Are conduction electrons localized in space?

[crazy_photon]

I don't think people should be confusing temperature to the question about the proper boundary conditions. Finite-size effects may or may not be important even at T=0. It is true that at higher temperatures electrons start to lose their phase coherence, but this is mainly due to increased inelastic scattering. In this case quantum mechanical interference effects such as weak localization are lost. The whole idea of electron wavefunction becomes then very blurred, but it does not mean that suddenly they turn from delocalized to localized. I guess even at T=0 electron motion can be described as diffusive in dirty systems and one can derive Ohm's law. This does not require high temperature in itself.

This post probably did not make much sense, but the whole thread is quite a mess :)

I agree with you on one thing - that it didn't make much sense (at least to me) :)

First off, the question wasn't about boundary conditions -- it was about localization versus delocalization. Boundary conditions were sucked into the argument...

I don't understand when you say that electrons lose their phase-coherence due to inelastic scattering. Any scattering event would cause decoherence -- the difference between elastic versus inelastic is just a matter of energy transfer (electron-electron is elastic, electron-phonon is inelastic because there's transfer of energy to the lattice and back). Both scattering mechanisms would cause decoherence, but only one (inelastic) would be responsible for establishing thermal equilibrium with the lattice. Shall we say that this reasoning frees us from (unnecessarily) bringing scattering events to answer the main question?

There's also no 'suddenly' here -- its a very gradual process, at least for initially.

If by 'dirty' systems you mean system without translational periodicity then i totally agree with you, but the question is about metals - i.e. (nearly) defect-free lattices. And if you are talking about metals (with periodicity) and near-zero temperature -- things are nothing like Ohms law.

One thing I agree with you is the role of the wavefunction in this argument -- its the decoherence (i.e. scattering) that allows classical models to treat electrons as point particles. but that has been said already (at the very least by myself).

 

[saaskis]

I agree with you on one thing - that it didn't make much sense (at least to me) :)

First off, the question wasn't about boundary conditions -- it was about localization versus delocalization. Boundary conditions were sucked into the argument...

You're probably right, I have lost track of what people are arguing about.

I don't understand when you say that electrons lose their phase-coherence due to inelastic scattering. Any scattering event would cause decoherence -- the difference between elastic versus inelastic is just a matter of energy transfer (electron-electron is elastic, electron-phonon is inelastic because there's transfer of energy to the lattice and back).

I must disagree with you here. For example, elastic mean free path and dephasing length can be very different length scales in a mesoscopic structure. One can see clear interference effects even if there is a lot of elastic scattering in the structure, e.g. static impurities. This also leads to UCF.

There's also no 'suddenly' here -- its a very gradual process, at least for initially.

Yes, I meant that simply by increasing temperature the electrons do not become entirely different entities. This was stated a little unclearly.

If by 'dirty' systems you mean system without translational periodicity then i totally agree with you, but the question is about metals - i.e. (nearly) defect-free lattices. And if you are talking about metals (with periodicity) and near-zero temperature -- things are nothing like Ohms law.

Yes, of course, in a perfect metal static conductivity is infinite. I just meant that it is not the temperature in itself that causes the electron motion to become diffusive and "more classical", but dephasing caused by e.g. phonons. And of course the number of phonons increases rapidly when increasing temperature.

 

[conway]

I still don't see it, especially on context with the photoelectric effect. For example, if you look at the photoemission Hamiltonian, where exactly is the potential well model "good" here?
Zz.

Like you said, if we really want to get into this in detail someone should start another thread. I was really just commenting on the usefulness of the potential well model for metals in general. I might have just as well taken the Josephson junction as an example.

 

[ZapperZ]

Like you said, if we really want to get into this in detail someone should start another thread. I was really just commenting on the usefulness of the potential well model for metals in general.

Actually, you didn't. You specifically brought it up in the context of the photoelectric effect.

Even without going into detail, the photoelectric effect, i.e. the naive version of it, assume the existence of "free electrons" in the conduction band. For ALL photon energies above the work function, you will get photoelectrons, i.e. it is a continuous range of photon energy.

Yet, a "potential well" will have discrete energy levels. It means that as you increase the photon energy, you'll get some photoelectrons at one photon energy, but none for another range of photon energy. In fact, if you look at the energy distribution of the photoelectrons, you'll see sharp peaks corresponding to each of the potential well energy levels! We see no such thing. What we see instead is a continuous, broad distribution of energy of the photoelectrons coming from the conduction band. This is NOT what one would expect out of an infinite potential well.

