Calculating Energy Levels for Lattice Using Analytic Continuation?

[trogvar]

When energy levels for a lattice are constructed the Bloch wave vector is evaluated along the edges of the irruducible zone. Like $\Gamma$ - $X$ - $M$ path for a square lattice.

I wonder why the calculation is NOT performed for values within the zone? And how the energy corresponding to an arbitrary vector within the zone but not laying at the boundary can be obtained from a band-gap diagrams (plotted for example in $\Gamma$ - $X$ - $M$ coordinates)

Thanks in advance

 

[sam_bell]

You're quite free to plot the band structure along any path you like. Much of the time we're only concerned about the behavior at the min and max of the bands, which only requires plotting in a few directions. You might want to know the band gaps, effective masses, and anisotropies near theses extrema. If, for some reason, you're especially concerned about what's happening off the band path, you'll just have to do a calculation of that system yourself (or call authors). Incidentally, there is a theorem that says it's possible to reconstruct the band structure from the band eigenvalues only at the gamma point. From k.p theory I think. You can look it up.

 

[trogvar]

Thank you!

That's quite interesting that we can reconstruct the band structure from only information at $\Gamma$

Still people plot the diagrams in the particular paths along irreducible zones. You say

Much of the time we're only concerned about the behavior at the min and max of the bands, which only requires plotting in a few directions.

Why these are more important?

Thanks again

 

[raul_l]

Why these are more important?

These give you the most elementary information about the material such as whether it is a metal, semiconductor or insulator (determined by band gap, which is the energy difference between the maximum of the valence band and the minimum of the conduction band) and whether it has a direct or indirect gap (determined by whether the minimum and maximum occur at the same k-point). This is what the band structure is mainly used for. If calculated using the density-functional theory (usually the case) it would be risky trying to determine more subtle effects from the band structure due to the inaccuracies of the method.

 

[Gokul43201]

When energy levels for a lattice are constructed the Bloch wave vector is evaluated along the edges of the irruducible zone. Like $\Gamma$ - $X$ - $M$ path for a square lattice.

I wonder why the calculation is NOT performed for values within the zone?

The $\Gamma$-point is at the zone center, and the $\Gamma-X$ line, for instance, does span a region of reciprocal space "within the zone".

 

[Dr Transport]

That's quite interesting that we can reconstruct the band structure from only information at $\Gamma$

The technique is called analytic continuation, and it only works if you have a large number of wave functions at the center of zone. You get a decent approximation if you have 16 or more wave functions and all the coupling constants between them.