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A new theory for interpreting electron conductance quantization in multiwalled carbon nanotubes without defects
ARTICLE IN PRESS Physica B 404 (2009) 1544–1545
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
A new theory for interpreting electron conductance quantization in multiwalled carbon nanotubes without defects M.A. Grado-Caffaro , M. Grado-Caffaro M.A. Grado-Caffaro and M. Grado-Caffaro-Scientific Consultants, C/ Julio Palacios 11, 9-B, 28029-Madrid, Spain
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 November 2008 Received in revised form 11 January 2009 Accepted 12 January 2009
A novel theory for determining the quantized electrical conductance through a multiwalled carbon nanotube without defects is proposed. Starting from establishing the conductance of a given single layer (single-walled nanotube) of the multiwall tube under a standing-wave approach and by taking into account the mutual interaction with the remaining layers, a closed expression for the quantized conductance of the multi-layer tube is found. & 2009 Elsevier B.V. All rights reserved.
Keywords: Multiwalled carbon nanotube Single-walled carbon nanotube Quantized electrical conductance Standing wave
1. Introduction The full elucidation of the mechanisms underlying quantum transport through multiwalled carbon nanotubes (from now on, MWCNTs) represents a great challenge given that a number of questions still remain open. Indeed, in contrast to single-walled carbon nanotubes (from now on, SWCNTs) where the main problems concerning electrical conduction have been solved, important research efforts for understanding the physics of MWCNTs are needed; in particular, we refer to charge-carrier transport. The conception of an MWCNT as a quantum electrical conductor constitutes a relevant example of quantized conductance through molecular wires. For instance, Apostol [1] has investigated new aspects related to quantized conductance in nanowires under the action of electric and magnetic fields by means of special procedures which can be used in relation to MWCNTs. In this context, the main issue is the determination of the quantized conductance through an MWCNT. Frank et al. [2] measured several values of conductance by means of an experimental set-up in which a single MWCNT protruding from a nanotube bundle, which supplied one contact, was lowered slowly into mercury which provided the other contact. In addition, the tube was cleaned of adhering material. Lin and Shung [3] and Frank et al. [2] pointed out that the conductance should scale with the number of layers; Grado-Caffaro and Grado-Caffaro calculated the conductance of an MWCNT with defects [4,5] and the
conductance of a perfect MWCNT [6] with results in agreement with experimental data [2]. In contrast, approaches as in Refs. [7–9], elaborated on wrong grounds, have failed. Given that the conductance of an SWCNT without scattering is twice the value of the fundamental conductance quantum, one can think (following Refs. [2,3]) that the conductance of an MWCNT is equal to the number of layers multiplied by 2G0 where G0 is the fundamental conductance quantum. This fact tells us that the electron conductance through an MWCNT is quantized taking values which are even multiples of G0. Nevertheless, this result disagrees with experimental observations [2] by which conductance values near 1G0, 2G0 and 3G0 were measured. In view of this, one can say that the conductance should be any multiple of the fundamental conductance quantum. This result will be obtained in the present work for an MWCNT without defects by starting from the knowledge of the conductance through an imperfect single layer with elastic scattering and neglecting quantummechanical interference. In the context of ballistic transport [2,4–6], we shall calculate the conductance of a single shell showing that this conductance coincides with the maximum conductance, that is, 2G0. Since the effective or actual conductance equals the maximal conductance in series with the wire conductance, then the total conductance is equal to the fundamental conductance quantum.
