# Donor energy levels in a doped semiconductor

I simply want to know why, in an n-type semiconductor, we have this "new" kind of energy level called the "donor level", apart from the valence and conduction and band, and , more importantly, why is the donor level near the top of the band gap?

My (possibly arm-wavy) qualitative explanation is that the donor electrons are "free" , so you would naturally expect them to be in a higher energy level than the valence electrons...

• The donor level is near the top of the band gap because that is where it has to be to be able to get an electron from it to the conduction band with ~$kT$ energy near room temperature. There are plenty of other known impurity levels throughout the gap, with only the ones near the valance band (acceptors) also useful to semiconductor devices. Well, sometimes mid-gap states are deliberately put in things like power electronics to make sure they can turn off. Apr 8, 2021 at 13:31

Since small energy gaps are generally associated with large dielectric constants, it is almost always the case that the binding energy of an electron to a donor impurity is small compared with the energy gap of the semiconductor. Since this binding energy is measured relative to the energy of the conduction band levels from which the bound impurity level is formed, we conclude that donor impurities introduce additional electronic levels

at energies $$\epsilon_d$$ which are lower than the energy $$\epsilon_c$$ at the bottom of the conduction band by an amount that is small compared with the energy gap $$E_g$$

A similar argument can be applied to acceptor impurities, whose valence is one less than that of the host atoms (e.g., gallium in germanium). Such an impurity can be represented by the superimposition of a fixed charge - e on top of a host atom, along with the presence of one less electron in the crystal. The missing electron can be represented as a bound hole, attracted by the excess negative charge representing the impurity, with a binding energy that is again small on the scale of the energy gap,$$E_g$$ In terms of the electron picture this bound hole will be manifested as an additional electronic level at an energy $$\epsilon_a$$ lying slightly above the top of the valence band. The hole is bound when the level is empty. The binding energy of the bole is just the energy $$\epsilon_a - \epsilon_v$$ necessary to excite an electron from the top of the valence band into the acceptor level, thereby filling the hole in the vicinity of the acceptor and creating a free bole in the valence band.

Knowledge gained from: "Solid State Physics:Neil W. Ashcroft, N. David Mermin"

• The problem with this explanation is that there are impurity levels all across the gap (from different elements). Those close to the conduction band are donors, those near the valence band are acceptors, and the rest are mid-gap states. There are far more mid-gap elements (and some elements are amphoteric) than acceptor/donor elements. Apr 8, 2021 at 13:17
• @JonCuster I am sorry I think I was ignorant about this. Apr 8, 2021 at 14:10

One approach is to consider the translation from a single covalent bond to a bulk semiconductor with multiple covalent bonds. The picture below shows the energy diagram.

The left image shows the molecular orbitals as the linear combinations of atomic orbitals. A single Si-Si bond (blue) would fill the highest occupied molecular orbital (HOMO) with two electrons. Consider the case for Al (green) to the left of Si on the periodic table (a p-type dopant). It has one fewer valence electron than Si. It also has a lower ionization energy than Si. The molecular orbital configuration is skewed, with the HOMO in the Si-Al bond containing an empty hole. Consider the case for P (red) to the right of Si on the periodic table (an n-type dopant). It has one more valence electron than Si. It also has a higher ionization energy than Si. the molecular orbital configuration is skewed, with the lowest unoccupied molecular orbital (LUMO) in the Si-P bond containing an extra electron.

Translate this single bond picture to the band structure. The HOMOs in Si become the valence band (VB) and the LUMOs become the conduction band (CB). The image shown has the case for one p-type or one n-type dopant.

The electrons in these pictures are bound electrons, not free electrons. A free electron is produced when the electron from the valence band moves to conduction band. This process is the same as an intra-molecular excitation of an electron from a HOMO to a LUMO. In p-type semiconductors, the valence electrons move from the valance band to the HOMO orbital between the intrinsic and dopant. This produces extra holes in the VB.