Are conduction electrons localized in space?

[crazy_photon]

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

Well, thank you for sharing all of your insights. Perhaps you can suggest me some books to start from? BTW, i figured you're doing research on nanotubes/nanowires from your earlier posts - can you share your insights as to how the boundary conditions change the problem there?

Also, when you referred to 'E-k diagram', in physics that's commonly referred to as dispersion relation, just thought I'll point it out in case you'd like to read my post with 'technical explanation' at your leisure.

 

[ZapperZ]

Yes, that nasty, mean-spirited attitude you refer to is certainly part of the culture of professional physicists. I've encountered it before and I don't understand the reason for it. It's certainly nothing to be proud of.

Or maybe you misinterpret someone questioning the rational for your actions as being "ridiculing".

When your funding request has been denied because your feelings were hurt when you are asked to show the reasons why something "... maybe interesting, but is it IMPORTANT?", then come back and tell me that this is due to a ridiculing.

And note that you didn't even address my rebuttal on why I considered my questions on the need for such a model as NOT out of line here. I spent time and effort explaining myself, and all I got was this nasty one-line attack. Have you ever considered that maybe YOU are doing the very same thing that you are criticizing?Zz.

 

[conway]

Or maybe you misinterpret someone questioning the rational for your actions as being "ridiculing".


Zz.

Perhaps. I think I can usually tell, but you never know.

 

[Manchot]

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

I don't think it's a trivial question at all. As I mentioned before, while it is certainly the case that Bloch states are delocalized, the semiclassical Boltzmann transport equation can also be used to model electron densities.

 

[jensa]

Let me try to rephrase, what I interpret part of the question to be, in a different way. We know that for very small samples we enter the mesoscopic regime where, due to boundary effects, energy levels are quantized but more importantly the energy eigenstates are (under ideal conditions) well described by standing waves. This is precisely the situation that has been asked about in this thread. Now we "know" that when the size of the sample increases boundary effects become no longer important and we may consider only the bulk part of the Hamiltonian, the eigenstates of which (provided a single particle picture is still relevant) are plane waves, Bloch waves or other more complicated states (depending on how difficult you want to make the problem). The question now is WHY do the boundaries play a lesser role when the size of the system increases? And what determines the characteristic length scale at which the boundaries become unimportant?

Obviously when the size increases the level splitting becomes smaller eventually leading to a continuum of states. This, however does not explain why the bound states can be effectively replaced by plane waves (or bloch waves). In fact it seems quite counter-intuitive since these states are highly delocalized and thus should "feel" the boundaries no matter how far away they are. As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified. Of course I could be wrong, the fact that the experts did not provide this explanation makes me wonder.@Manchot

You may, or may not, know this but I can say that the dynamical equation which governs the wigner function (wigner transform of the density matrix) reduces to the Boltzmann equation in the classical limit. More generally the dynamical equation which governs the wigner transform of the keldysh part of the single particle GF is, in the quasiclassical limit, known as the quantum Boltzmann equation and also reduces to the classical Boltzmann equation in the classical limit (The scattering integral is obtained from the self energy which of course depends on the particular scattering processes involved). The basis states used in this derivation are plane waves. While this is a technical comment, if you want to find out the relationship between delocalized states and the Boltzmann equation, this is probably where you should look.

 

[ZapperZ]

I don't think it's a trivial question at all. As I mentioned before, while it is certainly the case that Bloch states are delocalized, the semiclassical Boltzmann transport equation can also be used to model electron densities.

Actually, unless I'm forgetting my solid state physics classes, the Boltzmann transport equation is purely classical, very much like the Drude model. They both consider free electron gas acting as classical ideal gas. So the comparison here with Bloch wavefunction isn't entirely kosher.

Zz.

 

[JoAuSc]

As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified. Of course I could be wrong, the fact that the experts did not provide this explanation makes me wonder.

Thank you, I think you've provided the best answer so far. I'll look into what you said to Manchot about the relationship between quantum mechanics and the Boltzmann equation.

