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Abstract
We study numerically plasmonic localization and enhancement of the photo-magnetic excitation in metal-garnet nanostructures with surface plasmons excited at the opposite interfaces of the metallic layer. Contrary to intuitive expectations, the resonance at the metal/air interface results in the highest photo-magnetic excitation efficiency. Our findings highlight the key role of the phase shift between the electric field components inside the garnet layer. A comprehensive analysis of the optical losses reveals the importance of absorption in the photo-magnetic dielectric. These results provide a fruitful perspective for future experiments on metal-dielectric photo-magnetic systems towards all-optical magnetization switching at the nanoscale.
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1. Introduction
The last decade has witnessed a rapid progress of magneto-plasmonics as a bridge research area between magnetism and nanooptics [1–4]. One of its brightest perspectives involves possibilities to control the efficiency of light-matter interaction in a non-invasive way at an ultrafast timescale [5]. In particular, recent advances in employing optical methods for data recording highlight a promising direction for magneto-plasmonic development [6,7]. The integration of these fields with ultrafast magnetism opens a plethora of unique capabilities allowing for high-density all-optical magnetic recording using localization of the light below the diffraction limit. All-optical magnetic recording with ultrashort laser pulses represents an attractive alternative to the magnetic field- or current-based control of magnetic bits [8–11]. It overcomes the inherent speed limitations of electronic systems and enables manipulation of spins on timescales inaccessible by other means.
Recently, the potential of sub-diffractional localization of the opto- and photo-magnetic phenomena in metal-dielectric magneto-plasmonic systems was theoretically analyzed [12–14] and experimentally demonstrated [15,16]. Of particular interest is the photo-magnetic effect based on resonant optical excitation of the strongly anisotropic ions using linear polarization of light, which is known through ultrafast non-dissipative photo-magnetic recording [10]. There, a resonantly excited, light-induced effective field of the photo-magnetic anisotropy $H_{L}$ with a picosecond lifetime drives large-angle magnetization precession in Co-doped YIG (YIG:Co). Notably, in the near-infrared range YIG:Co films exhibit a number of absorption peaks, the most pronounced ones of which are close to $\lambda =1.14~\mu$m and $\lambda =1.3~\mu$m [17–19]. They originate in the $^4\rm {T}_1\!\rightarrow \!^4\rm {T}_2$ and $^5\rm {E}\!\rightarrow \!^5\rm {T}_2$ transitions of the Co-ions at octahedral and tetrahedral sites, respectively. Owing to the high transparency of the garnets in the near-infrared range (multiple d-d resonances on Co ions around $1.14~\mu$m and $1.3~\mu$m are deep inside the bandgap of the YIG matrix), the energy efficiency of optical control approaches 100%, as each absorbed photon contributes to the nascent magnetic anisotropy [20].
The specific efficiency of this excitation can be strongly enhanced by employing plasmonic field localization in Au/YIG:Co magneto-plasmonic systems [16]. There, an amplification of the precession amplitude at the surface plasmon-polariton (SPP) resonance at about $1.3~\mu$m at the interface between the YIG:Co film and the metallic grating was shown to originate in the interference of the incident and SPP electromagnetic fields in the YIG:Co layer. The opposite phase of the photo-induced precession in Au/YIG:Co systems and bare YIG:Co garnets indicates the importance of a delicate balance of various contributions to the photo-magnetic anisotropy. As such, various plasmonic excitations in metal-dielectric bilayers [21,22] provide a promising playground for engineering the optical field and the photo-induced anisotropy in garnets. In particular, in this work we perform comparative analysis of the photo-magnetic excitation efficiency when SPPs are excited at the opposite interfaces of the metallic film. We further analyze the impact of optical loss of the dielectric layer and investigate the possibility of replacing Au with other noble metals. Our results provide a fruitful perspective for future experimental studies in metal-dielectric photo-magnetic systems towards all-optical magnetization switching at the nanoscale.
