# Electronic Band Structure of a Single-Walled Carbon Nanotube by the Zone-Folding Method

Electronic Band Structure of a Single-Walled Carbon Nanotube by the Zone-Folding Method

This Demonstration calculates and plots the tight-binding (TB) electronic band structure of a single-walled carbon nanotube (SWNT) of any chirality according to the zone-folding (ZF) method introduced by Saito and Dresselhaus (neglect of curvature effects valid for large diameter tubes). The chiral indices are pairs of integers that specify uniquely the geometry of the SWNT (diameter , chiral angle, length of the unit cell along the tube axis , number of graphene unit cells inside the SWNT unit cell, etc.).

(n,m)

(n,m)

d

t

|T|

The ZF band structure of a SWNT is obtained starting from the TB electronic dispersion relation of a single graphitic sheet or graphene for -orbitals (shown as a contour plot of the well-known analytical expression from the literature over the 2D graphene Brillouin zone). The graphene electronic dispersion is given by a simple analytical relation for the electronic energy ,=, with the phase factor given by . In this expression, is the TB hopping parameter, is the on-site energy parameter, is the overlap parameter, is the lattice parameter of graphene, and stand for valence and conduction bands, respectively, and are the wavevector components along the and directions in the 2D Brillouin zone of graphene. The parameters and are expressed in electron-volt units (eV), whereas is given in non-dimensional units. This relation can be derived by diagonalization of the 2×2 Bloch Hamiltonian for the biatomic graphene unit cell, as shown in the works by Saito and Dresselhaus.

π

v,c

E

2D

k

x

k

y

ϵ±tw,

k

x

k

y

1±sw,

k

x

k

y

w(,)=

k

x

k

y

f(,)(,)

k

x

k

y

*

f

k

x

k

y

w(,)=

k

x

k

y

1+4cosacosa+4a

3

k

x

2

k

y

2

2

cos

k

y

2

t

ϵ

s

a

v

c

(,)

k

x

k

y

x

y

t

ϵ

s

The graphene electronic band structure is then sectioned along the directions given by the SWNT 1D Brillouin zones or cutting lines, whose spacing and length are related to the chiral indices of the nanotube, so that one finally gets the SWNT electronic band structure as μ+, where and are the basis wavevectors in the SWNT BZ. The spacing between cutting lines is inversely proportional to the SWNT diameter (), while the length of the lines is inversely proportional to the length of the SWNT unit cell along the tube axis (); is the azimuthal band quantum number, which labels each cutting line or, conversely, each SWNT band, whereas is the axial quantum number, which is continuous inside a given cutting line. The irreducible number of bands is equal to the irreducible number of cutting lines, which is given by the number of graphene unit cells inside the SWNT unit cell. Usually they are indexed as or equivalently by . Additionally, is cyclically periodic in .

(n,m)

v,c

E

2D

K

1

k

z

K

2

||

K

2

K

1

K

2

||=

K

1

2/d

t

||=2π/|T|

K

2

μ

k

z

N

μ=0,1,…,N-1

μ=-N/2+1,…,0,…,N/2

μ

±N

When a cutting line crosses the graphene BZ special points and , the SWNT is metallic () and the SWNT bands cross at the Fermi level (energy at 0 eV); otherwise the SWNT is semiconducting () with a nonzero energy gap at the Fermi level.

K

K'

mod(n-m,3)=0

mod(n-m,3)≠0

From the characteristic features of the band plots for any given chirality one can verify the close relationship existing between geometry and electronic structure in SWNTs.

(n,m)