Therefore, how in the world is this representation of a metal even close to being useful when it predicts something entirely different than what we get?

Zz.

 

[conway]

Even without going into detail, the photoelectric effect, i.e. the naive version of it, assume the existence of "free electrons" in the conduction band. For ALL photon energies above the work function, you will get photoelectrons, i.e. it is a continuous range of photon energy.

Yet, a "potential well" will have discrete energy levels. It means that as you increase the photon energy, you'll get some photoelectrons at one photon energy, but none for another range of photon energy...

Zz.

I understand what you're saying and it's a natural mistake for people to make. Yes, in the one-dimensional potential well, the energy levels get farther and farther apart the more electrons you add. But for the 2-d well, the geometry exactly compensates for this sparseness. Go to 3-d and the the density of energy levels actually increases the more electrons you add. For practical sizes, you can consider it a continuum.

 

[ZapperZ]

I understand what you're saying and it's a natural mistake for people to make. Yes, in the one-dimensional potential well, the energy levels get farther and farther apart the more electrons you add. But for the 2-d well, the geometry exactly compensates for this sparseness. Go to 3-d and the the density of energy levels actually increases the more electrons you add. For practical sizes, you can consider it a continuum.

That still doesn't work!

Look at as 3D standing wave rectangular waveguide. If you connect a spectrum analyzer to it, you'll see various modes that can be sustained in in. Make it larger to get more modes in it, and you can still detect "ripples" in the spectrum signifying the location of each mode. In fact, if I have a good enough resolution (and spectrum analyzers nowadays have amazing resolutions as it is already), I can certainly detect such modes.

Note that this is just a consideration of the energy state. We haven't even looked at how one would get the band dispersion of an ordinary metal. How would you propose to get that our of such a model?

Zz.

 

[sokrates]

Any scattering event would cause decoherence --

You just proved that the operation principle of Resonant Tunneling Diodes collapses! (any many other hallmark experiments fall apart) What do you mean by saying ANY scattering event would cause decoherence?! This is a serious misconception. If what you said was true, the barriers in resonant tunneling diodes would randomize electron interference and we wouldn't get resonant tunneling when the barrier width is half wavelengths long! Because all the electrons would decohere upon hitting the barriers and they would act like particles which would never give you that negative differential resistance effect in the I-V curve. If you have an elastic scatterer with no internal degrees of freedom, this does not cause decoherence. In other words, if you can include your scatterer in your basic Hamiltonian (say by a large potential corresponding to an impurity) there is NO decoherence. This would correspond to an elastic, coherent scattering event. It occurred to me that you are seriously confused about decoherence (or what you mean by it) so you can check the operation of RTDs to get it right. I can send you a MATLAB code if you want to play with RTDs and see how they work.

 

[conway]

That still doesn't work!


Note that this is just a consideration of the energy state. We haven't even looked at how one would get the band dispersion of an ordinary metal. How would you propose to get that our of such a model?

Zz.

That would be pretty tough.

 

[jensa]

You just proved that the operation principle of Resonant Tunneling Diodes collapses! (any many other hallmark experiments fall apart) What do you mean by saying ANY scattering event would cause decoherence?! This is a serious misconception. If what you said was true, the barriers in resonant tunneling diodes would randomize electron interference and we wouldn't get resonant tunneling when the barrier width is half wavelengths long! Because all the electrons would decohere upon hitting the barriers and they would act like particles which would never give you that negative differential resistance effect in the I-V curve. If you have an elastic scatterer with no internal degrees of freedom, this does not cause decoherence. In other words, if you can include your scatterer in your basic Hamiltonian (say by a large potential corresponding to an impurity) there is NO decoherence. This would correspond to an elastic, coherent scattering event. It occurred to me that you are seriously confused about decoherence (or what you mean by it) so you can check the operation of RTDs to get it right. I can send you a MATLAB code if you want to play with RTDs and see how they work.

Please Sokrates, it is clear from the context that crazy photon is talking about random impurity scattering which does cause dephasing (and thus decoherence). Stop nitpicking and try to focus on the issues. It amazes me that you have yet to comment on the relevant posts by crazy photon and me. Do you agree or do you not? If not, then why? If yes then why are you giving crazy photon such a hard time??
You seem to have an infinite amount of time for "intellectual bashing" as you called it. Yet you have not even once addressed the original question with a constructive answer.