2. Methods
We used a designated software (COMSOL) package employing the finite element calculation method, which is a way of solving differential equations based on the variational principle [23]. COMSOL can be applied to solve a problem in nanophotonics in the frequency domain with irregular mesh. Figure 1(a) schematically describes the structure assumed in the numerical simulations. A photo-magnetic YIG:Co layer (8 $\mu$m-thick) is covered with a 50-nm thick Au grating with a period of 800 nm (ridge : gap = 700 nm : 100 nm). This value conveniently allowed for exciting the surface plasmon resonance at the Au/garnet interface at $1.3~\mu$m at about $25$ degrees of incidence in the experiment [16], close to the photo-magnetic resonance in YIG:Co. The sample was illuminated by p-polarized light incident from the grating side at an angle $\theta$ in the near-infrared range ($\lambda =1.1-1.4~\mu$m). A nonuniform triangular mesh was used, with the cell size down to 5 nm close to the metal interfaces (Fig. 1(b)) and increasing up to $\sim$50 nm away from them. The computation domain was laterally enclosed by the Floquet periodic boundaries so that the grating virtually continues infinitely, and the top and the bottom of the domain were truncated by the perfectly matched layers allowing sufficiently large margins (>$\lambda$) between the grating and the domain boundary. Owing to the translational symmetry in the in-plane direction perpendicular to the grating periodicity, the effective length of the metallic bars was infinitely large. For the simulations of the basic Au/YIG:Co system we used the refractive indices just like in our previous work [16] (namely, $n+i\kappa =2.01+i0.01$ for YIG:Co and from Ref. [24] for Au) while the data for other metals were taken from the literature (Ag and Cu [24], and Pt [25]).
Fig. 1. (a) Schematics of the simulated system. The two SPPs can be excited at the Au/air (blue) and Au/YIG:Co (red) interfaces. (b) Irregular mesh structure used in the simulations. Only a single golden bar is shown for brevity, whereas in the simulations three periods of the grating were modelled.
In the calculations, the system was irradiated at an angle of incidence $\theta = 25$° unless specified otherwise. The incident electric field strength was set to 1 V/m. The electric field map was recorded together with the optical transmittance and reflectance of the Au/YIG:Co structure. In line with our earlier results, we consider a situation where magnetization in the YIG:Co is aligned in-plane (for example, with the help of a sufficiently strong external dc magnetic field). In this geometry, we demonstrated [16] that the photo-induced anisotropy field $H_{L}$ exhibits a dominant perpendicular component which takes the following form:
3. Results
We begin with revisiting the previously obtained results on magneto-plasmonic Au/YIG:Co bilayers [16]. To that end, we simulate the spectra of reflectivity and photo-magnetic efficiency $\mathcal {E}$ in the vicinity of the SPP resonances. The simulations reveal two distinct resonances (Fig. 2(a)) corresponding to the SPP excitation at the Au/air and Au/YIG:Co interfaces. The dispersion of the SPP excitations closely follows the calculated trends (see inset of Fig. 2). Furthermore, the spectral shapes of $\mathcal {E}$ demonstrate a characteristic inverted feature in the vicinity of the SPP resonance at the Au/YIG:Co interface. The similarity of these results to those obtained in our previous work reinforces our confidence in the simulated results.
Fig. 2. Reflectivity (a) and photo-magnetic efficiency $\mathcal {E}$ spectra (b) averaged in the YIG:Co layer. The inset: SPP dispersion, calculated analytically for a flat interface (solid lines) and numerically (points).
We note the apparent dissimilarity of the two SPP resonances with respect to the average $\mathcal {E}$ in the garnet. Despite the SPP mode at the Au/YIG:Co interface is characterized by the spatial localization of the electric field closer to the dielectric, it yields smaller $\mathcal {E}$ inside the garnet layer than the SPP mode excited at the Au/air interface. In fact, it is rather the Au/air mode that can be characterized by the enhancement of the average photo-magnetic efficiency $\mathcal {E}$. Considering its values to the short wavelengths side of the resonance as a non-resonant background, the Au/air SPP provides a three-fold enhancement of $\mathcal {E}$ (Fig. 2(b)). On the contrary, the effect of the Au/YIG:Co SPP resonance is more subtle and, as we will see below, is inherently related to the phase shift $\varphi$ between the two components of the electric field.
