Ropes of Nanotubes

As discussed above, an isolated (n, n) carbon nanotube has two linearly dispersing conduction bands which cross at the Fermi level forming two "Dirac" points, as schematically presented in Fig. 19a. This linear band dispersion in a one-dimensional system gives rise to a finite and constant density of electronic states at the Fermi energy. Thus, an (n, n) tube is a metal within a. b

Fig. 19. Band crossing and band repulsion. (a) Schematic band structure of an isolated (n, n) carbon nanotube near the Fermi energy. (b) Repulsion of bands due to the breaking of mirror symmetry k along tube axis k along rope axis

Fig. 19. Band crossing and band repulsion. (a) Schematic band structure of an isolated (n, n) carbon nanotube near the Fermi energy. (b) Repulsion of bands due to the breaking of mirror symmetry the one-electron picture. The question of interest is: How does the electronic structure change when the metallic tubes are bundled up to form a closely packed two-dimensional crystal, as in the case of the (10,10) ropes. In the calculation, a large (10,10) rope is modeled by a triangular lattice of (10,10) tubes infinitely extended in the lateral directions. For such a system, the electronic states, instead of being contained in the 1-D Brillouin zone of a single tube, are now extended to a three-dimensional irreducible Brillouin zone wedge. If tube-tube interactions are negligibly small, the electronic energy band structure along any line in the wedge parallel to the rope axis would be exactly the same as the band dispersion of an isolated tube. In particular, at the fc-wavevector corresponding to the band crossing point, there will be a two-fold degenerate state at the Fermi energy. This allowed band crossing is due to the mirror symmetry of the (10,10) tube. For a tube in a rope, this symmetry is however broken because of intertube interactions. The broken symmetry causes a quantum level repulsion and opens up a gap almost everywhere in the Brillouin zone, as schematically shown in Fig. 19b.

The band repulsion resulting from the broken-symmetry strongly modifies the DOS of the rope near the Fermi energy compared to that of an isolated (10,10) tube. The calculated DOS is presented in Fig. 20a. Shown are the results for two cases: aligned and misaligned tubes in the rope. In both cases, there is a pseudogap of the order of 0.1 eV in the density of states. Examination of the electronic structure reveals that the system is a semimetal with both electron and hole carriers. The existence of the pseudogap in the rope makes the conductivity and other transport properties of the metallic rope

Fig. 20. (a) Calculated density of states for a rope of misaligned (10,10) carbon nanotubes (broken line) and aligned tubes (solid line). The Fermi energy is at zero. (b) Calculated joint density of states for a rope of misaligned (broken line) and aligned (solid line) (10,10) tubes. Results are in units of states per meV per atom [83,84]

Fig. 20. (a) Calculated density of states for a rope of misaligned (10,10) carbon nanotubes (broken line) and aligned tubes (solid line). The Fermi energy is at zero. (b) Calculated joint density of states for a rope of misaligned (broken line) and aligned (solid line) (10,10) tubes. Results are in units of states per meV per atom [83,84]

significantly different from those of isolated tubes, even without considering the effect of local disorder in low dimensions. Since the DOS increases rapidly away from the Fermi level, the carrier density of the rope is sensitive to temperature and doping. The existence of both electron and hole carriers leads to qualitatively different thermopower and Hall-effect behaviors from those expected for a normal metal. The optical properties of the rope are also affected by the pseudogap. As illustrated by the calculated Joint Density of States (JDOS) in Fig. 20b, there would be a finite onset in the infrared absorption spectrum for a large perfectly ordered (10,10) rope, where one can assume fc-conserving optical transitions. In the case of high disorder, an infrared experiment would more closely reflect the DOS rather than the JDOS. For most actual samples, the fraction of (10,10) carbon nanotubes (compared with other nanotubes of the same diameter) in the experimentally synthesized ropes appears to be small. However, the conclusion that broken symmetry induces a gap in the (n, n) tubes is a general result which is of relevance for tubes under any significant asymmetric perturbations, such as those due to structural deformations or external fields.

3.2 Crossed-Tube Junctions

The discussion of nanotube junctions in Sect. 2 is focused on the on-tube junctions, i.e., forming a junction by joining two half tubes together. These systems are extremely interesting, but difficult to synthesize in a controlled manner at this time. Another way to form junctions is to have two tubes crossing each other in contact [86] (Fig. 21). This kind of crossed-tube junction is much easier to fabricate and control with present experimental techniques.

