Conduction band splitting under spin/orbit coupling

[easytool]

Hi,

Does anyone have an intuitive idea of why it is always the valence bands split under spin/orbit coupling, but not conduction band? (or a much smaller splitting than valence band)

I know through tight-binding calculations, if I plug in numbers correctly, conduction bands always have tiny splitting, but intuitively why conduction band splitting is so much different from valence band?

Thanks!

 

[Amok]

Wild guess here, but wouldn't it have to do with larger angular momentum of the states composing the valence band?

 

[Darwin123]

Hi,

Does anyone have an intuitive idea of why it is always the valence bands split under spin/orbit coupling, but not conduction band? (or a much smaller splitting than valence band)

I know through tight-binding calculations, if I plug in numbers correctly, conduction bands always have tiny splitting, but intuitively why conduction band splitting is so much different from valence band?

Thanks!

Maybe a quasiclassical explanation will suffice.

The large splitting of the valence band is a relativistic effect. It is a little like the precession of an orbit.

It is because the electrons of the valence band are closer to the nucleus then the electrons of the conduction band. In a quasi-classical approximation, the electrons near the nucleus move faster than the electrons farther from the nucleus. Because the electrons are moving faster near the nucleus, the magnetic force on the electrons are greater near the nucleus. The magnetic force on the electrons causes the splitting in energy levels.

Here is a link and quote on this.

http://www.nd.edu/~djena/kdotp.pdf

“ What is spin-orbit interaction? First, we have to understand that it is a purely relativistic effect (which immediately implies there will be a speed of light c somewhere!). Putting it in words, when electrons move around the positively charged nucleus at relativistic speeds, the electric field of the nucleus Lorentz-transforms to a magnetic field seen by the electrons. The transformation is given by <variation on Lorentz force equation>
…
where the approximation is for v << c. To give you an idea, consider a Hydrogen atom - the velocity of electron in the ground state is v ≈ _c where _ = 1/137 is the fine structure constant, and the consequent magnetic field seen by such an electron (rotating at a radius r0 = 0.53°A) from the nucleus is - hold your breath - 12 Tesla! That is a very large field, and should have perceivable effects.

Spin-orbit splitting occurs in the band structure of crystal precisely due to this effect. Specifically, it occurs in semiconductors in the valence band, because the valence electrons are very close to the nucleus, just like electrons around the proton in the hydrogen atom. Furthermore, we can make some predictions about the magnitude of splitting - in general, the splitting should be more for crystals whose constituent atoms have higher atomic number - since the nuclei have more protons, hence more field!”

It is a little like the precession of the orbit of Mercury in GR. The reason the precession is so much larger for mercury is because Mercury is so close to the sun. Mercury moves faster, so relativistic effects become important.

I’ll bet you didn’t expect to see relativity in solid state physics!

 

[Useful nucleus]

Thank you, Darwin123! your answer was useful for me too!

 

[DrDu]

I am not sure whether the statement of the OP is true. Take silicon: The interesting bands at the gamma point are either formed from p or from s orbitals. The valence band is formed from p orbitals with l=1 so that under SO coupling two bands with j=3/2 and j=1/2 are formed. The lowest conduction band is also p so that I would expect a similar if not a larger splitting. However, SO-coupling is small in Si, and the effect isn't important anyhow, and I don't know it's size.
In Ge, where SO coupling is much stronger, the lowest conduction band is formed from s-orbitals with l=0 and only one band with j=1/2 arises. So the lowest conduction band can clearly not be split.

 

[easytool]

Thank you Darwin123 !

A good lesson I learned here: The eigenvectors from tight-binding calculation is as important as eigen-energies ! The latter gives the band structure, while the eigenvectors tell one the orbital composition.