In what follows, we take a closer look at the photo-magnetic efficiency $\mathcal {E}$ in the vicinity of the two resonances. In contrast to the bulk-averaged $\mathcal {E}(z)$ values discussed above, in Fig. 3 we show the depth profiles of $\mathcal {E}$ calculated at each wavelength in the range of interest. The depth profile can be characterized by $\mathcal {E}(z)=A +B\exp (-z/l_{\rm eff})$, where $l_{\rm eff}$ is the effective length with arbitrary constants A and B. The more general form for $l_{\rm eff}$ was employed in the previous work [16] though both of them yield very consistent results. Fig. 3(a) demonstrates the apparent reduction of the effective length towards longer wavelengths. This is further exemplified in Fig. 3(b), where the depth profiles are shown for a few selected wavelengths. The red curve calculated at the Au/YIG:Co SPP resonance closely resembles the result obtained previously [16] and exhibits a characteristic sign-changing behaviour. On the contrary, the blue curve calculated at the Au/air SPP resonance has a nearly exponential shape. We argue that this difference is related to the distinct phase behaviour at the two resonances. Indeed, Figs. 4(a)-b illustrate that $\mathcal {E}$ exhibits a typical plasmonic (Fano) resonant line shape in the vicinity of $\lambda =1.14-1.15~\mu$m (Au/air SPP resonance). In particular, we note that at $\lambda =1.14~\mu$m where $\mathcal {E}$ and its field component $|E_{\parallel }E_{\perp }|$ peak (highlighted with the vertical blue dashed line), $\cos \varphi$ experiences only small changes (Fig. 4(c)) resulting in the similar behaviour of $\mathcal {E}$ and $|E_{\parallel }E_{\perp }|$. This similarity is also retained in the depth profiles so that the SPP-induced exponential field decay holds in the $\mathcal {E}(z)$ profile, too (Fig. 3(b)). Later on, at $\lambda =1.145~\mu$m where the phase flips (i.e. phase variations are the strongest), the field component $|E_{\parallel }E_{\perp }|$ is strongly suppressed. Owing to the narrow line shape of the Au/air resonance, these variations of the two distinct components can be resolved, and thus the general line shape of $\mathcal {E}$ is largely determined by the spectral behaviour of the field component $|E_{\parallel }E_{\perp }|$. On the contrary, the quality factor of the Au/YIG:Co SPP resonance is lower, resulting in much broader spectral line shapes of all relevant quantities (Figs. 4(a)-4(c). In particular, due to the increased width, the phase variations (minimum of $\cos \varphi$) are overlapped with the SPP-enhanced field component $|E_{\parallel }E_{\perp }|$ at $\lambda \approx 1.27~\mu$m (see the vertical red dashed line). As such, an interplay of the increase and reduction in the field and phase components, respectively, results in a non-trivial spectral dependence and a non-exponential depth profile of $\mathcal {E}$ exemplified in Fig. 3(b).
Fig. 3. (a) False colour map of $\mathcal {E}(z,\lambda )$ inside the dielectric layer. (b) Characteristic $\mathcal {E}(z)$ profiles at a few selected wavelengths.
Fig. 4. Comparison of the spectral shapes of photo-magnetic efficiency $\mathcal {E}$ (a), electromagnetic field intensity $|E_{\parallel }E_{\perp }|(\lambda )$ (b), and the phase shift $\varphi (\lambda )$ (c) averaged over the dielectric layer.
4. Discussion
It is now seen that the enhancement and localization of the surface plasmon-induced photo-magnetic excitation in hybrid metal-dielectric systems cannot be reduced to the conventional plasmonic field effects. The characteristic phase dependence of $\mathcal {E}$ plays an important role in the overall excitation efficiency too, governing not only the magnitude of the effect but also its depth profile in the magnetic dielectric. This is not a coincidence but rather a consistent feature of plasmonic-assisted opto- and photo-magnetic effects, as discussed in Refs. [15,16]. Indeed, owing to the tensorial nature of those effects, phase relations play a key role in the resonant behaviour of the opto- and photo-magnetic susceptibility, thus introducing an important yet underestimated aspect of magneto-plasmonics beyond the conventional field enhancement.
We argue that the generality of our results allows to extend this consideration for the entire class of the metal-dielectric bilayers. In particular, we explore the potential of the localized photo-magnetic excitation in these systems from diverse points of view including dimensions, various optical losses, and different plasmonic metals.
We begin with discussing the impact of the geometry of our system. In Fig. 5 we compare plasmonic and photomagnetic properties of the three gratings with various gap widths. Firstly, it is seen that around the SPP resonance at the Au/air interface all three systems produce comparable results, although the strongest photomagnetic efficiency $\mathcal {E}$ correlates with the deepest reflectivity minimum and the strongest field enhancement in the $50$ nm system. The situation is totally different in the vicinity of the SPP resonance at the Au/YIG:Co interface (around $1.27~\mu$m). The $200$ nm-wide system exhibits almost no plasmonic behaviour at all while an increase from 50 to 100 nm in the gap width results in a noticeable shift of the SPP resonance. Still, it is seen that although the system with $100$ nm-wide gaps provides stronger field enhancement (due to the better coupling between the far-field radiation and the SPP mode), the photomagnetic SPP-induced resonance is stronger in the $50$ nm system, accompanied by the pronounced sign change in $\mathcal {E}$. This dissimilarity between the spectral dependencies of the photomagnetic efficiency $\mathcal {E}$ and field intensity $|E|^2$ further indicates the importance of numerical simulations of photomagnetic effects in a variety of well-known plasmonic systems.