 

[sokrates]

Please Sokrates, it is clear from the context that crazy photon is talking about random impurity scattering which does cause dephasing (and thus decoherence). Stop nitpicking and try to focus on the issues. It amazes me that you have yet to comment on the relevant posts by crazy photon and me. Do you agree or do you not? If not, then why? If yes then why are you giving crazy photon such a hard time??
You seem to have an infinite amount of time for "intellectual bashing" as you called it. Yet you have not even once addressed the original question with a constructive answer.

It wasn't clear, on the contrary, it was clearly misleading. Why didn't you wait for crazyPhoton to speak for himself, and to clarify the issue? Maybe he really meant what I say. I nexted this thread, I am sorry. No more answers from "the expert(!)" THere are far more knowledgeable people here than I am. So why is it so important for you to get an answer specifically from me? I am not an authority, people! Why is everyone expecting "constructive answers" from Sokrates? I don't own this forum and I am not in charge! I am reading the thread just like you are, learning things, sharing things, calm down, don't be so sensitive! Unlike people who criticize my way of communication, I never made a personal remark --- the worst thing I did was calling models "simplistic". I like to respond to things that matter the most, from my view. Sorry, I am not obliged you to read all of your posts ( I don't even know what you asked me, why don't you PM me instead next time?) and reply to them.

In my previous post, I tried to correct a scientific statement which was not true. And what was your purpose in your last post apart from criticizing my style? Anything related to physics? Oh, and I need to focus on issues? Hmm...

 

[crazy_photon]

Please Sokrates, it is clear from the context that crazy photon is talking about random impurity scattering which does cause dephasing (and thus decoherence). Stop nitpicking and try to focus on the issues. It amazes me that you have yet to comment on the relevant posts by crazy photon and me. Do you agree or do you not? If not, then why? If yes then why are you giving crazy photon such a hard time??
You seem to have an infinite amount of time for "intellectual bashing" as you called it. Yet you have not even once addressed the original question with a constructive answer.

Thank you jensa! I thought it was only me that saw it that way.

Sokrates, i would try to address you to the point that you raised (not the point of the thread which i would still love to discuss)... If you indeed want to talk about tunneling (resonant or not) - I wouldn't call it scattering. Scattering is a process where wavevector changes direction at random (if not its called reflection). In tunneling, wavevector becomes purely imaginary inside the barrier and hence causes 'decay'. If barrier is thin enough, like you say, then resonant effects can happen. It would be interesting to look at your code, I'm just very swamped right now. Regardless of the code though, i wouldn't call it scattering.

I was actually having second thoughts after what i have said about elastic scattering causing decoherence, and I think it is still true -- even though the energy is conserved and momentum direction is not being randomized - that doesn't matter. What IS being randomized is phase -- so if you have a scattering process that imposes random phase shift upon each scattering event - that would lead to decoherence of the wavefunction. now, I'm trying to read up on that phase shift... and see if i can learn whether this is indeed what happens. if you can shed some light on that - i'd be interested to hear about it.

I would also be really insterested in getting back to the original theme of the post -- or is jensa and myself are the only ones that feel it still hasn't been addressed properly?

 

[sokrates]

Thank you jensa! I thought it was only me that saw it that way.

Sokrates, i would try to address you to the point that you raised (not the point of the thread which i would still love to discuss)... If you indeed want to talk about tunneling (resonant or not) - I wouldn't call it scattering. Scattering is a process where wavevector changes direction at random (if not its called reflection). In tunneling, wavevector becomes purely imaginary inside the barrier and hence causes 'decay'. If barrier is thin enough, like you say, then resonant effects can happen. It would be interesting to look at your code, I'm just very swamped right now. Regardless of the code though, i wouldn't call it scattering.

I didn't know that nuance between scattering and reflection. Not that I think it's not true, but could you point me out to some reference that addresses the issue?

Regardless of this new point, what about the point, a previous poster, I think saaskis, raised
that the mean free path is a different length scale from the dephasing length?

Maybe I'll go totally astray here (correct me if I am wrong - I don't know a whole lot on this) but if elastic and coherent scattering were indeed impossible, then how would double-slit experiment work?

The electrons are scattering from the slits, right? And if their phase is randomized, how come do they show interference patterns after being scattered?

 

[sokrates]

so if you have a scattering process that imposes random phase shift upon each scattering event - that would lead to decoherence of the wavefunction. now, I'm trying to read up on that phase shift... and see if i can learn whether this is indeed what happens. if you can shed some light on that - i'd be interested to hear about it.

But this is not what you said previously:

Any scattering event would cause decoherence -- the difference between elastic versus inelastic is just a matter of energy transfer

 

[crazy_photon]

But this is not what you said previously:

YES! and i'ms still standing by every word of it (unless i find out phase shift is negligible).

