- #1

amjad-sh

- 246

- 13

- 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.

##\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.