- Email: [email protected]
Low- and high-field transport studies for semiconducting carbon nanotubes
ARTICLE IN PRESS
Physica E 34 (2006) 666–669 www.elsevier.com/locate/physe
Low- and high-field transport studies for semiconducting carbon nanotubes M.Z. Kausera,, A. Vermab, P.P. Rudena a
Department of ECE, University of Minnesota, Minneapolis, MN 55455, USA b School of ECE, Georgia Institute of Technology, Atlanta, GA 30332, USA Available online 19 April 2006
Abstract We calculate low- and high-field electron transport characteristics for semiconducting, single-wall, zigzag carbon nanotubes with diameters ranging from small to relatively large. The Boltzmann transport equation is solved directly using an iterative technique and indirectly by the ensemble Monte Carlo method. The basis for the transport calculation is provided by electronic structure calculations within the framework of a simple tight-binding model. Scattering mechanisms considered are due to the electron–phonon interactions involving longitudinal acoustic, longitudinal optic, and radial breathing-mode phonons. Results show significant effects of tube diameter on both low- and high-field transport characteristics. r 2006 Elsevier B.V. All rights reserved. PACS: 73.63.fg; 73.63.b; 73.22.f Keywords: Carbon nanotubes; Boltzmann transport; Ensemble Monte Carlo
1. Introduction Carbon nanotubes (CNTs) have demonstrated excellent potential to be used as basic building blocks in nanoscale devices [1,2]. Their electronic properties vary widely with tube chirality and diameter [3]. However, data on the transport properties of CNTs are still relatively sparse. Some theoretical studies have been made employing different methods, including analytical solution of the Boltzmann transport equation (BTE) [4] and Monte Carlo simulations [5–7]. Experimental work demonstrated the presence of strong longitudinal optic (LO) [8] and radial breathing-mode (RBM) [9] electron–phonon scattering in CNTs. Recent work on ensemble Monte Carlo (EMC) modeling of electron transport in CNTs has focused on the scattering of electrons due to longitudinal acoustic (LA), LO, and RBM phonons [6,7]. However, these studies only considered CNTs with relatively small diameter. Here we report on both low- and high-field transport properties for semiconducting, zigzag, single-wall CNTs Corresponding author. Tel.: +1 612 624 8545; fax: +1 612 625 4583.
E-mail address: [email protected] (M.Z. Kauser). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.055
with a range of diameters, from relatively small to large. The BTE is solved directly using an iterative technique (Rode’s method [10]) and indirectly by the EMC method in the low- and high-field regimes, respectively. Both steadystate and transient conditions are investigated for high fields. The common basis for both techniques is provided by electronic structure calculations in the framework of a tight-binding model. Scattering is due to electron–phonon interactions involving LA, LO, and RBM phonons. The two semiconducting zigzag CNT families, (3p+1,0) and (3p+2,0) with p a positive integer, are explored. Calculations are done for (n,0) CNTs, with n varying from 10 to 50 (corresponding to diameters d ¼ 0.77–3.86 nm), and for temperatures ranging from 100 to 600 K. 2. Model description 2.1. Electronic bandstructures The basis for the transport calculations is provided by the energy bandstructure obtained within the framework of the tight-binding model taking into account the corrections due to the curvature of the CNT [6]. For both zigzag CNT
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669
2.2. Phonon dispersions LA and LO phonon branches are obtained by fitting the phonon dispersions of graphene with polynomials and then applying the method of zone-folding to obtain the phonon dispersion relationships of the CNTs [6]. The radial breathing-mode phonons are evaluated for zigzag CNTs based on separate lattice dynamical calculations [7]. For this work only the lowest RBM phonon (m ¼ 0, where m is the phonon branch index) is considered. 2.3. Electron–phonon scattering Deformation potential induced electron and phonon scattering rates are evaluated by application of Fermi’s golden rule. Details of the LA, LO, and RBM phonon scattering rates and selection rules for Normal and Umklapp processes can be found in our earlier works [6,7]. The RBM-electron-interaction deformation potentials for (3p+1,0) and (3p+2,0) CNTs can be expressed as 3pþ2 D3pþ1 RBM ¼ A1 =d and DRBM ¼ A2 =d, where A1 ¼ 10.05 eV and A2 ¼ 8.05 eV are obtained from fit to recently reported results of ab initio calculations [11]. Scattering rates are evaluated for the lowest six and fourteen (two-fold spin degenerate) subbands for smalland large-diameter CNTs, respectively. 2.4. Transport calculations
