Unidirectional bound states in the continuum in Weyl semimetal nanostructures

Chen Zhao; Guangwei Hu; Yang Chen; Qing Zhang; Yongzhe Zhang; Cheng-Wei Qiu

doi:10.1364/PRJ.459383

Abstract

Recently studied bound states in the continuum (BICs) enable perfect localization of light and enhance light–matter interactions although systems are optically open. They have found applications in numerous areas, including optical nonlinearity, light emitters, and nano-sensors. However, their unidirectional nature in nonreciprocal devices is still elusive because such trapping states are easily destroyed when the symmetry of an optical system is broken. Herein, we propose nonreciprocal and dynamically tunable BICs for unidirectional confinement of light and symmetry-protected BICs at

Γ

-point by introducing antiparallel magnetism into the optical system. We demonstrate that such BICs can be achieved by using topological magnetic Weyl semimetals near zero-index frequency without any structural asymmetry, and are largely tunable via modifying the Fermi level. Our results reveal a regime of extreme light manipulation and interaction with emerging quantum materials for various practical applications.

1. INTRODUCTION

Bound states in the continuum (BICs), as perfect confinement of light in an open system, have been extensively studied both experimentally and theoretically [1 –3], spanning from quantum mechanics to photonics [4,5]. Such non-radiating eigenmodes are peculiar states that remain localized with lifetimes that diverge to infinity, although a leaky channel may be present. Various types of BICs that originate from diverse physical mechanisms have been proposed in numerous geometrical photonics systems, such as symmetry-protected BICs, Friedrich–Wintgen (FW) BICs, and accidental single-resonance BICs [6]. These BICs facilitate exciting applications including enhanced optical nonlinearity [7], high-efficiency light guiding, and optical sensing [8] and have been implemented in zero-index plasmonic metamaterials [9,10], photonic slabs [11], high-index Mie resonators [12], and others [13]; they usually have a topological nature [14] and significantly enhanced light–matter interactions [15 –17]. However, current investigations on BICs are restricted in reciprocal systems, and their mechanisms and behaviors in nonreciprocal systems are still elusive, as well as their applications to nonreciprocal devices. Nonreciprocal devices that exhibit different responses of the same transmission channel when their sources and detectors are interchanged have become fundamental building blocks in photonic systems [18 –20], and drive widespread applications with asymmetric wave manipulation [21] assisted by magnetic bias, nonlinearities, or external gains and losses [22,23]. Such applications usually require nonreciprocal eigenmodes with high quality factors ( $Q$ -factors) and large contrast ratios simultaneously [24], especially in the field of unidirectional or angle-selective sensing and emission [25]. Therefore, nonreciprocal BICs appear to be significantly important and need to be more explored.

However, nonreciprocal BICs are very challenging to realize in generic systems. One reason is that the system supporting stable BICs is invariant under $C_{2}^{y} T$ (rotation of $π$ about the $y$ axis followed by time reversal) and $σ_{y}$ (up–down mirror symmetry of the $x - z$ plane) operators (corresponding to the coordinates in Fig. 1) to achieve zero radiation loss [26]. Generally, the nonreciprocal system is more convenient to achieve by introducing an external magnetic field compared to some metamaterials with complex structural design, and time-reversal symmetry is broken in these magnetic systems, which means $T (\bar{\bar{ε}}) \neq \bar{\bar{ε}}$ , where $\bar{\bar{ε}}$ is the permittivity tensor of materials. So such BICs can hardly be supported. The other is that spectral nonreciprocity is imperative for unidirectional BICs, which have to break inversion symmetry of the band structure in $k$ space, i.e., $ω (k) \neq ω (- k)$ . Such implementation needs to break the rotation-time-reversal symmetry (rotation of $π$ about the $x$ axis followed by time reversal) [27,28], but usually degrades BICs and inevitably reduces their $Q$ -factors due to the destruction of geometric symmetry. Consequently, the light emission efficiency and isolation contrast in the system are relatively low, which limits its application in photonic circuits [29,30]. For asymmetric light manipulation operating at different frequencies, schemes in tuning unidirectional BICs are indispensable in designing the optical system. Although some literatures have reported dynamic BICs via photodoping or electric bias [31,32], the significant material losses and imperfection of the structure, such as increased roughness or resultant deformation, largely limit the achievement of tunable BICs. Hence, dynamically tunable BICs [33] with a distinguished response such as nonreciprocity remain largely unexplored.

