Metallicity Condition and Electronic Density of States in Achiral Single-Walled Carbon Nanotubes

hopping parameter t

2.9

onsite energy parameter ϵ

0.

energy window (x t)

1

2

3

4

5

index n

4

5

6

7

8

9

10

11

12

symmetry

armchair (n,n)

zigzag (n,0)

This Demonstration plots the electronic density of states (DOS) computed from the tight-binding method for zigzag (

n,m=0

) and armchair (

n,n=m

) single-walled carbon nanotubes (SWNTs). The spikes in the electronic DOS profile are called van Hove singularities (vHs) and originate from the different values of the azimuthal band quantum number

μ

. The total number of allowed

μ

values is determined by the number of graphene unit cells inside the SWNT cell, which is always

2n

for achiral SWNTs. By playing with the chiral pairs

(n,m)

, you can verify the metallicity condition for these achiral SWNTs:

n-m≡0(mod3)

for metallic SWNTs and

n-m≢0(mod3)

for semiconducting SWNTs. For metallic SWNTs the electronic DOS is nonzero at the Fermi level, whereas for semiconducting SWNTs, a gap occurs at the Fermi level. Armchair SWNTs are always metallic for any

n

. By changing the sliders related to the tight-binding parameters

t

and

ϵ

you can change the value of the energy at which the vHs occur and the asymmetry of the electronic structure with respect to the Fermi level (

E

F

).