- Email: [email protected]
A solvable model for electronic transport of a nanowire in the presence of effective impurities
Superlattices and Microstructures 59 (2013) 155–162
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
A solvable model for electronic transport of a nanowire in the presence of effective impurities Mohammad Mardaani ⇑, Hassan Rabani Department of Physics, Faculty of Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran Nanotechnology Research Center, Shahrekord University, 8818634141 Shahrekord, Iran
a r t i c l e
i n f o
Article history: Received 18 December 2012 Received in revised form 13 February 2013 Accepted 2 April 2013 Available online 22 April 2013 Keywords: Impurity Defect Tight-binding Transmission
a b s t r a c t We propose an analytical model to describe the electronic transport properties of a uniform chain including one or two on-site impurities located at arbitrary sites by using Green’s function technique within the tight-binding approach. We reduce some quasi one-dimensional periodic quantum wires to uniform chains including one or two effective on-site impurities with help of the renormalization procedure. Then, we employ the formalism to drive their electronic conductance analytically. For some interesting configurations, we provide a list for on-site energies of effective impurities and all needed quantities. Some defects in a conjugated polymer can also be described by the effective on-site impurities when the system is reduced into a uniform chain. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Today, advances in technology have enabled us to engineer and fabricate nanoscale systems. Therefore, the topic of electron transport through nanostructures and molecular devices has attracted considerable interest in the last few decades, both experimentally [1–4] and theoretically [4–12]. Many authors have studied the effect of defects and impurities on the electronic conductance of quantum wires. Their results often show that an individual defect or impurity can substantially modify the transport properties of the system. For example, the potential strength and the position of impurity influence on the conductance of confined quantum wires [13,14]. In addition to electric or magnetic atoms, the adatoms, vacancies and defects can be interpreted as effective impurities. Moreover, the
⇑ Corresponding author at: Department of Physics, Faculty of Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran. Tel./fax: +98 3814424419. E-mail addresses: [email protected] (M. Mardaani), [email protected] (H. Rabani). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.04.003
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presence of electrical impurities which affect the conductance [14], can cause interesting effects such as the quantum interference [15], antiresonance [13,16], Fano resonance [16,17], localization [18], metallic-semiconductor phase transition [2,3], disorder and dephasing [16,19] and shot noise [20]. These phenomena usually take place in the quasi 1D chains like backbone, comb-like, polymer, nanoribbon and molecular nanowires [16,19–22]. Some extra levels are introduced into the system energy band due to the existence of impurities or defects [13,19]. The quantum interferences in these levels cause the appearance of Fano or anti-resonances in the conductance spectrum. When the energy levels related to the defect are located at the gap area, some new conductance channels may be created that improve the tunneling conductance [14]. The effect of impurities on the electronic conductance of quantum wires has been investigated numerically in most cases, e.g., Refs. [15,21,22]. In this paper, we derive a solvable model to study the electronic transport of some configurations including one or two effective on-site impurities within the tight-binding approach. The model can be useful for testing the computational codes developed for more complicated and realistic systems. This study is based on Landauer’s formula that reveals conductance is proportional to transmission coefficient [5,8]. We provide a series of exact formulas related to some interesting structures. By employing the renormalization technique [23], we generalize the formalism for more complicated quasi 1D molecular wires. Moreover, we apply the formalism to investigate the effect of one or two defects in conjugated polymers. The paper is organized as follows. In Section 2, we drive all analytical formulas regarding the calculation of transport properties for a uniform chain in the presence of one or two on-site impurities. In Section 3, we use the model for two basic examples, i.e., (i) an infinite simple chain attached to a linear or cyclic system as an effective on-site impurity and (ii) a quasi 1D quantum wire including one defect or impurity which can be reduced to a uniform chain after the renormalization procedure. In this section, we summarize the important derived results in two tables. Finally, the results and concluding remarks are given in Section 4. 2. Theoretical framework We consider electrons in a 1D extended quantum wire within the nearest neighbor tight-binding approach. According to Fig. 1, we suppose the center wire includes N reduced sites with renormalized ~W . on-site energies of ~eW and the renormalized hopping terms between the nearest reduced sites of b The on-site and hopping energies of the left (right) lead as well as the contact hopping integral between the left (right) lead and the center wire are eL(R), bL(R) and bWL(R), respectively. Two on-site impurities are located at I and Jth sites with renormalized on-site energies of ~eI and ~eJ , respectively. The relation between the transmission coefficient and the determinant inverse ofthe (dimensionless) e I or two D e I;J impurities is Green’s function matrix of the system, [24], in the presence of one D N N
TðeÞ ¼
~ L Imr ~R 4Imr 2 : e IðI;JÞ DN
ð1Þ
~ LðRÞ is the dimensionless self-energy of the center wire due to the existence of the left (right) where r electrode. We discuss this quantity which depends on the parameters of unreduced (original) system in the next section.
