# Allowed wave vectors of CNT

1. Jun 9, 2009

### rejinisaac1

how does one find the allowed wave vectors in a CNT? I'm trying to use an sp3s* tight binding approach to compute the current through a CNT. While writing the off diagonal elements, is exp^{ik.R} a necesaary prefix? If so what value of k does one have to take?

2. Jun 9, 2009

### Born2bwire

I can't remember, but you can find a number of textbooks on the subject that will detail the process.

EDIT: My notes aren't in order about my sources but Nature Nanotechnology Vol 2 Ocober 2007 had a review article of CNT that has a large list of references. By the way, I thought graphene was sp2 bonded or is that only when it is in graphite?

Last edited: Jun 9, 2009
3. Jun 9, 2009

### rejinisaac1

i was referring to this book named 'physical properties of carbon nanotubes' by saito and dresselhaus. Page 47 of this book shows 2 vectors K1 and K2. The allowed vectors are shown parallel to K1. So are the values of K1 the allowed wave vectors of the in the first brillouin zone of the CNT under consideration? Sorry, couldn't attach the book as its around 14 MB.

4. Jun 9, 2009

### Born2bwire

This depends on the chirality of the CNT. However, it is my recollection that yes, the propagation only occurs along one direction. However, the wave numbers that are allowed are more than one and are discrete. Unfortunately, I don't have complete notes on the derivation although I went over it about 6 months ago.

If you can find a 3D picture of the band structure, I believe that you will see that the the metallic CNT only has the conduction and valence bands meeting at one point (well, I think it's four points actually but I think it only counts as one point when using the Brillouin vectors. So this single point accounts for the single direction of the k vector and the periodicity of the Brillioun (How do you spell this guy's name fer crying out loud?) lattice is the cause of the discrete wave numbers that are possible I believe.

But if you are just interested in calculating the current that is dependent upon the situation. The CNT's are capable of ballistic transport, under these conditions the CNT's have zero resistance but there is a contact resistance due to the fact that there are only two conducting bands from the pi-bands. This results in a contact resistance of 12.9 kiloohms. This is only useful under specific conditions. In addition, I believe Burke has developed a transmission line model for the current if you are interested in an AC excitation. That's Peter J. Burke of UC-Irvine.

5. Jun 15, 2009

### rejinisaac1

The equation of the allowed wave vectors for a CNT is given by

K= k*( K1/|K1|) + nK2

-pi/T< k <=pi/T

n=0,1,2,3......N

How does one construct K ? Run a loop of k from -pi/T to pi/T for a given value of n?