By Myron Fish Eddy

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**Example text**

97), we have 5(p+i)//^ _5(p+i)/ According to the present definition of the angle given by Eq. (98), we have However, we can always put 6{—kj) = 6{kj) — 7r for j = 1 , . . ,p in the expression of 5(p+i)" because of Eq. (97) since 0{-kj) always appears in a pair. By making the replacement ^(k)—>^(—k)—TT and 0(—k)-^^(k)+7r, we can immediately obtain g{p+i)f' ^ _5(p+i)/ =, ^(p+i)*. (107) To see the above explicitly, we have {s\R[ei-k)-27r]R-^[ei-ki)]\si) = {si\R[0{ki)]R-'[eik)]\sr, {sp\R[e{-kp)]R-'[e{k)]\s) = {s\R[e{-k)]R-'[e{kp)]\sp)*.

0 x 10"^ in units of e^/'KU and is normalized by the total number of vacancies having specified ANAB- The inset shows an example of a vacancy which causes a large deviation from the quantized conductance. After Ref. [85]. and S^^^ - - i (119) Ta^, fi^\VaVl3\ where Va and vp axe the velocity of channels a and /3. , (122) and T, = [{l-^VGie+iO)y'j^vl. (123) 42 T. Ando with Vjf = Vj6jf. ^^,_^,^^^^^,^^,y and ^i(a;, ^) =^i(x, -'^). We have introduced a cutoff function /c[/^(n)^fc]defined by in order to exclude contributions from states except in the vicinity of the Fermi level.

0 Fermi energy (units of e(1)) Fig. 33: Calculated conductance in units of e^/'nh as a function of the Fermi energy for CNs with vacancy I (a), IV (b), and II (c), where the energ^-^ is scaled by e(l), which corresponds to the bottom of the second conduction bands. After Ref. [84]. 5x10^ vacancies and demonstrated that such quantization is quite general [85]. Let NA and NB be the number of removed atoms at A and B sublattice points, respectively, and ANAB = NA—NB' Then, the numerical results show that for vacancies much smaller than the circumference, the conductance vanishes for lAiV^^I > 2 and quantized into one and two times of e^/nh for IAAT^BI = 1 and 0, respectively.