Electronic and Transport Properties of OnTube Structures

In this section, we discuss the electronic properties and quantum conductance of nanotube structures that are more complex than infinitely long, perfect nanotubes. Many of these systems exhibit novel properties and some of them are potentially useful as nanoscale devices.

2.1 Nanotube Junctions

Since carbon nanotubes are metals or semiconductors depending sensitively on their structures, they can be used to form metal-semiconductor, semiconductor-semiconductor, or metal-metal junctions. These junctions have great potential for applications since they are of nanoscale dimensions and made entirely of a single element. In constructing this kind of on-tube junction, the key is to join two half-tubes of different helicity seamlessly with each other, without too much cost in energy or disruption in structure. It has been shown that the introduction of pentagon-heptagon pair defects into the hexagonal network of a single carbon nanotube can change the helic-ity of the carbon nanotube and fundamentally alter its electronic structure [16,17,18,67,68,69,70,71]. This led to the prediction that these defective nano-tubes behave would as the desired nanoscale metal-semiconductor Schottky barriers, semiconductor heterojunctions, or metal-metal junctions with novel properties, and that they could be the building blocks of nanoscale electronic devices.

In the case of nanotubes, being one-dimensional structures, a local topo-logical defect can change the properties of the tube at an infinitely long distance away from the defect. In particular, the chirality or helicity of a carbon nanotube can be changed by creating topological defects into the hexagonal network. The defects, however, must induce zero net curvature to prevent the tube from flaring or closing. The smallest topological defect with minimal local curvature (hence less energy cost) and zero net curvature is a pentagon-heptagon pair [16,17,18,67,68,69,70,71]. Such a pentagon-heptagon defect pair with its symmetry axis nonparallel to the tube axis changes the chirality of a (n, m) tube by transferring one unit from n to m or vice versa. If the pentagon-heptagon defect pair is along the (n, m) tube axis, then one unit is added or subtracted from m. Figure 8 depicts a (8,0) carbon tube joined to a (7,1) tube via a 5-7 defect pair. This system forms a quasi-1D semiconductor/metal junction, since within the band-folding picture the (7,1) half tube is metallic and the (8,0) half tube is semiconducting.

Figures 9 and 10 show the calculated local density of states (LDOS) near the (8,0)/(7,1) junction. These results are from a tight-binding calculation for the n electrons [16]. In both figures, the bottom panel depicts the density of states of the perfect tube, with the sharp features corresponding to the van Hove singularities of a quasi-1D system. The other panels show the calculated LDOS at different distances away from the interface, with cell 1 being the closest to the interface in the semiconductor or side and ring 1 the closest to the interface in the metal side. Here, "cell" refers to a one unit cell of the tube and "ring" refers to a ring of atoms around the circumference. These results illustrate the spatial behavior of the density of states as it transforms from

Fig. 8. Atomic structure of an (8,0)/(7,1) carbon nanotube junction. The large light-gray balls denote the atoms forming the heptagon-pentagon pair [16]

cell 1

cell 1

. 1 1 1 . 1 : , N , , 1

' J 1 ' ' 1 ' ' I

1 . . n , :

10 -5 0 5 10 cell 2

: , M , , 1 . ,



i ' ' LM

A , ]

10 -5 0 5 10 cell 3

1 IUX-j

, . , , .

•Jm 1

10 -5 0 5 10 perfect (8,0) tube

h i f.tUr

, , 1 . 1 U, J


1 1 1



Fig. 9. Calculated LDOS of the (8,0)/(7,1) metal-semiconductor junction at the semiconductor side. From top to bottom, LDOS at cells 1, 2, and 3 of the (8,0) side. Cell 1 is at the interface [16]

that of a metal to that of a semiconductor across the junction. The LDOS very quickly changes from that of the metal to that of the semiconductor within a few rings of atoms as one goes from the metal side to the semiconductor side.

Fig. 10. Calculated LDOS of the (8,0)/(7,1) metal-semiconductor junction at the metal side. From top to bottom, the LDOS at rings 1, 2, and 3 of the (7,1) side. Ring 1 is at the interface [16]

As the interface is approached, the sharp van Hove singularities of the metal are diluted. Immediately on the semiconductor side of the interface, a different set of singular features, corresponding to those of the semiconductor tube, emerges. There is, however, still a finite density of states in the otherwise bandgap region on the semiconductor side. These are metal induced gap states [72] which decay to zero in about a few A into the semiconductor. Thus, the electronic structure of this junction is very similar to that of a bulk metal-semiconductor junction, such as Al/Si, except it has a nanometer cross-section and is made out of entirely the element carbon.