Fig. 1. Schematic diagram showing a reduced uniform quantum wire, including N sites embedded between two semi-infinite ~W , respectively. The parameter ~eIðJÞ simple chains. The renormalized on-site and hopping energies of the center wire are ~eW and b represents the renormalized on-site energy of the impurity or defect on the I(J)th site. Furthermore, eL(R) and bL(R) are the on-site and hopping terms of the left (right) lead, respectively as well as bWL(R) is the left (right) hopping integral of the contacts.
M. Mardaani, H. Rabani / Superlattices and Microstructures 59 (2013) 155–162
157
e I , is In the case of one impurity, the determinant expansion of the inverse Green’s function matrix, D N given by
e I ðr e I1 ðr e NI ð0; r e I2 ðr e NI ð0; r ~ L; r ~ R ; dI Þ ¼ 2ðnW dI Þ D ~ L ; 0Þ D ~ RÞ D ~ L ; 0Þ D ~ RÞ D N e I1 ðr e NI1 ð0; r ~ L ; 0Þ D ~ R Þ; D
ð2Þ
e n is determinant of the inverse Green’s function ~W ; dI ¼ ð~eI ~eW Þ=2b ~W and D where nW ¼ ðe ~eW Þ=2b matrix of an extended reduced chain with n sites which is given by [25]
e n ðr ~ L; r ~ R Þ ¼ Dn ðr ~L þ r ~ R ÞDn1 þ r ~ Lr ~ R Dn2 ; D
ð3Þ
here, Dn = (kn+1 kn1)/(k k1) is determinant inverse Green’s function matrix of an isolated of the 1=2 reduced chain with n sites in which k ¼ nW þ n2W 1 . e I;J , In the case of two impurities, determinant expansion of the inverse Green’s function matrix, D N reads
e I;J ðr e I1 ðr e JI ð0; r e I2 ðr e JI ð0; r ~ L; r ~ R ; dI ; dJ Þ ¼ 2ðnW dI Þ D ~ L ; 0Þ D ~ R ; dJ Þ D ~ L ; 0Þ D ~ R ; dJ Þ D N NI NI e I1 ðr ~ L ; 0ÞDJI1 ~ D NI1 ð0; rR ; dJ Þ;
ð4Þ
~W . By using the above formulas one can calculate the transmission coefficient where dJ ¼ ð~eJ ~eW Þ=2b of periodic quantum wires in the presence of one or two impurities. In order to use this model, after renormalization procedure, all the renormalized hopping energies of the reduced system should be the same. In the next section, we use this formalism for some simple systems. 3. Discussion with two illustrative examples An impurity can influence the local density of states and band energy as well as electronic conductance spectrum especially in 1D systems. For a T-shape junction constructing of a side group molecule and a wire [19], there are two possibilities. One is the eigenenergies of the attached molecule lie in the wire resonance area, the other in gap region. The former decreases conductance and causes the Fano and anti-resonances [19], the latter arises the tunneling conductance due to creating some extra routes [14,19]. Moreover, the transport properties of the system are highly sensitive to the shapes and positions of the attached molecule which leads to design of molecular switches or rheostats [9,19]. In this section, we present two simple examples, i.e., (i) a simple uniform chain in the presence of one effective on-site impurity and (ii) a polyacetylene (PA) molecule including one defect, one effective on-site impurity or a STM potential tip. For this purpose, we first specify the self-energies and renormalized tight-binding parameters presented in the model. Then, we calculate transmission coefficient by using the analytical relations and look for physical aspects appeared in the results. 3.1. A uniform chain including one effective impurity We investigate the influence of one effective impurity on the conductance of an infinite uniform chain. In this case, the on-site and hopping energies of the center wire in Fig. 1 take the following values
~eW ¼ eW ;
~W ¼ b ; b W
where eW and bW are constants and do not depend on the energy. Moreover, the left (right) wire selfenergy, for the simple chain lead, is
b2
r~ LðRÞ ¼ ~ WLðRÞ expðıhLðRÞ Þ; bW bLðRÞ
ð5Þ
where hL(R) = cos1[(e eL(R))/2bL(R)]. Here, eL(R), bL(R) and bWL(R) are the tight-binding parameters of the left (right) lead and contact, respectively. For the ideal chain (bL(R) = bWL(R) = bW and eL(R) = eW) including one effective impurity, the transmission coefficient in Eq. (1), is simplified as
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Table 1 The list of effective impurity on-site energies and the values of dI(e = eW) corresponding to configurations depicted in Fig. 2. The last column is the occupation number at impurity site calculated in the half-band case, when: M ¼ 6; M 1 ¼ M 2 ¼ 3; I ¼ 3; eW 0 ¼ 0, bW 0 ¼ 0:5 eV and bI = 0.4 eV. Here, I0 is the atom site number which connects the chain to the main wire in Fig. 2a. Also, M1 and M2 are the number of atoms for the attached chains in Fig. 2b. Moreover, bI is the hopping term between the attached subsystem and the main wire. dI(e = eW)
~eI ðeÞ
# Fig. 2a Fig. 2b
eW þ
b2I b0W
eW þ
b2I b0W
b2I
DI0 1 DMI0 DM
D
M 1
DM
1
1 2 DM1 þ DM2 b2I eW ; ~eW ¼ eW þ eeW 0
Fig. 2c Fig. 2d Fig. 2e Fig. 2f
TðeÞ ¼
1þ
2bW 0 bW b2I 2bW 0 bW
ðMI0 ÞðI0 1Þ M
2 M11 M12
b2I
2bW eW 0
nI 0.908 0.831 0.920
eW þ
b2I b0W
1
0.908
eW þ
b2I M eeW 0 2bW 0
Mb2I 2b ðe 0 þ2b W W W0 Þ
0.287
eW þ
2b2I b0W
b2I bW 0 bW
0.814
DM1 DM D M2 2
þ1 DM1 DM
!1
d2I
Re 1 n2W
:
ð6Þ
