Observation of localized magnetic plasmon skyrmions

Deng, Zi-Lan; Shi, Tan; Krasnok, Alex; Li, **; Alù, Andrea

doi:10.1038/s41467-021-27710-w

Observation of localized magnetic plasmon skyrmions

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Published: 10 January 2022

Volume 13, article number 8, (2022)
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Abstract

Optical skyrmions have recently been constructed by tailoring vectorial near-field distributions through the interference of multiple surface plasmon polaritons, offering promising features for advanced information processing, transport and storage. Here, we provide experimental demonstration of electromagnetic skyrmions based on magnetic localized spoof plasmons (LSP) showing large topological robustness against continuous deformations, without stringent external interference conditions. By directly measuring the spatial profile of all three vectorial magnetic fields, we reveal multiple π-twist target skyrmion configurations mapped to multi-resonant near-equidistant LSP eigenmodes. The real-space skyrmion topology is robust against deformations of the meta-structure, demonstrating flexible skyrmionic textures for arbitrary shapes. The observed magnetic LSP skyrmions pave the way to ultra-compact and robust plasmonic devices, such as flexible sensors, wearable electronics and ultra-compact antennas.

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Introduction

Skyrmions - topologically stable three-dimensional (3D) vector field configurations confined within a two-dimensional (2D) domain - have been prompting significant interest in a number of physical systems, including elementary particles¹, Bose-Einstein condensates², nematic liquid crystals³, and chiral magnets^4,5. Beyond elementary skyrmion, nested multiple skyrmions have also been demonstrated in magnetic materials, such as the skyrmionium^{6,7,Full size image}

The space-coiling cylinder can be modeled as a homogeneous rod with extremely anisotropic properties (Supplementary Fig. S3), with an anisotropic permittivity tensor with infinite azimuthal- and z-oriented components while staying finite in the radial direction. This extreme form of anisotropy supports quasi-static modal distributions, having a radially polarized electric E-field and an azimuthal surface current J, in contrast to conventional magnetic Mie resonators where E is typically parallel to J²⁹. Such exotic resonant mode supports extreme field enhancement (Fig. 2b), as high as 10⁵ in the geometry of Fig. 2a (inset). Figure 2c shows the linear dependence of the resonant frequency (k₀R) and resonant linewidths (Δk₀R) with the mode index, confirming its equidistant response. Figure 2d shows that the field confinement factor (resonant wavelength over cavity radius) of the fundamental mode linearly increases with n_r, indicating that the resonator compactness can be enhanced by densifying the space-coiling fineness. Figure 2e shows that increasing the meta-structure fineness also increases field enhancement and quality factor, following a quadratic trend. These features are fundamentally limited by the considered finite conductivity of the involved materials, implying a tradeoff between quality factors and material losses studied in Supplementary Figs. S4a, b. Our study shows that realistic metals can support these resonant responses over a wide frequency range, spanning GHz and THz frequencies, as shown in Supplementary Fig. S4c, d.

Now, we consider finite-thickness space-coiling meta-structures to identify the skyrmion nature of their vectorial field configurations. In the 2D geometry, the LSP field profiles only exhibit scalar (H-field) or 2D vectorial (E-field) properties. However, for the 3D space-coiling meta-structure with finite thickness, both H-field and E-field distributions manifest 3D vectorial configurations at the interface between the meta-structure and the surrounding background (Supplementary Fig. S5). With decreasing the 3D space-coiling meta-structure thickness, the resonant behavior and the localization of the field pattern are preserved, with a small frequency shift compared to the 2D scenario (Supplementary Fig. S6)³⁰. The topological properties of the 3D vectorial field configuration can be quantitatively evaluated by the skyrmion number¹⁴

$$S=\frac{1}{4\pi }\iint {{{{{\bf{h}}}}}}\cdot \left(\frac{\partial {{{{{\bf{h}}}}}}}{\partial x}\times \frac{\partial {{{{{\bf{h}}}}}}}{\partial y}\right)dxdy,$$

