Formation pathways for single-wall carbon nanotube multiterminal junctions

Inna Ponomareva¹, Leonid A Chernozatonskii¹, Antonis N Andriotis² and Madhu Menon^3,4

¹Institute of Biochemical Physics, Russian Academy of Sciences, Moscow 119991, Russian Federation
²Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, PO Box 1527, 71110 Heraklio, Crete, Greece
³Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA
⁴Center for Computational Sciences, University of Kentucky, Lexington,KY 40506-0045, USA

Email: iponomar@sky.chph.ras.ru, cherno@sky.chph.ras.ru, andriot@iesl.forth.gr, super250@pop.uky.edu and madhu@ccs.uky.edu

Received 23 June 2003
Published 30 September 2003

Contents

• 1. Introduction
• 2. T-junctions
  • 2.1. (9, 0)-(10, 0)-(9, 0) T-junction formation
  • 2.2. (5, 5)-(10, 0)-(5, 5) T-junction formation
  • 2.3. Energy considerations in the formation of T-junctions
• 3. Y-junctions
  • 3.1. (6, 6)-(6, 6)-(6, 6) Y-junction formation
  • 3.2. Transport properties
• 4. X-junctions
• 5. Summary
• Acknowledgments
• References

1. Introduction

Single-wall carbon nanotubes (SWCN) constitute molecularly perfect materials with many interesting properties. The electronic structure of these nanotubes can be either metallic or semiconducting, depending on both the diameter and chirality, which can be uniquely determined by the chiral vector (n,m), where n and m are integers [1]-[3].

Junctions formed by SWCN offer the possibility of their use in nanoscale electronic devices. The simplest way to connect two dissimilar nanotubes is proposed to be via the introduction of a heptagon and pentagon pair in an otherwise perfect hexagonal graphene sheet [4]-[10]. The resulting structure still contains three-fold coordination for all carbon atoms. Similarly, three-terminal SWCN junctions can also be constructed by the introduction of topological defects including pentagons and heptagons [11]-[15]. Among the three-terminal junctions formed by carbon nanotubes, the most interesting (from a device perspective) are the `T-' and `Y-junctions'. Earlier experimental observations of carbon nanotube Y-junctions [16, 17] did not attract much attention for electronics applications due mainly to the difficulties associated with their synthesis and the complexities of their structures. Very recently, however, experimentalists have succeeded in achieving controlled growth of Y-junctions [18, 19]. The conductance measurements on these Y-junctions have shown intrinsic nonlinear and asymmetric I-V behaviour at room temperature [20]. Theoretical calculations have supported these experimental findings [14, 15].

In an exciting development experimentalists have recently succeeded in creatingX-shaped molecular connections by welding achieved by electron-beam irradiation of crossing SWCNs [21]. On further irradiation with the electron beam they were selectively able to remove one of the arms of the X-junction to create Y- and T-junctions. This work demonstrated the feasibility of using controlled electron irradiation in tailoring the junction geometry to create the desired SWCN multiterminal junctions.

In this work we explore the formation of SWCN T-, Y- and X-junctions through the fusion of two SWCNs using tight-binding as well as classical molecular dynamics simulations. We note that similar modelling has been carried out for fullerene-fullerene [22] and fullerene-nanotube [23] coalescence. The highlights of our work will be published elsewhere [24]. Here we provide more details by performing a systematic investigation of nanotube-nanotube coalescence by considering energetically efficient topological sequences leading to the formation of T-, Y- and X-junctions. No structural defects (vacancies or interstitials) are present at any step of the process and the transformation is achieved only through topological defects. The total number of carbon atoms remain the same at each step. It should be noted, however, that the total number of possible pathways for multiterminal junction formation could be too numerous for an exhaustive analysis. We explore the lowest energy pathway for the process. Since the initial and final geometries involve only sp² bonding for all carbon atoms, it is reasonable to assume that having only sp² bonding during each of the intermediate steps of coalescence would provide the lowest energy pathways. The intermediate steps involve the formation of many topological defects in the form of pentagons, heptagons, octagons, etc. Furthermore, the defects must obey a generalization of the well known Euler formula specifying the local bond surplus at the junction [25]. In this scheme, each pentagon causes a bond surplus of  - 1, while heptagons and octagons contribute to bond surpluses of  + 1 and  + 2 each, respectively. According to the generalized Euler's rule, for all three-point junctions (such as T- and Y-junctions), the total bond surplus at each junction should be  + 6. For four-point junctions (such as X-junctions), on the other hand, this rule gives a bond surplus of  + 12. With these in mind, in the following, we propose a sequence of intermediate steps leading to the formation of T-, Y- and X-junctions in which all carbon atoms remain sp²-coordinated throughout.

