1. Introduction

Since the discovery of flat bands and unconventional superconductivity at the magic angle in twisted bilayer graphene by Cao et al. [1–3], a variety of related exotic phenomena including correlated insulating states [4,5] and ferromagnetism [6] has been investigated. The hybrid electron wavefunction and coupling between adjacent layers in these stacked bilayer and multilayer graphene structures can be actively controlled by altering the relative rotation angle, giving rise to the new research area of twistronics [7–10]. Analogous twistronic concepts have been extended to phonon and plasmon polaritons in layered low-dimensional photonic materials. Moiré configurations arise when a twist is introduced to tailor the enhanced light-matter interactions, inspiring the emergence and development of twistronics for photons [11]. Thus, there has been rapidly growing interest in the photonic responses of twisted stacks of van der Waals crystals with hyperbolic dispersion [12], such as bilayer α-phase molybdenum trioxide for extreme plasmonic polariton manipulation [13–16], twisted hexagonal boron nitride bilayers forming a soliton superlattice [17], atomic-level-thick twisted-bilayer graphene with plasmons for photonic crystals [18,19], and twisted bilayer transition metal dichalcogenides inducing spin-orbit interactions and moiré excitons [19–22], which have great potential in imaging, surface wave transmission, quantum optics, and chirality. More recently, attention has been concentrated on the deep subwavelength twist-stacked hyperbolic metasurfaces (HMSs) due to their low-loss plasmon polaritons and controllable hyperbolicity by twisting one stack with respect to the other [23–25]. Additionally, the manipulation of electromagnetic waves relies on the tunable and largely modified hyperbolic dispersion characteristics, leading to the topological transition of the iso-frequency contour (IFC) from a closed ellipse to an open hyperbola [26–28]. However, previous findings have so far been mostly limited to natural metamaterials and electric surface wave manipulation, with limitations on the frequency range, fabrication, and controlling of rotation angles.

In addition to natural metasurfaces mentioned above, structured metal HMSs have also shown great advantages owing to their relatively low transmission loss, simple micro-nano fabrication processes, and integration technology. For example, planarized metal-dielectric-metal and dielectric-metal structures featuring extreme anisotropy and unique hyperbolic dispersion, can support high-k modes, enhance photonic density of states, and provide powerful control of surface plasmons by altering the permittivity, permeability, or structural parameters [29–33]. Therefore, a Moiré hyperbolic metasurface (MHMS) is formed upon superimposing two HMS arrays at a relative rotation angle Δ$\theta$ [34], inducing Moiré effects including manipulation of electric/magnetic surface waves and flat bands analogous to a Lifshitz transition in electronics [35]. Actually, we can treat the metasurface as a homogeneous effective medium slab based on the effective medium theory, whose electromagnetic properties can be characterized by the effective permittivity and permeability tensors. An HMS can be divided into two kinds: the electric HMS and its magnetic counterpart. For the electric HMS, (${\varepsilon _{xx}} \times {\varepsilon _{yy}}\mathrm{\ < }\;\textrm{0}$), $\hat{\mu } = \textrm{1}$, and $\hat{{\boldsymbol \varepsilon }} = \left( {\begin{array}{cc} {{\varepsilon_{xx}}}&0\\ 0&{{\varepsilon_{yy}}} \end{array}} \right)$, where the subscripts xx and yy indicate the components parallel to the x- and y-axes, respectively. The dispersion relation for the transverse magnetic (TM) polarization can be determined by substituting the permittivity tensors into Maxwell’s equations by the following expression: $\frac{{k_x^2}}{{{\varepsilon _{yy}}}} + \frac{{k_y^2}}{{{\varepsilon _{xx}}}} = k_0^2$. In 2021, a periodic electric HMS array composed of metal patterns and dielectric slabs has been proposed and observed by Liu et al. in [36], which described twist-induced dispersion, electric surface wave propagation, and magic angles using an H-shaped MHMS. In contrast, for the magnetic HMS, the permeability tensors need to fulfill the condition: ${\mu _{\textrm{xx}}} \times {\mu _{yy}} \lt 0$, where $\hat{\mu } = \left( {\begin{array}{cc} {{\mu _{xx}}}&0\\ 0&{{\mu _{yy}}} \end{array}} \right)$, ${\varepsilon _{xx}}\textrm{ = }{\varepsilon _{yy}}\textrm{ = } \varepsilon $, and the dispersion relation for the transverse electric (TE) polarization can be written as: $\frac{{k_x^2}}{{{\mu _{yy}}}} + \frac{{k_y^2}}{{{\mu _{xx}}}} = \varepsilon k_0^2$. In 1967, Veselago proposed the concept of negative refractive index and predicted anomalous propagation of electromagnetic waves in materials with a negative permittivity and permeability [37]. The double-split ring resonator (DSRR) array with strong magnetic response was first discovered by Pendry et al. [38,39]. By 2018, Yang’s group extended the structured DSRR HMS to the magnetic paradigm, realizing the manipulation of magnetic surface plasmons at microwave frequencies [40]. The MHMSs, however, are currently restricted to electric HMS, and the realization of magnetic HMSs remains elusive despite the importance of moiré effects and magnetic response manipulation.

