Spin-dependent and tunable perfect absorption in a Fabry-Perot cavity containing a multi-Weyl semimetal

Jipeng Wu; Jipeng Wu; Rongzhou Zeng; Jiaojiao Liang; Di Huang; Xiaoyu Dai; Yuanjiang Xiang

doi:10.1364/OE.499381

1. Introduction

Light absorber, as a functional device that can effectively absorb incident light at the required wavelengths, has attracted broad interests due to its potential applications in optical switches and detectors, thermal emitters, and sensors [1,2]. For realizing perfect absorption, numerous approaches and compositions have been proposed. For instance, Wu et al. have demonstrated that the adjustable perfect absorption phenomenon can be realized by taking advantage of a graphene-hexagonal Boron Nitride hyper crystal [3]. Photonic crystal (PhC), a type of microstructure holding cyclical spatial modulation in dielectric constant, provide a platform to control light at the mesoscopic dimension since a given range of frequencies called photonic bandgap can be generated to prevent the transmission of light [4–6]. Taking advantage of PhC develops numerous available approaches to achieve perfect absorption phenomenon. For example, Wang et al. have proposed a graphene coated PhC to obtain the tunable and multichannel perfect absorption phenomenon by exciting the optical Tamm states [7]. Lin et al. have designed the asymmetric topological PhCs with graphene defect to acquire the tunable perfect absorption phenomenon by making use of the topological interface states [8]. Generally, such strategies focus on realizing the adjustable perfect absorption phenomenon of linear polarization. The perfect absorption phenomenon of circular polarization originates from the right hand circularly polarized (RCP) and left hand circularly polarized (LCP) incident waves. To support the difference in optical absorption between RCP and LCP waves, the chiral material should be involved [9]. Chirality develops a significant role in numerous fields, such as chemistry, biology, circular dichroism, and asymmetric transmission [10–12]. To acquire the perfect absorption phenomenon of circular polarization, metamaterials and metasurfaces with chirality have been widely presented, such as gamma-similar array [13], double-bar patterns [14], L-shaped patterns [15], liquid-crystal-loaded resonators [16], stereoscopic full metallic resonator array [17], and all-dielectric planar chiral metasurface [18]. Although metamaterials and metasurfaces show the superiority of flexible design, the relatively complex manufacturing processes may restrict their experimental discussions and practical applications. Therefore, it is still significant to develop other available and convenient chiral materials to discuss the tunable perfect absorption phenomenon of circular polarization.

Recently, a new topological material, called Weyl semimetal (WSM), has attracted multitudinous interests due to its outstanding optical features. WSM holds a linear dispersion analogous to graphene, thus generating a great deal of prominent optical behaviors, such as controllable plasmon resonance [19] and distinctive optical nonlinearity [20]. Unlike graphene, WSM possesses the three-dimensional Hamiltonian, which develops numerous distinctive optical performances, such as chiral anomalous and Hall effects [21,22]. Particularly, a couple of Weyl points carrying opposite chirality are present in the electronic dispersion relation of WSM, thus creating a couple of equivalent positive and negative magnetic charges in momentum space [23]. Because of the involvement of the equivalent magnetic charges, the optical behaviors of WSM should be characterized by taking advantage of the axion electrodynamics [24]. Popularly, the WSM material holds the Weyl nodes relevant topological charge n = 1, discovered in TaAs, TaP, and NbAs [25–27]. Very recently, it has been found that the topological charge can be enlarged to n = 2 and 3 [28,29]. The WSM material with higher topological charge is named multi-WSM (mWSM), which has been presented to achieve in condensed matter systems [30,31]. The double and triple WSM phases have also been achieved in HgCr₂Se₄, SrSi₂, Rb(MoTe)₃, and TI(MoTe)₃ [30,32,33]. The mWSM (n = 2, 3) holds anisotropic energy dispersion instead of the linear dispersion of signal WSM [28]. Besides that, the mWSM material almost holds the identical excellent optical properties as those of conventional WSM material [28,29]. Currently, the chiral properties of WSM and mWSM have also been extensively discussed, such as Faraday and Kerr effects [28,29,34–37], circular polarizer [38], circularly thermal radiation [39], and optical isolators [40,41]. However, there is still a lack of discussion on the perfect absorption phenomenon of circular polarization in WSM/mWSM-based system. Therefore, making use of the mWSM material with natural chirality develops feasible and new approaches to investigate the perfect absorption phenomenon of circular polarization without involving an external magnetic field.

In the paper, we theoretically design a PhC Fabry-Perot (FP) cavity with mWSM to discuss the perfect absorption phenomenon. Results indicate that the distinct refractive indices of RCP and LCP waves are available because of the nonzero off-diagonal term of mWSM, thus enabling the perfect absorption phenomenon of RCP and LCP waves at different FP resonant wavelengths. The distinct perfect absorption wavelengths of RCP and LCP waves imply the realization of spin-dependent perfect absorption phenomenon. The perfect absorption wavelength of RCP and LCP waves can be controlled flexibly by altering the Fermi energy, tilt degree of Weyl cones, Weyl nodes separation, topological charge, and thickness of the mWSM layer. Particularly, the linear adjustable perfect absorption wavelength with thickness of the mWSM layer enables the feasibility of determining the perfect absorption wavelength at different mWSM thicknesses precisely. Our findings develop effective and available strategies to realize the spin-dependent and regulable perfect absorption phenomenon without the external magnetic field, and might play practical applications in spin-dependent photonic devices.