You know what i enjoy (among lots of things in life) is to chat with a smart person, say by the blackboard and reason about things from basic principles, perhaps not knowing exactly the answers but coming up with such during the interaction and exchange of ideas. you know how i feel when i 'talk' to you? like I'm going through molasses that drags me more and more the more i try to reach the goal (which is answering the question raised by original post). perhaps that's not your intention and we just clash on the style differences, i don't know... what i do know that i came to this thread in attempt to learn something i didn't know about localization versus delocalization (on the basic level, which i think i understand and wanter re-confirmation) to perhaps more advanced level where i could gain some knowledge. I'm getting nothing except my every phrase turned back at me as a question.

I asked you to share something interesting about physics of nanostructures (when we were on the topic of boundary conditions) - denied! I tried to reason that ISW can be still 'savlaged' despite its simplicity to recover some real aspects of physics - denied! i asked to share about what books would you suggest reading on condensed matter physics - denied. shall we just quit or are you going to come back with another question on something within this post?

 

[crazy_photon]

I didn't know that nuance between scattering and reflection. Not that I think it's not true, but could you point me out to some reference that addresses the issue?

Regardless of this new point, what about the point, a previous poster, I think saaskis, raised
that the mean free path is a different length scale from the dephasing length?

Maybe I'll go totally astray here (correct me if I am wrong - I don't know a whole lot on this) but if elastic and coherent scattering were indeed impossible, then how would double-slit experiment work?

The electrons are scattering from the slits, right? And if their phase is randomized, how come do they show interference patterns after being scattered?

The terminology of scattering versus diffraction (the reason why you get interference after the slit) is explained in a number of texts. i just checked and beginning of chapter 10 in jackson talks about that (i'm sure there are other places). if by 'coherent scattering' you mean 'diffraction' then we are in agreement. but i never talked about coherent scattering, i only talk about elastic versus inelastic scattering.

as for addressing saaskis point, i must have overlooked it.. I've been busy answering your mirriad of questions :) By the way, what is UCF?

I know that mean-free path is classical concept (back to Drude in our context) while dephasing length is ? the length scale on which coherence is lost? in other words wave-like behavior is not there - in other words - particle-like picture - i.e. back to Drude? Seems like dephasing length is length scale beyond which Drude model would apply. So, they are of the same nature and i would then think of the order of the same length (scale) in the problem. since saaskis is talking about mesoscopic structures, maybe he can share something with us that contributes to this 'everything-goes-solid-state-thread'?

 

[saaskis]

as for addressing saaskis point, i must have overlooked it.. I've been busy answering your mirriad of questions :) By the way, what is UCF?

UCF means universal conductance fluctuations in a mesoscopic structure that is larger than elastic mean free path and smaller than dephasing length. For e.g. B=0 the electrons collide around in the conductor randomly, and we get a conductance that displays how these paths interfere. But when one increases the magnetic field, the paths of the electrons are different and one gets slighly different conductance. So after all, the scattering was not random. This is called the magnetofingerprint of the structure, and it is completely reproducible. These fluctuations are of the order of conductance quantum, as can be shown by e.g. random matrix theory. Dephasing suppresses the fluctuations, or actually it is the ratio of mean free path and dephasing length that matters. (I once gave a small presentation about UCF in the context of random matrix theory :) )

I know that mean-free path is classical concept (back to Drude in our context)

This is wrong, the life-time of quasiparticles can be calculated from first principles using quantum mechanics. But if I remember correctly, it is the transport time and not the actual lifetime that takes the place of mean free path in Drude formula. This is because backscattering suppresses conductivity much more than forward scattering.

in other words wave-like behavior is not there - in other words - particle-like picture - i.e. back to Drude? Seems like dephasing length is length scale beyond which Drude model would apply.

This is not exactly true. The structure can be ballistic even without coherence. I guess that when interference is negligible, one can resort to semiclassical Boltzmann equation.

since saaskis is talking about mesoscopic structures, maybe he can share something with us that contributes to this 'everything-goes-solid-state-thread'?

Well, you are talking about mesoscopic quantities. If we have a macroscopic block of metal at T=300 K, I don't really think there is anything interesting happening. As for the discussion about boundary conditions, I have yet to see how one calculates the bandstructure and energy gap of Si in an infinite potential well.