Similarly, semiconductor-semiconductor and metal-metal junctions may be constructed with the proper choices of tube diameters and pentagonheptagon defect pairs. For example, by inserting a 5-7 pair defect, a (10,0) carbon nanotube can be matched to a (9,1) carbon nanotube [16]. Both of these tubes are semiconductors, but they have different bandgaps. The (10,0)/(9,1)

junction thus has the electronic structure of a semiconductor heterojunction. In this case, owing to the rather large structural distortion at the interface, there are interesting localized interface states at the junction. Theoretical studies have also been carried out for junctions of B-C-N nanotubes [73], showing very similar behaviors as the carbon case, and for other geometric arrangements, such as carbon nanotube T-junctions, where one tube joins to the side of another tube perpendicularly to form a "T" structure [74].

Calculations have been carried out to study the quantum conductance of the carbon nanotube junctions. Typically these calculations are done within the Landauer formalism [75,76]. In this approach, the conductance is given in terms of the transmission matrix of the propagating electron waves at a given energy. In particular, the conductance of metal-metal nanotube junctions is shown to exhibit a quite interesting new effect which does not have an analog in bulk metal junctions [67]. It is found that certain configurations of pentagon-heptagon pair defects in forming the junction completely stop the flow of electrons, while other arrangements permit the transmission of current through the junction. Such metal-metal junctions thus have the potential for use as nanoscale electrical switches. This phenomenon is seen in the calculated conductance of a (12,0)/(6,6) carbon nanotube junction in Fig. 11. Both the (12,0) and (6,6) tubes are metallic within the tight-binding model, and they can be matched perfectly to form a straight junction. However, the conductance is zero for electrons at the Fermi level, EF. This peculiar effect is not due to a lack of density of states at EF. As shown in Fig. 11, there is finite density of states at EF everywhere along the whole length of the total system for this junction. The absence of conductance arises from the fact that there is discrete rotational symmetry along the axis of the combined tube. But, for electrons near EF, the states in one of the half tubes are of a different rotational symmetry from those in the other half tube. As an electron propagates from one side to the other, the electron encounters a symmetry gap and is completely reflected at the junction.

The same phenomenon occurs in the calculated conductance of a (9,0)/(6,3) metal-metal carbon nanotube junction. However, in forming this junction, there are two distinct ways to match the two halves, either symmetrically or asymmetrically. In the symmetric matched geometry, the conductance is zero at EF for the same symmetry reason as discussed above (Fig. 12). But, in the asymmetric matched geometry, the discrete rotational symmetry of the total system is broken and the electrons no longer have to preserve their rotational quantum number as they travel across the junction. The conductance for this case is now nonzero. Consequently, in some situations, bent junctions can conduct better than straight junctions for the nano-tubes. This leads to the possibility of using these metal-metal or other similar junctions as nanoswitches or strain gauges, i.e., one can imagine using some symmetry breaking mechanisms such as electron-photon, electron-phonon or

Fig. 11. Calculated results for the (12,0)/(6,6) metal-metal junction. Top: conductance of a matched tube (solid line), a perfect (12,0) tube (dashed line), and a perfect (6,6) tube (dotted line). Center: LDOS at the interface on the (12,0) side (full line) and of the perfect (12,0) tube (dotted line). Bottom: LDOS at the interface on the (6,6) side (full line) and of the perfect (6,6) tube (dashed line) [67]

Fig. 11. Calculated results for the (12,0)/(6,6) metal-metal junction. Top: conductance of a matched tube (solid line), a perfect (12,0) tube (dashed line), and a perfect (6,6) tube (dotted line). Center: LDOS at the interface on the (12,0) side (full line) and of the perfect (12,0) tube (dotted line). Bottom: LDOS at the interface on the (6,6) side (full line) and of the perfect (6,6) tube (dashed line) [67]

mechanical deformation to switch a junction from a non-conducting state to a conducting state [67].

Junctions of the kind discussed above may be formed during growth, but they can also be generated by mechanical stress [77]. There is now considerable experimental evidence of this kind of on-tube junction and device behavior predicted by theory. An experimental signature of a single pentagonheptagon pair defect would be an abrupt bend between two straight sections of a nanotube. Calculations indicate that a single pentagon-heptagon pair would induce bend angles of roughly 0-15 degrees, with the exact value depending on the particular tubes involved. Several experiments have reported sightings of localized bends of this magnitude for multiwalled carbon nanotubes [23,78,79]. Having several 5-7 defect pairs at a junction would allow

Electronic Properties, Junctions, and Defects of Carbon Nanotubes 127 (9,0)/(6,3) symmetric matching

