Trying to understand quantum-mechanical conduction

[FireBones]

Hello all,
I'm trying to put as precise a finger as I can on the quantum-mechanical description of conduction so I could explain it to, say, a curious 8th grader. The goal here is not to be able to perform calculations but give an explanation that does not suffer the errors of the Drude model. Unfortunately, the Wikipedia article for Quantum Conductivity is sorely lacking.

These are the questions I'm particularly interested in answering (at least to myself) before trying to frame an 8th-grader explanation.

I understand that electrons suffer no resistance in a perfect lattice due to diffraction, but that scattering occurs only due to imperfections in the lattice. At room temperature in a wire, these are mostly attributable to thermal vibrations of the ions.
My questions on this topic are:

A. How imperfections cause scattering? Do they disrupt phonons locally? If an electron's wavelength is too large to be scattered by a lattice where the ions are spaced, say 4 Angstroms apart, how does a missing ion or other imperfection leave the electron prone to scattering (after all, the wavelength of a slow electron is greater than 8 Angstroms, so why does a missing ion matter?)

B. What constitutes an electron-phonon interaction? If you had a movie camera and could document what this meant visually, what would it look like?

C. In the quantum-mechanical model, what causes heat dissipation? Perhaps the answers to "A" and "B" above will answer this question. Even in the semi-classical picture, it seems there is a bit of explaining to do since a moving electron interacting with a (moving) ion will not necessarily increase the speed of the ion, especially if the electron is glancing off the ion and the ion was moving toward the electron due to thermal vibrations.

D. What relevance do the vibrational modes induced by the electric field have on the transport of electrons with in the wire? Is it better to say these modes are induced by individual drifting electrons or due to the field itself caused by the charge separation in the wire? Does the fluctuating EM field due to these vibrations modulate and moderate the flow of electrons (like a string of revolving doors that facilitate the flow of electrons at specific speeds)?

E. Is any of the above discussion significantly different when comparing conduction in a normal-width circuit wire versus what is going on inside a filament of a lightbulb?

I'm hopeful the post-ers here will be gentle in trying to explain these things in a way that could be conveyed without resorting to the underlying wave mathematics. I'm writing a book on commonly mis-taught science and I'm really trying to explain this more accurate model in the chapter dealing with circuits, etc.

I'm willing to simply state outright certain things, like that an electron can propagate unhindered through a perfect lattice, without explaining diffraction or wave-particle duality. I just want to pick my battles in this regard and explain anything that is easy to explain.

Thanks so much!

 

[sokrates]

Hello all,
I'm trying to put as precise a finger as I can on the quantum-mechanical description of conduction so I could explain it to, say, a curious 8th grader. The goal here is not to be able to perform calculations but give an explanation that does not suffer the errors of the Drude model. Unfortunately, the Wikipedia article for Quantum Conductivity is sorely lacking.

These are the questions I'm particularly interested in answering (at least to myself) before trying to frame an 8th-grader explanation.

I suppose you are talking about "electronic" transport by description of conduction.

I understand that electrons suffer no resistance in a perfect lattice due to diffraction, but that scattering occurs only due to imperfections in the lattice. At room temperature in a wire, these are mostly attributable to thermal vibrations of the ions.

The reason electrons do not suffer from any resistance in a perfect lattice is not diffraction. The reason is there is no scattering, or more technically, there's nothing that "mixes" states in the Hamiltonian.

A. How imperfections cause scattering? Do they disrupt phonons locally? If an electron's wavelength is too large to be scattered by a lattice where the ions are spaced, say 4 Angstroms apart, how does a missing ion or other imperfection leave the electron prone to scattering (after all, the wavelength of a slow electron is greater than 8 Angstroms, so why does a missing ion matter?)

You seem to be confusing two different scattering mechanisms. You can check #2 in the following thread: https://www.physicsforums.com/showthread.php?t=353902

An imperfection causes a "scattering potential". It puts an "off-diagonal" term to your Hamiltonian, thereby mixing states. Normally, states are decoupled in the Hamiltonian (if you choose the eigen-energies as your basis set) but an imperfection "breaks" this symmetry.
The scattering rates might be high or low, depending on the wavevector of the electron and the scattering potential.

B. What constitutes an electron-phonon interaction? If you had a movie camera and could document what this meant visually, what would it look like?

The exact treatment goes as deep as Quantum Field Theory. In practice, usually a treatment like Fermi's Golden Rule under Born Approximation is used. There's no visual picture. It's a scattering problem. You can check Feynman diagrams at best, but I am told they are not to be taken literally.

C. In the quantum-mechanical model, what causes heat dissipation? Perhaps the answers to "A" and "B" above will answer this question. Even in the semi-classical picture, it seems there is a bit of explaining to do since a moving electron interacting with a (moving) ion will not necessarily increase the speed of the ion, especially if the electron is glancing off the ion and the ion was moving toward the electron due to thermal vibrations.

