# Qualitative band theory

1. Apr 21, 2014

### Ichimaru

Question Statement

Polyacetylene can be modelled naively as a one dimensional chain of carbon atoms each separated by a lattice constant 'a'. Taking the electrons in such a system to be nearly free and applying a weak periodic perturbation we can derive a dispersion relation giving a curve such as the one shown on page 6 here: http://web.mit.edu/course/6/6.732/www/new_part1.pdf

Now with more detail: Polyacetylene actual has an alternating structure of double bond (length 0.9a) then single bond (length 1.1a). This gives it a lattice constant of 2a and a basis of (0) and (0.9a). How does this affect the shape and values of the dispersion curve? What are the differing electrical properties of the naive and more detailed models?

Attempt at solution

This seems like it would be similar to the introduction of a diatomic unit cell for phonons in a crystal lattice. When considering phonons the introduction of a second atom in the unit cell gives an extra branch in the dispersion relation. However I don't understand how this applies in the case of electron bands. We would have two bands anyway as a result of the periodic potential (in the tight binding model there will be two complete bands, in our weak binding model there are only two at the Brillouin sone boundaries). So what would the effect be?

Thanks very much for any help!

2. Apr 22, 2014

### DrDu

The simplest tight binding model considers just one p orbital on each atom ( this is called a Hueckel model in chemistry). If bonds were equal there would be only one atom per unit cell and only one tight binding band. With a two atom basis, there will be two bands, but, as long as bond lengths are equal, the two bonds would touch at the Brillouin boundary. In fact, you only have reduced the size of the brillouin zone and folded back part of the band. Only when bond lengths become unequal, the two bands will be separated at the Brillouin zone boundary. Now try to consider the corresponding weakly perturbed free electron model.