Optically induced electron spin currents in the Kretschmann configuration

We investigate electron spin currents induced optically via plasmonic modes in the Kretschmann configuration. By utilizing the scattering matrix formalism, we take the plasmonic mode coupled to an external laser drive into consideration and calculate induced magnetization in the metal. The spatial distribution of the plasmonic mode is inherited by the induced magnetization, which acts as an inhomogeneous effective magnetic field and causes the Stern-Gerlach effect to drive electron spin currents in the metal. We solve the spin diffusion equation with a source term to analyze the spin current as a function of the spin diffusion length of the metal, the frequency, and the incident angle of the external drive.

Figure 1

Optically driven electron spin current in a trilayer system (Kretschmann configuration), where a metal film ($\epsilon ={\epsilon }^m$) with a thickness of $d$ is sandwiched by two dielectrics, the air (${\epsilon }^a=1$) and prism (${\epsilon }^p=1.5^2$). A laser drive field is incident on the metal from the prism side, and surface plasmon modes can be excited on the air-metal interface when the incident angle exceeds the critical angle (${\theta }_{\mathrm{in}}>{\theta }_{\mathrm{c}}\equiv arcsin\sqrt{{\epsilon }^a/{\epsilon }^p}$). In the excited plasmonic modes, the electric field is circulating, which induces an orbiting motion of the electron gas in the metal and results in inhomogeneous magnetization. The electron spin experiences the Stern-Gerlach effect under the magnetization, and the electron spin current is driven in the direction perpendicular to the air-metal interface.

Figure 2

Electromagnetic field in each layer is expanded by the eigenmodes as in (9) and (10). We set the amplitudes of the fields with a polarization of $\lambda =s,p{E}_{\lambda }^{\sigma \tau }$, where $\sigma =\pm$ and $\tau =a,m,p$ specify the direction of the $z$ components of the wave vectors and the medium, respectively (i.e., the permittivity in each layer is ${\epsilon }^{a,m,p}$). The frequency $\omega$ and the wave number in the direction parallel to the interfaces ${\mathit{k}}_{\parallel }\left(\right|{\mathit{k}}_{\parallel }{|}^2<{\omega }^2{\epsilon }^p/c^2)$ are free parameters. We impose the field continuity (boundary conditions) at $z=\pm d/2$ in order to derive simultaneous equations determining the scattering matrix of this trilayer system (12).

Figure 3

The field distribution in the trilayer system. The metal region ($-d/2\le z\le d/2$) is indicated by the dashed gray lines. (a) The incident angle is smaller than the critical angle between the prism and the air layers. We can confirm that the incident field decays in the metal layer and is transmitted to the other side. (b) The critical angle of incidence ${\theta }_{\mathrm{in}}={\theta }_{\mathrm{c}}$. At the critical angle, the electromagnetic field in the air layer is a single plane wave, and the magnetic field distribution is uniform in the $z$ direction. (c) The incident angle is the SP angle ${\theta }_{\mathrm{sp}}$ at which the $x$ component of the incident wave vector matches that of the SP. It is clear that the SP is excited at the interface between the metal and air layers. The electromagnetic energy of the SP is compressed at the interface, and the field has an exponential profile in the air region, unlike the plane wave at the critical angle. In order to generate these plots, we set the incident frequency $\omega =c{k}_0=2.88\times {10}^{15}{\mathrm{s}}^{-1}(k_0=2\pi /655{\mathrm{nm}}^{-1})$, the incident amplitude $E_{p+}^p=1.0\mathrm{V}/\mathrm{m}, d=20\mathrm{nm}$, and ${\omega }_{\mathrm{p}}=2\pi \times 2.1\times {10}^{15}{\mathrm{s}}^{-1}$ (the plasma frequency of gold).

Figure 4

Magnetization at the metal surface $z=+d/2$ induced by the plasmonic modes as a function of the wave number in the $x$ direction and the frequency of the incident field. We can see two peaks in this color map because there are two types of plasmonic excitations in our system. One is the SP at the air-metal interface which mainly contributes to the upper peak. The other is the SP at the metal-prism interface, contributing to the lower peak. The light dispersion in the air and prism, $\omega =ck/\sqrt{{\epsilon }^a}$ and $\omega =ck/\sqrt{{\epsilon }^p}$, and the surface plasma frequencies at the two interfaces, ${\omega }_{\mathrm{p}}/\sqrt{1+{\epsilon }^a}$ and ${\omega }_{\mathrm{p}}/\sqrt{1+{\epsilon }^p}$, are indicated by dashed lines. Here, we set $E_{p+}^p=\sqrt{Z_0\times 100\mathrm{mW}/{\mathrm{cm}}^2}\approx 0.62\times {10}^3$ and ${\omega }_{\mathrm{p}}=2\pi \times 2.1\times {10}^{15}{\mathrm{s}}^{-1}$ (the plasma frequency of gold), and we define $k_{\mathrm{p}}\equiv {\omega }_{\mathrm{p}}/c$. Note that we set the upper bound of this plot range at $5\mathrm{nA}/\mathrm{m}$ because the peaks are so large that other data become invisible if we plot the full range.

Figure 5

Induced magnetization at the metal surface $z=+d/2$ as a function of the incident angle. It is clear that it peaks at the SP angle of incidence (${\theta }_{\mathrm{in}}={\theta }_{\mathrm{sp}}\approx {43}^{\circ }$). We set the incident frequency $\omega =c{k}_0=2.88\times {10}^{15}{\mathrm{s}}^{-1}(k_0=2\pi /655{\mathrm{nm}}^{-1})$, and all other parameters are the same as in the previous figures.

Figure 6

Optically generated spin current $j_s$ at $z=0$ in the Kretschmann configuration. (a) Color map of $j_s$ as a function of the incident frequency $\omega$ and the incident angle ${\theta }_{\mathrm{in}}$ for a fixed spin diffusion length ${\lambda }_s$. The gray dashed line indicates the SP angle ${\theta }_{\mathrm{sp}}\left(\omega \right)$. We can see that the peak of the spin current density is around the dashed curve, which implies the spin current is driven by the SP. The peak slightly deviates from the SP angle condition around the right top region, where the spontaneous diffusion is comparable to the induced diffusion. (b) Color map of $j_s$ as a function of the incident frequency $\omega$ and the spin diffusion length ${\lambda }_s$ at the SP angle of incidence ${\theta }_{\mathrm{in}}={\theta }_{\mathrm{sp}}$. (c) Color map of $j_s$ as a function of the incident angle ${\theta }_{\mathrm{in}}$ and the spin diffusion length at a given frequency. To generate these plots, we set ${\rho }_0=12.8\times {10}^{-8}$ (the resistivity of gold), and other parameters are the same as in the previous figures. Note that we set the lower bounds of the plot ranges at $-0.1\mathrm{mA}/{\mathrm{m}}^2$ so that the peaks can be seen clearly.