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Physics Letters A 374 (2010) 761–764
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Physics Letters A
www.elsevier.com/locate/pla
Electronic transport for armchair graphene nanoribbons with a potential barrier
Benliang Zhou a , Benhu Zhou a , Wenhu Liao a , Guanghui Zhou a,b,∗
a
b
Department of Physics and Key Laboratory (Educational Ministry) for Low-Dimensional Structures and Quantum Manipulation, Hunan Normal University, Changsha 410081, China
International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, China
a r t i c l e
i n f o
Article history:
Received 11 September 2009
Received in revised form 5 November 2009
Accepted 23 November 2009
Available online 27 November 2009
Communicated by R. Wu
PACS:
78.40.Ri
78.67.-n
73.22.-f
a b s t r a c t
We theoretically investigate the electronic transport properties through a rectangular potential barrier
embedded in armchair-edge graphene nanoribbons (AGNRs) of various widths. Using the Landauer
formula and Dirac equation with the continuity conditions for all segments of wave functions at the
interfaces between regions inside and outside the barrier, we calculate analytically the conductance and
Fano factor for the both metallic and semiconducting AGNRs, respectively. It is shown that, by some
numerical examples, at Dirac point the both types of AGNRs own a minimum conductance associated
with the maximum Fano factor. The results are discussed and compared with the previous relevant works.
Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.
Keywords:
AGNRs
Electronic transport
Scattering matrix
Recently, graphene has attracted intensive research attention
due to the successful fabrication experiments [1–6]. Several abnormal phenomena have been observed [2,3], such as half integer
quantum Hall effect, nonzero Berry’s phase and minimum conductance. These unusual transport properties may result in the novel
applications in graphene-based nanodevices. The energy band in
graphene can be described by a two-dimensional Dirac-like equation at Dirac points of the honeycomb lattice Brillouin zone [7,8],
and the dispersion relation E = ±|h̄k| v F is linear around the Dirac
points where v F ≈ 106 m/s is the Fermi velocity [9]. The massless
Dirac fermions in graphene manifest several quantum electrodynamical phenomena in the low-energy range such as the Klein
paradox phenomenon [10], which describes relativistic electron
tunneling through a high potential barrier. It has predicted that the
electron can approach the perfect transparency for a very high barrier while the transmission probability of the conventional nonrelativistic tunneling decays exponentially with the increasing of the
barrier height. Very recently, Katsnelson et al. [6] have proposed
an experimental realization of the prediction of the Klein paradox
by using electrostatic barriers in a two-dimensional monolayer or
bilayer graphene. Meanwhile, GNRs of various widths made by the
mechanical [2,3] and the epitaxial growth methods [4,11] have also
*
Corresponding author at: Department of Physics and Key Laboratory (Educational Ministry) for Low-Dimensional Structures and Quantum Manipulation, Hunan
Normal University, Changsha 410081, China.
E-mail address: [email protected] (G. Zhou).
0375-9601/$ – see front matter Crown Copyright
doi:10.1016/j.physleta.2009.11.068
© 2009 Published
been extensively studied [8,12–18]. However, few works have considered the electron tunneling through the barrier embedded in
GNRs.
In this Letter we investigate the properties of massless Dirac
fermions tunneling through a rectangular potential barrier embedded in AGNRs, where the barrier may be produced by the electric
field effect of a thin insulator substrate [1–3,6,20] and has been
studied in the previous works [6,19]. By using the Landauer formula and the Dirac equation with the continuity conditions for
wave functions at the two interfaces between regions inside and
outside the barrier, the conductance and Fano factor (the ratio of
noise power and mean current) dependence on the electron energy
and barrier range are calculated for both metallic and semiconducting AGNRs. Some interesting transport properties are predicted
for the system, and the results are discussed and compared with
the previous similar works [6,12–19].
The system considered in this Letter is an AGNR containing a
rectangular potential barrier, where the width of the AGNR is labeled as W and its edges are parallel to x-axis. The potential is
modeled as
V (x, y ) =
V 0,
0,
0 < x < D, 0 < y < W ,
otherwise.
(1)
The system can be separated into three regions by the employed
potential marked as I , II and III, and the electron wave function in
each region can be formally assumed to be:
by Elsevier B.V. All rights reserved.