We note that for the infinite chain including one effective impurity, there is not any dependence on I. In Table 1, we list the formulas of ~eI for some interesting configurations shown in Fig. 2. Here, we overlook unnecessary algebraic details. In this table, Dn belongs to the attached chain, W0 , that is a function of nW 0 . For a simple chain including one impurity located at its I-th site, the local density of states at this site can be obtained by the following expression
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2W LDOSI ¼ : pbW 1 n2W þ d2I
ð7Þ
Therefore, the occupation number at site I is
nI ¼
Z
eF
LDOSI de;
ð8Þ
eW 2bW
where eF is Fermi energy of the system. In metallic state of a simple chain (half-band case), when eW = 0, bW = 1 eV, we can numerically compute the integral in Eq. (8) for all configurations of Fig. 2. The results are listed in the last column of Table 1. As can be seen, for the cases of Fig. 2c and e, the occupation number at the impurity site gets the maximum and minimum values, respectively. When dI is independent of energy, one can analytically obtain the following formula in the half-band case
dI nI ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ d2I
ðeW ¼ 0; bW ¼ 1 eVÞ:
ð9Þ
Figures 3a and b present the transmission coefficients and local density of states at impurity site as functions of energy for the configurations of Fig. 2a, d and e. Here, we take all on-site energies in the system equal to zero. Also, the hopping energies in the ideal main wire, attached structure and contact(s) between them are set to 1, 0.5 and 0.4 eV, respectively. Furthermore, for all configurations we choose M = 6 and for Fig. 2a I0 = 3. It can be seen that the variation of T with respect to energy is similar to the variation of LDOSI. The transmission curves for the cases of Fig. 2a and d, are symmetric around zero energy and have some dips corresponding to quasi-eigenenergies of the corresponding attached system. While the transmission for multi-contact case (Fig. 2e) is asymmetric and its value in all energies except in dips is smaller than in two other cases due to more electron scattering.
...
b
...
...
a
159
...
M. Mardaani, H. Rabani / Superlattices and Microstructures 59 (2013) 155–162
...
c
e
d
...
f
...
Fig. 2. Shows some systems that each of which can be converted to a simple chain including one effective impurity. The number of atoms in the attached chains for (a and b) as well as in the rings for (d–f) are M.
Fig. 3. (a) Transmission and (b) local density of states at impurity site vs. energy for three configurations of Fig. 2a, d and e. The used parameters are the same as parameters used in Table 1.
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3.2. A reduced PA molecule including one defect or effective impurity As the second example we consider a reduced PA molecule including one defect [14] or impurity (Fig. 4a and b) which is embedded between two simple metallic leads. As we have shown in our previous works [26,27], within the tight-binding approach, the Hamiltonian of some conjugated polymers like PPP and PPy can be converted to the Hamiltonian of PA-like chains (see Table 2). We use the reduced PA phrase for the system which is convertible to a chain with alternative effective hopping energies. By using the renormalization method [23], such a system can be easily changed to a reduced uniform chain containing one effective on-site impurity. Therefore, the effective on-site and hopping energies of the new chain, respectively, are [24]
~eW ¼ eW þ
b2W þ b02 W ; e eW
ð10Þ
and 0 ~W ¼ bW bW ; b e eW
ð11Þ
where eW, bW and b0W are the renormalized on-site, single and double hopping terms of the reduced PA, respectively. We list these parameters for PA, PPP and PPy in Table 2. In this case where the reduced PA is attached to two metallic chains at its ends, the left (right) self-energy becomes
r~ LðRÞ ¼ ~
b2WR
bW bLðRÞ
expðıhLðRÞ Þ
b2W : ~ bW ðe eW Þ
ð12Þ
The last term in the above equation guarantees that the effective Hamiltonian of the center system gets the Hamiltonian form of uniform chain. We note that, after the renormalization procedure, which converts the reduced PA to a uniform chain, the renormalized on-site energy of impurity is
~eI ¼ eI þ
2b2W
e eW
:
ð13Þ
In the presence of an external potential or a STM tip depicted in Fig. 4b, one can derive
~eI ¼ eI þ
b2W þ b02 W : e eW
ð14Þ
By using the presented relations in this section, the extension of formalism for the structures including two effective impurities is straightforward. Fig. 5 displays log(T) as a function of energy for a structure depicted in Fig. 4a, when a benzene ring plays the role of effective impurity, for three different positions of benzene ring (I = 2, 4, 6). Here, the length of PA molecule is 11 unit cells, all on-site energies in the system are taken zero, the hopping energies for the single and double bands of PA are chosen b = 0.836 eV and b0 = 1.164 eV like Ref. [14], respectively; and finally for the leads and contacts we set bL(R) = 1 eV, bWL(R) = 0.8 eV and
Fig. 4. Model systems with PA molecule connected to two metallic leads. (a) PA molecule includes one defect on the hopping which is interpreted as an effective impurity on the Ith site. (b) A STM tip effects only on the on-site energy of the Ith site.
