a r X i v : 0 8 0 6 . 4 5 3 1 v 2 [ c o n d  m a t . m e s  h a l l ] 1 6 S e p 2 0 0 8
Analytical Tightbinding Approach for Ballistic Transportthrough Armchair Graphene Ribbons:Exact Solutions for Propagation throughSteplike and Barrierlike Potentials
Yu. Klymenko
∗
Space Research Institute of NAS and NSA of Ukraine, Kyiv, 03187, Ukraine
O. Shevtsov
†
National Taras Shevchenko University of Kyiv, 03022, Kyiv, Ukraine
(Dated: September 16, 2008)Based on a tightbinding approximation, we present analytical solutions for the wavefunctionand propagation velocity of an electron in armchair graphene ribbons. The derived expressionsare used for computing the transmission coeﬃcients through steplike and barrierlike potentials.Our analytical solutions predict a new kind of transmission resonances for onemode propagationin semiconducting ribbons. Contrary to the Klein paradox in graphene, this approach shows thatbackscattering for gapless mode is possible. In consistence with a higher order
k
·
p
method, thebackscattering probabilities vary with the square of the applied potential in the lowenergy limit.We also demonstrate that gaplessmode propagation through a potential step in armchair ribbonscan be described by the same throughstep relation as that for an undimerized 1D chain of identicalatoms.
PACS numbers: 72.10.d,73.63.b,73.63.NmKeywords: Armchair graphene ribbon, tightbinding model, energy dispersion, step potential, barrier potential, transmission coeﬃcient, resonance, lowenergy limit, Klein paradox in graphene.
I. INTRODUCTION
Recently graphene sheet, a monolayer of covalentbond carbon atoms forming a dense honeycomb crystal, has beenobtained experimentally
1,2
. Even at room temperature submicron graphene structures act as a highmobility electronor hole conductors
1
. This phenomenon has brought to life a promising ﬁeld of carbonbased nanoelectronics, wheregraphene ribbons (GRs) could be used as connections in nanodevices.The presence of armchair and zigzagshaped edges in graphene has strong implications for the spectrum of
π
electrons
3,4,5,6,7
and drastically changes the conducting properties of GRs. Especially, a zigzag edge provides localizededge states close to the Fermi level (
E
F
= 0), which leads to the metallic type of conductivity. In contrast, anylocalized state does not appear in an armchair GR. An armchair GR can be easily made to be either metallic orsemiconducting by controlling the width of the current channel.Most of the intriguing electronic properties of GRs were ﬁrst predicted by the tightbinding model
4,5,6,8,9,10,11
.These features are also well reproduced in the
k
·
p
method
12,13
based on the decomposition of linear Schr¨odingerequations in the vicinity of zeroenergy point. The commonly used
k
·
p
equation is a twodimensional analog of therelativistic Dirac equation
1,3,13,14
, which is a certain approximation of general tightbinding Schr¨odinger equations.The continuum Dirac description of the electronic states near
E
=
E
F
has been shown to be quite accurate bycomparing with a numerical solution in the nearestneighbor tightbinding model
3,14
. Particularly, in the frameworkof the Dirac approach, perfect electron transmission is predicted through any high and wide potential barriers, knownas the Klein paradox in graphene
15
.Recent analytical studies based on the relativistic Dirac equation have considered transmission through barriers
15
,graphene wells
16,17
, doublebarriers
18
, quantum dots
19
, superlattices
20
, and
n
−
p
junctions
21
. However, transportproperties of these structures in the tightbinding model has not been analyzed systematically yet. Considerableattention has been paid to the eﬀects of conductance quantization in GRs in the absence of scattering, where thequantization steps clearly indicate the number of propagating states crossing the Fermi level
11
.In this paper, we develop an exact model of electron transport through armchair GRs based on the nearestneighbortightbinding approximation. (In contrast to the zigzag GRs, the armchair GR spectrum does not have a special bandof edge states complicating the description.) We obtain simple analytical expressions for normalized wavefunctionsand propagation velocity of an electron wave in armchair ribbons. No similar explicit solutions have been obtained forarmchair GRs yet. In a tightbinding study by Zheng
