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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

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71966
10.1038/s41598-024-71966-3
Article
Achieving a comparable transverse magneto-optical Kerr effect by spin–orbit field driven magnetoplasmonic
Asteraki Mohammad Hassan
Farzad Mahmood Hosseini hosseinif@shirazu.ac.ir

https://ror.org/028qtbk54 grid.412573.6 0000 0001 0745 1259 Physics Department, College of Science, Shiraz University, Shiraz, 71946-84795 Iran
10 9 2024
10 9 2024
2024
14 2109311 3 2024
2 9 2024
© The Author(s) 2024
2024
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In this study, we propose and simulate a magnetoplasmonics heterostructure that utilizes spin–orbit fields to generate an internal magnetic field and create a significant magneto-optical effect. Our approach offers a new way to overcome the challenges of using permanent magnets or magnetic coils in conventional magnetoplasmonics, such as high-power consumption and non-scalability. We demonstrate that it is possible to create an appropriate amount of magnetic field using spin–orbit fields induced by the spin-Hall effect, such that the consumption power becomes reasonable and the dimensions could be miniaturized. This approach will be an important development in the field of magneto-optics, as it can lead to enhanced transverse magneto-optical Kerr effect in the present of surface plasmon polaritons. The proposed nanostructure consists of a ferromagnetic film adjacent to a heavy metal layer, both sandwiched between two noble metal films, and deposited on a dielectric prism. The strength of the Kerr signal strongly depends on the thickness of the ferromagnetic layer, with the maximum effect observed at a thickness of 5nm. This concept has potential for various nanophotonic and spintronic applications, particularly for developing high-speed active plasmonic devices for ultrafast light modulation.

Keywords

Magnetic multilayers
Magneto-optical effect
Magnetoplasmonic
Spin Hall effect
Spin–orbit field
Surface plasmon polaritons
Transverse magneto-optical Kerr effect
Subject terms

Magneto-optics
Nanophotonics and plasmonics
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pmcIntroduction

Surface plasmon polaritons (SPPs) are collective oscillations of electric charge excited in a metal–dielectric interface under phase-matching conditions and propagate along the interface. One of the exciting properties of SPPs is their ability to confine electromagnetic fields beyond the diffraction limit. Recent advances in plasmonic systems illustrate that those are promising candidates to be used in miniaturized photonic and plasmonic devices. However, far more engineering of characteristics is required in practice when the SPPs are used in active. These characteristics, such as wave vector, propagation length, and penetration depth, depend on the optical properties of the propagation platform. In other words, manipulation of the optical properties of the platform leads to the variation of the SPPs characteristics. Therefore, many methods, such as temperature controlling1, applying an external electric field2, utilizing a pulsed laser3, and use of active materials like graphene or ferroelectrics4,5, could be taken to tune the SPPs features properly. Each mentioned mechanism has advantages and disadvantages depending on the interested application. Besides, by applying an external magnetic field, one can also modulate the optical properties of the materials (magneto-optical effect (MOE)) along with exploiting some other attractive features such as optical nonreciprocity6 and high-speed switching7. Magnetic manipulation of SPP characteristics (magnetoplasmonics) is nonreciprocal, making it an important candidate for designing an optical isolator. The switching speed of magnetic-based systems has a fundamental role in high-speed devices such as data storage devices which operate at ultrafast frequencies. Despite all these benefits, applying an external magnetic field is accompanied by some cost–benefit challenges in relatively high-power consumption and limiting the downscaling (non-scalability). These difficulties produce crucial barriers for future photonic circuits, which look forward to smaller dimensions and less energy consumption. To the best of our knowledge, this issue has not yet been investigated so far in the literature reports for magnetoplasmonics nanostructures, and only the effects of the external magnetic field on the magnetoplasmonic feature of the structures have been investigated8–15. In the first attempt, Yongkang Gong et al.16 proposed a design to modulate the reflected light by applying the magnetic field induced by an electric current in a magnetoplasmonic multi-layer structure. The drawback of this task is a large amount of current for generating a sufficient magnetic field to reach the magnetization of the ferromagnetic layer to its saturation. This plan is not practically performed because increasing the electric current up to 120mA causes imperial thermal effects so that the structure tends to be damaged. On the other hand, by applying such electric current the MOEs not to be operated well.