Inelastic scattering causes heat dissipation. If the electron is not exchanging any energy with its surroundings there will be no dissipation. A physical example I like is a single wall carbon nanotube resistor, where extremely high densities of current flows through the tube, without burning it. This is "proof" that there's no "inelastic" scattering in the tube, limiting heat dissipation.

D. What relevance do the vibrational modes induced by the electric field have on the transport of electrons with in the wire? Is it better to say these modes are induced by individual drifting electrons or due to the field itself caused by the charge separation in the wire? Does the fluctuating EM field due to these vibrations modulate and moderate the flow of electrons (like a string of revolving doors that facilitate the flow of electrons at specific speeds)?

Drifting electrons picture is very different than free electrons picture. I don't quite understand what you mean by the induced vibrational modes, but what I am sure is that they cannot be related to the drift of the electrons.
The drift is an emerging phenomenon out of a very chaotic and collective behavior of an ensemble of electrons, so individual electromagnetic effects would probably be washed out in between. Think of drift as a random walk, while moving 2 microns, you go through 20 scattering events. Think of the "ballistic" picture like a free flight between the contacts, where the device length is on the order of hundreds of nanometers.

Just to make a distinction => Ballistic (resistance-less) Quantum Transport and Drift-Diffusion are two ends. They are the top-down and the bottom-up pictures of current flow.

E. Is any of the above discussion significantly different when comparing conduction in a normal-width circuit wire versus what is going on inside a filament of a lightbulb?

The filament of a light bulb is a prime example of a drift-diffusion picture I outline above, and it has almost nothing to do with Quantum world. It can be very well understood by a Drude treatment. Whereas a 1 micron long carbon nanotube is on the other end of things, where Drude fails miserably, but a quantum transport model is doing a good job.

I'm hopeful the post-ers here will be gentle in trying to explain these things in a way that could be conveyed without resorting to the underlying wave mathematics. I'm writing a book on commonly mis-taught science and I'm really trying to explain this more accurate model in the chapter dealing with circuits, etc.

I'm willing to simply state outright certain things, like that an electron can propagate unhindered through a perfect lattice, without explaining diffraction or wave-particle duality. I just want to pick my battles in this regard and explain anything that is easy to explain.

Thanks so much!

There's a plethora of references and top quality lectures in the following link:
http://nanohub.org/topics/ElectronicsFromTheBottomUp

I also recommend:
http://nanohub.org/resources/6580
and:
http://nanohub.org/courses/cqt
and:
http://nanohub.org/resources/5279

 

[FireBones]

You seem to be confusing two different scattering mechanisms. You can check #2 in the following thread: https://www.physicsforums.com/showthread.php?t=353902

I had read that thread before, and was saddened to see that DrDru had not ever gotten around to expounding more on the "interactions" the electrons had with phonons, which is what I am interested in understanding as those interactions cause most of the resistance in a current-carrying wire.

The reason electrons do not suffer from any resistance in a perfect lattice is not diffraction. The reason is there is no scattering, or more technically, there's nothing that "mixes" states in the Hamiltonian.

Thanks, that sounded odd when I wrote it [after all, it is the scattering that is measured to determine the lattice structure, right?]

Anyways, pursuant to my efforts to give an 8th-grader friendly explanation..., can you explain why a perfect lattice causes no scattering?

Inelastic scattering causes heat dissipation.

But in the context of electrons bouncing off gyrating ions, what does an "inelastic collision" look like? Does it mean some of the energy is transiently absorbed by the ion, perhaps increasing the energies of the bound electrons? I've heard some reference to "vibrating electrons" in these atoms, but I"m not sure what that would even really mean.

In the macroscopic world, I can understanding inelastic collisions more easily, where energy of motion is transferred to microscopic states, so the momenta of the colliding items is conserved but some of the energy of motion is transferred to non-net-directional motion of small particles...but such an understanding seems less obvious when the things being collided are an electron and an ion.

Drifting electrons picture is very different than free electrons picture. I don't quite understand what you mean by the induced vibrational modes, but what I am sure is that they cannot be related to the drift of the electrons.

Perhaps it has nothing to do with anything...but (paring things down to a single drifting electron) if an electron moves a bit, it will tug on the nearby ions. The inter-ionic forces overcome this as the ion moves out of equilibrium. If we think of the ions as attached to one another with springs, any small oscillation will set up a vibration propagating through the material. These vibrations affect the electrical field inside the conductor, and I thought it might affect the flow of the electron itself.

The filament of a light bulb is a prime example of a drift-diffusion picture I outline above, and it has almost nothing to do with Quantum world. It can be very well understood by a Drude treatment.

Why does Drude work well for a filament? Is it due to the increased temperature causing more scattering events?

