Abstract
We present a systematic study on the optical and magnetooptical properties of Ni/SiO_{2}/Au dimer lattices. By considering the excitation of orthogonal dipoles in the Ni and Au nanodisks, we analytically demonstrate that the magnetoplasmonic response of dimer lattices is governed by a complex interplay of near and farfield interactions. Nearfield coupling between dipoles in Ni and lowloss Au enhances the polarizabilty of single dimers compared to that of isolated Ni nanodisks. Farfield diffractive coupling in periodic lattices of these two particle types enlarges the difference in effective polarizability further. This effect is explained by an inverse relationship between the damping of collective surface lattice resonances and the imaginary polarizability of individual scatterers. Optical reflectance measurements, magnetooptical Kerr effect spectra, and finitedifference timedomain simulations confirm the analytical results. Hybrid dimer arrays supporting intense plasmon excitations are a promising candidate for active magnetoplasmonic devices.
Introduction
Noblemetal nanoparticles are widely used in plasmonics because their high electrical conductivity supports the excitation of lowloss localized surface plasmon resonances (LSPRs)^{1}. The ensuing optical response of metal nanoparticles can be tuned by variation of their size, shape, or arrangement^{2,3}. Strong enhancements of the optical field at the surface of metal nanoparticles and in their immediate vicinity are exploited, for instance, in biological and chemical sensors^{4,5}, photovoltaics^{6}, and optoelectronics^{7}. Nanoparticles made of ferromagnetic metals also support the excitation of LSPRs^{8,9,10,11}. Since plasmon resonances and magnetooptical activity are strongly linked in ferromagnetic nanoparticles, their magnetooptical spectra can be tailored by employing design rules known from plasmonics. Conversely, nanoscale ferromagnets enable active control of light via magnetization reversal in an external field. Both effects are relevant for technology and are studied in the field of magnetoplasmonics^{12,13,14}.
Large ohmic losses in ferromagnetic metals lead to significant damping of plasmon resonances. To overcome this limitation, hybrid structures comprising ferromagnetic and noble metals have been explored as magnetoplasmonic systems. Examples include, Au/Co/Au trilayers^{15}, nanosandwiches^{16}, and nanorods^{17}, coreshell Co/Ag or Co/Au nanoparticles^{18,19} and nanowires^{20}, and Au/Ni nanoring resonators^{21}. Contacting subwavelength ferromagnetic elements and noble metals results in materials that can be considered as optical alloys. Various noncontacting realizations have also been investigated. Dimers of two metal nanodisks that are separated by a dielectric layer are particularly attractive as they allow for a strong redistribution of the optical nearfield^{22}. In vertical dimers containing noble and ferromagnetic metals, this effect has been exploited to enlarge the magnetooptical response via an increase of the optical field in the ferromagnetic layer^{23} or induction of magnetooptical activity on the lowerloss noble metal^{24,25}.
Another way to circumvent large ohmic losses in ferromagnetic nanoparticles involves the excitation of collective plasmon modes. In periodic arrays of metal nanoparticles, radiative coupling between LSPRs and diffracted waves in the array plane produces narrow and intense surface lattice resonances (SLRs)^{26,27,28,29,30}. Lowloss SLRs in arrays of noble metal nanostructures are used in several contexts, including sensing^{31,32,33}, lasing^{34,35}, and metamaterials^{36,37}. In ferromagnetic nanoparticle arrays, SLRs enhance the magnetooptical activity and provide versatility in the design of magnetooptical spectra via the tailoring of lattice symmetry or particle shape^{38,39}. Checkerboard patterns of pure Ni and Au nanodisks have also been studied^{40}. In this hybrid approach, farfield diffractive coupling between the different particles enhances the magnetooptical response via the excitation of lowloss SLRs and the induction of magnetooptical activity on the Au nanodisks.
Here, we report on tunable magnetoplasmonics in lattices of Ni/SiO_{2}/Au dimers (Fig. 1). Our structures combine two aforementioned approaches, namely, the integration of noble and ferromagnetic metals in vertical dimers^{23,24,25} and ordering of magnetooptically active elements in periodic arrays^{38,39,40}. Because the noble metal and ferromagnetic constituents of our lattices interact via optical nearfields within dimers and farfields between dimers, the hybrid arrays provide a rich playground for the design of optical and magnetooptical effects. First, we present an analytical model to evaluate the effect of dimer polarizability and lattice periodicity on the magnetoplasmonic properties of our system. Next, we compare model calculations and experiments on dimer arrays with different lattice constants. As reference, we discuss experiments on arrays with Au and Ni nanodisks.
