Electronic structure of carbon nanotubes 林永昌

Classification of Carbon Nanotubes a. armchair (θ=30°, (n,n)=(5,5) ) b. zigzag (θ=0°, (n,0)=(9,0) ) c. chiral (0°<|θ|<30°, (n,m)=(10,5) )

Chiral vector: Ch Diameter: dt Chiral angle: θ Translation vector: T

Bounding Between Carbon Atoms H H H H c c c c c c c H H H sp Hybridization: Acetylene, HC CH sp Hybridization: Acetylene, HC CH sp 2 : Polyacetylene, (HC CH ) n sp 2 : Polyacetylene, (HC CH ) n sp 3 : Methane, (CH 4 ) n sp 3 : Methane, (CH 4 ) n

Tight Binding Method for a Crystalline Solid 1.Bloch ’ s Equation: 1.Bloch ’ s Equation: (1) (2) (3) (4) (5) (6) Transfer integral matrices (7) Overlap integral matrices 2.Schrodinger Equation: 2.Schrodinger Equation: 3.Secular Equation: 3.Secular Equation:

Electronic Structure of Polyacetylene H H H H C C C C C C C C C C H H H H H H (A)(B) -Secular equation (8) (9) (10) (11) (12) (13) (14) (15) E + (k): bounding π energy band E - (k) : antibounding π* energy band unit cell

ε 2p =0 t = -1 s = 0.2 ka/π E E+E+ E-

π-Bands of Two-Dimensional Graphit We consider only nearest-neighbor interactions. We consider only nearest-neighbor interactions. (16) (17) (18) (19) (20) (21) (22) (23)

The reciprocal lattice const is 4π/√3a (22)ε 2p =0, s =0 (*)

E Energy dispersion relations 2D graphite

Electronic Structure of Single-Wall Nanotubes Zone-folding of energy dispersion relations: K2K2 K1K1 w Y w’w’ Γ K’K’ M K M K’K’ E μ : 1D energy dispersion kyky kxkx (24) (25) (26) (27) (28) (29) (30)

C h =(4, 2), T=(4, -5), N=28, K 1 =(5b 1 +4b 2 )/28, K 2 =(4b 1 -2b 2 )/28 Ch T

Energy Dispersion of Armchair and Zigzag Nanotubes Armchair : Ch=(n, n) Armchair : Ch=(n, n) Zigzag : Ch=(n, 0) Zigzag : Ch=(n, 0)

Dispersion of chiral nanotubes: Ch=(n, m) Dispersion of chiral nanotubes: Ch=(n, m) (5,5) (9,0) metallic (10,0)semiconducting metallic armchair zigzag

Density of States, Energy gap Density of states: Density of states: Energy gap: Energy gap:

metallic semiconducting