2. Theory Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.01.020
We start from the above-mentioned fact that, in the absence of scattering, the conductance of an SWCNT equals 4e2/h which
ARTICLE IN PRESS M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1544–1545
comes from the existence of two spins per conduction band together with that there are two bands (two conducting channels), e2/h being the value of the conductance corresponding to each conducting mode (one band and one spin) where e is the absolute value of the electron charge and h is the Planck constant (note that G0 ¼ 2e2/h). Therefore, we can write Gmax ¼ 2G0. On the other hand, we must determine the so-called wire conductance which will be denoted by Gw. To get this end, we conceive the tube as a quasi-one-dimensional wire represented by a longitudinal ideal quantum box so that the width of this box coincides with the length of the tube. Under these conditions, the conduction electrons behave as standing waves whose wavelength is quantized obeying the relation namely nln ¼ 2l where ln is the quantized electron de Broglie wavelength, l is the length of the tube, and n ¼ 1,2,y. In addition, we have that ln ¼ h/(mvn) where m is the free-electron mass and vn denotes the magnitude of the quantized electron velocity. Now we consider the following expressions namely l ¼ vntn (where tn is the quantized motion time), sn ¼ e2Nntn/(mlA) where sn denotes quantized conductivity, Nn is the quantized number of conduction electrons, and A is the pffiffiffi cross-sectional area of the tube so A5l by virtue of the quasione-dimensionality of the tube. On the other hand, by the exclusion principle, one has that Nn ¼ 2n [6,10]. Combining all the above formulae and the relation for the quantized conductance namely Gn ¼ snA/l, it follows that Gn ¼ 2G0. This value (which is Gw) does not depend on n, as expected, so that Gw ¼ 2G0. Since, for an imperfect tube with scattering and neglecting quantum-mechanical interference, the effective conductance is the series combination of Gmax and Gw (see, for instance, Refs. [11,12]), we have that the effective conductance of a single layer is given by G ¼ GmaxGw/(Gmax+Gw) ¼ G0. We recall that, here, scattering consists of elastic collisions [2]. The interwall coupling in an MWCNT can be taken into account by assuming that the net contribution from a single layer to the conductance is given by the aforementioned effective conductance. In other words, with respect to the interaction between layers, each layer behaves as an imperfect SWCNT with scattering. For an MWCNT with m layers, the conductance is the parallel combination of the effective conductances of all the shells of the MWCNT so that the resulting conductance is Gm ¼ G0m where m ¼ 1,2,y. This result agrees well with experimental data [2] except for the 0.5G0 value measured in Ref. [2]. This value was attributed to a defect in the tube [2]. In this respect, notice that our formulation deals with MWCNTs without defects. Then, we have deduced that the conductance through an MWCNT is quantized. The fact that this quantization consists of that the conductance is any integer multiple of the fundamental conductance quantum appears as a very significant fact in contrast to expect that the conductance should be even multiples of G0. At this point, we recall that the mutual interaction between layers is equivalent to regard each layer as an imperfect tube (but without defects) with elastic scattering and negligible quantum interference. It is clear that this fact is crucial to understand why the relation Gm ¼ G0m (where m is any strictly positive integer) takes place instead of Gm ¼ 2G0m. An issue that becomes very relevant here is the ballistic nature of the electronic transport through MWCNTs (see Refs. [2,13]). In this respect, it has been shown that MWCNTs are ballistic
1545
conductors at room temperature [13]. In fact, in Ref. [13], a momentum mean free-path larger than 30 mm has been measured. This value is much larger than the typical length (around 4 mm) of an MWCNT so that the ballistic nature of the tube is manifest. Another question to be considered is the relationship between conductance quantization and ballistic transport: it is well-known that conductance quantization implies ballistic transport [2,11]. In other words, ballistic transport is a necessary (but not sufficient) condition for conductance quantization. Therefore, diffusive transport is excluded since this type of transport gives rise to non-quantized conductance. Since LGmax ¼ lGw (where L denotes mean free path) and Gmax ¼ Gw, it follows that L ¼ l which corroborates that there is scattering (here, scattering consists of elastic collisions [2,13]). However, we can say that here carrier transport is quasi-ballistic so that, although the condition L4l does not occur, when L ¼ l can be considered as a limit case in which ballistic or at least quasi-ballistic regime takes place [12,13]. In this case, given that Gw ¼ GmaxT/(1T) (where T is the transmission probability), it follows that T ¼ 12.
