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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)