Thanks for your help and the links. I wish there were some way of visualizing electron-phonon interactions.

 

[sokrates]

Thanks, that sounded odd when I wrote it [after all, it is the scattering that is measured to determine the lattice structure, right?]

Anyways, pursuant to my efforts to give an 8th-grader friendly explanation..., can you explain why a perfect lattice causes no scattering?

Sorry if I was being overly technical. I didn't mean to. I know that one of the bigger challenges of a specialist is to be able to tell things in an 8-th grader friendly way.

Let me try this again, shall I?

To answer your question, let's ask this first: What is scattering? Scattering is a "collision" that disrupts the free flight of an electron and deflects it. If the electron is a small ball moving freely in space (let's resort to the semi-classical picture here), a scattering event is an hindrance in the way, it could be another small ball that collides with the original, it could be a very large balloon (ionized impurity) or it could be a rough surface along the way (surface roughness). Now why does a perfect crystal have no resistance could be answered this way: On the contrary, why would it see any resistance? Because to have resistance, we must put something that makes it harder for the ball to pass through right? If there's nothing like that the ball just moves in a straight line. In technical terms, it evolves 'unitarily'.. with fixed momentum and energy.

The lattice is typically probed by X-Ray diffraction analysis but that's a little different than electronic conduction.

But in the context of electrons bouncing off gyrating ions, what does an "inelastic collision" look like? Does it mean some of the energy is transiently absorbed by the ion, perhaps increasing the energies of the bound electrons? I've heard some reference to "vibrating electrons" in these atoms, but I"m not sure what that would even really mean.

Very good questions.. Yes, some of its energy is instantly absorbed by the lattice if it's an inelastic phonon scattering. I have never heard the term vibrating electrons, could be related. The bound electrons would be typically sitting at much lower energies ( remember we have a conduction band where a soup of electrons are freely moving around and we have these tightly bound electrons that are deep down in energy).. So typically they don't have any interaction with what's going on above..

In the macroscopic world, I can understanding inelastic collisions more easily, where energy of motion is transferred to microscopic states, so the momenta of the colliding items is conserved but some of the energy of motion is transferred to non-net-directional motion of small particles...but such an understanding seems less obvious when the things being collided are an electron and an ion.

Your observations hold true in the microsopic world also. Electrons are scattering off of other electrons, too. And in the Quantum world, phonons also show particle properties, so all you can imagine for billiard balls is analogously happening in the small world, too.

Perhaps it has nothing to do with anything...but (paring things down to a single drifting electron) if an electron moves a bit, it will tug on the nearby ions. The inter-ionic forces overcome this as the ion moves out of equilibrium. If we think of the ions as attached to one another with springs, any small oscillation will set up a vibration propagating through the material. These vibrations affect the electrical field inside the conductor, and I thought it might affect the flow of the electron itself.

I think these vibrations you describe are phonons themselves.

Why does Drude work well for a filament? Is it due to the increased temperature causing more scattering events?

Yes, a filament is probably a few cm's long (much much longer than a mean-free path, the average distance an electron travels before scattering, typically tens of nanometers) so there's lots of scattering. When there's lots of scattering, Ohm's law and Drude model work very well, because they describe the "collective" behavior of these complicated motion of electrons, in a statistical way..

Thanks for your help and the links. I wish there were some way of visualizing electron-phonon interactions.

The momentum is conserved, as well as energy. So resorting to the particle picture of phonons, maybe we can visualize a couple of things. But these analogies should not be taken too seriously.

 

[jambaugh]

Ultimately the dissipation of energy for electrons flowing in a material is going to be coupling between phonon modes (heat) and the electrons' momenta. Remember that even in a perfect crystal phonons alter the spacing. Also due to the potential the electron wavelengths are not constant since they experience the background electric field (inducing the current flow).

In effect you can think of the periodicity of the crystal as creating a conducting region where the electron's wavelength is "in resonance" with the atom spacing. But the region's boundary is "rubbery" due to phonon interaction and the electrons are accelerated into the boundary due to the potential. They hence "bounce" of the boundaries exchanging energy with phonons.

 

[FireBones]

Sokrates, thanks for continued indulgence...

If there's nothing like that the ball just moves in a straight line. In technical terms, it evolves 'unitarily'.. with fixed momentum and energy.

But that "straight line" might have a lattice ion in its way...which puts me back where we started...why do those ions never seem to get in the way in a perfect structure...but do find themselves in the way when the structure is not perfect?

Very good questions.. Yes, some of its energy is instantly absorbed by the lattice if it's an inelastic phonon scattering.

But just because the energy is "absorbed" by the lattice does not mean it is absorbed in a way that we would consider thermal energy. For example, if a pool ball of mass X hits a stationary pool ball of mass X without any spin, the second ball "absorbs" the energy and moves, but it moves straight ahead. The absorbed energy is macroscopic, directed kinetic energy rather than vibrations at the molecular level.