3. Conclusions We have determined Gw under a standing-wave approach so that the conduction electrons behave as standing waves confined in a quasi-one-dimensional ideal potential well. In this respect, we wish to remark the manifest usefulness of the particle-in-a-box model. The precedent study has led to results in excellent agreement with experimental observations [2]. As a matter of fact, it has been found that the electron conductance of an MWCNT without defects is any integer multiple of the conductance quantum. Interwall coupling has been taken into account by considering the net conductance of a single shell as the resulting conductance from Gmax in series with Gw. We have found that Gw ¼ 2G0 so Gw ¼ Gmax. Consequently, the effective or net conductance of a single layer reads G ¼ G0 which gives rise to the quantized conductance of an MWCNT namely Gm ¼ G0m. This result tells us that the interwall coupling gives rise to halving the expected conductance obtained by simply multiplying 2G0 by the number of layers. Halving 2G0 due to a possible spin-chirality coupling was suggested in Ref. [2]. However, we have derived this result without invoking considerations upon spin and its possible coupling to helicity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
M. Apostol, Phys. Lett. A 372 (2008) 5093. S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Science 280 (1998) 1744. M.F. Lin, K.W.K. Shung, Phys. Rev. B 51 (1995) 7592. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 18 (2004) 761. M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 116 (2005) 409. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 19 (2005) 967. P. Delaney, M. di Ventra, S.T. Pantelides, Appl. Phys. Lett. 75 (1999) 3787. S. Sanvito, Y.K. Kwon, D. Toma´nek, C.J. Lambert, Phys. Rev. Lett. 84 (2000) 1974. Y.G. Yoon, P. Delaney, S.G. Louie, Phys. Rev. B 66 (2002) 327. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 18 (2004) 501. S. Datta, Electronic Transport Properties of Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. T. Du¨rkop, B.M. Kim, M.S. Fuhrer, J. Phys. Condens. Matter 16 (2004) R553. C. Berger, Y. Yi, Z.L. Wang, W.A. de Heer, Appl. Phys. A 74 (2002) 363.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
A new theory for interpreting electron conductance quantization in multiwalled carbon nanotubes without defects M.A. Grado-Caffaro , M. Grado-Caffaro M.A. Grado-Caffaro and M. Grado-Caffaro-Scientific Consultants, C/ Julio Palacios 11, 9-B, 28029-Madrid, Spain
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 November 2008 Received in revised form 11 January 2009 Accepted 12 January 2009
A novel theory for determining the quantized electrical conductance through a multiwalled carbon nanotube without defects is proposed. Starting from establishing the conductance of a given single layer (single-walled nanotube) of the multiwall tube under a standing-wave approach and by taking into account the mutual interaction with the remaining layers, a closed expression for the quantized conductance of the multi-layer tube is found. & 2009 Elsevier B.V. All rights reserved.
Keywords: Multiwalled carbon nanotube Single-walled carbon nanotube Quantized electrical conductance Standing wave
1. Introduction The full elucidation of the mechanisms underlying quantum transport through multiwalled carbon nanotubes (from now on, MWCNTs) represents a great challenge given that a number of questions still remain open. Indeed, in contrast to single-walled carbon nanotubes (from now on, SWCNTs) where the main problems concerning electrical conduction have been solved, important research efforts for understanding the physics of MWCNTs are needed; in particular, we refer to charge-carrier transport. The conception of an MWCNT as a quantum electrical conductor constitutes a relevant example of quantized conductance through molecular wires. For instance, Apostol [1] has investigated new aspects related to quantized conductance in nanowires under the action of electric and magnetic fields by means of special procedures which can be used in relation to MWCNTs. In this context, the main issue is the determination of the quantized conductance through an MWCNT. Frank et al. [2] measured several values of conductance by means of an experimental set-up in which a single MWCNT protruding from a nanotube bundle, which supplied one contact, was lowered slowly into mercury which provided the other contact. In addition, the tube was cleaned of adhering material. Lin and Shung [3] and Frank et al. [2] pointed out that the conductance should scale with the number of layers; Grado-Caffaro and Grado-Caffaro calculated the conductance of an MWCNT with defects [4,5] and the