Fig. 21. AFM image of a crossed SWNT device (A). Calculated structure of a crossed (5,5) SWNT junction with a force of 0nN (B) and 15 nN (C) [86]

When two nanotubes cross in free space, one expects that the tubes at their closest contact point will be at a van der Waals distance away from each other and that there will not be much intertube or junction conductance. However, as shown by Avouris and coworkers [87], for two crossed tubes lying on a substrate, there is a substantial force pressing one tube against the other due to the substrate attraction. For a crossed-tube junction composed of SWNTs with the experimental diameter of 1.4 nm, this contact force has been estimated to be about 5 nN [87]. This substrate force would then be sufficient to deform the crossed-tube junction and lead to better junction conductance.

In Fig. 21, panel A is an AFM image of a crossed-tube junction fabricated from two single-walled carbon nanotubes of 1.4 nm in diameter with electrical contacts at each end [86]. Panels B and C show the calculated structure corresponding to a (5,5) carbon nanotube pressed against another one with zero and 15 nN force, respectively. Because of the smaller diameter of the (5,5) tube, a larger contact force is required to produce a deformation similar to that of the experimental crossed-tube junction. The calculation was done using the ab initio pseudopotential density functional method with a localized basis [86]. As seen in panel C, there is considerable deformation, and the atoms on the different tubes are much closer to each other. At this distance, the closest atomic separation between the two tubes is 0.25 nm, significantly smaller than the van der Waals distance of 0.34 nm.

For the case of zero contact force (panel B in Fig. 21), the calculated intra-tube conductance is virtually unchanged from that of an ideal, isolated metallic tube, and the intertube conductance is negligibly small. However, when the tubes are under a force of 15 nN, there is a sizable intertube or junction conductance. As shown in Fig. 22, the junction conductance at the

1.00

-10 1 Energy

Fig. 22. Calculated conductance (expressed in units of e2/h) of a crossed (5,5) carbon nanotube junction with a contact force of 15 nN on a linear (top) and log (bottom) scale. The dashed (dotted-dashed) curve corresponds to the intra-tube (intertube) conductance [86]

Fermi energy is about 5% of a quantum unit of conductance G0 = 2e2/h. The junction conductance is thus very sensitive to the force or distance between the tubes.

Experimentally, the conductance of various types of crossed carbon nanotube junctions has been measured, including metal-metal, semiconductor-semiconductor, and metal-semiconductor crossed-tube junctions. The experimental results are presented in Fig. 23. For the metal-metal crossed-tube junctions, a conductance of 2 to 6% of G0 is found, in good agreement with the theoretical results. Of particular interest is the metal-semiconductor case in which experiments demonstrated Schottky diode behavior with a Schot-tky barrier in the range of 200-300 meV, which is very close to the value of 250 meV expected from theory for nanotubes with diameters of 1.4 nm [86].

3.3 Effects of Long-Range Disorder and External Perturbations

The effects of disorder on the conducting properties of metal and semiconducting carbon nanotubes are quite different. Experimentally, the mean free path is found to be much longer in metallic tubes than in doped semiconducting tubes [19,20,21,24,88]. This result can be understood theoretically if the disorder potential is long range. As discussed below, the internal structure of the wavefunction of the states connected to the sublattice structure of graphite lead to a suppression of scattering in metallic tubes, but not

-10 1 Energy

Fig. 22. Calculated conductance (expressed in units of e2/h) of a crossed (5,5) carbon nanotube junction with a contact force of 15 nN on a linear (top) and log (bottom) scale. The dashed (dotted-dashed) curve corresponds to the intra-tube (intertube) conductance [86]

§ MS +

i * . I i I i j i

Ef"

EF"

Ef"

Fig. 23. Current-voltage characteristics of several crossed SWNT junctions [86] (see text)

in semiconducting tubes. Figure 24 shows the measured conductance for a semiconducting nanotube device as a function of gate voltage at different temperatures. [88]. The diameter of the tube as measured by AFM is 1.5 nm, consistent with a single-walled tube. The complex structure in the Coulomb blockade oscillations in Fig. 24 is consistent with transport through a number of quantum dots in series. The temperature dependence and typical charging energy indicates that the tube is broken up into segments of length of about 100 nm. Similar measurements on intrinsic metal tubes, on the other hand, yield lengths that are typically a couple of orders of magnitude longer [19,20,21,24,88].