Actually, I thought it was quite a good model to choose for the problem posed by the OP: namely, whether electrons in a metal are localized. The simple model of a potential well would seem to potentially offer some useful insights on this question. I haven't seen any of the naysayers offer any explanation as to why it might be inadequate.

Ditto.

Perhaps it would be clearer if I asked a couple of related questions:

1. How can we reconcile the image of electrons as particles bouncing around like billiard balls with a certain average collision time with the image of wavefunctions which are spread across the solid, if the wavefunctions are indeed that delocalized?
2. Are conduction electrons typically in one of the basis states or in a superposition? Assume that the electron-electron interaction is negligible, or if that's a looney assumption please let me know why.

Now my query has no reference to any model whatsoever, and people can stop complaining that it's too simplistic.

(For the record, I checked my Kittel text (Introduction to Solid State Physics, 8th ed.). Chapter 6, on the free electron fermi gas, starts on pg. 133. On pg. 134, he introduces the 1D infinite square well, bounded at 0 and L, and lists the eigenstates and energies, including the fermi energy. On pg. 137, when he starts with three dimensions, he lists the sine wave solutions to the 3D ISW but then says that it's convenient to use periodic boundary conditions and thereafter uses plane wave states. I mention this because many here doubt that the infinite square well could be used to model a solid, for example:

There's no such thing as modeling the problem with an infinite potential well. Dynamics -- and boundary conditions are two different things. These should not be mixed up.

I'm not doubting that the periodic boundary conditions make things pedagogically simpler, as ZapperZ said. Btw, I should note here that in my previous post, I said that only the ISW had a non-zero fermi energy. I was assuming that the plane wave model had boundaries infinitely far away and had a finite number of particles, which in retrospect was a silly assumption.)

A classical delusion of a beginner...

Just because all the zeroth order elementary QM textbooks start with the amusingly simplistic particle in a box problem to show the beginner that the levels will come out to be quantized DOES NOT mean that it is applicable to the HUGELY complicated solid structures where the picture is NOTHING like that "TOY" model.

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

So... wait... my "simplistic" model can't be used for my "trivial" question?

Even if what you say is true, you're being a jerk. You insult me and the person you're replying to numerous times. You assume I'm making "a 'valiant' attempt to come up with an extremely 'versatile' model nobody has thought before", rather than using the simplest model mentioned in my textbook to illustrate a quite general question. You say the model is suitable for the beginning of a QM book for layman (I didn't know laymen knew what the infinite square well was). You say it's at the level of freshmen. You say "if you think you can get away with that, you are deeply mistaken". You assume that what I'm doing is almost an insult to solid state physics, as if one could insult a subject by misunderstanding it.

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

 

[ZapperZ]

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

Using your model:

1. Derive Ohm's Law
2. Derive the temperature dependent resistivity of conductors

Shall I go on?

I don't quite get this fascination with "localization" or "non-localization". This is the only criteria you are going by in which a model for a conductor would be considered to be valid? How about being able to match some observed behavior? When will that come in? Next week?

You will note that I had already asked about this already. We are talking about standard conductors here, in which the properties are VERY well-known. ANY proclaimed model to be considered must (i) show some resemblance to a few observed properties AND (ii) claim superior pedagogical simplicity over the more extensive description (see example I gave about the wave picture).

I am still not quite sure why are barking up this tree.

Zz.

 

[JoAuSc]

I don't quite get this fascination with "localization" or "non-localization". This is the only criteria you are going by in which a model for a conductor would be considered to be valid?

Forget the models. I mentioned the infinite square well model as an example of a model where electrons are delocalized, but that doesn't mean I'm wedded to it. I used it to show people my basis for believing that electrons were spread across the solid. For that purpose, I assumed the ISW model would show that aspect as well as any other, but without the extra details needed for this model to be valid in other ways. Maybe I was wrong to assume those extra details would have nothing to do with whether an electron's wavefunction was spread across the solid or just in a small region, but I don't know. I could've used the plane wave model, but it didn't come to mind.

That having been said, let me restate my question: "Are conduction electrons localized in space?" Feel free to answer based on what you know about real metals or realistic models.