Electronic Properties, Junctions, and Defects of Carbon Nanotubes 127 (9,0)/(6,3) symmetric matching

10 -5 0 5 10

(9,0)/(6,3) asymmetric matching

12 —T—I—I—I—I—I—r—I—I—j—I—I—I—I—J—

Fig. 12. Calculated conductance of the (9,0)/(6,3) junction - matched system (solid line), perfect (6,3) tube (dotted line), and perfect (9,0) tube (dashed line) [67]

Fig. 12. Calculated conductance of the (9,0)/(6,3) junction - matched system (solid line), perfect (6,3) tube (dotted line), and perfect (9,0) tube (dashed line) [67]

the joining of tubes of different diameters and add complexity to the geometry. The first observation of nonlinear junction-like transport behavior was made on a rope of SWNTs [22], where the current-voltage properties were measured along a rope of single-walled carbon nanotubes using a scanning tunneling microscopy tip and the behavior shown in Fig. 13 was found in some samples. At one end of the tube, the system behaves like a semimetal showing a typical I-V curve of metallic tunneling, but after some distance at the other end it becomes a rectifier, presumably because a defect of the above type has been introduced at some point on the tube. A more direct measurement was carried out recently [23]. A kinked single-walled nanotube lying on several electrodes was identified and its electrical properties in the different segments were measured. The kink was indicative of two half tubes of different chiralities joined by a pentagon-heptagon defect pair. Figure 14 shows the measured I-V characteristics of a kinked nanotube. The inset is the I-V curve for the upper segment showing that this part of the tube is a metal; but the I-V curve across the kink shows a rectifying behavior indicative of a metal-semiconductor junction.

Fig. 13. Current-voltage characteristic measured along a rope of single-walled carbon nanotubes. Panels A, B, C, and D correspond to successive different locations on the rope [22]

400 300

1 200 100

400 300

1 200 100


-1-1— - —

T -



Upper segment ^

• • ;

- 1° -50


« i

-10 -5 0 5 V(mV)


Across the kink



Fig. 14. Measured current-voltage characteristic of a kinked single-walled carbon nanotube [23]

2.2 Impurities, Stone-Wales Defects, and Structural Deformations in Metallic Nanotubes

An unanswered question in the field has been why do the metallic carbon nanotubes have such long mean free paths. This has led to consideration of the effects of impurities and defects on the conductance of the metallic nanotubes. We focus here on the (10,10) tubes; however, the basic physics is the same for all (n, n) tubes. In addition to tight-binding studies, there are now first-principles calculations on the quantum conductance of nanotube structures based on an ab initio pseudopotential density functional method with a wavefunction matching technique [80,81]. The advantages of the ab initio approach are that one can obtain the self-consistent electronic and geometric structure in the presence of the defects and, in addition to the conductance, obtain detailed information on the electronic wavefunction and current density distribution near the defect.

Several rather surprising results have been found concerning the effects of local defects on the quantum conductance of the (n, n) metallic carbon nanotubes [81]. For example, the maximum reduction in the conductance due to a local defect is itself often quantized, and this can be explained in terms of resonant backscattering by quasi-bound states of the defect. Here we discuss results for three simple defects: boron and nitrogen substitutional impurities and the bond rotation or Stone-Wales defect. A Stone-Wales defect corresponds to the rotation of one of the bonds in the hexagonal network by 90°, resulting in the creation of a quite low energy double 5-7 defect pair, without changing the overall helicity of the tube.

Figure 15 depicts several results for a (10,10) carbon nanotube with a single boron substitutional impurity. The top panel is the calculated conductance as a function of the energy of the electron. For a perfect tube, the conductance (indicated here by the dashed line) is 2 in units of the quantum of conductance, 2e2/h, since there are two conductance channels available for the electrons near the Fermi energy. For the result with the boron impurity, a striking feature is that the conductance is virtually unchanged at the Fermi level of the neutral nanotube. That is, the impurity potential does not scatter incoming electrons of this energy. On the other hand, there are two dips in the conductance below EF. The amount of the reduction at the upper dip is one quantum unit of conductance and its shape is approximately Lorentzian. In fact, the overall structure of the conductance is well described by the superposition of two Lorentzian dips, each with a depth of 1 conductance quantum. These two dips can be understood in terms of a reduction in conductance due to resonant backscattering from quasi-bound impurity states derived from the boron impurity.

