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DANIEL ORLIKOWSKI et al. PHYSICAL REVIEW B 63 155412FIG. 2. Resonant tunnelingpeaks for finite-sized 10,10nanotubes as calculated with the-orbital left panels and Charlierright panels models showingboth off- and on-resonance devices.The solid lines mark theconductance spectra while the dottedlines denote the LDOS. Nanotubeswith different number N ofunit cells are shown: a N6; bN6; and c N7, and L,R0.5.ing peaks. Numerical investigations show that this effect isquite small for temperatures up to 300 K, so that these effectscan safely be ignored.It is also evident from Fig. 2 that both positions andheights of the resonant peaks depend on the nanotube length.The off-resonant 10,10 tubes consisting of 10 and 12atomic layers, respectively 20 atoms per atomic layer,show a peak height of one conductance unit, while the onresonanttube of 14 atomic layers is characterized by a peakheight of about two units. In fact, if one considers increasingthe length of the armchair nanotubes layer-by-layer, then everythird nanotube shows a peak at the Fermi level. Numerically,we have checked this all the way out to 1000 atomiclayers (240 nm. This result is also consistent with previousstudies of the band gap for finite-length armchairnanotubes. 52 These calculations showed that the gap in theeigenvalue spectrum oscillates between large and small valuesas tube length is increased, with the small-gap case correspondingto on-resonance systems. This is because forsmall values of the coupling , the eigenvalues are closeto the scattering states, which therefore show a similarbehavior.Physically, one can attribute both the peak height andperiodicity of the on-resonance devices to the electronicFIG. 3. Conductance spectrafor nanotubes with different helicitiesas calculated with theCharlier model: a 9,0 zigzagtubes 18 atoms per layer; b7,4 chiral 62 atoms per layer;and c 9,6 chiral nanotubes 114atoms per layer. The solid linesmark tubes consisting of N10unit cells; dotted line, N11; andthe dashed N12, respectively,with L,R 0.5.155412-4
RESONANT TRANSMISSION THROUGH FINITE-SIZED . . . PHYSICAL REVIEW B 63 155412FIG. 4. Conductance spectra for short, finitesizedsemiconducting nanotubes, which display aresonant peak at the Fermi level that decays asthe nanotube length increases. Here a shows thefull spectra for 17,0 nanotubes: dashed line, N5; dotted line, N10; thin-solid line; N25,long-dashed, N30; thick solid line, infinte tube.In b, we display the decay of the peak as afunction of nanotube length for different semiconductornanotubes: circles, 10,0 tubes;squares, 10,5; diamonds, 14,0; triangles,17,0. As shown in c, there is good scaling inthe decay of this central peak G* as a function ofthe scaled nanotube length x, which is defined asx(NN max )/N 1/2 , where N max is the lengthwhere the maximum conductance is attained andN 1/2 is the length where the conductance peakheight takes on half of its maximal value. For allruns, L,R 0.5.properties of the nanotubes. The band structure of armchairnanotubes consist of two non-degenerate bands that cross theFermi level at k2/3a o , with lattice constant a o . Finitetubes sample k’s in units of 2/L, where L is the nanotubelength. Clearly, only when L3Na o with integer N can oneprobe the Fermi level exactly where the two bands overlap.Otherwise, for incommensurate lengths, one probes the otherregions of the band structure that contain no overlap, 53 andtherefore give rise to peaks of height of unity. This thendirectly accounts for the peak pattern shown in Fig. 2.Similar considerations account for the peak heights anddistribution with respect to tube length for nanotubes withdifferent symmetry, as illustrated in Fig. 3. For instance, theband structure of the (n,0) metallic zigzag tubes consists oftwo degenerate bands crossing the Fermi level at the-point. Hence all the peaks will have a height of two. Sincethe -point cannot be probed for tubes of finite length, wesee no peaks at the Fermi level for the tubes as a function oflength. As the tubes become longer, the peaks simply crowdin toward E0.The behavior of the metallic chiral tubes is qualitativelysimilar to either the armchair and/or zigzag tubes. The bandstructure of (n,m) chiral tubes, for which the greatest commondivisor gcd(n,m)3, is qualitatively similar to that ofthe armchair tubes with two nondegenerate bands crossing atk2/(3T), where T is the length of a unit cell typicallyan irrational number, so that one can expect the resonanttunneling peaks at the Fermi level to have a height of twoand a repeat length of 3T. For a detailed discussion of thenanotube band structure and unit cells, see Ref. 1. Indeed,that is what we observe in the simulations except that incontrast to the armchair nanotubes, some of the other peaksFIG. 5. Conductance spectraas the nanotube length becomesvery large as calculated with the-orbital model for the 5,5 armchairnanotube left panels andthe 9,0 zigzag tubes right panels.The nanotube lengths are aN10; b N100; and c andd are for N1000, respectively.For a–c, L,R 0.5 while L,R2.0 for d. As is increased,the peak widths become wider andbegin to overlap.155412-5
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