Are conduction electrons localized in space?

[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.