How can I identify TRIM points in the Brillioun zone?

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In summary, the conversation discusses the concept of TRIM (time reversal invariant momentum) points in the Brillioun zone. These points are related by a reciprocal lattice vector and are considered special high symmetry points. They are relevant in the context of topological insulators and are located on the boundary of the BZ where the wave function can change its parity. TRIM points are a subset of the high-symmetry points on the surface of the BZ.
  • #1
dipole
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Hello, I'm trying to find a good reference for how to find or calculate or know which points in the Brillioun zone are "TRIM" (time reversal invariant momentum) points? If anyone is familiar with this topic and could perhaps post a reference or two it would be of great help.

Thanks!
 
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  • #2
At first look time reversal transforms k into -k. For these two points in the BZ to be equivalent, they have to be related by a reciprocal lattice vector G, e.g. -k = k + G or
k = (-)G/2. That would be pretty much all high-symmetry points on the surface of the BZ.
 
  • #3
Thanks for the reply M Quack. I am under the impression that TRIM points are somehow a subset of high symmetry points, that they are "special" high symmetry points.

If it helps at all, TRIM points come up in the context of topological insulators, they are points where the wave function can change its parity I believe.
 
  • #4
Thanks, that is exactly what my quick scan of Google threw up. Unfortunately none of the papers I looked at were very clear on the definition of TRIM, so I ... improvised.

You know that the BZ is limited by planes in reciprocal space half way to the next reciprocal lattice point. Therefore any point k on the line between the origin and a neighbor reciprocal lattice G point AND on the BZ boundary will fulfill k=G/2.

For some directions the BZ boundary intersects before the half way point. Therefore TRIM points are a subset of the high-symmetry points.
 

1. What are TRIM points in Brillouin zone?

TRIM (Time-reversal invariant momentum) points refer to special points in the Brillouin zone, which is a representation of the reciprocal space of a crystal lattice. These points are associated with high symmetry and play an important role in the band structure of materials.

2. How are TRIM points determined in the Brillouin zone?

TRIM points are determined by the symmetry of the crystal lattice. Each crystal system has a specific set of symmetry operations, such as reflections, rotations, and translations, which can be used to identify the TRIM points in the Brillouin zone.

3. What is the significance of TRIM points in Brillouin zone?

TRIM points are significant because they correspond to points of high symmetry in the crystal lattice, where the energy bands are well-defined. These points can be used to simplify calculations and analyze the electronic properties of materials.

4. Can TRIM points be used to study anisotropic materials?

Yes, TRIM points can be used to study anisotropic materials, which have different properties in different directions. By analyzing the electronic structure at TRIM points, scientists can gain insight into the anisotropy of a material and its potential applications.

5. How are TRIM points related to the band structure of materials?

TRIM points play a crucial role in the band structure of materials. They are used to identify the symmetry of the energy bands and determine the direction and magnitude of the energy dispersion. The band structure at TRIM points can provide valuable information about the electronic properties of a material.

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