Calculating Energy Levels for Lattice Using Analytic Continuation?

In summary: Why these are more important?The band structure is more important than the wave vector when it comes to determining the properties of materials. For instance, the band structure can tell you whether a material is a metal, semiconductor, or insulator, and whether it has a direct or indirect gap. Additionally, the band structure can also give you information about the material's properties such as its band gap and its anisotropies.
  • #1
trogvar
2
0
When energy levels for a lattice are constructed the Bloch wave vector is evaluated along the edges of the irruducible zone. Like [itex]\Gamma[/itex] - [itex]X[/itex] - [itex]M[/itex] 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 [itex]\Gamma[/itex] - [itex]X[/itex] - [itex]M[/itex] coordinates)

Thanks in advance
 
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  • #2
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.
 
  • #3
Thank you!

That's quite interesting that we can reconstruct the band structure from only information at [itex]\Gamma[/itex]

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

sam_bell said:
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
 
  • #4
trogvar said:
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.
 
  • #5
trogvar said:
When energy levels for a lattice are constructed the Bloch wave vector is evaluated along the edges of the irruducible zone. Like [itex]\Gamma[/itex] - [itex]X[/itex] - [itex]M[/itex] path for a square lattice.

I wonder why the calculation is NOT performed for values within the zone?
The [itex]\Gamma[/itex]-point is at the zone center, and the [itex]\Gamma-X[/itex] line, for instance, does span a region of reciprocal space "within the zone".
 
  • #6
trogvar said:
That's quite interesting that we can reconstruct the band structure from only information at [itex]\Gamma[/itex]

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.
 

What is an irreducible Brillouin zone?

An irreducible Brillouin zone is a unit cell in reciprocal space that represents the smallest region of the Brillouin zone that contains all of the necessary information about the symmetry and electronic properties of a crystal lattice.

Why is the concept of an irreducible Brillouin zone important in solid state physics?

The irreducible Brillouin zone is important because it simplifies the analysis of the electronic and vibrational properties of a crystal by reducing the calculations to a single, representative unit cell. This allows for a more efficient and accurate description of the crystal's behavior.

How is the irreducible Brillouin zone determined?

The irreducible Brillouin zone is determined by considering all possible symmetry operations and translations of a crystal lattice. The resulting unit cell is then translated to span the entire Brillouin zone, creating a reduced set of points that represent the full symmetry of the crystal.

What is the relationship between the irreducible Brillouin zone and the reciprocal lattice?

The irreducible Brillouin zone is a subset of the reciprocal lattice, which is the Fourier transform of the real space lattice. The reciprocal lattice is used to describe the periodicity of the crystal in reciprocal space, and the irreducible Brillouin zone is a way to simplify the description of this periodicity.

How does the irreducible Brillouin zone affect the band structure of a crystal?

The irreducible Brillouin zone determines the boundary conditions for the electronic wavefunctions in a crystal, which in turn affects the band structure. By considering only a single unit cell in reciprocal space, the band structure can be more accurately calculated and interpreted, leading to a better understanding of the electronic properties of the crystal.

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