Unified description of the dc conductivity of monolayer and bilayer graphene at finite densities based on resonant scatterers

Abstract

We show that a coherent picture of the dc conductivity of monolayer and
bilayer graphene at finite electronic densities emerges upon considering that
strong short-range potentials are the main source of scattering in these two
systems. The origin of the strong short-range potentials may lie in adsorbed
hydrocarbons at the surface of graphene. The equivalence among results based on
the partial-wave description of scattering, the Lippmann-Schwinger equation,
and the T-matrix approach is established. Scattering due to resonant impurities
close to the neutrality point is investigated via a numerical computation of
the Kubo formula using a kernel polynomial method. We find that relevant
adsorbate species originate impurity bands in monolayer and bilayer graphene
close to the Dirac point. In the midgap region, a plateau of minimum
conductivity of about $e^2/h$ (per layer) is induced by the resonant disorder.
In bilayer graphene, a large adsorbate concentration can develop an energy gap
between midgap and high-energy states. As a consequence, the conductivity
plateau is supressed near the edges and a "conductivity gap" takes place.
Finally, a scattering formalism for electrons in biased bilayer graphene,
taking into account the degeneracy of the spectrum, is developed and the dc
conductivity of that system is studied.

Full text

Page 1

A unified description of the dc conductivity of monolayer and bilayer graphene based
on resonant scatterers
Aires Ferreira1, J. Viana-Gomes1, Johan Nilsson2, E. R. Mucciolo3, N. M. R. Peres1, and A. H. Castro Neto4
1Department of Physics and Center of Physics,
University of Minho, P-4710-057, Braga, Portugal
2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden
3Department of Physics, University of Central Florida, Orlando, Florida 32816, USA and
4Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
(Dated: October 21, 2010)
We show that a coherent picture for the dc conductivity of monolayer and bilayer graphene emerges
from considering that strong short-range potentials are the main source of scattering in these two
systems. The origin of the strong short range potentials may lie in adsorbed hydrocarbons at
the surface of graphene. The equivalence between results based on the partial wave description of
scattering, the Lippmann-Schwinger equation, and the T−matrix approach is established. Scattering
due to resonant impurities close to the neutrality point is investigated via a numerical computation
of the Kubo formula using a kernel polynomial method. We find that realistic adsorbates originate
impurity bands in monolayer and bilayer graphene close to the Dirac point. In the midgap region, a
plateau of minimum conductivity of about e2/h (per layer) is induced by the resonant disorder. In
bilayer graphene, a large adsorbate concentration can develop an energy gap between midgap states
and high energy states. As a consequence, the conductivity plateau is supressed near the edges
and a “conductivity gap” takes place. Finally, a scattering formalism for electrons in biased bilayer
graphene, taking into account the degeneracy of the spectrum, is developed and the dc conductivity
of that system is studied.
PACS numbers: 81.05.ue, 72.80.Vp, 78.67.Wj
I.INTRODUCTION
In his famous book,1Peierls noted that in three di-
mensions the first Born approximation cannot be used
to deal with short-range potentials in general, even when
the potential is not too strong. The reason lies on the
fact that the first Born approximation overestimates the
value of the scattering cross section, and modifies the
dependence of this latter quantity on the energy of the
incoming particle relatively to the exact result. The fun-
damental reason why this effect takes place has its roots
on the modification of the wave function in the region
where the potential is finite: even for moderate poten-
tials, the wave function in the region of the potential is
strongly deformed relatively to the plane wave used in
the first Born approximation.
Since his main concern was nuclear physics, Peierls
did not address the validity of the first Born approxi-
mation in systems of reduced dimensions. Contrary to
nuclear physics, some condensed matter systems impose
dimensional constraints on the electronic motion—a di-
rect consequence of the lattice structure of the given
solid. Electrons moving in graphene2–5face the most dra-
matic dimensional constraint, being forced to move along
a strictly two-dimensional plane formed by a honeycomb
lattice of carbon atoms. In bilayer graphene, electrons
are also confined to move in two-dimensions. Since this
latter system is a stacking of two graphene sheets, the
electrons may, additionally, hop between the layers.
Scattering cross sections in condensed matter physics
are of ultimate importance for the interpretation of the
dc transport in solids, specially in what concerns the
effect of localized impurities. These can be either de-
scribed by short-range or long-range potentials. Follow-
ing Peirels,1the correct interpretation of the conductiv-
ity of a metal at low temperatures may require a descrip-
tion of electronic scattering by impurities beyond the first
Born approximation—this is specially true if the impuri-
ties give rise to strong short-range potentials.
In systems such as monolayer and bilayer graphene,
where the electronic density can be tuned between zero
and ∼ 1014cm−2,6computing the correct dependence of
the cross section on the Fermi energy is a crucial ingredi-
ent for a meaningful interpretation of the data. Since the
early days of graphene physics,2,3it became clear that the
conductivity of monolayer graphene shows slightly sub-
linear dependence on electronic density. On the other
hand, the conductivity of bilayer graphene shows, con-
sistently, a robust linear dependence on the backgate po-
tential. Both monolayer and bilayer graphene-based de-
vices use sheets from flakes produced in exactly the same
manner, i.e., via exfoliation of graphite. (More recently,
graphene has been isolated via epitaxial growth on SiC7
and chemical vapor depositions on metal surfaces.8,9) It
is by now believed that the main sources of scattering in
exfoliated graphene are introduced during the fabrication
process of the devices. (Chemically synthesized graphene
displays alternative scattering mechanisms.10)
The sources of disorder in graphene can vary. They can
be due to adsorbed chemical species, such as hydrogen
atoms or hydrocarbons molecules, random strain, such
as rippling11–13and scrolling,14and electrostatic random
potentials at the surface of the silicon oxide substrate,
arXiv:1010.4026v1 [cond-mat.mes-hall] 19 Oct 2010

