Generation of spin and charge current due to interference and SOC

[amjad-sh]

TL;DR Summary

In a concise way , my question is about how an incident scattering wave function of a certain electron moving in the z direction when it interfere with its reflected wave function(potential step problem)may create another wave function moving in another direction ,like x for example, or in another way, how I can interpret the physical meaning of this interference?

My question may include condensed matter physics concepts but my main question is related to quantum mechanics in general, that's why I posted it here.In fact I'm working on an condensed matter physics paper, where we are dealing with a two-metal system. The interface between the two metals is represented by a potential step. We inject a spin current in the left metal, where all the electrons injected are spin-polarized in the x-direction. The Hamiltonian of a single electron is:
$\hat{H}=\dfrac {p^2}{2m}- \dfrac{\partial_z^2}{2m}+V(z)+\gamma V'(z)(\hat z \times \vec{p})\cdot \vec{\sigma}$

The term "$\gamma V'(z)(\hat z \times \vec{p})\cdot \vec{\sigma}$" represents the Rashba spin orbit coupling.
$\dfrac {p^2}{2m}- \dfrac{\partial_z^2}{2m}$ represents the kinetic energy term. We also consider that $\hbar=1$.
$\vec{p}=p_x\vec{i}+p_y\vec{j}$
$V(z)=V\theta(z)$
$V'(z)=V\delta(z)$

$\vec{\sigma}$ are the pauli matrices.

The system is modeled as a uniform free electron gas.
The system is transnational invariant in the x-y plane.
So using Bloch theorem we can write: $\psi_{p,k}(\vec{r},z)=e^{i\vec{p}\cdot \vec{\rho}}\varphi_k(z)$. $\vec{\rho}=(x,y)$
Away from the interface the wave functions corresponding to the energy $\varepsilon=\dfrac{p^2+k^2}{2m}$ have the following form:
$\vec{\varphi}_{k\sigma}(z) = \begin{cases} (e^{ikz}+\hat{r_k}e^{-ikz})\chi_{\sigma} & \text{if } z < 0 \\ \hat{t_k}e^{ik'z}\chi_{\sigma} & \text{if } z > 0 \end{cases}$
$\hat{r}_k$ is the reflection coefficient.
$\hat{t}_k$ is the transmission coefficient.
$\chi_{\sigma}$ is the eigenfunction of $\sigma_x$

We emit a spin current with spin polarized in the x-direction and moving in the z-direction.
using the spin-continuity equation, $\dfrac{d\hat{s_x}}{dt}=\vec{\nabla} \cdot \vec{j}_{z,L}^{x,\sigma}$ for z<0 and z>0, where $\hat{s}_x=\psi^{\dagger}_{k\sigma}\sigma_x \psi_{k\sigma}$ is the magnetic moment density in the x-direction, and $j_{z,L}^{x,\sigma}$ is the spectral spin current generated from the left metal corresponding to one electron moving in the z direction and polarized in the x direction. It can be proved that $j_{z,L}^{x,\sigma}(z)=-\dfrac{i}{4m}[\vec{\varphi}_{\sigma}^{\dagger}(z)\sigma_x\partial_z\vec{\varphi}_{\sigma}(z)-(\partial_z\varphi_{\sigma}^{\dagger}(z))\varphi_{\sigma}(z)]$
The total spin current polarized in the x direction and moving in the z direction is defined to be $J_z^x(z)=\sum_{\vec{p},k} f_F(\varepsilon_{p,k}-\mu_{L,\sigma})j_{z,L}^{x,\sigma}(z)$, where F_f is the fermi distribution function that gives the average number of the electrons.
The spin loss is defined by$L_s=J_z^x(z<0)-J_z^x(z>0)$Now,in the other hand, we can also use the continuity equation to calculate the charge current generated in several directions.
$\dfrac{d\rho}{dt}=\vec{\nabla} \cdot \vec{j}_e^{\sigma}$(continuity equation of charge current), where $\rho$ is the charge density, $\rho=\psi_{k,\sigma}^{\dagger}e\psi_{k,\sigma}$ and $\vec{j}_e^{\sigma}$is the spectral charge current corresponding to one electron.
suppose we want to calculate the charge current created in y-direction, from the continuity equation we can prove that:
$j_{y,L}^{\sigma}(\vec{r},z)=\dfrac{-ie}{2m}[\vec{\psi}_{\sigma}^{\dagger}(\partial_y\vec{\psi}_{\sigma})-(\partial_y\vec{\psi}_{\sigma}^{\dagger})\vec{\psi}_{\sigma}]-\dfrac{e}{m}[\psi_{\sigma}^{\dagger}\hat{\Gamma}_y\vec{\psi}_{\sigma}]$ where $\hat{\Gamma}_y=\gamma mV'(z)\sigma_x$
This term $j_{y,L}^{N,\sigma}(z)=\dfrac{-ie}{2m}[\vec{\psi}_{\sigma}^{\dagger}(\partial_y\vec{\psi}_{\sigma})-(\partial_y\vec{\psi}_{\sigma}^{\dagger})\vec{\psi}_{\sigma}]$ the author of the paper named it the normal charge current, and the last term he named it, the anomalous charge current.
The anomalous charge current is just created at the interface and its physical origin is the "momentum dependence of the SOC term of the Hamiltonian".
The total normal charge current in the y-direction can be written as: $J_y^N(z)=\sum_{\vec{p},k}f_F(\varepsilon_{p,k}-\mu_{L,\sigma})j_{y,\sigma}^N(z)$