 

[crazy_photon]

UCF means universal conductance fluctuations in a mesoscopic structure that is larger than elastic mean free path and smaller than dephasing length. For e.g. B=0 the electrons collide around in the conductor randomly, and we get a conductance that displays how these paths interfere. But when one increases the magnetic field, the paths of the electrons are different and one gets slighly different conductance. So after all, the scattering was not random. This is called the magnetofingerprint of the structure, and it is completely reproducible. These fluctuations are of the order of conductance quantum, as can be shown by e.g. random matrix theory. Dephasing suppresses the fluctuations, or actually it is the ratio of mean free path and dephasing length that matters. (I once gave a small presentation about UCF in the context of random matrix theory :) )

This is wrong, the life-time of quasiparticles can be calculated from first principles using quantum mechanics. But if I remember correctly, it is the transport time and not the actual lifetime that takes the place of mean free path in Drude formula. This is because backscattering suppresses conductivity much more than forward scattering.

This is not exactly true. The structure can be ballistic even without coherence. I guess that when interference is negligible, one can resort to semiclassical Boltzmann equation.

Well, you are talking about mesoscopic quantities. If we have a macroscopic block of metal at T=300 K, I don't really think there is anything interesting happening. As for the discussion about boundary conditions, I have yet to see how one calculates the bandstructure and energy gap of Si in an infinite potential well.

the original question asked whether the conduction electrons were localized or delocalized. all the time i have been on this thread i have been thinking about that question (and issues that are around it). now, the corrections that happen on the mesoscopic scale or corrections due to weak localization or other particulars that you guys raise are interesting deviations but these deviations need to be considered on case-by-case basis -- and hence you have to go in detail defining your problem, etc etc. Case in point: weak localization that you mention is applicable when sufficient disorder is present (which was not what was being discussed). Don't get me wrong, I would love to learn more from you on the interesting corrections/additions/coherences that arise in say carbon nanotubes... effect of disorder etc... let me first understand basic metal... or if you do then educate me (and others that might still be reading this messy thread) as whether electrons are localized or delocalized ina metal? If you'd like to discuss particulars, maybe you can start a separate thread, say: 'localization versus delocalization in mesoscopic systems'? do you understand where I'm coming from?

I think there might be a language issue here, so i would ask you: what does it mean to you - localized versus delocalized? There are effects like weak localization, Anderson localization, dynamic localization. Are you implying localization in the context of a trapped excitation? If so, than that's not what i have been talking about (and i think that's not what original post was asking).

I'm thinking about consistent model that recovers both Bloch states and Drude picture in the two extremes. As Landau would say: theory that has a knob(s) on a scale from 0 to 1 that recovers known behaviors in the limits. what is that knob? what is that description? i don't see how we can find these answers by invoking mesoscopic structures, nanotubes, etc...

maybe the answer is in the solid state book, staring right at me and I'm just too stupid to see it? In such case, please point it out.

if you're talking about Si in an ISW -- then both lattice periodicity and ISW boundary conditions have to be taken into account. i think i already discussed that, but i'll just say that once potential length b becomes comparable to interatomic lattice spacing a, you'll start seeing the effect of boundary conditions in the appearance of energy gaps within the silicon 'bulk' like bands.

 

[saaskis]

let me first understand basic metal... or if you do then educate me (and others that might still be reading this messy thread) as whether electrons are localized or delocalized ina metal? If you'd like to discuss particulars, maybe you can start a separate thread, say: 'localization versus delocalization in mesoscopic systems'? do you understand where I'm coming from?

Bloch wave function in a perfect metal is extended throughout the structure, so I would call the states delocalized.

I think there might be a language issue here, so i would ask you: what does it mean to you - localized versus delocalized? There are effects like weak localization, Anderson localization, dynamic localization. Are you implying localization in the context of a trapped excitation? If so, than that's not what i have been talking about (and i think that's not what original post was asking).

I mean Anderson localization, I guess. The localization length depends on the Fermi wave length and the mean free path. In metals, the localization length turns out to be of the order of millimeters, which is much larger than a typical dephasing length. But the Anderson localization length is not that well defined in my opinion, and the size of the electron wave packet can be identified with it only heuristically. I might be wrong here.

I'm thinking about consistent model that recovers both Bloch states and Drude picture in the two extremes. As Landau would say: theory that has a knob(s) on a scale from 0 to 1 that recovers known behaviors in the limits. what is that knob? what is that description? i don't see how we can find these answers by invoking mesoscopic structures, nanotubes, etc...