Page 2

2
being originated from charged impurities.15–18
It is widely accepted that the electron-hole puddles15,19
forming in the material at the neutrality or Dirac
point are due to localized subsurface charged impurities.
Whether charged impurities are the limiting source of
scattering in doped graphene remains unclear. Indeed,
combined with charged scatterers, the resonant scatter-
ing mechanism due to adsorbed hydrocarbons20is cur-
rently ascending as one of the dominant processes limit-
ing the electronic mobility in graphene.21–24As we show
in Sec. IIIA, adsorbed hydrocarbons can effectively act
as strong short-range scatterers, which can be mimicked
by a model of vacancies.25–27
Since the sources of scattering are, most likely, intro-
duced during the fabrication process, they must be the
same for both monolayer and bilayer graphene. There-
fore, a consistent theoretical description of the conduc-
tivity of graphene at low temperatures has to be able to
describe the experimental curves of both monolayer and
bilayer graphene by invoking the same source of scatter-
ing. In this paper, we show that such consistent theo-
retical description can be achieved by considering strong
short-range potentials whose origin may lie in adsorbed
chemical species at the surface of the material. Instru-
mental to our description is the critical analysis devel-
oped by Peierls:1The calculation of the exact scattering
cross sections is essential for a correct interpretation of
the experimental data.
Before studying the dc conductivity for both mono-
layer and bilayer graphene, a task we defer to Sec. III, we
first survey the scattering theory for electrons in mono-
layer and bilayer graphene in Sec. II. This first step is
essential to make understandable the treatment of Sec.
III. We show that the calculation of the dc conductiv-
ity of both monolayer and bilayer graphene can be easily
done using the intuitive approach to scattering given the
partial-wave analysis. In addition, dc transport close to
the neutrality point is investigated by means of a numer-
ical calculation of the Kubo conductivity. In Sec. IV, we
consider scattering and dc transport in a biased graphene
bilayer. Conclusions are drawn in Sec. V. Several tech-
nical aspects of our results are given in the Appendix.
II.PARTIAL WAVE ANALYSIS OF STRONG
SHORT-RANGE POTENTIALS
As we discussed in the introduction, the calculation of
the dc conductivity of a metal requires the calculation
of the transport cross section as accurately as possible.
A well-established approach requires the computation of
the phase shifts induced in the scattered electron wave
function by the scattering potential. If the phase shifts
are known exactly, so is the cross section. Below, we set
the notation and introduce the central quantities needed
in this work by giving a concise presentation of the phase-
shift approach to scattering in the context of graphene
and its bilayer.28–31These results will later be used in
Sec. III. Also, and to the best of our knowledge, the
scattering theory for electrons in a biased graphene bi-
layer has not been developed so far in the literature, and
therefore it is presented in Sec. IV.
Scattering theory states that the large-distance wave
function of a particle in the presence of a scattering po-
tential (with cylindrical symmetry) must have the form
(in two-dimensions)
ψ(r) ? eikix+ f(θ)eikfr
√r
,
(1)
where ki= (ki,0) and kf = kf(cosθ,sinθ) are the mo-
mentum of the incoming and scattered waves, respec-
tively; clearly, for elastic scattering, we have ki= kf= k.
The scattering amplitude f(θ) can be written in terms of
the phase shifts δmassociated with the partial-wave ex-
pansion of the scattered wave function in the basis of
angular momentum states. In Eq. (1), the first term
represents the incoming particle, with the incoming mo-
mentum oriented along the x−axis, and the second one
represents the cylindrical scattered wave function.
As it stands, Eq. (1) holds for the two-dimensional
Schrödinger equation.32However, for both graphene and
graphene bilayer, the large distance behavior of the wave
function differs slightly, but significantly, from Eq. (1).
A.Electronic scattering in graphene
Figure 1:
zone of monolayer graphene. Left: the hexagonal lattice of
graphene, with the next nearest neighbor, δi, and the prim-
itive, ai, vectors depicted.The area of the primitive cell
is Ac = 3√3a2
Brillouin zone of graphene, with the Dirac points K and K?
indicated. Close to these points, the dispersion of graphene
is conical and the density of states is proportional to the ab-
solute value of the energy.
(Color online) Lattice structure and Brillouin
0/2 ? 5.1 Å2, and a0 ? 1.4 Å. Right: the
For graphene, the motion of the electrons in the
π orbitals is, at low energies, described by the two-
dimensional massless Dirac Hamiltonian, reading4
HK= vFσ · p,
(2)
where the Fermi velocity is defined as vF = 3ta0/(2?),
t is the hopping integral between the pz orbitals of two