In our work, with inspiration from the magnetic nanostructures used in magnetic random-access memories (MRAM)17, we used spintronics to generate a large amount of internal magnetic field. Spintronics is a field that considers the spin of electrons as a degree of freedom in addition to the electric charge, and it is advancing the development of microelectronic devices. The main goal of spintronics is to manipulate the magnetization vector (its magnitude and direction) in a thin ferromagnetic layer by a spin-polarized electric current. By passing an electric current through a ferromagnetic layer, the equal distribution of up and down electron spins can be changed and produce some majority and minority in these two directions. The reverse of this phenomenon can switch the magnetization direction correspondingly, which is called the spin transfer torque (STT) effect. Also, the Spin Hall effect (SHE) occurs in materials with high spin–orbit coupling; thus, spin separation is induced such that the two kinds of spin accumulate at the boundaries. As a consequence, a spin current (Js) is generated perpendicular to the direction of the electric current (Je)18. Both of these effects can apply some torques to the magnetization vector of a ferromagnetic layer. For example, in the structures consisting of heavy metal (HM) and ferromagnetic (FM) layers, owing to the strong spin–orbit interaction in HM, SHE happens and non-equilibrium spin accumulation is induced at the interface. The electron spins with a specific direction penetrate the FM layer and can transfer angular momentum and some torques to the magnetization vector of the FM layer, ultimately forced the FM layer magnetized along its direction. This phenomenon is called spin–orbit torque (SOT)19–23. In this article, we try to investigate a transverse magneto-optical Kerr effect (TMOKE) by manipulating the magnetization of the ferromagnetic layer with the SOT process. The internal magnetic field emanating from this task plays as an alternative to the external magnetic field, which is required for this effect.

This research can be considered as a first step for the development of photonic and plasmonic magnetic devices in order to making them smaller and more energy efficient with programmable optical functionalities.

Magnetization dynamics and spin–orbit fields (SOF)

The magnetization dynamics of the ferromagnetic layer can be analyzed using the Landau–Lifshitz–Gilbert equation (LLG) as follow:1 ∂m∂t=-γm×Heff+αm×∂m∂t

where m is the magnetization vector normalized to the corresponding saturation one, γ is the gyromagnetic constant, α represents damping, and Heff includes all possible contributions to the magnetic field such that:2 Heff=Hext+HCI+Hk+HFLSOT+HALSOT

where Hext is external magnetic fields, HCI is the current-induced magnetic field, and Hk indicates the effective perpendicular anisotropy field. HFLSOT and HALSOT are SOFs that refer to the field-like (FL) and anti-damping-like (AL) components, respectively. These two last magnetic fields are generated by spin current density due to spin Hall effect (SHE). This phenomenon originates from strong spin–orbit coupling of the electron in materials, which causes spin accumulation at the lateral boundaries with the directions of the spins being opposite at the opposition boundaries. The SHE can be quantitatively analyzed using the equation of JS=θSHσ×Je, where JS, Je, and σ represent the spin current density, charge current density, and spin moment, respectively. The spin Hall angle, θSH, can be defined as the conversion efficiency of the spin-to-charge current conversion ratio, i.e., θSH=JSJSJeJe.

The LLG relation shows that the magnetization vector brings about a damping precession around the effective magnetic field vector by applying an external magnetic field. If the frequency of the injected electric current is much smaller than that of the magnetic resonance (~ GHz), the quasi-static condition can be considered. Hence, the direction of magnetization becomes toward the effective magnetic field, consequently resulting in m×Heff=0. Therefore, in our study, the dynamic of magnetization is ignored.

Now, we turn our attention to a SOT effect and the corresponding related fields. According to our knowledge, in the geometry shown in Fig. 1, the value and orientation of the two types of SOF depend on the electric current density Je as follows19,22:3 HFLSOT=ξFLy^×JeHALSOT=ξALJe×y^×m

where y^ is the unit vector normal to the interface. ξFL and ξAL indicate the strength of each SOF.Fig. 1 Schematic of the SOFs, which is induced by a lateral current density J. The black arrow shows the direction of HFLSOT and HCI. Owing to spin–orbit interaction in the heavy metal layer (Ta). Some amount of electron spin accumulation is induced at the boundaries. Thus, the current density J produces a spin current Js perpendicular to its direction.

