# Band structure of a 2deg including spin orbit coupling

In summary: Your Name]In summary, the individual is facing a challenge in obtaining the energy eigenvalues of the Hamiltonian due to the presence of Dirac and step functions. They are seeking assistance and have shared two possible approaches - simplifying the equation using the properties of these functions or using numerical methods.
Homework Statement
In fact,I'm trying to obtain the band structure of a two dimensional electron gas with spin orbit coupling localized just in the interface.

The Hamiltonian is described by:
## \hat H=\dfrac{p^2}{2m}-\dfrac{\partial_z^2}{2m}+V(z)+\gamma V'(z)(\hat z \times \mathbf{p})\cdot \sigma##

##\mathbf{p}## is the two dimensional momentum in the x-y plane.
## V(z)## is the potential step, ##V(z)=V\theta(z)##
##\sigma## is the vector of pauli matrices
##\gamma## is the material dependent parameter which describes the strength of spin orbit coupling at the interface.
the first two terms of the hamiltonian describe the kinetic energy of the electron.
##\gamma V'(z)(\hat z \times \mathbf{p})\cdot \sigma## corresponds to SOC due to the gradiant of the potential barrier.
Relevant Equations
\\\
first of all, I tried to obtain the energy eigenvalues of the Hamiltonian, by using the equation ##det(\hat H -\lambda \hat I)=0##

##\gamma V\delta(z)(\hat z \times \mathbf{p}) \cdot \sigma=\gamma V\delta(z)(p_x\hat j-p_y\hat i)\cdot(\sigma_x\hat i + \sigma _y \hat j)=\gamma V\delta(z)\begin{pmatrix}
0 &-p_xi-p_y\\
ip_x-p_y & 0
\end{pmatrix}##

now ##\hat H=
\begin{pmatrix}
\frac{p^2}{2m} -\frac{\partial_z^2}{2m}+V\theta(z) &-\gamma V\delta(z)(p_xi+p_y)\\
\gamma V\delta(z) (ip_x-p_y) & \frac{p^2}{2m} -\frac{\partial_z^2}{2m}+V\theta(z)
\end{pmatrix}
##I stopped here, because I don't know how to solve ##det(\hat H -\lambda \hat I)=0## while dirac and step functions are present in the Hamiltonian.

Any help is appreciated.

Thank you for sharing your progress on obtaining the energy eigenvalues of the Hamiltonian. It seems like you have reached a roadblock in solving the equation ##det(\hat H -\lambda \hat I)=0## due to the presence of Dirac and step functions in the Hamiltonian.

One approach you could take is to use the properties of Dirac and step functions to simplify the equation and then solve it. For example, you could use the fact that the Dirac delta function is non-zero only at ##z=0## and the step function is non-zero only for ##z>0##. This could help you reduce the equation to a simpler form that can be solved using standard methods.

Another approach could be to use numerical methods to solve the equation. This could involve discretizing the Hamiltonian and then using techniques such as matrix diagonalization to obtain the energy eigenvalues.

## 1. What is the band structure of a 2deg system?

The band structure of a 2deg (two-dimensional electron gas) system refers to the energy levels of electrons in a two-dimensional material, such as a thin film or a semiconductor heterostructure. These energy levels are typically represented as a graph, with the energy on the y-axis and the momentum on the x-axis.

## 2. What is spin orbit coupling in a 2deg system?

Spin orbit coupling is a phenomenon that arises due to the interaction between an electron's spin and its orbital motion. In a 2deg system, this interaction can lead to a splitting of energy levels, resulting in a modification of the band structure. This can have important implications for the electronic properties of the material.

## 3. How does spin orbit coupling affect the band structure of a 2deg system?

Spin orbit coupling can cause a splitting of energy levels in a 2deg system, resulting in the formation of new bands. This can lead to changes in the electronic properties of the material, such as the density of states and the effective mass of electrons.

## 4. What are the applications of studying the band structure of a 2deg system?

The band structure of a 2deg system is important for understanding the electronic properties of materials and can have implications for various applications. For example, it can help in the design of new semiconductor devices, such as transistors and solar cells, and in the development of new materials for spintronics and quantum computing.

## 5. How is the band structure of a 2deg system experimentally determined?

The band structure of a 2deg system can be experimentally determined using various techniques, such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM). These techniques allow for the measurement of the energy and momentum of electrons in a material, which can then be used to reconstruct the band structure.

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