2. T-junctions

We begin our study by exploring the formation pathways for two SWCN T-junctions. The first case consists of a head-on fusion of a (10, 0) nanotube into the side wall of a (9, 0) nanotube, resulting in a (9, 0)-(10, 0)-(9, 0) T-junction. In the second case a (10, 0) nanotube fuses head on with the side wall of a (5, 5) nanotube to form a (5, 5)-(10, 0)-(5, 5) T-junction. In both cases the structures are relaxed at each step using the generalized tight-binding molecular dynamics (GTBMD) scheme of Menon et al [26].

2.1. (9, 0)-(10, 0)-(9, 0) T-junction formation

In figure 1 we show an 8-step process leading to the formation of a (9, 0)-(10, 0)-(9, 0) T-junction. In the first step, (1), a capped zig-zag (10, 0) nanotube is shown near a zig-zag (9, 0) nanotube wall containing defects. All defect rings are shown in red. The two SWCN clusters contain a total of 288 carbon atoms. Note that the defected (9, 0) nanotube contains 6 more carbon atoms than the corresponding defect-free nanotube. These defects, however, do not break the sp² atomic arrangement. In the second step (2), four new bonds are formed while two bonds break in each of the (9, 0) and (10, 0) nanotubes. These new bonds form sides of two octagonal defects that are situated between the two nanotubes. Additionally, there are 10 pentagons, 6 heptagons, and 2 enneagons (9-fold rings). In step 3, the neck is widened and the two octagons move more into the (9, 0) nanotube. This structure also contains 8 pentagons and 10 heptagons. In the next step (4), the two octagons have moved almost completely into the (9, 0) nanotube region. Additionally, there are 6 pentagons and 8 heptagons. In step 5, the two octagons move into the two `armpits' of the emerging T-junction. This structure also includes 4 pentagons and 6 heptagons. The two octagons are annihilated in the next step (6) and the structure is left with defects in the form of 6 pentagons and 12 heptagons. In step 7, the two octagons emerge again, but on the (10, 0) side of the junction. Additionally, the structure includes 4 pentagons and 6 heptagons. The next step (8) results in the final configuration in which, once again, the octagons are annihilated. The structure now contains topological defects in the form of 8 heptagons and 2 pentagons only. By keeping track of the defects it is easy to see that the bond surplus at every step (except, of course, step 1 which is prior to the junction formation) of the process remains at  + 6, confirming the generalization of Euler's rule. In table 1 we list the number and types of defects at each step as well as the relative energies obtained using the GTBMD scheme.

The process described above assumes the existence of topological defects in the form of pentagons and heptagons in the side walls of the (9, 0) nanotube prior to step 1. These defects could arise due to localized e-beam or ion-beam irradiation-induced localized heating/welding at the location of the junction point (see [21]). These defects can very well be initiated by the formation of a Stone-Wales [27] defect in the pristine nanotube followed by separation of pentagon-heptagon pairs as a result of the incorporation of extra carbon atoms. This process of defect formation and subsequent separation is illustrated in figure 2. In figure 2(a) we show the side view of a pristine (9, 0) SWCN. The three Stone-Wales rotations of adjacent bonds result in the structure shown in figure 2(b), where topological defects in the form of 6 pentagons, 2 heptagons and 2 octagons have been created (all shown in red). The presence of defects results in an increase in the local curvature and reactivity. The defected sites can then incorporate 6 extra carbon atoms (shown in blue in figure 2(c)) from external sources. The carbon atoms then rearrange themselves by separating the defects so that an energetically efficient all sp² configuration for all atoms can be achieved, as shown in figure 2(d). This then leads to the process in step 1 of figure 1.

There are two ways to introduce extra atoms into the system: (i) to introduce them into the cap of the (10, 0) nanotube or (ii) to introduce them into the wall of the (9, 0) nanotube. Introducing extra atoms in the cap of the (10, 0) tube leads to a very strained and, therefore, energetically unfavourable configuration with four-membered rings. So, the lower energy pathway is to introduce extra atoms into the wall of the (9, 0) tube as shown in figure 2.

