Question

4. From Graphene to Carbon Nanotube (30 pts) Let's say, we somehow measured a diameter (d) of the "zigzag-type" carbon nanotube with chirality vector-2bmý; d 1.02 nm (a) What is m? (20 pts) (b) Is this carbon nanotube metallic or semiconducting? Why? (10 pts) *m is a positive integer *The unit vector of graphene's lattice is given by *The lattice constant of graphene is 1.42Å. a+bý.

Answer 1

Chiral vectors is defined as bar C_in = n vec a_1 + m vec a_2 using the vectors vec a_1 and vec a_2 for the hexagonal lattice. Vec a_1 = (3/2 a_c c . squareroot 3/2 a_c c) vec a_2 = (3/2 a_c c, - squareroot 3/2 a_c c) a_c c = 1.42 bar A is lattice constant Length of the choral vector C_n is the peripheral Length of the nanotube. C_n = squareroot 3 a_c c squareroot n^2 + m n + m^2 For zigzag nanotube, n = 0 rightarrow C_n = squareroot m a_c c and diameter of the nanotube is d_t = C_n/pi = squareroot 3 m a_c c/pi rightarrow 1.02 times 10^- 9 = squareroot 3 m times 1.42 times 10^- 10/pi rightarrow m = 13 for metallic nanotube Vertical-bar n - m Vertical-bar /3 should be an integer, rightarrow Vertical-bar 13 - 0 Vertical-bar /3 Notequalto integer The