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Abstract
The topological magnetoelectric (TME) effect is a novel optical response from topological insulators. This effect shows that magnetic (electric) polarization can be induced by an applied electric (magnetic) field, and it is characterized by the fine structure constant. However, the TME effect is generally very weak and still a challenge to be observed in the experiment. In this paper, we showed that the far-field scattering of a core-shell topological cylinder can be tuned by the TME effect which was enhanced at the surface of plasmonic core around Fano resonance. The interference of broad dipolar mode and narrow quadrupole mode is changed with the topological magnetoelectric polarizability. We demonstrated the reversal of optical responses associated with the TME effect in both far-field and near field. Our results may offer an alternative way to observe the TME effect in topological insulators.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, topological insulator (TI) has attracted much attention in various physical systems [1–3]. One of the most interesting properties arising from TI is the topological magnetoelectric (TME) effect, which shows that magnetic polarization can be induced by applied electric field and vice versa [4]. Due to the TME effect, many exotic phenomena can be observed from the unusual magnetoelectric response of topological insulator. For instance, giant magneto-optical Kerr and Faraday effects are predicted when the time-reversal symmetry of topological insulator is broken [5]. Meanwhile, these topological phenomenona could be directly measured to determinate the precise values of three basic physical constants: e (fundamental electric charge), h (Planck's constant), and c (the speed of light) [6]. More recently, this concept of determination has been realized experimentally by observing the Faraday rotation angle from a 3D topological insulator [7].
However, the TME effect is generally rather weak since it is associated with the fine structure constant. Direct observation of the TME effect requires terahertz measurement techniques and heterostructure engineering [7,8]. Due to the extreme challenge in experiment, the observation of the TME effect has been limited only in the measurement of quantized Faraday angles and Kerr angles [9]. More recently, M. Mondal et al. have demonstrated the TME effect by tuning the chemical potential of Bi2Se3 films [10]. They suppress the bulk carrier density and shift the Fermi energy above the Dirac point to observe quantized Faraday angles. What's more, the TME effect can be anisotropic and detected by changing the film thickness of the topological insulators [11]. The optical response of nanoparticles, such as the optical absorption spectrum, is also found to be dependent on the topological magnetoelectric polarizability [12].
In this paper, we investigate the optical far-field scattering around Fano resonances by a plasmonic core and topological shell cylinder. Due to the inherent high sensitivity of Fano resonances, small charges of interfering modes by the TME effect could induce observable shifts of far-field scattering. The plasmonic core not only offers electric dipolar and quadrupole modes but also enhances the TME effect by localized surface plasmon. Hence, we can observe shifted light scattering consistent with the TME that can be mediated by the constructive and destructive interferences around Fano resonance. The proposed light scattering of nanoparticles dependent on the TME effect offers an alternative way to probe the electromagnetic characters of the topological insulator with far-field optical sensing technique [13–16].
2. Theoretical model
In this section, the Mie theory is extended to investigate the electromagnetic scattering of a core-shell topological cylinder. We consider a nano cylinder with the radius of core a and shell b illuminated by a TE polarized light. The permittivity of the dielectric shell (region 2) is ${\varepsilon _2}$, and the plasmonic core (region 1) is described by Drude model ${\varepsilon _1}(\omega ) = 1 - {\omega _\textrm{p}}^2/({\omega ^2} + i{\gamma _\textrm{d}}\omega )$, with the plasma frequency ${\omega _\textrm{p}}$ and the damping rate ${\gamma _\textrm{d}}$, respectively [17]. In this work, the TE polarized light ${{\bf E}_{\textrm{inc}}} = {\textrm{E}_0}{e^{ - ikx}}$ is incident perpendicularly to the cylinder, where ${\textrm{E}_\textrm{0}}$ is the amplitude and $k = \omega \sqrt {{\varepsilon _3}{\mu _3}}$ is the wavevector in the background medium (region 3). Here the time dependence $\textrm{exp} {\kern 1pt} {\kern 1pt} (i{\kern 1pt} \omega {\kern 1pt} {\kern 1pt} {\kern 1pt} t)$ is suppressed. Based on Mie theory, one can expand the incident wave by the vector cylindrical harmonics,
And the electric field and the magnetic flux density in region 3 can also be expressed as:
According to the topological field theory [19], the constitutive relations for topological nanoparticles should be modified as:
where $\bar{\alpha } = \Theta \alpha /\pi$ is an odd-numbered multiple of $\alpha$. The topological parameter $\Theta $ is called axion angle, and $\alpha = 1/137$ is the fine structure constant [20]. It has been verified that, the corresponding equations remain the same form as the topologically trivial case ($\bar{\alpha } = 0$), when the modified constitutive relations for the nontrivial case are applied in the Maxwell equations [21]. Then, the scattered coefficients ${a_n}$ and $b{}_n$ can be written in the matrix form by imposing the following boundary conditions at the interfaces r = a and r = b, respectively,And combining with the constitutive relations, we can obtain the following boundary conditions at r = a :
Note that the magnetic multipoles cannot be excited inside an ordinary dielectric cylinder by a TE incident light because of ${b_n} = 0$. However, for a TI cylinder, the magnetic multipoles can be excited due to the TME effect.
