Indirect absorption and phonons

[hokhani]

In an indirect absorption process that an electron jumps from the top of the valence band to the bottom of the conduction band,why a phonon must be involved to conserve the momentum? At these two points (top of the valence band and bottom of the conduction band) the electron momentum is zero so the momentum of electron would not change during the process and the momentum is conserved without the need of phonons.

 

[DrDu]

The crystal momentum of the electron is not zero, at least not both at the valence band maximum and the conduction band minimum. Conservation of crystal momentum requires that a phonon get's invoqued.
What is zero at the band extrema is the electronic group velocity.

 

[hokhani]

Conservation of crystal momentum requires that a phonon get's invoqued.

Thanks for the reply. Could you please explain more about Conservation of crystal momentum?

 

[Darwin123]

In an indirect absorption process that an electron jumps from the top of the valence band to the bottom of the conduction band,why a phonon must be involved to conserve the momentum? At these two points (top of the valence band and bottom of the conduction band) the electron momentum is zero so the momentum of electron would not change during the process and the momentum is conserved without the need of phonons.

The photon that was absorbed had nonzero momentum. The phonon is required to absorb the momentum of the initial photon.

 

[hokhani]

The photon that was absorbed had nonzero momentum. The phonon is required to absorb the momentum of the initial photon.

No, the photon momentum is too small (compared with the phonon momentum) to regard.

 

[Darwin123]

No, the photon momentum is too small (compared with the phonon momentum) to regard.

Why does the momentum of the phonon have to be small compared to the momentum of the photon?

The pseudomomentum of a phonon can have any value between 0 and the pseudomoment corresponding to the edge of the Brillouin zone. Indirect absorption often involves optical phonons. It can involve acoustical phonons but I don't think this is what you are talking about.

I'll use the word momentum rather than pseudomomentum in further discussion. The fine difference between the two probably doesn't matter for this topic.

An optical phonon has an energy that varies very little with momentum. Therefore, the energy of the optical phonon is almost constant. The energy of the optical phonon will generally be very small compared to the energy of the initial phonon. This is probably what you are thinking of.

The momentum of an optical phonon is only limited by the boundaries of the Brillouin zone. The momentum of an optical phonon can be much greater than the momentum of the initial phonon. The direction of propagation of the optical phonon can be in any direction.

 

[DrDu]

Thanks for the reply. Could you please explain more about Conservation of crystal momentum?

Better have a look at e.g. Ashcroft and Mermin.

 

[Cthugha]

Why does the momentum of the phonon have to be small compared to the momentum of the photon?

Ehm...are you mixing up something? hokhani claimed that the PHOTON momentum is small compared to the phonon momentum and is therefore typically not relevant, not the other way round.

The photon that was absorbed had nonzero momentum. The phonon is required to absorb the momentum of the initial photon.

That is pretty much never the case in semiconductors. You need really huge photon energies to get a substantial amount of photon momentum. Typically, the phonon will give you the momentum for the indirect transition and the photon will give you most of the energy. It is a bit more difficult if optical selection rules rule out transitions using optical phonons, but I doubt, these details are what hokhani is interested in.

Could you please explain more about Conservation of crystal momentum?

The Cardona/Yu also has a reasonable, but somewhat technical discussion of indirect absorption. For a more pedagogical introduction, Fox's "Optical properties of solids" has a good discussion on indirect absorption and crystal momentum, too. The already mentioned Ashcroft/Mermin has a very good appendix on what crystal momentum or quasimomentum are.

 

[Darwin123]

Ehm...are you mixing up something? hokhani claimed that the PHOTON momentum is small compared to the phonon momentum and is therefore typically not relevant, not the other way round.

He claimed that both the energy and momentum of the phonon were negligible. They can't both be negligible at the same time because of the phonon's dispersion relationship. The functional variation of energy with momentum is often referred to as dispersion.

I deduce that he was talking about optical phonons when he mentioned "indirect absorption." Although both types of phonons can take part in an indirect absorption, only the optical phonon shows itself as a discrete absorption band. This is because the optical phonon has a "minimum energy". The discrete bands in absorption and emission spectra are often referred to as "phonon replicas".

Phonon replicas always refer to features caused by indirect absorption with either optical phonons or local phonons.

The acoustical phonons can broaden the band edge absorption, but can't produce "phonon replicas". The acoustical phonon has no minimum energy. Therefore, their effect on band edge absorption can best be described as a "phonon tail".