I think you have misunderstood the term mesoscopic. "Mesos" means "middle", i.e. the borderline between the very small and the very large. Usually we of course mean both the borderline and what happens below it. If you have a perfect metal with full coherence, your length scales are infinite and your structure is mesoscopic, by definition!

Remember that it is all about length scales. Your theory should be able to tackle the whole complex dependence on the relative sizes of Fermi wavelength, elastic mean free path, dephasing length, energy relaxation length and the size of your structure.

if you're talking about Si in an ISW -- then both lattice periodicity and ISW boundary conditions have to be taken into account.

Yes, but the lattice periodicity is quite awkward to take into account in a structure that is not periodic. Take graphene, for example. If you terminate the lattice on zigzag edge, you always have zero energy states at the Dirac point, no matter how large your structure. But in a perfectly periodic structure, the density of states at the Dirac point should be zero.

 

[crazy_photon]

Bloch wave function in a perfect metal is extended throughout the structure, so I would call the states delocalized.

No - i disagree. That's misconception that is why so many people think its triviality ask these questions. Bloch wavefunction is (orthonormal) basis function in which electronic state can be represented - in momentum space and yes indeed - its delocalized. However that doesn't mean that a particular electronic state (which can easily be in superposition of these eigen-states) is also delocalized... that's the whole point of this thread. this has been talked about near the beginnings and mentioned by several people.

Remember that it is all about length scales. Your theory should be able to tackle the whole complex dependence on the relative sizes of Fermi wavelength, elastic mean free path, dephasing length, energy relaxation length and the size of your structure.

That is true, i agree! and i have been trying to talk about length scales and energy scales in the problem in several of my posts. nobody has ever commented on the content of those posts...

Yes, but the lattice periodicity is quite awkward to take into account in a structure that is not periodic. Take graphene, for example. If you terminate the lattice on zigzag edge, you always have zero energy states at the Dirac point, no matter how large your structure. But in a perfectly periodic structure, the density of states at the Dirac point should be zero.

Sorry, not that its not interesting to talk about hexagonal 2D lattices, why is there a need to bring up some specifics again?

Let me define a problem:

we have a perfectly-periodic (no impurity) 1D lattice of scale 'a' and bounding potential of scale 'b'. we have non-interacting electrons (so ignoring elastic scattering here) and electron-phonon scattering (inelastic scattering). we also have a temperature T that describes both electron and phonon distributions (assuming equilibrium). This is a toy model of a solid - true. But adopting such model can we now answer the question: are electrons in localized or delocalized states? And even more interestingly, what aspects of condensed-matter physics such model recovers (we agree that it omits plenty, like nanotubes for instance).

So, as a starting point, can we, within the constraints stated above, come to some agreements, for example:

1) electrons are definitely delocalized because they are described by Bloch states (i'm saying that's wrong, but I'm open for discussion)

2) electrons are definitely localized (in a sense of classical particles, there are no other localizations -- we have perfect lattice without external fields).

3) neither of the above: the relevant energy/length scale is ...

4) the constraints are not sufficient to talk answer the posed question.

Can we 'solve' this problem (which is in essence how i took the original post and therefore found it interesting to participate in this thread) first?

 

[saaskis]

we have a perfectly-periodic (no impurity) 1D lattice of scale 'a' and bounding potential of scale 'b'.

Umm... So your bounding potential is e.g. infinite potential well? But then the problem is not perfectly periodic, right?

And if your bounding potential is periodic, then why introduce a different length scale for lattice? The lattice usually represents the periodicity of the potential landscape, right?

we also have a temperature T that describes both electron and phonon distributions (assuming equilibrium). This is a toy model of a solid - true. But adopting such model can we now answer the question: are electrons in localized or delocalized states?

So is it absolutely necessary to introduce a finite temperature? At T=0, all the eigenstates up to Fermi level are occupied. End of story. At T>0, the states are occupied according to Fermi-Dirac distribution, or more precisely, the density matrix is not simply the pure ground state. The single-particle states are the same as before, in any case. Are you saying that due to T>0, electron wavefunction is smeared in the k-space and therefore it becomes a wave packet and localized? I don't think this makes sense.

1) electrons are definitely delocalized because they are described by Bloch states (i'm saying that's wrong, but I'm open for discussion)

At T=0 all the eigenstates up to Fermi level are occupied. If the problem is translationally invariant, there is no way to say whether the electron is here or there.

2) electrons are definitely localized (in a sense of classical particles, there are no other localizations -- we have perfect lattice without external fields).

It is pure metaphysics to talk about where the electron is, if we know that the wavefunction is extended.