2.2. (5, 5)-(10, 0)-(5, 5) T-junction formation

We next illustrate the formation of a (5, 5)-(10, 0)-(5, 5) T-junction by bringing a capped zig-zag (10, 0) nanotube near an armchair (5, 5) nanotube wall containing defects (shown in red). Once again, these defects could arise due to localized e-beam or ion-beam irradiation-induced localized heating/welding at the location of the junction point [21]. The two SWCN clusters here contain a total of 294 atoms. The 8-step process in which sp² coordination is maintained throughout for all the atoms in the junction region is shown in figure 3. In the first step (1) the capped end of the (10, 0) nanotube is near the defected region of the (5, 5) nanotube. Note that the (5, 5) nanotube with defects contains 2 more carbon atoms than the corresponding defect-free case, while preserving the sp² arrangement of carbon atoms in the nanotube. All structures are, once again, relaxed using the GTBMD scheme. In the second step (2) the inter-nanotube connectivity is through 4 bonds forming the sides of 4 octagons. This structure also contains defect rings in the form of 4 pentagons and 2 heptagons. In step 3, two of the octagons are annihilated, and, additionally, the structure contains 4 pentagons and 6 heptagons in the junction region. In step 4, the rearrangement of atoms causes the neck to widen. The defects are identical to step 3, but their positioning is different. The subsequent arrangements of defects as we proceed toward the final structure are: 6 pentagons and 12 heptagons (step 5); 4 pentagons, 6 heptagons and 2 octagons (step 6); and 2 pentagons and 8 heptagons (step 7). The final structure (step 8) contains 7 heptagons and no pentagons. Once again, it can easily be seen that the total bond surplus at each step (except step 1) is  + 6, confirming the generalized Euler rule. In table 1 we list the number and types of the defects at each step as well as the relative energies obtained using the GTBMD scheme.

In figure 4 we illustrate the process of defect formation and subsequent separation resulting in the defected (5, 5) nanotube in step 1. The side view of a pristine (5, 5) SWCN is shown in figure 4(a). A single Stone-Wales rotation of a C-C bond results in the structure shown in figure 4(b), where topological defects in the form of 2 pentagons and 2 heptagons have been created (all shown in red). This defected site can incorporate 2 external C atoms (shown in blue in figure 4(c)). An all sp² arrangement of C atoms can then be achieved by a separation of defects as shown in figure 4(d), leading to the process in step 1 of figure 3.

2.3. Energy considerations in the formation of T-junctions

More insights can be gained from a study of energies at various steps leading to the formation of the two T-junctions. The structures are fully relaxed at each step and the total energy is calculated using the GTBMD scheme. The relative energies at each step in the formation of the two T-junctions are plotted in figure 5. We have also calculated intermediate energy points (small circles and squares) that rule out the presence of large barriers between the steps. As seen in the figure, an increase in energy (indicating a barrier to the process in steps 1-3) is followed by a drop in energy (in steps 3-8) without any significant barrier. The activation barrier to the process, thus, is the overall increase in energy in steps 1-3. Note that topological defects already exist before step 1 and are shown in red in figures 1 and 3. As stated previously, these defects could arise due to localized e-beam or ion-beam irradiation-induced localized heating/welding at the location of the junction point (see [21]). Note that a static activation barrier for formation of a SW defect in a pristine nanotube has been reported to be between 8 and 12 eV by different groups using different techniques [28, 29]. We, however, note that the relevant activation barrier for a T-junction formation process should be dynamic rather than static. Indeed, the dynamic activation barrier for formation of a SW defect has been reported recently for the first time [30]. The value of 3.6 eV reported there for the dynamic activation barrier is considerably lower than the generally accepted value of 10 eV for the static activation barrier. The experimental process reported in [21] can, therefore, easily create the defects necessary for step 1 and lends strong support to the plausibility of the T-junction formation process proposed here. The subsequent contribution of about 5 eV is rather small and indicates the relative ease of the formation of the junction if topological defects at the junction site are already formed. The mechanism of SW defect formation during electron-beam irradiation and annealing of a bundle of nanotubes has been used to explain the coalescence of nanotubes [31]. We use the same mechanism and similar energetics for the initiation of the tip to the side-wall welding of nanotubes in the formation of T-junctions reported in our work. Once the initial barrier is overcome, the formation of the junction in steps 3-8 can be driven by the lowering of energy as shown in figure 5.

3. Y-junctions

In this section we study the formation pathways for SWCN Y-junctions by bringing a capped armchair (6, 6) nanotube near the side wall of another armchair (6, 6) nanotube at an angle. Since the number of atoms involved is rather large we perform the relaxation using Brenner's three-body reactive potential [32] and the GTBMD method is used to check the stability of the final structure only. Our goal is to merely illustrate energetically efficient plausible pathways in which all atoms maintain sp² coordination throughout. The overall qualitative conclusions are not expected to be different from the case when the GTBMD is used.