(1)

where ${{{{{\bf{h}}}}}}=\{{H}_{x},{H}_{y},{H}_{z}\}/|{{{{{\bf{H}}}}}}|$ is the local unit vector of the field, and the integrand ${{{{{\bf{h}}}}}}\cdot (\frac{\partial {{{{{\bf{h}}}}}}}{\partial x}\times \frac{\partial {{{{{\bf{h}}}}}}}{\partial y})$ is the skyrmion density. The skyrmion number is a topological invariant that characterizes the order of topological knots formed by field vectors, i.e., the number of times the field wraps around the unit sphere. The skyrmion number of the magnetic field profile calculated at the air interface (Supplementary Figs. S5a–c and S6b) is equal to 1, confirming a pure skyrmion field configuration.

Experimental observation of LSP skyrmions

To experimentally observe the skyrmion vectorial field configurations, we fabricated an ultra-thin space-coiling meta-structure (h = 0.016 mm, a = 1 mm, d = 1.5 mm, R = 30 mm, n_r = 20) over a printed circuit board (Fig. 3a). We scanned the near field of the resonant modes with a 3D scanning platform connected to a vector network analyzer (Fig. 3b). The measured response indeed manifests a multi-resonant spectrum with nearly equidistant sharp peaks (blue curve in Fig. 3c and Supplementary Fig. S18). This feature is consistent with the simulation results with the same geometry parameters (red curve in Fig. 3c). Since these modes are eigen-resonances of the meta-structure, we observe a strong excitation of the skyrmions with the near-equidistant multi-resonant spectrum for various near-fields or far-field sources (Supplementary Fig. S7), in stark contrast with previous skyrmions based on the interference of carefully tailored propagating surface plasmons^14,15,16.

**Fig. 3: Magnetic field nature and topological features of LSP skyrmions in an ultrathin space-coiling meta-structure.**

Because these modes resonate at a deeply subwavelength scale with large radial wavevectors k_r ≫ k₀, the z-component wavevector ${k}_{z}=-i\sqrt{{k}_{r}^{2}-{k}_{0}^{2}}$ is mostly imaginary, yielding strong field confinement far beyond the diffraction limit. The measured elementary skyrmion has a lateral size d_m = λ/100, and a half vertical size h_m = λ/400 (Fig. 3d, left-lower panel), leading to an extremely subwavelength mode volume V_m = 2πh_m(d_m/2)² = π(λ/2)³/10⁶. We stress that such tiny mode volume is obtained with just n_r = 20, limited by our fabrication and measurement setup. Further squeezing may be achieved by increasing n_r. We experimentally observe a relatively high Q-factor of 165 (Fig. 3c), yielding a Purcell factor exceeding 10⁷, promising for various applications requiring strong light-matter interactions. Both out-of-plane (Fig. 3d) and in-plane (Supplementary Fig. S8) magnetic fields along the radial direction have been simulated and measured, yielding excellent agreement, except around the center of the sample, where we observe additional field nulls for the lower modes m = 1, 2, due to strong coupling between localized EM fields and the magnetic loop probe.

Due to the axial symmetry of the magnetic field profile, the unit vector can be written as ${{{{{\bf{h}}}}}}(x,y,z)={\{\sin \varTheta (\rho )\cos \varphi ,\sin \varTheta (\rho )\sin \varphi ,\cos \varTheta (\rho )\}}^{T}$, where ρ and φ are coordinates in the polar system and $\varTheta (\rho )$ is the orientation angle of the unit vector. The skyrmion number of the ith radial mode lobe can be calculated in closed form as

$${S}_{i}=\frac{1}{4\pi }\int _{0}^{2\pi }d\varphi \int _{{\rho }_{i}}^{{\rho }_{i+1}}d\rho \frac{d\varTheta (\rho )}{d\rho }\sin \,\varTheta (\rho )=-\frac{1}{2}\,\cos \,\varTheta (\rho )\bigg|\begin{array}{c}{\rho }_{i+1}\\ {\rho }_{i}\end{array},$$

(2)

showing that the skyrmion number only depends on the initial and final states of $\varTheta (\rho )$. Figure 3e shows the unit vectors and $\cos \,\varTheta (\rho )$ distributions along the radial direction. According to Eq. (2), each radial lobe of the mode profile has a skyrmion number +1 or −1, representing an elementary skyrmion polarized in opposite directions. The accumulated total skyrmion number is 1 for odd modes and 0 for even modes, respectively, building a multiple-π-twist TS constructed by multiple elementary skyrmions⁹, with rich possibilities to implement various topological configurations of different orders¹⁰.