Figure 1.(a) Geometry of the plasmonic coating sanwiched structure. (b) Antiparallel-magnetism configuration with MWS coating. Unidirectional quasi-BICs and their corresponding radiation leakage in left (blue) and right (red) propagation are presented. BICs are achieved in both propagation directions in (a), while only the right propagating BIC (red) is in (b). (c) Angular-frequency dispersions of MWS in different amounts of magnetization. Red, yellow, green, blue, and purple lines correspond to different $ε_{a}$ when $2 b$ equals $q$ , $3 q$ , $5 q$ , $10 q$ , and $16 q$ , respectively, where $q = 1 \times 10^{8} m^{- 1}$ . The inset is the dependence of $ε_{a}$ on $2 b$ at ENZ frequency (vertical dotted line). (d) Angular-frequency dispersions of permittivity and effective permittivity in narrow frequency range.

In this paper, we propose a feasible paradigm toward unidirectional and dynamic BICs by introducing magnetism asymmetry from magnetic Weyl semimetal (MWS) slabs as shown in Fig. 1(b). By incorporating novel MWS into the system, large nonreciprocity but lower losses can be achieved without external magnetic bias, compared to the conventional counterpart of magnetoplasmonic materials [34,35]. The major findings of this work are that properly pairing MWS in an antiparallel-magnetism configuration could establish unidirectional quasi-BICs and a symmetry-protected quasi-BIC at $Γ$ -point without breaking either $C_{2}^{y} T$ or $σ_{y}$ symmetry, and dynamically tunable quasi-BICs are attainable through chemical doping. Our results promise the extreme manipulation of light in emerging quantum and low-dimensional nanomaterials [36 –38], taking advantage of superior properties in MWS.

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2. OPTICAL RESPONSE OF MAGNETIC WEYL SEMIMETAL

As a class of emerging low-dimensional quantum materials, MWS simultaneously supports intrinsic optical nonreciprocity and magneto-optic (MO) plasmonics characteristics [39,40]. The large MO response is endowed by its unique topological electronic bands and is easily tunable by the chemical doping and its intrinsic momentum. With net zero chirality, pairs of topologically protected gapless crossing points, known as Weyl nodes, exist at the corners of the Brillouin zone in conical reciprocal space, resulting in monopoles and anti-monopoles with different Berry curvatures [41,42]. These paired Weyl nodes exhibit opposite chirality and are separated by a wave vector $b$ in the first Brillouin zone or by an energy offset $ℏ b_{0}$ in the energy spectrum. Direct observations of Weyl nodes, Fermi arcs, and their relation are necessary to confirm the existence of an MSW state. Recently, comprehensive evidence on the MWS state in ${Co}_{3} {Sn}_{2} S_{2}$ was reported [42], verifying that its Fermi-arc connectivity varies with surface termination. Following this work, more experimetal realizations of MWS and its physical properties and effects associated with Weyl nodes were pursued [43,44]. Stimulated by the extensive demand on the application of MWS in nanophotonics, its optical model and electrodynamic response have been explored in previous works [45 –48].

A concrete expression of its optical response can be described by the additional axion term in the electromagnetic action and has been applied in studying magnetic surface plasmon polaritons [45,46] and nonreciprocal modes in waveguides [47]. The constitutive relation of MWS can be written as [45] $D = ε_{d} E + \frac{i e^{2}}{π ℏ ω} 2 b \times E - \frac{i e^{2}}{π ℏ ω c} 2 b_{0} B .$ (1)

