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Fractional conductance quantization in multiwalled carbon nanotubes with disorder
ARTICLE IN PRESS
Optik
Optics
Optik 116 (2005) 409–410 www.elsevier.de/ijleo
SHORT NOTE
Fractional conductance quantization in multiwalled carbon nanotubes with disorder M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029-Madrid, Spain Received 19 November 2004; accepted 14 February 2005
Abstract A close relationship is established for describing fractional conductance quantization in multiwalled carbon nanotubes with disorder in the framework of the particle-in-a-box model. Our results agree well with experimental data in the context of nanotubes with defects. r 2005 Elsevier GmbH. All rights reserved. Keywords: Fractional conductance quantization; Multiwalled carbon nanotubes; Disorder
The theoretical determination of the electrical conductance of multiwalled carbon nanotubes (MWCNTs) constitutes a great challenge because the above determination represents the elucidation of a central problem within the context of nanoscience and nanotechnology given that the potential applications of MWCNTs should configure exciting future perspectives. Recently, theoretical descriptions upon the subject in question have been developed [1,2] within the framework of fractional conductance quantization. In the present note, we will establish a mathematical relationship for the conductance in an MWCNT with defects. We start from the consideration of an MWCNT with disorder (disorder will be regarded as defects in general) so that we will conceive the motion of the delocalized electron shared by all the carbon atoms as the origin of a certain kind of disorder; at this point, note that the delocalized electron in question is one of the four valence electrons corresponding to each carbon atom
(see Refs. [1–3]). The preceding considerations in conjunction with the assumption of the particle-in-abox model [1,3] permit to regard a single-electron device so that we can write the following relationship which gives electron drift mobility in MWCNTs [3]: mn ¼
0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.02.011
ðn ¼ 1; 2; . . .Þ,
(1)
where e is the electron charge, h is Planck constant, a is the tube length, and n is a quantum number that serves for numbering the layers of the tube [1–4] (conductance should scale with the number of layers). Now we consider the conductivity relative to a single electron as follows: em pffiffiffiffi sn ¼ eZmn ¼ n A5a , (2) aA where Z ¼ 1=ðaAÞ is spatial electron density and A stands for the cross-sectional area of the tube. On the other hand, the conductance reads
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro).
4ea2 ð2n þ 1Þh
Gn ¼
sn A . a
(3)
ARTICLE IN PRESS 410
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 116 (2005) 409–410
Then, by combining formulae (1)–(3), one gets Gn ¼
2G 0 , 2n þ 1
(4)
where G 0 ¼ 2e2 =h is the fundamental conductance quantum. Note that Gn is quantized and fractional so that we have G 1 ¼ 23G 0 , in a reasonable agreement with the first main conductance plateau observed in Ref. [5] (0.5G0 or 1G0); in fact, the 0.5G0 value was attributed by the authors of Ref. [5] to be due to a defect in the tube (for more details, see Ref. [5]). Finally, we define an averaged conductance as follows: N X ¯ ¼ 1 G Gn N n¼1
(5)
for an MWCNT of N layers and assuming that each mode contributes equally to the total conductance. Then, by Eqs. (4) and (5), one has ¯ ¼ 2G 0 G N
N X n¼1
1 2n þ 1
(6)
so that, by using Stolz theorem, from formula (6) it follows: ¯ ðN!1Þ ¼ 2G 0 lim G
N!1
¼ 2G 0 lim
N!1
PNþ1
P N n¼1 N þ1N
1 n¼1 2nþ1
1 ¼ 0. 2N þ 3
1 2nþ1
ð7Þ
Result (7) tells us that, for MWCNTs with a relatively ¯ conductance large number of layers, then the G approaches zero so that a quasi-insulating behaviour is achieved. In conclusion, we have calculated the quantized conductance in MWCNTs so that the values taken by this conductance are found to be fractional (fractions of G0). When nb1, we have that G n G 0 =n, in good agreement with Ref. [6]. In addition, the average ¯ has been evaluated giving rise to a conductance G quasi-insulating behaviour when the number of layers is sufficiently high.
References [1] 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. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance in multiwalled carbon nanotubes: a semiclassical theory, Mod. Phys. Lett. B 18 (2004) 761–767. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multiwalled carbon nanotubes, Optik 115 (2004) 45–46. [4] M.F. Lin, K.W.-K. Shung, Magnetoconductance of carbon nanotubes, Phys. Rev. B 51 (1995) 7592. [5] S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Carbon nanotube quantum resistors, Science 280 (1998) 1744–1746. [6] M.A. Grado-Caffaro, M. Grado-Caffaro, Quantization of the electric conductivity in carbon nanotubes, Act. Pass. Electron. Components 24 (2001) 165–168.