Gaussian distribution. Standard EMC methodology and algorithms are applied [12,13]. 3. Results and discussion 3.1. Low-field transport The low-field mobility as a function of diameter for different temperatures is plotted in Fig. 1. It can be seen that for a particular temperature the low-field mobility increases with increasing CNT diameter. This is due to a reduction in the scattering rate, which decreases as a combined effect of decreasing effective mass and increasing linear mass density with increasing tube diameter. Another notable feature is that the mobility of (3p+1,0) CNT is larger than that of (3p+2,0) CNT for a particular p. In addition, for a particular CNT the mobility decreases with increasing temperature as m / T b , where b varies from 0.75 to 0.89 for the range of CNTs considered in this work. These calculated low-field mobility values are consistent with previous theoretical [4] and experimental [14] work. 3.2. High-field transport 3.2.1. Steady state Fig. 2 shows the velocity vs. electric field plots at room temperature for small diameter CNTs, i.e. (10,0) and (11,0). Results for relatively large diameter CNTs, i.e. (49,0) and (50,0), are shown in the inset. As can be seen, (3p+2,0) CNTs have a smaller velocity at the same electric field compared to the (3p+1,0) CNTs for the same p. This difference however gets progressively smaller with increasing diameter. Both families show saturation of the electron velocity at intermediate electric fields. This phenomenon
d (nm) 1.0
15 µ (x104 cm2V-1s-1)
families the lowest-energy subbands are two-fold degenerate (fourfold including spin) and correspond to indices n1 ¼ 7(2p+1). The second and third subbands correspond to indices n2 ¼ 72p and n3 ¼ 7(2p+2) for (3p+1.0) and n2 ¼ 7(2p+2) and n3 ¼ 72p for (3p+2.0) CNTs, respectively, and have the same level of degeneracy. The intersubband energy differences decrease with increasing tube diameter. The effective mass of the lowest conduction subband is smaller for the (3p+1,0) group compared to that of the (3p+2,0) group for small diameter tubes. However, they become comparable for both groups for large diameters. This has a strong effect on low-field mobility, as will be discussed later. A relatively large difference between the effective masses of the second and the lowest subband is obtained for small diameter (3p+1,0) CNTs compared to (3p+2,0) CNTs. However the difference between the effective masses of the third and the lowest subband is smaller for (3p+1,0) CNTs compared to (3p+2,0) CNTs.
667
1.5
2.0
2.5
3.0
3.5
100 K 300 K 600 K
10
5
The low-field transport properties of different CNTs for different temperatures are calculated by solving the steadystate BTE using Rode’s iterative technique [10]. Only the lowest conduction band is considered for the calculation. All the intrasubband electron scattering mechanisms (m ¼ 0 LA, RBM, and LO phonons) are included. EMC simulations are performed for electrons injected into the nanotubes at time t ¼ 0 with an equilibrium
0 10
20
30 n
40
50
Fig. 1. Low-field mobility as a function of diameter or the zigzag index n at three different temperatures. Solid and dashed lines represent (3p+1,0) and (3p+2,0) CNTs, respectively.
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669
668
7
3.5
(10,0) CNT
8 kV cm-1 6 Velocity (x107 cm s-1)
Velocity (x107 cm s-1)
3.0 (11,0) CNT
2.5
6
2.0
(49,0) CNT
5
1.5
(50,0) CNT
4 3
1.0
2
0.5
T= 300 K
0
100
4 25 kV cm-1
T= 300 K
3 2
1 0
0.0
5
150
0
10
20
200
30
250
40
50
300
Electric field (kV cm-1) Fig. 2. Average steady-state drift velocity vs. electric field at 300 K for (10,0) and (11,0) CNTs (main panel) and for (49,0) and (50,0) CNTs (inset).