conductance of a perfect MWCNT [6] with results in agreement with experimental data [2]. In contrast, approaches as in Refs. [7–9], elaborated on wrong grounds, have failed. Given that the conductance of an SWCNT without scattering is twice the value of the fundamental conductance quantum, one can think (following Refs. [2,3]) that the conductance of an MWCNT is equal to the number of layers multiplied by 2G0 where G0 is the fundamental conductance quantum. This fact tells us that the electron conductance through an MWCNT is quantized taking values which are even multiples of G0. Nevertheless, this result disagrees with experimental observations [2] by which conductance values near 1G0, 2G0 and 3G0 were measured. In view of this, one can say that the conductance should be any multiple of the fundamental conductance quantum. This result will be obtained in the present work for an MWCNT without defects by starting from the knowledge of the conductance through an imperfect single layer with elastic scattering and neglecting quantummechanical interference. In the context of ballistic transport [2,4–6], we shall calculate the conductance of a single shell showing that this conductance coincides with the maximum conductance, that is, 2G0. Since the effective or actual conductance equals the maximal conductance in series with the wire conductance, then the total conductance is equal to the fundamental conductance quantum.
2. Theory Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.01.020
We start from the above-mentioned fact that, in the absence of scattering, the conductance of an SWCNT equals 4e2/h which
ARTICLE IN PRESS M.A. Grado-Caffaro, M. Grado-Caffaro / Physica B 404 (2009) 1544–1545
comes from the existence of two spins per conduction band together with that there are two bands (two conducting channels), e2/h being the value of the conductance corresponding to each conducting mode (one band and one spin) where e is the absolute value of the electron charge and h is the Planck constant (note that G0 ¼ 2e2/h). Therefore, we can write Gmax ¼ 2G0. On the other hand, we must determine the so-called wire conductance which will be denoted by Gw. To get this end, we conceive the tube as a quasi-one-dimensional wire represented by a longitudinal ideal quantum box so that the width of this box coincides with the length of the tube. Under these conditions, the conduction electrons behave as standing waves whose wavelength is quantized obeying the relation namely nln ¼ 2l where ln is the quantized electron de Broglie wavelength, l is the length of the tube, and n ¼ 1,2,y. In addition, we have that ln ¼ h/(mvn) where m is the free-electron mass and vn denotes the magnitude of the quantized electron velocity. Now we consider the following expressions namely l ¼ vntn (where tn is the quantized motion time), sn ¼ e2Nntn/(mlA) where sn denotes quantized conductivity, Nn is the quantized number of conduction electrons, and A is the pffiffiffi cross-sectional area of the tube so A5l by virtue of the quasione-dimensionality of the tube. On the other hand, by the exclusion principle, one has that Nn ¼ 2n [6,10]. Combining all the above formulae and the relation for the quantized conductance namely Gn ¼ snA/l, it follows that Gn ¼ 2G0. This value (which is Gw) does not depend on n, as expected, so that Gw ¼ 2G0. Since, for an imperfect tube with scattering and neglecting quantum-mechanical interference, the effective conductance is the series combination of Gmax and Gw (see, for instance, Refs. [11,12]), we have that the effective conductance of a single layer is given by G ¼ GmaxGw/(Gmax+Gw) ¼ G0. We recall that, here, scattering consists of elastic collisions [2]. The interwall coupling in an MWCNT can be taken into account by assuming that the net contribution from a single layer to the conductance is given by the aforementioned effective conductance. In other words, with respect to the interaction between layers, each layer behaves as an imperfect SWCNT with scattering. For an MWCNT with m layers, the conductance is the parallel combination of the effective conductances of all the shells of the MWCNT so that the resulting conductance is Gm ¼ G0m where m ¼ 1,2,y. This result