Figure 1 shows the spin separation of the electrons propagating through the Ta layer and the spin accumulation along the + ẑ direction at its boundary with the CoFeB layer and -ẑ direction on the opposite boundary. The directions of HFLSOT and HCI inside the CoFeB layer are both in the ẑ direction, according to the direction of current density (see also Eq. (3)). The direction of HALSOT depends on the initial magnetization of the ferromagnetic layer, but it’s definitely perpendicular to the ẑ direction.

According to the measurement results in Reference22, we analyzed and compared the current dependency of HFLSOT and HALSOT values and also the corresponding dependence of HCI in a Ta/CoFeB bilayers, see the diagrams in Fig. 2.Fig. 2 The magnitude of different magnetic fields (HFLSOT, HALSOT, and HCI) in terms of electric current in Ta/CoFeB bilayers. The experimental data reported in22 has been used for SOF parameters (ξFL and ξAL) to draw this diagram.

In order to use the total capacity of the magnetic property of the ferromagnetic layer in magneto-optical structures, it is necessary to saturate the magnetization in this layer. The ferromagnetic layer (CoFeB) reaches its saturation in the field of about 300Oe24. From the information in Fig. 2, look at the current dependency of HFLSOT, this magnetic field can be obtained from a current density value of 9 × 1011 A.m-2. Considering the cross-sectional area of the Ta layer in our structure (4 nm × 20 μm), therefore, the corresponding electric current for this situation is ~ 70 mA. In Reference16, which have only used the Orsted field of electric current (HCI), they reported that the average magnetic field in the ferromagnetic layer (Co) is about 2mT (20 Oe) when using HCI produced by the current of about 90mA injected into the gold layer in their investigated magnetoplasmonic structure. This value is matched well with dark line in Fig. 2. They also notified; this magnetic field could not be saturate the magnetization of the exploited sample. When by our geometry, since HFLSOT and HCI are in the same direction, HFLSOT can also contribute to transverse magneto-optical Kerr effect. The magnitude of HFLSOT is several times larger than of HCI, and it can be prepared the saturation magnetization for the ferromagnetic layer. To the best of our knowledge, Reference22 is the only source that has so far obtained spin–orbit fields (SOF) experimentally in CoFeB/Ta bilayer from two methods of electrical transport and optical measurement. Accordingly, we used their measured results in our simulation.

The use of spin–orbit fields is one of the most important topics in the field of spintronics in order to changing the magnetization of the magnetic free layer without applying any external magnetic fields. Therefore, it is possible to use HFLSOT and HCI and suitable ferromagnetic layer (here, CoFeB) to design a structure with optimum manipulation of magnetization in order to enhance the magneto-optical effect.

Magnetoplasmonics

In the presence of a magnetic field, the non-diagonal terms of the dielectric tensor of a magnetic material become non-zero. The values of these terms depend on the magnitude and direction of the magnetization. The MOEs originate from these non-diagonal elements in the relative permittivity tensor. The general form of this tensor could be represented as:4 ε=εxxεxy(mz)εxz(my)-εyx(mz)εyy-εyz(mx)-εzx(my)εzy(mx)εzz

where mi (i = x, y, z) refers to three magnetization components, the non-diagonal elements (magneto-optical constant) are tiny for diamagnetic and paramagnetic materials. Therefore, ferromagnetic materials with relatively large non-diagonal elements are necessary for magneto-optical systems.

In magnetoplasmonic structures, a noble metal and a dielectric material are also present in addition to a ferromagnet. In this article's proposed structure, a heavy metal layer is further required to support and accomplish the spin–orbit torque effect. Therefore, it is necessary to have an appropriate heterostructure containing ferromagnetic material, noble metal, dielectric, and heavy metal layers.

With an optimized design for the proposed multi-layer structure, as depicted in Fig. 3, the thicknesses of different layers, means Au/SiO2/Ta/CoFeB/SiO2/Au, are 15, 2, 4, 6, 2, and 5 nm, respectively. In addition, two thin layers of SiO2 act as an insulator, so the electric current through the heavy metal layer does not penetrate the adjacent layers.Fig. 3 Schematic of the proposed structure in which the spin–orbit torques are able to modulate the surface plasmon polaritons. At a special angle of incident light (from the cylindrical prism), SPPs are ultimately excited at the last (Au/air) interface and propagated along the x direction. By applying an electric current through the Ta-layer, the spin–orbit torques change the magnetization of the CoFeB-layer, and as a result, the SPP modulation is performed.