 

[ZapperZ]

Forget the models. I mentioned the infinite square well model as an example of a model where electrons are delocalized, but that doesn't mean I'm wedded to it. I used it to show people my basis for believing that electrons were spread across the solid. For that purpose, I assumed the ISW model would show that aspect as well as any other, but without the extra details needed for this model to be valid in other ways. Maybe I was wrong to assume those extra details would have nothing to do with whether an electron's wavefunction was spread across the solid or just in a small region, but I don't know. I could've used the plane wave model, but it didn't come to mind.

But see, this what I really, really do not understand. If your intention was to show that electrons in metals are delocalized, then what is the problem with looking at Chapter 1 of Ashcroft and Mermin, adopt the free electron plane wave model, and go home? Aren't the simplistic plane-wave solution already show that the electrons are delocalized? It is so easy and so obvious because this is QM 101. Why do we bother with the infinite square well model?

Do you now see why I was utterly puzzled with this? You are going from A to B, not directly, but rather in a rather circuitous manner that I find rather unnecessary. I've tried several times to understand the rational behind wanting to do it this way, I haven't seen any.

That having been said, let me restate my question: "Are conduction electrons localized in space?" Feel free to answer based on what you know about real metals or realistic models.

They are not. Based simply on the bloch wavefunction, one can already see that it can't.

Zz.

 

[JoAuSc]

But see, this what I really, really do not understand. If your intention was to show that electrons in metals are delocalized, then what is the problem with looking at Chapter 1 of Ashcroft and Mermin, adopt the free electron plane wave model, and go home? Aren't the simplistic plane-wave solution already show that the electrons are delocalized? It is so easy and so obvious because this is QM 101. Why do we bother with the infinite square well model?

Do you now see why I was utterly puzzled with this? You are going from A to B, not directly, but rather in a rather circuitous manner that I find rather unnecessary. I've tried several times to understand the rational behind wanting to do it this way, I haven't seen any.

I see why you were puzzled.

They are not. Based simply on the bloch wavefunction, one can already see that it can't.

Zz.

Could you elaborate? Does this assume that each electron exists in a definite energy rather than a superposition of different Bloch waves?

 

[ZapperZ]

Could you elaborate? Does this assume that each electron exists in a definite energy rather than a superposition of different Bloch waves?

Er.. we were talking about nonlocalization, no?

All you need to do is to see if you can find <r>, i.e. the expectation value of the position. As with plane-wave function, you'll end up with the same situation.

Zz.

 

[crazy_photon]

As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified.

This is the same explanation that i gave in my posts, i.e. kT (thermal quanta) in comparison with energy separation.

I can elaborate:

for a given size there's energy separation between the modes allowed by the boundary conditions (if size of the solid is very small, then we have to talk about lattice periodicity instead). So, there's a 'grid' of allowed states defined by the boundary conditions (ISW). If kT overlaps with N of those states, then 'electron' occupies those states and hence (by Fourier transform) starts being localized (in real space) -- this is the high-temperature limit of Boltzmann, Drude, etc... (billiard-ball models). However as we keep lowering the temperature kT decreases until only few or even one of the k states overaps with it - then we are in highly delocalized regime. Another way of saying the same thing: at high temperature electrons lose coherence very fast, however as temperature is lowered, coherent effects start to dominate, i.e. if electron is in well defined k-state its delocalized in real space.

From this also follows that electrons are *not* delocalized in Drude model (as some suggest here). In classical approximations they are treated as billiard balls (localized in space). The confusion with plane wave might be from the fact that its just the basis in which localized state can be represented.

 

[sokrates]

Even if what you say is true, you're being a jerk. You insult me and the person you're replying to numerous times. You assume I'm making "a 'valiant' attempt to come up with an extremely 'versatile' model nobody has thought before", rather than using the simplest model mentioned in my textbook to illustrate a quite general question. You say the model is suitable for the beginning of a QM book for layman (I didn't know laymen knew what the infinite square well was). You say it's at the level of freshmen. You say "if you think you can get away with that, you are deeply mistaken". You assume that what I'm doing is almost an insult to solid state physics, as if one could insult a subject by misunderstanding it.