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Table 2 The renormalized on-site and hopping energies of some conjugated polymers in terms of carbon on-site energy (eC), carbon– carbon single (b) and double (b0 ) hopping bond integrals. Here, eN and bCN are the nitrogen on-site energy and carbon-nitrogen band hopping integral, respectively. #
Polyacetylene (PA)
eW
eC
Poly(p-phenylene) (PPP)
b0W
0
b
bb ðeeC Þ2 b2
bW
b
b
eC þ
b02 ðeeC Þ ðeeC Þ2 b2 02
þ
þ
b2 ðeeC Þ ðeeC Þ2 b02
0 2
bb ðeeC Þ2 b02
Polypyrrole (PPy) 2
02
CÞ eC þ ebCNeN þ ðebeðeÞ2eb 2 C
b
2
02
bb
CN eeN þ ðeeC Þ2 b2
b
Fig. 5. log(T) as a function of energy for three different positions (I = 2, 4, 6) of an attached benzene molecule to an extended PA molecule with the length of N = 11. Here, we choose: eW = 0, b = bI = 0.836 eV, b0 = 1.164 eV, bL(R) = 1 eV and bWL(R) = 0.8 eV.
bI = 0.836 eV. There is an antiresonance around e = 1 eV in each curve which is caused from existence of the benzene ring. Here, the important result, as has also been shown numerically in Ref. [14] is the tuning of transmission at the zero energy by the variation of benzene ring position. 4. Conclusions We demonstrate an analytical approach based on the nearest neighbor tight-binding approximation to study the electronic conductance of a 1D quantum wire in the presence of one or two effective impurities located at arbitrary sites. The formalism can be applied on any system which is convertible to a reduced uniform chain including one or two effective on-site impurities. We obtain the effective on-site energies, local density of states at impurity site and transmission coefficient in terms of incoming electron energy and the system tight-binding parameters for some selected configurations (Table 1). The occupation number on the impurity site can be determined by using the value of the system Fermi energy. The formalism is also useful to analyze the effect of one or two defects on the electronic transport properties of some conjugated polymers such as polyacetylene, poly(p-phenylene) and polypyrrole (Table 2). Acknowledgments We acknowledge the Iranian Nanotechnology Initiative for its partial financial support. This work has also been supported by Shahrekord University through a research fund.
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References [1] N.J. Tao, Nat. Nanotechnol. 1 (2006) 173–181. [2] L. Liu, S. Ryu, M.R. Tomasik, E. Stolyarova, N. Jung, M.S. Hybertsen, M.L. Steigerwald, L.E. Brus, G.W. Flynn, Nano Lett. 8 (2008) 1965–1970. [3] D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, S.V. Morozov, P. Blake, M.P. Halsall, A.C. Ferrari, D.W. Boukhvalov, M.I. Katsnelson, A.K. Geim, K.S. Novoselov, Science 323 (2009) 610–613. [4] D.K. Ferry, S.M. Goodnick, Transport in Nanostructures, Cambridge University Press, Cambridge, 1997. [5] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [6] E.N. Economou, Green’s Functions in Quantum Physics, Springer-Verlag, New York, 1990. [7] W. Barford, Electronic and Optical Properties of Conjugated Polymers, Oxford University Press, New York, 2005. [8] N.A. Zimbovskaya, M.R. Pederson, Phys. Rep. 509 (2011) 1–87. [9] P. Sautet, C. Joachim, Phys. Rev. B 38 (1988) 12238–12247. [10] A. Onipko, Y. Klymenko, L. Malysheva, Mater. Sci. Eng. C 8–9 (1999) 273–281. [11] A. Onipko, Phys. Rev. B 59 (1999) 9995–10006. [12] H. Rabani, M. Mardaani, Solid State Sci. 14 (2012) 1100–1105. [13] X.R. Wang, Y. Wang, Z.Z. Sun, Phys. Rev. B 65 (2002) 193402. [14] D. Nozaki, H.M. Pastawski, G. Cuniberti, New J. Phys. 12 (2010) 063004. [15] T. Hansen, G.C. Solomon, D.Q. Andrews, M.A. Ratner, J. Chem. Phys. 131 (2009) 194704. [16] D. Nozaki, C.G. da Rocha, H.M. Pastawski, G. Cuniberti, Phys. Rev. B 85 (2012) 155327. [17] A. Chakrabarti, Phys. Lett. A 366 (2007) 507512; A. Chakrabarti, Phys. Rev. B 74 (2006) 205315. [18] H. Cheraghchi, S.M. Fazeli, K. Esfarjani, Phys. Rev. B 72 (2005) 174207. [19] D. Nozaki, H. Sevinçli, S.M. Avdoshenko, R. Gutierrez, G. Cuniberti, arXiv:1203.5269v3. [20] J.H. Ojeda, M. Pacheco, L. Rosales, P.A. Orellana, Org. Electron. 13 (2012) 1420–1429. [21] J.O. Vasseur, P.A. Deymier, L. Dobrzynski, J. Choi, J. Phys.: Condens. Matter 10 (1998) 8973–8981. [22] V. Mujica, M.A. Ratner, Chem. Phys. 326 (2006) 197–203. [23] H. Rabani, M. Mardaani, Solid State Commun. 152 (2012) 235–239. [24] M. Mardaani, H. Rabani, A. Esmaeili, Solid State Commun. 151 (2011) 928–932. [25] M. Mardaani, K. Esfarjani, Physica E 25 (2004) 119–130. [26] M. Mardaani, H. Rabani, J. Magn. Magn. Mater. 331 (2013) 28–32. [27] M. Mardaani, S. Ansary, Iran. J. Phys. Res. 12 (2012) 45–50.