Are you saying that in the case of the lattice, any absorbed kinetic energy always ends up contributing to microscopic energy states because the lattice is bound, so whatever instantaneously-directed energy-of-motion is given to the ion, it will eventually have to vibrate back?

One issue I have with this [but I could see a statistical argument addressing it] is that it is possible for such a collision to slow down the thermally-agitated ion (depending on the angle of striking and direction). Thus, at least in these cases, the vibration of the ion is lessened.

Your observations hold true in the microsopic world also. Electrons are scattering off of other electrons, too. And in the Quantum world, phonons also show particle properties, so all you can imagine for billiard balls is analogously happening in the small world, too.

But my point is that in the macroscopic world there are two sets of elements: the large items that are hitting each other, and separate, smaller entities that can absorb some of the kinetic energy in a non-directionally coherent way (i.e. thermal energy). In the case of electrons smashing into other items, I don't see what these microscopic entities are.

Put another way, if one electron hit another electron, I don't see how such a collision could be inelastic because there are no third-party particles around to absorb the kinetic energy. You have one electron with a particular energy and momentum and another electron with a given energy and momentum...there is nothing else around to vibrate, no ensemble of particles to individually move randomly but with 0 net momentum.

I think these vibrations you describe are phonons themselves.

Yes, that is what I thought...and I was wondering if these vibrations somehow facilitated the motion of the electrons (like the next poster suggested).

Yes, a filament is probably a few cm's long (much much longer than a mean-free path, the average distance an electron travels before scattering, typically tens of nanometers) so there's lots of scattering. When there's lots of scattering, Ohm's law and Drude model work very well, because they describe the "collective" behavior of these complicated motion of electrons, in a statistical way.

But won't there still be far less scattering than Drude would suggest? And would you not still get the same errors [mean-free path too long, resistance going as square root of temperature], etc?

 

[FireBones]

Ultimately the dissipation of energy for electrons flowing in a material is going to be coupling between phonon modes (heat) and the electrons' momenta. Remember that even in a perfect crystal phonons alter the spacing. Also due to the potential the electron wavelengths are not constant since they experience the background electric field (inducing the current flow).

In effect you can think of the periodicity of the crystal as creating a conducting region where the electron's wavelength is "in resonance" with the atom spacing. But the region's boundary is "rubbery" due to phonon interaction and the electrons are accelerated into the boundary due to the potential. They hence "bounce" of the boundaries exchanging energy with phonons.

Jambaugh, YES...this is the sort of picture I was hoping for, where the phonons are considered "in resonance" with the electrons...and this was the reason behind their being able to move unhindered.

Could you please, please elaborate on these things and this picture [pretending you were talking to an 8th grader]...perhaps if you could addressing the specific A,B,...,E I had asked in the OP. The picture you draw is what I was hoping to get at in describing the interactions between electrons and phonons.

 

[sokrates]

But that "straight line" might have a lattice ion in its way...which puts me back where we started...why do those ions never seem to get in the way in a perfect structure...but do find themselves in the way when the structure is not perfect?

Let me be clearer on what I mean by a perfect lattice: A perfect lattice has UNIFORM charge density all over, so if there are ionized impurities they are equally distributed and the electric field the electron sees is uniform spatially.. So there's nothing in the way, for such an electron. An extra charge simply changes the overall field, which is usually called the "mean field" that all the other electron see.

There are highly advanced theories for "disordered" structures where an isolated impurity distorts this uniform distribution and leads to interesting effects. But a crystalline substance is usually ASSUMED to be perfectly uniform.

But just because the energy is "absorbed" by the lattice does not mean it is absorbed in a way that we would consider thermal energy. For example, if a pool ball of mass X hits a stationary pool ball of mass X without any spin, the second ball "absorbs" the energy and moves, but it moves straight ahead. The absorbed energy is macroscopic, directed kinetic energy rather than vibrations at the molecular level.

In the microscopic world also, there are ELASTIC and INELASTIC scattering events. An electron might hit a BALLOON (a charged impurity) and scatter WITHOUT changing the MAGNITUDE of its momentum, this is an elastic scattering event. Whereas in an inelastic collision, the overall energy between the colliders would STILL be conserved but the electron could lose some of its energy by EMITTING a phonon, or gain some energy by ABSORBING it. It is usually much harder to absorb than to emit.

I don't quite see what you mean by macroscopic energy..

Are you saying that in the case of the lattice, any absorbed kinetic energy always ends up contributing to microscopic energy states because the lattice is bound, so whatever instantaneously-directed energy-of-motion is given to the ion, it will eventually have to vibrate back?

