# Why is the energy for an electron-hole pair for $\rm CdTe$ bigger than the band gap energy?

Let's take the indirect semiconductors Si, Ge & Diamond. All these semiconductors are indirect, meaning that the maximum of the valence band is not directly under the minimum of the conduction band. This is how we can explain that the band gap for Si is $$1.12$$ eV, whereas the average required energy to cerate an electron-hole pair is $$3.65$$ eV (basically, the energy difference goes into phonons, heat).

But then, for the direct semiconductor $$\rm CdTe$$ (Cadmium-Tellur), why is the required energy for the creation of an $$e/h$$ pair $$4.43$$ eV, whereas the band gap is $$1.44$$ eV!? Very similar numbers also hold for GaAs (reference: Kolanoski-Wermes "Particle Detectors. Fundamentals and Applications. 2020. p. 261").

I thought that for direct semiconductors, we don't have phonon excitations.. But okay, I was probably wrong here.

• Context matters. Why do you believe this, given a wide range of papers on, say, solar cell performance? Mar 3, 2021 at 16:46
• @JonCuster Well, it's not a matter of belief. I quoted numbers for CdTe and GaAs. The context is Particle Physics/Detector Physics.
– user248824
Mar 3, 2021 at 16:51
• @MathIsFun - well, the context absolutely does matter. The issue with particle detectors is you look at how much current (e-h pairs) you get out of the device for the total electronic stopping that occurs in the active volume. Because the energy deposited along a particle track is quite high, the carrier densities are high enough that there is significant in-device recombination of the free carriers. Ballpark is you divide total electronic stopping by the bandgap, then divide that by a factor of order 3-10 that you must figure out based on geometry, particle, energy, etc. Mar 3, 2021 at 17:32
• Notice all the numbers are ~3x the band gap, direct or indirect. Mar 3, 2021 at 21:29
• Another way to consider it: high energy particles can transfer energy well above the band gap to electrons. Those highly energetic electrons can create further electron-hole pairs, as well as rapidly thermalize through electron-electron scattering or phonon production. Lots of energy goes to things other than just making e-h pairs at the band gap energy. Very hard to model definitively, various hand waving factors are used. Mar 4, 2021 at 2:46

Here is an image of the photoluminescence from bulk CdTe layers in a heterostructure device take from this reference https://doi.org/10.1063/1.4803911. Hopefully we can agree that the energy of the emitted light is in the range 1.4-1.5eV, which agrees with your band gap value.

So we can also say that electron holes pair exist in the material with this energy difference, therefore there is some non-zero oscillator strength coupling the electron and hole states, therefore the material will also absorb and emit at this energy. In fact that’s a thermodynamic requirement via Kirchoff’s law.

I think you are reading about using CdTe as a particle detector, this will have certain constraints and limits of applicability, and may explain why the higher value is used as a figure of merit. But if we characterise the band structure, it absorbs and emits fine around 1.45eV

It could also be that the 4.4eV in the table is the electron affinity: the energy to promote an electron from the conduction band to the vacuum level. Which I guess would make sense if using the photoelectric effect to detect particles. • Okay, really interesting.. :) I didn't know it might be context-dependent. I updated my post with a Table taken from the book, that might help for the context.
– user248824
Mar 3, 2021 at 19:11

The exciton creation energy should be the direct band gap energy minus the binding energy, which is small. Here the value of 1.6 eV is given for the exciton energy in CdTe: https://www.nextnano.com/manual/nextnano3/tutorials/exciton_1D.html

The band gap is the minimum amount of energy needed to produce an e/h pair (1.44 eV in CdTe). But if you input 1.44 eV the energy can also be used to heat up the lattice. Therefore, on AVERAGE 4.43 eV are consumed per produced e/h pair, because from one event to the next different amounts of energy are dissipated into e/h creation and elsewhere, most notably into lattice oscillations (phonon creation). This, to first order, has nothing to do with whether a material is an indirect or direct semiconductor. The latter means that in addition to energy transfer also (crystal) momentum transfer is required for a transition from valence to conduction band.

The interaction with matter and high energy photon generate two kinds of effect: charge carriers and heat (phonon). Some of photon energy converted as heat. Thus, average energy for ehp creation energy is higher than the energy gap. The energy ratio consumed in charge carriers and phonon is constant for specific material and this ratio is called as Fano factor.