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Transfer-matrix simulations of electronic transport in single-wall and multi-wall carbon nanotubes
Carbon 43 (2005) 717–726 www.elsevier.com/locate/carbon
Transfer-matrix simulations of electronic transport in single-wall and multi-wall carbon nanotubes A. Mayer
*
Laboratoire de Physique du Solide, Faculte´s Universitaires Notre-Dame de la Paix, Rue de Bruxelles 61, B-5000 Namur, Belgium Received 17 May 2004; accepted 20 October 2004 Available online 8 December 2004
Abstract We present simulations of electronic transport in single-wall and multi-wall carbon nanotubes, which are placed between two metallic contacts. We consider situations where the electrons first encounter a singe-wall nanotube (corresponding to either the inner or the outer shell of the (10, 10)@(15, 15)@(20, 20) and (10, 10)@(20, 10)@(20, 20) nanotubes), before encountering the multi-wall structures. The role of this two-step procedure is to enforce the electrons to enter a single shell of the multi-wall nanotubes, and we study how from that point they get redistributed amongst the other tubes. Because of reflections at the metallic contacts, the conductance of finite armchair nanotubes is found to depend on the length of the tubes, with values that alternate between three separate functions. Regarding the transport in multi-wall nanotubes, it is found that the electrons keep essentially propagating in the shell in which they are initially injected, with transfers to the other tubes hardly exceeding one percent of the whole current. In the case where the three tubes are conducting, these transfers are already completed after four nanometers. The conductance and repartition of the current present then oscillations, which are traced to the band structure of the nanotube. The transfers between the shells and the amplitude of these oscillations are significantly reduced when the intermediate tube is semiconducting. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; D. Transport properties
1. Introduction Carbon nanotubes have many properties of interest for practical applications. Because of their high aspect ratio, they are characterized as field emitters by low extraction fields, high current densities and long operating times. In general, the current–voltage characteristics of carbon nanotubes are found to follow a Fowler– Nordheim-type tunneling law [1–4] with an emitter work function around 5 eV depending on the type of nanotube. They are also characterized by a high mechanical resistivity, which makes them excellent candidates for the reinforcement of materials [5]. As recipient of smal*
Tel.: +32 817 247 20; fax: +32 817 247 07. E-mail address: [email protected] URL: http://www.fundp.ac.be~amayer
0008-6223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2004.10.040
ler molecules, they offer wide possibilities both for the storage of these molecules and the study of their properties in these particular confinement conditions [6]. The electronic transport properties of carbon nanotubes are another interesting aspect of these structures (see [7–13]). Disregarding hybridization effects that are important only when the diameter is smaller than one nanometer [14], it is well known that a (n, m) nanotube is either metallic or semiconducting depending on whether n m is an integer multiple of 3 or not [15,16]. In principle, this makes it possible to tailor the electronic properties of individual nanotubes. The investigation of systems composed of several nanotubes (either fitted in a multi-wall structure or connected in a junction) is not less challenging. It is known from previous studies that the coupling between atoms in the neighboring shells of multi-wall structures is relatively
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small [9,17], but impacts the properties of the individual tubes [18–21]. The way electrons are transported in these composite systems and the way this transport may depend on external factors is still open to investigations. In a recent paper [22], we developed a local pseudopotential that accounts for the properties of p electrons in carbon nanotubes. Using a transfer-matrix technique for the resolution of the Schro¨dinger equation [23–28], one can then address large composite systems and, in particular, investigate how an electronic signal that is injected into a single shell of a multi-wall nanotube is influenced in its propagation by the neighboring shells and whether significant spreading to those shells occurs. The main ideas of our model are described in Section 2. Section 3 then presents simulations of electronic transport in the (10, 10)@(15, 15)@(20, 20) and (10, 10)- @(20, 10)@(20, 20) nanotubes. The systems under consideration include the metallic electrodes as well as a section of a single-wall nanotube, which precedes the multi-wall structure and enforces the incident flux of electrons to enter a single tube of the device. Our results are discussed in Section 4. Because of reflections at the metallic contacts, the conductance of finite armchair nanotubes is found to depend on the length of the tubes, with values that alternate between three separate functions. Regarding the transport in multiwall structures, it is found that in the case where the three tubes are conducting the transfers of current between neighboring shells are already completed after four nanometers. The conductance and repartition of the current present then oscillations, which are traced to the band structure of the nanotube. The transfers between the layers and the amplitude of these oscillations are significantly reduced when the intermediate tube is semiconducting. Provided one can prepare the electrons to enter a given shell of multi-wall nanotubes, these results show that it may be possible to use them as independent conduction channels.
2. Methodology In these simulations of transport, the nanotube stands in an intermediate region (Region II) between two identical regions (Regions I and III) of constant potential, which are representative of the flat metallic contacts. The structure in Region II consists actually of the three following parts: (i) 16 units of either the (10, 10) or (20, 20) nanotubes (this first part is periodic), (ii) the connection between the single-wall and the multiwall nanotubes (this second part contains two basic units of each tube), (iii) up to 60 units of the multi-wall nanotube (this third part is periodic when the (10, 10)@(15, 15)@(20, 20) tube is considered). The role of the 16 units of the single-wall nanotube is to enforce the electrons to enter a single shell of the multi-wall
structure to which it is connected. See Fig. 1 for a representation of the whole system. To compute the potential energy in Region II, we use the local pseudopotential developed in Ref. [22]. This potential was fitted to reproduce the band-structure properties of the p electrons in graphite and isolated graphene sheets. It was shown to be well suited to carbon P3 nanotubes.2 Its expression is given by V ðrÞ ¼ i¼1 Ai expðai r Þ, where r is the distance to the nucleus. The coefficients A1, A2 and A3 are 10.607, 29.711 and 98.911 eV respectively, while the parameters a1, a2 and a3 take the value of 0.12126, 1.9148 and 0.60078 r2 Bohr respectively. To compute the electronic scattering from Region I to Region III, we use the transfer-matrix technique presented with more details in Ref. [23–27]. Using cylindrical coordinates, the wave function in Regions I and III is III; expanded in terms of basis states WI; m;j ðrÞ and Wm;j ðrÞ defined by J m ðk m;j qÞ expðim/Þ I=III; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wm;j ðq; /; zÞ ¼ RR 2 R1 2 0 dqq½J m ðk m;j qÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2m ð1Þ E k 2m;j z ; exp i h2 where the radial wave vectors km,j are solutions of J 0m ðk m;j RÞ ¼ 0 and E is defined as the kinetic energy in Regions I and III (those two regions are described as perfect metals with a constant potential energy 25 eV lower than the vacuum level). R refers to a confinement radius, which is 0.34 nm larger than the radius of the multi-wall nanotube (this is indeed sufficient to obtain results independent of its particular value). The ± signs refer to the propagation direction relative to the z-axis, which is oriented from Region I to Region III. In order to achieve a high level of convergence, we considered m values ranging from 100 to 100 and took a cut-off energy Ecut-off of 425 eV in Region II (Ecut-off is actually the parameter that limits the number of km,j used in Region qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi II through the relation k m;j 6 2m E ). The calculah2 cut-off tions presented in this paper take account of the rotational and reflection symmetries of the tubes. Applying the transfer-matrix technique, one then obtains scattering solutions of the form X r2Region I I; Wþ ¼ WI;þ S þ m;j m;j þ ðm0 ;j0 Þ;ðm;jÞ Wm0 ;j0 r2Region III
¼
X
m0 ;j0 III;þ S þþ ðm0 ;j0 Þ;ðm;jÞ Wm0 ;j0 ;
ð2Þ
m0 ;j0
corresponding to single incident states WI;þ m;j with a unit amplitude. Total current densities result from the contribution of every solution associated with a propagative state in the region of incidence. The conductance G(E) associated with a given value of the energy E is obtained using
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Fig. 1. Potential energy in the xz-plane at the junction between (20, 20) and (10, 10)@(15, 15)@(20, 20) nanotubes. The region z 6 0 contains N periodic repetitions of the (20, 20) nanotube as well as the connection between the two tubes (which consists of two units of each tube). The region z P 0 contains the periodic repetitions of the (10, 10)@(15, 15)@(20, 20) nanotube. Only N = 2 units of the periodic part of the (20, 20) nanotube are shown here, while N = 16 units are considered in the simulations.
GðEÞ ¼
dI 2e2 ¼ dV h
where vI=III;ðm;jÞ
X
vIII;ðm0 ;j0 Þ þþ 2 j S ðm0 ;j0 Þ;ðm;jÞ j ; ð3Þ v I;ðm;jÞ ðm0 ;j0 Þ;ðm;jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mh 2m E k 2m;j are the group velocities h2
in Regions I and III and the summation only includes propagative states [29]. In the zero-bias limit, we only consider contributions associated with electrons at the Fermi level EF.