The calculated results thus show that boron behaves like an acceptor with respect to the first lower subband (i.e., the first subband with energy below the conduction states) and forms two impurity levels that are split off from the top of the first lower subband. These impurity states become resonance states or quasi-bound states due to interaction with the conduction states. The impurity states can be clearly seen in the calculated LDOS near the boron impurity (middle panel of Fig. 15). The two extra peaks correspond to the two quasi-bound states. The LDOS would be a constant for a perfect tube in the region between the van Hove singularity of the first lower subband and that of the first upper subband. Because a (n, n) tube with a substitutional impurity still has a mirror plane perpendicular to the tube axis, the defect states have

Fig. 15. Energy dependence of the calculated conductance, local density of states, and phase shifts of a (10,10) carbon nanotube with a substitutional boron impurity [81]

definite parity with respect to this plane. The upper energy state (broader peak) in Fig. 15 has even parity and the lower energy state (narrower peak) has odd parity, corresponding to s-like and p-like impurity states, respectively.

The conductance behavior in Fig. 15 may be understood by examining how electrons in the two eigen-channels interact with the impurity. At the upper dip, an electron in one of the two eigen-channels is reflected completely (99.9%) by the boron impurity, but an electron in the other channel passes by the impurity with negligible reflection (0.1%). The same happens at the lower dip but with the behavior of the two eigen-channels switched. The bottom panel shows the calculated scattering phase shifts. The phase shift of the odd parity state changes rapidly as the energy sweeps past the lower quasi-bound state level, with its value passing through n/2 at the peak position of the quasi-bound state. The same change occurs to the phase shift of the even parity state at the upper impurity-state energy. The total phase shift across a quasi-bound level is n in each case, in agreement with the Friedel sum rule. The picture is that an incoming electron with energy exactly in resonance with the impurity state is being scattered back totally in one of the channels but not the other. This explains the exact reduction of one quantum of conductance at the dip. The upper-energy impurity state has a large binding energy (over 0.1 eV) with respect to the first lower subband and

Fig. 16. Calculated conductance, local density of states and phase shifts of a (10,10) carbon nanotube with a substitutional nitrogen impurity [81]

hence is quite localized. It has an approximate extent of —10 A, whereas the lower impurity state has an extent of -250 A.

The results for a nitrogen substitutional impurity on the (10,10) tube are presented in Fig. 16. Nitrogen has similar effects on the conductance as boron, but with opposite energy structures. Again, the conductance at the Fermi level is virtually unaffected, but there are two conductance dips above the Fermi level just below the first upper subband. Thus, the nitrogen impurity behaves like a donor with respect to the first upper subband, forming an s-like quasi-bound state with stronger binding energy and a p-like state with weaker binding energy. As in the case of boron, the reduction of one quantum unit of conductance at the dips is caused by the fact that, at resonance, the electron in one of the eigen-channels is reflected almost completely by the nitrogen impurity but the electron in the other channel passes by the impurity with negligible reflection. The LDOS near the nitrogen impurity shows two peaks corresponding to the two quasi-bound states. The phase shifts of the two eigen channels show similar behavior as in the boron case.

For a (10,10) tube with a Stone-Wales or double 5-7 pair defect, the calculations also find that the conductance is virtually unchanged for the states at the Fermi energy. Thus these results show that the transport properties of the neutral (n, n) metallic carbon nanotube are very robust with respect to

-0.5 0 0.5 E, eV

Fig. 17. Energy dependence of the calculated conductance, local density of states, and phase shifts of a (10,10) carbon nanotube with a Stone-Wales defect [81]

these kinds of intra-tube local defects. As in the impurity case, there are two dips in the quantum conductance in the conduction band energy range, one above and one below the Fermi level. These are again due to the existence of defect levels, and the reduction at the two dips is very close to one quantum of conductance for the same reason, as discussed above. The symmetry of the Stone-Wales defect in this case does not cause mixing between the n and n* bands, and these two bands remain as eigen channels in the defective system. The lower dip is due to a complete reflection of the n* band and the upper dip is due to complete reflection of the n band. This implies that the conductance of the nanotube, when there are more than one double 5-7 pair defect, would not sensitively depend on their relative positions, but only on their total numbers, as long as the distance between defects is far enough to be able to neglect inter-defect interactions. The analysis of the phase shifts show that the lower quasi-bound state is even with respect to a mirror plane perpendicular to the tube axis, while the upper quasi-bound state is odd with respect to the same plane.

The conductance of nanotubes can also be affected by structural deformations. Two types of deformations involving bending or twisting the nanotube structure have been considered in the literature. It was found that a smooth bending of the nanotube does not lead to scattering [54], but formation of a local kink induces strong a-n mixing and backscattering similar to that discussed earlier for boron impurity [82]. Twisting has a much stronger effect [54]. A metallic armchair (n, n) nanotube upon twisting develops a bandgap which scales linearly with the twisting angle up to the critical angle at which the tube collapses into a ribbon [82].

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