Fig. 5. Reflectance (a), electromagnetic field intensity $|E|^2$ and photo-magnetic efficiency $\mathcal {E}$ (b) spectra calculated for various gap widths of the metallic (Au) grating.
Another attractive perspective for further enhancing the performance of plasmonic systems consists in engineering their losses [26–28]. We thus proceed with the analysis of the impact of the optical losses with studying the role of optical absorption in the dielectric garnet. The imaginary part of the complex refractive index of $n+i\kappa =2.01+i0.01$ in the infrared region is largely caused by the Co doping [18], thus allowing for a gentle tuning of the loss function at the fabrication stage. In Figs. 6(a)–6(b), we show the reflectivity, field intensity and photo-magnetic efficiency calculated for a few different values of $\kappa$. It is seen in Fig. 6(a) that, just as expected, engineering losses in the dielectric has a significant impact on the quality factor of the SPP resonance at the Au/YIG:Co interface while the other SPP (at the Au/air interface) remains unaffected. On the contrary, the photo-magnetic efficiency $\mathcal {E}$ (solid lines in the panel b) is enhanced even at the Au/air SPP resonance. Furthermore, we note that the sign change around the Au/YIG:Co SPP resonance appears only if the losses are below the threshold. Indeed, the negative part of $\mathcal {E}$ is present for $\kappa \leq 0.004$ only. This sign change is responsible for the overall phase flip of the photo-magnetically induced spin precession with respect to that in a bare garnet, as it was observed in Ref. [16]. Yet again we note the apparent dissimilarity of the electromagnetic field intensity and photo-magnetic excitation spectra, which holds over an order of magnitude variation of the optical losses. Further, reduction of $\kappa$ from $0.01$ to $0.001$ results in an increase of the effective photo-magnetic length from $3.5$ to $32~\mu$m at $\lambda \approx 1.14~\mu$m (Au/air SPP) and up to $1.7~\mu$m at $\lambda \approx 1.27~\mu$m (Au/YIG:Co SPP), as it is seen from the depth profiles of the photomagnetic efficiency (Fig. 6(c)). This effect originates in the twofold impact of the optical loss function in the dielectric. Apart from the faster decay of the electric field in the garnet, the higher losses are responsible for the faster phase variation with depth, also contributing to the reduction of the effective length.
Fig. 6. Reflectance, electromagnetic field intensity $|E|^2$ and photo-magnetic efficiency $\mathcal {E}$ spectra calculated for various metals (a-c) and those for various optical losses in the dielectric (d-f). Cu/YIG:Co data are not shown.
An alternative approach towards engineering the optical properties of plasmonic metal-dielectric bilayers consists in employing various plasmonic metals. Although in the infrared range there are quite a few materials with good plasmonic properties, we outline Au, Ag and Cu as the most commonly considered plasmonic metals. We do not consider Al here since due to its high frequency of the interband transitions its value is mostly relevant for the UV plasmonics [29,30]. On the other hand, owing to the high potential of Pt/garnet systems for spintronics and optomagnetism [31–34], we consider Pt as a potential substitute for the conventionally plasmonic metals as well. In Figs. 6(d)-f, we summarize the results of our simulations of the photo-magnetic excitation in metal-dielectric systems with various plasmonic metals. The spectral positions of the SPP resonances are very similar due to the large negative real part of their dielectric function $\varepsilon ^{\prime }_m\ll -1$ compared with that of air or YIG:Co resulting in very similar values of the SPP wavevector $k_{\rm SPP}$ predominantly governed by the permittivity of air or YIG:Co. We also note that the results on a Cu/YIG:Co bilayer are practically identical to those on Au/YIG:Co which is why we do not show them here. Yet, we conclude that copper offers no improvement in performance as compared to gold. Silver, however, despite exhibiting slightly smaller depth of the resonance minimum in reflectance (Fig. 6(d)), does offer an improvement in the total optical field intensity (Fig. 6(e)). However, the increase of the peak photo-magnetic efficiency is marginal: at $\lambda =1.14~\mu$m (metal-air SPP) it does not exceed 10% while at the other, metal-YIG resonance ($\lambda =1.27~\mu$m) the SPP-induced variations for Ag are not as strongly pronounced as for Au. The negative SPP-induced peak is hardly seen on a general decreasing trend common for all metal-dielectric systems. In the Pt case we, on the contrary, deal with a metal with worse plasmonic properties than Au. The SPP-driven negative peak at $\lambda =1.27~\mu$m almost entirely disappears while the metal-air SPP peak at $\lambda =1.14~\mu$m is approximately half as strong. Yet, the non-resonant contribution between the two resonances in Pt-YIG bilayers is the strongest. One possible reason behind this is the increased transmittance of the Pt layer away from the resonances, so that more light intensity is capable of reaching the dielectric medium. This is in line with the lowest reflectance of Pt-YIG bilayer among all investigated metals (see Fig. 6(d)).
