Band Structure

Of Graphene Sheets and Carbon Nanotubes

Sarah, John, DougiePhys 571 - Spring 2004

Nanotube Structures

a) armchair carbon nanotube

b) zig-zag tube

c) chiral tube

a) armchair (n,n)

b) zig-zag (n,0)

c) chiral (n,m)

Armchair carbon nanotube

Twisted nanotubes slice allowed energy states for electrons at an angle

2/3 semiconductors

1/3 are semi-conductor when n is multiple of 3 (zig-zag) or n-m is a multiple of 3 (chiral)

limits electrons to a select few slices of graphite’s energy states

always metallic

Lattice of Graphene

carbon atoms are located at corner

lines indicate the chemical bonds

primitive lattice vectors a1, a2

unit-cell shaded

Reciprocal Lattice of Graphene

1st Brillouin zone shaded

primitive lattice vectors b1, b2

Graphite Band Gapfrom the top

Graphite Band Gapcross-section

conduction band

valence band

Electrical properties of a material depend on separation between the collection of energy states that are :

valence states filled by electrons (red)

conduction states that are empty and available for electrons to hop into (blue)

Metals conduct electricity easily because there are so many electrons with easy access to adjacent conduction states.

In semiconductors, electrons need an energy boost from light or an electrical field to jump the gap to the first available conduction state.

Graphite is a semi-metal that just barely conducts, because without these external boosts, only a few electrons can access the narrow path to a conduction state

1-D bandstructure of nanotubes

Armchair nanotubes (n,n)

Zig-zag nanotubes (n,0)

Chiral nanotubes (n,m)

always metalbandstructure resembles that of graphite (a)

2/3 are metal, bandstructure (a)1/3 are semiconducting, n multiple of 3, (b)

2/3 are metal, bandstructure (a)1/3 are semi-conducting, n-m multiple of 3, (b)

armchair(5,5)

zig-zag(9,0)

zig-zag(10,0)