Band structure of a 2deg including spin orbit coupling

[amjad-sh]

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

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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.

 

[mark726]

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.

I hope these suggestions are helpful. Good luck with your research!