# Band structures with non-primitive cells

How does one interpret band structures (i.e. E,k solution pairs for a given Hamiltonian) for a non-primitive cell? I was looking at these slides: http://www.tcm.phy.cam.ac.uk/castep/CASTEP_talks_06/clark2.pdf

Slide 9 has the "normal" silicon band structure obtained with a primitive cell. Slide 11 uses a cubic cell and obtains a different band structure. Do they somehow contain equivalent information? I know zones fold over, but the non-primitive band structure actually seems to have a direct band gap at Gamma, while Si is known to have an indirect gap. How was this information lost?

Dear, erst
I have some problem like you ...
I've calculated the band structure of si by mathematica.
I could plot non-primitive bandstructure (slide 11). (by psudopotential coefficients)
It is true.

I couldn't calculate the slide 9.

I ussing thease deffinitions for non-primitive ...

pos = {0, 0, 0};
pos = {0, 1/2, 0};
pos = {0, 1/2, 1/2};
pos = {1/2, 1/2, 1/2};
];
pos];

I don't know why the gamma point=(0,0,0) has 4 eigenvalues in the slide 11 in valence bands but 2 eigenvalues in the slide 9. D-:
R={0..5,0.5,0.5} in the slide 11 is the L point in the slide 9.

Sincerely,

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A larger unit cell in real space gives a smaller Brillouin zone in reciprocal space.

If, for example, the real space cell is 4 times larger than the primitive cell, then the Brillouin zone will be 4 times smaller.

Since it must contain the same information and describe the same number of electrons, that means you will get 4 times more bands.
At the BZ boundary, these bands will meet without gap. In fact, the extra band can be obtained by taking the larger BZ and folding the bands inward at the BZ boundary of the smaller BZ.

What is lost are the selection rules, e.g. it looks as if you could have a transition from an "unfolded" to a "folded" band. This is because you loose some information about the periodicity when you go from a small real space cell to a larger, non-primitive one.

Thank you very much M Quack for your post

I could calculate the primitive cell band structure by your guidance.

forgive me because of my poor English