# Exciton binding energy

## Main Question or Discussion Point

Hey!

Can someone help me calculate the binding energy of an exciton? I have been told to use the Rydberg energy but isn't the Rydberg energy derived using the kinetic energy of an atom in a circular orbit about a positve core (hydrogen atom), and the coulomb potential between the two? In an exciton there is no circular orbit, so how can the Rydberg energy be valid in this case??

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ZapperZ
Staff Emeritus
Because it has the SAME form of potential, i.e. a coulombic central potential with no screening term (at least in the less complex model). Thus, you have an equivalent of a hydrogenic atom.

Zz.

Dr Transport
Gold Member
Think about the solution of the hydrogen atom, an electron and a proton. An exciton is an electron-hole pair, bound by the very same potentials as a hydrogen atom. The difference betwen the two systems energies is in the reduced mass.

Gokul43201
Staff Emeritus
Gold Member
One more thing to keep in mind here. You can use the Rydberg approximation to calculate the exciton binding energies only if the screening (and hence, the effective dielectric constant) is large enough that the "Bohr radius" of the exciton is large compared to a lattice spacing. This allows you to approximate the permittivity of the medium at the position of the exciton with the macroscopic dielectric constant.

Dr Transport
Gold Member
Gokul43201 said:
One more thing to keep in mind here. You can use the Rydberg approximation to calculate the exciton binding energies only if the screening (and hence, the effective dielectric constant) is large enough that the "Bohr radius" of the exciton is large compared to a lattice spacing. This allows you to approximate the permittivity of the medium at the position of the exciton with the macroscopic dielectric constant.
Excellent point and one that gets overlooked many times.

Hi guys!

is there anybody know, what is the suitable meaning for "exciton binding energy"?

thank you...

In a semiconductor at least it's the binding energy between a hole-electron pair within the solid. (As long as the lattice spacing is << the "Bohr radius" as Gokul says.)

It's typically tiny, ~meV, with the absorption spectrum for excitons sitting just below the bandgap in a semiconductor.

Hi!

One more thing to keep in mind here. You can use the Rydberg approximation to calculate the exciton binding energies only if the screening (and hence, the effective dielectric constant) is large enough that the "Bohr radius" of the exciton is large compared to a lattice spacing. This allows you to approximate the permittivity of the medium at the position of the exciton with the macroscopic dielectric constant.
I thought the limitation of the model was more with the effective mass, which is a valid concept only for relatively delocalized electron. When the electron is localized, we need a lot of k points to construct the wave function and we must use the k points far from k = 0, where the band is no more parabolic (and hence the effective mass concept loose it validity).

But you make a good point. Thanks for pointing that. Have you a reference where it is clearly written?

TP

Thanks for pointing that. Have you a reference where it is clearly written?
Ok. There is a word on that (localization vs macroscopic dielectric constant usage) in Ashcroft/Mermin, p. 579.

In Marder, p. 593, one first demand spatial extend for a valid utilization of the effective mass concept. Then one use Bohr model to predict radius, and from its large value, conclude that it is consistent with the picture of an electron and a hole sitting in a classical background dielectric.

TP

How many dimensions are you working in? The binding energy in lower dimensional systems can be huge, eg in a carbon nanotube you can have a binding energy on the order of half an eV, compared to the band gap in many of about an eV, while in bulk semi-conductors it is often quite small. Also, what densities are you looking at are you worried about band gap renormalization?