Solid State Curiosity: Infraband Non-Radiative Transitions

[armandowww]

Is a conductive movement of an electron inside a band a particular infraband non-radiative transition?

 

[inha]

Your question is a bit confusing but if I'm reading you right you mean if an electron's motion in a particular band is a non-radiative transition within the band? Then I'd say no since transition is a word reserved for interband transitions.

 

[armandowww]

Your comprehension looks fine. I thought so, me too. But dealing with vibrations you understand other transitions. There are several of them which violate crystal momentum conservation, that is vertical properties of one-photon interband transitions you just referred to.

 

[ZapperZ]

No, inha is correct. Your question IS very confusing, which is why I did not respond in the first place in the hope that you would put some EFFORT into an explanation. Why does conduction requires a transition? The charge carriers are ALREADY in the conduction band, which means these are mobile carriers and have empty states of the SAME ENERGY to moves about throughout the material. So why is there a need for any kind of transition?

Zz.

 

[Gokul43201]

I think Armando's question is one relating to terminology. Maybe he is looking for some equivalent or generalization to the term "gapless excitation" ?

 

[armandowww]

People you leave me speechless. Thanks Gokul, I was talking about this subject on a terminologic level. If an electron moves around within a band do you got some kind a "transition"? Your way of saying gapless excitation seems good: for fixed n (band index) E(k) is a single-valued continuous function, so if k varies you got a gapless "transition". Is it clear?

 

[ZapperZ]

People you leave me speechless. Thanks Gokul, I was talking about this subject on a terminologic level. If an electron moves around within a band do you got some kind a "transition"? Your way of saying gapless excitation seems good: for fixed n (band index) E(k) is a single-valued continuous function, so if k varies you got a gapless "transition". Is it clear?

If that's the case, then you still haven't answered my question. What is the connection between "conduction" and such gapless transition? You solve the band structure for a simple metal with already a built-in assumption of superposition of plane Bloch waves, which means that the charge carriers described by such wavefunction is already non-localized. This is why I don't understand why one needs such a transition to "explain" conduction.

Zz.

 

[armandowww]

dear Zz, I don't need of any transition. I put down this post just for curiosity. My question was confusing but I think I efforted it: when I talk about E(k), I'm already in the speech juice. It's useless your invocation to Bloch Theorem: I took it for granted.
Instead, in the semiclassical theory of conduction, in the classical limits (due to the Ehrenfest Theorem and the correspondence principle) you can deal with electrons like particles in the k-space, so your electronic states are subject to quantum statistics, Pauli exclusion principle... You justify also Ohm law.
So, finally, if you see electrons moving within band, with some energy variations, why don't you talk about transition?
If your answer is no, say it and that shuold be enogh!
Regards

 

[ZapperZ]

dear Zz, I don't need of any transition. I put down this post just for curiosity. My question was confusing but I think I efforted it: when I talk about E(k), I'm already in the speech juice. It's useless your invocation to Bloch Theorem: I took it for granted.
Instead, in the semiclassical theory of conduction, in the classical limits (due to the Ehrenfest Theorem and the correspondence principle) you can deal with electrons like particles in the k-space, so your electronic states are subject to quantum statistics, Pauli exclusion principle... You justify also Ohm law.
So, finally, if you see electrons moving within band, with some energy variations, why don't you talk about transition?
If your answer is no, say it and that shuold be enogh!
Regards

Eh?

Can you show me a text or a source that indicates of "electrons moving within band, with some energy variations"? Maybe if I get to read that, I might have a clearer picture of what you are referring to.

Furthermore, what is even more confusing is why, if you're doing a "semiclassical theory of conduction ... in the classical limit", that you simply don't just use the Boltzmann Transport equation. I don't need to invoke any semiclassical treatment of conduction to get Ohm's Law. I can use the classical Drude model. There are no intra-band transition of any kind here.

I can't say yes or no to something I don't understand.

Zz.

 

[armandowww]

I'm sorry but your point of view is too far by the mine. Maybe we belong to two different schools of thinking. My textbook refers to Ziman, Principles of the theory of solids, 1969.
As ultimate attempt of my explanation: the real meaning of crysal momentum takes place in its variation. If you introduce effective mass, a crystal momentum variation is equal to the electron linear momentum variation that contains effective mass instead of electron mass.
So the dynamical equation in this MY theory can relate a external force, electrical field times the charge or Lorentzian force (or both of them), with these variations.
Now group velocity is zero only in critical points; energy variations are always involved, in conductive subject...
I am on the edge of the embarrassment.