762
B. Zhou et al. / Physics Letters A 374 (2010) 761–764
n
Ψ I (x, y ) = e ikx x ψn ( y ) +
ΨII (x, y ) =
n
rn n e −ikx x ψn ( y ),
n
amn e
iqm
x x
ψm ( y ) +
m
ΨIII (x, y ) =
bmn e
−iqm
x x
(−kx + ikn )e ikn y /ε +
=
ψm ( y ),
m
m
t n n e
iknx x
ψn ( y ).
+
(2)
In the above wave functions, ψn (ψn ) stands for a four-component
vector (φ nA , φ nB , −φ An , −φ B n ) T in region I (III) without the barrier,
n
n (μ = A,
while ψm in region II with the barrier, where φμ
and φμ
B is the sublattice index) are the nth mode component for the two
valleys, respectively. knx (qm
x ) are the nth (mth) mode longitudinal
wavevectors in regions I and II, respectively. And the coefficients
rn n , amn , bmn and tn n are determined by the continuity conditions
at the two interfaces. The wave function in region I satisfies the
Dirac equation [8]
0
⎜ −kx − ik y
⎜
⎝
0
0
⎛
−kx + ik y
0
0
0
0
0
0
kx − ik y
⎞
φA
⎟
ε ⎜
⎜ φB ⎟ ,
=
γ a ⎝ −φ A ⎠
−φ B
amn (−q x + ikm )e ikm y /(ε − V 0 )
bmn (q x + ikm )e ikm y /(ε − V 0 ),
amn e ikm y +
m
bmn e ikm y ,
m
m
amn (−q x + ikm )e ikm y e iqx D /(ε − V 0 )
+
=
bmn (q x + ikm )e ikm y e −iqx D /(ε − V 0 )
m
m
n
tn n (−kx + ikn )e ikn y e ikx
n
m
amn e ikm y e iqx
D
+
D
/ε ,
bmn e ikm y e −iqx
m
D
=
n
tn n e ikn y e ikx
D
.
n
m
(7)
(3)
√
Unlike the second-order derivative Schrödinger equation, one only
needs to match the wave function but not its derivative, because
the Dirac equation employed here is first-orderly derivative [6,9,
19]. Using the orthogonality
of the wave functions in region I ,
∗ ∗
∗
i.e., φnA φnA dτ = φnB φnB dτ = δnn and (e ikn y ) e ikn y dτ = δnn ,
we can obtain the transmission coefficient tn n by solving Eq. (7).
Therefore, the total transmission probability T and the Fano factor F are followed by summing over the modes:
T=
p
|tn n |2 ,
n ,n=1
(−kx + ik y )φ B = ε φ A /(γ a),
F=
(−kx − ik y )φ A = ε φ B /(γ a),
p
n ,n=1
(kx + ik y )φ B = ε φ A /(γ a),
(kx − ik y )φ A = ε φ B /(γ a).
(4)
By using the boundary (edge) conditions [9], one obtains φ A =
γ a(−kx + ikn )e ikn y /ε , φ B = e ikn y , φ A = −γ a(kx − ikn )e −ikn y /ε , and
φ B = −e −ikn y , where the discrete kn satisfies [9] kn = nπ / W −
4π /(3a) (n labels the mode order) and k2x + kn2 = (ε /γ a)2 . Assuming the structure of the AGNR in this Letter is the same as that in
Ref. [18], then W satisfies W = ( p + 1)a/2 (p is the number of carbon atoms along the zigzag-edge and denotes the width of AGNR)
and n just takes values of 1, 2, . . . , p. Thus each ψn ( y ) in region I
can be written as
ψn ( y ) = γ a(−kx + ikn )e ikn y /ε , e ikn y ,
T
γ a(kx − ikn )e−ikn y /ε, e−ikn y ,
(5)
while ψm ( y ) in region II can be given in the similar way
e ikm y , γ a(q x − ikm )e −ikm y /(ε − V 0 ), e −ikm y
rn n e ikn y =
m
where γ = 3t /2 (t 2.7 eV is the nearest hopping energy),
a = 0.246 nm is the C–C distance along the transversal direction
of AGNR, and ε is the electron energy. Therefore, the following set
of equations can be obtained from Eq. (3):
ψm ( y ) = γ a(−q x + ikm )e ikm y /(ε − V 0 ),
n
m
⎞⎛
⎞
φA
⎟
0
φB ⎟
⎟⎜
⎝
⎠
kx + ik y ⎠ −φ A
−φ B
0
0
e ikn y +
rn n (kx + ikn )e ikn y /ε
n
m
n
⎛
T
,
(6)
2
where q x = (ε − V 0 )2 /(γ a)2 − km
. Applying the continuity conditions for wave functions at the two interfaces, i.e., Ψ I = ΨII at
x = 0 and ΨII = ΨIII at x = D, and combining Eqs. (2), (5) and (6),
one obtains the following equations for the transmission and reflection coefficients:
|
t n n
2
| | 1−|
t n n
|
2
p
|tn n |2 .