Here, in this work, we propose a scheme that is able to manipulate magnetic surface waves and realize precise control of magic angles at two types of topological transition frequencies through tailoring interlayer coupling in twisted bilayer DSRR arrays. To investigate surface plasmon propagation in this magnetic MHMS, we calculate the first four energy bands and IFCs of the monolayer and stacked bilayer magnetic HMSs in the first Brillouin zone. Besides, strong terahertz magnetic dipole modes are also simulated to study the underlying physical mechanism. Then, Fourier transform (FT) of the magnetic field distributions in the twisted HMS bilayers is performed to analyze the twist-controlled dispersion engineering of the polaritons, where two kinds of transition occur, from being elliptical to hyperbolic and from a hybrid ellipse to a hyperbola. Moreover, we study the evolution process of flat bands in the two transition regions when the mutual rotation angle increases from 0° to 45°. The so-called magic angle occurs at a specific rotation angle, which means that the topological transitions are not only present at every single frequency when the top layer is rotated with respect to the bottom layer but also appear at a specific angle with the change of excitation frequency. Finally, we observe frequency-dependent magnetic field enhancement effects and the local spatial distribution reaches a maximum at the flat band, which shows great consistence with the calculated transition frequency. The spacer thickness dependence of the magnetic MHMS has also been discussed to control the dispersion characteristics. The proposed magnetic MHMS enables new degrees of freedom to manipulate magnetic surface plasmons for various applications enabled by the concept of twistronics for light, such as anomalous wave propagation, quantum technologies, and planarized plasmonic devices with strong magnetic light–matter interactions.

2. Structure design

Previous works on photonic responses and moiré effects in MHMSs, however, are mostly concerned with the electric type and atomically-thin condensed matter, whereas magnetic MHMSs with enhanced light-matter interactions and actively-controlled dispersion still need to be designed and further investigated. We know that there exists a magnetic surface mode when the two media have opposite permeability values at an interface, and a DSRR array with resonant circular surface currents can support magnetic surface wave propagation. Considering these properties, we here propose a magnetic MHMS, as shown in the middle of Fig. 1, which consists of two closely stacked DSRR arrays with the top layer twisted counterclockwise with respect to the bottom layer by $\mathrm{\Delta}\theta$, and where the induced current is produced by circular surface or displacement electric currents located at the center of the metal patterns. The inset at the top left depicts the details of the double-layer unit cell, where the golden and white regions indicate a gold film (which can be regarded as a perfect electric conductor in the terahertz region) and lossless dielectric substrate with a permittivity of 2.96, respectively. The structure is designed with p = 80 µm, a = 62 µm, w = 6 µm, and g = 8 µm, and the thicknesses of the dielectric slab and the gold patterns are 10 µm and 0.35 µm, respectively. In particular, enhanced magnetic response and dispersion engineering are driven by the spacer thickness and mutual rotation between two adjacent layers, which will provide new degrees of freedom for the manipulation of magnetic plasmons compared with a single magnetic HMS. In addition, as depicted at the top right in Fig. 1, the evolution of the surface states of the magnetic MHMS includes two types of topological transition: from elliptical to flat and finally to hyperbolic, and from hybrid modes to flat lines and then to a hyperbola. Therefore, the coupling between interlayer eigenmodes will lead to a dramatical change in the dispersion curves and magnetic surface plasmons, which can be applied to manipulate magnetic surface waves for non-diffraction propagation, anomalous diffraction, and negative refraction.