has been demonstrated earlier to be due to LA and LO phonon emission and the resulting streaming motion [4,6]. The saturation in electron velocity has been observed in high-voltage experiments [15]. The saturation electric field, Fsat, is vastly different for CNTs with different diameter. Fsat is found to decrease with increasing tube diameter, as seen in Fig. 2. As the energy required for the intersubband phonon emission decreases with diameter, electrons with smaller effective mass gain enough energy at relatively low field to emit phonons and enter streaming motion [6]. Another important noticeable transport phenomenon is the negative differential mobility (NDM). As can be seen from Fig. 2, (3p+1,0) CNTs show a relatively more pronounced NDM than (3p+2,0) CNTs for small-diameter tubes, whereas it is more or less similar and pronounced for both CNT families at larger diameter. Due to the reduction of the intersubband energy differences with tube diameter the higher subbands become more readily occupied for large diameter tubes. 3.2.2. Transients Fig. 3 shows the average transient velocity vs. time for various CNTs at room temperature. It can be seen that due to lower effective mass of the bigger tubes, the peak velocity increases with increasing tube diameter. The oscillation is more pronounced and the settling time is longer in larger diameter tubes. This is due to lower damping, because the LA and RBM scattering rates, which act as the damping mechanism, decrease with increasing tube diameter. 4. Conclusions Charge transport properties of different diameter zigzag CNTs have been investigated. These transport properties are not only a function of the diameter but also of the
(10,0) CNT (11,0) CNT (49,0) CNT (50,0) CNT
1 0 0.0
0.2
0.4 0.6 Time (ps)
0.8
1.0
Fig. 3. Average transient drift velocity vs. time for (10,0), (11,0), (49,0), and (50,0) CNTs at room temperature. (10,0) and (11,0) CNTs are under 25 kV cm1 whereas (49,0) and (50,0) CNTs are under 8 kV cm1 electric field.
family to which the CNTs belong, i.e. (3p+1,0) and (3p+2,0). In general, (3p+1,0) CNTs are superior in terms of low-field mobility and high-field velocity. Similarly, larger diameter CNTs are superior to smaller diameter CNTs. Acknowledgements Access to the facilities of the Minnesota Supercomputing Institute is gratefully acknowledged. This work is partially supported by NSF-ECS. References [1] R. Martel, T. Schmidt, H.R. Shea, T. Hertel, Ph. Avouris, Appl. Phys. Lett. 73 (1998) 2447. [2] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317. [3] R. Saito, M.S. Dresselhaus, G. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [4] V. Perebeinos, J. Tersoff, Ph. Avouris, Phys. Rev. Lett. 94 (2005) 086802. [5] G. Pennington, N. Goldsman, Phys. Rev. B 68 (2003) 045426. [6] A. Verma, M.Z. Kauser, P.P. Ruden, J. Appl. Phys. 97 (2005) 114319; A. Verma, M. Z. Kauser, B. W. Lee, K. F. Brennan, P. P. Ruden, in: Proceedings of the 27th International Conference on Physics of Semiconductors, ICPS, Flagstaff, AZ, 2004, AIP Conf. Proc. 772 (2005) 1049. [7] A. Verma, M.Z. Kauser, P.P. Ruden, Appl. Phys. Lett. 87 (2005) 123101. [8] A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, H. Dai, Phys. Rev. Lett. 92 (2004) 106804; M. Freitag, V. Perebeinos, J. Chen, A. Stein, J.C. Tsang, J.A. Misewich, R. Martel, Ph. Avouris, Nanoletters 4 (2004) 1063. [9] B.J. LeRoy, S.G. Lemay, J. Kong, C. Dekker, Nature 432 (2004) 371. [10] D.L. Rode, Low-field transport in semiconductors, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, 10, Academic Press, New York, 1975.
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669 [11] M. Machon, S. Reich, H. Telg, J. Maultzsch, P. Ordejon, C. Thomsen, Phys. Rev. B 71 (2005) 035416. [12] K. Hess, Monte Carlo Device Simulation: Full Band and Beyond, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991. [13] C. Jacoboni, P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer, Wein, New York, 1989.
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[14] P.L. McEuen, M.S. Fuhrer, H. Park, IEEE Trans. Nanotechnol. 1 (2002) 78. [15] J.Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. U¨stu¨nel, S. Braig, T.A. Arias, P.W. Brouwer, P.L. McEuen, Nanoletters 4 (2004) 517.