Page 3

3
adjacent carbon atoms, and a0 ≈ 1.4 Å is the carbon-
carbon distance in graphene (see Fig. 1). The vector σ
is written in terms of Pauli’s matrices as σ = (σx,σy)
and p is the momentum operator.
The eigenfunctions of the Hamiltonian in Eq. (2) have,
in Cartesian coordinates, the explicit form
?
with θk= arctan(ky/kx) and A denoting the total area
of the system. The energy eigenvalues corresponding to
the eigenfunction in Eq. (3) are E = ±vF?k.
the latter follows the density of states per spin, ρ(E) =
2|E|/(π√3t2). The probability density current reads33
J = vF؆
Ψ±(r) =
1
√2A
1
±eiθk
?
eik·r,
(3)
From
±σΨ±.
(4)
For the study of scattering, it is more convenient to recast
the Hamiltonian in Eq. (2) in cylindrical coordinates r
and θ as
HK= −ivF?
?
0
ˆL−
0
ˆL+
?
,
(5)
where the operatorsˆL±= e±iθ(∂r± ir−1∂θ) act as ris-
ing/lowering operators, according to the following result
ˆL±[Cm(kr)eiθm] = ∓kCm±1(kr)eiθ(m±1).
In Eq. (6), the function Cm(kr) stands for Jm(kr) and
Ym(kr), the first kind and second kind Bessel functions,
respectively, and for the Hankel functions of first kind
H(1)
tion Km(kr) we have
(6)
m and second kind H(2)
m . For the modified Bessel func-
ˆL±[Km(kr)eiθm] = −kKm±1(kr)eiθ(m±1).
In cylindric coordinates, the radial probability density
current reads
(7)
Jr= vF؆
±σrΨ±,
(8)
where σris defined as
σr=
?
0e−iθ
0eiθ
?
.
(9)
The tangential component of the probability density cur-
rent reads Jθ = vFΨ†
and where diag(i,−i) represents a diagonal matrix. Let
us now derive, for massless Dirac electrons in two dimen-
sions, the equivalent of the asymptotic wave function in
Eq. (1). To that end, we note that a state having the
form
?
is also an eigenstate of the Hamiltonian in Eq. (5). We
start by assuming that the asymptotic (large r) behavior
±σθΨ±, with σθ = σrdiag(i,−i),
Ψm(r,θ) =
1
√2A
Jm(kr)
±ieiθJm+1(kr)
?
eimθ
(10)
of the wave function in the angular momentum channel
m has the form (from here on, we consider only E > 0)
?
πAkr
×ei(mθ+δm),
an ansatz inspired by the fact that the Dirac equation
for graphene is a set of two coupled first-order differen-
tial equations and in the asymptotic limit of the Bessel
functions at large r:34
?
?
with λm= mπ/2+π/4. Using Eq. (11), we write the total
wave function as an expansion in partial waves, reading
Ψm(r,θ) ?
1
?
cos(kr − λm+ δm)
ieiθsin(kr − λm+ δm)
?
(11)
Jm(x) =
2
πxcos(x − λm),
2
πxsin(x − λm),
(12)
Ym(x) =