2. Theoretical model and method

To study the perfect absorption phenomenon enabled by FP resonance, a FP cavity is constructed, which is composed of the one-dimensional PhC containing a mWSM defect, as shown in Fig. 1. The cavity can also be regarded as a sandwich composition that consists of a mWSM layer and two identical PhCs (BA)⁵. The PhC (BA)⁵ is composed of the ZnS and MgF₂ layers, which hold relative dielectric constants of ε_B= 5.0625 and ε_A= 1.9044, respectively [35,42]. The thickness of the B and A layers are denoted as d_B and d_A, respectively. The thicknesses d_B and d_A are set to d_B=λ₀/(4n_B) and d_A=λ₀/(4 nA) to support the Bragg condition, where λ₀= 10.5 µm denotes the central wavelength of the PhC, and n_B and n_A indicate the refractive index of the B and A layers, respectively. The C layer represents the mWSM layer, which possesses the thickness of d_m. Furthermore, to realize the perfect absorption phenomenon assisted by FP resonance, the perfect electrical conductor (PEC) substrate is exploited to suppress transmission.

Fig. 1. Schematic denoting the FP cavity composed of the one-dimensional PhC containing a mWSM defect.

Download Full Size | PDF

Popularly, to characterize the two discrete multi-Weyl nodes holding contrary chirality χ=±1 and topological charge n = 1, 2, and 3, the low-energy available Hamiltonian can be introduced, expressed as [28]

(1)$$H_n^\chi = \chi \hbar \left\{ {{\alpha _n}k_ \bot ^n\left[ {\cos \left( {n{\phi _k}} \right){\sigma _x} + \sin \left( {n{\phi _k}} \right){\sigma _y}} \right] + {\nu _F}k_z^\chi {\sigma _z}} \right\} + {C_\chi }\hbar {\nu _F}k_z^\chi - \chi {Q_0},$$

where ${k_ \bot } = \sqrt {k_x^2 + k_y^2} $, $k_z^\chi = {k_z} - \chi Q$, ${\phi _k} = {\tan ^{ - 1}}({{k_y}/{k_x}} )$, $\left( {\begin{array}{ccc} {{\sigma_x}}&{{\sigma_y}}&{{\sigma_z}} \end{array}} \right)$ indicates the Pauli matrix that supports the pseudo-spin exponents, $\hbar $ displays the reduced Planck constant, ${\nu _F}$ shows the Fermi velocity, ${C_\chi }$ denotes the tilt degree of Weyl cones, and k_x, k_y and k_z display the momentum components toward the x, y and z directions, respectively. Thanks to the broken time-reversal symmetry (TRS) and inversion symmetry, the momentum space and energy space with 2Q and ± Q₀ are present along the z axis in the discrete Weyl nodes, respectively. Here, ${\alpha _n} = {\nu _ \bot }/k_{mm}^{n - 1}$ with ${\nu _ \bot }$ representing the effective velocity of the quasiparticles in the x-y plane, where k_mm indicates a material-based parameter possessing the dimension of momentum. The parameter ${C_\chi }$ controls the phase change from type-Ι to type-ΙΙ mWSMs. Here, we focus on investigating the type-Ι mWSM material, i.e., $|{{C_\chi }} |< 1$, which supports the point type Fermi surface at the Weyl node.

Particularly, for the TRS breaking mWSM possessing two discrete multi-Weyl nodes along the z axis, a 3 × 3 tensor can be exploited to represent its conductivity, denoted as [28,36]

(2)$$\overline{\overline \sigma } = \left( {\begin{array}{ccc} {{\sigma_{xx}}}&{{\sigma_{xy}}}&0\\ {{\sigma_{yx}}}&{{\sigma_{yy}}}&0\\ 0&0&{{\sigma_{zz}}} \end{array}} \right),$$

where ${\sigma _{xy}}$ and ${\sigma _{yx}}$ display the Hall conductivities with ${\sigma _{xy}} ={-} {\sigma _{yx}}$, ${\sigma _{xx}}$ and ${\sigma _{yy}}$ represent the longitudinal conductivities with ${\sigma _{xx}} = {\sigma _{yy}}$ perpendicular to the Weyl nodes separation, and ${\sigma _{zz}}$ denotes the longitudinal conductivity toward the Weyl nodes separation. Considering the linearly polarized wave striking our proposed composition with normal incidence, the conductivity ${\sigma _{zz}}$ is not involved in the calculation of the reflection and transmission coefficients. By taking advantage of the Kubo formula and the wavefunctions of the valence and conduction bands, the conductivity of type-Ι and type-ΙΙ mWSMs can be acquired in low temperature limit (T = 0 K) [28]. In particular, in the case of type-Ι WSMs with $C = {C_ + } ={-} {C_ - } < 1$, the analytical formulas of the conductivity components ${\sigma _{xx}}$ and ${\sigma _{xy}}$ are represented as [28]

