Band Structure

1. Band Structure • Of Graphene Sheets and Carbon Nanotubes Sarah, John, Dougie Phys 571 - Spring 2004

2. 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)

3. Armchair carbon nanotube limits electrons to a select few slices of graphite’s energy states always metallic 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)

4. 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

5. Graphite Band Gap from the top conduction band Graphite Band Gap cross-section valence band

6. 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)

7. 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

8. 1-D bandstructure of nanotubes Armchair nanotubes (n,n) always metal bandstructure resembles that of graphite (a) Zig-zag nanotubes (n,0) 2/3 are metal, bandstructure (a) 1/3 are semiconducting, n multiple of 3, (b) Chiral nanotubes (n,m) 2/3 are metal, bandstructure (a) 1/3 are semi-conducting, n-m multiple of 3, (b)

9. armchair (5,5) zig-zag (9,0) zig-zag (10,0)