Hmm... There are certain degrees of freedom in a highly idealized crystal where scattering takes place. If there is no scattering within the device, the electron does not exchange any energy with the surroundings (this is one of the simplest current flow pictures, sometimes called the "Landauer" picture of current flow). If there are phonons within the energy range of transport, energy exchange with phonons are also possible. If what you mean by an "ION" is a charged stationary object, it usually doesn't have a degree of freedom to vibrate, think of these as big, stationary scatterers, like heavy balloons. But lattice ATOMs are usually NEUTRAL, so they don't have these Coulombic effects, and they can exchange energy with an incoming electron. This interaction between the lattice and the electron is modeled by the phonon model.

Is this making more sense or going the other way?

One issue I have with this [but I could see a statistical argument addressing it] is that it is possible for such a collision to slow down the thermally-agitated ion (depending on the angle of striking and direction). Thus, at least in these cases, the vibration of the ion is lessened.

What exactly are you referring to when you say an ion in the lattice?

But my point is that in the macroscopic world there are two sets of elements: the large items that are hitting each other, and separate, smaller entities that can absorb some of the kinetic energy in a non-directionally coherent way (i.e. thermal energy). In the case of electrons smashing into other items, I don't see what these microscopic entities are.

But momentum and energy is always conserved. And in the microscopic world there are no "unknown" degrees of freedom so EVERYTHING must be accounted for. I always thought that this makes things rather neat, because you know EXACTLY how the energy is distributed for a scattering event. Of course, the scattering amplitudes are treated quantum mechanically, so that the probabilistic nature of things is accounted for.

Put another way, if one electron hit another electron, I don't see how such a collision could be inelastic because there are no third-party particles around to absorb the kinetic energy. You have one electron with a particular energy and momentum and another electron with a given energy and momentum...there is nothing else around to vibrate, no ensemble of particles to individually move randomly but with 0 net momentum.

Usually electron-electron interactions are considered to conserve momentum, for the reason you outline above - i.e, the lost momentum is picked up by the other. But do remember that this holds for the COLLECTIVE picture and NOT for a single electron... A binary collision event between two electrons could easily change their individual momentum vectors, thus energy. But things are usually considered within this collective picture, so e-e interactions usually play a small role in big devices. But very interesting effects can arise when the mean-field picture fails for small structures, where the mere presence of an electron inside the device prohibits one another to jump in. This is called the Coulomb Blockade regime, and is a special, rather delicate transport phenomenon.

Yes, that is what I thought...and I was wondering if these vibrations somehow facilitated the motion of the electrons (like the next poster suggested).

Phonons are distributed according to Bose-Einstein statistics. To facilitate the motion of electrons a phonon must be ABSORBED by an electron, but absorption is usually LESS LIKELY to happen because of the rates of emission vs. absorption that depend on the statistics of phonons.

What jambaugh is referring to is the "Bloch wavefunctions" where the wavefunctions of electrons are indeed in some sort of resonance with the lattice. But due to the really complicated and technical details of this subject matter, people usually assume a FREE environment an electron moves, without seeing the complicated nuclear potential around it. Of course, to MODEL the same phenomena, electron is assigned a DIFFERENT mass, which is usually called the effective mass to first order. It's really hard to talk about Bloch wavefunctions without getting overly mathematical, so an effective mass model, where the electron sees an AVERAGE, MEAN FIELD works well most of the time.

But won't there still be far less scattering than Drude would suggest? And would you not still get the same errors [mean-free path too long, resistance going as square root of temperature], etc?

Why would there by less scattering in an extremely turbulent medium like the light bulb filament than Drude suggests? On the other hand, the extreme rates of scattering is causing the radiation and heat. Doing justice in an extremely complicated problem like that well exceeds the capabilities of modern quantum transport theories. Top down works much better than bottom-up especially in these cases. As I said, Quantum Transport would work for a 100 nanometer carbon nanotube but not for a 2 centimeter light bulb filament.

So my concluison is, the classical conduction theory, where the electrons have an effective mobility (the ratio between the electric field inside the conductor to the average velocity the electrons "drift") works really well in the example of a light bulb filament.

But of course, the filament is probably deliberately fabricated as a "disordered" material, where the nice periodicity of a crystal structure is lost. Other effects might kick in in such cases, but to explain light bulbs, we can assume that the filament is a single crystal highly resistive conductor.

 

[jambaugh]

Could you please, please elaborate on these things and this picture [pretending you were talking to an 8th grader]...perhaps if you could addressing the specific A,B,...,E I had asked in the OP. The picture you draw is what I was hoping to get at in describing the interactions between electrons and phonons.

Let me see... with eighth graders I'd first start with the analogy for conducting bands a description of walking along railroad tracks. You can think of the cross ties as the periodicity of an atomic lattice and then given the electrons are moving via QM as waves they are sort of like a group of fellows trying to walk down a railroad track. If you've ever tried it you have to walk with a certain (usually awkward) stride to just match the spacing of the rails otherwise you find it quite cumbersome. Electrons traveling at higher momenta will be like men with shorter legs trying to match the spacing of the rails. Certain spacings will just match so those momenta will be able to walk freely through the periodic lattice.