Here, $ε_{d}$ denotes the major permittivity function of the Dirac semimetal. The first term denotes the major dielectric response, the second represents the anomalous Hall effect, manifesting itself as an MO effect, and the third describes the chiral magnetic effect, stimulated by an applied magnetic field. Without considering external magnetic bias on MWS in an equilibrium state, the third term can be neglected. Such schemes are experimentally realized in ${EuCd}_{2} {As}_{2}$ [46,48]. Further, we assume an unperturbed magnetic permeability (i.e., $μ = 1$ ), and crystal orientation is chosen such that $b$ is along $z$ direction, i.e., $b = b z$ . Thus, the permittivity tensor of MWS can be written as $\bar{\bar{ε}} = (\begin{matrix} ε_{d} & i ε_{a} & 0 \\ - i ε_{a} & ε_{d} & 0 \\ 0 & 0 & ε_{d} \end{matrix}),$ (2)where the off-diagonal component is $ε_{a} = \frac{2 b e^{2}}{4 π^{2} ℏ ω ε_{0}}$ [49].

Moreover, applying the random phase approximation and considering the interband electronic transitions in the Dirac cone, the diagonal term is derived through Kubo formalism as the following form [47]: $ε_{d} = ε_{b} - \frac{2 r_{s} g}{3 π Ω^{2}} + \frac{r_{s} g}{6 π} [\ln (\frac{4 Λ^{2}}{| Ω^{2} - 4 |}) + i π G (\frac{Ω}{2})],$ (3)where $ε_{b}$ is the background permittivity accounting for contributions from all bands below the Dirac cone, $G (E) = n (- E) - n (E)$ , with $n (E)$ being the Fermi distribution function [46], and $Ω = ℏ (ω + i τ) / E_{F}$ is the normalized complex frequency, with $E_{F}$ being the Fermi level and $τ^{- 1}$ being the Drude damping rate. Also, $g$ is the number of Weyl points, which is two in our case; $r_{s} = e^{2} / 4 π ε_{0} ℏ v_{F}$ and $Λ = E_{c} / E_{F}$ , where $E_{c}$ is the cutoff energy. Importantly, such an MO response is fundamentally different from the well-known magnetized plasma under external magnetic biases, although both share similar forms of permittivity tensors in Eq. (2). The MWS obtains a magnetic effect from the pseudo-vector $2 b$ without an external magnetic field, playing the role of an effective magnetic field. Such an intrinsic MO response is solely due to the electronic structure of MWS. Following Ref. [50], we use parameters $ε_{b} = 6.2$ , $Λ = 3$ , and $E_{F} = 0.3 eV$ . We observe an obvious magneto-plasmonic response with the robust and low-loss epsilon near zero (ENZ) point near the plasma frequency, as shown in Fig. 1(c). The solid and dashed black lines are, respectively, the real and imaginary parts of $ε_{d}$ , while the colored solid lines represent $ε_{a}$ in different $2 b$ . The enlarged plot of permittivity dispersion around the plasma frequency when $2 b = 1 \times 10^{8} m^{- 1}$ is shown in Fig. 1(d). The permittivity components and associated nonreciprocal response can be manipulated through changing $E_{F}$ , which is very important for dynamically tunable BICs as shown later. Though the imaginary part of epsillon is unavoidable at this ENZ point, destorying BIC, the $Q$ -factor around this point is still relatively high, thus forming a quasi-BIC, which possesses a high contrast ratio and can be applied in photonics devices as well.