To summarize these observations, we conclude that there are two key parameters of the photo-magnetic excitation in metal-dielectric bilayers, namely, the average magnitude quantified by $\mathcal {E}$, and the effective length $l_{\rm eff}$. As it can be seen from Fig. 6, $\mathcal {E}$ can hardly be improved by choosing a different metal while being quite sensitive to the losses in the dielectric. Thus, reduction of the losses has a pronounced twofold effect: it results in an increase of both the magneto-refractive excitation efficiency $\mathcal {E}$ and the effective length $l_{\rm eff}$. In particular, we found a sixfold increase in $l_{\rm eff}$ (Fig. 6(c)) at the lowest investigated level of the optical losses in YIG:Co ($\kappa =0.001$) at the Au/YIG:Co SPP resonance around $\lambda \approx 1.27~\mu$m, as compared to the $\kappa =0.01$ case. At the other, Au/air SPP resonance, one order of magnitude smaller optical losses in the dielectric yields one order of magnitude larger effective length. As such, we conclude that reducing the optical losses in the dielectric is unlikely to be a viable option to maintain subdiffractional localization of the photo-magnetic excitation in plasmonic metal-dielectric bilayers.
Finally, we discuss localization of the photo-magnetic excitation in the alternative approach, that is, varying the optical properties of the metal (Fig. 6(f)). At the metal/air SPP resonance around $\lambda \approx 1.14~\mu$m, the variations of the effective length are negligible: for all metals $l_{\rm eff}$ remains in the range of $3.5-4~\mu$m. The photo-magnetic excitation at the other SPP resonance is more material-dependent: although the difference between Au, Ag and Cu is only marginal, Pt offers an increase of $l_{\rm eff}$ up to $1.4~\mu$m. While this value is still smaller than the characteristic spatial scale in a bare garnet, the conventional plasmonic metals such as Ag and Au remain unsurpassed for their localization degree. As such, we conclude that the most promising direction for future photo-magneto-plasmonic research should deal with the dielectric properties of the photo-magnetic system rather than plasmonic properties of the metallic layer. Further, experimental verification of the amplification of the photo-magnetic excitation at the Au/air SPP resonance as compared to the Au/YIG:Co one remains an interesting challenge in the nearest future.
5. Conclusions
In summary, we performed numerical simulations of the SPP-driven photo-magnetic excitation in a variety of metal-dielectric hybrid systems. We found that, contrary to the intuitive expectations, although the electric field intensity at the Au/YIG SPP resonance is stronger than that at the Au/air SPP resonance, the Au/air SPP resonance demonstrates the higher potential for the SPP-assisted photo-magnetic magnetization control than that of Au/YIG SPP. We investigated the impact of the optical loss in the dielectric and the comparative efficiency of employing various plasmonic metals. Our results provide a fruitful perspective for the future optimization of the metal-dielectric nanostructures towards high-density all-optical photo-magnetic recording.
Funding
Narodowe Centrum Nauki (DEC-2019/35/B/ST3/00853); Fundacja na rzecz Nauki Polskiej (POIR.04.04.00-00-413C/17).
Acknowledgments
This work was funded by the Foundation for Polish Science (Grant No. POIR.04.04.00-00-413C/17). I.R. thankfully acknowledges partial funding from the National Science Center Poland (Grant No. DEC-2019/35/B/ST3/00853).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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