The vectorial nature of the skyrmion modes can be observed in our real-space measurements of all three magnetic field components, shown in Fig. 4. The in-plane fields are along the radial direction, H_x and H_y reveal a nodal-line profile along their perpendicular axis (y axis and x axis, respectively). At the same time, the out-of-plane component (H_z) has only radial variations, in good agreement with our simulations. The mode lobes are distributed purely along the radial direction, in stark contrast to conventional WGM modes with multiple azimuthal modes. The three field components form a hedgehog-like vector configuration (bottom panel of Fig. 4), which is the direct signature of Neel-type skyrmions³¹. This outcome is also confirmed by the skyrmion density and skyrmion number extraction from the field patterns (Supplementary Figs. S11 and S13a).

**Fig. 4: Real-space imaging of the vectorial magnetic skyrmion configurations.**

In contrast to the full skyrmion supported by the H-field, the E-field profile shows a skyrmion configuration with an extra-π/2-twist, yielding a total skyrmion number of 1/2 (Supplementary Fig. S9)^{32,5a–c. Even though the geometries are drastically different, their EM response manifests similar near-equidistant multi-resonant spectra. The spatial profiles of all vectorial magnetic field components of these modes for different shapes are directly measured by our near-field scanning technique (as shown in Supplementary Figs. S15–S17), which agree well with the simulation results, manifesting arbitrarily shaped skyrmion textures indicated by the extracted skyrmion number from the arbitrarily shaped mode profile (Supplementary Fig. S14a). Based on the three field components, we reconstruct their vectorial field configurations in the right panels of Fig. 5a–c, showing multiple nested rings adapting their shapes, with multiple-π-twists from the geometry center to the periphery. Such arbitrarily shaped skyrmion configurations have been long-sought in various systems⁵, providing a unique way to realize flexible skyrmions for applications in various technological areas. In addition to continuous shape deformations, we also studied the effect of abrupt defects introduced within the meanderline geometry (Supplementary Note 7 and Fig. S19), particularly in the forms of gaps or shorts in the line. These results show that, as long as the defect does not abruptly modify the field continuity, the structure’s skyrmion nature is preserved.}

**Fig. 5: LSP skyrmions with robust properties against continuous shape deformations of the space-coiling meta-structure.**

Discussion

In this work we show magnetic LSP skyrmions in a space-coiling meta-structure. The skyrmions stem from the eigen-resonances of the tailored meta-structure, and hence they do not require external interference in the illumination. We observe multiple-π-twist TS vectorial configurations in real-space through near-field scanning, yielding extremely subwavelength features, down to λ³/10⁶. The magnetic LSP skyrmion shows large topology stability against continuous deformations of the space-coiling geometry, and manifests an overall stable multi-resonant spectrum and flexible skyrmionic textures with arbitrary shape. Although our proof-of-principle has been demonstrated in the microwave regime, we envision exciting opportunities in various frequency ranges, from near-DC to THz regimes. Our findings offer an ideal platform to support the next revolution of information processing with inherent advantages in compactness, stability, and precision for potential applications, including miniaturized spectroscopy, THz sources and microwave photonics.

Methods

Numerical simulations

The SCS spectra of the 2D space-coiling cylinder and the EM spectral response of the space-coiling metasurface were calculated with finite element simulations implemented by COMSOL Multiphysics. In simulations, a plane wave with TM polarization was used to illuminate the PEC space-coiling cylinder. The SCS was extracted by integrating the out-going Poynting vector along a closed surface enclosing the structure. EM spectral responses of the ultra-thin space-coiling metasurface were calculated by probing the magnetic field at its central point above the structure. All field components within a square area enclosing the space-coiling meta-structure were extracted to calculate the skyrmion densities and skyrmion numbers.