 

Fig. 1. Schematic of the proposed magnetic MHMS, where the structural parameters of the split-ring resonator (inset at the top left) are designed as: p = 80 µm, a = 62 µm, w = 6 µm, and g = 8 µm, and the thicknesses of the dielectric slab and gold film are 10 µm and 0.35 µm, respectively. The inset at the top right describes two types of topological transition of the magnetic surface plasmons when the superimposed top layer is rotated relative to the bottom layer with a counterclockwise rotation angle $\mathrm{\Delta}\theta$.

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3. Basic physical properties

3.1 Dispersion characteristics of the monolayer and bilayer magnetic HMSs

Understanding the physical mechanism behind the magnetic resonators and their MHMS configurations is both of scientific interest and application importance, where the emergent unique hyperbolic dispersion characteristics and enhanced magnetic plasmons are driven by the strong anisotropy of the metal patterns and the coupling magnetic dipoles. Besides, since the thicknesses of the magnetic monolayer and bilayer $(\mathrm{\Delta}\theta=\theta^{\circ})$ are much smaller than the wavelength, they can be treated as a homogeneous anisotropic effective-medium slab with a specific period. Then, using the eigenmode solver of the commercial software Computer Simulation Technology (CST) and assuming the metal pattern of the unit cells as perfect electric conductors, we obtain the k-space properties including 3D band diagrams and the first four IFCs in the first Brillouin zone, as presented in Figs. 2(a) and 2(b), respectively. The IFCs of the magnetic metasurfaces could realize continuous topological transition by altering the structural parameters, which are calculated via the equation $k=\pi\times\theta/180p$ For the first IFC of the single metasurface, we observe a transition process from an ellipse (below 0.69 THz) to a dispersion curve (from 0.69 to 0.87 THz) which is analogous to an elliptical shape except at the edge of the Brillouin zone (k_x is close to ± π/p), to a flat line about 0.87 THz), and finally to a hyperbola (above 0.87 THz). Similarly, the IFCs of the second energy band in the first Brillouin zone experience a topological transition from elliptical (below 0.99 THz), to analogously elliptical (from 0.99 to 1.25 THz), to flat (about 1.25 THz), and finally to hyperbolic (above 1.25 THz). Differently, the single-layer HMS does not exist flat bands in the frequency ranges of the third and fourth IFCs, and the magnetic surface waves propagate in different directions as the surface plasmon group velocity is perpendicular to the IFCs ($\bar {{{\boldsymbol V}_g}} = {\nabla _{\bar {\boldsymbol k} }}\omega $). Then we discuss how the multi-physics interactions between light and matter in the stacked bilayer structure can tailor their photonic response and engineer light dispersion in extreme ways. As shown in Fig. 2(b), for the first four energy bands, the directions of topological transition are along the x-axis, and the frequency ranges decrease obviously, which originates from the magnetic dipole coupling between two adjacent magnetic arrays.

 

Fig. 2. Dispersion properties of the monolayer and double-layer magnetic HMSs. (a, b) The first four energy band diagrams and their IFCs in the first Brillouin zone of the single layer and bilayer structures, respectively. (c) Normalized transmittance spectra when the incident waves are polarized along the x and y axes. The corresponding surface current distributions on the surface of the unit cell are plotted for the magnetic dipoles at the resonance valleys.