3. Application: simulation of transport in single-wall and multi-wall carbon nanotubes 3.1. Initial filtering of the incident states and transport in single-wall nanotubes
Fig. 2. Conductance of sections of (5, 5) (solid), (10, 10) (dashed), (15, 15) (dot-dashed), (20, 20) (dotted) and (25, 25) (solid) nanotubes, for electrons at the Fermi level. One basic unit measures 0.24595 nm.
Since the electrons are provided by the metallic contact in Region I and one wants them to enter a single shell of the multi-wall nanotubes considered here-after, it is necessary to include N periodic repetitions of the corresponding single-wall nanotube between the metallic contact and these multi-wall structures. This indeed removes the electronic states that would otherwise be injected from the beginning in the other shells. The way transfers to those shells occur once the electrons are in the multi-wall nanotube is the subject of the next sections. In order to address the transport properties of singlewall nanotubes, we represented in Fig. 2 the conductance of sections of individual (5, 5), (10, 10), (15, 15), (20, 20) and (25, 25) nanotubes, as a function of their length. Besides accounting for the finite size of the tubes, these results also account for reflections at the metallic contacts. The figure reveals that there is a transition where the conductance changes from that characterizing the lower contact to that characterizing the tube itself. The length of this transition increases with the radius
of the tube. We explain this behavior by the larger number of states that wide tubes can accommodate (before the selection of particular combinations due to bandstructure effects). In all cases N = 16 units are sufficient to filter the non-propagating states, which justifies the choice of that value in the rest of this paper. A deeper analysis also reveals that the filtered solutions belong to the group of basis states characterized by m = . . ., 2n, n, 0, n, 2n, . . . when a (n, n) nanotube is considered [28]. This is due to the fact that the band crossing that characterizes the Fermi level of these tubes is associated with this group. The bands associated with the other groups do not cross the Fermi level and can therefore only contribute to evanescent solutions at this energy. As only propagative states are wanted in the situations considered here-after, this observation means that we only need to consider the group of basis states associated with m = . . ., 2n, n, 0, n, 2n, . . . where n characterizes here the rotational symmetry of these
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situations (n = 5 for the simulations involving the (10, 10)@(15, 15)@(20, 20) tube, and n = 10 for those involving the (10, 10)@(20, 10)@(20, 20)). Besides the transition associated with the filtering of non-propagating states, Fig. 2 also reveals oscillations whose period (32 a, with a = 0.24595 nm the length of each basic unit) is consistent with the band crossing appearing at wave vectors kz close to 23 p=a (similar oscillations were also reported by Krompiewski et al. [30], Rochefort et al. [31] and Buia et al. [32]). In all cases the conductance is smaller than the theoretical value of 2 (2e2/h) that characterizes infinite nanotubes. Other authors gave an explanation to this phenomenon, invoking the backscattering of p orbitals due to their strong coupling with the metallic contacts (for symmetry reasons p*orbitals are less affected by this process) [34–36]. In the framework of our model, this reduction in the conductance is attributed to reflections that arise at the metallic contacts and introduce a modulation of the conductance. Furthermore, the metallic contacts can only accommodate a part of the basis states that built the wave function in Region II (Ecut-off is 25 eV in Regions I and III, while it is 425 eV in Region II). The summation that defines the conductance in Eq. (3) thus contains less terms than that characterizing an infinite nanotube, which explains for the conductance never reaching the maximal value of 2 (2e2/h). These reflection effects and the deviation of the wave vectors kz from the ideal value of 23 p=a also explain for the conductance values being not repeated exactly when going from one cycle to the next. This effect is already visible in Fig. 2 for the (5, 5) nanotube, where these curvature-induced deviations are more pronounced than for the other tubes. This evolution in the amplitudes is
actually present in all tubes, as demonstrated for the (25, 25) nanotube in Figs. 3 and 4, where larger tube dimensions are considered. Figs. 3 and 4 represent the conductance of a (25, 25) nanotube, for electron energies respectively 0.05 and 0 eV below the Fermi level. The potential energy in Regions I and III is here reduced to minus the cut-off energy used in Region II in order to suppress any effect due to the truncation of Eq. (3). We also suppressed contributions associated with states that do not belong to the group characterized by m = . . ., 50, 25, 0, 25, 50, . . . as they only contribute to evanescent solutions. Fig. 3 is a clear illustration of the modulation that characterizes the conductance of finite nanotubes placed between two contacts (the free-propagating Regions I and III). Similar long-wavelength oscillations were reported in the conductance of nanotubes placed in parallel or concentric contact [32]. Long-range oscillations in the conductance were also reported by Kong et al. [37] and related to standing waves. The periodicity of the modulation observed in Fig. 3 agrees with the separation between the actual values of kz at the energy 2p considered (k 2z k 1z ¼ 1:75 108 m1 ¼ 146a ). As the contributions to the conductance get out of phase after a few hundred basic units, the range where this periodicity is well defined is however limited. In Fig. 4, 2p k 2z k 1z ¼ 9:85 106 m1 ¼ 2600a is so small that the coherence between the contributions to the conductance is lost before the first cycle is completed. The figure however illustrates the second point of interest: the fact that the conductance values actually lie on three different curves. This phenomenon was observed by Chibotaru et al. [33] in their study of the conductance of point-contacted nanotubes. It is given a mathematical justification
Fig. 3. Conductance of sections of a (25, 25) nanotube, for electrons 0.05 eV below the Fermi level. The length of the tube changes from 1 to 500 basic units (one basic unit measures 0.24595 nm). The potential energy in Regions I and III is lowered to minus the cut-off energy used in Region II in order to accommodate the same number of basis states. Only states associated with m = . . ., 50, 25, 0, 25, 25, . . . (i.e., those associated with the band crossing at the Fermi level) are represented.
Fig. 4. Conductance of sections of a (25, 25) nanotube, for electrons at the Fermi level. The length of the tube changes from 1 to 2000 basic units (one basic unit measures 0.24595 nm). The potential energy in Regions I and III is lowered to minus the cut-off energy used in Region II in order to accommodate the same number of basis states. Only states associated with m = . . ., 50, 25, 0, 25, 25, . . . (i.e., those associated with the band crossing at the Fermi level) are represented.
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Once the electrons provided by the lower metallic contact have been filtered by the 16 units of either the (10, 10) or (20, 20) nanotube, one can study how they are transported in the multi-wall (10, 10)@(15, 15)@ (20, 20) structure and how they get redistributed amongst the other shells. The connection between the 16 units of the single-wall nanotube and the periodic part of the multi-wall structure is achieved through a connection made of two basic units of each tube (see again Fig. 1). This connection also contains the contact barrier that adjusts the two Fermi levels. From the transfer matrices associated with the basic unit of each tube and their connection, it is easy using the layer-addition algorithm [27] to compute the scattering through any number of these units. We represented in Fig. 5 the conductance of the whole structure (that depicted in Fig. 1), as calculated at the upper metallic contact for different lengths of the multi-wall nanotube (the units are counted from the limit between the junction and the periodic part of the multi-wall nanotube). In order to avoid at this stage effects associated with the cut-off of the wave-function in the upper metallic contact, we gave Region III a poten-
tial energy equal to minus the cut-off energy used in Region II. This indeed makes the underlying physical processes more transparent (reflections at the upper contact are still present). Effects resulting from the truncation of the wave function in Region III will be presented later. The oscillations of the conductance in Fig. 5 are similar to those already reported in Fig. 3 and find the same explanation. Indeed one can check that the period of these oscillations is consistent with the separation be2p tween the kz values at the Fermi level (42a , this value resulting from overlaps between the pseudopotentials of the different shells). As the potential energy in Region I involves a restriction in the components of the wave function used in Eq. (3), the conductance never reaches the value of 2 (2e2/h) that would characterize the singlewall nanotube in the first part of Region II if it was infinite. We represented in Fig. 6 the conductance associated with the system where electrons first encounter the (10, 10) nanotube before entering the multi-wall structure. Although there are still three independent curves, it appears that the terms that build the wave function are directly out of phase and there is no well-defined periodicity. We explain this difference with the previous situation by the fact that the symmetry of the (10, 10) nanotube involves the introduction of states that are exponentially decreasing in the (20, 20) nanotube (i.e., those with m = . . ., 30, 10, 10, 30, . . .). As the incident states are in a coherent state, this results in constraints on the coefficients of the propagative states, which fail to be in an easy phase relation at the beginning of the multi-wall structure (thus the lack of well-defined periodicity).