Basically, for the field analysis of these magnetoplasmonic structures, we consider the electric field of incident light to have only x and y components (TM mode) therefore, the corresponding amplitudes of the electric and magnetic fields in each layer (i) are deduced from Maxwell's equations as follows:5 Hz,i=AieikSPPxe-kiy+BieikSPPxekiy

6 Eζ,i=eikSPPx-iωεi2+εxy2[kiεi∓ikSPPεxyiAie-kiy∓ikSPPεxyi±kiεiBiekiy]

where Ai and Bi (i = 1, 2, …, 8) are different constants for various layers, kSPP is SPP wave vector, ki2=kSPP2-k02εi indicates the transverse part of wave vector where light incident wave vector is k0, ω is the incident light frequency, and ζ refers to x (up signs) and y (down signs) components of the electric field. It is noted that, the only assumption that has been considered is the TM polarization of the incident light. This assumption makes only the non-diagonal εxy components of the dielectric tensor (Eq. (4)) become effective, which are also non-zero due to the magnetization component in the z direction. Therefore, as it is clear from (6), the Ex and Ey fields in magnetic material depend only on the non-diagonal component of εxy, which is originated from the magnetization in the ẑ direction. By applying the boundary conditions at infinity and continuity of Hz and Ex at each interface, the Ais and Bis are determined numerically. So, the distribution of the electromagnetic field inside the multilayer structure can be calculated. We will perform this task to obtain the dispersion relation of the proposed magnetoplasmonic structure in a simplified model.

Passive study of the structure

We use the finite element method with COMSOL MULTIPHYSICS 5.6 software to simulate the optical transmission and reflection from the structure considered in Fig. 3 in a two-dimensional (2D) model. Periodical boundary conditions have been applied to the boundaries. It should be mentioned that the experimental data published in Johnson-Christy25 work have been used to calculate the permittivity of the Au layer. The value of the magnetic layer permittivity tensor elements is taken from the corresponding data in Ref.24. The relative permittivity and its dispersion for SiO2 and heavy metal (Ta) layers are also considered from εSiO226 and εTa27, respectively. Without applying any current, we call this situation the study under passive condition, the reflectivity of this structure with respect to the incident angle is plotted in Fig. 4a.Fig. 4 (a) Reflectivity (black) and Transmissivity (red) of the proposed magnetoplasmonic structure in wavelength λ = 632.8 nm. (b) The dependence of reflectivity on wavelength and incident angle.

As it is shown, beyond the critical angle, the phenomenon of total reflection occurs and any decrease in the intensity of the reflected light is used to excite SPPs. Initially, the intensity of SPPs is low, but as we get closer to the θSPP, more light is used to excite them, and finally, the intensity of SPPs reaches its maximum in the θSPP = 48.0°. Figure 4b shows the three-dimensional dependency of reflectivity on the incident angles for different wavelengths. The surface plasmon resonance is optimally excited in the region where the reflection is minimum (dark blue area). Therefore, the SPPs excitations are weaker for wavelengths less than ~ 500 nm and more than ~ 750 nm. Additionally, the angle at which SPPs are strongly excited, θSPP, shifts slightly to larger angles as the wavelength increases, look at the shape behavior of the contours in Fig. 4b.

Figure 5 indicates the electric field distribution across the structure when the SPP is strongly excited by the incident light. The presence of surface plasmon polaritons (SPPs) at the interface of the bottom Au layer adjacent to the air produces a strong and confined electric field. This enhancement caused the electric field penetrates through the other Au layer interface into the ferromagnetic layer (CoFeB), as clearly shown in the inset of Fig. 5. By changing the dielectric function of this layer (CoFeB), originates from its magnetization, leads a variation in the properties of SPPs. Therefore, any possible controlling of the magnetization, here performed by the SOT effect, could be regarded as an active control over SPP’s features and consequently contribute to the reflectivity.Fig. 5 Simulated spatial distribution of Ex across the proposed structure for TM-mode incident light with wavelength λ = 632.8 nm at minimum reflectivity (θSPP under passive conditions, without any magnetic field. The amplitude of the field is normalized to the amplitude of the incident light.

