle spectral function relates to the Green’s
function through A
k ,E=−Im G
k ,E /.
23
Figure
2
a is A
k+,E for k
y
=0, V
0
=1, n
imp
/N=10%, and various
k
x

in unit of a
−1

curves from the left to the right in the
figure ranging from 0.0
n
x
=0 to 0.098
n
x
=56.Inthe
absence of disorders, the spectral function A
0

k ,E is a
delta function, reflecting that the wave vector k is a good
quantum number and has all its spectral weight precisely at
the energy E=E
k
. In the presence of disorders, the transla-
tional symmetry is broken and the spectral function is broad-
ened, resulting from the disorder scattering effect. The
widths of the spectral function is given by Im 
E that mea-
sures the elastic relaxation lifetime

e
,

e
=

−2 Im 
E
. There-
fore, the elastic scattering relaxation time is then akin to
FIG. 1.
Color online
a Real part of self-energy as a function
of energy.
b Imaginary part of self-energy as a function of energy.
The squares and circles are our numerical calculations for the dis-
order concentration n
imp
/N=10% and V
0
=0.5 and V
0
=2.0, respec-
tively. Dashed lines are for the SCBA results for the same disorder.
Energy is measured in the units of t.
FIG. 2.
Color online Single-particle spectral function A
k


+,E plotted as a function of energy E at several k points
from left
to right k=0.000, 0.014, 0.028, 0.042, 0.056, 0.070, 0.084, and
0.098
or n
x
=0−56 along the k
x
direction. Model parameters are
n
imp
/N=10% and V
0
=1.0. Inset: the energy dependence of single-
particle relaxation time

e
for V
0
=0.5−2.0 and n
imp
/N=10%.
BRIEF REPORTS PHYSICAL REVIEW B 82, 153405
2010
153405-2