Fig. 5. Conductance of a system made of sixteen units of a (20, 20) nanotube connected by four additional units to a (10, 10)@ (15, 15)@(20, 20) nanotube. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
Fig. 6. Conductance of a system made of sixteen units of a (10, 10) nanotube connected by four additional units to a (10, 10)@(15, 15)@(20, 20) nanotube. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cutoff energy used in Region II.
in Appendix A. As explained there, the fact that the conductance defines three supporting functions is the consequence of the reflections at the metallic contacts and the wave vectors kz being close to 2p . The variations of these 3a functions are related to the deviations of kz from that ideal value. In order to reduce the appearance of specific curvature effects, we consider in the rest of this paper structures at least as large as the (10, 10) nanotube. 3.2. Transport in a (10, 10)@(15, 15)@(20, 20) nanotube
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We represented in Fig. 7 the repartition of the current in the three shells of the (10, 10)@(15, 15)@(20, 20) nanotube, for the two cases considered so-far (injection into either the (20, 20) or (10, 10) nanotube). The figure shows that the most part of the current keeps propagating in the shell in which it was injected, with transfers to the other shells hardly exceeding one percent. The fraction of the current that exists in these other shells at the beginning of the representation agrees with the values one would obtain with single-wall nanotubes. Indeed the radial part of the wave function, although exponentially decreasing outside these tubes, assumes finite values. In the case of an isolated (10, 10) nanotube, one can check that the fraction of the current at the position of the (15, 15) and (20, 20) tubes are 4.31 · 104 and 4.65 · 109 respectively. In the case of an isolated (20, 20) nanotube, the corresponding fractions at the place of the (15, 15) and (10, 10) tubes are 2.82 · 104 and 1.74 · 109 respectively. The fraction of the current in the shell opposite to that in which it was initially injected tends to increase exponentially with the length of the multi-wall nanotube. After typically 16 units (4 nm), the redistribution of the current is completed and the fraction of the current in each shell oscillates around fixed values (0.999, 103 and 105 for each tube, with peak deviations of 0.99, 102 and 104 respectively). These oscillations are more pronounced in the case where the electrons are injected into the (20, 20) nanotube and exhibit the same periodicity as the conductance (around 42a). In the framework of our model, the fact that the current remains essentially in the shell in which it is injected can be explained by the selection of the basis states WI;þ m;j ðrÞ that enter the multi-wall structure. Because of the initial propagation through a single-wall nanotube, only a filtered combination of these states enter the multi-wall structure. Since the m values that characterize
the basis states of a (n, n) nanotube are separated by integer multiples of n, they extend over a wider range in the (20, 20) nanotube than in the (10, 10). Several factors then contribute to reduce the coupling between combinations associated with different shells: (i) the relevant m values are not the same, (ii) coupling between wave function components whose m values are not separated by a multiple of 5 is impossible, (iii) for the allowed coupling elements the overlap between states situated on different shells is small. Since coupling between different shells is only possible for a restricted part of the wave function, the fraction of the current in each tube remains essentially constant. 3.3. Transport in a (10, 10)@(20, 10)@(20, 20) nanotube Let us now consider transport in the (10, 10)@(20, 10)@(20, 20) nanotube, where the intermediate shell is semiconducting. The radius of this shell is comparable with that of the (15, 15) (1.035 instead of 1.017 nm) and the whole structure is characterized by a 10-fold symmetry. As the (20, 10) shell is incommensurate with the two others, the (10, 10)@(20, 10)@(20, 20) nanotube has no basic unit and we refer here-after by this term to the length of the basic cell of the (10, 10) and (20, 20) tubes (0.24595 nm). As for the previous simulations, we consider N = 16 units of either the (10, 10) or the (20, 20) nanotube and a connection between the two structures that contains four units. The plane z = 0 sets the limit with the bulk part of the (10, 10)@(20, 10)@(20, 20) structure. This situation is depicted in Fig. 8. One can compute the conductance and repartition of the current in the (10, 10)@(20, 10)@(20, 20) nanotube, as a function of its length. It turns out that the conductance essentially exhibits the same features as those observed previously (repartition of its values on three
Fig. 7. Fraction of the current running in the (10, 10) (solid), (15, 15) (dashed) and (20, 20) (dot-dashed) shell of a (10, 10)@(15, 15)@(20, 20) nanotube, in situations where electrons are initially injected into the (20, 20) (left) or (10, 10) (right) shell. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
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Fig. 8. Potential energy in the xz-plane at the junction between (20, 20) and (10, 10)@(20, 10)@(20, 20) nanotubes. The region z 6 0 contains N periodic repetitions of the (20, 20) nanotube as well as the connection between the two tubes (which consists of two ‘‘units’’ of each tube). The region z P 0 contains the bulk part of the (10, 10)@(20, 10)@(20, 20) nanotube. Only N = 2 units of the periodic part of the (20, 20) nanotube are shown here, while N = 16 units are considered in the simulations.
Fig. 9. Fraction of the current running in the (10, 10) (solid), (20, 10) (dashed) and (20, 20) (dot-dashed) shell of a (10, 10)@(20, 10)@(20, 20) nanotube, in situations where electrons are initially injected into the (20, 20) (left) or (10, 10) (right) shell. The units are counted from the limit between the junction and the bulk part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
separate curves). As the (20, 10) nanotube involves exponentially decreasing solutions in all groups of basis states, there is no easy phase relation at the entrance of the nanotube and the conductance fails to exhibit a well-defined periodicity. The fraction of the current in each tube is represented in Fig. 9, where the ‘‘basic units’’ are counted from the plane z = 0 of Fig. 8. One can see that the intermediate semiconducting tube is responsible for a sharp reduction in the transfers between the different shells. Starting from the shell in which the current is injected, the repartition of the current is typically 0.9995, 5 · 104 and 5 · 108 with peak deviations around 0.999, 103 and 106. These transfers are already completed after two units. The peak value of the fraction of the current that leaves the shell in which it is initially injected is thus reduced by one order of magnitude (compared to the situation where the three tubes are conducting), while the peak value of the trans-
fer to the opposite shell is reduced by two orders of magnitude. On average the fraction of the current in the intermediate tube turns out to be slightly smaller than that characterizing isolated tubes (which is consistent with the fact that the gap tends to reduce any value the wave function may have in the intermediate shell). This impacts the overlap that states situated in the (10, 10) and (20, 20) tubes may have, which explains the drastic reduction in the transfers between these tubes. The fact that electrons get to cross the (20, 10) nanotube may require an explanation, as it is often believed that no state exist in the gap of a semiconducting material (except sometimes for gap states related to impurities). This statement is actually only true for infinite materials, which can only accommodate propagative solutions (exponential solutions explode in one direction and are therefore forbidden). In the finite structures
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considered in this paper, evanescent states however exist in the gap and it it through these evanescent states that the transfers arise. The fact that only evanescent states exist in the semiconducting tube explains the absence of interference-related variations of the current in this tube. The incommensurability of this tube with the two others is responsible for the erratic variations observed in the last tube (especially in the left part of Fig. 9). 3.4. Effects due to the truncation of the wave function in the upper contact When Region III is given the same potential energy as Region I (i.e., a value 25 eV lower than the vacuum level), only a part of the wave function can enter the upper contact and contribute to the conductance. The remaining part is reflected repeatedly between the upper and lower contact until it finally gets projected on a wave function component for whom transition to the metallic contacts is possible. Although the conductance is not drastically affected by this process, the expansion of the wave function in Region III is however not sufficient to provide the distribution of the current with an accuracy better than 103 and one misses the physical description presented in the previous sections. Independently of the nature of the intermediate tube, the projection process typically changes the current repartition from those calculated previously to 0.99, 0.01 and 0.001. Although these values still express the fact that the current essentially remains in the shell in which it was injected, they are basically related to the limits of the wave-function expansion in Region III.
4. Discussion Our results can be compared with those obtained by Roche et al. [9]. With their tight-binding Hamiltonian and for multi-wall nanotubes with commensurate shells, they also observed a fast evolution of the current to a situation where each tube carries a fixed ratio of the total current. These final ratios are not accounted for by simple geometric arguments, since their initial outer shell keeps carrying more current than what would be expected from a simple comparison between the radii of the tubes. They found that the electronic transport depends on whether the tubes are commensurate or not, a ballistic regime being obtained with commensurate tubes while the transport in incommensurate tubes tends to be diffusive. Besides differences in the methodology, a significant change with previous studies [7–11] consists in the fact that electrons are introduced in the multi-wall nanotube after an initial propagation through a single shell of this structure. Our starting electronic states are therefore eigenstates of a given shell of the nanotube and have a
defined symmetry. Our initial conditions are thus different from those corresponding to a wave packet that is localized on a given atom and has no specific symmetry. There is however no disagreement between our results and those found elsewhere. In Ref. [38], Kim et al. investigated the conductance of telescoping nanotubes and showed that these tubes have to be characterized by common symmetries in order for the conductance to be significant. They also indicated that the transmission between telescoping tubes is essentially due to the p*electrons, which according to Ref. [34–36] may have a poor transmission to the metallic contacts. The influence that reduced symmetries or defects may have on our results should be quantified in future publications. This paper point to the interest of exploiting both the symmetry of the nanotubes and that of the initial states and suggest that the outer and inner shells of multi-wall nanotubes may be considered with a good accuracy as independent for the conduction of current. Incidently our results support the observation that the external shells of multi-wall nanotubes start burning first when high currents are run through (the current is initiated in the outer shell).