Theoretical calculations have been carried out to examine the effects of long-range external perturbations [88]. In the calculation, to model the perturbation, a 3-dimensional Gaussian potential of a certain width is centered on one of the atoms on the carbon nanotube wall. The conductance with the perturbation is computed for different Gaussian widths, but keeping the integrated strength of the potential the same. Some typical tight-binding results are presented in Fig. 25. The solid lines show the results for the conductance of a disorder-free tube, while the dashed and the dot-dashed lines are, respectively, for a single long-range (<=0.348 nm, A V = 0.5 eV) and a short-range (<7=0.116 nm, A V = 10 eV) scatterer. Here A V is the shift in the on-site energy at the potential center. The conduction bands (i.e., bands crossing the Fermi level) of the metallic tube are unaffected by the long-range scatterer, unlike the lower and upper subbands of both the metallic and semiconducting tubes, which are affected by both long- and short-range scatterers. All

Fig. 24. Conductance vs. gate voltage Vg for a semiconducting single-walled carbon nanotube at various temperatures. The upper insert schematically illustrates the sample geometry and the lower insert shows dl/dV vs. V and Vg plotted as a gray scale [88]
Fig. 25. Tight-binding calculation of the conductance of a (a) metallic (10,10) tube and (b) semiconducting (17,0) tube in the presence of a Gaussian scatterer. The energy scale on the abscissa is 0.2 eV per division in each graph [88]

subbands are influenced by the short-range scatterer. The inset shows an expanded view of the onset of conduction in the semiconducting tube at positive E, with each division corresponding to 1 meV. Also, the sharp step edges

sheet. There are two carbon atoms per unit cell (lower right inset). The dispersions of the states near Ef are cones whose vertices are located at the corner points of the Brillouin zone. The Fermi circle, defining the allowed k vectors, and the band dispersions are shown in (b) and (c) for a metallic and a semiconducting tube, respectively [88]

sheet. There are two carbon atoms per unit cell (lower right inset). The dispersions of the states near Ef are cones whose vertices are located at the corner points of the Brillouin zone. The Fermi circle, defining the allowed k vectors, and the band dispersions are shown in (b) and (c) for a metallic and a semiconducting tube, respectively [88]

in the calculated conductance of the perfect tubes are rounded off by both types of perturbations.

Both the experimental and theoretical findings strongly suggest that longrange scattering is suppressed in the metallic tubes. One can actually understand this qualitatively from the electronic structure of a graphene sheet [89,90]. The graphene structure has two atoms per unit cell. The properties of electrons near the Fermi energy are given by those states near the corner of the Brillouin zone (Fig. 26). If we look at the states near this point and consider them in terms of a k-vector away from the corner K point, then they can be described by a Dirac Hamiltonian. For these states, the wavefunctions can be written in terms of a product of a plane wave component (with a vector k) and a pseudo-spin which describes the bonding character between the two atoms in the unit cell. The interesting result is that this pseudo-spin points along k. For example, if the state at k is bonding, then the state at —k is antibonding in character. Within this framework, one can work out the scattering between the allowed states in a carbon nanotube due to long-range disorder, i.e., disorder with Fourier components V(q) such that q ^ K. This, for example, will be the case for scattering by charged trap states in the substrate (oxide traps). In this case, the disorder does not couple to the pseudo-spin portion of the wavefunction, since the disorder potential is approximately constant on the scale of the interatomic distance. The resulting matrix element between states is then [89,90]: \{k'\V(r)\k}\2 = \V(k — k')|2 cos2[(1/2)6k,k'], where 6k,k>

is the angle between the initial and final states. The first term in V(k — k') is the Fourier component at the difference in k values of the initial and final envelope wavefunctions. The cosine term is the overlap of the initial and final spinor states.

For a metallic tube Fig. 26b, backscattering in the conduction band corresponds to scattering between k and —k. Such scattering is forbidden, because the molecular orbitals of these two states are orthogonal. In semiconducting tubes, however, the situation is quite different Fig. 26c. The angle between the initial and final states is less than n, and scattering is thus only partially suppressed by the spinor overlap. As a result, semiconducting tubes should be sensitive to long-range disorder, while metallic tubes should not. However, short-range disorder which has Fourier components q ~ K will couple the molecular orbitals together and lead to scattering in all of the subbands. These theoretical considerations agree well with experiment and with the detailed calculations discussed above. Long-range disorder due to, e.g., localized charges near the tube, breaks the semiconducting tube into a series of quantum dots with large barriers, resulting in a dramatically reduced conductance and a short mean free path. On the other hand, metallic tubes are insensitive to this disorder and remain near-perfect 1D conductors.

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