# The k direction in a k.p model ?

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• lichen1983312
In summary, the conversation discusses a calculation based on a k.p model of GaN proposed by S. Chuang, which involves an 8 by 8 Kane model with basis functions. The 8 by 8 Hamiltonian contains the first order of k and the reciprocal space and high symmetry points are represented as pictures. The question is how to define the direction of kx and ky in the Hamiltonian and how to express this direction in terms of kx and ky for drawing an E-k curve along ##\Gamma \to M##. The conversation also mentions that the Hamiltonian is invariant under certain group transformations, and suggests looking into the little groups of the wavevector to ensure invariance under the corresponding transformations.
lichen1983312
I am trying to do some calculation based on a k.p model of GaN proposed by S. Chuang [Phys. Rev. B, 54, 2491]. It is a 8 by 8 Kane model with basis functions:

The 8 by 8 Hamiltonian contain first order of k is

where ##{k_ \pm } = {k_x} \pm i{k_y}##

the reciprocal space and high symmetry points is pictures as [Phys. rev. B, 53, 10173]

My question is: How is the direction of kx and ky defined in the 8 by 8 Hamiltonian?
Put is in another way. If I want to draw a E-k curve along ##\Gamma \to M##, how do I express this direction in terms of kx and ky ?

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• basis function.png
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• Hamiltonian 8by8.png
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##k=(0,\lambda,0)^T## with ##\lambda=0## corresponding to ##\Gamma## and ##\lambda=k_\mathrm{M}## corresponding to M?

DrDu said:
##k=(0,\lambda,0)^T## with ##\lambda=0## corresponding to ##\Gamma## and ##\lambda=k_\mathrm{M}## corresponding to M?
Thanks
That is what is said in the graph, which is taken from a different article.

But if I only look at the Hamitonian and the basis functions, how can I tell?

##\Gamma##, M , ##\Sigma##, means that the hamiltonian is invariant under a specific group of rotations and reflections, respectively, which also leave k invariant. These groups are called the "little groups of the wavevector". This are sub groups of the point group under which ##\Gamma## is invariant. Look up which elements they contain in case of the Point Group of GaN and make sure that for the coice of k, your Hamiltonian is invariant under the corresponding transformations.

DrDu said:
##\Gamma##, M , ##\Sigma##, means that the hamiltonian is invariant under a specific group of rotations and reflections, respectively, which also leave k invariant. These groups are called the "little groups of the wavevector". This are sub groups of the point group under which ##\Gamma## is invariant. Look up which elements they contain in case of the Point Group of GaN and make sure that for the coice of k, your Hamiltonian is invariant under the corresponding transformations.
Thanks very much, I will look into it.

## 1. What is the k direction in a k.p model?

The k direction in a k.p model refers to the direction of the wavevector, which represents the momentum of an electron in a solid material. In the k.p model, the direction of the wavevector is used to describe the energy bands and electronic properties of semiconductors and insulators.

## 2. How is the k direction related to the band structure in a k.p model?

The k direction is directly related to the band structure in a k.p model. The band structure is a graph that shows the energy levels of electrons in a material as a function of the wavevector. By varying the k direction, we can determine the shape and behavior of the energy bands.

## 3. Can the k direction change in a k.p model?

Yes, the k direction can change in a k.p model. This is because the k direction is a vector quantity that can be manipulated by changing the direction and magnitude of the wavevector. By changing the k direction, we can study different properties and behaviors of electrons in a material.

## 4. How does the k direction affect the electronic properties of a material in a k.p model?

The k direction plays a crucial role in determining the electronic properties of a material in a k.p model. By changing the k direction, we can control the energy levels and band structure of electrons, which in turn affects properties such as conductivity, mobility, and optical properties.

## 5. Are there any limitations to using the k direction in a k.p model?

Yes, there are some limitations to using the k direction in a k.p model. This model is most accurate for materials with a small bandgap and a simple crystal structure. It also does not take into account interactions between electrons, which can play a significant role in certain materials.

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