5. Conclusions We presented simulations of electronic transport in single-wall and multi-wall carbon nanotubes, using a local pseudopotential for the representation of carbon atoms and a transfer-matrix technique for the resolution of the Schro¨dinger equation. This methodology accounts for three-dimensional aspects of the potential energy and includes the metallic contacts. The simulations show that the reflections at the metallic contacts are responsible for length-dependent variations in the conductance of finite armchair nanotubes. The sequence of values obtained by changing the length of these tubes progressively turns out to alternate between three separate curves, whose variations are traced to the deviation of the wave vectors kz from their ideal value of 2p . In conditions where the wave-function 3a components are in a coherent phase relation, the conductance turns out to exhibit a well-defined periodicity. Except in one case, this phase coherency was lost from the beginning in situations where single-wall nanotubes are connected to multi-wall structures. To study the transport of electrons in multi-wall nanotubes, we considered situations where electrons first encounter 16 units of a single-wall nanotube before entering the whole structure. This initial filtering process results in the electrons entering a single shell of the nanotube, in a state specific to that shell. The simulations revealed that from this point electrons essentially keep propagating in the shell in which they were
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injected, with transfers to the other shells hardly exceeding one percent. The peak value of these transfers are reduced by one order of magnitude when the intermediate shell is semiconducting. These simulations thus show that provided one can prepare the electrons to enter a given shell of multi-wall nanotubes, it may be possible to use them as independent conduction channels.
Acknowledgments This work was supported by the National Fund for Scientific Research (FNRS) of Belgium. The author acknowledges the use of the Namur Scientific Computing Facility and the Belgian State Interuniversity Research Program on Quantum size effects in nanostructured materials (PAI/IUAP P5/01). We acknowledge Ph. Lambin for useful discussions.
Appendix A. Influence of reflections at the metallic contacts on the conductance of armchair nanotubes We justify here the appearance of three different curves in the conductance of armchair finite-size nanotubes as the result of reflections at the metallic contacts. We also show that the variations of these curves are related to the deviation of kz from the ideal value of 2p . 3a Let us consider first the ideal situation where the wave vector kz takes the ideal value of 2p . Using the 3a BlochÕs theorem, we can expand the wave function as X 2p 2p W ¼ ei3az cj eiðj a Þz ; ðA:1Þ j
where the lateral dependence of the wave function is contained in the coefficients cj. In the case of an infinite nanotube, it is trivial to show 2p 2p that Wðz þ aÞ ¼ ei 3 WðzÞ, Wðz þ 2aÞ ¼ ei 3 WðzÞ and 2p Wðz þ 3aÞ ¼ WðzÞ. Since the phase factors ei 3 are physically unimportant and because there are two solutions at the Fermi level, one will obtain the same conductance 2 2ð2eh Þ in the three cases. To introduce differences in the conductance, it is necessary to consider reflections at the metallic contacts. The wave function at the upper contact consists then of the part that crossed the n periods of the nanotube directly, plus the part that crossed the nanotube after two reflections (and three crossings of the tube), plus the part that crossed the nanotube after four reflections (and five crossings of the tube), etc. If we consider three nanotubes with a length of a, 2a and 3a, the factor that precedes the summation in Eq. (A.1) is given by 2p
2p
2p
WðaÞ ¼ ei3aa þ R2 ei3a3a þ R4 ei3a5a þ 2p
2p
¼ ei 3 þ R2 þ ei 3 R4 þ ;
ðA:2Þ
725
2p
2p
2p
Wð2aÞ ¼ ei3að2aÞ þ R2 ei3a3ð2aÞ þ R4 ei3a5ð2aÞ þ 2p
2p
¼ ei 3 þ R2 þ ei 3 R4 þ ; 2p
2p
ðA:3Þ
2p
Wð3aÞ ¼ ei3að3aÞ þ R2 ei3a3ð3aÞ þ R4 ei3a5ð3aÞ þ ¼ 1 þ R2 þ R4 þ ;
ðA:4Þ
where R is the reflection coefficient at the metallic contacts. We now have three physically different expressions for the wave function at z = a , 2a and 3a that give rise to three different values of the conductance (provided R 5 0). Note that if half-integer units were considered, one would obtain the same expressions except for a minus sign (Wð2a1 Þ ¼ Wð2aÞ, Wð2a3 Þ ¼ Wð3aÞ and Wð2a5 Þ ¼ WðaÞ). Note also that when R = 1, W(a) = W(2a) = 0 (stationary-wave condition). We can also check that W(4a) = W(a), W(5a) = W(2a), W(6a) = W(3a), etc. so that we have three different values that are actually repeated exactly when going from one cycle to the next. To introduce differences when going from one cycle to the next, the kz must be slightly different from 2p . This 3a is indeed always the case because of curvature effects or as the result of doping. Eq. (A.1) must then be replaced by X X 1 2 2p 2p 2p 2p W ¼ eið3aþdkz Þz c1j eiðj a Þz þ eið3aþdkz Þz c2j eiðj a Þz j
¼e
idk 1z z
j
W1ideal þ e
where W1ideal dk 1z and dk 2z
idk 2z z
W2ideal ;
ðA:5Þ
W2ideal
and satisfy Eqs. (A.2)–(A.4). When are small, there is no reason to expect the coefficients c2j and c2j to be orthogonal and the preceding phase factors will appear as mixed quantities when computing the conductance. Assuming for simplicity c1j and c2j to be equal, the factor preceding the summation in the wave-function expansion will be given by one of the three different functions: WðzÞ
z¼aþi 3a i2p 3
¼ e ðeid
k 1z z
2
1
2
þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2p
2
þ ei 3 R4 ðeidkz 5z þ eidkz 5z Þ þ ; WðzÞ
z¼2aþi 3a i2p 3
¼
e
1
2
ðA:6Þ
1
2
ðeidkz z þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2p
2
þ ei 3 R4 ðeidkz 5z þ eidkz 5z Þ þ ; WðzÞ
z¼3aþi 3a
¼
1
2
1
ðA:7Þ 2
1ðeidkz z þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2
þ R4 ðeidkz 5z þ eidkz 5z Þ þ ;
ðA:8Þ
where the quantities dk 1z and dk 2z are here responsible for variations of these functions when going from one cycle to the next. Substituting dk 1z ¼ dk dk/2 and dk 2z ¼ dk þ dk/2, these expressions turn out to involve factors of the form 2eidkð1þ2nÞz cosðdk2 ð1 þ 2nÞzÞ. As G is proportional to the square of the wave function, this involves a periodicity of the form cos(dk (1 + 2n)z) in
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A. Mayer / Carbon 43 (2005) 717–726
agreement with that observed in Fig. 3. This substitution also explains why the conductance reaches its maximum when z is small (the influence of dk 1z and dk 2z is then negligible and one actually gets the results associated with the ideal value 2p ). When z takes higher values, the differ3a ent factors in Eqs. (A.6)–(A.8) get out of phase, which explains for the decreasing intensities and the loss of periodicity.
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[16] Dresselhaus MS, Dresselhaus G, Eklund PC, Saito R. In: Shinar JS, Vardeny ZV, Kafafi ZH, editors. Optical and Electronic Properties of Fullerenes and Fullerene-Based Materials. New York: Marcel Dekker; 2000. p. 236–50. [17] Hansson A, Stafstro¨m S. Phys Rev B 2003;67:075406–11. [18] Saito R, Dresselhaus G, Dresselhaus MS. J Appl Phys 1993;73:494–500. [19] Kwon Y-K, Toma´nek D. Phys Rev B 1998;58:16001–4. [20] Sanvito S, Kwon Y-K, Toma´nek D, Lambert JC. Phys Rev Lett 2000;84:1974–7. [21] Uryu S. Phys Rev B 2004;69:075402–11. [22] Mayer A. Carbon 2004;42:2057–66. [23] Mayer A, Vigneron J-P. Phys Rev B 1997;56:12599–607. [24] Mayer A, Vigneron J-P. J Phys Condens Matter 1998;10:869– 81. [25] Mayer A, Vigneron J-P. Phys Rev B 1999;60:2875–82. [26] Mayer A, Vigneron J-P. Phys Rev E 2000;61:5953–60. [27] Mayer A, Vigneron J-P. Phys Rev E 1999;59:4659–66. [28] Mayer A, Vigneron J-P. Phys Rev E 1999;60:7533–40. [29] Bu¨ttiker M, Imry Y, Landauer R, Pinhas S. Phys Rev B 1985;31:6207–15. [30] Krompiewski S, Martinek J, Barnas J. Phys Rev B 2002;66: 073412–5. [31] Rochefort A, Salahub DR, Avouris P. J Phys Chem B 1999;103: 641–6. [32] Buia C, Buldum A, Lu JP. Phys Rev B 2003;67:113409–12. [33] Chibotaru LF, Compernolle S, Ceulemans A. Phys Rev B 2003; 68:125412–42. [34] Choi HJ, Ihm J, Yoon Y-G, Louie SG. Phys Rev B 1999;60: R14009–11. [35] Taylor J, Guo H, Wang J. Phys Rev B 2001;63:245407–19. [36] Palacios JJ, Pe´rez-Jime´nez AJ, Louis E, SanFabia´n E, Verge´s JA. Phys Rev Lett 2003;90:106801_1–1_4. [37] Kong J, Yenilmez E, Tombler TW, Kim W, Dai H, Laughlin RB, et al. Phys Rev Lett 2001;87:106801_1–1_4. [38] Kim D-H, Chang KJ. Phys Rev B 2002;66:155402–6.