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

Being insulted is something that is "perceived". I certainly did not attempt to give that impression so I am not going to apologize for being harsh. Oh, by the way, I am not insulted by you calling me a JERK - that's usually how INTELLECTUAL discussions end for the losing side.

Finally, I am very happy that you took some time by YOURSELF, going through Kittel or Wikipedia to sustain your arguments and to fill the gaps in your knowledge instead of perfunctorily stating what you know about a very ADVANCED concept and waiting to be cherished for that. I am sorry but I am not your high school teacher. This is the whole point. Do a little homework before coming up with brilliant ideas. No, I don't want to be "extra" kind because your attitude is wrong. You start by PROPOSITIONS and MODELS instead of very mild suggestions or questions on a topic that you obviously did not spend as much time.

You are mixing up two issues once again, by cutting and pasting my irrelevant posts. Let me clarify what I said, if you care to read it carefully this time:

Electrons in a conduction band are OBVIOUSLY delocalized, that is the whole point of free-electron and almost-free electron models. That part is obvious. [[ Did you honestly know what localization meant before ZapperZ's posts, by the way, I am just asking?]] HOWEVER, the concept of current flow and QUANTUM MECHANICAL models that get you to OHM'S LAW are NOT obvious. Not trivial. Even Ohm's law breaks down when L goes to zero. How does it break down ? What happens at the nanoscale? Have you ever thought of taking L to zero in Ohm's Law? Does the conductor become resistanceless? So if it does, what is the difference between ballistic conductors and superconductors? These questions require a few courses in theoretical physics departments. Can you capture that simply with an infinite square well, or some back-of-the-envelope sanity checks? If you can, please let me know.

If you can't and if what you are GENUINELY interested in is to LEARN, start by saying something like " This is something I DON'T KNOW, can I use a zeroth order such and such model to understand this?" instead of being forceful and clinging to arguments that don't hold. If you think I didn't answer your question in detail, why don't you spend time on clarifying PHYSICS rather than calling people names?

If you do that, I promise you everbody in this forum (including the JERKS) will say
"Look, you can't answer your question with that model, and here's why..."

 

[crazy_photon]

I've been seeing your style of responses and let me try to emulate you a little (just for fun, shall we?).

I think you're the one that needs to go back (waaaaay back) and retake the beginning solid state class where things like Ohms Law (direct consequence of Drude model) stems from the classical description of charges colliding as billiard balls - hence highly localized in space. Even my 9th grader sister knows that! :tongue: You don't have to believe her though, just take some solid-state text that you might have and read it again... perhaps its been too long?

As for shrinking the dimensions to zero - that's when quantum mechanics has to come in (forget your V=IR!) and all things including boundary conditions and lattice periodicity start quantizing available states in k-space leading to for example discontinuous jumps in macroscopic observables.

So, as I have asked once, I'll repeat myself - from your posts you sound like an expert on condensed-matter physics on the nanoscale. So, can you please enlighten us (instead of asking questions back) as to what DOES happen to say resistivity on the nanoscale and from what physical principles does it follow from? I'd like to learn from the expert instead of being called names... again...

 

[sokrates]

I've been seeing your style of responses and let me try to emulate you a little (just for fun, shall we?).

I think you're the one that needs to go back (waaaaay back) and retake the beginning solid state class where things like Ohms Law (direct consequence of Drude model) stems from the classical description of charges colliding as billiard balls - hence highly localized in space. Even my 9th grader sister knows that! :tongue: You don't have to believe her though, just take some solid-state text that you might have and read it again... perhaps its been too long?

Did I say anything that contradicts that? It's interesting that you reached that conclusion. Maybe you think I jumped from kinder-garden to graduate school without seeing the "classical description of billiard balls as a model of resistivity" ( whatever that is )

So, as I have asked once, I'll repeat myself - from your posts you sound like an expert on condensed-matter physics on the nanoscale. So, can you please enlighten us (instead of asking questions back) as to what DOES happen to say resistivity on the nanoscale and from what physical principles does it follow from? I'd like to learn from the expert instead of being called names... again.