Modeling
We start our analysis by calculating the optical and magnetooptical response of an individual plasmonic nanoparticle based on the modified long wavelength approximation (MLWA)^{41}. The absorption and emission properties of a metal nanoparticle are described by its volume polarizability α_{e}′, which relates the induced polarization P to the incident electric field E_{i}. If the particle is small compared to the wavelength of incident light, the electric field inside the particle E_{1} is approximately uniform. Following classical electrodynamics, the electric field inside the nanoparticle is given by E_{1} = E_{i} + E_{d}, where E_{d} is the depolarization field. E_{d} can be calculated by assigning a dipole moment dp = PdV to each volume element dV of the nanoparticle and calculating the retarded depolarization field dE_{d} generated by dp in the nanoparticle center^{42}. This gives
Here, L_{d} is the depolarization factor describing interactions between polarizable volume elements inside the particle^{43}. The nanodisks that we consider in our study can be approximated as ellipsoids^{41,44}. For ellipsoidal particles, Eq. 1 can be solved analytically. This gives
The three terms in Eq. 2 include static (L) and dynamic (D) depolarization factors that account for the particle shape and a radiative reaction correction (ik^{3}V/6π)^{42}. To calculate L and D, we use the integrals given in refs^{41,42}. The net dipole moment of an ellipsoidal particle (dp = PdV) can be written as
where ε_{d} and ε_{m} are the permittivity of the particle and surrounding medium, respectively, α_{e} is the particle polarizability (α_{e} = α_{e}′V), and V is the particle volume. Combining Eqs 1 and 3 gives
The permittivity of a particle changes in the presence of a large external magnetic field or spontaneous magnetization. In our experiments, we use perpendicular magnetic fields of ±400 mT to saturate the magnetization of Ni nanodisks along the zaxis (the saturation field of the Ni nanodisks is approximately 300 mT). The permittivity tensor for this configuration contains two offdiagonal components^{45}
where m_{z} is the perpendicular magnetization normalized to 1 and Q is the Voigt magnetooptical constant. We use tabulated data from ref.^{46} to calculate the permittivity of Ni. Because the fieldinduced diamagnetic moment of Au is small (m_{z} ≈ 0) compared to the magnetization of Ni, we set the offdiagonal terms of ε_{d} to zero for this material. We use optical constants from ref.^{47} to calculate the permittivity of Au.
Following Eq. 4, nonzero offdiagonal components in ε_{d} lead to offdiagonal terms in the polarizability tensor. Macroscopically, this produces a rotation and ellipticity in the polarization of reflected (magnetooptical Kerr effect) or transmitted (Faraday effect) light. For nanoparticles, the microscopic origin of magnetooptical activity can be understood by considering the excitation of two orthogonal LSPRs. One of the LSPRs, which can be described as electric dipole p, is driven by the incident electric field E_{i}. For linearly polarized light at normal incidence, the induced dipole is oriented inplane along E_{i}. If the nanoparticle exhibits perpendicular magnetization (m_{z}), a second electric dipole is induced orthogonal to E_{i} and m_{z} by spinorbit coupling. The amplitude and phase relations of the two excited dipoles determine the rotation and ellipticity of light polarization upon reflection or transmission^{10}. In our study, the incident electric field is oriented along the xaxis, the magnetization of Ni is saturated by a perpendicular magnetic field, and the spinorbit induced dipole is oriented along y (Fig. 1(b)). Hereafter, we refer to the directly excited dipole (p_{x}) as optical dipole. The dipole along the orthogonal direction (p_{y}) is labeled as magnetooptical dipole.