(13)
Ψ(r,θ) =
∞
?
m=−∞
imΨm(r,θ).
(14)
Exploiting of the relation
eikr cosθ=
∞
?
m=−∞
imeimθJm(kr),
(15)
we obtain
Ψ(r) ?
1
√2A
?1
1
?
eikx+
1
√2A
?
1
eiθ
?
f(θ)eikr
√r,
(16)
with the scattering amplitude reading
?
f(θ) =
2i
πk
∞
?
m=−∞
eimθeiδmsinδm.
(17)
It is a simple exercise to show that the first term in
Eq. (16) corresponds to a flux Jx= vF/A (and Jy= 0),
whereas to the second term corresponds a radial flux
Jr = vF|f(θ)|2/(rA) (and Jθ = 0).
to the usual definition of differential cross section, σ(θ),
it follows that
Then, according
σ(θ) = |f(θ)|2.
(18)
Before we turn to scattering in bilayer graphene, it will
be useful, for later use, to introduce other asymptotic
forms of the Bessel functions Jm(x), Ym(x), and Km(x),
in addition to those already given in Eqs. (12) and (13).
For large x, we have34
?π
For x ? 1, the asymptotic forms read34
Jm(x) = (x/2)mΓ−1(m + 1),
Km(x) =
2xe−x.
(19)
(20)

Page 4

4
Y0(x) = 2π−1lnx,
(21)
Ym(x) = −π−1Γ(m)(x/2)−m,m = 1,2,... , (22)
and
K0(x) = −lnx,
(23)
Km(x) = 2−1Γ(m)(x/2)−m,m = 1,2,... , (24)
where Γ(x) is the gamma function. We now consider
scattering in bilayer graphene.
B. Electronic scattering in bilayer graphene
Bilayer graphene has four atoms per unit cell, with the
two honeycomb sheets arranged according to a Bernal
stacking, as shown in Fig. 2. Two of the atoms belonging
to each of the layers are on top of each other (the atoms
A1 and B2, in Fig. 2), allowing for interlayer hopping.
Such process is represented by a hopping parameter t⊥≈
0.5 eV.35,36The other two carbon atoms, termed A2and
B1in Fig. 2, are not coupled to the carbon atoms of the
other layer, in accordance with the assumptions of the
minimal model for electronic motion in bilayer graphene.
The band structure of bilayer graphene has four bands,
but the low-energy physics (|E| ? t⊥) can be described
by an effective model of only two bands,35–37where the
atoms linked by t⊥are projected out since they describe
high-energy bands – the dimmer of atoms A1 and B2,
linked by t⊥, form a two-level system with energy levels
±t⊥. Additionally, the atoms in the two sheets can be
made non-equivalent by applying an electric field perpen-
dicular to the sheets, inducing in this way a gap in the
spectrum (the electrostatic potential difference between
the two layers is 2V ).35–38
The derivation of the effective Hamiltonian is straight-
forward. We write the full Hamiltonian as