(8)
n ,n=1
The conductance G of the system can be derived via the Landauer
formula relation G = g 0 T with g 0 = 4e 2 /h, where the factor 4 accounts for both the spin and valley degeneracy. It is noted that
only the propagating modes contributed to the system transport
at the vicinity of Γ point are considered for simplicity, while the
effect of evanescent waves is not taken into account since it is usually negligible [9].
In what follows we present some numerical examples of the
calculated conductance and Fano factor for the system. In this
Letter, we take the metallic AGNR W = 6a (about 1.48 nm with
p = 11) and the semiconducting AGNR W = 7a (about 1.72 nm
with p = 13) as examples without losing the generality. The barrier potential is set to V 0 = 3γ (about 6.9 eV).
In Fig. 1, we present the dependence of conductance G (in units
of 4e 2 /h) and Fano factor on electron energy ε (in units of γ )
for a metallic AGNR with W = 6a. The range of potential barrier
is set to D = 5a. First of all, as shown in Fig. 1, we notice that
the conductance for a AGNR is different from that for an infinitive graphene which supports perfect transmission regardless of
the barrier height (Klein tunneling [6]). In contrast, the minimum
conductance (about 1) appears around ε = 3 which is associated
with a maximum Fano factor (about 1). It is worthwhile to note
that the Fermi energy may increase from 0 in region I to 3 in region II due to the existence of the potential barrier V 0 . Therefore,
assuming ε = 3 corresponding to the Dirac point seems reasonable. This phenomenon is also very different from that in Ref. [19],
where the minimum conductance and the maximum Fano factor
at the Dirac point are 4e 2 /(π h) and 1/3, respectively. Furthermore,
B. Zhou et al. / Physics Letters A 374 (2010) 761–764
Fig. 1. The electron energy dependence of the conductance and Fano factor for the
metallic AGNR, where W = 6a, D = 5a and V 0 = 3γ .
Fig. 2. The electron energy dependence of the conductance and Fano factor for the
semiconducting AGNR, where W = 7a, D = 5a and V 0 = 3γ .
the series of tunneling peaks in Fig. 1 may be from the chiral nature of the relativistic particles in the AGNR [12–19]. However, one
should notice that the ratio of width to length for AGNR in this
Letter (about 1) is much smaller than that in Ref. [19], where this
ratio is much larger than 5.
For comparison with the transport property for a metallic
AGNR, in Fig. 2 we plot the energy dependence of the conductance
(in units of 4e 2 /h) and Fano factor for a semiconducting AGNR of
W = 7a with the same V 0 and D as those in Fig. 1. The transport
property for the semiconducting AGNR, as shown in Fig. 2, has a
similar feature as that for the metallic one (shown in Fig. 1). However, due to the zero energy gap between conduction and valence
bands the average conductance for the metallic AGNR is larger than
that for semiconducting one. Moreover, the minimum conductance
and the corresponding Fano factor at the Dirac point converts to
0.8 and 1.1, respectively. Therefore, it is relatively more difficult for
the Dirac particles to tunnel through a barrier in a semiconducting
AGNR than that in a metallic one, owing to a nearly 0.3γ energy
gap for semiconducting AGNR. It should be pointed out that the
nonzero minimum conductance, as shown in Fig. 2, may due to
the conservation of pseudospin and the chiral nature of the relativistic particles in the AGNR. Besides, the fewer tunneling peaks in
Fig. 2 compared with Fig. 1 can be observed because of the fewer
propagating modes [19] in the semiconducting AGNR with a energy gap.
