1. Introduction

Patterned ferromagnetic/plasmonic metal films can exhibit strong plasmonic and magneto-plasmonic activity resulting in large enhancement of magneto-optical effects in the presence of localized or propagating surface plasmons [1–3]. Recently, the magneto-plasmonic research has also investigated to what extent pure ferromagnetic patterned films [4–6] and nanoparticles [7, 8] can host surface plasmons and increase the magneto-optic response. The technological perspectives of magneto-plasmonic structures are large: the potential capability to trap light locally (near field enhancement) in magneto-plasmonic nanostructures [4, 9, 10] opens new routes in the research of magnetism, optomagnetism and ultrafast all-optical switching [3, 11]. In addition, investigations of plasmon-induced spin wave excitations in magnetic systems can extend the field of magneto-plasmonics research to magnonics [12–14] and spintronics [15].

In this work we focus on the magneto-plasmonic response of pure ferromagnetic films around the geometrical threshold, i.e. the formation of magnetic clusters that can physically touch each other. Particularly, we address periodically nano-perforated cobalt films: films evolving from an island-like array to a connecting network forming an anti-dot pattern. The geometrical transition also defines the plasmonic transition from propagating to localized modes and thus is expected to strongly influence the magneto-optic response. The plasmonic properties of such transition structures can be interpreted with the concept of percolation theory. Percolation theory describes the transition of isolated particles or clusters usually located inside a host matrix, to a kind of a connected network [16]. The rise of nanotechnology has facilitated the preparation of composite multicomponent materials in physics, chemistry and materials science and therefore, it has offered to percolation theory many fields of applications. The physical properties of such structures like electrical and thermal conduction are very interesting since around the percolation threshold a system can behave either as a dielectric or as a conducting medium. This behaviour can also be reflected in the optical properties: in a dielectric-metal transition the change from a positive real part of the dielectric function ϵ(ω) for the dielectric to a negative real part of ϵ(ω) of the metal will drastically change the optical properties.

The studies on the optical properties on similar hole-to-island structural evolution patterns have shown that the lack of continuity affects the system in a non-trivial way, that may induce not only continuous shifts [5] but also changes on the overall spectral shape of the optical response [17]. Those cases have been qualitatively explained by considering an effective dielectric tensor [18]. Furthermore, a critical behavior of the effective dielectric function was revealed demonstrating that the hole-to-island structural evolution is a topological analog of the percolation threshold at the transitional pattern. In addition, the critical scaling of the plasmonic modes in similar metallic nanonstructures has been discussed [17, 19]. Here we show the effect of surface plasmon resonance on the magneto-optical Kerr effect with respect to the transition from islands (dots) to mebranes (anti-dots) nanostructures composed of thin cobalt layers.

2. Results and discussion

2.1. Sample preparation

All nanostructures were obtained using lithography by self-assembly of colloidal nanospheres. [20, 21] Polystyrene nanospheres with a diameter of a = 470 nm have been deposited under appropriate conditions on 0.5 mm thick quartz subtrates. Self-assembly leads to the formation of a hexagonal closed-packed array of nanospheres. Plasma etching was subsequently used in order to shrink the diameter of the spheres [6, 22]. By varying the etching time it was possible to vary d from d = a = 470 nm to d = 300 nm. A 50 nm Co film was subsequently deposited over each colloidal mask. The last stage of preparation involved cleaning with toluene to remove the polystyrene mask. Figure 1 shows scanning electron microscopy (SEM) images of the resulting Co nanostructures. Since no protective overlayer was deposited, the Co layers can be oxidized at the surface as it can be seen in Fig. 1: the different color at the rim of the holes for the samples with d = 300, 373 nm for example, indicate the presence of a self passivating cobalt oxide rather than metallic cobalt. We have measured directly after the preparation of the sample in order to minimize the effect of oxidation.

