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Electronic transport for armchair graphene nanoribbons with a potential barrier
Physics Letters A 374 (2010) 761–764
Contents lists available at ScienceDirect
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 = ±|¯hk| 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.
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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.
Contents lists available at ScienceDirect
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 = ±|¯hk| 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
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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.
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