Contents lists available at SciVerse ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
A solvable model for electronic transport of a nanowire in the presence of effective impurities Mohammad Mardaani ⇑, Hassan Rabani Department of Physics, Faculty of Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran Nanotechnology Research Center, Shahrekord University, 8818634141 Shahrekord, Iran
a r t i c l e
i n f o
Article history: Received 18 December 2012 Received in revised form 13 February 2013 Accepted 2 April 2013 Available online 22 April 2013 Keywords: Impurity Defect Tight-binding Transmission
a b s t r a c t We propose an analytical model to describe the electronic transport properties of a uniform chain including one or two on-site impurities located at arbitrary sites by using Green’s function technique within the tight-binding approach. We reduce some quasi one-dimensional periodic quantum wires to uniform chains including one or two effective on-site impurities with help of the renormalization procedure. Then, we employ the formalism to drive their electronic conductance analytically. For some interesting configurations, we provide a list for on-site energies of effective impurities and all needed quantities. Some defects in a conjugated polymer can also be described by the effective on-site impurities when the system is reduced into a uniform chain. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Today, advances in technology have enabled us to engineer and fabricate nanoscale systems. Therefore, the topic of electron transport through nanostructures and molecular devices has attracted considerable interest in the last few decades, both experimentally [1–4] and theoretically [4–12]. Many authors have studied the effect of defects and impurities on the electronic conductance of quantum wires. Their results often show that an individual defect or impurity can substantially modify the transport properties of the system. For example, the potential strength and the position of impurity influence on the conductance of confined quantum wires [13,14]. In addition to electric or magnetic atoms, the adatoms, vacancies and defects can be interpreted as effective impurities. Moreover, the
⇑ Corresponding author at: Department of Physics, Faculty of Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran. Tel./fax: +98 3814424419. E-mail addresses: [email protected] (M. Mardaani), [email protected] (H. Rabani). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.04.003
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presence of electrical impurities which affect the conductance [14], can cause interesting effects such as the quantum interference [15], antiresonance [13,16], Fano resonance [16,17], localization [18], metallic-semiconductor phase transition [2,3], disorder and dephasing [16,19] and shot noise [20]. These phenomena usually take place in the quasi 1D chains like backbone, comb-like, polymer, nanoribbon and molecular nanowires [16,19–22]. Some extra levels are introduced into the system energy band due to the existence of impurities or defects [13,19]. The quantum interferences in these levels cause the appearance of Fano or anti-resonances in the conductance spectrum. When the energy levels related to the defect are located at the gap area, some new conductance channels may be created that improve the tunneling conductance [14]. The effect of impurities on the electronic conductance of quantum wires has been investigated numerically in most cases, e.g., Refs. [15,21,22]. In this paper, we derive a solvable model to study the electronic transport of some configurations including one or two effective on-site impurities within the tight-binding approach. The model can be useful for testing the computational codes developed for more complicated and realistic systems. This study is based on Landauer’s formula that reveals conductance is proportional to transmission coefficient [5,8]. We provide a series of exact formulas related to some interesting structures. By employing the renormalization technique [23], we generalize the formalism for more complicated quasi 1D molecular wires. Moreover, we apply the formalism to investigate the effect of one or two defects in conjugated polymers. The paper is organized as follows. In Section 2, we drive all analytical formulas regarding the calculation of transport properties for a uniform chain in the presence of one or two on-site impurities. In Section 3, we use the model for two basic examples, i.e., (i) an infinite simple chain attached to a linear or cyclic system as an effective on-site impurity and (ii) a quasi 1D quantum wire including one defect or impurity which can be reduced to a uniform chain after the renormalization procedure. In this section, we summarize the important derived results in two tables. Finally, the results and concluding remarks are given in Section 4. 