Finally, the conductance (in units of 4e 2 /h) as a function of the
barrier range D (in units of a) for the both metallic and semiconducting AGNRs are illustrated in Fig. 3 with the incident electron
763
Fig. 3. The barrier range dependence of the conductance for the metallic AGNR with
W = 6a (solid line) and semiconducting AGNR with W = 7a (dashed line), where
ε = 2γ and V 0 = 3γ .
energy ε = 2γ . As shown in Fig. 3, some regularly periodical tunneling peaks appear for the metallic AGNR when D > 7.4 (the solid
line), while the semiconducting AGNR demonstrates irregularly periodical tunneling peaks (the dashed line) due to the quantum
interference between different propagating modes when D > 6.1.
Physically, this phenomenon can be understood as: when particle
traveling satisfies conventional resonance condition q x D = nπ , it
can pass through the barrier region via resonant tunneling because of the phase accumulation. Generally, the Dirac particles
tunnel more easily through a shorter range potential than a wider
one. However, when the potential range D gets bigger than certain value, the periodical influence on the conductance turns up.
The similar regularly periodical tunneling peaks can be obtained
in a non-chiral zero-gap semiconductor [6], therefore, the metallic AGNR may behave as the traditional semiconductors in some
aspects.
In conclusion, using the Dirac equation with the continuity of
the wave functions at the two interfaces, we have theoretically calculated the conductance and Fano factor by using the Landauer
formula for both the metallic and semiconducting AGNRs containing a rectangular potential barrier. It has been shown that at the
Dirac point both the metallic and semiconducting AGNRs own a
minimum conductance associated with a maximum Fano factor.
The Dirac particles can tunnel more easily through a barrier in the
metallic AGNR than in the semiconducting one. The behavior of
the conductance as a function of the barrier range shows regular
periodical oscillating for the metallic AGNR and irregular for the
semiconducting AGNR, respectively. These results may be helpful
to deeply understand the transport in the nanoribbons and design
the graphene-based nanodevices.
References
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V.
Grigorieva, A.A. Firsov, Science 306 (2004) 666.
[2] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197.
[3] Y.B. Zhang, Y.W. Tan, H.L. Stormer, P. Kim, Nature 438 (2005) 201.
[4] C. Berger, Z.M. Song, X.B. Li, X.S. Wu, N. Brown, C. Naud, D. Mayou, T.B. Li, J.
Hass, A.N. Marchenkov, E.H. Conrad, P.N. First, W.A. Heer, Science 312 (2006)
1191.
[5] T. Ohta, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Science 313 (2006) 951.
[6] M.I. Katsnelson, K.S. Novoselov, A.K. Geim, Nature Phys. 2 (2006) 620.
[7] P.R. Wallace, Phys. Rev. 71 (1947) 622.
[8] L. Brey, H.A. Fertig, Phys. Rev. B 73 (2006) 235411.
[9] A.H.C. Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod.
Phys. 81 (2009) 109.
[10] O. Klein, Z. Phys. 53 (1929) 157.
[11] C. Berger, Z. Song, T. Li, X. Li, A.Y. Ogbazghi, R. Feng, Z. Dai, A.N. Marchenkov,
E.H. Conrad, P.N. First, W.A. de Heer, J. Phys. Chem. B 108 (2004) 19912.
[12] K. Nakada, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 54 (1996)
17954.
764
[13]
[14]
[15]
[16]
[17]
B. Zhou et al. / Physics Letters A 374 (2010) 761–764
K. Wakabayashi, M. Fujita, H. Ajiki, M. Sigrist, Phys. Rev. B 59 (1999) 8271.
M. Ezawa, Phys. Rev. B 73 (2006) 045432.
N.M.R. Peres, A.H. Castro Neto, F. Guinea, Phys. Rev. B 73 (2006) 195411.
Y.W. Son, M.L. Cohen, S.G. Louie, Phys. Rev. Lett. 97 (2006) 216803.
D. Gunlycke, C.T. White, Phys. Rev. B 77 (2008) 115116.
[18] H.X. Zheng, Z.F. Wang, T. Luo, Q.W. Shi, J. Chen, Phys. Rev. B 75 (2007)
165414.
[19] J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, C.W.J. Beenakker, Phys. Rev.
Lett. 96 (2006) 246802.
[20] K. Wakabayashi, Y. Takane, M. Sigrist, Phys. Rev. Lett. 99 (2007) 036601.