A. How imperfections cause scattering? Do they disrupt phonons locally? If an electron's wavelength is too large to be scattered by a lattice where the ions are spaced, say 4 Angstroms apart, how does a missing ion or other imperfection leave the electron prone to scattering (after all, the wavelength of a slow electron is greater than 8 Angstroms, so why does a missing ion matter?)

The imperfections are like gaps in the periodicity of the track rails and a fellow otherwise able to pace along will hit a pint where he steps short stumbling on the rails. With regard to phonons that's a matter of treating the railroad track analogy as if it were a rubber rail with sliding cross-ties. The change in spacing (imperfection) may cause the walking electron to "misstep" so they are pushing on the cross-tie inducing a vibration in the rail. In my earlier post though as I explained it you don't even need initial imperfections. Existent phonons (vibrations of the railroad) will alter the matching of the "walking" electrons and the spacing of the atoms (cross-ties) so there will be random interaction.

As to locality these interactions will actually not be totally localized. The rail walker analog might better be change to a military squad double-timing it along the railroad.

B. What constitutes an electron-phonon interaction? If you had a movie camera and could document what this meant visually, what would it look like?

Picture a sine wave (sine(x)+1) corresponding to a snapshot of a moving electron's wave-function and a cosine wave (cos(Lx)+1) of slightly larger (or smaller) wavelength (L = 1+/- d) corresponding to the periodicity of positive nuclei in an atomic lattice. Remembering that there is some elasticity to the atomic spacing you note that in the region near the origin the peaks of the electron's sine wave are just inside the peaks of the atomic spacing. They thus will want to attract the nuclei toward the center. (or if the spacing is smaller they want to pull the spacing apart). This is how the quantum wave-function of an electron can couple to the contraction-expansion waves of a phonon in the atomic lattice.

First order approximations dealing with quantum electrons moving through a lattice treat the spacing as fixed. However we know this is not truly the case and in fact electron waves can induce forces on the atom spacing which cause phonon waves. Add three dimensions and you can get electrons scattering off the atomic lattice itself producing phonons. (Picture typical Feynman diagram of electron in, electron+phonon out.)

C. In the quantum-mechanical model, what causes heat dissipation? Perhaps the answers to "A" and "B" above will answer this question.

Yes exactly. Coupling to phonons is coupling to quantized lattice vibrations which is exactly what the heat is in the atomic solid in which the electrons are traveling. Heat at the quantum level of a solid is the black-body phonons just as it is black body photons in a vacuum.

D. What relevance do the vibrational modes induced by the electric field have on the transport of electrons with in the wire? Is it better to say these modes are induced by individual drifting electrons or due to the field itself caused by the charge separation in the wire? Does the fluctuating EM field due to these vibrations modulate and moderate the flow of electrons (like a string of revolving doors that facilitate the flow of electrons at specific speeds)?

There is that too but it can get a little material dependent. In a conductor the outer atomic electrons are already of sufficient energy to be in the conducting band (defined by average atomic spacing and momentum wavelength of electrons of a given energy e.g. tallness of walkers along the three-dimensional semi-random railroad track). But the atoms still have inner electrons which are repelled by the moving outer ones just as the nuclei are attracted. There will be some electromagnetic (photon) component to the phonon modes. As well as independent photon coupling. However typically in a conduction situation the photons can't propagate freely and their contribution is taken care of quantitatively in the description of the phonons.

In short don't worry too much about it.

E. Is any of the above discussion significantly different when comparing conduction in a normal-width circuit wire versus what is going on inside a filament of a lightbulb?

I'm not sure how you mean the filament of a light-bulb differs from a "normal-width" circuit wire. The main issue as I see it is that the hotter the conducting material the more randomized over time the atomic spacing and hence the more conducting electrons are likely to scatter off not-quite resonant spacing. (Try walking along the train tracks when they're rubber and somebody keeps jumping up and down on them). You generally see a decrease in conductivity due to this (but you can also see an opposite increase in conductivity with temp in materials there is a sparsity of electrons with enough energy to conduct and the heat increases that number.)

Well I hope these analogies and qualitative points help. There's a great deal I glossed over and just plain left out (and a lot I don't understand myself) but it should give your 8th graders a picture of the context of quantum theory and conduction. If they had a little more math under their belts you could show a lot more of the action by going into the Fourier domain of the atomic lattice. My exposition really doesn't address for example super-conductivity but that's a far more sophisticated quantum phenomenon.

 

[DrDu]

Much has been laid out in detail here, already. Just a few remarks.