 

[ZapperZ]

I'm sorry but your point of view is too far by the mine. Maybe we belong to two different schools of thinking. My textbook refers to Ziman, Principles of the theory of solids, 1969.
As ultimate attempt of my explanation: the real meaning of crysal momentum takes place in its variation. If you introduce effective mass, a crystal momentum variation is equal to the electron linear momentum variation that contains effective mass instead of electron mass.
So the dynamical equation in this MY theory can relate a external force, electrical field times the charge or Lorentzian force (or both of them), with these variations.
Now group velocity is zero only in critical points; energy variations are always involved, in conductive subject...
I am on the edge of the embarrassment.

Ugh! This is getting worse! Gokul, where are you?!

I know of Ziman's text. I still want to know where is the indication that one needs some kind of a transition, radiative or not, to allow for conduction to occur. Somehow, we have been all over the charts here in trying to get that.

Since you brought up the crystal momentum, look at the disperson curve of a simple "free" electron gas in your "semiclassical" model (use the Sommerfeld model of metals if you want). The E versus k shows parabolic disperson in the conduction band. The band is filled up to the Fermi energy. Let's look at what's going on there. Practically ALL of the conduction property of the metal takes place at and very near the Fermi energy. But these electrons ALREADY have a momentum, the Fermi momentum, k_F. It is already moving with a certain momentum. This was my original point, that in deriving the band structure, you already set it up in such a way that these electrons are not localized. An application of a potential difference can easily redirect these momentum into a particular direction, depending on the crystalographic orientation. Again, no "transition".

If we really insist on tying in "gapless" excitation here, just an application a spread in the occupation number due to scatting with phonons would be sufficient. But these are continuous fluctuation, i.e. electrons continually scatter in an out, creating an equilibrium in the number of states occupied above the Fermi level. But I have the impression that isn't what you're asking for.

Zz.

 

[Modey3]

"People you leave me speechless. Thanks Gokul, I was talking about this subject on a terminologic level. If an electron moves around within a band do you got some kind a "transition"? Your way of saying gapless excitation seems good: for fixed n (band index) E(k) is a single-valued continuous function, so if k varies you got a gapless "transition". Is it clear?"

E(k) may be a single valued contiuous funtions, but the values of k are discrete. Only in large lattices do energy bands develope, which means that there are lots and lots of k-states. Now with that in mind, electons are dynamically switching k-states, but the statical distribution stays the same. I think your idea of "state switching" isn't right. In a lattice, all the electrons occupy the same space (real space), which is the space of the entire lattice. When averaged out over the entire lattice the electron density should be uniform. So for an electron to "move around in a band" doesn't imply conduction because the energy distribution still stays the same.

Modey3

 

[Physics Monkey]

If I may, armandowww, you are talking about the semi-classical equations of motion for Bloch electrons in a lattice, right? Now, in the absence of a magnetic field these equations tell you that a uniform electric field will cause the average crystal momentum of wavepackets to increase linearly with time. In this picture, the wavepacket is "gliding" along the energy band as it smoothly transitions between different k states (here imagined continuous). This sounds to me like the picture that you have in your head, armandowww. This picture is ok, but it is important to remember that you don't actually get the usual conduction laws from a perfect crystal (think here Bloch oscillations). Only after introducing the relaxation time into the semi-classical equations of motion do you get a DC conductivity. This is, I think, what ZapperZ is alluding to. After a certain (very short) time, the crystal momentum of the wavepacket is fixed hence the wavepacket stops gliding along the band. Thus in this simplest picture, DC conductivity is actually associated with the absence of your transitions, rather than with the presence of them.

 

[armandowww]

Modey3... I think the discrete property is not a good motive here. Thanks Physics Monkey, your exposition is my favourite. You have got my comprehension. But I don't understand why Zz thought of my idea as criminal. He is still keeping at thinking I am right and I want so. I'm just searching for learning...

 

[ZapperZ]

Modey3... I think the discrete property is not a good motive here. Thanks Physics Monkey, your exposition is my favourite. You have got my comprehension. But I don't understand why Zz thought of my idea as criminal. He is still keeping at thinking I am right and I want so. I'm just searching for learning...

No, I don't think you are "right", and that is why I am still puzzled by your "question". This is where you have to consider the possiblity that not only am I unable to clearly understand what you're asking, but also that you may not fully comprehend my querries to you.

But since it seems that you could understand everyone else's responses here except mine, then I'm must not be doing something right here, so I'll let everyone else handle the rest of this.

Zz.

 

[armandowww]

I didn't want hurt you Zz. Sorry again. But Physics Monkey point of view has turned on light. What I was missing was 2 basical concepts: wavepacket with k well defined, and vibrations, impurities, imperfections and boundary surfaces, whose role is to break simmetries. Without them, in the k-space (through which I ever thought electrons move) this model reaches to some wrong conclusions like k periodic or constant, or to Pauli exclusion principle violation (never let k be the same for 10^22 fermions!). Viscous terms are as important as basical is this replacement: delta k instead of k!
Now... is here that is possible to find k meaning. That hit me cause makes me really happy. In fact it links up effective mass too!
Maybe is my English to trouble all of you.