agrees well with experimental data [2] except for the 0.5G0 value measured in Ref. [2]. This value was attributed to a defect in the tube [2]. In this respect, notice that our formulation deals with MWCNTs without defects. Then, we have deduced that the conductance through an MWCNT is quantized. The fact that this quantization consists of that the conductance is any integer multiple of the fundamental conductance quantum appears as a very significant fact in contrast to expect that the conductance should be even multiples of G0. At this point, we recall that the mutual interaction between layers is equivalent to regard each layer as an imperfect tube (but without defects) with elastic scattering and negligible quantum interference. It is clear that this fact is crucial to understand why the relation Gm ¼ G0m (where m is any strictly positive integer) takes place instead of Gm ¼ 2G0m. An issue that becomes very relevant here is the ballistic nature of the electronic transport through MWCNTs (see Refs. [2,13]). In this respect, it has been shown that MWCNTs are ballistic
1545
conductors at room temperature [13]. In fact, in Ref. [13], a momentum mean free-path larger than 30 mm has been measured. This value is much larger than the typical length (around 4 mm) of an MWCNT so that the ballistic nature of the tube is manifest. Another question to be considered is the relationship between conductance quantization and ballistic transport: it is well-known that conductance quantization implies ballistic transport [2,11]. In other words, ballistic transport is a necessary (but not sufficient) condition for conductance quantization. Therefore, diffusive transport is excluded since this type of transport gives rise to non-quantized conductance. Since LGmax ¼ lGw (where L denotes mean free path) and Gmax ¼ Gw, it follows that L ¼ l which corroborates that there is scattering (here, scattering consists of elastic collisions [2,13]). However, we can say that here carrier transport is quasi-ballistic so that, although the condition L4l does not occur, when L ¼ l can be considered as a limit case in which ballistic or at least quasi-ballistic regime takes place [12,13]. In this case, given that Gw ¼ GmaxT/(1T) (where T is the transmission probability), it follows that T ¼ 12.
3. Conclusions We have determined Gw under a standing-wave approach so that the conduction electrons behave as standing waves confined in a quasi-one-dimensional ideal potential well. In this respect, we wish to remark the manifest usefulness of the particle-in-a-box model. The precedent study has led to results in excellent agreement with experimental observations [2]. As a matter of fact, it has been found that the electron conductance of an MWCNT without defects is any integer multiple of the conductance quantum. Interwall coupling has been taken into account by considering the net conductance of a single shell as the resulting conductance from Gmax in series with Gw. We have found that Gw ¼ 2G0 so Gw ¼ Gmax. Consequently, the effective or net conductance of a single layer reads G ¼ G0 which gives rise to the quantized conductance of an MWCNT namely Gm ¼ G0m. This result tells us that the interwall coupling gives rise to halving the expected conductance obtained by simply multiplying 2G0 by the number of layers. Halving 2G0 due to a possible spin-chirality coupling was suggested in Ref. [2]. However, we have derived this result without invoking considerations upon spin and its possible coupling to helicity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
M. Apostol, Phys. Lett. A 372 (2008) 5093. S. Frank, P. Poncharal, Z.L. Wang, W.A. de Heer, Science 280 (1998) 1744. M.F. Lin, K.W.K. Shung, Phys. Rev. B 51 (1995) 7592. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 18 (2004) 761. M.A. Grado-Caffaro, M. Grado-Caffaro, Optik 116 (2005) 409. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 19 (2005) 967. P. Delaney, M. di Ventra, S.T. Pantelides, Appl. Phys. Lett. 75 (1999) 3787. S. Sanvito, Y.K. Kwon, D. Toma´nek, C.J. Lambert, Phys. Rev. Lett. 84 (2000) 1974. Y.G. Yoon, P. Delaney, S.G. Louie, Phys. Rev. B 66 (2002) 327. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 18 (2004) 501. S. Datta, Electronic Transport Properties of Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. T. Du¨rkop, B.M. Kim, M.S. Fuhrer, J. Phys. Condens. Matter 16 (2004) R553. C. Berger, Y. Yi, Z.L. Wang, W.A. de Heer, Appl. Phys. A 74 (2002) 363.