That is pretty much never the case in semiconductors. You need really huge photon energies to get a substantial amount of photon momentum.

What you just said is never the case for optical phonons. Comparison of the dispersion functions for acoustical and optical phonons shows that there is a large difference.

An acoustical phonon has an energy that increases with pseudomomentum. An acoustical phonon with zero momentum has zero energy. The energy of the acoustical phonon reaches a maximum at the edge of the Brillouin zone. Hence, a small change in momentum makes a large change in energy.

An optical phonon decreases with pseudomomentum. An optical phonon with zero pseudomomentum has a maximum allowable energy. The energy of the optical phonon is at a minimum at the edge of the Brillouin zone. Hence, what you said is not valid for optical phonons. A small change in momentum makes a small change in energy.

For absorption, photoluminescence and Raman spectra, the optical phonons make the largest difference. However, there is a type of spectroscopy called Brillouin spectroscopy which involves acoustical phonons. Optical phonons don't make a difference in Brillouin spectra. I hypothesized that he wasn't talking about Brillouin spectra.

The boundaries of the Brillouin zone are a pseudomomenta cut-off for both acoustical and optical phonons. The phonon energy of both types of phonons at the edge of the zone are very close together. This energy is far less than the energy of a photon of light. However, the pseudomomentum at the edge of the Brillouin zone is much larger than the momentum of a photon of light.

This is why the energy of the phonon is usually much smaller than the energy of a light photon. However, the momentum of a phonon can be very large! The range of phonon momenta range from 0 to the magnitude of a reciprocal lattice vector. So there will always be a phonon with sufficient momentum to match the momentum of the incident photon.

I think it would be clearer if you look at a graph showing the dispersion relationship of both optical and acoustical phonons. Right now, I am looking at Figure 7 and Figure 8 in:
"Introduction to Solid State Physics" by Charles Kittel 7th ed. (Wiley, 1996) page 105.

 

[Cthugha]

Wait, now I am sure you misunderstand something. Let's see whether we can find out where we talk past each other.

He claimed that both the energy and momentum of the phonon were negligible. They can't both be negligible at the same time because of the phonon's dispersion relationship. The functional variation of energy with momentum is often referred to as dispersion.

The latter is of course correct, but he did not claim that both are negligible. In fact, he did not claim that ANY of these to is negligible for the phonon. He just correctly claimed that photon momentum is negligible and (incorrectly) assumed that a band minimum/maximum implies zero momentum (instead of zero group velocity) and therefore no phonons should be needed. This was the misconception at hand.

What you just said is never the case for optical phonons. Comparison of the dispersion functions for acoustical and optical phonons shows that there is a large difference.

Why do you come up with phonons? In the sentence you quote, I am talking about photons, not phonons. Photons never carry significant momentum in the regimes important in semiconductors. The difference between acoustic and optical (and possibly even transversal and longitudinal) phonons is perfectly clear.

An acoustical phonon has an energy that increases with pseudomomentum. An acoustical phonon with zero momentum has zero energy. The energy of the acoustical phonon reaches a maximum at the edge of the Brillouin zone. Hence, a small change in momentum makes a large change in energy.

An optical phonon decreases with pseudomomentum. An optical phonon with zero pseudomomentum has a maximum allowable energy. The energy of the optical phonon is at a minimum at the edge of the Brillouin zone. Hence, what you said is not valid for optical phonons. A small change in momentum makes a small change in energy.

For absorption, photoluminescence and Raman spectra, the optical phonons make the largest difference. However, there is a type of spectroscopy called Brillouin spectroscopy which involves acoustical phonons. Optical phonons don't make a difference in Brillouin spectra. I hypothesized that he wasn't talking about Brillouin spectra.

The boundaries of the Brillouin zone are a pseudomomenta cut-off for both acoustical and optical phonons. The phonon energy of both types of phonons at the edge of the zone are very close together. This energy is far less than the energy of a photon of light. However, the pseudomomentum at the edge of the Brillouin zone is much larger than the momentum of a photon of light.

Yes, this is trivial. How does that go against what I said? I said in indirect optical transitions most of the energy comes from the photon, most of the momentum comes from the phonon.

This is why the energy of the phonon is usually much smaller than the energy of a light photon. However, the momentum of a phonon can be very large!

This makes photon momentum negligible. This is what hokhani said and you objected to.

The range of phonon momenta range from 0 to the magnitude of a reciprocal lattice vector. So there will always be a phonon with sufficient momentum to match the momentum of the incident photon.

