Delocalizing Electrons: Time and Diffusion Lengths in Conduction Bands

[yigal]

Recently there was a discussion concerning whether an electron in the conduction band was localized or delocalized. The back and forth arguments seemed inconclusive, difficult to follow (too brief too couched in jargon/unrealistic models) and ended in something close to insults back and forth. Let's assume for a moment that the electrons in the conduction band are completely delocalized. In thermoluminescence kinetics, an electron can be trapped in a state created by a defect and the energy level of the trapped electron is in the forbidden band. The electron is considered localized (at least in the very near vicinity of the defect). Following excitation, (usually either thermal or optical ) the electron receives sufficient energy and is ejected to the conduction band. Thermoluminescence theory then assumes that the electron will become a delocalized entity. Any ideas on the time or diffusion length in the delocalization process ? Have any realistic calculations been carried out ? On semi-conductors, on insulators (very large band gaps).
Please try to give an answer in as intuitive a manner as possible (I realize that this may be difficult - but do your best).

 

[Enthalpy]

The available states in the bands are fully delocalized, because they're states with one defined energy and momentum, so position is unknown. An electron being somewhat localized must have several energy states at the same time.

Now, if the conduction band is occupied to a certain level, all you can say is "so many electrons occupy these levels". You can't distinguish the electrons, nor can you say "I caught this one because it was at that place". All you can say is "I caught one but ignore which". In this sense, telling whether a bunch of electrons are localized is futile.

Though, if at some time you know the position of an electron, you can tell it with some probability within a limited time, yes. This is because it occupies many states at once, and their interference define a limited (but fuzzy) volume. So the more states are available, with energy far separated, and enough excess energy available from the photon, the better the electron stays localized for some time.

 

[yigal]

Thank you
I realize that electrons are usually considered indistinguishable - your answer is a bit helpful -you seem to have said that an electron raised from a localized state (in the forbidden band) to the CB can be said to occupy a certain "fuzzy volume" for a certain amount of equally "fuzzy" time. This seems to me to indicate that for a fuzzy amount of time this particular electron is distinguishable from the others. Is this fuzzy thinking ?
Imagine an insulator with ONE single electron in a localized state in the FB (is this a possible statement ?) This electron is now ejected into the CB. How long and what distance will it travel before it is delocalized.
I'm sorry if this make no sense in the world of QM

 

[Enthalpy]

I had to think more about it and am not sure of the answer.

Presently I believe the answer about electron localization depends on the spectrum of incoming light. That is, if the light is very monochromatic, the electron can't be precisely localized. Only if the light spectrum is broad enough can the electron occupy enough states, separated enough, that its position can be precise for some time.

It may also need the light pulse to be short enough, so that you know when the electron was emitted... Because if you search for an electron emitted at a random date, you can't say how localized it has been. This is consistent with a spectrum wide enough.

Together with the previous ones, you need light intense enough that the electron has good chances to be emitted within the short time.

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If the electron's energy is moderate enough (not trivial) that its mass is constant, that is, near the minimum of the conduction band as opposed to a hot carrier, the situation isn't very different from an electron in vacuum! You have one photon, one electron (with a negative energy as it's bound to a deep level), an interaction and the electron is knocked away. Except the mass isn't the vacuum one, the electron behaves in the conduction band as if it were in vacuum. No special drama here.

You just have to know when it was emitted, with what energy in what direction (!) and apply the usual uncertainty computations. Then, shocks in the semiconductor will let forget the initial parameters, and this differs from vacuum, more so if the conduction band is populated.

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One single deep level: absolutely sensible. For instance gold in silicon is controlled to 10¹⁰cm^-3 because its effect is so strong. In a MOS channel of 20nm*20nm*100nm you have 0.4*10^-6 atoms of Au. In 1µm³ you have 100 atoms of Au.

I suspect deep levels in silica are extremely scarce, because fibres are transparent over 100km distance.