 

Fig. 1 Scanning electron microscopy images for seven different Co films deposited on quartz substrates. The center to center distance is a = 470 nm. Only two samples with d = 373 nm and d = 300 nm present a continuous coverage. The rest of the samples have an island like shape. The percolation limit between isolated islands and connected islands lies around d values of d = 425, 413, 405 and 373 nm.

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The samples with d = 373 nm and d = 300 nm represent a continuous coverage while the rest exhibit an island-like pattern. The given hole diameters in samples with non-continuous coverage have been calculated from the etching time and verified by the SEM images. It is important to notice that all samples should be membranes with continuous Co coverage, since all diameters are smaller than the lattice parameter. The non-continuity is induced during the deposition process. The high directionality of the evaporation beam leads to the formation of bridges between the spheres [23, 24] and facilitates our purposes to create structures around the percolation limit.

2.2. Optical and magneto-optical characterization

Figure 2(a) presents the experimental transmission spectra for the nanostructures presented in Fig. 1. The overall transmission decreases as the hole diameter decreases. This can be attributed to the increasing Co coverage as the diameter of the holes decreases. However, the spectra present qualitatively different features that require further attention. From the spectrum corresponding to sample with d = 300 nm, the one with more Co coverage (membrane like), we can see the typical features encountered in systems exhibiting extraordinary transmission: two well pronounced transmission maxima, and two transmission minima whose spectral position correspond to that expected from Bragg orders [25].

 

Fig. 2 (a) Experimental transmission spectra for the Co nanostructured thin films with different hole diameters. Clear extraordinary transmission peaks are observed for the continuous structure with d = 300 nm while there is gradually change of the shape of the spectra at the percolation limit. (b) Calculated transmission spectra for Co nanostructured thin films with different hole diameters as indicated. Filled symbols are for continuous membranes, and the open symbols are for the same structure but with a 60 nm gap breaking the bridges, resembling in such a way the islands.

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These transmission features are related to the periodicity of the pattern, and should not change position as a function of the holes diameter except for the small effects that the polydispersity of the nanospheres produce. It is indeed seen that the features remain, but are washed out as the Co amount decreases. The maxima between the Bragg features observed in d = 300 nm are the typical manifestation of extraordinary optical transmission in metallic samples, and have been associated to the excitation of Bragg-plasmons, i.e. lattice modes with a plasmonic character [6, 26, 27]. These maxima change position as the plasmon characteristics change when the holes diameter increases [5]. But for the rest of the samples there is an overall shape modification especially for the low energy part of the spectrum. This sudden change is evolving near the percolation limit since it is related to the disruption of the continuity of the cobalt over the quartz substrate.

To shed light into the experimental findings we have performed numerical simulations using both scattering matrix approaches [28, 29] and Finite Element [30] of connected and disconnected networks with the same nominal hole diameters. The purpose is to establish if there is a qualitative difference in the spectra due to the metal continuity. Transmission spectra are presented in Fig. 2(b) for membranes (filled symbols) with different hole diameters but keeping the same lattice parameter a = 470 nm, together with spectra corresponding to the very same structures, but with the metallic bridges broken by a 60 nm gap to mimic the discontinuity of the actual samples (open symbols). It has to be noted that the same gap size is used for all the simulated structures. The first apparent effect of the metal discontinuity can be seen in a marked difference in the low-energy part of the spectra, while the difference is smaller, if any, in the high-energy part. We will see, through calculations of the spatial distribution of the electromagnetic field, that the dissimilarity between the low-energy and high-energy part can be understood by light scattering in the broken regions and the generation of optical hot-spots.

In the low energy region, the spectral behaviour for continuous and broken membranes is different, not only quantitatively but also qualitatively. Maxima associated with the extraordinary transmission phenomena are present for unbroken membranes. In addition, the maxima are blue-shifted as the diameter of the hole increases in a similar way as in [5]. However, for broken membranes, the behaviour is the opposite. Minima appear which are red-shifted when the diameter of the holes increases. Both (maxima and minima) exhibit a sharp feature when the resonance spectrally overlaps with the lattice resonance. The high energy regions of the two systems exhibit a more similar behavior, presenting quantitative differences. The case of touching spheres, d = 470 nm, gives the same result with and without the presence of the gap.

