# 7. Magnetoplasmonics

Magnetoplasmonics combines the fields of magnetism and plasmonics in order to create a total new class of optical devices. As light is and electromagnetic wave, it consists both of an electric and a magnetic field component, which are perpendicular to each other and to the propagation direction of the wave. Typically, in plasmonic structures the electric field component is responsible for the coupling to the electron cloud in the nanostructures, as the intrinsic magnetic response of conventional materials is too slow to allow coupling to the magnetic field component of an incident wave at optical frequencies. In electromagnetic metamaterials on the other hands, it is possible to couple through the electric field of an incident wave to resonant electric modes that can induce artificial optical magnetism at the frequency of the incident wave.

One of the most important properties of localized surface plasmon resonances (LSPRs) and propagating surface plasmon polaritons (SPPs) is that they give rise to large enhancement of the electromagnetic near fields around the nanostructures. This property is often used in plasmonic biosensors such as refractive index sensing or surface-enhanced raman scattering (SERS) amongst others. Also in the field of magnetoplasmonics localized field enhancement plays a crucial role, as it allows to enhance the interaction between plasmon resonances and external magnetic fields.

The interaction of materials with electromagnetic waves is described by the Maxwell equations, in which the material response is introduced by means of the electric permittivity $\epsilon$ (also called the dielectric function of the material) and the magnetic permeability $\mu$ which describe the interaction of the material with electric and magnetic fields respectively. Both quantities are frequency dependent function which are in general complex functions which can be written as:

(1) $\epsilon(\omega) = \epsilon'(\omega) + i \epsilon''(\omega)$
(2) $\mu(\omega) = \mu'(\omega) + i \mu''(\omega)$

They also closely relate to the refractive index, such that

(3) $n(\omega) = n'(\omega) + i n''(\omega) = \sqrt{\frac{\mu(\omega)}{\mu_{0}}\frac{\epsilon(\omega)}{\epsilon_{0}}}$

in which $\epsilon(\omega)$ and $\mu(\omega)$ represent the relative permittivity and permeability, while $\epsilon_0$ and $\mu_0$ are the permittivity and permeability of vacuum. The material response to an incident electromagnetic wave can then be described in terms of the electric displacement $D(\omega)$ and the magnetic induction $B(\omega)$:

(4) ${\bf D} = \epsilon {\bf E}$
(5) ${\bf B} = \mu {\bf H}$

For high frequency waves in the visible and near-infrared wavelength range $\mu(\omega)$ can generally be considered unity, implying that there is no magnetic response in this range.

For isotropic materials, one can write the dielectric tensor of a material as

(6) $\epsilon = \begin{pmatrix} \epsilon_{xx} & 0 & 0 \\ 0 & \epsilon_{yy} & 0 \\ 0 & 0 & \epsilon_{zz} \\ \end{pmatrix}$

in which $\epsilon_{xx} = \epsilon_{yy} = \epsilon_{zz}$, while for anisotropic materials the different values of $\epsilon(\omega)$ are different along different symmetry axes of the material. For magnetic materials however, the dielectric tensor becomes more complicated as it also contains off-diagonal elements:

(7) $\epsilon = \begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \\ \end{pmatrix} = \begin{pmatrix} \epsilon & -i Q \epsilon m_z & i Q \epsilon m_y \\ i Q \epsilon m_z & \epsilon & -i Q \epsilon m_x \\ - i Q \epsilon m_y & i Q \epsilon m_x & \epsilon \\ \end{pmatrix}$

These off-diagonal terms are defined by the frequency dependent magnetic factor $Q(\omega) = Q'(\omega) + i Q''(\omega)$, the dielectric function $\epsilon(\omega)$ (diagonal terms) and the magnetization of the material $m$. As such, these terms can be switched on/off by applying a magnetic field along the desired direction. It can also be seen that terms in the XY-plane (resp. YZ, XZ) are activated when the medium is magnetized along the Z-direction (resp. X, Y). In classical terms this can be understood by means of the Lorentz force acting on a moving electron

(8) $\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)$

where $v$ is the velocity of the electron. As we take the cross product of the velocity and the magnetic induction, we can see that this interaction is the strongest when both of them are perpendicular to each other. In quantum mechanical terms, this effect is described by means of the spin-orbit interaction which defines the interaction of a particle’s pin with its motion.

It is exactly due to these off-diagonal terms that an electromagnetic wave changes its polarization state while traveling through a magnetized medium. This principle is used in so-called magneto-optical Kerr effect (MOKE) measurements in which the magnetization state of a material is probed by looking at the change in the polarization state of a transmitted (Faraday configuration) or reflected wave (Kerr configuration). Although MOKE has been a well-established characterization tool in magnetic research for decades, nowadays it has become the central measurement technique in the field of magnetoplasmonics. Nowadays instead of using a single wavelength to just measure magnetization loops, full spectral scans have become common in order to study the interaction of plasmons with magnetic fields. (Note: A detailed description of MOKE will soon be made available on a separate page.)

As plasmon resonances give rise to large localized field enhancements in the near field of a nanostructure, also the interaction with externally applied magnetic fields is strongly enhanced. This results in large increases in observed Kerr rotation when measuring at the plasmon resonance wavelength. In fact, by tuning the plasmonic properties of a nanostructure it is possible to tailor the Kerr response both in sign and amplitude over very wide spectral ranges. This could be applied in various sorts of optical devices such as optical modulators, which can in this case be controlled by means of the external magnetic field.

Of course it is also possible to use the reverse mechanism and to exploit plasmon resonances to define the magnetization state of a material, which has already been realist in ultrafast pulsed measurements. In future applications this might even lead to optically controlled magnetic memories, which could potentially be scaled down further beyond what is possible today.