For a core-shell cylinder, interference can be observable in differential scattering spectra, such as the backward scattering (BS) and the forward scattering (FS) [22]:
The scattering cross section is defined by the ratio of the total power scattered by the core-shell cylinder to the incident power per unit area on the core-shell cylinder or the ratio of scattering field to incident field, which is given by the following formula [23]:
3. Numerical results and discussions
Here we apply the above model to a realistic nanostructure. Without loss of generality, we consider a core-shell cylinder with plasmonic core and TI shell under TE incidence, as is shown in Fig. 1. And the cylinder is placed in air (region 3). The radius of the core is a = 17 nm, whereas the shell’s radius is b = 27 nm. The other parameters are as follows, ${\varepsilon _2} = 2.2$, ${\omega _\textrm{p}} = 4.306\textrm{ }ev$ and ${\gamma _\textrm{d}} = 0.0014\textrm{ }ev$. Both background (air) and nanostructure materials are assumed to be non-magnetic. The topological insulator materials can be found in various systems, such as Bi1-xSbx, Bi2Se3, Sb2Te3 [1], and these kinds of materials have been realized experimentally by many methods, such as laser ablation technique [24]. And this kind of core-shell cylinder with topological insulator materials can be fabricated with the recent developments of nano-fabrication technologies [25]. What’s more, optimal designs of the core-shell cylinder are possible with different materials, such as metal, semiconductors, or dielectrics [26].
Fig. 1. Schematic view of the nanostructure placed along the z axis. A TE wave with wave vector $k$ is incident perpendicularly to the core-shell cylinder, which consists of a plasmonic core (marked with 1) and topological insulator (TI) shell (marked with 2). The spatial distribution of energy flow and corresponding lines in the near field around Fano resonance are projected above the cylinder.
In this work, we investigate the topological magnetoelectric effect (TME) in topological insulator shells around quadrupole resonance. The plasmonic core can give rise to both dipole and quadrupole modes, which will lead to Fano resonance when these two modes constructively or destructively interfere with each other. Meanwhile, the localized surface plasmons in the topological insulator shell will be greatly affected by the TME effect. Due to the TME effect, surface Hall currents are produced at the outer surface and inner surface of the topological insulator shell [21]. The TME effect can be manifested in the scattering coefficients and far-field scattering of the core-shell nanostructure.
As is shown in Fig. 2, the scattering coefficients ${a_1}$ and ${a_2}$ redshift when the axion angle $\Theta $ increases. This redshift is more distinct at the plasmon resonances of dipole and quadrupole modes. Similar resonance frequency shifts can also be observed by dielectric perturbations or rising temperatures in other optical systems [27,28]. In addition, the far-field forward scattering ($\theta = {0^ \circ }$) and backward scattering ($\theta = {180^ \circ }$) also show corresponding redshifts. Importantly, the overlap of broad dipole mode and narrow quadrupole mode gives rise to Fano resonance at around $\lambda$ = 500 nm, see Fig. 2(b). Note that the narrow quadrupole mode is common for weakly dissipative nanostructures, which comes from the interplay of the dissipative and radiative damping [29,30]. The TME effect can tune the location of Fano resonance as well. Generally, the TME is too weak to be observed for non-resonant cases. However, Fano resonance has inherent sensitivity to tiny variations in the system because the resonance comes from the interference between different modes, such as electric dipole mode and electric quadrupole mode. Figure 2(c) and 2(d) show the modification of the TME effect on the scattering characters around Fano resonance. By changing the axion angle $\Theta $, the scattering coefficient ${a_1}$ remains almost the same magnitude while ${a_2}$ undergoes a sharp resonance and reaches its maximal at $\Theta = 21\pi$. Around this point, the magnitudes of ${a_2}$ are comparable to that of ${a_1}$, and the phases of ${a_2}$ also flip over leading to the switch of constructive destructive interference between ${a_1}$ and ${a_2}$. The interaction between broad resonance ${a_1}$ and narrow resonance ${a_2}$ gives rise Fano resonance [22,31]. The interaction can be manipulated by the axion angle of the nanostructure. Hence this topology-induced Fano resonance offers an additional way to explore Fano resonance.
Fig. 2. (a) and (c) shows the Mie scattering coefficients ${a_1}$ (electric dipole) and ${a_2}$ (electric quadrupole) versus the incident wavelength $\lambda$. (b) and (d) shows the differential scattering spectra QFS (forward scattering) and QBS (backward scattering) for different axion angles. In (c) and (d), the incident wavelength is 500 nm, and the black dash line shows the Fano fit of forward scattering.