Optik
Optics
Optik 116 (2005) 409–410 www.elsevier.de/ijleo
SHORT NOTE
Fractional conductance quantization in multiwalled carbon nanotubes with disorder M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 9-B, 28029-Madrid, Spain Received 19 November 2004; accepted 14 February 2005
Abstract A close relationship is established for describing fractional conductance quantization in multiwalled carbon nanotubes with disorder in the framework of the particle-in-a-box model. Our results agree well with experimental data in the context of nanotubes with defects. r 2005 Elsevier GmbH. All rights reserved. Keywords: Fractional conductance quantization; Multiwalled carbon nanotubes; Disorder
The theoretical determination of the electrical conductance of multiwalled carbon nanotubes (MWCNTs) constitutes a great challenge because the above determination represents the elucidation of a central problem within the context of nanoscience and nanotechnology given that the potential applications of MWCNTs should configure exciting future perspectives. Recently, theoretical descriptions upon the subject in question have been developed [1,2] within the framework of fractional conductance quantization. In the present note, we will establish a mathematical relationship for the conductance in an MWCNT with defects. We start from the consideration of an MWCNT with disorder (disorder will be regarded as defects in general) so that we will conceive the motion of the delocalized electron shared by all the carbon atoms as the origin of a certain kind of disorder; at this point, note that the delocalized electron in question is one of the four valence electrons corresponding to each carbon atom
(see Refs. [1–3]). The preceding considerations in conjunction with the assumption of the particle-in-abox model [1,3] permit to regard a single-electron device so that we can write the following relationship which gives electron drift mobility in MWCNTs [3]: mn ¼
0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.02.011
ðn ¼ 1; 2; . . .Þ,
(1)
where e is the electron charge, h is Planck constant, a is the tube length, and n is a quantum number that serves for numbering the layers of the tube [1–4] (conductance should scale with the number of layers). Now we consider the conductivity relative to a single electron as follows: em pffiffiffiffi sn ¼ eZmn ¼ n A5a , (2) aA where Z ¼ 1=ðaAÞ is spatial electron density and A stands for the cross-sectional area of the tube. On the other hand, the conductance reads
Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro).
4ea2 ð2n þ 1Þh
Gn ¼
sn A . a
(3)
ARTICLE IN PRESS 410
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 116 (2005) 409–410
Then, by combining formulae (1)–(3), one gets Gn ¼
2G 0 , 2n þ 1
(4)
where G 0 ¼ 2e2 =h is the fundamental conductance quantum. Note that Gn is quantized and fractional so that we have G 1 ¼ 23G 0 , in a reasonable agreement with the first main conductance plateau observed in Ref. [5] (0.5G0 or 1G0); in fact, the 0.5G0 value was attributed by the authors of Ref. [5] to be due to a defect in the tube (for more details, see Ref. [5]). Finally, we define an averaged conductance as follows: N X ¯ ¼ 1 G Gn N n¼1
(5)
for an MWCNT of N layers and assuming that each mode contributes equally to the total conductance. Then, by Eqs. (4) and (5), one has ¯ ¼ 2G 0 G N
N X n¼1
1 2n þ 1
(6)
so that, by using Stolz theorem, from formula (6) it follows: ¯ ðN!1Þ ¼ 2G 0 lim G
N!1
¼ 2G 0 lim
N!1
PNþ1
P N n¼1 N þ1N
1 n¼1 2nþ1
1 ¼ 0. 2N þ 3
1 2nþ1
ð7Þ
Result (7) tells us that, for MWCNTs with a relatively ¯ conductance large number of layers, then the G approaches zero so that a quasi-insulating behaviour is achieved. In conclusion, we have calculated the quantized conductance in MWCNTs so that the values taken by this conductance are found to be fractional (fractions of G0). When nb1, we have that G n G 0 =n, in good agreement with Ref. [6]. In addition, the average ¯ has been evaluated giving rise to a conductance G quasi-insulating behaviour when the number of layers is sufficiently high.
References [1] 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. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Fractional conductance in multiwalled carbon nanotubes: a semiclassical theory, Mod. Phys. Lett. B 18 (2004) 761–767. [3] M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical evaluation of electron mobility in multiwalled carbon nanotubes, Optik 115 (2004) 45–46. [4] M.F. Lin, K.W.-K. Shung, Magnetoconductance of carbon nanotubes, Phys. Rev. B 51 (1995) 7592. [5] S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Carbon nanotube quantum resistors, Science 280 (1998) 1744–1746. [6] M.A. Grado-Caffaro, M. Grado-Caffaro, Quantization of the electric conductivity in carbon nanotubes, Act. Pass. Electron. Components 24 (2001) 165–168.