2. Theoretical framework We consider electrons in a 1D extended quantum wire within the nearest neighbor tight-binding approach. According to Fig. 1, we suppose the center wire includes N reduced sites with renormalized ~W . on-site energies of ~eW and the renormalized hopping terms between the nearest reduced sites of b The on-site and hopping energies of the left (right) lead as well as the contact hopping integral between the left (right) lead and the center wire are eL(R), bL(R) and bWL(R), respectively. Two on-site impurities are located at I and Jth sites with renormalized on-site energies of ~eI and ~eJ , respectively. The relation between the transmission coefficient and the determinant inverse ofthe (dimensionless) e I or two D e I;J impurities is Green’s function matrix of the system, [24], in the presence of one D N N
TðeÞ ¼
~ L Imr ~R 4Imr 2 : e IðI;JÞ DN
ð1Þ
~ LðRÞ is the dimensionless self-energy of the center wire due to the existence of the left (right) where r electrode. We discuss this quantity which depends on the parameters of unreduced (original) system in the next section.
Fig. 1. Schematic diagram showing a reduced uniform quantum wire, including N sites embedded between two semi-infinite ~W , respectively. The parameter ~eIðJÞ simple chains. The renormalized on-site and hopping energies of the center wire are ~eW and b represents the renormalized on-site energy of the impurity or defect on the I(J)th site. Furthermore, eL(R) and bL(R) are the on-site and hopping terms of the left (right) lead, respectively as well as bWL(R) is the left (right) hopping integral of the contacts.
M. Mardaani, H. Rabani / Superlattices and Microstructures 59 (2013) 155–162
157
e I , is In the case of one impurity, the determinant expansion of the inverse Green’s function matrix, D N given by
e I ðr e I1 ðr e NI ð0; r e I2 ðr e NI ð0; r ~ L; r ~ R ; dI Þ ¼ 2ðnW dI Þ D ~ L ; 0Þ D ~ RÞ D ~ L ; 0Þ D ~ RÞ D N e I1 ðr e NI1 ð0; r ~ L ; 0Þ D ~ R Þ; D
ð2Þ
e n is determinant of the inverse Green’s function ~W ; dI ¼ ð~eI ~eW Þ=2b ~W and D where nW ¼ ðe ~eW Þ=2b matrix of an extended reduced chain with n sites which is given by [25]
e n ðr ~ L; r ~ R Þ ¼ Dn ðr ~L þ r ~ R ÞDn1 þ r ~ Lr ~ R Dn2 ; D
ð3Þ
here, Dn = (kn+1 kn1)/(k k1) is determinant inverse Green’s function matrix of an isolated of the 1=2 reduced chain with n sites in which k ¼ nW þ n2W 1 . e I;J , In the case of two impurities, determinant expansion of the inverse Green’s function matrix, D N reads
e I;J ðr e I1 ðr e JI ð0; r e I2 ðr e JI ð0; r ~ L; r ~ R ; dI ; dJ Þ ¼ 2ðnW dI Þ D ~ L ; 0Þ D ~ R ; dJ Þ D ~ L ; 0Þ D ~ R ; dJ Þ D N NI NI e I1 ðr ~ L ; 0ÞDJI1 ~ D NI1 ð0; rR ; dJ Þ;
ð4Þ
~W . By using the above formulas one can calculate the transmission coefficient where dJ ¼ ð~eJ ~eW Þ=2b of periodic quantum wires in the presence of one or two impurities. In order to use this model, after renormalization procedure, all the renormalized hopping energies of the reduced system should be the same. In the next section, we use this formalism for some simple systems. 3. Discussion with two illustrative examples An impurity can influence the local density of states and band energy as well as electronic conductance spectrum especially in 1D systems. For a T-shape junction constructing of a side group molecule and a wire [19], there are two possibilities. One is the eigenenergies of the attached molecule lie in the wire resonance area, the other in gap region. The former decreases conductance and causes the Fano and anti-resonances [19], the latter arises the tunneling conductance due to creating some extra routes [14,19]. Moreover, the transport properties of the system are highly sensitive to the shapes and positions of the attached molecule which leads to design of molecular switches or rheostats [9,19]. In this section, we present two simple examples, i.e., (i) a simple uniform chain in the presence of one effective on-site impurity and (ii) a polyacetylene (PA) molecule including one defect, one effective on-site impurity or a STM potential tip. For this purpose, we first specify the self-energies and renormalized tight-binding parameters presented in the model. Then, we calculate transmission coefficient by using the analytical relations and look for physical aspects appeared in the results. 3.1. A uniform chain including one effective impurity We investigate the influence of one effective impurity on the conductance of an infinite uniform chain. In this case, the on-site and hopping energies of the center wire in Fig. 1 take the following values