22
of the electronic structure in armchair ribbons, the derivedanalytical form of wavefunction has not been used for describing charge transport in GRs.On the basis of the analytical solutions for the wavefunction and propagation velocity of an electron in an inﬁnitearmchair GR, we describe the scattering problem through steplike and barrierlike proﬁles of site energy along the
2ribbon in closed form. Solving the corresponding equations for the scattering amplitudes, we obtain exact transmissioncoeﬃcients in the atomistic (tightbinding) model analytically. For the energy region close to the Fermi level wecompare our analytical expressions with their analogs obtained in the continuous (Dirac) model
15
. Our generalconclusion is that the validity of the relativistic approach based on the standard
k
·
p
equation
13
is overestimated forthe description of charge transport in armchair GRs. Speciﬁcally, for onemode propagation in semimetal GRs, thedeviation of transmission coeﬃcients
T
from the unity near the zeroenergy point is proportional to the square of thestep (barrier) magnitude, which contradicts the known Klein paradox in graphene
15
. It is worth noting that similardeviations of transmission coeﬃcients are intrinsic for the extended
k
·
p
equation, based on higher order
k
·
p
termsin the lowenergy limit. Such highorder approximations lead to predicting small backscattering due to the eﬀects of trigonal warping
12
of the bands, destroying perfect transmission in the channel.The paper is organized as follows. In the framework of the nearestneighbor tightbinding formalism we calculateenergy dispersion (Sec.II) and eigenstates (Sec.III) of electrons in the honeycomb graphene lattice. The lattice, in
contrast to Ref.
22
, is considered as a set of rectangular elementary cells of four nonequivalent atoms (Fig.1). Thisassumption simpliﬁes the description of electron transport processes in GRs in the presence of steplike or barrierlikesiteenergy proﬁles. Following Ref.
23
, in SectionIVwe derive boundary conditions for inﬁnite GRs with armchairedges. By using a combination of early found eigenstates of the honeycomb lattice, we obtain normalized wavesolutions for electrons in armchair GRs and formulate the quantization law for the transverse component
k
m
of thewave vector. As a result, the full
π
electron band spectra of graphene ribbon breaks into a set of subbands, whosefeatures are discussed in SectionV.In SectionVI, we ﬁnd the group velocity and propagation direction of an electron in an armchair ribbon. It is shownthat the group velocity is proportional to sin
θ
, where
θ
is the electron phase shift between two neighboring unit cellsof the ribbon. In sectionsVIIandVIII,the transmission probabilities through steplike and barrierlike potentials are
calculated. Based on the exact solutions, in SectionIX, we demonstrate that the transmission coeﬃcients exhibit thenew kind of resonance occurring when cos
θ
= cos¯
θ
, where¯
θ
is the intercell electron phase shift in the region wherethe potential exists. The number and positions of the transmission resonances are shown to be diﬀerent in the regions
k
m
<
2
π/
3 and
k
m
>
2
π/
3. As for onemode propagation in semimetal GRs (
k
m
= 2
π/
3), our theory predicts nounit propagation. In SectionX,by expanding the expressions for cos
θ
, cos¯
θ
in the vicinity of
E
= 0, we explain whythe Klein paradox does not hold in armchair GRs.In Sec.XI, we discuss throughstep and throughbarrier probabilities and provide some examples. Conclusions arepresented in Sec.XII. AppendicesA(on electron ﬂux in the tightbinding model) andB(on throughstep transmission
in a linear chain of identical atoms) describe some theoretical tools used in the paper. AppendixCcontains sometechnical details related to obtaining our approximate solution from the Dirac solution
15
.
II. DISPERSION RELATION FOR HONEYCOMB GRAPHENE LATTICE
We consider the graphene honeycomb lattice as a set of rectangular elementary cells of four atoms
α
=
l,λ,ρ,r
(seeFigure1). By taking the tightbinding representation for molecular orbitals