To your question A: Even an object much smaller than the wavelength of the electrons can lead to scattering. There is a analogous scattering for light called Mie scattering.
Obviously you cannot resolve the shape of the object that scatters, but at least you can detect its presence.

To your question B: There is a very visual picture for the scattering of electrons from phonons (especially for the scattering from long wave phonons): A phonon will break the translational invariance of the lattice, there being periodical increase and decrease of the positive charge of the ion cores. Then an electron may suffer Bragg reflection from these accumulations of positive and negative charge. The Bragg condition on the wave-length of the electron will ensure momentum conservation in that process.

For short wave phonons, the effect is very much like the scattering from defects. The phonons will create an increase or decrease of charge on the scale of the lattice spacing at which the electron may get scattered.

 

[FireBones]

Sokrates,
Thanks for your continued help. It appears I should have been more explicit as to the situation I was most interested in and the over-arching understanding of that situation.

I am not interested in nano-tubes. I'm looking at the situation of a pedestrian copper wire (or tungsten filament, which I admit may differ). In my understanding, the outer electrons of atoms in such metals form a type of electron cloud about the ions comprising the inner-shell electrons and positive nuclei of copper atoms.

These ions are relatively large and relatively massive compared to the electrons that are free to move under the effect of the electric field generated by large-scale charge-separation in the closed circuit.

Let me be clearer on what I mean by a perfect lattice: A perfect lattice has UNIFORM charge density all over, so if there are ionized impurities they are equally distributed and the electric field the electron sees is uniform spatially.. So there's nothing in the way, for such an electron. An extra charge simply changes the overall field, which is usually called the "mean field" that all the other electron see.

I think this statement was based on an "atom-lattice" view rather than an "ion-lattice" view, right? You are not claiming that the field immediately outside the last filled shell of an ion is the same as the field elsewhere, right? You are not claiming that an electron can go through such an ion due to the field contributions of neighboring ions, right?

In the microscopic world also, there are ELASTIC and INELASTIC scattering events. An electron might hit a BALLOON (a charged impurity) and scatter WITHOUT changing the MAGNITUDE of its momentum, this is an elastic scattering event. Whereas in an inelastic collision, the overall energy between the colliders would STILL be conserved but the electron could lose some of its energy by EMITTING a phonon, or gain some energy by ABSORBING it. It is usually much harder to absorb than to emit.

I don't quite see what you mean by macroscopic energy.

Thanks for the note on electrons emitting photons in these collisions. Is this a case of Bremsstrahlung?

To explain better the rest of my concerns about inelastic collisions and "macroscopic" energy, etc. I see Energy-of-motion as being in two categories: macroscopic, coherent, directed motion (kinetic energy) and incoherent, non-directed, microscopic motion (thermal energy). The "macroscopic" motion has momentum while the microscopic motion of an ensemble has no net momentum (the individual moving particles, of course, have momentum, but there is no coherent direction, so the net momentum of the collection is zero.)

Now, kinetic energy does not have to be "macroscopic." A neutron traveling through free space in a line has kinetic energy and momentum.

Anyways, so my concern was simply that when an electron strikes an ion, I was having trouble seeing where the increase in thermal energy came from. What microscopic particles are going to increase in their incoherent random motion? If the energy lost is in the form of a photon, all well and good...but otherwise it was hard to see (at high temperatures) where non-coherent motion would be increased. At low temperatures, of course, you could say the lattice was not moving much and the electron striking the lattice ion would cause it to move...but at room temperature the lattice ions are already moving at high speeds and, in fact, could actually be slowed down rather than sped up if they were struck at a particular angle.

Why would there by less scattering in an extremely turbulent medium like the light bulb filament than Drude suggests?

Perhaps I should have indicated where I was put onto this path from the very beginning. I was following the description on page 253 of Pillai's Solid state Physics [http://books.google.com/books?id=Af...=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false] and when I speak of the problems with the Drude model I don't speak of problems arising from a lack of sufficient scattering events to allow for statistical analysis, but rather the more organic weaknesses of errors in prediction of mean free path and relation of resistance to temperature, etc.

 

[FireBones]

Thanks, Jambaugh, for this very helpful post! Your railroad tie analogy seems a better version of my rotating door picture [Think of an infinite line of rotating doors set up so that someone leaving one enters the next...as long as they are going a certain speed, they move along without difficulty, but go too fast or too slow and they will find resistance...and then I was thinking of impurities or temperature agitation as one rotating door being out of sync with the others so someone had a bit of a jiggle making the transition from one to another.

But your rubber railroad works better.

There is one thing I was hoping you could expand on and answer some questions about:

Yes exactly. Coupling to phonons is coupling to quantized lattice vibrations which is exactly what the heat is in the atomic solid in which the electrons are traveling.