Why should you want to have a phonon with momentum matching that of the incident photon in indirect band gap transitions? You want a phonon matching the momentum gap between valence band maximum and conduction band minimum.

I think it would be clearer if you look at a graph showing the dispersion relationship of both optical and acoustical phonons. Right now, I am looking at Figure 7 and Figure 8 in:
"Introduction to Solid State Physics" by Charles Kittel 7th ed. (Wiley, 1996) page 105.

Thanks, I know the dispersions very well. I work in a semiconductor physics group. ;)

 

[Darwin123]

Wait, now I am sure you misunderstand something. Let's see whether we can find out where we talk past each other.



The latter is of course correct, but he did not claim that both are negligible. In fact, he did not claim that ANY of these to is negligible for the phonon. He just correctly claimed that photon momentum is negligible and (incorrectly) assumed that a band minimum/maximum implies zero momentum (instead of zero group velocity) and therefore no phonons should be needed. This was the misconception at hand.



Why do you come up with phonons? In the sentence you quote, I am talking about photons, not phonons. Photons never carry significant momentum in the regimes important in semiconductors. The difference between acoustic and optical (and possibly even transversal and longitudinal) phonons is perfectly clear.



Yes, this is trivial. How does that go against what I said? I said in indirect optical transitions most of the energy comes from the photon, most of the momentum comes from the phonon.

I don't think that I said that. Maybe I was unclear. Let me try again.

The photon has a small, nonzero momentum. By small, I mean that the photon momentum relative to the center of the band is small compared to the band edge momentum.

When the photon is absorbed, some of its momentum goes into the conduction-band electron, some into the valence-band hole, and some into the phonon. If the photon energy is just slightly above the band gap energy, most of the momentum has to go to the phonon.

At this point, I haven't specified whether the phonon is an acoustical or optical phonon.

This makes photon momentum negligible. This is what hokhani said and you objected to.

I didn't say that the photon momentum was negligible. Hokani said that the photon momentum was negligible. That is why I asked him how he knew that the photon momentum was negligible.

Maybe it is a difference in what we mean by negligible. If you mean that the momentum is so small that it can't be conserved, then I have to disagree. If you mean that the phonon momentum is only a small fraction of the momentum of any excitation at the band edge, then we are in agreement.

The lines in the energy level diagram are vertical because the change in momentum is small compared to the maximum cut-off momentum that borders on the edge of the Brillouin zone. No matter how small a momentum is, it still has to be conserved.



Why should you want to have a phonon with momentum matching that of the incident photon in indirect band gap transitions? You want a phonon matching the momentum gap between valence band maximum and conduction band minimum.[/QUOTE]
Here is the problem.

There is no "momentum gap" between a valence band maximum and a conduction band minimum. The pseudo-momentum of a free-carrier at either extremum is zero. In a semiconductor, there is an "energy gap" between the valence band maximum and the conduction band minimum.

I am thinking of the case where the light wave is precisely resonant with the energy gap. If a light wave is precisely resonant with the band edge, the energy of the photon equals the difference in energy between the conduction band minimum and the valence band maximum. If an electron-hole pair is formed, then the conservation of energy is satisfied. However, the momentum of the electron and the momentum of the hole is zero. The energy of the photon is positive, not zero. If the only excitations that formed were the electron and the hole, then conservation of momentum is not satisfied.

In the case of precise resonance with the energy gap, one way to conserve momentum would be by the simultaneous absorption of a phonon with a momentum that is precisely the negative of the photon momentum. There are other combinations with phonons that could work. However, both the energy and the momentum have to be conserved.

I am not sure what posters hear mean about "group velocity." I conjecture that they are trying to explain a process where the photon energy is slightly above band gap.

The free-carriers don't have much kinetic energy even if the photon energy is slightly above band gap. Kinetic energy in a carrier is proportional to the square of the momentum. Thus, the carriers can't account for much momentum. The phonon has to account for most of the momentum from the initial photon.

The group velocity is proportional to the partial derivative of energy with momentum. I don't want to get into the pre-factor, which you can easily calculate. However, the slope of the energy versus momentum curve is the group velocity. So the energy of a free-carrier band at the extrema is at either a maximum or minimum. If the excess energy goes into the kinetic energy of the free carrier, then very little momentum is going into the free carrier.