Ψ
=
∞
n,m
=
−∞
α
ψ
n,m,α

n,m,α
,
we come to the set of linear Schr¨odinger equations
−
Eψ
n,m,l
=
ψ
n,m,λ
+
ψ
n,m
+1
,λ
+
ψ
n
−
1
,m,r
,
−
Eψ
n,m,λ
=
ψ
n,m,l
+
ψ
n,m,ρ
+
ψ
n,m
−
1
,l
,
−
Eψ
n,m,ρ
=
ψ
n,m,λ
+
ψ
n,m,r
+
ψ
n,m
−
1
,r
,
−
Eψ
n,m,r
=
ψ
n,m,ρ
+
ψ
n,m
+1
,ρ
+
ψ
n
+1
,m,l
,
(1)with respect to wave function components
ψ
n,m,α
=
Ψ

n,m,α
. Here

n,m,α
is 2p
z
orbital of
α
th atom in
{
n,m
}
elementary cell,
E
≡
E/

β

is the electron energy in units of

β

,
β <
0 is a transfer integral between thenearestneighbor carbon atoms. Siteenergies of carbons equal zero and serve as a reference.We look for a solution of Eq. (1) in the form
ψ
n,m,α
=
φ
α
e
ik
n
n
+
ik
m
m
, α
=
l,λ,ρ,r,
(2)where the dimensionless wave numbers
k
n
(
≡
3
k
x
a
C
−
C
),
k
m
(
≡√
3
k
y
a
C
−
C
), and cofactors
φ
α
are unknown coeﬃcients,
k
x
and
k
y
are the components of the wave vector along
n
and
m
directions respectively. The lengths 3
a
C
−
C
and
3
ñ
l
ë
r l r r l
n,mn,m
1
n,m+
1
n+
1,m
n
1,m
a
CC
3
a
CC
nm
3
a
CC
ñë
FIG. 1: Sketch of 2D honeycomb lattice. The central shadowed block corresponds to an elementary cell
{
n,m
}
consisting of 4carbon atoms, labeled as
l,λ,ρ,r
. The lengths 3
a
C
−
C
and
√
3
a
C
−
C
are the translation periods in
n
and
m
directions,
a
C
−
C
isC
−
C bond length.
√
3
a
C
−
C
are the translation periods of the graphene sheet as depicted on the Fig.1,where
a
C
−
C
is the CC bondlength.Plugging in the solutions(2) into the system (1) gives the system of linear equations
Eφ
l
+
1 +
e
ik
m
φ
λ
+
e
−
ik
n
φ
r
= 0
,
1 +
e
−
ik
m
φ
l
+
Eφ
λ
+
φ
ρ
= 0
,φ
λ
+
Eφ
ρ
+
1 +
e
−
ik
m
φ
r
= 0
,e
ik
n
φ
l
+
1 +
e
ik
m
φ
ρ
+
Eφ
r
= 0
.
(3)A nontrivial solution of Eq. (3) demands a zero determinant. It leads to the dispersion relationcos
k
n
=
f
(
E,k
m
)
, f
(
E,k
m
) =
E
2
−
1
−
4cos
2
k
m
2
2
8cos
2
k
m
2
−
1
,
(4)deﬁning the nondimensional wave number
k
n
in terms of
k
m
and electron energy
E
. It follows from Eq.(4) that
E
2
= 1
±
4cos
k
m
2cos
k
n
2+ 4cos
2
k
m
2
,
(5)where index
±
distinguishes two branches of the dispersion relation.
III. WAVE SOLUTION IN GRAPHENE LATTICE
If the relation (4) holds, system (3) becomes degenerated, and we can express the coeﬃcients
φ
l
,
φ
λ
,
φ
ρ
in termsof
φ
r
, omitting the fourth equation in (3). The result can be written as follows
φ
α
=
C
φ
α
, α
=
l,λ,ρ,r,
(6)where
φ
l
=
−
E
2
−
1
e
−
ik
n
+ 4cos
2
k
m
2
E
E
2
−
1
−
4cos
2
k
m
2
,
φ
λ
=
1 +
e
−
ik
m
1 +
e
−
ik
n
E
2
−
1
−
4cos
2
k
m
2
,
φ
ρ
=
−
1 +
e
−
ik
m
E
2
+
e
−
ik
n
−
4cos
2
k
m
2
E
E
2
−
1
−
4cos
2
k
m
2
,
φ
r
= 1
.
4
l r l r r l
n,mn,m
1
n,m+
1
n+
1,m
n
1,m
nm
n,
1
¹
n,
r
ë ñ
l r l
ë ë ë
ë ññññ
FIG. 2: Sketch of inﬁnite graphene ribbon with armchair edges. The solidframed block is the unit cell
n
of GR.
Exploiting the energy dispersion relation (4), one can see that

