# Band structure

1. Oct 4, 2006

### juan perez

hi,
I am beginning study electronic structure and I do not understand formely what is a band, what mean the lines that appear when a band structure calculation is made?, where are the bands?.

thanks

2. Oct 11, 2006

### juan perez

interpretation bands

Somebody can help me to interpret the graphic of energy versus k , y do not understand the meaning of the lines, each line is a band?.

thnaks

3. Oct 13, 2006

### Staff: Mentor

4. Dec 22, 2006

### flowerew

5. Dec 22, 2006

### Dr Transport

No, nothing wrong, it is accessible.

Band structure is a collective phenomenon. Think about a particle in a well, you have very specific defined energy states. As you add more electrons and more potential wells, there is an interaction between these electrons and the calculated states vary somewhat. Now because electrons are fermions and cannot occupy the same energy except for opposite spin states, the energy states are all altered or more simply smeared, these are the bands you see calculated.

As an exercise, calculate the energies in an infinite well, then calculate the energies in a finite well. They will be differnt. Now add a second finite potential well and find the energies, add a third etc.... For each compare the energies, you should start to see slight variances about a center energy for each level. (This is a computational exercise that will challenge anyone out there but a fun one never the less and I might spend some time doing it for grins.)

6. Jan 13, 2007

### zeta

Think back to quantum 101 when you calculated the electron energy levels for an isolated hydrogen atom. They were exactly that, discrete levels, which your instructor summarily drew on the board, labelled n=0, n=1, etc etc. Everything was nice, the sun was shining, the birds were singing. Your professor, ever the harbinger of doom then said something like 'helium' and then 'interactions'. However you smiled goofily, assuming he referred to a mixer later that day with balloons; you planned to drop by because you were sweet on a girl called mary. A new set of lines appears on the board, labelled n=0, n=1, etc etc. He even labelled the orbits 4p4d etc and then he says it: 'spin'. This is when you notice out of the corner of your eye a rain cloud forming outside and he draws from the nice discrete levels a set of spidery lines. You hear 'spin splitting', he snaps the chalk, you laugh, although now nervously sensing your impending demise under a sea of symbols and myriad forthcoming interactions. You are now detached from time and space, the rain is falling, the mixer is off. With great alacrity the professor cries 'orbital splitting', more lines go up, another interaction that takes us farther from the simple coulombic electron-nucleus. you weep and then to end the lecture he throws up the spin-orbit interaction, a veritable dose of arsenic to your aspirations for an A. Through the sobs and tears, the diabolical laughter of the now triumphant professor, a sea of broken souls around him, you see that what was once a utopian series of single lines is approaching a set of bands.

The epilogue to our story is solid state physics 401 where you find that in reality the hero of our story, the electron has an energy which belongs to a band in a practical material composed of more than a few atoms. You learn about slater, kohn and sham, you look up some papers on PROLA. Despite the fact that you swore group theory was as useful as a paper umbrella in a hurricane, you can see how Mattheis and others used it to simplify a band structure calculation.

I hope my semi autobiographical rant has helped :)

Last edited: Jan 13, 2007