1. Introduction

Magneto-optical effects in magnetic dielectric crystals and films have been proven to be very valuable for light modulation, sensorics, magnetometry etc. [1–3]. To make values of the magneto-optical effects sufficiently large for these applications different ways can be used. The most straightforward approach is to find a proper chemical composition of magnetic constituents. However, this approach encounters some limitations and has been mainly exhausted for the last decades. That was the reason to develop some advanced approaches based on a designed nanostructuring of a magnetic medium [4–21]. Main concept was to localized light with in the magnetic medium to increase the effective length of interaction between photons and spins.

There is a fairly large number of works devoted to the Faraday rotation (FR) effect enhancement in magnetic dielectric films combined with plasmonic functionalities [6–13]. Among different types of the magnetoplasmonic structures, nanocomposites of noble metal nanoparticles combined with an iron-garnet film are of particular importance since, on the one side, they demonstrate an increased Faraday rotation angle of light polarization and, on the other side, they can be easily fabricated without necessity of using complex nanolithography facilities. Moreover, the magneto-plasmonic nanocomposites are very promising materials for different applications [13–17].

Up to now, the Faraday effect has been enhanced by several times in magnetoplasmonic nanocomposites if compared with the same bare magnetic film. Thus, article [7] demonstrates an increase of FR by less than 2 times. A more significant FR enhancement in a magnetoplasmonic nanocomposite was achieved in [9] and [6]: the FR gain reached 2.4 and 3.7 times, respectively. The greatest FR enhancement of about 8 times was demonstrated in [8,22].

Here we demonstrate an original approach in the magnetoplasmonic nanocomposite design which is based on the author’s method of the iron-garnet layer formation with the gradient of the thickness along the sample plane (Fig. 1(a), inset). This approach allows to achieve a record level of the Faraday effect enhancement as high as 20 times with respect to the same magnetic film but without metal nanoparticles. It is shown that interaction of different localized plasmonic modes with spins of bismuth-substituted iron-garnet lies in the origin of this huge enhancement. Optimal ratio of the gold nanoparticle radius and the thickness of the iron garnet film is found.

 

Fig. 1. (a) Distribution of BiIG film thickness h_BiIG along the gradient, inset – the scheme of the magnetoplasmonic nanocomposite GGG/Au_(NP)/BiIG_{(grad h)}; (b) Surface morphology of self-assembled Au_(NP) (REM-106 Selmi microscope, electron beam incidence angle is π/4 to the surface plane), inset – size distribution of Au_(NP) (columns - experimental data, line - approximation by the Gaussian function).

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2. Samples preparation and characterization

To obtain a magneto-plasmonic nanocomposite, a thin Au film with a thickness of h_eff = 5 nm was deposited on the gadolinium-gallium garnet Gd₃Ga₅O₁₂ (GGG) substrate (surface (111) crystallographic orientation) by thermal vacuum deposition at a residual gas pressure about 5·10⁻⁴ Pa. The nanoparticles (Fig. 1(b)) were obtained by annealing of the Au film in the air at a temperature of 950°C during 10 min [23]. The distribution of the self-assembled plasmonic Au nanoparticles is fairly well approximated by the Gaussian function with the most probable size (diameter) d₀ = 135 nm and standard deviation σ = 45 nm (Fig. 1(b), inset). After that, a layer of bismuth-substituted iron-garnet (BiIG) of composition Bi_2.0Gd_1.0Fe_3.8Al_1.2O₁₂ was deposited above by the method of ion-reactive sputtering of the target in an Ar(25%) + O2(75%) atmosphere. A special deposition technique was used to obtain a BiIG film with a thickness gradient along the sample [24]. After the deposition, the magnetic film was crystallized by annealing in air at a temperature of 680°C during 20 min.

The study of the BiIG layer thickness distribution along the gradient was carried out on a witness sample using a Linnik microinterferometer MII-4 with a digital processing unit (Fig. 1(a)). Optical and magneto-optical properties of the obtained nanocomposite GGG/Au_(NP)/BiIG_{(grad h)} at different values of the BiIG thickness were studied using an automated spectral magneto-polarimeter based on a KFK-3 spectrophotometer.