(3)$$\textrm{Re} ({{\sigma_{xx}}} )= \left\{ {\begin{array}{cc} {0,}&\omega < {\omega_1}\\ {\sigma_\omega^n({1/2 - {\kappa_d}} ),}&{\omega_1} < \omega < {\omega_2}\\ {\sigma_\omega^n,}&\omega > {\omega_2} \end{array}, } \right.$$

(4)$${\mathop{\rm Im}\nolimits} ({{\sigma_{xx}}} )={-} \frac{{\sigma _\omega ^n}}{{4\pi }}\left\{ \begin{array}{l} \xi \ln \left[ {\frac{{|{\omega_2^2 - {\omega^2}} |}}{{|{\omega_1^2 - {\omega^2}} |}}} \right] + \frac{8}{{{{|C |}^2}}}{\left( {\frac{\mu }{{\hbar \omega }}} \right)^2}\\ - {\left( {\frac{\mu }{{\hbar \omega }}} \right)^3}\zeta ({\omega ,\textrm{ }|C |,\textrm{ }\mu } )\ln \left[ {\frac{{|{{\omega_2} - \omega } |({{\omega_1} + \omega } )}}{{|{{\omega_1} - \omega } |({{\omega_2} + \omega } )}}} \right]\\ + \frac{6}{{{{|C |}^3}}}{\left( {\frac{\mu }{{\hbar \omega }}} \right)^2}\ln \left[ {\frac{{|{\omega_2^2 - {\omega^2}} |\omega_1^2}}{{|{\omega_1^2 - {\omega^2}} |\omega_2^2}}} \right] + 4\ln \left[ {\frac{{|{\omega_c^2 - {\omega^2}} |}}{{|{\omega_2^2 - {\omega^2}} |}}} \right] \end{array} \right\},$$

(5)$$\textrm{Re} ({\sigma_{xy}^{ac}} )= \textrm{sgn}(C )\sigma _\mu ^n\left\{ \begin{array}{l} \frac{{ - 1}}{{2{{|C |}^2}}}\ln \left[ {\frac{{|{\omega_2^2 - {\omega^2}} |\omega_1^2}}{{|{\omega_1^2 - {\omega^2}} |\omega_2^2}}} \right]\\ + \left( {\frac{\mu }{{2\hbar \omega {{|C |}^2}}} + \frac{{\hbar \omega }}{{8\mu }}\frac{{1 - {{|C |}^2}}}{{{{|C |}^2}}}} \right)\ln \left[ {\frac{{|{{\omega_2} - \omega } |({{\omega_1} + \omega } )}}{{|{{\omega_1} - \omega } |({{\omega_2} + \omega } )}}} \right] - \frac{1}{{|C |}} \end{array} \right\},$$

(6)$$\textrm{Re} ({\sigma_{xy}^{dc}} )= \sigma _Q^n + \sigma _\mu ^n\left[ {\frac{2}{C} + \frac{1}{{{C^2}}}\ln \left( {\frac{{1 - C}}{{1 + C}}} \right)} \right],$$

(7)$$\textrm{Re} ({{\sigma_{xy}}} )= \textrm{Re} ({\sigma_{xy}^{ac}} )+ \textrm{Re} ({\sigma_{xy}^{dc}} ),$$

(8)$${\mathop{\rm Im}\nolimits} ({{\sigma_{xy}}} )= \textrm{sgn}(C )\left\{ {\begin{array}{cc} {0,}&\omega < {\omega_1}\\ {3\sigma_\omega^n{\kappa_o},}&{\omega_1} < \omega < {\omega_2}\\ {0,}&\omega > {\omega_2} \end{array}.} \right.$$

Here the signs Im and Re display the imaginary and real parts respectively, and

(9)$${\kappa _d} = \left( {\frac{{2\mu }}{{\hbar \omega }} - 1} \right)\left[ {\frac{3}{{8|C |}} + \frac{1}{{8{{|C |}^3}}}{{\left( {\frac{{2\mu }}{{\hbar \omega }} - 1} \right)}^2}} \right], $$

(10)$$\zeta ({\omega ,\textrm{ }|C |,\textrm{ }\mu } )= \frac{4}{{{{|C |}^3}}} + 3{\left( {\frac{{\hbar \omega }}{\mu }} \right)^2}\left( {\frac{1}{{{{|C |}^3}}} + \frac{1}{{|C |}}} \right), $$

(11)$$\sigma _\omega ^n = {e^2}n\omega /({6h{\nu_F}} ), $$

(12)$$\sigma _Q^n = {e^2}nQ/({\pi h} ), $$

(13)$$\sigma _\mu ^n = {e^2}\mu n/({{h^2}{\nu_F}} ), $$

(14)$$\xi \textrm{ = 2 + 1/}({2{{|C |}^3}} )+ 3/({2|C |} ), $$

(15)$${\kappa _o} = \frac{1}{{{{|C |}^2}}}\left( {\frac{1}{8} - \frac{\mu }{{2\hbar \omega }} + \frac{{{\mu^2}}}{{2{\hbar^2}{\omega^2}}}} \right) - \frac{1}{8}, $$

where ${\omega _c} = {k_c}{\nu _F}$ indicates the cutoff frequency with ${k_c}$ representing the ultraviolet momentum cutoff toward the z direction, $\hbar {\omega _{1,2}} = 2\mu /({1 \pm |C |} )$ denote the two photon energy bounds in the mWSM, $\mu $ shows the Fermi energy, $h = 2\pi \hbar $ displays the Planck constant, e represents the charge of electron, Q shows the Weyl nodes separation, and $\omega $ displays the angular frequency.