To investigate the impact of additional layers such as Ta and CoFeB on the excitation of surface plasmon polaritons (SPPs), it is necessary to compare the Ex of a simple metal/dielectric structure with the corresponding field for the offered multilayer structure. The cross sections of these plasmonic structures and the corresponding variation of Ex across each structure are shown in Fig. 6. Here, in our proposed structure, several thin film layers (SiO2/Ta/CoFeB/SiO2) are sandwiched between two Au layers with the same total thickness of simple metal/dielectric structure. The comparison shows that the presence of thin layers inside the Au layer of a simple structure does not significantly change the electric field distribution, along the SPP propagation (Ex), across the structure. Therefore, from the SPP excitation point of view, it is possible to effectively approximate the structure as a magnetic-metal/dielectric structure. In reality, it is equivalent to a simple metal/dielectric structure with additional magnetic properties in its metal layer.Fig. 6 Normalized x-component of the SPP electric field distribution with respect to its maximum (at y = 0) across the multilayer structure compared to the corresponding distribution of SiO2/Au structure for λ = 632.8 nm.

Active study; Transverse magneto-optical Kerr effect

According to above discussion in the previous section it is possible to obtain an equivalent relation for the SPPs dispersion in the presence of an external magnetic field for the proposed magneto-plasmonic structure. For this goal, we considering a general case with three components of magnetic field (magnetization) for a magnetic-metal/dielectric interface. By using Eq. (5) and Eq. (6) for the electric and magnetic fields components (TM mode) in magnetic-metal and dielectric layers and relevant boundary conditions at their interface, and general form of dielectric tensor (Eq. (4)), one can be achieved the following SPP dispersion relation:7 kSPP=k0ε1ε2ε1+ε21-εxyε12ε12-ε22-ε1ε2+oεxy2

where ε1 and εxy are diagonal and off-diagonal components of the dielectric tensor of the magnetic layer and ε2 is dielectric constant of the other layer. The higher orders of εxy are omitted since the MO activity is a small perturbation in the system (|εxy|<<|ε2|). As it is clear from Eq. (7), the SPP dispersion relation depends on the εxy. This means that only the z-component of the magnetic field (or magnetization) can affect the SPP dispersion relation. It is reminded that, this component of magnetic field in our structure is due to sum of the two fields, HFLSOT and HCI, see these fields in Fig. 1. The noteworthy point that can be deduced from this relation is a prominent inherent non-reciprocally feature of the structure. In other words, the SPP wave vector values are different when the direction of the magnetic field is reversed. So, a modulation has happened in the plasmon wavevector induced by the magnetic field. On the other hand, this modulation can also emerge in the transverse magneto-optical Kerr effect (TMOKE) through the modulation of the reflectivity. To put it another way, the z component of the magnetization causes the TMOKE, which can be greatly enhanced by the SPP excitation8,11–13,15,28. Here, the strength of the TMOKE signal can be characterized using the δ parameter, which is the relative change in the light reflectivity when the magnetization direction is switched from its direction to the opposite one over the reflected light under zero magnetization condition. Also, since the switching of the magnetization is done by reversing the direction of applied electric current density (J) into the Ta-layer the Kerr signal (δ) is defined as follows:8 δ=ΔRR0=RJ+-RJ-RJ=0

where R(J+), R(J-) are the light reflectivity for opposite directions of current density and R(J = 0) indicates the reflectivity under zero current. Now, we are in a place that to study the effect of current density (in turn it generates a magnetic field and magnetization in the CoFeB layer by SOT process) in our proposed magneto-plasmonic structure.

Figure 7a shows the angular dependency of the reflectivity for current density in + x and -x direction and the reflectance difference (ΔR) in three points. The maximum value of ΔR occurred at the angle where the reflectivity falls rapidly from the maximum to its minimum. This feature is also shown in the blue diagram of Fig. 7b. This figure compares two quantities, ΔR and Kerr signal δ, as a function of light incidence angle. The reflectivity is also shown in the background to facilitate this comparison. As mentioned before, SPPs have raised for the angles greater than the critical angle, and this causes a change in light reflection (ΔR). However, ΔR decreases near the θSPP despite the increase in SPP intensity. As sketched in Fig. 7b the Kerr signal (δ) is very small (~ 10–4) for incident angles smaller than the angle at which the maximum ΔR is occurred. After this angle, δ increases rapidly and reaches its maximum (0.26) at θSPP. Furthermore, δ experiences an immediately falling for angles slightly larger than θSPP and then growths quickly up to its initial value and tends to constant value.Fig. 7 (a) Comparison of the reflectance difference (ΔR) in three points (insets) at reflectivity diagram (indicated by dark arrows) and (b) reflectance difference (ΔR) and Kerr signal (δ) as a function of the incidence angle for the 5 nm thickness of the CoFeB layer.