If dimers are formed from Au and Ni nanodisks, their optical nearfields couple. To describe this effect, we consider the electric field at each dipole position as the sum of the incident electric field and the scattered field from the dipole in the other disk. This results in two coupled equations
Here, E_{i1} and E_{i2} define the incident electric field at the Ni and Au nanodisks (including a phase difference), and G is a dyadic Green’s function describing how the electric field that is produced by one dipole propagates to the other^{48}. G is given by
where R is a vector connecting the dipoles in the two disks, R is its amplitude, and k = 2nπ/λ, with n the refractive index of the spacer layer and surrounding medium. Since electric dipoles are excited in the dimer plane, they mostly couple along the zaxis. Consequently, R ⊗ R in Eq. 7 is approximately zero. The optical and magnetooptical spectra of dimers are defined by dipole excitations along x and y. Considering nearfield coupling between the Ni and Au nanodisks (Eq. 6), the effective dipole moment along these axes can be written as
where α_{xx}, α_{yy}, and α_{xy} are the diagonal and offdiagonal components of the polarizability tensor (α) of a single Ni/SiO_{2}/Au dimer. We note that while offdiagonal components are absent in the polarizability matrix of Au, a magnetooptical dipole is induced on the Au nanodisk (p_{Au,y}) because of nearfield coupling to p_{Ni,y} (Eq. 6). The lowerloss Au nanodisk thus contributes to the magnetooptical activity of the dimer^{24,25}.
If dimers are ordered into a periodic array, the electric field at each lattice position is a superposition of the incident radiation and dipolar fields from other dimers. The optical and magnetooptical response of a periodic dimer array thus depend on the polarizability of single dimers (α) and their twodimensional arrangement. To take farfield coupling between dimers into account, we define an effective lattice polarizability^{27,28,49}
where S is the lattice factor. For an infinite array, this parameter is given by^{50,51}
where r_{j} is the distance between dimers and θ_{j} is the angle between the effective dipole moment and the vector connecting the dimers. For a twodimensional lattice under normal incidence radiation, we can thus write
where S_{x} and S_{y} are the lattice factors for radiation along x and y. Since α_{xx,yy} >> α_{xy}, the diagonal components of the effective lattice polarizability only depend on the diagonal terms of α and S. The offdiagonal components of α_{eff} contain more intricate parameter relations. By carrying out matrix operations (see Supplementary Note 1), we find
and
The effective dipole moments of the dimer lattice are thus given by
Equation 13 reveals a complex relationship between the polarizability of the dimers and their periodic arrangement. Because magnetooptical dipoles (p_{y}) are excited orthogonal to the optical dipoles (p_{x}), the polarizability and lattice factor along the yaxis also affect α_{eff,xy}^{39}.
For linearly polarized light at normal incidence, the optical reflectance and magnetooptical activity are linked simply to the effective lattice polarizability. In this geometry, the reflectance of a periodic plasmonic array is proportional to the scattering cross section^{44}
and thus
The magnetooptical Kerr angle Φ of a dimer lattice is defined as the amplitude ratio of the magnetooptical (p_{eff,y}) and optical (p_{eff,x}) dipoles
Following Eqs 16 and 17, it is possible to extract a quantity that is proportional to p_{eff,y} by multiplying the Kerr angle Φ by the square root of the optical reflectance R
Results and Discussion
To experimentally explore near and farfield coupling in dimer arrays, we fabricated periodic lattices of Ni/SiO_{2}/Au dimers on glass substrates using electronbeam lithography^{52}. The lower Au and upper Ni nanodisks of the dimers have a diameter of ~120 nm and ~110 nm, respectively, and both disks are 15 nm thick. The two metals are separated by 15 nm SiO_{2}. The lattice constants along x and y are 400 nm, 450 nm, or 500 nm. For comparison, we also fabricated arrays of pure Au and Ni nanodisks. The Au nanodisks have the same size as in the dimers. Because the optical reflectance from pure Ni nanodisks is small, we decided to increase their diameter and thickness to ~130 nm and 18 nm. In addition, we fabricated samples with randomly distributed dimers and nanodisks to characterize the optical and magnetooptical response without SLRs. All measurements were conducted with the nanoparticles immersed in indexmatching oil (n = 1.52). The creation of a homogeneous refractiveindex environment enhances the efficiency of farfield coupling between scatterers and, thereby, the excitation of collective SLR modes. More experimental details are given in the Methods section.