By taking advantage of the conductivity components of mWSM, the relevant dielectric constants can be described as [28,36]

(16)$${\varepsilon _{xx}} = {\varepsilon _b} + \frac{{i{\sigma _{xx}}}}{{\omega {\varepsilon _0}}},\quad {\varepsilon _{xy}} = \frac{i}{{\omega {\varepsilon _0}}}({{\sigma_{xy}} + \sigma_Q^n} ). $$

Here ${\varepsilon _b}$ represents the equivalent dielectric constant of the background medium, and ${\varepsilon _0}$ denotes the dielectric constant of the air. In the follow discussions, the relevant parameters mentioned above are set to ${\nu _F} = {10^6}{\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{m/s}$ and ${\varepsilon _b} = 1$ [28].

Notably, because of the nonzero off-diagonal term γ =ε_xy/i, the RCP and LCP lights in the mWSM layer hold distinct refractive indices, which is absent in the isotropic and nonmagnetic materials [41]. Therefore, the RCP and LCP waves will separate from each other during propagation, and distinct reflection behaviors are present. Particularly, for the linearly polarized wave hitting our designed composition with normal incidence, the reflection coefficient of circularly polarized waves can be obtained and written as [35,43]

(17)$${r_ + } = \frac{{{M_{21}}}}{{{M_{11}}}},\quad {r_ - } = \frac{{{M_{43}}}}{{{M_{33}}}}. $$

Here the marks + and – represent the RCP and LCP waves, respectively. The element M is a 4 × 4 transfer matrix, which can be expressed as [35,43]

(18)$$M = {({{D^{(0)}}} )^{ - 1}}{({{M_A}{M_B}} )^5}{M_m}{({{M_B}{M_A}} )^5}{D^{(22)}}, $$

where ${M_{(l)}}$ and ${D^{(l)}}$ denote the block diagonal characteristic matrix and the dynamical matrix for the l-th layer respectively, which can be written as [35,43]

(19)$${M_{(l)}} = \left[ {\begin{array}{cccc} {\cos ({\varphi_ +^{(l)}} )}&{(i/N_ +^{(l)})\sin ({\varphi_ +^{(l)}} )}&0&0\\ {iN_ +^{(l)}\sin ({\varphi_ +^{(l)}} )}&{\cos ({\varphi_ +^{(l)}} )}&0&0\\ 0&0&{\cos ({\varphi_ -^{(l)}} )}&{(i/N_ -^{(l)})\sin ({\varphi_ -^{(l)}} )}\\ 0&0&{iN_ -^{(n)}\sin ({\varphi_ -^{(l)}} )}&{\cos ({\varphi_ -^{(l)}} )} \end{array}} \right], $$

(20)$${D^{(l)}} = \left[ {\begin{array}{cccc} 1&1&0&0\\ {N_ +^{(l)}}&{ - N_ +^{(l)}}&0&0\\ 0&0&1&1\\ 0&0&{N_ -^{(l)}}&{ - N_ -^{(l)}} \end{array}} \right], $$

where $\varphi _ \pm ^{(l)} = \frac{\omega }{c}N_ \pm ^{(l)}{d^{(l)}}$ denote the phase change of RCP and LCP waves propagating through the l-th layer respectively, $N_ \pm ^{(l)} = \sqrt {\varepsilon _{xx}^{(l)} \pm i\varepsilon _{xy}^{(l)}} $ represent the refractive index of RCP and LCP waves in the l-th layer respectively, ${d^{(l)}}$ denotes the thickness of the l-th layer, and c indicates the speed of light in vacuum. Generally, for the isotropic materials A and B, their dielectric constants can be regarded as

(21)$$\overline{\overline {{\varepsilon _A}}} = \left[ {\begin{array}{ccc} {{\varepsilon_A}}&0&0\\ 0&{{\varepsilon_A}}&0\\ 0&0&{{\varepsilon_A}} \end{array}} \right],\quad \overline{\overline {{\varepsilon _B}}} = \left[ {\begin{array}{ccc} {{\varepsilon_B}}&0&0\\ 0&{{\varepsilon_B}}&0\\ 0&0&{{\varepsilon_B}} \end{array}} \right]. $$

Therefore, in the designed structure, the reflection of RCP and LCP waves can be obtained and denoted as R₊=|r₊|² and R₋=|r₋|², respectively. In the proposed symmetric FP cavity, the two PhCs (BA)⁵ layers act as two total reflection layers, and the mWSM layer serves as the resonant defect layer. Generally, the zero reflection phenomenon can be obtained when the FP resonance is present, thus supporting the feasibility of realizing perfect absorption phenomenon in the proposed structure [7,44].