Transfer-matrix simulations of electronic transport in single-wall and multi-wall carbon nanotubes A. Mayer
*
Laboratoire de Physique du Solide, Faculte´s Universitaires Notre-Dame de la Paix, Rue de Bruxelles 61, B-5000 Namur, Belgium Received 17 May 2004; accepted 20 October 2004 Available online 8 December 2004
Abstract We present simulations of electronic transport in single-wall and multi-wall carbon nanotubes, which are placed between two metallic contacts. We consider situations where the electrons first encounter a singe-wall nanotube (corresponding to either the inner or the outer shell of the (10, 10)@(15, 15)@(20, 20) and (10, 10)@(20, 10)@(20, 20) nanotubes), before encountering the multi-wall structures. The role of this two-step procedure is to enforce the electrons to enter a single shell of the multi-wall nanotubes, and we study how from that point they get redistributed amongst the other tubes. Because of reflections at the metallic contacts, the conductance of finite armchair nanotubes is found to depend on the length of the tubes, with values that alternate between three separate functions. Regarding the transport in multi-wall nanotubes, it is found that the electrons keep essentially propagating in the shell in which they are initially injected, with transfers to the other tubes hardly exceeding one percent of the whole current. In the case where the three tubes are conducting, these transfers are already completed after four nanometers. The conductance and repartition of the current present then oscillations, which are traced to the band structure of the nanotube. The transfers between the shells and the amplitude of these oscillations are significantly reduced when the intermediate tube is semiconducting. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; D. Transport properties
1. Introduction Carbon nanotubes have many properties of interest for practical applications. Because of their high aspect ratio, they are characterized as field emitters by low extraction fields, high current densities and long operating times. In general, the current–voltage characteristics of carbon nanotubes are found to follow a Fowler– Nordheim-type tunneling law [1–4] with an emitter work function around 5 eV depending on the type of nanotube. They are also characterized by a high mechanical resistivity, which makes them excellent candidates for the reinforcement of materials [5]. As recipient of smal*
Tel.: +32 817 247 20; fax: +32 817 247 07. E-mail address: [email protected] URL: http://www.fundp.ac.be~amayer
0008-6223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2004.10.040
ler molecules, they offer wide possibilities both for the storage of these molecules and the study of their properties in these particular confinement conditions [6]. The electronic transport properties of carbon nanotubes are another interesting aspect of these structures (see [7–13]). Disregarding hybridization effects that are important only when the diameter is smaller than one nanometer [14], it is well known that a (n, m) nanotube is either metallic or semiconducting depending on whether n m is an integer multiple of 3 or not [15,16]. In principle, this makes it possible to tailor the electronic properties of individual nanotubes. The investigation of systems composed of several nanotubes (either fitted in a multi-wall structure or connected in a junction) is not less challenging. It is known from previous studies that the coupling between atoms in the neighboring shells of multi-wall structures is relatively
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small [9,17], but impacts the properties of the individual tubes [18–21]. The way electrons are transported in these composite systems and the way this transport may depend on external factors is still open to investigations. In a recent paper [22], we developed a local pseudopotential that accounts for the properties of p electrons in carbon nanotubes. Using a transfer-matrix technique for the resolution of the Schro¨dinger equation [23–28], one can then address large composite systems and, in particular, investigate how an electronic signal that is injected into a single shell of a multi-wall nanotube is influenced in its propagation by the neighboring shells and whether significant spreading to those shells occurs. The main ideas of our model are described in Section 2. Section 3 then presents simulations of electronic transport in the (10, 10)@(15, 15)@(20, 20) and (10, 10)- @(20, 10)@(20, 20) nanotubes. The systems under consideration include the metallic electrodes as well as a section of a single-wall nanotube, which precedes the multi-wall structure and enforces the incident flux of electrons to enter a single tube of the device. Our results are discussed in Section 4. Because of reflections at the metallic contacts, the conductance of finite armchair nanotubes is found to depend on the length of the tubes, with values that alternate between three separate functions. Regarding the transport in multiwall structures, it is found that in the case where the three tubes are conducting the transfers of current between neighboring shells are already completed after four nanometers. The conductance and repartition of the current present then oscillations, which are traced to the band structure of the nanotube. The transfers between the layers and the amplitude of these oscillations are significantly reduced when the intermediate tube is semiconducting. Provided one can prepare the electrons to enter a given shell of multi-wall nanotubes, these results show that it may be possible to use them as independent conduction channels.
2. Methodology In these simulations of transport, the nanotube stands in an intermediate region (Region II) between two identical regions (Regions I and III) of constant potential, which are representative of the flat metallic contacts. The structure in Region II consists actually of the three following parts: (i) 16 units of either the (10, 10) or (20, 20) nanotubes (this first part is periodic), (ii) the connection between the single-wall and the multiwall nanotubes (this second part contains two basic units of each tube), (iii) up to 60 units of the multi-wall nanotube (this third part is periodic when the (10, 10)@(15, 15)@(20, 20) tube is considered). The role of the 16 units of the single-wall nanotube is to enforce the electrons to enter a single shell of the multi-wall
structure to which it is connected. See Fig. 1 for a representation of the whole system. To compute the potential energy in Region II, we use the local pseudopotential developed in Ref. [22]. This potential was fitted to reproduce the band-structure properties of the p electrons in graphite and isolated graphene sheets. It was shown to be well suited to carbon P3 nanotubes.2 Its expression is given by V ðrÞ ¼ i¼1 Ai expðai r Þ, where r is the distance to the nucleus. The coefficients A1, A2 and A3 are 10.607, 29.711 and 98.911 eV respectively, while the parameters a1, a2 and a3 take the value of 0.12126, 1.9148 and 0.60078 r2 Bohr respectively. To compute the electronic scattering from Region I to Region III, we use the transfer-matrix technique presented with more details in Ref. [23–27]. Using cylindrical coordinates, the wave function in Regions I and III is III; expanded in terms of basis states WI; m;j ðrÞ and Wm;j ðrÞ defined by J m ðk m;j qÞ expðim/Þ I=III; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wm;j ðq; /; zÞ ¼ RR 2 R1 2 0 dqq½J m ðk m;j qÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2m ð1Þ E k 2m;j z ; exp i h2 where the radial wave vectors km,j are solutions of J 0m ðk m;j RÞ ¼ 0 and E is defined as the kinetic energy in Regions I and III (those two regions are described as perfect metals with a constant potential energy 25 eV lower than the vacuum level). R refers to a confinement radius, which is 0.34 nm larger than the radius of the multi-wall nanotube (this is indeed sufficient to obtain results independent of its particular value). The ± signs refer to the propagation direction relative to the z-axis, which is oriented from Region I to Region III. In order to achieve a high level of convergence, we considered m values ranging from 100 to 100 and took a cut-off energy Ecut-off of 425 eV in Region II (Ecut-off is actually the parameter that limits the number of km,j used in Region qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi II through the relation k m;j 6 2m E ). The calculah2 cut-off tions presented in this paper take account of the rotational and reflection symmetries of the tubes. Applying the transfer-matrix technique, one then obtains scattering solutions of the form X r2Region I I; Wþ ¼ WI;þ S þ m;j m;j þ ðm0 ;j0 Þ;ðm;jÞ Wm0 ;j0 r2Region III
¼
X
m0 ;j0 III;þ S þþ ðm0 ;j0 Þ;ðm;jÞ Wm0 ;j0 ;
ð2Þ
m0 ;j0
corresponding to single incident states WI;þ m;j with a unit amplitude. Total current densities result from the contribution of every solution associated with a propagative state in the region of incidence. The conductance G(E) associated with a given value of the energy E is obtained using
A. Mayer / Carbon 43 (2005) 717–726
719
Fig. 1. Potential energy in the xz-plane at the junction between (20, 20) and (10, 10)@(15, 15)@(20, 20) nanotubes. The region z 6 0 contains N periodic repetitions of the (20, 20) nanotube as well as the connection between the two tubes (which consists of two units of each tube). The region z P 0 contains the periodic repetitions of the (10, 10)@(15, 15)@(20, 20) nanotube. Only N = 2 units of the periodic part of the (20, 20) nanotube are shown here, while N = 16 units are considered in the simulations.