We first discuss the optical and magnetooptical response of randomly distributed dimers and nanodisks (Fig. 2). A filling fraction of 5% was chosen for these samples to approximately match those of periodic arrays (7% for a = 400 nm, 5% for a = 500 nm). Because of the low filling fraction, randomly distributed dimers and nanodisks can be considered as noninteracting and, consequently, their optical spectra represent the properties of individual nanoparticles. Figure 2(a) compares reflectance spectra of randomly distributed Ni/SiO_{2}/Au dimers and Au and Ni nanodisks. Nearfield coupling between the Au and Ni disks of dimers redshifts the LSPRinduced reflectance maximum. The LSPR wavelength of a dimer is measured at ~860 nm, while those of the Au and Ni nanosdisks are recorded at ~790 nm and ~720 nm, respectively. The LSPR linewidth of the dimer structure is also larger than that of the Au nanodisk because of dipolar coupling to a higherloss excitation in Ni. Figure 2(b) shows the magnetooptical Kerr angle of the dimer and Ni nanodisk. From data in Fig. 2(a,b) we also extract \({\boldsymbol{\Phi }}\sqrt{R}\), which is proportional to the magnetooptical dipole amplitude p_{y} (Eq. 18). For the dimer structure (red line), p_{y} is the vector sum of a spinorbit induced magnetooptical dipole in Ni (p_{Ni,y}) and the dipole moment that it produces on Au (p_{Au,y}). The values of p_{y} for the dimer and Ni nanodisk are similar at ~800 nm, despite the latter containing ~70% more Ni. This result confirms that the Au nanodisk of a dimer contributes to the magnetooptical activity. We also note that p_{y} of the dimer structure decays more strongly below the resonance wavelength. This effect is caused by a weakening of the nearfield coupling strength at shorter wavelengths, i.e., a decrease of p_{Au,y}, as illustrated by calculations of the dyadic Green’s function describing dipolar coupling inside the dimer (Supplementary Note 2).
To further delve into the details of nearfield coupling in our magnetoplasmonic dimers, we present calculations of p_{x}^{2} and p_{y} of single nanodisks and dimers in Fig. 2(d,f). By plotting data in this format, the results can be compared directly to the experimental spectra of Fig. 2(a,c). We also show the calculated magnetooptical Kerr angle (p_{y}/p_{x}) in Fig. 2(e). In all cases, the wavelengths and lineshapes of plasmon resonances agree well. Main features such as a redshift of the dimer LSPR are thus reproduced. In the calculations, we can separate how dipole moments in the Au and Ni nanodisks contribute to the optical and magnetooptical response of dimers. Taking the phase difference between excitations in Au and Ni along x and y (ϕ_{x}, ϕ_{y}) into account, the optical and magnetooptical dipoles of dimers are given by
Analyzing the results of Fig. 2(f), we find that, in dimers, the maximum magnetooptical dipole strength in Au is about 75% compared to that of Ni. The strong p_{Au,y} is explained by the large polarizability of Au, enabling p_{Ni,y} to effectively induce a magnetooptical dipole moment on Au. The calculations thus confirm the big impact of p_{Au,y} on the magnetooptical activity of single dimers.
We now consider the optical and magnetooptical response of dimer lattices. In farfield measurements, SLRs with an asymmetric lineshape and narrow RayleighWood anomalies emerge from radiative coupling between LSPRs and diffracted waves in the array plane^{26,27,28,29,30}. The anomalies appear at the diffracted orders (DOs) of the array, which are defined by
Here, θ_{k} is the angle of the k^{th} diffracted order, θ_{i} is the angle of incidence, λ is the wavelength, n is the refractive index of the embedding medium, and a is the lattice constant. For normal incident light (θ_{i} = 0°), a RayleighWood anomaly associated with the passing of a DO is measured in reflectance or transmittance spectra when kλ = na. This corresponds to a transition from an evanescent to a propagating lattice mode if sinθ_{k} = ±1 in Eq. 20. For a twodimensional lattice and normal incident light, the DO wavelengths (λ_{DO}) can be calculated using \(\sqrt{({p}^{2}+{q}^{2})}{\lambda }_{{\rm{DO}}}=na\), where p and q indicate the order of diffraction along x and y. If the DO of the array and the LSPRs of individual nanodisks overlap, an asymmetric SLR comprising optical and plasmonic components is formed. The excitation of a hybrid SLR mode enhances the optical field at the nanodisks. In ferromagnetic nanodisk arrays, this effect enhances the magnetooptical activity^{38,39}.