3. Results and discussions

Because of the nonzero off-diagonal term of the mWSM layer, the unequal dielectric constants of RCP and LCP waves in mWSM are present, thus supporting their distinct resonant wavelengths. To support the perfect absorption phenomenon of RCP and LCP waves induced by FP resonance, the dielectric constant of RCP and LCP waves in mWSM should be expressed as positive real numbers. The dielectric constant of mWSM as a function of wavelength is plotted in Fig. 2. The tilt degree of Weyl cones, Fermi energy, Weyl nodes separation and topological charge are optimized to 0.5, 0.35 eV, 0.09 nm^-1 and 2, respectively. It is found that the imaginary part of ε_xx and γ are absent in the wavelength range about from 8 to 14 µm. The real part of ε_xx and γ exist and denote as positive numbers. Furthermore, the real part of ε_xx is greater than the real part of γ. As a result, the dielectric constant of RCP and LCP waves in mWSM are positive real numbers. Therefore, the wavelength range from 8 to 14 µm can be made use of to investigate the spin-dependent perfect absorption phenomenon induced by FP resonance. In the following discussion, we focus on studying the realization and regulation of the spin-dependent perfect absorption phenomenon.

Fig. 2. The real and imaginary parts of ε_xx and γ as a function of wavelength.

Download Full Size | PDF

Notably, the co-polarized and cross-polarized reflection coefficients of circular polarization are present when the nonzero off-diagonal term of mWSM is considered [41]. Due to the cross-polarized reflection coefficients of circular polarization, the FP resonant condition of RCP and LCP waves cannot be directly described by using the similar resonant condition of linear polarization [7,35,44]. To demonstrate the feasibility of FP resonance in our designed structure, we firstly discuss the realization of FP resonance for the transverse magnetic- (TM-) polarized wave. At this situation, the off-diagonal term is ignored by taking the Weyl nodes separation as zero. The FP resonance of the TM polarized light is present at the condition of $r_1^2\,\exp ({ - 2i\phi } )= 1$ [7,44], where r₁ denotes the reflection coefficient induced by the TM polarized wave hitting the PhC (BA)₅ from the defect layer, and $\phi = \sqrt {{\varepsilon _{xx}}} {k_0}{d_m}$ represents the phase change of the TM polarized wave propagating through the defect layer. The reflection of the FP cavity and the mWSM-PhC (BA)₅ for the TM polarized wave are denoted as R_FP and R₁=|r₁|², respectively. It is seen in Fig. 3(b) that the phase of $r_1^2\,\exp ({ - 2i\phi } )$ is equal to 0 at three different wavelengths of 9.264, 10.73, and 12.55 µm. However, it is found in Fig. 3(a) that only the middle wavelength meets the condition of R₁= 0.95, almost equal to 1. For other two wavelengths of 9.264 and 12.55 µm, the reflection R₁ is far less than 1. Therefore, the FP resonance only appears at the wavelength of 10.73 µm. It is also shown in Fig. 3(a) that the reflection of the FP cavity reaches to 0 at the wavelength of 10.73 µm. It is well demonstrated that the zero reflection can be obtained by taking advantage of the FP resonance. Furthermore, when the off-diagonal term is considered, the RCP and LCP waves hold distinct refractive indices, thus resulting in that the resonant wavelength of RCP and LCP waves separate from each other. As seen in Fig. 3(c), when the Weyl nodes separation is set to 0.09 nm^-1, the zero reflection of RCP and LCP waves are situated at the wavelength of 9.93 and 11.31 µm, respectively.

Fig. 3. The zero reflection phenomenon enabled by FP resonance. (a) denotes the reflection of the FP cavity and the mWSM-PhC (BA)₅ for the TM polarized wave as a function of wavelength. (b) denotes the phase of ${r_1}{r_1}\exp ({ - 2i\phi } )$ as a function of wavelength. (c) represents the reflection R₊ and R₋ for the FP cavity as a function of wavelength. Other parameters are equivalent to those of Fig. 2.

