# How many hot electrons are in solar cells that do not thermalize?

When an electron in a solar cell is excited beyond the conduction band edge, it will most likely thermalize which can only be counted as a heat loss. But in a few cases the electron will directly recombine back to the valence band, correct? My question: How "often" does this happen?

I heard of the future technology Hot-carrier solar cells (HCSC). They aim at using all of these thermalization energy, correct?

You are asking about recombination dynamics in semiconductors. This is a vast topic!

But let's make some assumptions:

• A lot of hot-carrier research has be done in the III-V semiconductor family. So let's restrict ourselves to GaAs.
• Undoped semiconductor
• Let's assume our material quality is so good that we can ignore recombination via defects.
• Let's restrict ourselves to talk about the recombination dynamics of electrons rather than holes.

Given these assumption, what are the remaining interband processes cause an electron in the conduction band to recombine with a hole in the valence band?

Radiative recombination is proportional to the concentration of electrons $$n$$ and holes $$p$$.

$$R_{rad} = B n p$$

1. Auger recombination

Auger recombination of electrons is a three particle process. An electron in the conduction band will recombine with a hole in the valence band and transfer the energy to a second hole (the third particle in the process).

$$R_{Auger} = C_{p} n p^2$$

To illustrate the point radiative recombination is the top process and Auger recombination is the bottom in the diagram.

Aside: Auger recombination (and it's inverse mechanism impact ionisation) is interesting from a hot-carrier solar cell perspective because it keeps energy within the electronic system. Moreover, Auger recombination reduces electron density in exchange for increasing the average energy (i.e. temperature) of the hole distribution.

Let's write the recombination rate as the sum of these processes,

$$R = B n p + C_{p} n p^2$$

Values for GaAs are $$B=7.2 \times 10^{-10}\text{cm}^3\text{s}^{-1}$$ and $$C_p=10^{-30}\text{cm}^6\text{s}^{-1}$$

Now, if you can estimate the electron and hole density you can calculate the recombination rate.

Let's assume that $$n=p$$ because the semiconductor is undoped, the recombination rates of the radiative (blue) and Auger (purple) are shown in the plot,

The recombination rate is very much dependent on the carrier density generated in the device.

This is a simplification of a very complicated problem. There are a number of additional details not considered

1. Here we consider the ensemble rate. A better approach would be to consider the rate of carriers at different parts in k-space.
2. We are using coefficients for derived from experiment at room temperature. Probably they will have some temperature dependence or at least have been pushed beyond their limit of applicability.
• Why should a carrier recombine if it is in equilibrium with the surrounding lattice? – AskingBecauseIHaveQuestions Oct 24 '20 at 15:03
• There are different types of equilibrium and equilibrium between different systems. That’s a good questions to ask. There is not enough room here, so will update the answer. – boyfarrell Oct 24 '20 at 15:15
• I wish I could book you for a voice chat where things probably would be much easier to explain, because you seem like a real expert, haha. :D I am not a physics student, but teacher for physics (and special education, so physics is limited in my curriculum). Perhaps this limits my knowledge. I hope my questions are not too confusing. But I don't find answers in textbooks or papers. For example, a good explanation on the Urbach tail... And I am still wondering, how many hot electrons in non-hot carrier solar cells directly recombine to the valence band... :/ – AskingBecauseIHaveQuestions Oct 24 '20 at 15:54
• Happy to help. You have not accepted any of the answers to any of the questions I’ve helped you with. I can update this answer with some more information a bit later – boyfarrell Oct 26 '20 at 10:05
• Thanks! I thought I'd give other people chances to answer, since you are the only one who answered them so far. With this, I feel like you answered the second one about the HCSC, but not the first, with respect to hot-carriers in "normal" solar cells. There is also a lot I don't understand about the answer, for example why the CB and VB look inverted in the figure. If it makes you happy I can of course accept all of your answers. – AskingBecauseIHaveQuestions Oct 26 '20 at 18:42