Figure 3(a–c) show optical reflectance spectra for square arrays of dimers and Au and Ni nanodisks with lattice constants of 400 nm, 450 nm, and 500 nm. For these lattices, RayleighWood anomalies are observed at λ_{DO} = 610 nm, 680 nm, and 760 nm, respectively, in agreement with λ_{DO} = 1.52a. For the Au nanodisk array with a = 400 nm, the DO is positioned at the lower tail of the LSPR. In this case, the SLR mode is broad and almost symmetric. For a = 450 nm and a = 500 nm, the LSPR of the Au nanodisks and DO overlap more, causing narrower and asymmetric Fanolike SLRs. The evolution of the spectral line shapes with lattice constant shown in Fig. 3(b) corresponds to previously published data on Au and other noble metal nanostructure arrays^{27,28,29}. For all dimer and Ni nanodisk lattices in this study, the DOs overlap with their broader LSPRs. Consequently, clear RayleighWood anomalies are measured and these sharp features are followed by asymmetric SLR reflectance peaks. Because the LSPRs of individual dimers and nanodisks are different, hybridization of these modes with diffracted waves in the array plane produces SLRs with different lineshapes, resonance wavelengths, and intensities. For all particle types, the excitation of a SLR mode significantly increases the reflectance in comparison to randomly distributed dimers and nanodisks (Fig. 2(a)). In arrays with a = 400 nm, the reflectance increases by a factor ~10 for the Ni nanodsiks and a factor ~40 for the dimers and Au nanodisks. The induced optical dipoles in particle lattices (\({p}_{\mathrm{eff},x}\propto \sqrt{R}\)) are therefore significantly enhanced near the SLR wavelength.
To analyze how excitations in the Au and Ni nanodisks of dimers contribute to the optical response of a periodic array, we consider the effective lattice polarizability along the incident electric field (α_{eff,xx} in Eq. 12). Parameter α_{eff,xx} depends on the polarizability of individual dimers α_{xx} and the lattice factor S_{x}. In Fig. 4 we plot the real and imaginary parts of 1/α_{xx} and S_{x} for different lattice parameters. Data for the inverse polarizability of Ni and Au nanodisks are shown also. The effective polarizability of a nanoparticle lattice is resonantly enhanced when the real part of the denominator in Eq. 12, 1/α_{xx}−S_{x}, becomes zero. This condition corresponds to a crossing of the Re(1/α_{xx}) and Re(S_{x}) curves in Fig. 4(a–c). The intensity and linewidth of the resulting SLR modes depend on the slope with which Re(1/α_{xx}) and Re(S_{x}) cross and the imaginary values of these parameters. For large Im(1/α_{xx})− Im(S_{x}) (Fig. 4(d–f)), the SLRs are damped strongly. Since S_{x} solely depends on the lattice geometry, single particles only affect the excitation of SLRs through their inverse polarizability. Because Im(1/α_{xx}) can be written as −Im(α_{xx})/α_{xx}^{2}, it is approximated by −1/Im(α_{xx}) close to the resonance condition (Re(α_{xx}) ≈ 0). For a dimer without gain α_{xx} is positive and Im(1/α_{xx}) is negative. Consequently, the lattice factor S_{x} contributes to the damping of SLR modes if Im(S_{x}) is positive. In contrast, negative Im(S_{x}) counteracts the ohmic losses of individual nanoparticles, enabling the excitation of more narrow and intense SLRs. Because Im(S_{x}) changes sign from positive to negative at the DOs of a lattice, stronger SLR excitations are generated when the Re(1/α_{xx}) and Re(S_{x}) curves cross at λ > λ_{DO}.