~eW ¼ eW ;
~W ¼ b ; b W
where eW and bW are constants and do not depend on the energy. Moreover, the left (right) wire selfenergy, for the simple chain lead, is
b2
r~ LðRÞ ¼ ~ WLðRÞ expðıhLðRÞ Þ; bW bLðRÞ
ð5Þ
where hL(R) = cos1[(e eL(R))/2bL(R)]. Here, eL(R), bL(R) and bWL(R) are the tight-binding parameters of the left (right) lead and contact, respectively. For the ideal chain (bL(R) = bWL(R) = bW and eL(R) = eW) including one effective impurity, the transmission coefficient in Eq. (1), is simplified as
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Table 1 The list of effective impurity on-site energies and the values of dI(e = eW) corresponding to configurations depicted in Fig. 2. The last column is the occupation number at impurity site calculated in the half-band case, when: M ¼ 6; M 1 ¼ M 2 ¼ 3; I ¼ 3; eW 0 ¼ 0, bW 0 ¼ 0:5 eV and bI = 0.4 eV. Here, I0 is the atom site number which connects the chain to the main wire in Fig. 2a. Also, M1 and M2 are the number of atoms for the attached chains in Fig. 2b. Moreover, bI is the hopping term between the attached subsystem and the main wire. dI(e = eW)
~eI ðeÞ
# Fig. 2a Fig. 2b
eW þ
b2I b0W
eW þ
b2I b0W
b2I
DI0 1 DMI0 DM
D
M 1
DM
1
1 2 DM1 þ DM2 b2I eW ; ~eW ¼ eW þ eeW 0
Fig. 2c Fig. 2d Fig. 2e Fig. 2f
TðeÞ ¼
1þ
2bW 0 bW b2I 2bW 0 bW
ðMI0 ÞðI0 1Þ M
2 M11 M12
b2I
2bW eW 0
nI 0.908 0.831 0.920
eW þ
b2I b0W
1
0.908
eW þ
b2I M eeW 0 2bW 0
Mb2I 2b ðe 0 þ2b W W W0 Þ
0.287
eW þ
2b2I b0W
b2I bW 0 bW
0.814
DM1 DM D M2 2
þ1 DM1 DM
!1
d2I
Re 1 n2W
:
ð6Þ
We note that for the infinite chain including one effective impurity, there is not any dependence on I. In Table 1, we list the formulas of ~eI for some interesting configurations shown in Fig. 2. Here, we overlook unnecessary algebraic details. In this table, Dn belongs to the attached chain, W0 , that is a function of nW 0 . For a simple chain including one impurity located at its I-th site, the local density of states at this site can be obtained by the following expression
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n2W LDOSI ¼ : pbW 1 n2W þ d2I
ð7Þ
Therefore, the occupation number at site I is
nI ¼
Z
eF
LDOSI de;
ð8Þ
eW 2bW
where eF is Fermi energy of the system. In metallic state of a simple chain (half-band case), when eW = 0, bW = 1 eV, we can numerically compute the integral in Eq. (8) for all configurations of Fig. 2. The results are listed in the last column of Table 1. As can be seen, for the cases of Fig. 2c and e, the occupation number at the impurity site gets the maximum and minimum values, respectively. When dI is independent of energy, one can analytically obtain the following formula in the half-band case
dI nI ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ d2I
ðeW ¼ 0; bW ¼ 1 eVÞ:
ð9Þ
Figures 3a and b present the transmission coefficients and local density of states at impurity site as functions of energy for the configurations of Fig. 2a, d and e. Here, we take all on-site energies in the system equal to zero. Also, the hopping energies in the ideal main wire, attached structure and contact(s) between them are set to 1, 0.5 and 0.4 eV, respectively. Furthermore, for all configurations we choose M = 6 and for Fig. 2a I0 = 3. It can be seen that the variation of T with respect to energy is similar to the variation of LDOSI. The transmission curves for the cases of Fig. 2a and d, are symmetric around zero energy and have some dips corresponding to quasi-eigenenergies of the corresponding attached system. While the transmission for multi-contact case (Fig. 2e) is asymmetric and its value in all energies except in dips is smaller than in two other cases due to more electron scattering.
...
b
...
...
a
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...
c
e
d
...
f
...
Fig. 2. Shows some systems that each of which can be converted to a simple chain including one effective impurity. The number of atoms in the attached chains for (a and b) as well as in the rings for (d–f) are M.
Fig. 3. (a) Transmission and (b) local density of states at impurity site vs. energy for three configurations of Fig. 2a, d and e. The used parameters are the same as parameters used in Table 1.