I never said I am an expert. I was repeatedly saying that I am a Ph.D student. And I am sorry I just don't feel like spoon-feeding you after a heated debate where people are calling others "jerks" because they can't handle an intellectual discussion. I have extensively posted in this thread and if you care to read what I have posted carefully you'll see plenty of references where you could start learning what happens to resistivity when L goes to zero. I hinted the answers quite a lot of times. If you are truly interested in learning, PM me and I'll help you.

 

[crazy_photon]

From reading your post #49: you say (equivalently) that there are Bloch states in the conduction band and hence electron is delocalized. What you probably should say is that available electronic states (in nearly-free electron model) are spanned by the eigen-states (Bloch states) -- and that DOES NOT necessarily mean that electrons are localized - I talk about that in my posts which I don't think you're reading... (at least skipping the physics part of it). So, one thing I'll grant you is that you know about Bloch states. What's really suprising is that in the same paragraph when you mention Bloch states (maybe without even knowning it?) you say this jem: "models that get you to OHM'S LAW are NOT obvious. Not trivial" - If Bloch states are chapter 3-4 of most solid states books, then Ohms Law (Drude model) is chapter 1 (at least in A&M). So, did you learn it backwards and just haven't gotten to the first chapter yet?

So we have at least two people that say in this thread that the "localized vs delocalized" depends on the temperature ('coherence' is another way of saying it). And there are at least two other people that say that electrons are DEFINITELY delocalized.

I don't see scientific discussion going here, so should we just ask moderators to close this thread? Alternatively, can the proponents of 'definitely delocalized' please state the ground of their objections? There are plenty of scientific arguments from the other point of view.

 

[sokrates]

From reading your post #49: you say (equivalently) that there are Bloch states in the conduction band and hence electron is delocalized. What you probably should say is that available electronic states (in nearly-free electron model) are spanned by the eigen-states (Bloch states) -- and that DOES NOT necessarily mean that electrons are localized - I talk about that in my posts which I don't think you're reading... (at least skipping the physics part of it). So, one thing I'll grant you is that you know about Bloch states. What's really suprising is that in the same paragraph when you mention Bloch states (maybe without even knowning it?) you say this jem: "models that get you to OHM'S LAW are NOT obvious. Not trivial" - If Bloch states are chapter 3-4 of most solid states books, then Ohms Law (Drude model) is chapter 1 (at least in A&M). So, did you learn it backwards and just haven't gotten to the first chapter yet?

I think this is going to be my one of my last posts in this thread. You are totally confused by what I try to convey. Maybe I haven't been clear.

Drude model is a CLASSICAL model. It was proposed in 1900, even before QM was established! Do you consider that the state-of-the-art derivation of OHM's LAW? Since the models your camp has been proposing have been related to Quantum Mechanics (particle in a box, etc..), which are inherently Quantum Mechanical, I was implying that a QUANTUM MECHANICAL DERIVATION of OHM'S LAW is NOT trivial when I said "models that get you to OHM'S LAW are NOT obvious. Not trivial". I edited my post to include that later on.

You can check KUBO formula, Non-equilibrium Green's Function Method (NEGF) etc... for a Quantum Mechanical description of Ohm's law. You'll see that it doesn't come out so easily. And let me tell you this: Temperature is NOT the only factor in the transition from the QM world to the classical world. What about electron-electron interactions that wipe out the off-diagonal elements in the electron density matrix? Strongly correlated systems have those interactions even at very low temperatures.

So a humble suggestion: Wait at least a few minutes and think before you mention Drude Model, Boltzmann Equation, Infinite Square Wells, Localization, and Bloch waves in the SAME POST without explicitly stating (and understanding) what you are talking about.

 

[crazy_photon]

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!

But, without entering into a pissing match, I just ask you again (i think 3rd time):
please answer the question as to why do you think/know that electrons are definitely delocalized? opposite camp offered interpretation in terms of thermal quanta that defines the relevant energy for the problem. can you please bring something to the discussion table instead of stating that its 'OBVIOUS'?

 

[conway]

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]

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]

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]

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]

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]

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]

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...