Physica E 34 (2006) 666–669 www.elsevier.com/locate/physe
Low- and high-field transport studies for semiconducting carbon nanotubes M.Z. Kausera,, A. Vermab, P.P. Rudena a
Department of ECE, University of Minnesota, Minneapolis, MN 55455, USA b School of ECE, Georgia Institute of Technology, Atlanta, GA 30332, USA Available online 19 April 2006
Abstract We calculate low- and high-field electron transport characteristics for semiconducting, single-wall, zigzag carbon nanotubes with diameters ranging from small to relatively large. The Boltzmann transport equation is solved directly using an iterative technique and indirectly by the ensemble Monte Carlo method. The basis for the transport calculation is provided by electronic structure calculations within the framework of a simple tight-binding model. Scattering mechanisms considered are due to the electron–phonon interactions involving longitudinal acoustic, longitudinal optic, and radial breathing-mode phonons. Results show significant effects of tube diameter on both low- and high-field transport characteristics. r 2006 Elsevier B.V. All rights reserved. PACS: 73.63.fg; 73.63.b; 73.22.f Keywords: Carbon nanotubes; Boltzmann transport; Ensemble Monte Carlo
1. Introduction Carbon nanotubes (CNTs) have demonstrated excellent potential to be used as basic building blocks in nanoscale devices [1,2]. Their electronic properties vary widely with tube chirality and diameter [3]. However, data on the transport properties of CNTs are still relatively sparse. Some theoretical studies have been made employing different methods, including analytical solution of the Boltzmann transport equation (BTE) [4] and Monte Carlo simulations [5–7]. Experimental work demonstrated the presence of strong longitudinal optic (LO) [8] and radial breathing-mode (RBM) [9] electron–phonon scattering in CNTs. Recent work on ensemble Monte Carlo (EMC) modeling of electron transport in CNTs has focused on the scattering of electrons due to longitudinal acoustic (LA), LO, and RBM phonons [6,7]. However, these studies only considered CNTs with relatively small diameter. Here we report on both low- and high-field transport properties for semiconducting, zigzag, single-wall CNTs Corresponding author. Tel.: +1 612 624 8545; fax: +1 612 625 4583.
E-mail address: [email protected] (M.Z. Kauser). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.055
with a range of diameters, from relatively small to large. The BTE is solved directly using an iterative technique (Rode’s method [10]) and indirectly by the EMC method in the low- and high-field regimes, respectively. Both steadystate and transient conditions are investigated for high fields. The common basis for both techniques is provided by electronic structure calculations in the framework of a tight-binding model. Scattering is due to electron–phonon interactions involving LA, LO, and RBM phonons. The two semiconducting zigzag CNT families, (3p+1,0) and (3p+2,0) with p a positive integer, are explored. Calculations are done for (n,0) CNTs, with n varying from 10 to 50 (corresponding to diameters d ¼ 0.77–3.86 nm), and for temperatures ranging from 100 to 600 K. 2. Model description 2.1. Electronic bandstructures The basis for the transport calculations is provided by the energy bandstructure obtained within the framework of the tight-binding model taking into account the corrections due to the curvature of the CNT [6]. For both zigzag CNT
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669
2.2. Phonon dispersions LA and LO phonon branches are obtained by fitting the phonon dispersions of graphene with polynomials and then applying the method of zone-folding to obtain the phonon dispersion relationships of the CNTs [6]. The radial breathing-mode phonons are evaluated for zigzag CNTs based on separate lattice dynamical calculations [7]. For this work only the lowest RBM phonon (m ¼ 0, where m is the phonon branch index) is considered. 2.3. Electron–phonon scattering Deformation potential induced electron and phonon scattering rates are evaluated by application of Fermi’s golden rule. Details of the LA, LO, and RBM phonon scattering rates and selection rules for Normal and Umklapp processes can be found in our earlier works [6,7]. The RBM-electron-interaction deformation potentials for (3p+1,0) and (3p+2,0) CNTs can be expressed as 3pþ2 D3pþ1 RBM ¼ A1 =d and DRBM ¼ A2 =d, where A1 ¼ 10.05 eV and A2 ¼ 8.05 eV are obtained from fit to recently reported results of ab initio calculations [11]. Scattering rates are evaluated for the lowest six and fourteen (two-fold spin degenerate) subbands for smalland large-diameter CNTs, respectively. 2.4. Transport calculations