So, my question here is why these interactions (always/most of the time) increase these vibrations rather than decrease them. I have this lattice of ions jiggling around and an impurity (thermal or otherwise) causes an "interaction" between an electron and a phonon... isn't it the case that this interaction could just as easily cause the lattice to slow down rather than speed up its lattice vibrations?

If the heating of a wire is due to these interactions, then why (in sum) do these interactions tend to increase the quantized lattice vibrations rather than decrease them?

Just to take a dumb, super-simplified, classical snap shot... you have an electron tooling along "in sync" with the vibrations and an ion is a bit out of place from where it should be due to thermal agitation, let's say it is moving a bit too fast toward the electron or it is further out of equilibrium than it should be and the interaction with the electron helps push it back before it gets even further out of its equilbrium and is smacked back by intra-lattice forces, etc. [Sorry if these are bad examples...I'm just saying that a given push or pull to something out of sync seems as likely to slow things down than speed them up.]

Thanks again for your help, guys.

 

[jambaugh]

So, my question here is why these interactions (always/most of the time) increase these vibrations rather than decrease them. I have this lattice of ions jiggling around and an impurity (thermal or otherwise) causes an "interaction" between an electron and a phonon... isn't it the case that this interaction could just as easily cause the lattice to slow down rather than speed up its lattice vibrations?

Exactly right. But this doesn't occur in a coherent fashion. So in the absence of an external field over time there will be a thermal equilibrium with electrons and phonons moving about randomly and randomly exchanging energy. That's thermodynamics. But you then add a the external field trying to move the electron one way. This field will add energy to the system in the form of uniform electron motion some of which then gets randomized into the phonons modes as heat. That's resistance and resistive heating.

If the heating of a wire is due to these interactions, then why (in sum) do these interactions tend to increase the quantized lattice vibrations rather than decrease them?

You now have to quantify the entropy and you'll see that it gets maximized. Remember the key point is the uniform electron motion gets scrambled into randomized electron motion and lattice vibration.

But the reverse effect is how a thermocouple works. Given a heat flux through a conductor the phonons will on average be moving more in one direction and so they will push the electrons in that direction. The coupling is different for different metals so you can get a net potential from heat flowing to or from a junction of dissimilar metals (or even better oppositely doped semiconductors). That's called the Seeback effect.

Dually an electron current can push the phonons in a preferred direction and pump heat along the conductor in the Peltier effect. That's how those small electronic refrigerators work.

See the wikipedia article on thermoelectricity http://en.wikipedia.org/wiki/Thermoelectric_effect" .

 

[FireBones]

Let me be clearer on what I mean by a perfect lattice: A perfect lattice has UNIFORM charge density all over, so if there are ionized impurities they are equally distributed and the electric field the electron sees is uniform spatially.. So there's nothing in the way, for such an electron. An extra charge simply changes the overall field, which is usually called the "mean field" that all the other electron see.

Hi Sokrates, I hope you have time to respond to the question I had about this.

As a short review about this topic [since it has been several months]...

My understanding was that the ions formed a diffraction grating for the free electron wave, but as long as the lattice of ions were truly, perfectly symmetric [and infinite], no resistance was encountered.

You said that diffraction had nothing to do with it, but rather that no resistance occurred because there was no scattering. You later indicated the reason was that there was a uniform charge density, so there was nothing to scatter the electrons.

However, I think you and I were meaning different things by a "perfect lattice." I refer to a perfect lattice of ions, say in a copper wire, and I think you were referring to uncharged atoms.

I have seen several references that perfect ion lattices cause no resistance, and I'm trying to understand what symmetries they have in mind without appealing to the Schrodinger equation.

Those references include "Principles of the solid state" [H.V. Keer], Introductory solid state physics [Myers], "Electrical resistance of metals" [Maeden], and several others. It is often just stated without any explanation that does not resort to Schrondinger.

I'm trying to understand how this could be explained purely by wave interference and geometry, without resorting to band theory or eigenstates.

I'm looking for something like "If an electron is at A and we look at all possible paths where that electron hits an ion, complete destructive interference occurs due to symmetry."

Or, even something following the non-technical presentation of QED by Feynman where we add together the arrows for all paths that hit the various ions and get an arrow of zero length.

 

[FireBones]

Qualitative methods in physical kinetics and hydrodynamics [Krainov] offers this explanation: "An ideal lattice does not offer any resistance to a current of conduction because of the coherence of the scattered electron waves from the ions of an ideal lattice."

This seems like something I could pursue, but wouldn't it be _incoherence_ of those waves that cause them to cancel [that is to say, if you take the set of all waves representing scattered photons, they combine destructively, so no such electrons can be found...]

[Edit] I found something useful here...he says elsewhere "the electrons...interfere with one another when scattered from an ideal lattice such that the rescattered wave is identical to the incident wave." That gives me something to consider at least...anyone want to amplify or flesh that out?