# Why do electrons move towards a vacant position (hole) in a crystal lattice?

Why do electrons in a crystal lattice move towards the vacant position? Aren't electrons stable in their current position?

Consider first an intrinsic semiconductor crystal at zero temperature. The crystal will be charge neutral. All states in the valence bands (and at lower energies) are occupied with electrons, all higher energy states above the band gap are unoccupied.

Suppose now that we remove one electron from the valence band, thereby creating a ‘hole’. A ‘hole’ in a crystal lattice means that there is a net positive charge.

Let us now put the previously removed electron into a conduction band state. The crystal as a whole will now again be charge neutral, but there exists a positive charge in the valence band, and a negative charge in the conduction band. Both can freely move through the crystal. As a result of their opposite electric charge they will attract each other (Coulomb interaction), so the electron will tend to move towards the hole and vice versa.

What you say is not always true. Check that the conduction band is actually "full of holes", because there are many vacant places above the CB. However, the electrons do not use to go higher.

The reason is simpler: all particles seek the minimum energy. If a hole is created in the valence band, then there is an available place where the electron can have less energy. The electron will go there, unless it gains more energy.

In thermal equilibrium and $$T=0K$$, all electrons have the minimum energy, so it will definitely recombine down.

When you raise the temperature, electrons can gain thermal energy to get to the CB. Electrons can also jmp by gaining energy due to photons or other sources.

But, despite energy fluctuations, the principle of minimum energy keeps acting.

• Why holes have less energy. Isn't they are in the same energy level with electrons Feb 23, 2019 at 11:24
• No, I didn't say that. Electrons have less energy in lower levels. That's why they fill the lower bands first. Holes have the opposite energy of electrons, and that's why holes tend to live upside, in the empty levels above the conduction band. Feb 23, 2019 at 11:41
• Even if electrons and holes do not recombine (i.e. the electron does not change its energy), they attract each other via the Coulomb interaction. The resulting bound state is the well-known exciton. Also, electrons and holes can indeed exist at the same energy (e.g. in the Fermi sea of a metal). Feb 23, 2019 at 11:43

Usually electrons in conduction bands are quite delocalized and to speak about their position is not well founded in the QM formalism required to describe them. Holes in conduction band are also useless in the context of this discussion.

There is a case where the concept of filling a vacant position has a direct root in experimental evidence. that is the case of localized electrons around defects of the crystalline structure. In that case, electronic states of the crystal may show a strong localization, resulting in a localized electron which could be described, as a first approximation, as an electron in a potential well. This is the case, for example, of the so-called F-centers (see wikipedia for more ). On a slightly larger scale, quantum dots are also used to localize electrons.enter link description here

Edited after sending

The reason for electron localization in the case of an F-center or a quantum dot can be related to an energetic advantage of a localized electronic state, in the presence of defects which break the symmetry of the crystal. The effect was studied in detail in the case of the so-called Anderson's localization, but it turns out that it is a quite widespread phenomenon.

• You certainly make a valid point about the difficulty of the concept within QM. However excitons do exist, so there certainly is some sense in which electrons are attracted to holes. Feb 23, 2019 at 14:44
• Certainly excitons do exist. However de-localized excitons exist as well (they are known at least since the middle eighties), thus showing that elementary excitations and localized electronic states are related but not coinciding phenomena. Feb 23, 2019 at 14:51
• Bloch states are constructed from localized states. But I think we're on the same page. Feb 23, 2019 at 15:34
• I want to add that we come close to the concept of localized electronic charges by considering suitable superpositions of Bloch states. A localized electron around a defect can also be seen as being in a superposition of Bloch states. We may choose different basis states for the description of electrons in a solid, but saying ‘electrons in the conduction band are quite delocalized’ is a bit imprecise in my view. Feb 23, 2019 at 22:37
• @flaudemus By combining Bloch states it is possible to localize electronic charge but not beyond the limit of maximally localized Wannier functions (MLWF). And MLWF for conduction band in absence of defects are quite delocalized as compared with the case where a defect is present. Feb 24, 2019 at 11:28