# What is the meaning of element-rich environment in calculating formation energy?

I want to use the formation energy formula used by the article in the picture below. The authors have used the term S or Sn-rich environment. What is the meaning of this? How it is related to the calculation of formation energy?

Reference paper : https://doi.org/10.1063/1.5022151

How rich the environment is in $$\ce{Sn}$$ vs $$\ce{S}$$ affects the chemical potential $$\mu_\ce{Sn}$$ as described in Section IIIA of the linked paper:

... $$\mu_\ce{Sn}$$ for the Sn-rich environment is equal to $$E_\ce{Sn}$$, where $$E_\ce{Sn}$$ is the total energy of Sn per atom in the diamond crystal structure.

... $$\mu_\ce{Sn}$$ can be calculated for the S-rich environment as $$\mu_\ce{Sn}= E_\ce{SnS}-E_\ce{S}$$.

(Note $$E_\ce{SnS}$$ and $$E_\ce{S}$$ in the quote above are the energy per formula unit and atom respectively). As your image shows, the formation energy depends on $$\mu_\ce{Sn}$$, so it also depends on how rich the environment is in $$\ce{Sn}$$ vs $$\ce{S}$$.

Richness here is just referring to how abundant a given element is in the surroundings. This won't change the simulations of the monolayer, but it does affect the formation energy. This is because in an Sn-rich environment, you are not just producing a lone Sn atom when you remove it from SnS. Rather, you are implicitly incorporating it into bulk metallic Sn, which has different energetics. For an S-rich environment, they assume the Sn is instead producing a site of SnS at the expense of a site from bulk S. So depending on the environment, the overall formation process is different.

If you aren't concerned about the environment effects, you could use the same definition for the chemical potential as was used for the doping transition metal (e.g. the energy of one isolated atom). They actually do this earlier in the linked paper, where they compute vacancy formation energies for SnS either using the environment affected chemical potentials or just the energy of a lone atom.

• What does mean Sn or S-rich environment ? How to put this in calculations of formation energy ? – Chi Kou Apr 17 at 10:17
• @ChiKou I have updated the answer to clarify what they seem to mean by richness. Provided you have energies per atom for bulk Sn and S, you can simply plug these into the formation energy formula. It doesn't affect the process of simulating the monolayer. – Tyberius Apr 17 at 14:16
• Thank you very much @Tyberius for your useful answer. Just one more question : As I have understand from your answer and if I am not concerned with environment effect. I can replace the chemical potentials above in the formula by just the energy of isolated Sn and TM atoms, and this will lead to the correct formation energy. Is it right ? – Chi Kou Apr 17 at 16:18
• @ChiKou that is my understanding. They seem to do something similar with the vacancy formation energies they compute at the beginning of Section IIIA. – Tyberius Apr 17 at 16:28
• Yes thanks a lot. – Chi Kou Apr 17 at 17:07