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On ballistic transport in carbon nanotubes
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 601–602 www.elsevier.de/ijleo
On ballistic transport in carbon nanotubes M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029 Madrid, Spain Received 27 December 2006; accepted 30 April 2007
Abstract We investigate theoretically the ballistic regime exhibited by conduction electrons in multiwalled carbon nanotubes in relation to the conductance quantization in these tubes. Starting from the fact that electron drift mobility is quantized in multiwall tubes, essential aspects related to both ballistic and diffusive regimes are discussed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Ballistic transport; Multiwalled carbon nanotubes; Conductance quantization; Drift mobility; Diffusive regime
1. Introduction Ballistic electronic transport in carbon nanotubes constitutes a relevant feature that is related to a number of essential aspects of the physics underlying electron transport through the aforementioned tubes. In fact, theoretical studies have shown that carbon nanotubes present ballistic conductance of electrons so that these tubes behave as waveguides for the electrons [1–7]. It is known that the mean free path of conduction electrons is much longer than the length of a given nanotube; scattering relies on elastic collisions [5–7]. Conductance quantization takes place in multiwalled carbon nanotubes (MWCNTs) [5–10], electronic transport in these tubes being notoriously ballistic so that ballistic regime, as we have said above, is exhibited by carbon nanotubes in general [1–7]. As a matter of fact, it has been shown that the ballistic regime is a necessary condition for conductance quantization (see, for example, [6,7]); in other words, if in a certain system electrical conductance Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.04.003
is quantized, then electronic transport in that system is ballistic. Consequently, the diffusive regime is excluded in an MWCNT. In general terms, problems related to both ballistic and diffusive regimes are very relevant in order to explore crucial aspects of electron transport in nanostructures. The present paper is devoted to characterize ballistic regime in MWCNTs, starting from considerations relative to electron drift mobility. In fact, electron drift mobility in an MWCNT was found to be quantized [8]; hence this fact will be taken into account, in the following, to show that conductance quantization implies ballistic transport. Quantized conductance through an MWCNT scales with the number of layers of the tube so that the quantum number involved in the quantization in question serves for numbering or labelling the layers of the MWCNT [8–11]. Within this context, we emphasize that electron-mobility quantization in an MWCNT arises from the fact that electron velocity in an MWCNT is quantized [5,8–10]. In addition, we remark that the fact that electron mobility is quantized leads to conductivity quantization; hence this quantization gives rise, in turn, to conductance quantization, conductance being independent of tube diameter and length [7,9,10].
ARTICLE IN PRESS 602
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 601–602
2. Theoretical considerations
3. Conclusions
Let us consider a biased MWCNT; once Fermi velocity is calculated in this tube, it has been shown that the corresponding electron drift mobility is quantized, obeying the following relationship [8]:
We have developed new ideas to state theoretically that ballistic transport is required as a necessary (but not sufficient) condition for conductance quantization. The key result in order to establish the aforementioned condition is formula (5) or its simplified version (6) so that, from these formulae, it follows that the squared diffusion length is proportional to T; in addition, we see that l n / a2 at a given temperature and for any layer of the tube. On the other hand, from Eq. (5) or (6) it is inferred that l n ! 0 as theoretically n ! 1; in fact, diffusion length decreases roughly as a linear function of 1/n (see relationship (6)) as n increases. Finally, we wish to emphasize that l n ba is equivalent to saying that the mean free path of conduction electrons is much longer than the tube length.