Download Full Size | PDF

Based on the above discussion, the perfect absorption phenomenon of RCP and LCP waves are present at distinct wavelengths of 9.93 and 11.31 µm, respectively. Notably, the dielectric constant of mWSM shows the dependence on Fermi energy and tilt degree of Weyl cones. By varying the Fermi energy and tilt degree, the dielectric constant can be regulated, thus affecting the perfect absorption wavelength induced by FP resonance. To further illustrate the effect of Fermi energy and tilt degree on the perfect absorption phenomenon of circularly polarized waves, Fig. 4 indicates the absorption of RCP and LCP waves at various Fermi energies and tilt degrees. Figures 4(a) and 4(c) display the absorption of RCP and LCP waves at distinct Fermi energies, respectively. It is seen that the perfect absorption phenomenon of RCP and LCP waves can be realized at distinct Fermi energies, and their relevant perfect absorption wavelengths can be manipulated with the changing Fermi energy. With the increasing Fermi energy, the perfect absorption wavelength of RCP and LCP waves experience red and blue shifts, respectively. Similarly, it is shown in Figs. 4(b) and 4(d) that the perfect absorption wavelength of RCP and LCP waves undergo red and blue shifts with the enlarging tilt degree, respectively. The decreasing perfect absorption wavelength difference between LCP and RCP waves with the increasing Fermi energy and tilt degree are obvious in Figs. 4(e) and 4(f), respectively. As the Fermi energy increases from 0.25 to 0.55 eV, the perfect absorption wavelength difference between LCP and RCP waves decreases from 1.742 to 0.3 µm. With the increasing tilt degree from 0.3 to 0.7, the perfect absorption wavelength difference between LCP and RCP waves decreases from 2.284 to 0.1 µm. The reason is chiefly as follows. With the increasing Fermi energy and tilt degree, the off-diagonal term γ is mainly affected, and shows a decrease. The refractive index difference of the mWSM layer between LCP and RCP waves $\Delta N = \sqrt {{\varepsilon _{xx}} + \gamma } - \sqrt {{\varepsilon _{xx}} - \gamma } $ decreases with the decreasing off-diagonal term, resulting in the reduced resonant wavelength difference between LCP and RCP waves. Particularly, when the off-diagonal term γ disappears, the resonant wavelength of RCP and LCP waves will overlap due to their identical refractive indices, which has been discussed in Figs. 3(a) and 3(b). In a word, altering Fermi energy and tilt degree of Weyl cones reveal effective approaches to realize the spin-dependent adjustable perfect absorption phenomenon.

Fig. 4. Tunable perfect absorption phenomenon with Fermi energy and tilt degree. (a) and (c) represent the absorption of RCP and LCP waves at various Fermi energies, respectively. (b) and (d) display the absorption of RCP and LCP waves at distinct tilt degrees, respectively. (e) and (f) show the dependence of the perfect absorption wavelength difference between LCP and RCP waves on Fermi energy and tilt degree, respectively. Other parameters are equivalent to those of Fig. 3.

Download Full Size | PDF

Furthermore, the dielectric constant of mWSM can also be manipulated by altering the Weyl nodes separation and topological charge. To further illustrate the effect of Weyl nodes separation and topological charge on the perfect absorption phenomenon of circularly polarized waves, Fig. 5 plots the absorption of RCP and LCP waves at various Weyl nodes separations and topological charges. Figures 5(a) and 5(c) exhibit the absorption of RCP and LCP waves at distinct Weyl nodes separations, respectively. It is found that the perfect absorption phenomenon of RCP and LCP waves are existent at different Weyl nodes separations. With the increasing Weyl nodes separation, the perfect absorption wavelength of RCP and LCP waves exhibit blue and red shifts respectively, which is contrary to the change trend in Fig. 4. The increasing perfect absorption wavelength difference between LCP and RCP waves with the increasing Weyl nodes separation is obvious in Fig. 5(e). As the Weyl nodes separation increases from 0.06 to 0.12 nm^-1, the perfect absorption wavelength difference between LCP and RCP waves increases from 0.24 to 2.484 µm. The reason is that the off-diagonal term γ increases with the increasing Weyl nodes separation. Owing to the enlarged off-diagonal term γ, the refractive index difference of the mWSM layer between RCP and LCP waves increases, thus developing the growing resonant wavelength difference between RCP and LCP waves. As shown in Figs. 5(b) and 5(d), the perfect absorption phenomenon of RCP and LCP waves are present at different topological charges. As the topological charge increases, the perfect absorption wavelength of RCP and LCP waves undergo red shifts. The reason is chiefly as follows. With the increasing topological charge, the diagonal component ε_xx and off-diagonal term γ show enlargement. It is worth noting that the impact of topological charge on the diagonal component ε_xx is relatively greater than the impact on the off-diagonal term γ. Furthermore, the diagonal component ε_xx decreases with the increasing wavelength. Therefore, as the topological charge increases, the wavelength should be increased to restrict the growing refractive index of RCP and LCP waves to support the FP resonance. It is also shown in Fig. 5(f) that with the increasing topological charge from 1 to 3, the perfect absorption wavelength difference between LCP and RCP waves increases from 0.955 to 1.54 µm. Particularly, when the topological charge reaches to 3, a new perfect absorption phenomenon of LCP wave emerges at small wavelength. The major reason is that the diagonal component ε_xx and off-diagonal term γ increase significantly with the increasing topological charge, thus supporting the other resonant mode at the small wavelength. In a word, the spin-dependent adjustable perfect absorption phenomenon can be obtained by varying the Weyl nodes separation and topological charge.

Fig. 5. Adjustable perfect absorption phenomenon with Weyl nodes separation and topological charge. (a) and (c) indicate the absorption of RCP and LCP waves at various Weyl nodes separations, respectively. (b) and (d) show the absorption of RCP and LCP waves at different topological charges, respectively. (e) and (f) show the dependence of the perfect absorption wavelength difference between LCP and RCP waves on Weyl nodes separation and topological charge, respectively. Other parameters are equivalent to those of Fig. 3.