3. GENERATION OF UNIDIRECTIONAL QUASI-BIC AND SYMMETRY-PROTECTED QUASI-BIC IN NONRECIPROCAL SYSTEM

Having established the optical responses of MWS, we discuss more details of unidirectional quasi-BICs in our structure composed of a planar dielectric slab covered by MWS, where the direction of magnetism is perpendicular to the incidence plane (Voigt configuration). A leaky mode could exist inside this dielectric core with the complex propagation constant $β$ , and the disappearance of radiation shows a feature of quasi-BICs when the dielectric core is covered by plasmonic materials [Fig. 1(a)] [9]. However, unidirectional quasi-BICs appear in the antiparallel-magnetism case, which means the directions of $2 b$ in upper and lower MWS layers are antiparallel to each other [Fig. 1(b)], where the radiation leakage along positive- $x$ direction is perfectly forbidden, while that along the negative direction is open. Considering our Voigt configuration under transverse-magnetically polarized oblique incidence with an angle of $θ$ , no polarization conversion is allowed. The dispersion relation in MWS layer is $k_{x}^{2} + k_{y m}^{2} = k_{0}^{2} ε_{eff}$ , where $k_{y m}$ is the $y$ component of the wave vector in MWS. $ε_{eff} = \frac{ε_{d}^{2} - ε_{a}^{2}}{ε_{d}}$ denotes effective permittivity. When $ε_{d}$ is close to zero without considering intrinsic loss, $ε_{eff}$ tends to become infinite while $ε_{a}$ remains almost unchanged, leading to a singular point in the reflection spectrum. Figures 2(a) and 2(b) show the reflection spectrum in parallel- and antiparallel-magnetism cases, where the vanishing feature of resonances suggests the existence of BICs. The off- $Γ$ BIC arises from the coupling between extreme plasmonic resonance in the coating and Fabry–Perot resonance of the slab [9]. In our cases, considering the reflection at the interface between the MWS and dielectric, the condition of Fabry–Perot resonance is obtained as $k_{0} d_{c} \sqrt{ε_{c} - \sin^{2} θ} + φ_{r} = n π,$ (4)where $\sin θ = k_{x} / k_{0}$ . $ε_{c}$ and $d_{c}$ are permittivity and thickness of the dielectric slab. Here, $φ_{r}$ is the extra phase introduced by reflection coefficients on two interfaces of the core layer because it contains the off-diagonal component in the permittivity tensor. In the parallel-magnetism case, $e^{i φ_{r}} = r_{+} r_{-}$ , where $r_{\pm}$ represent reflection coefficients on lower (+) and upper sides (−) of the dielectric slab, respectively, and they are the function of effective wave vector $k_{\pm}$ in the MWS layer given by $k_{\pm} = \frac{k_{y m} \pm \frac{i k_{x} ε_{a}}{ε_{d}}}{ε_{eff}} .$ (5)

Figure 2.Reflection spectra for structure with (a) parallel and (b) antiparallel magnetism when the thickness of MWS is 0.1 μm; $ε_{c} = 5$ and $d_{c} = 1 μm$ . Reflection spectra for structure with infinite thickness of MWS layers at two sides with (c) parallel and (d) antiparallel magnetism when $ε_{d} = 1$ and $ε_{a} = 0.2$ . The white dashed line denotes ENZ frequency.

Therefore, the Fabry–Perot resonance is symmetric regardless of propagation directions [Fig. 2(c)]. Such results are obtained from an ideal model in which a dielectric slab is covered by infinite MWS with $ε_{d} = 1$ , which contains a magnetic response outside the cavity. However, in the antiparallel-magnetism case, the extra phase becomes two different forms as $e^{i φ_{r +}} = r_{+} r_{+}$ and $e^{i φ_{r -}} = r_{-} r_{-}$ , where the sign represents forward ( $+ k_{x}$ ) and backward ( $- k_{x}$ ) propagation, respectively, resulting in asymmetric distribution of the reflection spectrum [Fig. 2(d)]. Such an extra phase influences the performance of unidirectional quasi-BICs because its eigenfrequency is not exactly at Fabry–Perot resonance but close to it. So the resonance cannot be perfectly coupled, thus forming the quasi-BIC, as further explicitly discussed in Appendix A. The white line in Fig. 2(d) represents the frequency when $ε_{d} = 0$ , defined as ENZ frequency, corresponding to plasma frequency. It intersects with Fabry–Perot resonance only in the $+ k_{x}$ region, leading to a unidirectional quasi-BIC emerging in this case.