GðEÞ ¼
dI 2e2 ¼ dV h
where vI=III;ðm;jÞ
X
vIII;ðm0 ;j0 Þ þþ 2 j S ðm0 ;j0 Þ;ðm;jÞ j ; ð3Þ v I;ðm;jÞ ðm0 ;j0 Þ;ðm;jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mh 2m E k 2m;j are the group velocities h2
in Regions I and III and the summation only includes propagative states [29]. In the zero-bias limit, we only consider contributions associated with electrons at the Fermi level EF.
3. Application: simulation of transport in single-wall and multi-wall carbon nanotubes 3.1. Initial filtering of the incident states and transport in single-wall nanotubes
Fig. 2. Conductance of sections of (5, 5) (solid), (10, 10) (dashed), (15, 15) (dot-dashed), (20, 20) (dotted) and (25, 25) (solid) nanotubes, for electrons at the Fermi level. One basic unit measures 0.24595 nm.
Since the electrons are provided by the metallic contact in Region I and one wants them to enter a single shell of the multi-wall nanotubes considered here-after, it is necessary to include N periodic repetitions of the corresponding single-wall nanotube between the metallic contact and these multi-wall structures. This indeed removes the electronic states that would otherwise be injected from the beginning in the other shells. The way transfers to those shells occur once the electrons are in the multi-wall nanotube is the subject of the next sections. In order to address the transport properties of singlewall nanotubes, we represented in Fig. 2 the conductance of sections of individual (5, 5), (10, 10), (15, 15), (20, 20) and (25, 25) nanotubes, as a function of their length. Besides accounting for the finite size of the tubes, these results also account for reflections at the metallic contacts. The figure reveals that there is a transition where the conductance changes from that characterizing the lower contact to that characterizing the tube itself. The length of this transition increases with the radius
of the tube. We explain this behavior by the larger number of states that wide tubes can accommodate (before the selection of particular combinations due to bandstructure effects). In all cases N = 16 units are sufficient to filter the non-propagating states, which justifies the choice of that value in the rest of this paper. A deeper analysis also reveals that the filtered solutions belong to the group of basis states characterized by m = . . ., 2n, n, 0, n, 2n, . . . when a (n, n) nanotube is considered [28]. This is due to the fact that the band crossing that characterizes the Fermi level of these tubes is associated with this group. The bands associated with the other groups do not cross the Fermi level and can therefore only contribute to evanescent solutions at this energy. As only propagative states are wanted in the situations considered here-after, this observation means that we only need to consider the group of basis states associated with m = . . ., 2n, n, 0, n, 2n, . . . where n characterizes here the rotational symmetry of these
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A. Mayer / Carbon 43 (2005) 717–726
situations (n = 5 for the simulations involving the (10, 10)@(15, 15)@(20, 20) tube, and n = 10 for those involving the (10, 10)@(20, 10)@(20, 20)). Besides the transition associated with the filtering of non-propagating states, Fig. 2 also reveals oscillations whose period (32 a, with a = 0.24595 nm the length of each basic unit) is consistent with the band crossing appearing at wave vectors kz close to 23 p=a (similar oscillations were also reported by Krompiewski et al. [30], Rochefort et al. [31] and Buia et al. [32]). In all cases the conductance is smaller than the theoretical value of 2 (2e2/h) that characterizes infinite nanotubes. Other authors gave an explanation to this phenomenon, invoking the backscattering of p orbitals due to their strong coupling with the metallic contacts (for symmetry reasons p*orbitals are less affected by this process) [34–36]. In the framework of our model, this reduction in the conductance is attributed to reflections that arise at the metallic contacts and introduce a modulation of the conductance. Furthermore, the metallic contacts can only accommodate a part of the basis states that built the wave function in Region II (Ecut-off is 25 eV in Regions I and III, while it is 425 eV in Region II). The summation that defines the conductance in Eq. (3) thus contains less terms than that characterizing an infinite nanotube, which explains for the conductance never reaching the maximal value of 2 (2e2/h). These reflection effects and the deviation of the wave vectors kz from the ideal value of 23 p=a also explain for the conductance values being not repeated exactly when going from one cycle to the next. This effect is already visible in Fig. 2 for the (5, 5) nanotube, where these curvature-induced deviations are more pronounced than for the other tubes. This evolution in the amplitudes is
actually present in all tubes, as demonstrated for the (25, 25) nanotube in Figs. 3 and 4, where larger tube dimensions are considered. Figs. 3 and 4 represent the conductance of a (25, 25) nanotube, for electron energies respectively 0.05 and 0 eV below the Fermi level. The potential energy in Regions I and III is here reduced to minus the cut-off energy used in Region II in order to suppress any effect due to the truncation of Eq. (3). We also suppressed contributions associated with states that do not belong to the group characterized by m = . . ., 50, 25, 0, 25, 50, . . . as they only contribute to evanescent solutions. Fig. 3 is a clear illustration of the modulation that characterizes the conductance of finite nanotubes placed between two contacts (the free-propagating Regions I and III). Similar long-wavelength oscillations were reported in the conductance of nanotubes placed in parallel or concentric contact [32]. Long-range oscillations in the conductance were also reported by Kong et al. [37] and related to standing waves. The periodicity of the modulation observed in Fig. 3 agrees with the separation between the actual values of kz at the energy 2p considered (k 2z k 1z ¼ 1:75 108 m1 ¼ 146a ). As the contributions to the conductance get out of phase after a few hundred basic units, the range where this periodicity is well defined is however limited. In Fig. 4, 2p k 2z k 1z ¼ 9:85 106 m1 ¼ 2600a is so small that the coherence between the contributions to the conductance is lost before the first cycle is completed. The figure however illustrates the second point of interest: the fact that the conductance values actually lie on three different curves. This phenomenon was observed by Chibotaru et al. [33] in their study of the conductance of point-contacted nanotubes. It is given a mathematical justification
Fig. 3. Conductance of sections of a (25, 25) nanotube, for electrons 0.05 eV below the Fermi level. The length of the tube changes from 1 to 500 basic units (one basic unit measures 0.24595 nm). The potential energy in Regions I and III is lowered to minus the cut-off energy used in Region II in order to accommodate the same number of basis states. Only states associated with m = . . ., 50, 25, 0, 25, 25, . . . (i.e., those associated with the band crossing at the Fermi level) are represented.
Fig. 4. Conductance of sections of a (25, 25) nanotube, for electrons at the Fermi level. The length of the tube changes from 1 to 2000 basic units (one basic unit measures 0.24595 nm). The potential energy in Regions I and III is lowered to minus the cut-off energy used in Region II in order to accommodate the same number of basis states. Only states associated with m = . . ., 50, 25, 0, 25, 25, . . . (i.e., those associated with the band crossing at the Fermi level) are represented.
A. Mayer / Carbon 43 (2005) 717–726
721
Once the electrons provided by the lower metallic contact have been filtered by the 16 units of either the (10, 10) or (20, 20) nanotube, one can study how they are transported in the multi-wall (10, 10)@(15, 15)@ (20, 20) structure and how they get redistributed amongst the other shells. The connection between the 16 units of the single-wall nanotube and the periodic part of the multi-wall structure is achieved through a connection made of two basic units of each tube (see again Fig. 1). This connection also contains the contact barrier that adjusts the two Fermi levels. From the transfer matrices associated with the basic unit of each tube and their connection, it is easy using the layer-addition algorithm [27] to compute the scattering through any number of these units. We represented in Fig. 5 the conductance of the whole structure (that depicted in Fig. 1), as calculated at the upper metallic contact for different lengths of the multi-wall nanotube (the units are counted from the limit between the junction and the periodic part of the multi-wall nanotube). In order to avoid at this stage effects associated with the cut-off of the wave-function in the upper metallic contact, we gave Region III a poten-
tial energy equal to minus the cut-off energy used in Region II. This indeed makes the underlying physical processes more transparent (reflections at the upper contact are still present). Effects resulting from the truncation of the wave function in Region III will be presented later. The oscillations of the conductance in Fig. 5 are similar to those already reported in Fig. 3 and find the same explanation. Indeed one can check that the period of these oscillations is consistent with the separation be2p tween the kz values at the Fermi level (42a , this value resulting from overlaps between the pseudopotentials of the different shells). As the potential energy in Region I involves a restriction in the components of the wave function used in Eq. (3), the conductance never reaches the value of 2 (2e2/h) that would characterize the singlewall nanotube in the first part of Region II if it was infinite. We represented in Fig. 6 the conductance associated with the system where electrons first encounter the (10, 10) nanotube before entering the multi-wall structure. Although there are still three independent curves, it appears that the terms that build the wave function are directly out of phase and there is no well-defined periodicity. We explain this difference with the previous situation by the fact that the symmetry of the (10, 10) nanotube involves the introduction of states that are exponentially decreasing in the (20, 20) nanotube (i.e., those with m = . . ., 30, 10, 10, 30, . . .). As the incident states are in a coherent state, this results in constraints on the coefficients of the propagative states, which fail to be in an easy phase relation at the beginning of the multi-wall structure (thus the lack of well-defined periodicity).