4ea2 , ð2n þ 1Þh
mn ¼
(1)
where e is electron charge, a is tube length, h is Planck’s constant, and the quantum number n ¼ 1,2,y coincides with the index for labelling the layers of the tube according to the fact that the quantized conductance scales with the number of layers [7,9–11]. Starting from expression (1), we will show that quantized conductance implies ballistic conductance. To obtain this result, we consider the following relation: kTmn , (2) e where Dn, k, and T denote diffusion coefficient, Boltzmann constant, and absolute temperature, respectively. Furthermore, we can write
Dn ¼
Dn ¼
l 2n , tn
(3)
where ln stands for diffusion length and tn denotes relaxation time. We also have that etn mn ¼ , (4) m where m is the free-electron mass. By combining formulae (1)–(4), it follows that ln ¼
4a2 ðkTmÞ1=2 . ð2n þ 1Þh
(5)
By taking into account that the typical values of a belong to the range of values between 1 and 10 mm approximately [7]and also that 5pnp30 [7], then, by using formula (5), the reader can easily verify that (at room temperature) l n ba, that is to say, electron transport in an MWCNT is ballistic. Given the above range of the number of shells in a typical MWCNT, one has 2n þ 1 2n; hence Eq. (5) becomes ln
2a2 ðkTmÞ1=2 . nh
(6)
References [1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart & Winston, New York, 1976. [2] C.T. White, D.H. Robertson, J.W. Mintmire, in: P. Jena, S. Behera (Eds.), Clusters and Nanostructured Materials, Nova, New York, 1996, pp. 231–237. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Quantization of the electric conductivity in carbon nanotubes, Act. Pass. Electron. Comp. 24 (2001) 165–168. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, On the size of small single-walled carbon nanotubes, Optik 116 (2005) 459–460. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, A theoretical analysis on the Fermi level in multiwalled carbon nanotubes, Mod. Phys. Lett. B 18 (2004) 501–503. [6] S. Datta, Electronic Transport Properties in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [7] S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Carbon nanotube quantum resistors, Science 280 (1998) 1744–1746. [8] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multiwalled carbon nanotubes, Optik 115 (2004) 45–46. [9] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance quantization in multiwalled carbon nanotubes with disorder, Optik 116 (2005) 409–410. [10] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance in multiwalled carbon nanotubes: a semiclassical theory, Mod. Phys. Lett. B 18 (2004) 761–767. [11] M.F. Lin, K.W.-K. Shung, Magnetoconductance of carbon nanotubes, Phys. Rev. B 51 (1995) 7592.
Optik
Optics
Optik 119 (2008) 601–602 www.elsevier.de/ijleo
On ballistic transport in carbon nanotubes M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029 Madrid, Spain Received 27 December 2006; accepted 30 April 2007
Abstract We investigate theoretically the ballistic regime exhibited by conduction electrons in multiwalled carbon nanotubes in relation to the conductance quantization in these tubes. Starting from the fact that electron drift mobility is quantized in multiwall tubes, essential aspects related to both ballistic and diffusive regimes are discussed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Ballistic transport; Multiwalled carbon nanotubes; Conductance quantization; Drift mobility; Diffusive regime
1. Introduction Ballistic electronic transport in carbon nanotubes constitutes a relevant feature that is related to a number of essential aspects of the physics underlying electron transport through the aforementioned tubes. In fact, theoretical studies have shown that carbon nanotubes present ballistic conductance of electrons so that these tubes behave as waveguides for the electrons [1–7]. It is known that the mean free path of conduction electrons is much longer than the length of a given nanotube; scattering relies on elastic collisions [5–7]. Conductance quantization takes place in multiwalled carbon nanotubes (MWCNTs) [5–10], electronic transport in these tubes being notoriously ballistic so that ballistic regime, as we have said above, is exhibited by carbon nanotubes in general [1–7]. As a matter of fact, it has been shown that the ballistic regime is a necessary condition for conductance quantization (see, for example, [6,7]); in other words, if in a certain system electrical conductance Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.04.003
is quantized, then electronic transport in that system is ballistic. Consequently, the diffusive regime is excluded in an MWCNT. In general terms, problems related to both ballistic and diffusive regimes are very relevant in order to explore crucial aspects of electron transport in nanostructures. The present paper is devoted to characterize ballistic regime in MWCNTs, starting from considerations relative to electron drift mobility. In fact, electron drift mobility in an MWCNT was found to be quantized [8]; hence this fact will be taken into account, in the following, to show that conductance quantization implies ballistic transport. Quantized conductance through an MWCNT scales with the number of layers of the tube so that the quantum number involved in the quantization in question serves for numbering or labelling the layers of the MWCNT [8–11]. Within this context, we emphasize that electron-mobility quantization in an MWCNT arises from the fact that electron velocity in an MWCNT is quantized [5,8–10]. In addition, we remark that the fact that electron mobility is quantized leads to conductivity quantization; hence this quantization gives rise, in turn, to conductance quantization, conductance being independent of tube diameter and length [7,9,10].