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Therefore, in order to analyze the underlying magnetic coupling modes of the metasurfaces, we have numerically calculated the transmittance curves under x- and y-polarized incidence with CST time-domain solver, as depicted in Fig. 2(c). The transmission spectra of the monolayer behave like Lorentzian profiles which are symmetrically-shaped lines when the incident waves are polarized in the x and y directions. The contributions to the radiation power are dominated by magnetic dipoles resonating in the x (1.30 THz) and y directions (2.22 THz), respectively. Actually, the interlayer electromagnetic coupling introduces additional resonance valleys (0.91 THz and 1.82 THz), which introduce weaker magnetic dipoles with a dipole moment along the x and y directions, respectively. In this case, twist-stacked magnetic metasurfaces may support more exotic photonic responses excited by interlayer coupling and mode hybridization. Although fascinating magnetic surface propagation phenomena can be predicted by the IFCs, it is difficult to calculate the dispersion properties of the aperiodic structures $(\mathrm{\Delta}\theta)\neq\theta^{\circ})$. Thus, we try to perform the FT of the H_z distributions to obtain the IFCs of the twisted MHMS in the following part.

3.2 Two types of topological transition process

Here we perform a simulation of the magnetic field profiles for the surface waves that are excited through the near field of a displacement electric current along the y-axis in the middle of the DSRR pattern. By applying a 2D FT to the magnetic field distributions, we obtain the IFCs of the dispersion of the bilayer metasurface at various frequencies. In order to mimic the magnetic response in the MHMS and engineer the polaritons at will in frequency and space, we have simulated the z-oriented magnetic field distributions of the MHMS with a relative rotation angle $(\mathrm{\Delta}\theta=0^{\circ}\hbox{-}45^{\circ})$ at 20 µm above the top surface, where two types of topological transition can be observed at each rotation angle. As shown in Fig. 3(a), with the H_z distribution for Δ$\theta$ = 0° taken as an example, various wavefront geometries emerge with an increase of frequency. For the first type of frequency-dependent topological transition, the wavefronts of the magnetic surface plasmons exhibit a split X-shaped trajectory at 0.52 THz (the group velocity of the surface plasmons is positive because $\bar {{{\boldsymbol V}_g}} = {\nabla _{\bar {\boldsymbol k} }}\omega $), an X-shaped pattern with a very smaller angle at 0.60 THz, a diffraction-less collimated path at 0.64 THz, and a convergent path at 0.67 THz (the group velocity of the surface waves is negative), showing the redirection of the SPP propagation. In contrast, the second type of transition phenomenon appears in the frequency range of 0.74-0.82 THz; the propagation of the magnetic surface waves varies from straight-like (0.74 THz) to X-shaped (0.79 THz), then to flat (0.80 THz), and finally to concave (0.82 THz).

 

Fig. 3. Two types of topological transition of magnetic surface plasmons $(\mathrm{\Delta}\theta=0^{\circ})$. (a) Simulated H_z distributions at 20 µm above the top surface when the frequency changes from 0.52 to 0.67 THz (type 1 topological transition) and from 0.74 to 0.82 THz (type 2 topological transition). (c) Numerically calculated dispersion contours via FT at the frequencies corresponding to (a). The white and red arrows denote the directions of the wave vectors.

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The dispersion curves obtained by FT of the near-field H_z distributions are illustrated in Fig. 3(b). We can see that the IFC shapes change from a closed ellipse (0.52 THz) to an ellipse-like shape (0.60 THz), to flat lines (0.64 THz), and eventually to an open hyperbola (0.67 THz), yielding a topological transition that is governed by the coupling between the eigenmodes that are individually supported by each magnetic HMS. The flat polariton band necessarily emerges at a topological transition, supporting enhanced photonic density of states and low-loss polariton propagation, and these findings can be applied to manipulate magnetic polaritons and design planarized and integrated devices for focusing, non-diffraction propagation, and imaging. The white and red arrows denote the associated Poynting vectors which are drawn on the basis of the rule that the transmission direction of surface waves is perpendicular to the IFC, which shows great agreement with the simulated H_z distributions in Fig. 2(a). However, the dispersion curves vary from hybrid ellipse and hyperbola shapes (0.74 THz) to ellipse-like curves with an open angle at the boundary of the Brillouin zone (0.79 THz), to straight lines (0.80 THz), and finally to a hyperbola (0.82 THz). Therefore, at 0.74 THz, the magnetic polariton trajectory is similar to a flattened line as a result of the coupling between the hyperbolic and elliptical dispersions. To investigate the twist-induced dispersion engineering of the two types of topological transition, we will further discuss the magic angle, defined as the rotation angle where the dispersion behaves like a flat line.

