Advanced Solid State Physics

Figure 114: Dispersion relationship for a semiconductor with the energy levels of excitons

The concentration of the created excitons in an insulator depends on the intensity of the incoming
light. When we have a look at fig. 117 we see that at low temperatures and low concentrations, which
means low intensity of light, a free exciton gas is generated. The material still remains to be an
insulator, because excitons are uncharged particles and are uninvolved by an external electric field.
When the intensity of light gets bigger, more and more excitons are created and they start to overlap,
which has influence on the screening of the electrons (Mott criterion in fig. 117). So there are no more
bound states and the result is a gas of unpaired electrons and holes, which means that the insulator
becomes a semiconductor.

14.1.3 Frenkel excitons

In contrast to a Mott Wannier exciton, a Frenkel exciton is localized on an atom or molecule in a
crystal. For example solid krypton has a band gap of 11. 7 eV but the lowest atomic transition in the
solid is at 10. 17 eV. So it is possible to excite an atom to a higher state without bringing electrons
from the valence band to the conduction band. When an exciton is created in an ideal molecular
crystal at one site, it can move to another site, because in a perfect crystal all the sites are equivalent.
That’s the working principle of organic solar cells.

It is possible to describe this movement with a hamiltonian:

Hψj=E 0 ψj+t(ψj− 1 +ψj+1) (238)

The exciton has an energy at a specific site and there is also some coupling to the neighboring sites.
This is the same problem as for a linear chain of phonons, spinons or the tight binding model. The
solutions of such problems are plane waves, which should be already known from the tight binding
model:

ψk =

∑

j

eijkaψj (239)

Ek = E 0 + 2t·cos(ka) (240)