Why do dopant levels lie in the energy band gap of semiconductor?

Question

Dopants create impurity levels in the band gap, this is what we were taught in semiconductor classes. And indeed that's what's seen in experiments (for example https://www.globalsino.com/EM/page2777.html). Is this a restriction imposed by physics - that these levels must be inside the bandgap? Or is this just a coincidence (which I don't think so)? For example, if I throw a donor into silicon, why can't the extra electron's energy level lie above the conduction band edge - i.e., sit comfortably inside the conduction band of Si?

A common treatment of this issue (e.g., in Sze's Physics of Semiconductor Devices) uses the hydrogen-atom model as a first approximation, replacing electron mass with effective mass in the lattice. But this already assumed the "free" electron's energy level, i.e., the energy of an electron moving freely in the conduction band of silicon, is above the energy of the electron bonded to the donor atom (phosphorous, arsenic, etc). I don't see how this assumption is justified.

It seems I missed some critical reasoning here, maybe something related to the energy levels in a crystalline lattice. Thank you in advance for pointing it out or suggesting some references where I may find the answer.

Answer 1

Since you mention the hydrogen-like model for impurity states, it is important to keep in mind that while we often care about the discrete, bound states of hydrogen (or other atoms), there are also infinitely many unbound (scattering) states with energy $E>E_0$, where $E_0$ is the energy of a free electron (typically taken to be $E_0=0$).

The transport in the conduction band of a pure semiconductor is often mathematically analogous to that of a free electron (at least close to the band edge), with the free electron mass replaced by the effective mass, and the conduction band minimum as the potential energy (what I called $E_0$ above). In the case of a donor impurity in the semiconductor, the analysis of the energy states is therefore similar to that of a hydrogen atom in vacuum.

The bound states of the impurity + lattice system are the inter-band states we associate with the impurity, while the unbound states are the entire conduction band. In other words, the impurity does affect the conduction band states, but there are still a whole band of states up there with a dense, almost continuous energy distribution. The more impactful consequence of adding impurities are the discrete states residing in the band gap.

On a more practical point, if impurities were to contribute additional intra-gap states, these are still vastly outnumbered by the conduction and valence bands that already exist without impurities. Therefore, they have little impact in terms of carrier statistics in a semiconductor. You might also be aware that trap-assisted recombination depends strongly on the energy level of the trap state, with mid-gap states having the most influence on the recombination lifetime. Intra-band trap states also have little impact in that regard.