[...]... and have been a subject of an extensive study A CN is a few concentric tubes of twodimensional (2D) graphite consisting of carbon-atom hexagons arranged in a helical fashion about the axis The diameter of CNs is usually between 20 and 300 A and their length can exceed 1 /im The distance of adjacent sheets or walls is larger than the distance between nearest neighbor atoms in a graphite sheet and therefore... discussed and the total absence of 4 T Ando Fig, 3: (a) The lattice structure of a 2D graphite sheet and various quantities |a| = |6| =a (b) The reciprocal lattice vectors and the first Brillouin zone backward scattering is pointed out except for scatterers with a potential range smaller than the lattice constant Further, the conductance quantization in the presence of lattice vacancies, i.e., strong and... pz orbital of a carbon atom located at the origin, jR^ = naa-hn66+Ti, and RB—riaa-hTibb with integer Ua and n^ Let —70 be the transfer integral between nearest-neighbor carbon atoms and choose the energy origin at that of the carbon pz level Then, we have eipAiRA) •• Carbon nanotubes ^ 2 •2 1 c »-1 L U H Y V- EF \ H M Wave Vector Fig 4: Calculated band structure of a two-dimensional graphite along K... Single-wall nanotubes usually exhibit large charging effects presumably due to nonideal contacts Carbon nanotubes Fig 1: Some examples of transmission micrograph images of carbon nanotubes [1] The diameter is 67, 55, and 65 A from left to right Fig 2: A computer graphic image of a single-wall armchair nanotube [37-41] In this article we shall mainly discuss electronic states and transport properties of nanotubes... eY,9{r-RA )a{ RA )a{ RA)'^FA{r) = -'yoY2l9{r-RA )a{ RA)b{RA-Ti)+[FB{r) - {rr—)FB{r) + ] (11) I R -A Noting that j:9{r-RA )a{ RA)aiRAy^(^l J), (12) we immediately obtain the left hand side of Eq (11) as eFA{r) As for right hand side, we should note first that '£g{r-RA )a{ RA)b{RA-riy RA « \ ^ ,^ e-'^^e-'^ ""^ j (13) This immediately leads to the conclusion that the first term in the right hand side of Eq (11) vanishes... one-dimensional Brillouin zone On the other hand, an armchair nanotube is always metallic We have prua = ria - 2n6 = 0 and pruh = 2na -nh = 3m, which gives ma = 0 and m^ = 1, and /x = 1 and T = a Thus, the conduction and valence bands cross each other always at ko = ±27r/ 3a In a tight-binding model, the periodic boundary condition il){ri-L) = il){r) is converted into II ^A{ RA+L) = XIJA{RA)^, ipsiRB'^L)... nanotubes, zigzag with {na^rih) = (m,0) and armchair with (no,nfe) = (2m,m), as shown in Fig 7 A zigzag nanotube is metallic when m is divided by three and semiconducting otherwise We have prua = na-2nb = m and prrih = 2na — nt = 2m, which give ma = l and mi, = 2, and // = 0 and T = \/ 3a When a zigzag nanotube is metallic, two conduction and valence bands having a linear dispersion cross at the r point... perpendicular to the axis, the higher order term was shown to cause the appearance of a small band-gap except in armchair nanotubes and a shift of the wave vector corresponding to e = 0 in armchair nanotubes [51] 3 Optical properties 3.1 Dynamical conductivity We shall consider the optical absorption of CN with an Aharonov-Bohm flux using the linear response theory We first expand electric field E^{0^ ui) and... bands with the same band index n as is seen from Eq (62) Since all the conduction bands are specified by different n's, there is no transition within conduction bands and within valence bands At a band edge fc=0, in particular, the wave function is an eigen function of ax and therefore transitions between valence and conduction bands having the same index n are all allowed An exception occurs for bands... CNs are dominated by those of a single layer CN Single-wall nanotubes are produced in a form of ropes [3,4] The purpose of this article is to give a brief review of recent theoretical study on electronic and transport properties of carbon nanotubes Figure 1 shows a transmission micrograph image of multi-wall nanotubes and Fig 2 a computer graphic image of a single-wall nanotube Carbon nanotubes can . we assume that the total wavefunction is written as t/;^(HA) = exp(iK-H^)Ff (HA) + e^^exp(ix'.HA)Ff'(BA), '^B{RB) = -uje^'^exp{iK'RB)Fl^{RB) + exp{iK''RB)F^'{RB) backward scattering except for scatterers with a potential range smaller than the lattice constant, a conductance quantization in the presence of short-range and strong scatterers such as lattice. SQ with the wave function So = 0, (25) for the K point. Other Landau levels are at e = en with wave function ^'^ = ;^C'''%?"'")' ^n = ssain)^M (n =