Gaussian distribution. Standard EMC methodology and algorithms are applied [12,13]. 3. Results and discussion 3.1. Low-field transport The low-field mobility as a function of diameter for different temperatures is plotted in Fig. 1. It can be seen that for a particular temperature the low-field mobility increases with increasing CNT diameter. This is due to a reduction in the scattering rate, which decreases as a combined effect of decreasing effective mass and increasing linear mass density with increasing tube diameter. Another notable feature is that the mobility of (3p+1,0) CNT is larger than that of (3p+2,0) CNT for a particular p. In addition, for a particular CNT the mobility decreases with increasing temperature as m / T b , where b varies from 0.75 to 0.89 for the range of CNTs considered in this work. These calculated low-field mobility values are consistent with previous theoretical [4] and experimental [14] work. 3.2. High-field transport 3.2.1. Steady state Fig. 2 shows the velocity vs. electric field plots at room temperature for small diameter CNTs, i.e. (10,0) and (11,0). Results for relatively large diameter CNTs, i.e. (49,0) and (50,0), are shown in the inset. As can be seen, (3p+2,0) CNTs have a smaller velocity at the same electric field compared to the (3p+1,0) CNTs for the same p. This difference however gets progressively smaller with increasing diameter. Both families show saturation of the electron velocity at intermediate electric fields. This phenomenon
d (nm) 1.0
15 µ (x104 cm2V-1s-1)
families the lowest-energy subbands are two-fold degenerate (fourfold including spin) and correspond to indices n1 ¼ 7(2p+1). The second and third subbands correspond to indices n2 ¼ 72p and n3 ¼ 7(2p+2) for (3p+1.0) and n2 ¼ 7(2p+2) and n3 ¼ 72p for (3p+2.0) CNTs, respectively, and have the same level of degeneracy. The intersubband energy differences decrease with increasing tube diameter. The effective mass of the lowest conduction subband is smaller for the (3p+1,0) group compared to that of the (3p+2,0) group for small diameter tubes. However, they become comparable for both groups for large diameters. This has a strong effect on low-field mobility, as will be discussed later. A relatively large difference between the effective masses of the second and the lowest subband is obtained for small diameter (3p+1,0) CNTs compared to (3p+2,0) CNTs. However the difference between the effective masses of the third and the lowest subband is smaller for (3p+1,0) CNTs compared to (3p+2,0) CNTs.
667
1.5
2.0
2.5
3.0
3.5
100 K 300 K 600 K
10
5
The low-field transport properties of different CNTs for different temperatures are calculated by solving the steadystate BTE using Rode’s iterative technique [10]. Only the lowest conduction band is considered for the calculation. All the intrasubband electron scattering mechanisms (m ¼ 0 LA, RBM, and LO phonons) are included. EMC simulations are performed for electrons injected into the nanotubes at time t ¼ 0 with an equilibrium
0 10
20
30 n
40
50
Fig. 1. Low-field mobility as a function of diameter or the zigzag index n at three different temperatures. Solid and dashed lines represent (3p+1,0) and (3p+2,0) CNTs, respectively.
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669
668
7
3.5
(10,0) CNT
8 kV cm-1 6 Velocity (x107 cm s-1)
Velocity (x107 cm s-1)
3.0 (11,0) CNT
2.5
6
2.0
(49,0) CNT
5
1.5
(50,0) CNT
4 3
1.0
2
0.5
T= 300 K
0
100
4 25 kV cm-1
T= 300 K
3 2
1 0
0.0
5
150
0
10
20
200
30
250
40
50
300
Electric field (kV cm-1) Fig. 2. Average steady-state drift velocity vs. electric field at 300 K for (10,0) and (11,0) CNTs (main panel) and for (49,0) and (50,0) CNTs (inset).
