Nature of band gap in conjugate polymers

Electronic band gap, is theoretically defined as the energy difference between two electronic bands. In reality, the operational definition is associated with optical spectroscopy measurements. Normally the absorption onset for the magnitude of  alpha~105 cm is used to determine the band gap. [Note1] The nature of the band gap refers to the nature of excitation that is associated with the band gap energy. In most ionic crystals, charge transfer from the oxidant atom to the reductant atom is the dominant excitation, unless there is some other special strong excitation, e..g polaronic excitation to destroy charge order.

Then the question is what’s the nature of the band gap in conjugate polymers? It is definitely not C->H or C->C, which occur at much higher energy. In fact, it has to do with the energy gap created by Brillouin zone folding.

Taking the simplest case of poly-acetylene, with C-C C=C alternation. If all the bond length is the same, the pi electron system forms a single band. Since this band is only half full, it should be a good conductor, like graphite (or graphene).

In fact poly-acetylene is not a conductor. This is because the 1D system is very susceptible to Peierls distortion, in which the C-C=C chain dimerize. So the primitive cell is doubled, Brillouin zone becomes half size. Then the zone folding occurs, a gap appears at the zone boundary, which is the band gap we observe.

This can be generalized to the case that every primitive cell can have N C-C=C units. If N=1, it is a conductor. Otherwise, it is a insulator with a small band gap (normally <2 eV).

As show in the figure, if the energy bands are half filled, the gap at the center will be the band gap observed.

[Note1] This however does not preclude much weaker absorption with energy less the the band gap energy. Taking a simple example as Fe2O3, the optical band gap is decided by the charge transfer energy between O->Fe, which is about 2.5 eV. On the other hand, the Fe3+ on-site excitation is about 1.4 and 1.9 eV, with energy less than the band gap. Because the on-site excitation is about 1000 times weaker than the charge transfer, it is not used to determine the band gap. Interestingly, the red color is from the on-site excitation!