The integration of Au into Ni/SiO_{2}/Au dimers, enlarges the polarizability of dimers in comparison to Ni nanodisks. Consequently, Im(1/α_{xx}) is smaller and SLR modes are less damped. Figure 4(d–f) illustrate the large difference between Im(1/α_{xx}) of dimers and Ni nanodisks at relevant SLR wavelengths. To put some numbers on the resonant enhancement of the effective polarizability in our lattices, we compare the values of p_{x} in Fig. 2(a) and p_{eff,x} in Fig. 3(a,c). For single Ni/SiO_{2}/Au dimers and larger Ni nanodisks we extract α_{xx}(dimer)/α_{xx}(Ni disk) ≈ 1.5. In square lattices of the same particles α_{eff,xx}(dimer array)/α_{eff,xx}(Ni disk array) ≈ 3.1.
In Fig. 4(a) multiple crossings between Re(1/α_{xx}) and Re(S_{x}) are calculated for dimer and nanodisk arays with a lattice constant of 400 nm. However, only one of them, observed at λ = 690 nm for Ni nanodisks, λ = 780 nm for Au nanodisks, and λ = 805 nm for Ni/SiO_{2}/Au dimers, coincides with a situation where Im(1/α_{xx}) − Im(S_{x}) is small (Fig. 4(d)). Consequently, one intense SLR mode is expected for these lattices, in agreement with the experimental spectra of Fig. 3(a–c). Similar observations can be made for square arrays with lattice constants of 450 nm and 500 nm. The anticipated wavelengths of lowloss SLR modes for all particle types and lattice constants are indicated by vertical lines in Fig. 4. Coupling between the diagonal (1,1) DO and LSPRs produces an additional SLR in lattices with a = 500 nm. However, since Im(1/α_{xx}) is large at the wavelength of this mode, it appears much more damped in reflectance measurements.
Another feature in the experimental reflectance spectra of Fig. 3(a–c) that can be explained by considering Fig. 4 is the dependence of SLR wavelength on lattice constant. Because the slope of Re(1/α_{xx}) is particularly large for Au nanodisks, the crossing point between Re(1/α_{xx}) and Re(S_{x}) only shifts slightly with increasing a. In the experimental curves, this results in a minor inconsistency (λ_{SLR}(a = 400 nm) > λ_{SLR}(a = 450 nm)), which we attribute to sampletosample variations in the shape or size of the nanodisks. In contrast, smaller slopes of Re(1/α_{xx}) for dimers and Ni nanodisks result in stronger tuning of the SLR wavelength with lattice constant.
To calculate reflectance spectra of the different lattices (p_{eff,x}^{2}), we insert data for 1/α_{xx} and S_{x} from Fig. 4 into Eqs 12 and 14. The results are shown in Fig. 3(d–f). While our model calculations reproduce the main spectral features of the experimental curves, the resonances are more narrow. We attribute this discrepancy to inevitable imperfections in the experiments. For instance, we use a Gaussian beam with a finite wavelength range to excite our samples, while monochromatic plane waves are assumed in the calculations. Also, a finite distribution in the size and shape of the dimers and nanodisks (see Fig. 1(a)) broadens the experimental resonances. We also note that the RayleighWood anomalies appear more broad in the calculations. This effect is caused by the finite size of the lattice (30 × 30 particles) that we used in the calculation of S.