where the columns of the Hamiltonian are labeled by the
four atoms in the unit cell. In ascending order, this la-
beling is B1, A2, B2, and A1. The operator ˆ π is defined
as ˆ π ≡ vF(ˆ px+ iˆ py). The eigenproblem H|ψ? = E|ψ?
can be written as
?
It follows from Eq. (26) that
H =

V
0 −V
0
ˆ π†
00ˆ π
0ˆ π†
−V
−t⊥
ˆ π
0
−t⊥
V

≡

?
HL
H†
HLH
HH
LH
?
,
(25)
HL
H†
HLH
HH
LH
??|ϕ?
|χ?
?
= E
?|ϕ?
|χ?
?
.
(26)
HL|ϕ? + HLH(E − HH)−1H†
LH|ϕ? = E|ϕ?,
(27)
Figure 2: (Color online) Lattice structure of graphene bilayer.
The atoms represented by the letters A1and B1lie on the bot-
tom graphene layer, whereas atoms A2 and B2 are in the top
layer. The electrons can hop between layer via a perpendicu-
lar hopping parameter t⊥ between the carbon atoms termed
A1 and B2 (connected by dashed lines). The Brillouin zone
of bilayer graphene is the same as that of monolayer graphene
(see Fig. 1).
and considering that t⊥? (V,|E|), we have HBL|ϕ? =
E|ϕ?, with37
HBL= V σz−V
t2
⊥
To keep things simple, we consider in what follows the
case V = 0; later we will discuss the case V ?= 0. In
cylindric coordinates, the Hamiltonian (28) is written as
?ˆ πˆ π†
0
0
−ˆ π†ˆ π
?
+
1
t⊥
?
0(ˆ π)2
0(ˆ π†)2
?
(28)
.
HBL= −v2
F?2
t⊥
?
0
ˆL2
0
−
ˆL2
+
?
,
(29)
and the eigenfunctions (regular at the origin) can be writ-
ten as
?
to which corresponds the eigenvalues E = ±v2
From this latter result follows the density of states per
spin, ρ(E) = t⊥/(π√3t2).
It is important to stress two differences between the
Hamiltonians in Eqs. (2) and (29) regarding boundary
conditions and the nature of the scattering states. To
be concrete, let us assume that electron is subjected to
a potential well of the form V (r) = V0θ(R − r). In the
case of the Dirac Hamiltonian, the boundary conditions
at r = R imply the continuity of the two components
of the spinors, whereas for the bilayer Hamiltonian we
have to impose the continuity of both the components
of the spinors and their first derivative. The second as-
pect is related to the fact that elastic scattering con-
serves energy. Then, since in bilayer graphene we have
E = ±v2
E > 0, as in any scattering process, there are two admis-
sible solutions: a real solution, k =√t⊥E/(vF?), and
Ψm(r,θ) =
1
√2A
Jm(kr)
∓e2iθJm+2(kr)
?
eimθ,
(30)
F?2k2/t⊥.
F?2k2/t⊥, and keeping the energy constant, say