Fig. 5. Conductance of a system made of sixteen units of a (20, 20) nanotube connected by four additional units to a (10, 10)@ (15, 15)@(20, 20) nanotube. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
Fig. 6. Conductance of a system made of sixteen units of a (10, 10) nanotube connected by four additional units to a (10, 10)@(15, 15)@(20, 20) nanotube. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cutoff energy used in Region II.
in Appendix A. As explained there, the fact that the conductance defines three supporting functions is the consequence of the reflections at the metallic contacts and the wave vectors kz being close to 2p . The variations of these 3a functions are related to the deviations of kz from that ideal value. In order to reduce the appearance of specific curvature effects, we consider in the rest of this paper structures at least as large as the (10, 10) nanotube. 3.2. Transport in a (10, 10)@(15, 15)@(20, 20) nanotube
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We represented in Fig. 7 the repartition of the current in the three shells of the (10, 10)@(15, 15)@(20, 20) nanotube, for the two cases considered so-far (injection into either the (20, 20) or (10, 10) nanotube). The figure shows that the most part of the current keeps propagating in the shell in which it was injected, with transfers to the other shells hardly exceeding one percent. The fraction of the current that exists in these other shells at the beginning of the representation agrees with the values one would obtain with single-wall nanotubes. Indeed the radial part of the wave function, although exponentially decreasing outside these tubes, assumes finite values. In the case of an isolated (10, 10) nanotube, one can check that the fraction of the current at the position of the (15, 15) and (20, 20) tubes are 4.31 · 104 and 4.65 · 109 respectively. In the case of an isolated (20, 20) nanotube, the corresponding fractions at the place of the (15, 15) and (10, 10) tubes are 2.82 · 104 and 1.74 · 109 respectively. The fraction of the current in the shell opposite to that in which it was initially injected tends to increase exponentially with the length of the multi-wall nanotube. After typically 16 units (4 nm), the redistribution of the current is completed and the fraction of the current in each shell oscillates around fixed values (0.999, 103 and 105 for each tube, with peak deviations of 0.99, 102 and 104 respectively). These oscillations are more pronounced in the case where the electrons are injected into the (20, 20) nanotube and exhibit the same periodicity as the conductance (around 42a). In the framework of our model, the fact that the current remains essentially in the shell in which it is injected can be explained by the selection of the basis states WI;þ m;j ðrÞ that enter the multi-wall structure. Because of the initial propagation through a single-wall nanotube, only a filtered combination of these states enter the multi-wall structure. Since the m values that characterize
the basis states of a (n, n) nanotube are separated by integer multiples of n, they extend over a wider range in the (20, 20) nanotube than in the (10, 10). Several factors then contribute to reduce the coupling between combinations associated with different shells: (i) the relevant m values are not the same, (ii) coupling between wave function components whose m values are not separated by a multiple of 5 is impossible, (iii) for the allowed coupling elements the overlap between states situated on different shells is small. Since coupling between different shells is only possible for a restricted part of the wave function, the fraction of the current in each tube remains essentially constant. 3.3. Transport in a (10, 10)@(20, 10)@(20, 20) nanotube Let us now consider transport in the (10, 10)@(20, 10)@(20, 20) nanotube, where the intermediate shell is semiconducting. The radius of this shell is comparable with that of the (15, 15) (1.035 instead of 1.017 nm) and the whole structure is characterized by a 10-fold symmetry. As the (20, 10) shell is incommensurate with the two others, the (10, 10)@(20, 10)@(20, 20) nanotube has no basic unit and we refer here-after by this term to the length of the basic cell of the (10, 10) and (20, 20) tubes (0.24595 nm). As for the previous simulations, we consider N = 16 units of either the (10, 10) or the (20, 20) nanotube and a connection between the two structures that contains four units. The plane z = 0 sets the limit with the bulk part of the (10, 10)@(20, 10)@(20, 20) structure. This situation is depicted in Fig. 8. One can compute the conductance and repartition of the current in the (10, 10)@(20, 10)@(20, 20) nanotube, as a function of its length. It turns out that the conductance essentially exhibits the same features as those observed previously (repartition of its values on three
Fig. 7. Fraction of the current running in the (10, 10) (solid), (15, 15) (dashed) and (20, 20) (dot-dashed) shell of a (10, 10)@(15, 15)@(20, 20) nanotube, in situations where electrons are initially injected into the (20, 20) (left) or (10, 10) (right) shell. The units are counted from the limit between the junction and the periodic part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
A. Mayer / Carbon 43 (2005) 717–726
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Fig. 8. Potential energy in the xz-plane at the junction between (20, 20) and (10, 10)@(20, 10)@(20, 20) nanotubes. The region z 6 0 contains N periodic repetitions of the (20, 20) nanotube as well as the connection between the two tubes (which consists of two ‘‘units’’ of each tube). The region z P 0 contains the bulk part of the (10, 10)@(20, 10)@(20, 20) nanotube. Only N = 2 units of the periodic part of the (20, 20) nanotube are shown here, while N = 16 units are considered in the simulations.
Fig. 9. Fraction of the current running in the (10, 10) (solid), (20, 10) (dashed) and (20, 20) (dot-dashed) shell of a (10, 10)@(20, 10)@(20, 20) nanotube, in situations where electrons are initially injected into the (20, 20) (left) or (10, 10) (right) shell. The units are counted from the limit between the junction and the bulk part of the multi-wall nanotube. The potential energy in Region III is equal to minus the cut-off energy used in Region II.
separate curves). As the (20, 10) nanotube involves exponentially decreasing solutions in all groups of basis states, there is no easy phase relation at the entrance of the nanotube and the conductance fails to exhibit a well-defined periodicity. The fraction of the current in each tube is represented in Fig. 9, where the ‘‘basic units’’ are counted from the plane z = 0 of Fig. 8. One can see that the intermediate semiconducting tube is responsible for a sharp reduction in the transfers between the different shells. Starting from the shell in which the current is injected, the repartition of the current is typically 0.9995, 5 · 104 and 5 · 108 with peak deviations around 0.999, 103 and 106. These transfers are already completed after two units. The peak value of the fraction of the current that leaves the shell in which it is initially injected is thus reduced by one order of magnitude (compared to the situation where the three tubes are conducting), while the peak value of the trans-
fer to the opposite shell is reduced by two orders of magnitude. On average the fraction of the current in the intermediate tube turns out to be slightly smaller than that characterizing isolated tubes (which is consistent with the fact that the gap tends to reduce any value the wave function may have in the intermediate shell). This impacts the overlap that states situated in the (10, 10) and (20, 20) tubes may have, which explains the drastic reduction in the transfers between these tubes. The fact that electrons get to cross the (20, 10) nanotube may require an explanation, as it is often believed that no state exist in the gap of a semiconducting material (except sometimes for gap states related to impurities). This statement is actually only true for infinite materials, which can only accommodate propagative solutions (exponential solutions explode in one direction and are therefore forbidden). In the finite structures
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considered in this paper, evanescent states however exist in the gap and it it through these evanescent states that the transfers arise. The fact that only evanescent states exist in the semiconducting tube explains the absence of interference-related variations of the current in this tube. The incommensurability of this tube with the two others is responsible for the erratic variations observed in the last tube (especially in the left part of Fig. 9). 3.4. Effects due to the truncation of the wave function in the upper contact When Region III is given the same potential energy as Region I (i.e., a value 25 eV lower than the vacuum level), only a part of the wave function can enter the upper contact and contribute to the conductance. The remaining part is reflected repeatedly between the upper and lower contact until it finally gets projected on a wave function component for whom transition to the metallic contacts is possible. Although the conductance is not drastically affected by this process, the expansion of the wave function in Region III is however not sufficient to provide the distribution of the current with an accuracy better than 103 and one misses the physical description presented in the previous sections. Independently of the nature of the intermediate tube, the projection process typically changes the current repartition from those calculated previously to 0.99, 0.01 and 0.001. Although these values still express the fact that the current essentially remains in the shell in which it was injected, they are basically related to the limits of the wave-function expansion in Region III.
