I'll post a video here later today. This structure provides a perfect example of both lattice structure and band calculations*. If you understand how this structure is characterized, mathematically, then I think you can understand all crystal structures. Additionally, if you understand to some degree the band (E vs k) calculation for this structure, and why there is a 2x2 matrix, then I think you will have the potential to understand a wide range of band calculations.
*Band calculation means how you get from an atomic "orbital" to a periodic crystal eigenstate, and the energies of those crystal eigenstates, E(k), relative to that of the originating atomic orbital.
Here is part 2. This completes the calculation and also discusses the concept of Brillouin Zone, which is very important! (Why do we invoke a special zone? What is the reason?)
I'm a little confused about why you notated your lattice points the way you did in the first video. Why is the yellow dot n=-1 m=1 and the blue point n=-1 m= 0. I would think that those two point should have equal and opposite m values seeing as they are the same distance away from the (0,0) point.
ReplyDeleteI see what you mean, but think about this: what are the Bravais lattice generating vectors?
Delete(I think the 00, 0-1 and 1-1 set makes an equilateral triangle, and 1,-1 is not farther away.)
I'm thinking that in your second video you might have made a mistake regarding your Eigen vectors.I think your Eigen vectors for you 2x2 matrix with E and gamma are upside down. I'm not sure if that matters but shouldn't the i be on top?
ReplyDeleteI am not sure if this is relevant to your question, but you can multiply any eigenvector by any complex scalar and it is still an eigenvector belonging to the same eigenvalue.
Deleteor, I may have made a mistake there?
Delete