Are conduction electrons localized in space?


by JoAuSc
Tags: conduction, electrons, localized, space
conway
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#55
May30-09, 05:51 AM
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The delocalization property as seen in the potential well model is significant in helping to explain the photoelectric effect. Since the electron is spread over the whole volume of the metal, it can interact with incoming radiation over a very large cross-section. There is no need to postulate that the e-m energy is concentrated in little "clumps" called photons.

Yes, the delocalization of the electrons is important. But if it's such an obvious property of metals, then why do so many textbooks continue to use the ATOMIC cross-section in calculating the "expected" minimum time for photo-emission to occur? This incorrect argument is frequently used as a kind of "nail-in-the-coffin" clincher to prove that light must be a particle.

I
ZapperZ
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#56
May30-09, 06:33 AM
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Quote Quote by conway View Post
The delocalization property as seen in the potential well model is significant in helping to explain the photoelectric effect. Since the electron is spread over the whole volume of the metal, it can interact with incoming radiation over a very large cross-section. There is no need to postulate that the e-m energy is concentrated in little "clumps" called photons.

Yes, the delocalization of the electrons is important. But if it's such an obvious property of metals, then why do so many textbooks continue to use the ATOMIC cross-section in calculating the "expected" minimum time for photo-emission to occur? This incorrect argument is frequently used as a kind of "nail-in-the-coffin" clincher to prove that light must be a particle.

I
Er... this is completely OFF TOPIC. "Clumps" of energy can ALSO be delocalized, because it has nothing to do with such quanta having a particular location! A photon was never defined with definite size!

I suggest you create another thread to voice your disagreement with the photon picture. Or better yet, do a search on here and see all the tons of discussion that had been done on this already.

Zz.
sokrates
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#57
May30-09, 08:29 AM
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Quote Quote by crazy_photon View Post
Kubo linear response gives you Ohms law - but there's no quantization in that. Why do you bring up non-equilibrium greens functions? for a linear response? I don't get you, sorry!?

What are you talking about? What do you mean there's no quantization in Kubo formula?

NEGF and Kubo are the precious few quantum mechanical models that can go to Ohm's Law. People need to remember this very simple fact. A strictly quantum mechanical transport theory is VERY DIFFICULT, no matter what route you choose. This is where I started from in the discussion. Don't get lost in details. This is the bottom-line and it's enough.

I mentioned NEGF because it is another formalism that gives you Ohm's law starting from FIRST PRINCIPLES.

So that's the idea, get it? Derive Ohm's law from first principles. Not from billiard balls, or the Drude formula. Separate problems, almost completely independent topics.

Why do you bring up linear response? Off-topic, NEGF can handle non-linear (high-bias) systems as well. And interesting, you have heard of Kubo (yet you are confused with its roots) but you have never heard of NEGF.

This has gone completely off-topic and you and me have come to a point that we are not contributing anything. I am tired of entangling what you say, because it usually comes as a mess of highly theoretical concepts and you are confusing people who may be following the discussion.

Since there's already plenty of posts in this thread that give answers to your final questions, I am stopping to pollute the forum with this. And hey, don't take it seriously, calm down alright? : ) There's no table, no matches and challenges, it's okay! Believe it or not, my purpose is none other than learning or sharing.
conway
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#58
May30-09, 09:20 AM
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Quote Quote by ZapperZ View Post
Er... this is completely OFF TOPIC. "Clumps" of energy can ALSO be delocalized, because it has nothing to do with such quanta having a particular location! A photon was never defined with definite size!

I suggest you create another thread to voice your disagreement with the photon picture. Or better yet, do a search on here and see all the tons of discussion that had been done on this already.

Zz.
I'm just saying the potential well model is good for some things.
saaskis
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#59
May30-09, 09:55 AM
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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 :)
ZapperZ
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May30-09, 10:06 AM
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Quote Quote by conway View Post
I'm just saying the potential well model is good for some things.
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? And how this somehow connects to photons not being "clumps" of energy is completely lost on me.

Just for your info, I did my postdoc in photoemission spectroscopy. This is not meant to impress, but simply as baseline info that this is the area of my expertise. If you look at the spectral function of a metallic quasiparticle, you see no such boundary condition. See, for example, T. Valla et al. Phys. Rev. Lett. 83, 2085 (1999).

Zz.
crazy_photon
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#61
May30-09, 12:00 PM
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Quote Quote by saaskis View Post
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
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#62
May30-09, 12:20 PM
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Quote Quote by crazy_photon View Post
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.
Quote Quote by crazy_photon View Post
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.
Quote Quote by crazy_photon View Post
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.
Quote Quote by crazy_photon View Post
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
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#63
May30-09, 02:20 PM
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Quote Quote by ZapperZ View Post
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
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#64
May30-09, 04:34 PM
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Quote Quote by conway View Post
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
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#65
May30-09, 04:54 PM
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Quote Quote by ZapperZ View Post

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
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#66
May30-09, 07:10 PM
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Quote Quote by conway View Post
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
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#67
May30-09, 07:13 PM
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Quote Quote by crazy_photon View Post
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
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#68
May30-09, 08:16 PM
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Quote Quote by ZapperZ View Post
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
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#69
May30-09, 08:24 PM
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Quote Quote by sokrates View Post
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
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#70
May30-09, 09:12 PM
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Quote Quote by jensa View Post
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
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#71
May30-09, 09:26 PM
P: 34
Quote Quote by jensa View Post
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
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#72
May30-09, 09:34 PM
P: 480
Quote Quote by crazy_photon View Post
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?


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