3. Results and discussion

3.1 Optical and plasmonic properties of the magneto-plasmonic nanocomposite

The optical transmittance spectra of a GGG/Au thin film (before annealing) and self-assembled plasmonic nanoparticles GGG/Au_(NP) (after annealing) are presented in Fig. 2(a). It can be seen that after annealing the transmittance spectrum exhibits a spectral minimum associated with the absorption of the incident radiation energy upon resonant excitation of the LPR dipole mode in self-assembled Au_(NP) nanoparticles (d-mode) [25–27]. In this case the resonant wavelength is λ_LPR = 660 nm.

 

Fig. 2. Optical and plasmonic properties of the GGG/Au_(NP)/BiIG_{(grad h)} nanocomposite at different stages of synthesis: (a) transmittance spectra of a GGG/Au gold film (before annealing) and self-assembled GGG/Au_(NP) nanoparticles (after annealing); (b) transmittance spectra of the GGG/Au_(NP)/BiIG_{(grad h)} nanocomposite at different thicknesses of the BiIG layer (h_BiIG is indicated in the legend); inset – the results of theoretical analysis; (c) derivative of the transmittance dT/dλ at different thickness of the BiIG layer (h_BiIG is indicated in the legend), inset – the results of theoretical analysis; (d) the spectral position of various LPR modes as a function of h_BiIG (points – experimental data, lines – theoretical analysis, inset – a scheme of the theoretical model).

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It is seen that after deposition of the BiIG layer over plasmonic nanoparticles, the dipole d-mode of the LPR is shifted from λ_LPR = 660 nm to the infrared part of the spectrum at wavelength 760 nm and larger (Fig. 2(b)). Besides, a second local minimum appears in the spectra at around 650 nm, which is due to the excitation of an additional “high-frequency” LPR mode [28].

It can be seen that with an increase in the BiIG layer thickness, both LPR modes experience a significant “red” shift, while the efficiency of excitation of the “high-frequency” mode increases, and the “low-frequency” dipole mode decreases. Therefore, we can conclude that the additional “high-frequency” mode arises as a result of interaction of the neighboring plasmon dipoles through the BiIG magnetic dielectric layer, i.e. this mode, in its physical essence, is the mode of the coupled dipole-dipole oscillations (the d-d-mode). The emergence of a similar mode after the deposition of an iron-garnet layer was observed earlier in [12].

The d-d-mode resonance gets more pronounced with an increase of the BiIG layer thickness, since a thicker BiIG layer fills the interparticle space more efficiently and the garnet film enhances interaction of the magnetic components of the electromagnetic field of the resonating plasmonic dipoles. However, it should be noted that such increase is observed only up to the BiIG layer thickness of 160–180 nm. It is associated with the limited penetration depth of the optical field of resonating plasmonic particles into the surrounding dielectric medium (near-field interaction).

The d-d-modes excitation is less efficient than the d-modes excitation because the d-modes are exciting in all nanoparticles but the d-d-modes are exciting only in some particles that are locate near each other and capable to forming a coupled dipole–dipole state.

To analyze the resonance shift of various LPR modes with a change in the thickness of the BiIG layer, Fig. 2(c) shows the dependence of the derivative of the transmittance spectrum dT/dλ: vanishing dT/dλ corresponds to the resonant wavelengths. Variation of the BiIG layer thickness changes the resonance wavelengths λ_LPR (h_BiIG) for both LPR modes (Fig. 2(d)). It should be noted that with an increase of the BiIG thickness above 160 nm, the shift of resonance practically does not occur, which is due to the limited penetration depth of the localized plasmon field into the magnetic medium. Thus, the effective interaction (penetration depth of the near field) of the resonating plasmon subsystem with the magneto-optical BiIG layer is achieved at scales not exceeding h_NF = 2.5·r_NP [29], where r_NP is the average radius of Au nanoparticles.