ARTICLE IN PRESS 602
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 601–602
2. Theoretical considerations
3. Conclusions
Let us consider a biased MWCNT; once Fermi velocity is calculated in this tube, it has been shown that the corresponding electron drift mobility is quantized, obeying the following relationship [8]:
We have developed new ideas to state theoretically that ballistic transport is required as a necessary (but not sufficient) condition for conductance quantization. The key result in order to establish the aforementioned condition is formula (5) or its simplified version (6) so that, from these formulae, it follows that the squared diffusion length is proportional to T; in addition, we see that l n / a2 at a given temperature and for any layer of the tube. On the other hand, from Eq. (5) or (6) it is inferred that l n ! 0 as theoretically n ! 1; in fact, diffusion length decreases roughly as a linear function of 1/n (see relationship (6)) as n increases. Finally, we wish to emphasize that l n ba is equivalent to saying that the mean free path of conduction electrons is much longer than the tube length.
4ea2 , ð2n þ 1Þh
mn ¼
(1)
where e is electron charge, a is tube length, h is Planck’s constant, and the quantum number n ¼ 1,2,y coincides with the index for labelling the layers of the tube according to the fact that the quantized conductance scales with the number of layers [7,9–11]. Starting from expression (1), we will show that quantized conductance implies ballistic conductance. To obtain this result, we consider the following relation: kTmn , (2) e where Dn, k, and T denote diffusion coefficient, Boltzmann constant, and absolute temperature, respectively. Furthermore, we can write
Dn ¼
Dn ¼
l 2n , tn
(3)
where ln stands for diffusion length and tn denotes relaxation time. We also have that etn mn ¼ , (4) m where m is the free-electron mass. By combining formulae (1)–(4), it follows that ln ¼
4a2 ðkTmÞ1=2 . ð2n þ 1Þh
(5)
By taking into account that the typical values of a belong to the range of values between 1 and 10 mm approximately [7]and also that 5pnp30 [7], then, by using formula (5), the reader can easily verify that (at room temperature) l n ba, that is to say, electron transport in an MWCNT is ballistic. Given the above range of the number of shells in a typical MWCNT, one has 2n þ 1 2n; hence Eq. (5) becomes ln
2a2 ðkTmÞ1=2 . nh
(6)
References [1] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart & Winston, New York, 1976. [2] C.T. White, D.H. Robertson, J.W. Mintmire, in: P. Jena, S. Behera (Eds.), Clusters and Nanostructured Materials, Nova, New York, 1996, pp. 231–237. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Quantization of the electric conductivity in carbon nanotubes, Act. Pass. Electron. Comp. 24 (2001) 165–168. [4] M.A. Grado-Caffaro, M. Grado-Caffaro, On the size of small single-walled carbon nanotubes, Optik 116 (2005) 459–460. [5] M.A. Grado-Caffaro, M. Grado-Caffaro, A theoretical analysis on the Fermi level in multiwalled carbon nanotubes, Mod. Phys. Lett. B 18 (2004) 501–503. [6] S. Datta, Electronic Transport Properties in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [7] S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Carbon nanotube quantum resistors, Science 280 (1998) 1744–1746. [8] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multiwalled carbon nanotubes, Optik 115 (2004) 45–46. [9] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance quantization in multiwalled carbon nanotubes with disorder, Optik 116 (2005) 409–410. [10] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance in multiwalled carbon nanotubes: a semiclassical theory, Mod. Phys. Lett. B 18 (2004) 761–767. [11] M.F. Lin, K.W.-K. Shung, Magnetoconductance of carbon nanotubes, Phys. Rev. B 51 (1995) 7592.