To better illustrate the unidirectional quasi-BIC, we performed an eigenmode analysis in the antiparallel-magnetism case [Fig. 1(b)]. Complex dispersion can be found in the pole of the scattering matrix of the source-free Maxwell equations, which can be derived into the following form: $Ψ_{(k_{x})} Ψ_{(k_{x})} e^{i k_{y c} d_{c}} - Φ_{(k_{x})} Φ_{(k_{x})} e^{- i k_{y c} d_{c}} = 0,$ (6)and the dispersion in the parallel-magnetism case is as follows: $Ψ_{(k_{x})} Ψ_{(- k_{x})} e^{i k_{y c} d_{c}} - Φ_{(k_{x})} Φ_{(- k_{x})} e^{- i k_{y c} d_{c}} = 0 .$ (7)

Here, $Ψ$ and $Φ$ are the functions of $k_{x}$ , and $k_{y c}$ is the $y$ component of the wave vector in the central dielectric layer. A concrete expression and detailed derivation can be found in Appendix B. With careful examination by comparing these two equations, one can easily prove that Eq. (6) does not follow spatial-inversion symmetry. In particular, several eigenmodes of interest are plotted in Fig. 3(a), which are responsible for the sharp resonant reflection features in Fig. 2(b), and they are asymmetrically distributed in $+ k_{x}$ and $- k_{x}$ regions. Three symmetric modes are labeled as forward mode I (blue), backward mode (green) in the $+ k_{x}$ region, and forward mode II (red) in the $- k_{x}$ region, and only $β$ of forward mode I is non-zero at the ENZ frequency, which is possible to support quasi-BIC. Compared to Fig. 2(d), the Fabry–Perot resonance is determined by forward mode I and backward mode branches, which are in opposite slopes in the narrow bandwidth a little below the ENZ frequency. Forward mode II dramatically departs from its location in the parallel-magnetism case, remaining only in the high frequency branch, and the other half disappears. Moreover, the symmetry-protected quasi-BIC exists at $Γ$ -point in the antiparallel-magnetism case, determined by antisymmetric modes (yellow) in two wave vector regions, while it vanishes in the parallel-magnetism case. The underlying physics is that the antiparallel-magnetism case preserves $C_{2}^{y} T$ and $σ_{y}$ symmetry because the magnetic field, being a pseudovector, gains an additional sign flip under mirror symmetry operation. As the time-reversal symmetry is broken in the system, the permittivity tensor of MWS shown in Eq. (2) under a rotation operator along the $y$ axis becomes $C_{2}^{y} {\bar{\bar{ε}}}_{(x, y, z)} = {\bar{\bar{ε}}}_{(- x, y, - z)}^{*}$ . Then, combining the $T$ operation, we obtain $C_{2}^{y} T {\bar{\bar{ε}}}_{(x, y, z)} = {\bar{\bar{ε}}}_{(- x, y, - z)}^{*} = {\bar{\bar{ε}}}_{(x, y, z)} .$ (8)

Figure 3.(a) Angular-frequency dispersions of normalized $Re (β)$ for some eigenmodes related to the quasi-BIC. The antisymmetric mode has antisymmetric field distribution, shown in Appendix C. Angular-frequency dispersions of normalized $Im (β)$ of (b) forward mode I and (c) other eigenmodes. (d) Radiative $Q$ -factor of unidirectional quasi-BIC (blue) and symmetry-protected quasi-BIC (yellow), and radiation loss contrast between two modes in two opposite directions. (e) Time-snapshot of the magnetic field profile at the frequency close to ENZ frequency. (f) Time-snapshot of magnetic field intensity and electric field.

It is one of the sufficient conditions for symmetry-protected BICs [26] existing in nonreciprocal systems, and the detailed derivation can be found in Appendix E.