The spectral shift of the plasmon resonance dipole mode with increasing of the BiIG layer thickness can be described within the framework of the quasi-static approximation of the polarizability of a spherical metal nanoparticle (NP) with radius r_NP in a dielectric shell with radius r_s (see the inset in Fig. 2(d)). In an environment with a dielectric constant ɛ_env, the extinction coefficient for a spherical particle is described by an expression [26,30,31]:

Here ${\mathrm{\varepsilon }_{\textrm{NP}}} = \mathrm{\varepsilon }_{\textrm{NP}}^{\prime} + i\mathrm{\varepsilon }_{\textrm{NP}}^{^{\prime\prime}}$ is the complex dielectric constant of the metallic NP, ${\mathrm{\varepsilon }_\textrm{s}}$ is the dielectric constant of the shell, f = 1–(r_NP/r_s)³ is the core-shell volume parameter.

In the case when r_NP << λ the spectral dependence for ɛ_NP is determined by the expression [33]:

Here ${\mathrm{\varepsilon }_\textrm{M}} = \mathrm{\varepsilon }_\textrm{M}^{\prime} + i\mathrm{\varepsilon }_\textrm{M}^{^{\prime\prime}}$ is the frequency (spectral) dependence of the complex dielectric constant for a bulk metal, due to the band-to-band transitions. In Eq. (4) ω = 2πс/λ is the frequency of the exciting radiation, and the plasmon self-frequency ω_p is defined as ${\mathrm{\omega }_\textrm{p}} = {\left( {\frac{{n{e^2}}}{{{\mathrm{\varepsilon }_0}m}}} \right)^{1/2}}$, where ɛ₀ is the dielectric constant, n is the concentration of electrons in the metal, e/m is the specific charge of an electron. The electron collision frequency ω_τ is determined from the relation: ω_τ = v_F/l + 2v_F/r_NP, where v_F – is the Fermi velocity, l is the electron mean free path.

To analyze the plasmonic d-d-mode let us consider a model of the dipole-dipole interaction of neighboring plasmonic nanoparticles, taking into account the influence of the dielectric shell. In the case of parallel orientation of neighboring dipoles with a dipole moment P (within the framework of the model we assume that the dipoles are the same), located at a distance a from each other, the resulting dipole moment of the particle is determined by the expression:

Expressing the dipole moment from (5), we obtain:

In Eq. (6) g’ is the complex polarizability factor of a dipole in an external field E₀ in the in-phase oscillations with a neighboring dipole. Let us express g’ from (6) as:

Substituting the obtained value for the polarizability g’ in Eq. (1), we obtain the extinction spectrum upon excitation of the coupled LPR mode.

In expressions 1–3 and 5–7, the effect of the substrate is taken into account when calculating the dielectric constant of the shell ɛ_s, based on the effective medium model. It gives а good agreement between model and experiment for spectral location of various plasmonic resonance modes.

The total spectra of extinction were obtained as the sum of expressions for single dipole oscillations (d-mode) and for coupled dipole-dipole oscillations (d-d-mode).

Equation (1) provides a transmittance spectra T = 1 – Q_ext and their derivatives dT/dλ which are in a good correspondence with experimental data for different thicknesses of the BiIG film (in the model – the shell thickness: r_NP – r_s) (insets to Fig. 2(b), 2(c)).

Comparison of the theoretical model and experimental results for transmittance and its derivative plotted on one graph are shown in Fig. 1(S) (Supplement 1).

These results of calculate2d resonance wavelengths for the d-mode and d-d-mode with respect to the BiIG film thickness (Fig. 2(d), solid lines) are in an excellent agreement with the experimental data (Fig. (d), circles).

3.2 Enhancement of the Faraday effect in the magnetoplasmonic nanocomposite

To study the magneto-optical properties of the GGG/Au_(NP)/BiIG_{(grad h)} nanocomposite the rotation angle Θ_TR of the polarization plane of light was measured in the spectral range of 460–980 nm for the sample magnetically saturated out-of-plane (the external magnetic field was H₀ = 170 mT, while the saturation magnetic field is H_S ≈ 80 mT). The measurements were carried out with the magnetic field vector oriented both along the light wavevector (Θ_TR(H+)), and opposite to it (Θ_TR(H-)).