Sample preparation and field measurements

The space-coiling meta-structure samples were fabricated with printed circuit board technology by printing 0.016-mm-thick copper space-coiling patterns with air gap a = 1 mm, spiral pitch d = 1.5 mm, and turn number n_r = 20 on top of a 0.2-mm-thick dielectric substrate with relative permittivity ε_r = 3.5. The EM spectral response was measured with an integrated microwave system (Linbou NFS03) composed of a vector network analyzer (Agilent PNA N5222A) and a 3D scanning platform. Two magnetic (loop) antennas were connected to the two ports of the vector network analyzer. One antenna was placed at the bottom central point of the sample as a point source. The other antenna was set up on a 3D moving platform to probe the spatial magnetic field configuration. Measurement of all vectorial components (H_x, H_y, H_z) of the magnetic fields were realized by the probe antenna facing in the corresponding directions (x, y, z). By mounting the loop antenna on a 3D scanning platform, the spatial field patterns over a given area were obtained through point-by-point measurements.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

A.A. acknowledges the Air Force Office of Scientific Research, the Simons Foundation, and the National Science Foundation; Z.-L.D. acknowledges the National Natural Science Foundation of China (NSFC) (Grant 62075084), Guangdong Basic and Applied Basic Research Foundation (2020A1515010615), the Fundamental Research Funds for the Central Universities (21620415), Guangzhou Science and Technology Program (202102020566) and the China Scholarship Council (Grant 201906785011). X.L. acknowledges the Guangdong Provincial Innovation and Entrepreneurship Project (Grant 2016ZT06D081).

Author information

Alex Krasnok
Present address: Department of Electrical and Computer Engineering, Florida International University, Miami, FL, 33174, USA

Authors and Affiliations

Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, **an University, Guangzhou, 510632, China
Zi-Lan Deng, Tan Shi & ** Li
Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY, 10031, USA
Zi-Lan Deng, Alex Krasnok & Andrea Alù
Physics Program, Graduate Center, City University of New York, New York, NY, 10016, USA
Andrea Alù

Authors

Zi-Lan Deng
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Tan Shi
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Alex Krasnok
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** Li
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Andrea Alù
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Contributions

Z.-L.D., A.K., X.L. and A.A. conceived the idea. Z.-L.D. carried out the theoretical design and simulation of the space-coiling meta-structure. Z.-L.D. and T.S. performed the experimental measurements. Z.-L.D., A.A., X.L., and A.K. proposed the physical concept and analyzed the data. Z.-L.D. initiated the draft with inputs from A.A., X.L. and A.K. All authors contributed to discussions about the manuscript.

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Correspondence to Zi-Lan Deng, ** Li or Andrea Alù.

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Deng, ZL., Shi, T., Krasnok, A. et al. Observation of localized magnetic plasmon skyrmions. Nat Commun 13, 8 (2022). https://doi.org/10.1038/s41467-021-27710-w

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Received: 08 October 2021
Accepted: 06 December 2021
Published: 10 January 2022
DOI: https://doi.org/10.1038/s41467-021-27710-w
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Observation of localized magnetic plasmon skyrmions

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Abstract

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A brief report on nanophotonics and metamaterials landscape in India

Surface Plasmon Excitation: Theory, Configurations, and Applications

Laser-Induced Periodic Surface Structures (LIPSS)

Introduction

Experimental observation of LSP skyrmions

Discussion

Methods

Numerical simulations

Sample preparation and field measurements

Data availability

References

Acknowledgements

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Observation of localized magnetic plasmon skyrmions

From

Abstract

Similar content being viewed by others

A brief report on nanophotonics and metamaterials landscape in India

Surface Plasmon Excitation: Theory, Configurations, and Applications

Laser-Induced Periodic Surface Structures (LIPSS)

Introduction

Experimental observation of LSP skyrmions

Discussion

Methods

Numerical simulations

Sample preparation and field measurements

Data availability

References

Acknowledgements

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