Importantly, the imaginary part of an eigenmode, i.e., $Im (β)$ , vanishes for forward mode I [Fig. 3(b)], indicating the disappearance of energy leakage. The radiative $Q$ -factor (see the definition in Ref. [51]) of this mode (blue) is plotted in Fig. 3(d), diverging near the ENZ frequency. Moreover, $Im (β)$ of other eigenmodes are displayed in Fig. 3(c), where only antisymmetric eigenmodes converge to zero point. That supports the symmetry-protected quasi-BIC at $Γ$ -point, while the other two modes are all leaky. The radiative $Q$ -factors of symmetry-protected quasi-BICs (yellow) are lower than those of unidirectional quasi-BICs [Fig. 3(d)], and decay more rapidly, indicating that they are sensitive to tiny external perturbations. The unidirectional feature of unidirectional quasi-BIC can be obviously reflected from the radiation loss contrast (red), defined as $\frac{Im (β_{+})}{Im (β_{-})}$ , where $β_{\pm}$ represent the wave vector of eigenmodes in the $+ k_{x}$ and $- k_{x}$ regions, respectively. The maximum nonreciprocity is achieved at the ENZ frequency when quasi-BIC propagates in $+ k_{x}$ direction without any radiation loss. The field distributions near this BIC are shown in Figs. 3(e) and 3(f), corresponding to the incident angle of $\sin θ = 0.778$ . A resonant localized magnetic field in out-of-plane direction exists within the dielectric slab, consistent with the non-radiating nature of the mode, while the electric field is mainly directed along the orthogonal direction, marked by the white arrows. Actually, this strong oscillation of the electric field works as orthogonal electric dipoles to excite volume-plasmon resonance in MWS layers. Only such resonance coupling with Fabry–Perot resonance can achieve the accidental BICs in this transversely homogenous waveguide. Therefore, such a unidirectional quasi-BIC is an FW BIC, which is not limited by the symmetry restrictions mentioned above.

4. TUNABILITY ON UNIDIRECTIONAL BIC

We now discuss the dynamic unidirectional quasi-BICs via tailoring the tunability of MWS based on two avenues available, by the argument of breaking the reciprocal response. First, one could change its intrinsic magnetization value in two MWS layers to break the time-reversal symmetry to generate unidirectional quasi-BICs. As shown in Figs. 4(a)–4(c), we vary the magnitude of its intrinsic magnetism in the lower MWS layer while keeping the upper MWS layer unchanged. A pair of asymmetric quasi-BICs still occur with different wave vectors [Fig. 4(a)], quite as expected due to the parallel magnetism. However, only one quasi-BIC exists in the $+ k_{x}$ region when the MO response of the lower MWS disappears [Fig. 4(b)] or becomes enlarged [Fig. 4(c)]. Moreover, such a unidirectional quasi-BIC in forward direction carries a larger momentum and moves to a larger $k_{x}$ direction in the antiparallel-magnetism case. With too large magnetization, no unidirectional quasi-BIC is found, as the Fabry–Perot resonance is not supported near the ENZ frequency [Fig. 4(d)]. Notably, the symmetry-protected BIC still exists in the antiparallel-magnetism case when $C_{2}^{y} T$ and $σ_{y}$ symmetry is preserved [Fig. 4(d)], while it turns into a quasi-symmetry-protected BIC when such symmetry is broken [Fig. 4(c)]. In general, via tailoring the distance between Weyl nodes in momentum space in MWS independently without altering $ε_{d}$ , we could tune quasi-BICs of both forward and backward directions in a designer manner.

Figure 4.(a)–(d) Reflection spectra with different intrinsic magnetization values in two MWS layers. The insets are the corresponding schematic models. (e) Angular-frequency dispersions of the bulk MWS in different $E_{F}$ . Solid and dashed lines correspond to $Re (ε_{d})$ and $Im (ε_{d})$ , respectively. (f) Dependence of $ω_{p}$ on $E_{F}$ . (g), (h) Reflection spectra with and without losses for $E_{F} = 0.36 eV$ and $E_{F} = 0.4 eV$ , corresponding to ENZ frequency of $ω_{p} = 4.961 \times 10^{14} rad / s$ and $ω_{p} = 6.892 \times 10^{14} rad / s$ , marked by horizontal dashed lines. In this case, the model is the same as the antiparallel-magnetism configuration in Fig. 2(b) except for $d_{c} = 0.75 μm$ , with magnetization value being $2 b = q$ in upper layer and $2 b = - q$ in lower layer.