4. Moiré effects in magnetic MHMS

4.1 Magic angle properties in magnetic MHMS

Recent studies of MHMSs composed of twisted bilayer metasurfaces have demonstrated that dispersion engineering of the coupled system will be dramatically altered due to strong light-matter interactions emerging for certain “magic angles” between the adjacent monolayer structure. Here, we propose that a similar concept can be extended to the magnetic MHMS, particularly to the twist-induced dispersion response and magnetic polaritons. As shown in Fig. 4(a), two layers of magnetic DSRR arrays are superimposed with a relative twist angle $\mathrm{\Delta}\theta$ between them, and the individual layers have the same parameters and are stacked closely to each other in the z-direction with a total size of 2480 µm × 2480 µm. To observe the drastic dispersion modification and magnetic surface wave propagation, we have numerically calculated the magnetic field distributions and their corresponding IFCs, and the simulated results show that there exist two types of topological transition at an arbitrary rotation angle. Therefore, we summarize in Fig. 4(b) the rotation-independent critical angles of the proposed MHMS where the dispersion curves undergo a topological transition. Our images reveal that the two fitted curves (black dotted lines) of the magic angles separating the regions of elliptical dispersion and hyperbolic dispersion, and the magic angle changes linearly with frequency. Thus, the designed MHMS provides a new degree of freedom for controlling the propagation of magnetic surface waves: the flat bands could be realized not only by altering the excitation frequencies but also by changing the rotation angles. Therefore, to further verify the twist-induced dispersion manipulation, the field distributions and their FT curves for the magnetic MHMS with different $\mathrm{\Delta}\theta$ will be numerically studied as follows.

 

Fig. 4. Engineering of topological transition magic angles. (a) Front view of the magnetic MHMS, where the top layer is rotated counterclockwise relative to the bottom layer with a rotation angle $\mathrm{\Delta}\theta$. (b) Maps of the magic angle properties of type 1 (left) and type 2 (right) topological transition. The black dotted lines refer to the magic angles, which separate the hyperbolic (the orange areas) and elliptical (the blue areas) dispersion regimes. (c, d) Twist-induced dispersion modification and corresponding H_z distributions at 20 µm above the top surface when the frequencies are 0.63 THz and 0.79 THz, respectively.

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As shown in Figs. 4(c) and 4(d), by twisting one layer with respect to the other at a certain frequency, it is possible to induce and control the dispersion curves and the transmission of magnetic surface waves, resulting in strong magnetic interactions between light and matter. At 0.63 THz, we observe that the propagation trajectories can be easily manipulated by merely altering the mutual rotation angles, and their IFCs experience a topological transition from an ellipse-like (Δ$\theta$ = 0°) curve to titled straight lines (Δ$\theta$ = 15°), and finally to distorted hyperbolic curves (Δ$\theta$ = 30° and 45°). Similarly, the transmissions exhibit a V-shaped trajectory (Δ$\theta$ = 0°) to a flat line (Δ$\theta$ = 15°), and finally to a convergent path (Δ$\theta$ = 30° and 45°) at 0.79 THz. It can be seen that the FT patterns of the real part of the magnetic field distributions are analogous to the transition dispersion plotted in Fig. 4(c) but possess extremely stronger distortion, which will lead to the hybrid magnetic field distributions and shorten the propagation distance of magnetic surface waves. Therefore, the proposed magnetic MHMS enables drastic dispersion modification and highly tunable magnetic responses as their enhanced interlayer coupling originates from the relative displacement and rotation angle in a stacked bilayer, which can be applied to manipulate near-field focusing, imaging, and spatial multiplexing.