The rule of thumb in a direct gap semiconductor is this. The excess energy of the photon goes into the kinetic energy of the free carriers. "Excess energy" meaning energy above band gap. The momentum of the photon goes into a phonon.

I am not sure what posters are saying by "crystal momentum." I suspect that they are talking about the momentum of acoustical phonons. I also suspect that when the OP said phonons, he was thinking specifically about optical phonons.

Look at the dispersion curve of phonons. Where on the dispersion curve is there a low momentum cut off? I see a high momentum cut off. However, plug in some numbers. The high momentum cut off is very high.

A phonon at the Brillouin zone edge has about the same wavelength as the length of a primitive cell. That means that a phonon on the band edge has about the same momentum as an xray photon. So in terms of UV/visible/IR radiation, there is no upper limit to the momentum of a phonon. A phonon can have a momentum up to that that of an xray photon. There can always be a phonon with an momentum that matches that of the original photon.

A phonon at the Brillouin zone has about the same frequency as an far-IR photon. That means that the phonon can not absorb much of the energy from a visible or UV photon. Hence, the phonon can not account for much of the energy of the initial photon. Therefore, there is seldom a phonon with an energy equal to that of the original photon.

A phonon can always account for most of the momentum coming from visible or UV photons. However, a phonon does not always account for most of the energy coming from visible or UV photons.

I feel a ding from the moderators coming on. Honestly, fellows. I know solid state physics! Please don't ban me without a reason. If you think I am wrong, then tell me why.

 

[Cthugha]

I do not want to derail this thread and as I most of all do not want to confuse the poster who opened this thread, this is my final comment on where the misunderstanding might lie. Feel free to pm me, if you want to discuss this further.

Here is the problem.
[...]
There is no "momentum gap" between a valence band maximum and a conduction band minimum. The pseudo-momentum of a free-carrier at either extremum is zero. In a semiconductor, there is an "energy gap" between the valence band maximum and the conduction band minimum.
[...]
The rule of thumb in a direct gap semiconductor is this. The excess energy of the photon goes into the kinetic energy of the free carriers. "Excess energy" meaning energy above band gap. The momentum of the photon goes into a phonon.

Right, this is the rule of thumb for a direct band gap semiconductor and the above is correct for direct band gap materials. However, we are discussing indirect absorption and indirect band gap semiconductors here.

 

[Darwin123]

I do not want to derail this thread and as I most of all do not want to confuse the poster who opened this thread, this is my final comment on where the misunderstanding might lie. Feel free to pm me, if you want to discuss this further.
Right, this is the rule of thumb for a direct band gap semiconductor and the above is correct for direct band gap materials. However, we are discussing indirect absorption and indirect band gap semiconductors here.

Okay, I didn't know that we were discussing indirect band gap semiconductors. There is such a thing as indirect absorption in a direct gap semiconductor.

The OP said, "At these two points (top of the valence band and bottom of the conduction band) the electron momentum is zero". So I interpreted this as referring to a direct gap semiconductor. What I said was consistent with an indirect absorption process in a direct gap semiconductor.

Absorption in an indirect gap semiconductor is from center to edge. The momentum of the resulting conduction-electron is not zero. Therefore, what he said seemed inconsistent with an indirect gap semiconductor.

In the case of most indirect gap semiconductors, the electron jumps from the center of the Brillouin zone to the very edge of the Brillouin zone. This leaves a zero-momentum valence-hole and a conduction-electron with the maximum pseudomomentum. The conduction-electron has a momentum comparable to the momentum of an xray photon.

In the case of an indirect gap semiconductor, the photon momentum is always negligible compared to the phonon momentum.

This case seems self evident. The conduction electron has a lot of momentum. The photon had a very small momentum. Therefore, a phonon has to be emitted or absorbed to make up for the conduction electron.

I suppose there are indirect semiconductors where the valley is in the valence band. In that case, the phonon has to make up for the momentum of the valence-hole rather than the conduction=electron.

Either case:

1) In an indirect gap semiconductor (Si, Ge, etc.), a free-carrier is generated that has a lot of momentum. A phonon is necessary to make up for the momentum of the free-carrier.

2) In a direct gap semiconductor (GaAs, CdSe, etc.), both free-carriers have very little momentum. A phonon is necessary to make up for the momentum of the photon. Oops. I worked mostly with direct gap semiconductors. What I said before was correct for direct gap semiconductors.

The OP should have specified that he meant indirect bandgap semiconductors. Indirect absorption can refer to any photon absorption process where a phonon is also involved.