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Why is semiconductor band structure expressed in terms of momentum?

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Jane722
#1
Jan22-12, 12:21 PM
P: 3
I've been having a very difficult time understanding why band structure is expressed in terms of momentum/crystalline directions. I've included a picture of the band structure of silicon so that you can better understand my question.

I think I understand basic crystalline structures and the meaning of crystalline directions, but I don't understand what relevance this has to the band structure.

For example, Si is an indirect semiconductor because its maximum for the valence band is in the (000) direction and its minimum for the conduction band is in the (100) direction... so they don't line up.

But what does this really mean? When light hits silicon and excites an electron into the conduction band, does the electron begin to travel through the material in a different direction than the direction in which light struck the silicon?


I hope my question makes sense to you... Please help me if you can! I'm a chemist taking this type of physics for the first time and I feel lost...

Thank you so much for your help!!!
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band structure silicon.png  
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DrDu
#2
Jan23-12, 02:08 AM
Sci Advisor
P: 3,564
There is a theorem from group theory stating that the irreducible representations of a single electron moving in a periodic band structure are labeled by the crystal momentum k. The theorem is called Blochs theorem; probably you find something in Wikipedia.
This theorem also works in examples familiar from chemistry, like e.g. the benzene molekule. The lowest (pi) orbital has k=0, the next ones (usually labelled as E_{1g} correspond to k=+-1, the E_{2g} to k=+-2 and the highest one again to k=+-3 (there is only one orbital in that case as, in the language of solid state physics, it falls on the zone boundary).

The wavelength of visible (and also of UV) light is quite large in comparison with the typical range of variation of the electronic wavefunctions. This leads to the so called Franck Condon principle which states that the crystal momentum does not change in a direct electronic transition.
In molecular spectroscopy this principle is also used in the form that only electric dipole transitions need to be considered.
In a material with an indirect band gap, the excitation energy for absorption and emission is higher than the minimal energy difference between the ground and excited state. Furthermore, the absorption and emission wavelength will be different as the direct band gaps measured from the top of the valence band and from the bottom of the conduction band are different in general.

Furthermore there are transitions with lower wavelength in the course of which also phonons are emitted. That's also familiar from molecular spectroscopy: In the case of dipole forbidden transitions one observes a Franck Condon progression where the 0-0 vibrational line is missing.
Jane722
#3
Jan23-12, 08:59 AM
P: 3
Thank you for your reply!

I do have a question about your explanation. When you said, "In a material with an indirect band gap, the excitation energy for absorption and emission is higher than the minimal energy difference between the ground and excited state", were you referring to the need for the material to absorb a phonon in addition to a photon in order to excite a transition?

I really appreciate the help. :)

DrDu
#4
Jan23-12, 10:25 AM
Sci Advisor
P: 3,564
Why is semiconductor band structure expressed in terms of momentum?

Quote Quote by Jane722 View Post
Thank you for your reply!

I do have a question about your explanation. When you said, "In a material with an indirect band gap, the excitation energy for absorption and emission is higher than the minimal energy difference between the ground and excited state", were you referring to the need for the material to absorb a phonon in addition to a photon in order to excite a transition?

I really appreciate the help. :)
You are welcome!
I am originally a chemist, too, so that I can understand too well your problems.
To your question: No, I was refering only to the vertical transitions. However, now thinking about it, I cannot think of a simple emission process, so I should only have written that the excitation energy for absorption in a vertical transition is higher than the indirect band gap.



I just read your 1st post again:
"But what does this really mean? When light hits silicon and excites an electron into the conduction band, does the electron begin to travel through the material in a different direction than the direction in which light struck the silicon? "
I am not sure what you really mean here, but
in a vertical transition, the crystal momentum does not change.
The velocity of an electronic wavepacket is not proportional to k but to dE/dk. If both the valence and the conduction band have maximum at k=0, then the velocity of the excited electron will be 0.
In a non-vertical (phonon assisted) transition, also the velocity of the electron can change.
Jane722
#5
Jan25-12, 08:54 AM
P: 3
Thanks for understanding my situation too. I think I was just confusing concepts when I asked that question about the direction of an electron changing, so you can just neglect that statement. Your second post cleared up some more things for me - it was great!


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