The impact of the SPPs on the transverse magneto-optical Kerr effect can be investigated by comparing the results obtained from TE and TM polarizations of incident light. In the case of TE-polarization, the conditions for exciting the SPPs are not available. As shown in Fig. 8a, in the TE polarization reflection diagram, there is no valley that indicates the excitation of SPPs. In the diagram of Fig. 8b the reflectance difference (ΔR) for TE-polarization (red diagram) is about 4 orders of magnitude smaller than that of TM-polarization. It shows that the electric field enhancement by SPPs is very effective in the reflectance difference.Fig. 8 (a) Comparison of the reflectance between TM and TE polarizations for the incident light, and (b) reflectance difference (ΔR) as a function of the incidence angle for TM and TE polarization. The thickness of the CoFeB layer is 5 nm.

The effect of increasing the thickness of the ferromagnetic layer on the reflectivity is shown in Fig. 9a. As it is shown, increasing the thickness of the ferromagnetic layer shifts θSPP slightly to larger angles. It is expected that the influence of ferromagnetic layer (CoFeB) in reflectivity, is also increased by increasing its thickness. Conversely, a thicker CoFeB interlayer can take magnetoplasmonic system away from the optimum SPP excitation due to the optical properties of magnetic layer. Therefore, there is an optimal thickness for the ferromagnetic layer. This feature is studied in the simulation results presented in Fig. 9b, where the maximum value of δ occurs at a thickness of 5nm. After this value, the Kerr signal decreases because the SPP excitation effect becomes weaker.Fig. 9 (a) The effect of increasing the thickness of the ferromagnetic layer on the reflectivity and the variation of θSPP with respect to thickness of the ferromagnetic layer. (b) Evolution of the Kerr signal (δ) with respect to incident angle for different thicknesses of the CoFeB layer and dependence of the maximum value of changing in δ (Δδ) versus of the thickness of the CoFeB layer.

It is worth mentioning, in our simulation, it has been assumed that the magnetization of the ferromagnetic layer is oriented solely in the positive or negative direction of the z-axis when an electric current is injected into the heavy metal layer. However, in reality, this alignment may not be entirely in the z direction. The SOFs may not have enough power to saturate the magnetization, but the magnetization is switched in a saturated state is assumed in order to optimum the structure so that transverse magneto-optical Kerr effect is at its maximum value. As a result, it is predicted that the laboratory results may be have small deviations from the presented simulation results, but in practice it does not affect the optimization process. However, our task shows that the general behavior of the SOF in magnetoplasmonics effect.

Conclusion

We have proposed and demonstrated a novel magnetoplasmonic structure with appreciable MOEs in which spin accumulation effects generate the magnetic field. It is based on the fact that by injection of an electric current in a HM layer spin separation takes place and spin accumulates at the HM/FM interface, creates the torques to the magnetic dipoles in the FM layer, and finally can change its magnetization. Our study provides an approach to achieve active light control using SOF-driven magnetoplasmonics. In this method, the generated internal magnetic field has enough amount to saturate the magnetization without any use of a permanent magnet or an external magnetic coil. As a result, a significant amount of TMOKE signal with two orders of magnitude of enhancement could be achieved by applying a reasonable electric current. Thus, high-power consumption and non-scalability related problems can be avoided in this proposed structure. Here, we have focused on the TMOKE, moreover the SOFs can be also used to obtain other magneto-optical effects, such as the Faraday effect and the polar and longitudinal Kerr effects for corresponding suitable configurations. On the other hand, the presence of the ferromagnetic material can be modulated the wavevector of SPPs by the injection of a small electrical current. In this way, spin-magnetoplasmonic systems are potential candidates for developing high-speed active plasmonic devices with low-power consumption.

Acknowledgements

The authors would like to thank Dr. Mojtaba Ranjbar for the involved yet fruitful discussions about magnetic materials and spintronic.

Author contributions

M.H.A.: software, data curation. M.H.F.: methodology, investigation, conceptualization, writing—review, and editing. All authors discussed the results and contributed to the final manuscript.

Data availability

The calculated results during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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