Download Full Size | PDF

Finally, the impact of mWSM thickness on the perfect absorption phenomenon of RCP and LCP waves are also discussed. The absorption of RCP and LCP waves as functions of mWSM thickness and wavelength are denoted in Figs. 6(a) and 6(b), respectively. It is shown that the perfect absorption phenomenon of RCP and LCP waves can be realized at the whole thickness range. With the increasing thickness of the mWSM layer, the perfect absorption wavelength of RCP and LCP waves experience red shifts. The major reason is similar as that of Figs. 5(b) and 5(d). With the increasing thickness, the wavelength should be increased to restrict the increasing phases ${k_ \pm }{d_m}$ to support the FP resonance. What is more important, the perfect absorption wavelength of RCP and LCP waves increase linearly with the increasing thickness of the mWSM layer, which enables the accurate determination of perfect absorption wavelength at different mWSM thicknesses. It is also shown in Fig. 6(c) that with the increasing mWSM thickness from 2.6 to 3.8 µm, the perfect absorption wavelength difference between LCP and RCP waves increases from 1.14 to 1.5 µm.

Fig. 6. Tunable perfect absorption phenomenon with thickness of the mWSM layer. (a) and (b) display the absorption of RCP and LCP waves as functions of mWSM thickness and wavelength, respectively. (c) denotes the dependence of the perfect absorption wavelength difference between LCP and RCP waves on the thickness of the mWSM layer. Other parameters are equivalent to those of Fig. 3.

Download Full Size | PDF

It should be noted that although our study is only a theoretical work, the spin-dependent perfect absorber in our research is possible to be realized in experiments. Compared with the spin-dependent perfect absorber enabled by metamaterials and metasurfaces with chirality [14–18], the designed FP cavity is relatively easy to fabricate. By taking advantage of the current high-vacuum magnetron sputtering technique, the FP cavity containing the mWSM defect can be fabricated conveniently [45]. Compared with the similar FP cavity containing other magnetic material defects, such as bismuth iron garnet defect, the designed spin-dependent perfect absorber can be manipulated to the mesoscopic dimension since the external magnetic field is not required [46]. Furthermore, the Fermi energy, tilt degree of Weyl cones, Weyl nodes separation, topological charge, and thickness of the mWSM layer provide available approaches to regulate the perfect absorption phenomenon.

4. Conclusions

To sum up, a PhC FP cavity containing a mWSM defect has been presented to discuss the perfect absorption phenomenon. Results denote that the distinct refractive indices of RCP and LCP waves are present due to the nonzero off-diagonal term of mWSM, thus supporting the perfect absorption phenomenon of RCP and LCP waves at different FP resonant wavelengths. The distinct perfect absorption wavelengths of RCP and LCP waves indicate the realization of spin-dependent perfect absorption phenomenon. It is found that the perfect absorption wavelength of RCP and LCP waves can be regulated flexibly by altering the Fermi energy, tilt degree of Weyl cones, Weyl nodes separation, topological charge, and thickness of the mWSM layer. Particularly, the linear adjustable perfect absorption wavelength with thickness of the mWSM layer enables the feasibility of determining the perfect absorption wavelength at different mWSM thicknesses precisely. Our findings reveal effective and available approaches to realize the spin-dependent and regulable perfect absorption phenomenon without the involving external magnetic field, and might develop practical applications in spin-dependent photonic devices.

Funding

Natural Science Foundation of Hunan Province (2021JJ40168, 2023JJ40265); Education Department of Hunan Province (21B0534).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data of the results presented in this paper are available upon reasonable request from the authors.

References

1. S. Thongrattanasiri, F. H. L. Koppens, and F. J. G. De Abajo, “Complete optical absorption in periodically patterned graphene,” Phys. Rev. Lett. 108(4), 047401 (2012). [CrossRef]

2. Z. Zheng, Y. Zheng, Y. Luo, Z. Yi, J. Zhang, Z. Liu, W. Yang, Y. Yu, X. Wu, and P. Wu, “A switchable terahertz device combining ultra-wideband absorption and ultra-wideband complete reflection,” Phys. Chem. Chem. Phys. 24(4), 2527–2533 (2022). [CrossRef]

3. J. Wu, L. Jiang, J. Guo, X. Dai, Y. Xiang, and S. Wen, “Tunable perfect absorption at infrared frequencies by a graphene-hBN hyper crystal,” Opt. Express 24(15), 17103–17114 (2016). [CrossRef]

4. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998). [CrossRef]

5. M. Gryga, D. Ciprian, L. Gembalova, and P. Hlubina, “Sensing based on Bloch surface wave and self-referenced guided mode resonances employing a one-dimensional photonic crystal,” Opt. Express 29(9), 12996–13010 (2021). [CrossRef]

6. F. Wu, T. Liu, and S. Xiao, “Polarization-sensitive photonic bandgaps in hybrid one-dimensional photonic crystals composed of all-dielectric elliptical metamaterials and isotropic dielectrics,” Appl. Opt. 62(3), 706–713 (2023). [CrossRef]