As is shown in Fig. 2(d), a small variation of axion angles can turn the overall scattering from mainly forward scattering to mainly backward scattering. This typical asymmetric profile can be well fitted by the following Fano functional formula [32]:
with $\tau = 2(\delta - {\delta _0})/\Gamma $, where ${\delta _0}$ and $\Gamma $ are the position and width of the resonance, ${\sigma _0}$ and ${\sigma _{bg}}$ are the normalized scattering and background scattering, and f is the asymmetry parameter [33]. The dashed line shows the Fano fit of forward scattering, which well describes the asymmetric scattering profile. This means that the resonance is Fano resonance [32].Figure 3 shows the verification of the theoretical model by comparing the results from the finite element method (FEM) by using COMSOL Multiphysics V.5.4. In our extended Mie theory model, the topological constitutive relations are expressed by the boundary conditions, i.e., Eqs. (21–23). While these constitutive relations can be imposed into Maxwell's equations by directly modifying the built-in variables D and H in COMSOL. The simulation results by FEM are in accordance with that from Mie theory. We can see that the scattering of the topological nanoparticle is flipped over around the Fano resonance.
Fig. 3. Comparison of far-field scattering solutions by extendted Mie theory and commercial full-wave simulation by finite element method (FEM) for different axion angle. The two solutions are in good agreement. The other parameters are the same as those in Fig. 2(d).
In this paper, the Fano resonance comes from the interference of electric dipole and quadrupole modes. The interference can not only be observed in the topological cylinder's far-field scattering spectra but also be illustrated in the near field of the particle, such as the distribution of the energy flow, i.e., Poynting vector S. The energy flow reveals the re-emitted light from the nanoparticle. As can be seen in Fig. 4, the spatial distribution of energy flow (Fig. 4(a) for $\Theta = 0$) exhibits a typical blend of electric dipole and quadrupole patterns. Four vortex points appear at the boundary of the plasmonic core, which is the localized quadrupole plasmon and the upside two vortex points with middle saddle points form a pole of dipole. This localized surface plasmon boosts the topological magnetoelectric (TME) effect. In the topological shell small variance of topological parameter $\Theta $ can induce dramatic changes on the interference. When we tune the topological parameter to $\Theta = 19\pi$ (see Fig. 4(b) and 4(e)), the dipole mode interferes destructively with the quadrupole mode in the backward direction, while they interfere constructively in the forward direction. At the same time, the corresponding far-field shows a similar effect: the backward scattering is suppressed, and the forward scattering is enhanced. Interestingly, the interference reverses in the forward and backward directions when we slightly change the topological parameter $\Theta $ from $19\pi$ to $23\pi$ (see Fig. 4(c)). The handedness of the quadrupole vortex points reverses as well. The particle attracts much energy flow to it and transforms the path of energy flow from pass-by to pass-through the particle. The transition of the energy path around Fano resonance can be observed in many photonic systems [33,34]. What's more, the far-field scattering also flips over with the variation of topological parameter (see Fig. 4(f)).
Fig. 4. The near field energy flow diagram and corresponding far-field scattering cross section $\sigma$ for different axion angle $\Theta $ around Fano resonances. Both near filed and far-field flips over with the variation of topological parameter $\Theta $. The red circles represent vortex points, and the blue squares represent saddle points.
To demonstrate the TME effect, we show the field distributions for $\Theta = 19\pi$ in Fig. 5. For conventional dielectric cylinders ($\Theta = 0$) incident by TE waves, there is no distribution of ${D_z}$ and ${B_\varphi }$ inside the cylinder. However, due to the TME effect, ${D_z}$ is induced inside the cylinder by ${B_z}$. The nonzero ${B_\varphi }$ also comes from the TME effect, as is shown in Fig. 5. Interestingly, both ${D_z}$ and ${B_\varphi }$ are totally trapped in the cylinder. This trapping phenomenon cannot be observed in conventional dielectric cylinders or spheres [35,36]. What's more, the localized surface plasmon enhances the TME effect, which shows typical electric quadrupole field distributions.
Fig. 5. Field distributions at $\Theta \textrm{ = 19}\pi$. The nonzero values of ${E_\varphi }$, ${D_z}$ and ${B_\varphi }$ arise from the topological magnetoelectric effect. The black circles are the boundaries of the core-shell cylinder. The other parameters are the same as those in Fig. 2(d).
4. Conclusion
In conclusion, we have studied the far-field light scattering and near field energy distribution of plasmonic core and topological shell cylinder. We have demonstrated that the optical response can be tuned by the topological magnetoelectric effect around Fano resonance, which is enhanced by localized surface plasmon. Although the results in this paper are obtained from a core-shell cylinder, similar phenomena can be observed in other optical systems with Fano resonance, which possesses inherent sensitivity to fine variances. In addition, the core-shell nanostructure can be designed to realize broadband cloaking in the optical range without using exotic materials [37,38]. For more complicated structures, such as the cluster configuration of core-shell nanostructure, one can engineer optimized nanostructrues to obtain perfect absorption or unidirectional scattering [39–41]. Our results may have applications in measuring the topological magnetoelectric effect and determining the precise values of basic physical constants.
Funding
National Natural Science Foundation of China (12174281, 92050104); Suzhou Prospective Application Research Project (SYG202039).
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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