φ
α

= 1 for the realvalued
k
n
. Introducing a newfunction
e
iθ
=
−
E
2
−
1 + 4
e
ik
n
cos
2
k
m
2
E
E
2
−
1
−
4cos
2
k
m
2
=
−
1
±
2cos
k
m
2
e
ik
n
/
2
E ,
(7)we can get
φ
l
=
e
i
(
θ
−
k
n
)
,
φ
λ
=
±
e
−
i
(
k
m
+
k
n
)
/
2
,
φ
ρ
=
∓
e
−
i
(
k
m
+
k
n
)
/
2+
iθ
,
φ
r
= 1
.
(8)The choice of upper/lower signs in(7), (8) is determined by the sign in (5). Plugging in Eq. (8) into Eq. (6), and
then the resulting expression into (2) gives the expressions for electron eigenstates in the honeycomb lattice
ψ
n,m,α
(
k
n
,k
m
) =
C
e
ik
n
(
n
−
1)+
ik
m
m
+
iθ
, α
=
l,
±
e
ik
n
(
n
−
1
/
2)+
ik
m
(
m
−
1
/
2)
, α
=
λ,
∓
e
ik
n
(
n
−
1
/
2)+
ik
m
(
m
−
1
/
2)+
iθ
, α
=
ρ,e
ik
n
n
+
ik
m
m
, α
=
r.
(9)
IV. WAVE SOLUTION FOR ARMCHAIR GRS
Unlike the 2D honeycomb lattice, the electron wave function components
ψ
n,m,α
in GRs with armchair edges (seeFigure2) are the solution to the linear Schr¨odinger equations (1) only for inner elementary cells (1
< m
≤ N
). Forboundary cells with
m
= 1 or
m
=
N
+ 1, one needs to take into account that boundary carbons have only twoneighbors (see Fig.2). It demands the wave function to vanish on the set of absent sites
7,23
,
26
ψ
n,
0
,l
= 0
, ψ
n,
0
,r
= 0
, ψ
n,
N
+1
,l
= 0
, ψ
n,
N
+1
,r
= 0
.
(10)Similar boundary conditions were used in Refs
9,22
.To satisfy the relations (10), we represent the solution as a linear combination of the states (9),
ψ
n,m,α
(
k
n
) =
ψ
n,m,α
(
k
n
,k
m
)
−
ψ
n,m,α
(
k
n
,
−
k
m
)
.
Since the deﬁnition (7) implies that
θ
(
−
k
m
) =
θ
(
k
m
), we ﬁnally obtain
ψ
n,m,α
(
k
n
) =
C
′
e
ik
n
(
n
−
1)+
iθ
sin
k
m
m, α
=
l,
±
e
ik
n
(
n
−
1
/
2)
sin
k
m
(
m
−
1
/
2)
, α
=
λ,
∓
e
ik
n
(
n
−
1
/
2)+
iθ
sin
k
m
(
m
−
1
/
2)
, α
=
ρ,e
ik
n
n
sin
k
m
m, α
=
r,
(11)
5where
k
m
≡
k
mj
=
πj
N
+ 1
, j
= 1
,
2
,...,
N
,
(12)is the set of discretized transversal wave numbers, and the unknown constant
C
′
(
≡
2
iC
) can be found from thenormalization condition over a unit cell of the armchair GR,
N
m
=1