Of great interest is the comparison for the samples around the geometrical transition: the theoretical curve for the continuous membrane with d = 400 nm presents an equivalent shape to that of the sample with d = 373 nm (experiment) pointing to the fact that the experimental sample is still continuous. In this case the plasmonic resonance and the Wood anomaly are close enough to alter the shape from a rounded peak to a maximum with a sharp structure (as seen in both experiment and theory). In short, the distinct feature is a maximum in transmission for continuous membranes and a minimum for the islands (see for example the region from 1. 5 eV to 2 eV).

To summarize the optical characterization, the transmission curves exhibit different shapes at the percolation limit. The more pronounced difference appears at low frequencies. The extraordinary transmission peaks due to the coupling of the incident light to the surface plasmons though the lattice for the continuous membranes are replaced with minima in transmission for the broken island-type structures. The results of the numerical simulations fully agree with the experimental data in terms of the shape of the curves, revealing after all that the physics of the experiment is well captured by the theory. However, there are small deviations in the calculated intensities and the spectral position of the curve features. Multiple factors are behind these deviations. The most important one is the accuracy of the material parameters. We have used for our simulations the experimentally obtained values for Co from [31], but the fabrication process (grain size, humidity, temperature, etc.) affects the outcome. The oxidation of the Co layer is another major factor since in the simulations the optical and magneto-optical constants obtained from a pristine Co film were considered. The oxidation in our case leads to a shift of the experiment curves similarly to other published works, not only for Co, but other ferromagnetic metals [6, 32–34]. Besides that, the intensity will be also influenced by the surface oxidation. In the experiment, we observe higher transmission due to the fact that the oxidation reduces the metallic character of the films. Geometrical inhomogeneities of the pattern can also affect the comparison.

In Fig. 3(a) we present the experimental Kerr rotation spectra for the same samples as in Fig. 2(a). As one can readily see, the overall trend is that the Kerr rotation increases (the sign only reflects the direction of the Kerr rotation) with decreasing the hole diameter, as the amount of Co increases. However, the spectra present two main features for energies matching those appearing in the transmission spectra. In contrast to the sharp, qualitative change in the spectral shape found in the optical spectra for the broken and unbroken case, the Kerr spectra exhibit a more similar shape, with minima and maxima around the same positions (except for small spectral shifts). Quantitively, the maximum value of the Kerr rotation is seen for the cases of d = 300, 373 nm, more pronounced in the low energy region where the transmission curves exhibit large amplitude changes. The second maximum at the high energy region ≈ 3.2 – 3.3 eV is present for all samples being more pronounced for the d = 373 nm sample.

 

Fig. 3 (a) Experimental polar Kerr rotation spectra for islands and membranes of diameter size from d = 300 to 445 nm. The Kerr rotation increases with decreasing the hole diameter. Opposite to the sharp changes in the transmission curves the Kerr spectra exhibit a more similar shape for both broken and unbroken membranes. (b) Simulated polar Kerr rotation versus photon energy for islands and membranes of diameter size d = 300, 400, 470 nm. Filled symbols refer to continuous membranes, and the open symbols correspond to the same structure but with a 60 nm gap breaking the bridges similar to Fig. 2(b).