4. Discussion Our results can be compared with those obtained by Roche et al. [9]. With their tight-binding Hamiltonian and for multi-wall nanotubes with commensurate shells, they also observed a fast evolution of the current to a situation where each tube carries a fixed ratio of the total current. These final ratios are not accounted for by simple geometric arguments, since their initial outer shell keeps carrying more current than what would be expected from a simple comparison between the radii of the tubes. They found that the electronic transport depends on whether the tubes are commensurate or not, a ballistic regime being obtained with commensurate tubes while the transport in incommensurate tubes tends to be diffusive. Besides differences in the methodology, a significant change with previous studies [7–11] consists in the fact that electrons are introduced in the multi-wall nanotube after an initial propagation through a single shell of this structure. Our starting electronic states are therefore eigenstates of a given shell of the nanotube and have a
defined symmetry. Our initial conditions are thus different from those corresponding to a wave packet that is localized on a given atom and has no specific symmetry. There is however no disagreement between our results and those found elsewhere. In Ref. [38], Kim et al. investigated the conductance of telescoping nanotubes and showed that these tubes have to be characterized by common symmetries in order for the conductance to be significant. They also indicated that the transmission between telescoping tubes is essentially due to the p*electrons, which according to Ref. [34–36] may have a poor transmission to the metallic contacts. The influence that reduced symmetries or defects may have on our results should be quantified in future publications. This paper point to the interest of exploiting both the symmetry of the nanotubes and that of the initial states and suggest that the outer and inner shells of multi-wall nanotubes may be considered with a good accuracy as independent for the conduction of current. Incidently our results support the observation that the external shells of multi-wall nanotubes start burning first when high currents are run through (the current is initiated in the outer shell).
5. Conclusions We presented simulations of electronic transport in single-wall and multi-wall carbon nanotubes, using a local pseudopotential for the representation of carbon atoms and a transfer-matrix technique for the resolution of the Schro¨dinger equation. This methodology accounts for three-dimensional aspects of the potential energy and includes the metallic contacts. The simulations show that the reflections at the metallic contacts are responsible for length-dependent variations in the conductance of finite armchair nanotubes. The sequence of values obtained by changing the length of these tubes progressively turns out to alternate between three separate curves, whose variations are traced to the deviation of the wave vectors kz from their ideal value of 2p . In conditions where the wave-function 3a components are in a coherent phase relation, the conductance turns out to exhibit a well-defined periodicity. Except in one case, this phase coherency was lost from the beginning in situations where single-wall nanotubes are connected to multi-wall structures. To study the transport of electrons in multi-wall nanotubes, we considered situations where electrons first encounter 16 units of a single-wall nanotube before entering the whole structure. This initial filtering process results in the electrons entering a single shell of the nanotube, in a state specific to that shell. The simulations revealed that from this point electrons essentially keep propagating in the shell in which they were
A. Mayer / Carbon 43 (2005) 717–726
injected, with transfers to the other shells hardly exceeding one percent. The peak value of these transfers are reduced by one order of magnitude when the intermediate shell is semiconducting. These simulations thus show that provided one can prepare the electrons to enter a given shell of multi-wall nanotubes, it may be possible to use them as independent conduction channels.
Acknowledgments This work was supported by the National Fund for Scientific Research (FNRS) of Belgium. The author acknowledges the use of the Namur Scientific Computing Facility and the Belgian State Interuniversity Research Program on Quantum size effects in nanostructured materials (PAI/IUAP P5/01). We acknowledge Ph. Lambin for useful discussions.
Appendix A. Influence of reflections at the metallic contacts on the conductance of armchair nanotubes We justify here the appearance of three different curves in the conductance of armchair finite-size nanotubes as the result of reflections at the metallic contacts. We also show that the variations of these curves are related to the deviation of kz from the ideal value of 2p . 3a Let us consider first the ideal situation where the wave vector kz takes the ideal value of 2p . Using the 3a BlochÕs theorem, we can expand the wave function as X 2p 2p W ¼ ei3az cj eiðj a Þz ; ðA:1Þ j
where the lateral dependence of the wave function is contained in the coefficients cj. In the case of an infinite nanotube, it is trivial to show 2p 2p that Wðz þ aÞ ¼ ei 3 WðzÞ, Wðz þ 2aÞ ¼ ei 3 WðzÞ and 2p Wðz þ 3aÞ ¼ WðzÞ. Since the phase factors ei 3 are physically unimportant and because there are two solutions at the Fermi level, one will obtain the same conductance 2 2ð2eh Þ in the three cases. To introduce differences in the conductance, it is necessary to consider reflections at the metallic contacts. The wave function at the upper contact consists then of the part that crossed the n periods of the nanotube directly, plus the part that crossed the nanotube after two reflections (and three crossings of the tube), plus the part that crossed the nanotube after four reflections (and five crossings of the tube), etc. If we consider three nanotubes with a length of a, 2a and 3a, the factor that precedes the summation in Eq. (A.1) is given by 2p
2p
2p
WðaÞ ¼ ei3aa þ R2 ei3a3a þ R4 ei3a5a þ 2p
2p
¼ ei 3 þ R2 þ ei 3 R4 þ ;
ðA:2Þ
725
2p
2p
2p
Wð2aÞ ¼ ei3að2aÞ þ R2 ei3a3ð2aÞ þ R4 ei3a5ð2aÞ þ 2p
2p
¼ ei 3 þ R2 þ ei 3 R4 þ ; 2p
2p
ðA:3Þ
2p
Wð3aÞ ¼ ei3að3aÞ þ R2 ei3a3ð3aÞ þ R4 ei3a5ð3aÞ þ ¼ 1 þ R2 þ R4 þ ;
ðA:4Þ
where R is the reflection coefficient at the metallic contacts. We now have three physically different expressions for the wave function at z = a , 2a and 3a that give rise to three different values of the conductance (provided R 5 0). Note that if half-integer units were considered, one would obtain the same expressions except for a minus sign (Wð2a1 Þ ¼ Wð2aÞ, Wð2a3 Þ ¼ Wð3aÞ and Wð2a5 Þ ¼ WðaÞ). Note also that when R = 1, W(a) = W(2a) = 0 (stationary-wave condition). We can also check that W(4a) = W(a), W(5a) = W(2a), W(6a) = W(3a), etc. so that we have three different values that are actually repeated exactly when going from one cycle to the next. To introduce differences when going from one cycle to the next, the kz must be slightly different from 2p . This 3a is indeed always the case because of curvature effects or as the result of doping. Eq. (A.1) must then be replaced by X X 1 2 2p 2p 2p 2p W ¼ eið3aþdkz Þz c1j eiðj a Þz þ eið3aþdkz Þz c2j eiðj a Þz j
¼e
idk 1z z
j
W1ideal þ e
where W1ideal dk 1z and dk 2z
idk 2z z
W2ideal ;
ðA:5Þ
W2ideal
and satisfy Eqs. (A.2)–(A.4). When are small, there is no reason to expect the coefficients c2j and c2j to be orthogonal and the preceding phase factors will appear as mixed quantities when computing the conductance. Assuming for simplicity c1j and c2j to be equal, the factor preceding the summation in the wave-function expansion will be given by one of the three different functions: WðzÞ
z¼aþi 3a i2p 3
¼ e ðeid
k 1z z
2
1
2
þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2p
2
þ ei 3 R4 ðeidkz 5z þ eidkz 5z Þ þ ; WðzÞ
z¼2aþi 3a i2p 3
¼
e
1
2
ðA:6Þ
1
2
ðeidkz z þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2p
2
þ ei 3 R4 ðeidkz 5z þ eidkz 5z Þ þ ; WðzÞ
z¼3aþi 3a
¼
1
2
1
ðA:7Þ 2
1ðeidkz z þ eidkz z Þ þ R2 ðeidkz 3z þ eidkz 3z Þ 1
2
þ R4 ðeidkz 5z þ eidkz 5z Þ þ ;
ðA:8Þ
where the quantities dk 1z and dk 2z are here responsible for variations of these functions when going from one cycle to the next. Substituting dk 1z ¼ dk dk/2 and dk 2z ¼ dk þ dk/2, these expressions turn out to involve factors of the form 2eidkð1þ2nÞz cosðdk2 ð1 þ 2nÞzÞ. As G is proportional to the square of the wave function, this involves a periodicity of the form cos(dk (1 + 2n)z) in
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agreement with that observed in Fig. 3. This substitution also explains why the conductance reaches its maximum when z is small (the influence of dk 1z and dk 2z is then negligible and one actually gets the results associated with the ideal value 2p ). When z takes higher values, the differ3a ent factors in Eqs. (A.6)–(A.8) get out of phase, which explains for the decreasing intensities and the loss of periodicity.
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