Page 5

5
a pure imaginary one, k = i√t⊥E/(vF?).
bilayer graphene supports evanescent modes at the inter-
face r = R. This fact is essential to satisfy the boundary
conditions obeyed by the wave function.39
As in the case of the Dirac Hamiltonian, we have to de-
rive the form of the probability density current for elec-
trons described by the Hamiltonian in Eq. (29).
usual procedure33gives that any component J? of the
current has the form
Therefore,
The
J?= 2v2
F?
t⊥
ImΨ†ˆJ?Ψ,
(31)
where for ? = x,y we have
ˆJx= σx∂x+ σy∂y,
(32)
and
ˆJy= σy∂x− σx∂y.
(33)
For the radial component, ? = r, we have
?
and for the tangential component, ? = θ, we have
?
Taking into account that the Hamiltonian in Eq. (29)
forms a set of two coupled second-order differential equa-
tions, we assume that the asymptotic (large r) behavior
of the wave function in the angular momentum channel
m has the form
?
πAkr
×ei(mθ+δm).
Following the same procedure used to derive Eq. (16), we
can show that the large-r behaviour of the total electronic
wave function in graphene bilayer in the presence of a
potential has the form
?1
Using Eq. (31), we can show that the first term in
Eq. (37) corresponds to a flux Jx= 2v2
where v is the velocity of the particle, and that the
second term corresponds to a radial flux of the form
Jr = 2v2
still given by Eq. (17). As before, it follows that the
differential cross section is given by Eq. (18).
In Sec. III, we apply the this formalism to the case
of a potential well described by the potential V (r) =
V0θ(R − r) in the strong interacting regime V0? t. We
will see that the results are insensitive to the particular
form adopted for V (r) as long it corresponds to a strong
short-range potential.
ˆJr=
0e−2iθ(∂r+ ir−1∂θ)
0e2iθ(∂r− ir−1∂θ)
?
,
(34)
ˆJθ=
0
−ie−2iθ(∂r− ir−1∂θ)
0 ie2iθ(∂r+ ir−1∂θ)
?
,
(35)
Ψm(r,θ) ?
1
?
cos(kr − λm+ δm)
e2iθcos(kr − λm+ δm)
?
(36)
Ψ(r) ?
1
√2A
1
?
eikx+
1
√2A
?
1
e2iθ
?
f(θ)eikr
√r. (37)
F?k/(At⊥) ≡ v/A,
F?k|f(θ)|2/(rAt⊥) ≡ v|f(θ)|2/(Ar), with f(θ)
III.THE DC CONDUCTIVITY OF GRAPHENE
AND ITS BILAYER
As discussed in Sec. I, there is growing evidence that
the limiting scattering mechanism of the electronic mobil-
ity in graphene is due strong short-range potentials, likely
to be originated from adsorbed hydrocarbons. These ad-
sorbed atoms and/or molecules act as resonant scatter-
ers, giving rise to midgap states.20,40–42
A.Adsorbed atoms in graphene as strong
short-range scattering centers
The resonant scattering mechanism is easy to seize
by considering a simple model. Let us write the tight-
binding Hamiltonian of the π−electrons in graphene as
(spin index omitted)
?
where |A,Rn? represents the Wannier state at the unit
cell Rn; an equivalent definition holds for |B,Rn+ δi?,
where δiis one of three nearest-neighbor vectors in the
honeycomb lattice, as depicted in Fig. 1.
We now consider that an impurity is binding covalently
to a carbon atom at site Rn= 0. Such a situation adds
to the Hamiltonian in Eq. (38) a term of the form
H = −t
n,δi
|A,Rn??Rn+ δi,B| + H.c.,
(38)
Hrs= (Vad|ad??A,0| + h.c.) + ?ad|ad??ad|,
where Vadis the hybridization between the adatom (or a
carbon atom of a hydrocarbon molecule) and a given car-
bon atom of graphene, ?adis the relative (to graphene’s
carbon atoms) on-site energy of the electron in the
adatom, and |ad? is the ket representing the state of the
electron in the adatom. Taking the wave function to be
of the form
?
+ Cad|ad?,
the Schrödinger equation applied to the site Rn= 0 reads
(39)
|ψ? =
n
[A(Rn)|A,Rn? + B(Rn+ δ2)|B,Rn+ δ2?]
(40)
EA(0) − VadCad = −t[B(δ1) + B(δ2) + B(δ3)],(41)
(E − ?ad)Cad = VadA(0).
Solving for Cad, we obtain
(42)
− t[B(δ1) + B(δ2) + B(δ3)] =
?
E −
V2
ad
E − ?ad
?
A(0).
(43)
The resonant effect is included in the last term of
Eq. (43), which represents an effective local potential of
strength
Veff= V2
ad/(E − ?ad).
(44)