After establishing the optical response of different lattices, we now turn our attention to the magnetooptical activity of periodic Ni/SiO_{2}/Au dimer arrays. For comparison, we also discuss data for lattices with Ni nanodisks. Figure 5 shows the magnetooptical Kerr angle for square arrays with different lattice constants. Just like the optical reflectance measurements of Fig. 3(a–c), the magnetooptical Kerr spectra are shaped by RayleighWood anomalies (sharp minima) and SLR excitations (strong signal enhancements at λ > λ_{DO}). The magnitude of the Kerr effect is comparable for periodic arrays of Ni/SiO_{2}/Au dimers and Ni nanodisks. According to Eq. 17, the offdiagonal to diagonal polarizability ratio (α_{eff,xy}/α_{eff,xx}) determines the Kerr angle of a lattice. Because the diagonal polarizability of the dimer array is much larger than that of the Ni lattice, we conclude that the offdiagonal polarizability of the dimer array must be similarly enlarged. To substantiate this claim, we multiply the Kerr data of Fig. 5(a,b) with the square root of the reflectance spectra in Fig. 3(a,c). The resulting parameter \({\boldsymbol{\Phi }}\sqrt{R}\), shown in Fig. 5(c,d), is proportional to the effective magnetooptical dipole (p_{eff,y}) of the dimer and Ni lattices (Eq. 18). Alike the effective optical dipole p_{eff,x} (Fig. 3), the magnetooptical dipole p_{eff,y} of the Ni/SiO_{2}/Au dimer arrays is substantially stronger than that of pure Ni lattices. Thus, although the p_{y}′s of individual dimers and larger Ni nanodisks are similar (Fig. 2(c)), the effective magnetooptical dipole is enhanced much more when dimers are ordered into periodic arrays. This result can be understood by considering Eq. 13 for the offdiagonal polarizabilities of a nanoparticle array. The effective offdiagonal polarizabilities of an array are directly proportional to the offdiagonal polarizabilities of the individual nanoparticles, which, as stated earlier, are similar for dimers and Ni nanodisks. However, the effective offdiagonal polarizability is resonantly enhanced when the real part of the denominator in Eq. 13 becomes zero. For square lattices with α_{xx} = α_{yy} and S_{x} = S_{y}, this condition is met when the Re(1/α_{xx}) and Re(S_{x}) curves in Fig. 4 cross. Since resonances in α_{eff,xx} and α_{eff,xy} are determined by the same parameters in square arrays, the shapes of their optical and magnetooptical spectra are identical. Moreover, because Im(1/α_{xx}) is smaller for dimers than Ni nanodisks at the resonance wavelength, the magnetooptical Kerr angle is enhanced more by the excitation of an SLR mode in dimer arrays than in Ni lattices. Finally, we calculate the Kerr angle and magnetooptical dipole for both lattice types using the parameters of Fig. 4 and Eqs 12–14 and 17. Results are plotted in Fig. 5(e–h). The good agreements between the measured and calculated spectra demonstrate that our analytical model describes the physics of combined near and farfield coupling in hybrid dimer lattices.
To visualize the excitation of SLRs in dimer and Ni nanodisk arrays, we performed finitedifference timedomain (FDTD) simulations. Results for square arrays with a lattice constant of 400 nm are shown in Fig. 6. The data are obtained at λ = 780 nm for both particle types. At this wavelength, the magnetooptical Kerr angle is enhanced by the excitation of a collective SLR mode (see Supplementary Note 3). Strong optical dipoles are directly excited by the incident electric field E_{i} along the xaxis. Through spinorbit coupling in Ni nanodisks with perpendicular magnetization and near and farfield interactions between Ni and Au disks, magnetooptical dipoles are induced along the yaxis in both Ni and Au. In agreement with our experiments and model calculations, the simulated dipole moments along x and y are larger in Ni/SiO_{2}/Au dimer arrays than in Ni nanodisk lattices.
Finally, we consider the optical and magnetooptical response of rectangular dimer lattices with a_{x} ≠ a_{y}. Based on our model, the optical reflectance of rectangular lattices depends on α_{eff,xx}. Because the lattice factor S_{x} peaks when λ = 1.52a_{y}, the DO wavelengths are determined by the lattice constant along the yaxis. Consequently, only SLRs corresponding to this lattice period are expected in optical reflectance spectra. The same holds true for the magnetooptical response. While the denominator of α_{eff,xy} (Eq. 13) contains terms with S_{x} and S_{y}, the magnetooptical Kerr angle is given by α_{eff,xy}/α_{eff,xx} and thus
Since S_{y} peaks when λ = 1.52a_{x}, the SLRenhanced magnetooptical response depends on the lattice parameter along the xaxis. This crossdependence of the optical reflectance and magnetooptical Kerr angle on lattice constants a_{x} and a_{y}, which has been observed previously for pure Ni lattices^{38}, is experimentally confirmed for dimers. The model prediction that the magnetooptical dipole p_{eff,y} of dimer lattices depends on both S_{x} and S_{y} is also verified by measurements. Experiments and model calculations on rectangular lattices are summarized in Supplementary Note 4.
