# I Electrons and holes

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1. Oct 6, 2017

### Ranku

Holes have an effective mass. How is the mass of holes measured physically?

2. Oct 6, 2017

### ZapperZ

Staff Emeritus
Go back a bit. Tell me if you know how the effective mass of electrons are measured.

Zz.

3. Oct 6, 2017

### Ranku

All I'm familiar with is the formula for effective mass: ħ2/2Ab2 (A is amplitude, b is atomic/lattice spaceing). I guess I'm trying to get a sense of if it might be possible to measure the effectve mass of a hole in a straightforward way, just as we are able to measure the free-space mass charge ratio of the electron from Thomson's apparatus.
I am also wondering if the mass of a hole has an explicit character like the mass of a free-space electron which can collide and scatter.

4. Oct 6, 2017

### ZapperZ

Staff Emeritus
There's a fundamental issue here that you are not understanding.

The "effective mass" of even electrons in a solid is DIFFERENT than the bare mass. This is because of the many-body interactions that are within the solid, and what allows us to define something known as a "quasiparticle". The "electron" that you measure in a solid is an "electron quasiparticle" (quasi-electron), not a bare electron.

Therefore, these quasi-electrons and effective mass do NOT exist in "free space"! So already requiring that we should be able to measure holes in such a situation is meaningless. Holes (or quasi-holes) can, by definition, only exist inside a solid and as part of the many-body interaction or renormalization.

The effective mass is "measured" from the band structure of the material. It corresponds to the second derivative of the band dispersion (the E vs k diagram). Depending on whether you are above or below the Fermi level, the curvature of the dispersion will give you the effective mass of quasi-electrons or quasi-holes. There is no difference between the two as far as the math is concerned.

This is why I asked you about your knowledge of what an effective mass is. I think this might be something you need to look at more closely.

Zz.

5. Oct 6, 2017

### Ranku

I am indeed aware of the distinction between effective mass of an electron in solid mass and the bare mass of an electron, which is why I mentioned 'free-space' electrons.

Anyways, let me now confide the real motivation for asking the question, which I didn't mention in the beginning because it would have been outside the scope of this forum, as well as being entertainingly speculative.

The reason I asked this question is I was exploring a broad analogy between quasi-particles like holes and the nature of dark matter. Suppose dark matter is the absence of matter, analogous to the hole being the absence of electron, and if the absence of matter means that the mass of dark matter cannot be directy detectable, can it still possess a property like charge ?
As you clarified in detail, it is impossible for holes to exist in free space and therefore the question of somehow measuring its bare free-space mass simply doesn't arise. I guess that puts an end to any analogy between holes and dark matter, in terms of free-space dark matter being the lack of matter and yet possessing a property like charge.

6. Oct 6, 2017

### Drakkith

Staff Emeritus
We can't suppose that dark matter is the absence of matter, as the absence of matter is simply a vacuum. It also wouldn't fit observations, as we don't find large amount of dark matter out in the intergalactic voids. In fact we find very little dark matter in these voids.

7. Oct 6, 2017

### Ranku

But what if absence of matter doesn't have to be a vacuum - instead it is simply mass that is not material, say analogous to a purely spacetime object like a black hole, which too isn't material but possesses mass and probably charge (of course the analogy wouldn't extend to the origin of black holes)?

8. Oct 6, 2017

### ZapperZ

Staff Emeritus
And like you said, this is speculative and beyond what is allowed in the forum. You may not want to drag this out anymore than this.

Zz.

9. Oct 6, 2017

### rootone

I have a divide by zero issue with this concept.

10. Oct 12, 2017

### Henryk

Yes, you can measure the effective mass of holes (or electrons) directly. The easiest way is by cyclotron resonance.
You apply fixed magnetic field and measure absorption as a function of frequency. The cyclotron frequency is given by $\omega = \frac {eB} {m_{eff}}$.
The field has to be strong enough so that $\omega \tau >> 1$

11. Oct 14, 2017

### Ranku

Could you direct me to some literature, especially online, that has more experimental details of what you described?

12. Oct 17, 2017

### Henryk

13. Oct 18, 2017