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DANIEL ORLIKOWSKI et al. PHYSICAL REVIEW B 63 155412have a value that is between one and two. We attribute thisdifference to the proximity of the scattering states for thischiral nanotube which, when coupled to the leads whichserves to broaden the peaks, overlap and merge to formpeaks whose height varies between one and two units ofconductance. The electronic properties of metallic chiralnanotubes for which gcd(n,m)3 are qualitatively similarto the metallic zigzag tubes, with two doubly-degeneratebands crossing at the -point. And indeed, the conductancespectra for the 9,6 system resembles that of the n,0 tubes,as shown in Fig. 3.We have also examined transmission through finite-sizedsemiconducting nanotube systems. Long, semiconductingnanotubes display the expected behavior of isolated peaksoutside of the band gap. Surprisingly, we find that the verysmall nanotubes, consisting of less than about 40 atomic layers,are characterized by a central peak at the Fermi levelwhich decays as the length of the nanotubes increases in anon-monotonic fashion. For instance, Fig. 4a shows a typicalconductance signature for a 17,0 nanotube, while Fig.4b plots the decay of the central peak as a function ofnanotube length for different semiconducting systems. Theessential behavior consists of an initial increase in the heightof the central peak at about N5 –15 atomic layers, followedby a decay in the peak height. As shown in Fig. 4c,there is scaling behavior in the falloff of the central peakheight: by plotting the conductance G as a function of x(NN max )/N 1/2 , where N max denotes the number of layersof the peak maximum and N 1/2 represents the number oflayers where the conductance falls off to half its maximumvalue—i.e., it is a rough measure of the width or curvature ofthe central peak. We note that this display of metallic behaviorfor small systems is a characteristic of several semiconductingmaterial systems. For instance, both linear carbonchains 54 and very narrow silicon wires display metallicbehavior. 55Turning to the infinite limit, we note that down-and-upstep structure must be recovered as the nanotube length becomesvery large, irrespective of the nanotube helicity. Wehave investigated the approach to this limit, as shown in Fig.5, and find that all the nanotubes display the following generalfeatures. As the length increases, the number of resonantpeaks in any given energy range increases and the spacingFIG. 6. Here we show some statistical characteristics of theresonant tunneling peaks. The main figure shows the spacing betweenpeaks PS versus nanotube length L in nanometers: circles,5,5 armchair nanotubes; diamonds, 9,0 zigzag tubes. The insetshows typical, peak width at half maximum FWHM as a functionof nanotube length for 9,0 nanotubes for different ’s: circles: L,R 0.5; squares, 1.0; and triangles, 2.0.between any two given peaks decreases. At the same time,the width of these peaks decreases. These features are shownin Fig. 5 for both 5,5 armchair and 9,0 zigzag nanotubes.Because of the one-dimensional nature of the nanotubes, wehave been able to probe nanotube lengths that are within afactor of 2 for some of the experimental systems, showingthat the finite-size nature of the system persists at least out tohundreds of nanometers. Figure 6 shows some statistics forthe spacing between peaks and the peak widths. Both ofthese fall off exponentially, which has important implicationsfor the ac conductance, as we shall discuss later. 56 Figures5 and 6 also show the effects of varying , which parametrizesthe coupling to the leads. As increases, thepeaks broaden and begin to overlap, so that the conductancespectra becomes continuous and constant. As best can bedistinguished, changing has no effect on the number ofpeaks present, and otherwise simply shifts the peak width. ItFIG. 7. Shot noise of carbon nanotube devicesas obtained with the -model. The top panelshows the variance of the current (I) 2 as afunction of electron voltage, while the lowerpanel shows the Fano factor . Here a is for5,5 tubes, and b for 9,0 tubes; solid line—N10 layers; dotted line, N11 layers, whichrepresent on- and off-resonant devices for the respectivenanotubes.155412-6
RESONANT TRANSMISSION THROUGH FINITE-SIZED . . . PHYSICAL REVIEW B 63 155412FIG. 8. Schematic of the 5-7-7-5 defect, which forms via therotation of a C –C bond in the wall of a strained nanotube a andb. Upon annealing, the 5-7-7-5 defect may separate into two5-7 pairs, as shown in c.has been shown 57 that for a given energy level with wavefunction , the coupling is estimated to be2 2 C 2 N o ,where C is the matrix elements between the atoms in theconductor and lead, and N o the metallic density of state peratoms at the Fermi level. The latter quantity, of course, variessignificantly from atom to atom, so that one can expectthe length that recovers the infinite limit to vary tremendouslywhen different types of metals are used for the leads.For instance, Na, Mg, Cu, Ag, and Au all have total densityof states that are less than 7.0 at the Fermi level. The systemsthat use these metals for leads should therefore display theresonant tunneling behavior over very much longer lengthscales than metals like Mn, Fe, Pt, and K which are all characterizedby relatively high densities of states. 58As a final consideration of the pristine nanotubes, we considertheir shot noise, as given in Fig. 7. We have examinedboth on- and off-resonant devices for armchair tubes andtubes with a different helicity. In general, the same behavioris observed irrespective of the helicity of the nanotube.While the initial behavior of on-resonant devices is somewhatsteeper, the two types of devices track each other relativelywell. The basic statistics of the shot noise through thedifferent nanotubes appears to be similar as evidenced by theFano factor, which is mostly flat over a fairly large voltageregion with the shot noise being suppressed by a factor ofabout 1/4. This behavior is to be expected since, in the absenceof any correlated scattering events, the connectivity isprobably the major factor in determining the shot noise,which is the same for the different symmetry tubes. Note thatin this calculation we assume a temperature regime, such thatthe many-body Coulomb interactions do not play a majorrole. At extremely low temperatures of a few degrees Kelvin,FIG. 9. Formation of extended defects via the incorporation of aC 2 dimer into the walls of a strained carbon nanotube: a the 7-5-5-7 defect; b rotation of bond emanating from one of the pentagonsto form a defect with a single enclosed hexagon; c twoandd three-enclosed hexagons. Ultimately, the defect can wrapitself entirely about the circumference of a nanotube to form anelectronic heterojunction, as shown for a 17,0/8.8/17,0 quantumdot structure in e.it is likely that even further deviations from the classical shotnoise value would appear because of possible Luttinger liquideffects.B. Transmission through defective nanotubesWe have investigated the transmission through bent nanotubesand nanotubes with two sets of defects associated withstrained nanotubes. What is probably the most common defecton nanotubes under tension is initiated via the rotation ofa C –C bond in the nanotube wall, i.e., a Stone-Wales transformationfamiliar from C 45 60 that leads to the formation of a5-7-7-5 defect, as illustrated in Fig. 8. The formation of thisdefect has been investigated with classical, tight-binding, andab initio simulations 46 that show that it is energetically perferredon armchair nanotubes under tensions of 5% or more.Once nucleated, this defect undergoes either ductile or brittlebehavior depending on the temperature, rate of stress, andthe helicity of the nanotube. Ductile or plastic behavior istypically associated with armchair nanotubes, and involvesthe separation of the 5-7-7-5 defect into two 5-7 pairs,which then glide about the circumference of the nanotube.Brittle behavior, on the other hand, is typically associatedwith formation of large rings that subsequently open to forma crack and ultimately lead to the breakage of the nanotube.Zigzag tubes tend to display brittle behavior, while chiralnanotubes are expected to show some degree of ductility. 46155412-7
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