has been demonstrated earlier to be due to LA and LO phonon emission and the resulting streaming motion [4,6]. The saturation in electron velocity has been observed in high-voltage experiments [15]. The saturation electric field, Fsat, is vastly different for CNTs with different diameter. Fsat is found to decrease with increasing tube diameter, as seen in Fig. 2. As the energy required for the intersubband phonon emission decreases with diameter, electrons with smaller effective mass gain enough energy at relatively low field to emit phonons and enter streaming motion [6]. Another important noticeable transport phenomenon is the negative differential mobility (NDM). As can be seen from Fig. 2, (3p+1,0) CNTs show a relatively more pronounced NDM than (3p+2,0) CNTs for small-diameter tubes, whereas it is more or less similar and pronounced for both CNT families at larger diameter. Due to the reduction of the intersubband energy differences with tube diameter the higher subbands become more readily occupied for large diameter tubes. 3.2.2. Transients Fig. 3 shows the average transient velocity vs. time for various CNTs at room temperature. It can be seen that due to lower effective mass of the bigger tubes, the peak velocity increases with increasing tube diameter. The oscillation is more pronounced and the settling time is longer in larger diameter tubes. This is due to lower damping, because the LA and RBM scattering rates, which act as the damping mechanism, decrease with increasing tube diameter. 4. Conclusions Charge transport properties of different diameter zigzag CNTs have been investigated. These transport properties are not only a function of the diameter but also of the
(10,0) CNT (11,0) CNT (49,0) CNT (50,0) CNT
1 0 0.0
0.2
0.4 0.6 Time (ps)
0.8
1.0
Fig. 3. Average transient drift velocity vs. time for (10,0), (11,0), (49,0), and (50,0) CNTs at room temperature. (10,0) and (11,0) CNTs are under 25 kV cm1 whereas (49,0) and (50,0) CNTs are under 8 kV cm1 electric field.
family to which the CNTs belong, i.e. (3p+1,0) and (3p+2,0). In general, (3p+1,0) CNTs are superior in terms of low-field mobility and high-field velocity. Similarly, larger diameter CNTs are superior to smaller diameter CNTs. Acknowledgements Access to the facilities of the Minnesota Supercomputing Institute is gratefully acknowledged. This work is partially supported by NSF-ECS. References [1] R. Martel, T. Schmidt, H.R. Shea, T. Hertel, Ph. Avouris, Appl. Phys. Lett. 73 (1998) 2447. [2] A. Bachtold, P. Hadley, T. Nakanishi, C. Dekker, Science 294 (2001) 1317. [3] R. Saito, M.S. Dresselhaus, G. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. [4] V. Perebeinos, J. Tersoff, Ph. Avouris, Phys. Rev. Lett. 94 (2005) 086802. [5] G. Pennington, N. Goldsman, Phys. Rev. B 68 (2003) 045426. [6] A. Verma, M.Z. Kauser, P.P. Ruden, J. Appl. Phys. 97 (2005) 114319; A. Verma, M. Z. Kauser, B. W. Lee, K. F. Brennan, P. P. Ruden, in: Proceedings of the 27th International Conference on Physics of Semiconductors, ICPS, Flagstaff, AZ, 2004, AIP Conf. Proc. 772 (2005) 1049. [7] A. Verma, M.Z. Kauser, P.P. Ruden, Appl. Phys. Lett. 87 (2005) 123101. [8] A. Javey, J. Guo, M. Paulsson, Q. Wang, D. Mann, M. Lundstrom, H. Dai, Phys. Rev. Lett. 92 (2004) 106804; M. Freitag, V. Perebeinos, J. Chen, A. Stein, J.C. Tsang, J.A. Misewich, R. Martel, Ph. Avouris, Nanoletters 4 (2004) 1063. [9] B.J. LeRoy, S.G. Lemay, J. Kong, C. Dekker, Nature 432 (2004) 371. [10] D.L. Rode, Low-field transport in semiconductors, in: R.K. Willardson, A.C. Beer (Eds.), Semiconductors and Semimetals, 10, Academic Press, New York, 1975.
ARTICLE IN PRESS M.Z. Kauser et al. / Physica E 34 (2006) 666–669 [11] M. Machon, S. Reich, H. Telg, J. Maultzsch, P. Ordejon, C. Thomsen, Phys. Rev. B 71 (2005) 035416. [12] K. Hess, Monte Carlo Device Simulation: Full Band and Beyond, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991. [13] C. Jacoboni, P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer, Wein, New York, 1989.
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[14] P.L. McEuen, M.S. Fuhrer, H. Park, IEEE Trans. Nanotechnol. 1 (2002) 78. [15] J.Y. Park, S. Rosenblatt, Y. Yaish, V. Sazonova, H. U¨stu¨nel, S. Braig, T.A. Arias, P.W. Brouwer, P.L. McEuen, Nanoletters 4 (2004) 517.