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3.2. A reduced PA molecule including one defect or effective impurity As the second example we consider a reduced PA molecule including one defect [14] or impurity (Fig. 4a and b) which is embedded between two simple metallic leads. As we have shown in our previous works [26,27], within the tight-binding approach, the Hamiltonian of some conjugated polymers like PPP and PPy can be converted to the Hamiltonian of PA-like chains (see Table 2). We use the reduced PA phrase for the system which is convertible to a chain with alternative effective hopping energies. By using the renormalization method [23], such a system can be easily changed to a reduced uniform chain containing one effective on-site impurity. Therefore, the effective on-site and hopping energies of the new chain, respectively, are [24]
~eW ¼ eW þ
b2W þ b02 W ; e eW
ð10Þ
and 0 ~W ¼ bW bW ; b e eW
ð11Þ
where eW, bW and b0W are the renormalized on-site, single and double hopping terms of the reduced PA, respectively. We list these parameters for PA, PPP and PPy in Table 2. In this case where the reduced PA is attached to two metallic chains at its ends, the left (right) self-energy becomes
r~ LðRÞ ¼ ~
b2WR
bW bLðRÞ
expðıhLðRÞ Þ
b2W : ~ bW ðe eW Þ
ð12Þ
The last term in the above equation guarantees that the effective Hamiltonian of the center system gets the Hamiltonian form of uniform chain. We note that, after the renormalization procedure, which converts the reduced PA to a uniform chain, the renormalized on-site energy of impurity is
~eI ¼ eI þ
2b2W
e eW
:
ð13Þ
In the presence of an external potential or a STM tip depicted in Fig. 4b, one can derive
~eI ¼ eI þ
b2W þ b02 W : e eW
ð14Þ
By using the presented relations in this section, the extension of formalism for the structures including two effective impurities is straightforward. Fig. 5 displays log(T) as a function of energy for a structure depicted in Fig. 4a, when a benzene ring plays the role of effective impurity, for three different positions of benzene ring (I = 2, 4, 6). Here, the length of PA molecule is 11 unit cells, all on-site energies in the system are taken zero, the hopping energies for the single and double bands of PA are chosen b = 0.836 eV and b0 = 1.164 eV like Ref. [14], respectively; and finally for the leads and contacts we set bL(R) = 1 eV, bWL(R) = 0.8 eV and
Fig. 4. Model systems with PA molecule connected to two metallic leads. (a) PA molecule includes one defect on the hopping which is interpreted as an effective impurity on the Ith site. (b) A STM tip effects only on the on-site energy of the Ith site.
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Table 2 The renormalized on-site and hopping energies of some conjugated polymers in terms of carbon on-site energy (eC), carbon– carbon single (b) and double (b0 ) hopping bond integrals. Here, eN and bCN are the nitrogen on-site energy and carbon-nitrogen band hopping integral, respectively. #
Polyacetylene (PA)
eW
eC
Poly(p-phenylene) (PPP)
b0W
0
b
bb ðeeC Þ2 b2
bW
b
b
eC þ
b02 ðeeC Þ ðeeC Þ2 b2 02
þ
þ
b2 ðeeC Þ ðeeC Þ2 b02
0 2
bb ðeeC Þ2 b02
Polypyrrole (PPy) 2
02
CÞ eC þ ebCNeN þ ðebeðeÞ2eb 2 C
b
2
02
bb
CN eeN þ ðeeC Þ2 b2
b
Fig. 5. log(T) as a function of energy for three different positions (I = 2, 4, 6) of an attached benzene molecule to an extended PA molecule with the length of N = 11. Here, we choose: eW = 0, b = bI = 0.836 eV, b0 = 1.164 eV, bL(R) = 1 eV and bWL(R) = 0.8 eV.
bI = 0.836 eV. There is an antiresonance around e = 1 eV in each curve which is caused from existence of the benzene ring. Here, the important result, as has also been shown numerically in Ref. [14] is the tuning of transmission at the zero energy by the variation of benzene ring position. 4. Conclusions We demonstrate an analytical approach based on the nearest neighbor tight-binding approximation to study the electronic conductance of a 1D quantum wire in the presence of one or two effective impurities located at arbitrary sites. The formalism can be applied on any system which is convertible to a reduced uniform chain including one or two effective on-site impurities. We obtain the effective on-site energies, local density of states at impurity site and transmission coefficient in terms of incoming electron energy and the system tight-binding parameters for some selected configurations (Table 1). The occupation number on the impurity site can be determined by using the value of the system Fermi energy. The formalism is also useful to analyze the effect of one or two defects on the electronic transport properties of some conjugated polymers such as polyacetylene, poly(p-phenylene) and polypyrrole (Table 2). Acknowledgments We acknowledge the Iranian Nanotechnology Initiative for its partial financial support. This work has also been supported by Shahrekord University through a research fund.
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