Band theory of Conduction, what constititue resistance?

[I_am_learning]

In the band theory it is said that, in conductors, conduction bands are only half-filled. So, when Electric filed is applied, the electrons can easiliy jump into higher energy states of the same band and move freely.
If the electrons a so free to move, what constitute the resistance of the conductor?

 

[sokrates]

One line answer to your question is: Interaction with environment (lattice, other electrons, impurities, defects, etc...)

There are two main mechanisms that model the interaction of electrons with its environment:

1) Impurity ion scattering (if you have impurities or defects in the lattice)
2) Lattice scattering

Electrons interact with the lattice atoms while propagating through the lattice. With increasing temperature, the lattice jiggles even more wildly so this interaction (electron-phonon) becomes more pronounced.

Intuitively, you could think of electrons colliding with the lattice atoms and losing momentum in the process. This loss of momentum (not necessarily loss of energy) is the cause of resistance.

 

[I_am_learning]

If electrons are disturbed by the lattice atoms, they can't be called free!
in the quantum-mechanical view point, their wave-functions extends over the entire length of the conductor; We don't read about the lattice ions wave-function interfering with the electrons wave function!

This explanation of collision with lattice atoms, seems to be just like a poor method to get-way with the incomplete band-theory!
Of-course, I may be wrong, but I am wanting to hear a better explanation.

 

[sokrates]

If electrons are disturbed by the lattice atoms, they can't be called free!

Who says so? Are you saying delocalized electrons never feel any resistance? ...

in the quantum-mechanical view point, their wave-functions extends over the entire length of the conductor; We don't read about the lattice ions wave function interfering with the electrons wave function!

Lattice ions wave function?... Even in ab-initio methods, ions are treated as classical entities. So, no, you won't read about the lattice ions wave-function interfering with the electrons wave function even in a Quantum Transport context.

This explanation of collision with lattice atoms, ...

Well, this is not just an explanation. The "interaction" with the lattice IS the primary reason for resistance. As I said: No interaction, no resistance.

... to get-way with the incomplete band-theory

Incomplete band-theory? Do you mean to say free-electron model or something?

Of-course, I may be wrong, but I am wanting to hear a better explanation.

Then ask more relevant questions!

 

[carbon9]

Hi,

About the resistance of nanodevice, you can look at this site. I think you can find an explanation: http://nanohub.org/resources/2041/

Cheers

 

[DrDu]

To address the objections of "thecrictic" and "sokrates": Instead of collision with the lattice atoms I would speak of interaction with the lattice vibrations (phonons).

 

[sokrates]

To address the objections of "thecrictic" and "sokrates": Instead of collision with the lattice atoms I would speak of interaction with the lattice vibrations (phonons).

This is a semantic issue. The treatment of electron-phonon interaction conserves momentum (and energy if the collision is elastic) just like classical billiard balls.

The mathematics is the same. But to avoid imagining a "mechnanical" collision, some authors prefer the word "interaction".

 

[DrDu]

That is not quite true, as far as I know. The electron phonon interaction conserves crystal momentum k, but in general not the true momentum p. In so called Umklapp processes, momentum can be transferred to the crystal. However, at very low temperatures, this mechanism becomes inefficient. If the crystal is very pure, scattering at inhomogeneities is also unimportant. One then has to take into account scattering at the boundaries of the crystal.

 

[sokrates]

That is not quite true, as far as I know. The electron phonon interaction conserves crystal momentum k, but in general not the true momentum p. In so called Umklapp processes, momentum can be transferred to the crystal. However, at very low temperatures, this mechanism becomes inefficient. If the crystal is very pure, scattering at inhomogeneities is also unimportant. One then has to take into account scattering at the boundaries of the crystal.

Momentum in this context, is almost ALWAYS, the crystal momentum.

In an effective mass description, one NEVER knows the true momentum anyway... Crystal momentum doesn't mean the momentum of the lattice. It is the effective momentum of a propagating electron as if the electron is traveling through vacuum, instead of the very complicated nuclear potential landscape.

Surely, momentum can be transferred to the crystal (to phonons), just like a moving billiard ball's momentum can be transferred to a stationary billiard ball, after a binary scattering event.

I don't know why you bring up the Umklapp processes and how they are relevant. The original question is much more elementary.

 

[DrDu]

First again to my post #7. I didn't want to make a semantic distinction between collisisons and interactions but between the interaction with the (periodic) lattice on one side and with phonons on the other.

Refering to your answer #9, I disagree that momentum in this context is always the crystal momentum. The electric current is, cum grano salis, the total momentum and not the crystal momentum. Hence the effect of resistance is a decrease of total momentum while crystal momentum is strictly conserved, at least in an infinite lattice.
There exists some theorem due to Peierls stating that resistance due to electron phonon scattering can only arise due to Umklapp scattering.

 

[sokrates]

U-scattering mainly reduces thermal conductivity.

Resistance can arise due to many other e-p interactions.

ADP, ODP, POP, etc...

I'd like to see the theorem you are mentioning, could you give the name of the book?

 

[DrDu]

There's a nice interview of Hans Bethe by Mermin, where Bethe also mentions the theorem which stems from a time when he and Peierls where both doctorands with Sommerfeld:
http://www.ias.ac.in/resonance/Oct2005/
I suppose it can also be found in the book by Peierls on solid state physics.
It think to have it seen to be discussed at length in "electrons and phonons" by Ziman.
Basically, the phonons can only come in equilibrium with the lattice due to U processes. If there were no U processes, both the electrons and the phonons would flow with respect to the lattice, so that no resistance would arise.

 

[sokrates]

If there were no U processes, both the electrons and the phonons would flow with respect to the lattice, so that no resistance would arise.

Not quite true.
First of all, apart from any phonon discussion : Phonons are not the only sources of resistance, what about impurity ions? disorder? other carriers? surface roughness?
And remember there's also N-processes for phonon absorption (Check Kittel, for instance). So I am not at all sure whether the U-processes are exclusively responsible for phonon absorption. This doesn't sound right.edit:(I'll check the books and clarify it -- I am just very busy at the moment, thanks for the links!)/

 

[DrDu]

During the weekend, I found the book of Ziman at home on some dusty shelf and had a look at it. The subtitle says something about transport processes what makes it highly relevant on that subject. Apparently, in my post #10 I had mixed up some things quite a lot, so better forget abount everything but the Peierls theorem about what I wrote there.
Historically, the first calculation of the electrical resistance of a metal due to electron phonon scattering was due to Bloch. He simply assumed that the phonons are in equilibrium with the crystal. It was then pointed out by Peierls, that Umklapp processes are necessary to achieve this. As you remarked this does not mean that a decrease of electric current is not possible if only N-processes are possible. However, this decrease cannot be complete. Its like fireing bullets into a tube with water. When entering the water, the bullets will loose almost all of their velocity. However, if the friction between the water and the tube vanishes, then the water will start to move slowly and the bullets and the water will move slowly together without coming to a halt.

 

[PhaseShifter]

Not quite true.
First of all, apart from any phonon discussion : Phonons are not the only sources of resistance, what about impurity ions? disorder? other carriers? surface roughness?

Those should be rather obvious: The band structure is dependent on the periodicity of the lattice potential.

• Crystal defects such as vacancies, impurities,etc. introduce a non-periodic potential. These will scatter traveling electrons in a manner similar to to Compton scattering. The momentum gained by the defect will then be transferred through the lattice as phonons.
• Phonons themselves form a periodic perturbation of the lattice potential. Even this would be expected to alter the band structure (producing smaller sub-bands), but this is also a time-dependent perturbation, capable of transferring energy to and from electrons.