Finding Special K-Points in Cubic Structures - mechdude

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In summary, the conversation discusses the process of finding special k-points for a cubic structure, which involves looking at the Brillouin zone and using high-symmetry points and axes. These points have historical names and can be found by generating a regular grid and applying symmetry operations to find the irreducible special k-points.
  • #1
Mechdude
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Hi
im looking at a book and for a cubic structure they have stated about 20 special k-points are found, does anyone have a tutorial like resource for how this is done? or can anyone offer some insight?
mechdude.
ps see attached for what I am referring to.
 

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  • #2
This is not completely clear. How do you get from an unspecified grid (presumably regular) to high-symmetry points?

Usually you look at the Brillouin zone and then look for high-symmetry axes and high-symmetry points on the surface of the BZ. This was of course done decades ago, and all such points and axes carry names (historically). You have to look up books and tables to find the proper nomenclature for your system.
 
  • #3
They are probably referring to so-called "Monkhorst-Pack" special k-points. The regular 4x4x4 grid is first generated from the reciprocal lattice vectors as: (n1/4) b1 + (n2/4) b2 + (n3/4) b3 for ni = 0,1,2,3. Any points outside the 1st BZ are translated back in. Then equivalent points are found by applying all the symmetry operations of the corresponding crystal --48 for a perfect cube, less for the strained lattice. The remaining inequivalent points are the irreducible special k-points.
 

1. What is the purpose of finding special K-points in cubic structures?

Finding special K-points in cubic structures is important in solid state physics and materials science as it helps to accurately describe the electronic band structure of a material. This information is essential in understanding the material's electronic properties and behavior.

2. How are special K-points determined in cubic structures?

Special K-points are typically determined through a process called "Brillouin zone sampling". This involves dividing the Brillouin zone, which is a mathematical representation of the electronic structure of a material, into smaller sections and finding the K-points within those sections that accurately represent the overall electronic behavior of the material. This process is often done using computational methods.

3. What are the benefits of using special K-points in cubic structures?

Using special K-points in cubic structures allows for a more efficient and accurate description of the electronic band structure of a material. It also helps to reduce computational time and resources, as only a small number of K-points need to be calculated rather than the entire Brillouin zone.

4. Are all cubic structures the same when it comes to finding special K-points?

No, not all cubic structures are the same in terms of finding special K-points. The specific lattice parameters and symmetry of the cubic structure can affect the location and number of special K-points. Therefore, it is important to take into account the specific structure when determining the special K-points.

5. Can special K-points be used in other types of crystal structures?

Yes, special K-points can also be used in other types of crystal structures, such as hexagonal or tetragonal. However, the method for determining the special K-points may differ depending on the specific crystal structure. It is important to consult the literature or use computational methods to accurately determine special K-points in non-cubic structures.

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