Conclusions
We have experimentally and theoretically explored how plasmon resonances in hybrid Ni/SiO_{2}/Au dimer arrays compare to those of lattices that are made of Au or Ni nanodisks. Our results demonstrate that Ni/SiO_{2}/Au dimer arrays support more intense SLR modes than Ni lattices because the larger polarizability of individual dimer particles produces a stronger resonant enhancement of the effective lattice polarizability. The model that we present provides insight into the optical and magnetooptical response of ordered magnetoplasmonic dimers and offers clear directions on how to tailor the polarizability by material selection, variation of the particle size, or tuning of the lattice period or symmetry.
Methods
Sample preparation
We fabricated the samples on glass substrates using electronbeam lithography. After spincoating a polymethyl methacrylate (PMMA) layer and baking at 180 °C for 1 minute, the pattern was defined by exposing the resist layer to the electron beam. We developed the PMMA in a 1:3 methyl isobutyl ketone (MIBK):isopropanol (IPA) solution. Samples with pure Au or Ni nanodisks were fabricated by ebeam evaporation of a 15nmthick or 18nmthick film, followed by liftoff. For dimer samples, we first evaporated 1 nm Ti and 15 nm Au. After this, the samples were transferred to a magnetron sputtering system for the deposition of 15 nm SiO_{2} (rf sputtering from a SiO_{2} target). Finally, 15 nm of Ni was added and the stack was liftoff. We used SEM and atomic force microscopy to determine the nanodisk diameters.
Optical and magnetooptical characterization
Optical reflectance and magnetooptical Kerr effect measurements were conducted with a Kerr spectrometer (Fig. 7). The setup consisted of a broadband supercontinuum laser (SuperK EXW12 from NKT Photonics), polarizing and focusing optics, a photoelastic modulator (Hinds Instruments I/FS50), and a photodetector. The wavelength of the laser was tuned between 500 nm and 1000 nm. We used linear polarized light at normal incidence. During measurements, a ±400 mT field from an electromagnet switched the magnetization of the Ni nanodisks between the two perpendicular directions. The Kerr rotation (θ) and Kerr ellipticity (ε) were simultaneously recorded by lockin amplification of the modulated signal at 50 kHz and 100 kHz. From these data, we calculated the magnetooptical Kerr angle (Φ) using
Finitedifference timedomain simulations
Numerical simulations were carried out using finitedifference timedomain (FDTD) method. 400 nm × 400 nm unit cells comprising a vertical dimer made of 15nmthick Ni and Au nanodisks separated by 15 nm SiO_{2} (n = 1.5) or a single Ni disk of the same size were simulated. The disks diameters were set to 110 nm. Linearly polarized light was assumed to impinge along the sample normal from the Ni disk side. Periodic boundary conditions were applied at the edges of the simulation area. A uniform embedding medium with a dielectric constant of n = 1.5 was used. Broadband reflectivity spectra were obtained by placing an electric field monitor 2 μm above the nanoparticles. Distributions of nearfields shown in Fig. 6 were calculated near the SLR wavelength. Magnetooptical effects were introduced in the FDTD simulations via offdiagonal terms in the permittivity tensor of Ni, while an isotropic dielectric function was assumed for Au. Distributions of magnetooptical dipolar fields were obtained by subtracting results for two perpendicular magnetization directions in Ni. In the simulations, these magnetic configurations were implemented by using opposite signs for the offdiagonal terms in the Ni permittivity tensor.
Data Availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported by the Academy of Finland (Grant No. 316857) and the Aalto Centre for Quantum Engineering. Lithography was performed at the Micronova Nanofabrication Centre, supported by Aalto University. P.V. acknowledges support from the Spanish Ministry of Economy, Industry and Competitiveness under the Maria de Maeztu Units of Excellence Programme  MDM20160618 and from the European Union under the Project H2020 FETOPEN20162017 “FEMTOTERABYTE” (Project No. 737093).
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S.P., M.K. and S.v.D. designed and initiated the research. S.P. fabricated the samples, conducted the measurements, and performed the model calculations with help from M.K., N.M. and P.V. conducted the FDTD simulations. S.v.D. supervised the project. All authors discussed the results. S.P., M.K. and S.v.D. wrote the manuscript, with input from N.M. and P.V.
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Pourjamal, S., Kataja, M., Maccaferri, N. et al. Tunable magnetoplasmonics in lattices of Ni/SiO_{2}/Au dimers. Sci Rep 9, 9907 (2019). https://doi.org/10.1038/s41598019460582
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