ψ
n,m,l

2
+

ψ
n,m,r

2
+
N
+1
m
=1

ψ
n,m,λ

2
+

ψ
n,m,ρ

2
= 1
.
Then we ﬁnally get
C
′
= [2(
N
+ 1)]
−
1
/
2
.
V. BAND SPECTRA OF ARMCHAIR GRS
Due to the transverse momentum quantization (12), the full
π
electron band spectra of an armchair GR breaksinto a set of subbands connected with each mode
j
independently. Given
E
and
k
m
=
k
mj
, the dispersion relation(4) allows us to establish some important peculiarities of the
j
th part of the band spectra. Following the theory of a onedimensional crystal
24
with an arbitrary electronic structure of the elementary cell, each
j
th band of grapheneelectron spectra includes 4 subbands, symmetrically disposed with respect to
E
= 0. Real values of longitudinalwave number
k
n
∈
[0
,π
] deﬁne the region of propagating electron states (when

f
(
E,k
mj
)

<
1 in Eq.(4)). The cases
f
(
E,k
mj
)
>
1 and
f
(
E,k
mj
)
<
−
1 are related to the complex values of
k
m
=
iδ
and
k
m
=
π
+
iδ
, respectively. Theyrefer to the forbidden zones – the gaps between neighboring electron subbands and the regions above the highestsubband and below the lowest one. The subband boundaries correspond to
k
m
= 0 or
k
m
=
π
and satisfy thesolutions to
f
(
E,k
mj
) = 1 or
f
(
E,k
mj
) =
−
1, respectively. Since Eq.(4) deﬁnes only one
k
m
for each energy level
E
,the subbands of the
j
th band cannot overlap, however, they may touch each other along the frontiers with the same
k
m
(= 0
,π
).The results of the band spectrum modeling for armchair GRs with
N
= 10
−
12 are represented in Figure3.Similarshapes are observed for GRs with arbitrary number
N
. The common features of the band spectra are: (I) bands
¹
=11
j=
1234 56 7 8 91110
E
j=
1234 56 7 8 91110 12
E
¹
=12
¹
=10
j=
1234 56 7 8 910
E
k
n
=0
ð
k
n
=
k
n
=
k
n
=0
ð
FIG. 3: Examples of energy bands for armchair graphene ribbons with
N
= 10
−
12. The transverse momentum quantizationresults in subband series connected with each mode
j
independently. Gray (black) rectangles mark subbands, whose bottomcorresponds to
k
n
= 0 (
k
n
=
π
) and whose top – to
k
n
=
π
(
k
n
= 0), as depicted in the framed panel. The border betweenblack and gray rectangles in the positive and negativeenergy spectra determines the subband connecting
k
n
=
π
frontiers, thisis a degenerated solution to
f
(
E,k
mj
) =
−
1. The fundamental gap of a graphene ribbon is the minimal gap value ∆ betweenthe valence and conduction bands among all modes
j
= 1
,...,
N
.
are bounded by the energy interval

E
 ≤
3; (II) the band structure is symmetric with respect to
E
= 0; (III) if thenumber
N
+1 is divisible by 3, the mode
j
= 2(
N
+1)
/
3 (or
k
mj
= 2
π/
3) does not possess the energetic gap betweenpositive and negativeenergy bands, and, therefore, it possesses the semimetal type of electron conductivity
3,7,9,22
.For other
j
modes there is a gap∆ = 2
1
−
2cos
k
m
2
(13)between positive and negativeenergy subbands, which corresponds to a semiconducting ribbon whose gap decreaseswith increasing
N
.