Now my questions start here:
1- In the region z<0 it is proved in the paper that $J_y^N(z)$ and $J_x^N(z)$ are not zero, and the physical origin behind the generation of this charge current(the normal charge current) ,as mentioned in the paper, "is the interference between the incident wave and the reflected wave".In fact I am having a hard time grasping the intuition behind this. Can anybody illustrate it more to me, or at least give me another simpler example that contains the same physical idea?
2- There will be,also, a generated spin current, in z>0, that is"only" spin polarized in the z direction, as mentioned in the paper, the physical reason behind the spin current just being polarized in the z-direction is that the SOC term in the Hamiltonian just contains $\sigma_x$ and $\sigma_y$ so that the primary spin $\sigma_x$ can only rotate about the $\sigma_y$ internal magnetic field, creating spin polarized in the z direction. I also somehow had a trouble in understanding this,if somebody can simplify it more to me. What for example may happen if we generate a spin current in z<0 that is polarized in z-direction instead of x-direction?

I uploaded the paper if someone likes to have a look.
Thank you.

 

[eljose79]

Hello,

Thank you for sharing your paper and your questions. As a scientist specializing in quantum mechanics, I am happy to provide some insights and clarification on the concepts you are working with.

1. The interference between the incident and reflected wave is a common phenomenon in quantum mechanics. In simple terms, it occurs when two waves overlap and their amplitudes either reinforce or cancel each other out. In your system, the incident wave is spin polarized in the x-direction and as it encounters the potential step, part of it gets reflected back. This reflected wave has a different phase and amplitude compared to the incident wave, resulting in an interference pattern. This interference affects the spin and charge currents in the z<0 region, leading to the generation of the normal charge current. You can think of it as a superposition of two waves, where the resulting wave has a different amplitude and phase compared to the individual waves. A similar phenomenon can be observed in the famous double-slit experiment, where the interference between two waves creates an interference pattern on a screen.

2. The spin polarization in the z-direction in the z>0 region is a consequence of the Rashba spin-orbit coupling term in the Hamiltonian. As you mentioned, this term only contains $\sigma_x$ and $\sigma_y$, which means that it can only rotate the spin in the xy-plane. This results in a spin current that is polarized in the z-direction. If you were to generate a spin current in the z<0 region that is polarized in the z-direction, it would require a different type of spin-orbit coupling term in the Hamiltonian, such as the Dresselhaus term, which contains $\sigma_z$. In this case, the spin polarization would be in the xy-plane. Essentially, the type of spin-orbit coupling determines the direction of spin polarization.

I hope this helps to clarify the concepts in your paper. If you have any further questions, please feel free to ask.