Second, via controlling the $E_{F}$ of MWS, the ENZ frequency could be tuned in a broad range, thus tuning the unidirectional quasi-BIC in our case. The permittivity $ε_{d}$ with different $E_{F}$ is plotted in Fig. 4(e), and it shows the ENZ frequency appears to be approximately linearly blueshifted with smaller losses when increasing $E_{F}$ [Fig. 4(f)]. Such a characteristic makes our tunability on quasi-BICs come true. Since the loss is one major limitation in practical implementation in most systems [52], as a comparison, we calculate the reflection spectra with small intrinsic loss $γ = Im (ε_{d})$ [Figs. 4(i) and 4(j)], calculated by Eq. (3) and without loss [Figs. 4(g) and 4(h)], at $E_{F} = 0.36 eV$ and $E_{F} = 0.4 eV$ . Indeed, the asymmetric leaky modes change dramatically in both forward and backward directions with varying $E_{F}$ . The comparison between them shows that the unidirectional quasi-BIC carries larger momentum to satisfy the condition of Fabry–Perot resonance in the region when $E_{F}$ increases, while the symmetry-protected quasi-BIC still exists, unperturbed by varying the $E_{F}$ , protected by $C_{2}^{y} T$ symmetry as demonstrated above. Therefore, by carefully tuning the $E_{F}$ and setup of the model, we can obtain the unidirectional quasi-BIC in any appropriate frequency with different radiation angles. If losses are considered, though quasi-BICs are degraded to some extent with the decay of leaky modes and the blurring of spectra contrast, the physical mechanism remains evident as unidirectional quasi-BICs can also be observed at the original location. The effect of intrinsic losses in MWS on BICs is discussed in detail in Appendix F.

5. CONCLUSION

In conclusion, we realize unidirectional and dynamically tunable quasi-BICs in paired MWS slabs via tailoring their intrinsic MO responses. They show the extremely different radiation of forward and backward leaky modes—one mode perfectly closed despite being within the radiation channel, and the other open, which can be tuned by changing the geometric pairing and the $E_{F}$ of MWS. We discuss the symmetry requirement of symmetry-protected BICs in a nonreciprocal system. Our strategies neither rely on the discretized asymmetric lattice [14,52] to break the spatial-inversion symmetry, nor require the external magnetic bias, showcasing a new avenue of extreme wave manipulation with emerging exotic quantum materials.

Acknowledgment

Acknowledgment. Y. Z. acknowledges NSFC, and C. Q. acknowledges the A*STAR Pharos Program.

APPENDIX A: QUASI-BOUND STATES IN THE CONTINUUM

ω = 4.2167 \times 10^{14} rad / s

Figure 5.(a) Reflection and transmission spectrum dependence on transverse wave vector $k_{x}$ at frequency $ω = 4.2167 \times 10^{14} rad / s$ . (b) Reflection and transmission spectrum dependence on the frequency of incident light $ω$ at certain incident angle $\sin θ = 0.778$ . Solid and dashed lines represent the spectrum in linear coordinates and log scale, respectively.

Figure 6.Geometry of MWS sandwiched structure with (a) parallel magnetism and (b) antiparallel magnetism in the coatings.

APPENDIX B: DERIVATION OF SCATTERING MATRIX AND SOLUTION OF EIGENMODE

ε_{c}

e^{i (k \cdot r - ω t)}

n

T_{j}

M_{2}

Δ

\frac{Δ_{+}^{-} \cdot Γ_{-}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{-}^{+} \cdot Γ_{+}^{-} \cdot e^{- i k_{y m} d_{m}}}{Δ_{+}^{-} \cdot Γ_{-}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{-}^{+} \cdot Γ_{+}^{+} \cdot e^{- i k_{y m} d_{m}}} e^{2 i k_{y c} d_{c}} - \frac{Δ_{-}^{-} \cdot Γ_{+}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{+} \cdot e^{- k_{y m} d_{m}}}{Δ_{-}^{-} \cdot Γ_{+}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{-} \cdot e^{- i k_{y m} d_{m}}} = 0 .

