How to use the momentum matrix of the reduced k-points

The solution involves finding the irreducible representations formed by <j|, p and |i> and combining them to ensure total symmetry of the matrix element <j|p|i>. References for this topic are also requested.
  • #1
sandf
21
0
I have a set of k-points, e.g. k1,k2,k3,k4. and they are equivalent by symmetry.

Now I have calculated the momentum matrix element <i|p|j> at k1 point ONLY,

and then calculate the optical properties which, for example, depend on <i|p|j><j|p|i>

I have to make a summation on four k-points (k1,k2,k3,k4) for dk integration.

So, how to make this summation only on the basis of <i|p|j> of k1 point?

Can I simply do by multiplying <i|p|j> by the weight of k1 point? because these four k-points are equivalent.

If not, could you give me some references?Any help will be appreciated.

Youzhao Lan
 
Last edited:
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  • #2
I am no specialist on this, but I think that the p and |i> span a representation of the point group which is also responsible for the equivalence of the k points. So you have to find the irreducible representations spanned by p and |i> and find those combinations of irreducible representations formed from <j|, p and |i> that the total matrix element <j|p|i> is totally symmetric.
 
  • #3
Dr.Du, thanks for your help.

Lan
 

1. What is the momentum matrix of the reduced k-points?

The momentum matrix of the reduced k-points is a mathematical representation of the momentum space, which is the space where the momentum of a particle is described. It is used in solid state physics to understand the behavior of electrons in a crystal lattice.

2. How is the momentum matrix of the reduced k-points used?

The momentum matrix of the reduced k-points is used to calculate the energy levels of electrons in a crystal lattice, which is important in understanding the electronic properties of materials. It is also used in band structure calculations and in the study of electronic transport in materials.

3. How is the momentum matrix of the reduced k-points calculated?

The momentum matrix of the reduced k-points is calculated by taking into account the symmetry of the crystal lattice and the wavefunctions of the electrons. This involves solving the Schrödinger equation and using group theory to determine the symmetry operations of the crystal.

4. What is the significance of the momentum matrix of the reduced k-points?

The momentum matrix of the reduced k-points is significant because it allows us to understand the behavior of electrons in a crystal lattice and predict their properties. It also helps in designing new materials with desired electronic properties.

5. Can the momentum matrix of the reduced k-points be used for other materials besides crystals?

Yes, the momentum matrix of the reduced k-points can also be used for other periodic systems, such as 2D materials and superlattices. It can also be applied in non-periodic systems by approximating them as a periodic lattice.

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