4.2 Local magnetic field enhancement

In order to understand the local field enhancement phenomena of the proposed MHMS, a probe is placed at 160 µm to the left of the excitation source to monitor the magnetic intensities (at 20 µm above the surface of the top layer). As depicted in Fig. 5, it is obvious that the magnetic intensity reaches a maximum at the topological transition frequency, about 10⁴ A/m. We monitor the H_z distributions in the frequency range of 0-2 THz and finally retrieve the frequency range around the two kinds of topological transition as respectively represented in Figs. 5(a) and 5(b). For the single layer HMS, we observe obvious magnetic field enhancement at 0.87 THz and 1.30 THz which show good agreement with the simulated transition frequencies in Fig. 2 (0.87 THz and 1.25 THz). And for the bilayer MHMS when Δ$\theta$ = 0° and Δ$\theta$ = 15°, the frequencies where the photonic density of states are relatively large and the localization of surface waves is strong match well with the topological frequencies in Fig. 4(b). The designed magnetic MHMS exhibits exceptional capabilities in engineering the local-field distributions and will greatly increase the application range of magnetic waves in non-diffraction transmission, and offer new opportunities for developing plasmonic devices with higher sensitivity.

 

Fig. 5. Simulated enhanced local magnetic field phenomena for the monolayer HMS and double-layer MHMS with rotation angles Δ$\theta$ = 0° and Δ$\theta$ = 15°. (a, b) H_z distributions for the two types of topological transition, respectively.

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4.3 Tunable dispersion by changing the spacer thickness of the magnetic MHMS

Next, we investigate the spacer dependence in engineering the dispersion characteristics of the magnetic MHMSs. Figure 6 summarizes the change in the magnetic field distributions with respect to the distance (d) between the two HMS layers. When the inter-stack distance is relatively small (0 µm ≤ d <100 µm), the magnetic surface wave propagation changes from a convergent path (d = 0 µm) to diffraction-less transmission (d = 5 µm), then to a V-shaped trajectory (d = 20 µm), and finally to a V-shaped trace with a larger open angle (d = 50 µm). As the distance becomes larger (d ≥ 100 µm), the two magnetic HMS layers couple much more weakly to each other and the coupling finally disappears. The magnetic plasmons transmit along a V-shaped path when d = 100 µm, exhibiting the same phenomenon as that of the single layer structure. We have further numerically calculated the dispersion contours via FT, at small separations for d ≤50 µm, where the IFCs evolve from an open hyperbola to a closed ellipse, but for a larger separation (d = 100 µm), the IFC turns into an ellipse. Thus, in this work, we always keep the thickness of the spacer d = 0 µm to gain clearer magnetic plasmon propagation. The study on the distance dependence will provide a new method to manipulate the dispersion properties and engineer the magnetic light-matter interactions.

 

Fig. 6. Inter-stack distance dependence of the bilayer structure. (a) Side view of the proposed metasurface where the top and bottom layers are separated by d. (b) Simulated Hz distributions at different gap distances d = 0 µm, 5 µm, 20 µm, 50 µm, and 100 µm, and for the single magnetic HMS, respectively. (c) Dispersion contours via FT of the real parts of the magnetic field distributions when the distances are 0 µm, 5 µm, 50 µm, and 100 µm, respectively.

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5. Conclusion

In this work, we have demonstrated that magnetic moiré configurations can be realized using DSRR arrays. The proposed MHMS provides a new platform for highly tunable and sensitive magnetic responses, as its extremely enhanced magnetic light-matter interactions originate from the relative displacement or rotation angle in a stacked bilayer structure. By analyzing the transmission of the magnetic surface wave and their associated FTs (IFCs), we identify two types of topological transition with an increase of frequency. In addition, we investigate the magic angles of these two types of transition, magnetic surface wave engineering, magnetic field distribution enhancement, and inter-stack distance dependence, which present abundant magnetic moiré effects due to the interlayer coupling. Such unique magnetic MHMSs will greatly extend the application range of moiré physics in magnetic polariton propagation, and offer new opportunities for developing integrated circuits with tunable magnetic responses.