The Faraday rotation angle Θ_FR can be calculated then by (Fig. 3):

Here we avoid a non-Faraday input in the observed polarization rotation and take into account the Faraday rotation of the paramagnetic GGG substrate Θ_FR(GGG)(λ).

 

Fig. 3. (a-e) Spectra of the Faraday effect in the GGG/Au_(NP)/BiIG_{(grad h)} nanocomposite (black solid line) and the same magnetic film without nanoparticles GGG/BiIG (red dashed line) for different thicknesses of the BiIG layer: a – 206 nm; b – 160 nm; c – 136 nm; d – 100 nm; e – 78 nm. (f) The enhancement factor of the Faraday rotation at the d-mode and d-d-mode LPR versus thickness of the BiIG layer.

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Comparison of the Faraday rotation of the nanocomposite Θ_FR(BiIG-Au)(λ) (Fig. 3(a)-3(e), black solid line) and the same BiIG film but without nanoparticles Θ_FR(BiIG)(λ) (Fig. 3(a)-3(e), red dashed curve), reveals a significant enhancement of the Faraday rotation in the spectral vicinity of both LPR modes. It can be quantified by the enhancement factor (Fig. 3(f)):

As can be seen from Fig. 3, in the region of resonant excitation of various plasmonic modes in the magnetoplasmonic nanocomposite GGG/Au_(NP)/BiIG_{(grad h)}, the Faraday effect is enhanced. In this case, the magnitude of the enhancement factor is determined by the efficiency of the plasmonic mode excitation and the efficiency of the near-field interaction of the magnetic and plasmon subsystems in the composite.

The enhancement due to the LPR d-mode increases significantly with a decrease of the BiIG layer thickness and reaches giant value of η=21 for the BiIG thickness of h_BiIG = 78 nm. To the best of our knowledge this is the largest enhancement of the Faraday effect ever achieved by localized plasmon modes.

In the spectral region of the LPR d-d-mode excitation the Faraday effect enhancement is less pronounced: η is about 1.42–1.56.

4. Conclusions

In this work we have found a giant enhancement of the Faraday effect in magnetoplasmonic nanocomposite films based on Au nanoparticles and bismuth substituted iron garnet BiIG layer. It is due to the excitation of the single and collective localized plasmon resonances in the Au nanoparticles. The phenomenon is studied for different thicknesses of the BiIG layer. A decrease of the BiIG layer thickness below 160 nm, which corresponds to h_NF = 2.5·r_NP provides a spectral shift of the plasmonic resonances and increases the enhancement factor of the Faraday effect. For the BiIG layer thicker than 160 nm the magnetic layer thickness doesn’t influence the phenomenon anymore since it exceeds penetration depth of the near field of the localized plasmons into the magnetodielectric medium.

The giant enhancement of the Faraday rotation by d-mode was obtained due to the optimal ratio of parameters of the plasmonic and magnetic subsystems of the nanocomposite. It has become possible due to using an original approach of the magnetoplasmonic nanocomposite design based on a formation of the iron-garnet layer with the gradient of the thickness along the sample plane.

The largest enhancement exceeds 20 times for the thinnest BiIG film available for experimental studies. It is mediated by the single plasmonic dipole mode, while the collective dipole-dipole mode provides a much smaller increase of the Faraday rotation. The efficiency of d-d-mode excitation (and FR enhancement) can be increased by creating strictly periodic structures in which all nanoparticles can effectively form dipole-dipole pairs. In this case, the effects comparable in magnitude can be expected, or larger for d-d-modes. The effect of interaction between plasmonic d-d-mode and magnetic medium is very interesting and, with optimally selected parameters, can give a noticeable enhancement of the Faraday effect, therefore, its experimental study will be the topic of future scientific research.

The proposed magnetoplasmonic nanocomposite might serve as a building block for different nanophotonic structures including magnetophotonic and magnetoplasmonic crystals. The similar composite materials and structures with giant enhancement of the Faraday effect can be used for sensor magnetooptical systems with a high sensitivity.