# Analogy of a hydrogen atom in an $n$-type semiconductor

My professor told me that when an intrinsic semiconductor (ex-Silicon) is doped with a penta-valent atom (ex-Phosphorus) then the extra electron is quite far away from the nucleus of Phosphorus.I asked him about the calculation of energy of impurity levels created in the semiconductor then he told me the analogy of hydrogen atom. He said that as the extra electron is quite far away from nucleus of Phosphorus then we can assume the nucleus to have a positive charge $$+e$$. Now we should assume that the electron revolves in a circular orbit and then we should do the calculations as predicted by Bohr's model. He told me to use the effective mass of electron in semiconductor instead of the original mass.He told me that the energy required to go from impurity level to conduction band would be close to $$kT$$ where $$T$$ is the temperature of semiconductor.He himself did the calculations and the result was quite close to the experimental data provided.

Why the Bohr's model is so close to the real model?

• I'm not sure to have understood your question, however you know that inside a Silicon crystal each Silicon atom has 4 covalent bonds and they arrange as a tetrahedron. If you replace a Si atom with a P atom this will still form 4 covalent bonds but it has an extra valence electron which is not shared. If temperature is high enough, thermal energy is greater than the bonding energy and the electron is now free to move inside the crystal and the P atom becomes a P+ ion. Nov 10, 2019 at 13:22
• you may want to look up the entry for "exciton" in the wikipedia and refine your question based on what you read there Nov 10, 2019 at 16:44

## 1 Answer

The shallow donor defect in silicon is quite well described by the Schrödinger equation with the electron effective mass and the electric field reduced by the static dielectric constant. Only the 1s ground state somewhat deviates.

• you probably want to add the word "exciton" to your answer en.wikipedia.org/wiki/Exciton Nov 10, 2019 at 16:41
• @PaulYoung The exciton is a different but related subject. It is the solid state analogon of positronium, and should be describable with the Schrödinger equation similarly, using effective hole and electron mass and screened potential. Nov 10, 2019 at 19:07
• 100% agree ... but it is very related and I think reading the wiki article would help the OP refine their question (because I am not really that sure what the OP is confused about) Nov 10, 2019 at 20:30