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 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 twometal 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 spinpolarized in the xdirection. 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 xy 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 xdirection and moving in the zdirection.
using the spincontinuity 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 xdirection, 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 ydirection, 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 ydirection 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 any body 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 zdirection 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 zdirection instead of xdirection?
I uploaded the paper if someone likes to have a look.
Thank you.
In fact I'm working on an condensed matter physics paper, where we are dealing with a twometal 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 spinpolarized in the xdirection. 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 xy 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 xdirection and moving in the zdirection.
using the spincontinuity 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 xdirection, 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 ydirection, 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 ydirection 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 any body 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 zdirection 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 zdirection instead of xdirection?
I uploaded the paper if someone likes to have a look.
Thank you.
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