Ψ_{(k_{x})} = Δ_{-}^{-} \cdot Γ_{+}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{-} \cdot e^{- i k_{y m} d_{m}}, Ψ_{(- k_{x})} = Δ_{+}^{-} {\cdot Γ}_{-}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{-}^{+} {\cdot Γ}_{+}^{-} \cdot e^{- i k_{y m} d_{m}}, Φ_{(k_{x})} = Δ_{-}^{-} \cdot Γ_{+}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{+} \cdot e^{- i k_{y m} d_{m}}, Φ_{(- k_{x})} = Δ_{+}^{-} {\cdot Γ}_{-}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{-}^{+} {\cdot Γ}_{+}^{+} \cdot e^{- i k_{y m} d_{m}} .

Ψ_{(k_{x})} Ψ_{(- k_{x})} e^{i k_{y c} d_{c}} - Φ_{(k_{x})} Φ_{(- k_{x})} e^{- i k_{y c} d_{c}} = 0 .

Θ_{1}

\frac{Δ_{-}^{-} \cdot Γ_{+}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{-} \cdot e^{- i k_{y m} d_{m}}}{Δ_{-}^{-} \cdot Γ_{+}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{+} \cdot e^{- i k_{y m} d_{m}}} e^{2 i k_{y c} d_{c}} - \frac{Δ_{-}^{-} \cdot Γ_{+}^{-} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{+} \cdot e^{- i k_{y m} d_{m}}}{Δ_{-}^{-} \cdot Γ_{+}^{+} \cdot e^{i k_{y m} d_{m}} - Δ_{+}^{+} \cdot Γ_{-}^{-} \cdot e^{- i k_{y m} d_{m}}} = 0 .

Ψ_{(k_{x})} Ψ_{(k_{x})} e^{i k_{y c} d_{c}} - Φ_{(k_{x})} Φ_{(k_{x})} e^{- i k_{y c} d_{c}} = 0 .

From Eq. (B3), it is not difficult to prove that the equation stays same if the transverse wave vector is reversed, thus forming the symmetric eigenmode in this case, as shown below in Fig. 8 (a). However, the situation changes in Eq. (B5), which is attributed to the breaking of spatial inversion symmetry and results in the unidirectional quasi-BIC in this case.

APPENDIX C: MODE PROFILES IN ANTIPARALLEL-MAGNETISM CASE

+ k_{x}

Figure 7.Profile of each eigenmode in antiparallel-magnetism case. (a) Mode distribution of the real part of propagation constant, illustrated in the main text. (b)–(d) Magnetic field profile of the corresponding mode pointed out by the arrow in (a).

APPENDIX D: SYMMETRIC QUASI-BIC IN PARALLEL-MAGNETISM CASE

ε_{c} = 5.5

Figure 8.(a) Amplitude of reflection coefficients dependence on angular frequency and transverse wave vector in parallel-magnetism configuration with the thickness of MWS layers being 0.1 μm. The permittivity and thickness of the dielectric slab are $ε_{c} = 5.5$ and $d_{c} = 1 μm$ , respectively. (b) Angular-frequency dispersion of the real part of the wavenumber for some eigenmodes related to the quasi-BIC in (a). (c) Angular-frequency dispersion of the imaginary part of the wavenumber of the mode in blue in (b) at $+ k_{x}$ and $- k_{x}$ regions. (d) Dependence of the $Q$ -factor of the mode in blue on the angular frequency of incident light.

APPENDIX E: SYMMETRY REQUIREMENT FOR SYMMETRY-PROTECTED BIC IN NONRECIPROCAL SYSTEM

C_{2}^{y} T

y

{\bar{\bar{ε}}}_{l}

C_{2}^{y} T

σ_{y}

APPENDIX F: EFFECTS OF INTRINSIC LOSSES ON BICS

E_{F} = 0.3 eV

Figure 9.Imaginary part of diagonal element of permittivity $ε_{d}$ versus Fermi level. The inset is an enlarged plot of losses in a higher Fermi level range.

Figure 10.Reflection spectra of anti-magnetism case with different losses of (a) 0.001, (b) 0.005, and (c) 0.008, and the geometric parameter is the same as that in the main text. (d) Reflection spectrum at the transverse wave vector of $k_{x} = 0.81 k_{0}$ , where $k_{0}$ is a wave vector in air. This resonance caused by a leaky mode is near the frequency of BICs.

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