3.1. (6, 6)-(6, 6)-(6, 6) Y-junction formation

In figure 6 we illustrate the formation of an asymmetric (6, 6)-(6, 6)-(6, 6) Y-junction by bringing two armchair (6, 6) nanotubes together. This is a 21-step process in which all carbon atoms maintain their sp² coordination throughout. For brevity, however, we show only 8 of these steps in figure 6. The defect rings are shown in red. Note that, unlike in the case of T-junctions in section 2, the first step does not require any a priori existence of defects on the side wall of the nanotube. The initial fusion process may, however, require energetic assistance from external sources in the form of localized e-beam or ion-beam irradiation at the contact point. The two SWCN clusters here contain a total of 572 atoms and the number of atoms does not change as the fusion process proceeds through the 21 steps. The final structure contains 4 heptagons, 1 octagon and no pentagons. This structure is found to be stable under GTBMD relaxation. In table 2 we list the number and types of defects at each step. Once again, validity of the generalization of Euler's rule can be verified by keeping track of the defects at each step (except step 1) in the table.

3.2. Transport properties

We next investigate the transport properties of Y-junctions obtained in our simulations described in section 3.1. The quantum conductance of the SWCN Y-junction is calculated using a Green function embedding scheme [33]. The Green function is constructed using the same Hamiltonian as used for obtaining structural relaxations in section 2. This method has been used to obtain I-V characteristics of SWCNs in pristine form as well as in the presence of defects [33]. In particular, I-V characteristics obtained for SWCN Y-junctions using this method were in complete agreement with experimental results on these systems [14].

In figure 7 we show the I-V characteristics of the Y-junction obtained in the simulations shown in figure 6. The left (L), right (R) and stem (S) branches of the (6, 6)-(6, 6)-(6, 6) Y-junction are shown in the inset of figure 7. The current direction is taken to be positive when flowing towards the junction region and negative otherwise. We use a set-up in which both the left and right branches are grounded and the stem is biased at the bias voltage V_b, i.e.

In this set-up one measures the dependence of the currents I_L, I_R and I_S on the bias voltage V_b. A similar set-up has been used earlier by us to study the transport properties of Y-junctions with different radii and chiralities [15]. As seen in the figure, there is asymmetry in the I-V characteristics and no rectification. This is consistent with our finding reported earlier that no rectification at all is possible for asymmetric Y-junctions [15]. We have earlier shown that rectification in SWCN Y-junctions is determined by four factors;

4. X-junctions

We next study the formation pathways for SWCN X-junctions by bringing one (6, 6) armchair SWCN near another (6, 6) SWCN so that there is wall to wall contact. The two nanotube clusters used in the simulations contain a total of 608 atoms. Once again, since the number of atoms involved is rather large we perform relaxations using Brenner's three-body reactive potential [32], and the GTBMD scheme is used only to check the stability of the final structure. Also, as in the case of Y-junctions in section 3, the first step does not require any a priori existence of defects on the side walls of the nanotubes. The five-step process leading to the formation of a four-terminal X-junction is shown in figure 8. The defect rings are shown in red. The number of atoms do not change at each step and all the atoms maintain sp² coordination throughout. The final structure obtained in step 5 contains topological defects in the form of 12 heptagons only. This structure was also found to be stable under GTBMD relaxation. The number and types of defects at each simulation step are given in table 2. Since the X-junctions constitute four-terminal junctions, generalization of Euler's rule dictates that the bond surplus at each step must be  + 12. This can easily be verified by keeping track of defects at each step (except step 1) in table 2.

5. Summary

We have presented an atomistic picture of energetically efficient pathways for the formation of SWCN T-, Y- and X-junctions through a sequence of steps starting with one in which one SWCN is brought near the wall of another SWCN. During the entire process no structural defects are generated and the transformation is achieved through creation/annihilation of topological defects, while all carbon atoms retain their sp² coordination. Recent experimental advances in electron-beam welding techniques have greatly increased the plausibility of synthesizing the junctions as proposed in the simulations. We have also calculated the transport properties of a Y-junction thus formed.

Acknowledgments

The present work is supported through grants by NSF (ITR-0221916), DOE (00-63857), NASA (02-465679) and the Kentucky Science and Technology Corporation (KSTC). LC and IP acknowledge support from INTAS (00-237) and Russian Programs `Topical directions in condensed matter physics' and `Low-dimensional quantum structures'.