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In order to complement our experimental observations, we have performed calculations of the polar Kerr rotation spectra for the same geometrical parameters as for the optical analysis in Fig. 2(b). The data for Co used here were taken from [31] where the optical and magneto-optical constants of Co were experimentally determined, through a combination of ellipsometry and polar Kerr and Kerr ellipticity measurements. Dispersion, i.e. wavelength dependence, of the material parameters and the gyration of Co was taken into account. As shown in Fig. 3(b), the calculated Kerr rotation spectra indicate the same trend as the experimental data, pointing out that there is not a clear qualitative difference for broken and unbroken membranes. It is also evident that the sample with a hole diameter of d = 470 nm exhibits no difference at all. The samples d = 300 and 400 nm membranes in Fig. 3(b), are closely related to the transmission of the experimental antidot structures of d = 300 and d = 373 in Fig. 2(a). Generally, there is a quantitative modification of the Kerr rotation in the low energy region, but the features are similar in shape (minima stay minima irrespective of the state of the membrane), they stay in the same energy location and the spectra follow the (overall) trend: the smaller the hole the larger the Kerr rotation, due to the larger amount of Co in the system.

In order to reveal the origin of the differences found in the transmission and polar Kerr spectra, we will analyze an important factor in these kind of systems: the spatial distribution of the electromagnetic field in the structure. The localisation of the electromagnetic field is a key parameter that can define the optical response and the magneto-optical enhancement in metallic systems [4, 35, 36]. In Fig. 4 we present the spatial profile of the electromagnetic field distribution in a plane located 10 nm below the top surface of the membrane obtained for two energies that correspond to the locations of the maxima in transmission. In Fig. 4 (a) the profiles correspond to an incoming wave with an energy of 1.7 eV corresponding to the low energy region, whereas in Fig. 4(b) the energy is 2.7 eV and is located in the high energy range. In the case of the low energy range, and for d = 300, 400 nm hole diameters, there is a field enhancement in the direction parallel to the polarization axis (E_x) and the field localization occurs in the holes. However, for the broken membranes the spatial location varies, the field localization happens in the gaps that are perpendicular to the the polarization direction. In addition, the field in the holes (non-absorbing part of the membrane) decreases for broken geometries. The field within the metal presents differences but not as dramatic as for the holes. For the d = 470 nm sample the profiles are very similar. The bridges are so thin that the penetration in the metal (up to the skin depth ∼ 5 – 8 nm) is enough to give rise to hot spots in the unbroken membranes similar to the broken ones. The presence of these hot spots contributes to an effective decrease of the field that would travel through the membrane and thus eventually accounts for the transmission.

 

Fig. 4 Calculated intensity of the electric field |E²| 10nm below the upper surface for unbroken (u) and for broken (b) membranes, for hole diameter sizes of d = 300, 400, 470 nm. (a) Low energy spectral region, where the incoming plane wave has an energy of 1.7 eV. Clear differences between unbroken and broken membranes can be clearly seen. (b) High energy spectral region (2.7 eV), where differences are actually very small.

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The results of Fig. 3 have revealed a more quantitative than qualitative difference (shape) in the Kerr spectra between membranes and islands. The differences can be understood with the help of Fig. 4: the enhancement of the Kerr signal comes from the intensification of the electromagnetic field that probes the metal for d = 300, 400 nm hole diameters. However, the appearance of hot spots in the islands and in the d = 470 nm membrane in Fig. 4(a) changes the localization pattern but it does not result in an enhancement of the magneto-optic signal. The situation for the high energy range Fig. 4(b) is completely different: no hot spots are present for any of the geometries, and very subtle differences can be seen for broken and unbroken cases. This is the reason behind the very similar values for both transmission and Kerr rotation in this energy range.

3. Conclusions

The presented set of Co nanostructured thin films were prepared by nanosphere lithography, a method that enabled the fabrication of continuous, anti-dot structures as well as of broken, island-like arrays. The optical and magneto-optical measurements have revealed a qualitative difference in their spectral shape. Numerical simulations show that the effect of ’continuity’ is more pronounced in the low energy region of the optical spectra and that is supported by the spatial distribution of the electromagnetic field in the membrane. The formation of hot spots is the reason for the difference in both transmission and Kerr rotation for broken and unbroken systems. For very thin linking bridges the penetration of the field within the metal gives rise to hot spots before the membrane is actually broken and thus no noticeable difference can be seen for broken/unbroken systems.