References

[1]

Mintmire J W, Dunlap B I and White C T 1992 Phys. Rev. Lett. 68 631 
CrossRefPubMed

[2]

Saito R, Fujita M, Dresselhaus G and Dresselhaus M S 1992 Phys. Rev. B 46 1804 
CrossRef

[3]

Hamada N, Sawada S and Oshiyama A 1992 Phys. Rev. Lett. 68 1579 
CrossRefPubMed

[4]

Dunlap B I 1992 Phys. Rev. B 46 1933 
CrossRef

[5]

Chernozatonskii A 1992 Phys. Lett. A 170 37 
CrossRef

[6]

Iijima S, Ichihashi T and Ando Y 1992 Nature 356 776 
CrossRef

[7]

Lambin Ph, Fonseca A, Vigneron J, Nagy J B and Lucas A A 1995 Chem. Phys. Lett. 245 85 
CrossRef

[8]

Chico L, Crespi V H, Benedict L X, Louie S G and Cohen M L 1996 Phys. Rev. Lett. 76 971 
CrossRefPubMed

[9]

Saito R, Dresselhaus G and Dresselhaus M S 1996 Phys. Rev. B 53 2044 
CrossRef

[10]

Charlier J C, Ebbesen T W and Lambin Ph 1996 Phys. Rev. B 53 11108 
CrossRef

[11]

Scuseria G E 1992 Chem. Phys. Lett. 195 534 
CrossRef

[12]

Chernozatonskii L A 1992 Phys. Lett. A 172 173 
CrossRef

[13]

Menon M and Srivastava D 1997 Phys. Rev. Lett. 79 4453 
CrossRef

[14]

Andriotis A N, Menon M, Srivastava D and Chernozatonskii L 2001 Phys. Rev. Lett. 87 6802 
CrossRefPubMed

[15]

Andriotis A N, Menon M, Srivastava D and Chernozatonskii L 2002 Phys. Rev. B 65 165416 
CrossRef

[16]

Zhou D and Seraphin S 1995 Chem. Phys. Lett. 238 286 
CrossRef

[17]

Nagy P, Ehlich R, Biro L P and Gyulai J 2000 Appl. Phys. A 70 481 
CrossRef

[18]

Li J, Papadopoulos C and Xu J 1999 Nature 402 253 
PubMed

[19]

Satishkumar B C, Thomas P J, Govindaraj A and Rao C N R 2000 Appl. Phys. Lett. 77 2530 
CrossRef

[20]

Papadopoulos C, Rakitin A, Li J, Vedeneev A S and Xu J M 2000 Phys. Rev. Lett. 85 3476 
CrossRefPubMed

[21]

Terrones M, Banhart F, Grobert N, Charlier J-C, Terrones H and Ajayan P M 2002 Phys. Rev. Lett. 89 

[22]

Kim Y-H, Lee I-Ho, Chang K J and Lee S 2003 Phys. Rev. Lett. 90 

[23]

Zhao Y, Yakobson B I and Smalley R 2002 Phys. Rev. Lett. 88 185501 
CrossRefPubMed

[24]

Menon M, Andriotis A N, Srivastava D, Ponomareva I and Chernozatonskii L 2003 Phys. Rev. Lett. at press 

[25]

Crespi V H 1998 Phys. Rev. B 58 12671 
CrossRef

[26]

Menon M, Richter E and Subbaswamy K R 1996 J. Chem. Phys. 104 5875 
CrossRef

[27]

Stone A J and Wales D J 1986 Chem. Phys. Lett. 128 501 
CrossRef

[28]

Zhang P H, Lammert P E and Crespi V H 1998 Phys. Rev. Lett. 81 5346 
CrossRef

[29]

Zhao Q, Nardelli M B and Bernholc J 2002 Phys. Rev. B 65 144105 
CrossRef

[30]

Wei C, Cho K and Srivastava D 2003 Phys. Rev. B 67 115407 
CrossRef

[31]

Terrones M, Terrones H, Banhart F, Charlier J-C and Ajayan P M 2000 Science 356 776 

[32]

Brenner D W, Shenderova O A, Harrison J A, Stuart S J, Ni B and Sinnott S B 2002 J. Phys.: Condens. Matter 14 783 
IOPscience

[33]

Andriotis A N and Menon M 2001 J. Chem. Phys. 115 2737 
CrossRef

[34]

Andriotis A N, Srivastava D and Menon M 2003 Appl. Phys. Lett. 83 1674 
CrossRef