7. X. Wang, X. Jiang, Q. You, J. Guo, X. Dai, and Y. Xiang, “Tunable and multichannel terahertz perfect absorber due to Tamm surface plasmons with graphene,” Photonics Res. 5(6), 536–542 (2017). [CrossRef]

8. Y. C. Lin, S. H. Chou, and W. J. Hsueh, “Tunable light absorption of graphene using topological interface states,” Opt. Lett. 45(16), 4369–4372 (2020). [CrossRef]

9. Y. Ren, T. Zhou, C. Jiang, and B. Tang, “Thermally switching between perfect absorber and asymmetric transmission in vanadium dioxide-assisted metamaterials,” Opt. Express 29(5), 7666–7679 (2021). [CrossRef]

10. X. Li, L. Yu, L. Yu, Y. Dong, Q. Gao, Q. Yang, W. Yang, Y. Zhu, and Y. Fu, “Chiral polyaniline with superhelical structures for enhancement in microwave absorption,” Chem. Eng. J. 352, 745–755 (2018). [CrossRef]

11. J. Li, J. Li, Y. Yang, J. Li, Y. Zhang, L. Wu, Z. Zhang, M. Yang, C. Zheng, J. Li, J. Huang, F. Li, T. Tang, H. Dai, and J. Yao, “Metal-graphene hybrid active chiral metasurfaces for dynamic terahertz wavefront modulation and near field imaging,” Carbon 163, 34–42 (2020). [CrossRef]

12. M. Liu, E. Plum, H. Li, S. Duan, S. Li, Q. Xu, X. Zhang, C. Zhang, C. Zou, B. Jin, J. Han, and W. Zhang, “Switchable chiral mirrors,” Adv. Opt. Mater. 8(15), 2000247 (2020). [CrossRef]

13. B. Tang, Z. Li, E. Palacios, Z. Liu, S. Butun, and K. Aydin, “Chiral-selective plasmonic metasurface absorbers operating at visible frequencies,” IEEE Photonics Technol. Lett. 29(3), 295–298 (2017). [CrossRef]

14. L. Ouyang, W. Wang, D. Rosenmann, D. A. Czaplewski, J. Gao, and X. Yang, “Near-infrared chiral plasmonic metasurface absorbers,” Opt. Express 26(24), 31484–31489 (2018). [CrossRef]

15. X. T. Kong, L. K. Khorashad, Z. Wang, and A. O. Govorov, “Photothermal circular dichroism induced by plasmon resonances in chiral metamaterial absorbers and bolometers,” Nano Lett. 18(3), 2001–2008 (2018). [CrossRef]

16. D. Xiao, Y. J. Liu, S. Yin, J. Liu, W. Ji, B. Wang, D. Luo, G. Li, and X. W. Sun, “Liquid-crystal-loaded chiral metasurfaces for reconfigurable multiband spin-selective light absorption,” Opt. Express 26(19), 25305–25314 (2018). [CrossRef]

17. Z. Shen, X. Fang, S. Li, W. Yin, L. Zhang, and X. Chen, “Terahertz spin-selective perfect absorption enabled by quasi-bound states in the continuum,” Opt. Lett. 47(3), 505–508 (2022). [CrossRef]

18. H. Li, H. Zhou, G. Wei, H. Xu, M. Qin, J. Liu, and F. Wu, “Photonic spin-selective perfect absorptance on planar metasurfaces driven by chiral quasi-bound states in the continuum,” Nanoscale 15(14), 6636–6644 (2023). [CrossRef]

19. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). [CrossRef]

20. E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, “Coherent nonlinear optical response of graphene,” Phys. Rev. Lett. 105(9), 097401 (2010). [CrossRef]

21. A. A. Zyuzin and A. A. Burkov, “Topological response in Weyl semimetals and the chiral anomaly,” Phys. Rev. B 86(11), 115133 (2012). [CrossRef]

22. E. Liu, Y. Sun, N. Kumar, et al., “Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal,” Nat. Phys. 14(11), 1125–1131 (2018). [CrossRef]

23. P. Hosur, S. A. Parameswaran, and A. Vishwanath, “Charge transport in Weyl semimetals,” Phys. Rev. Lett. 108(4), 046602 (2012). [CrossRef]

24. F. Wilczek, “Two applications of axion electrodynamics,” Phys. Rev. Lett. 58(18), 1799–1802 (1987). [CrossRef]

25. S. Xu, I. Belopolski, N. Alidoust, et al., “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015). [CrossRef]

26. B. Lv, H. Weng, B. Fu, X. Wang, H. Miao, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015). [CrossRef]

27. B. Lv, N. Xu, H. Weng, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015). [CrossRef]

28. S. Ghosh, A. Sahoo, and S. Nandy, “Theoretical investigations on Kerr and Faraday rotations in topological multi-Weyl Semimetals,” arXiv, arXiv:2209.11217 (2022). [CrossRef]

29. A. Gupta, “Kerr effects in tilted multi-Weyl semimetals,” arXiv, arXiv:2209.07506 (2022). [CrossRef]

30. C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, “Multi-Weyl topological semimetals stabilized by point group symmetry,” Phys. Rev. Lett. 108(26), 266802 (2012). [CrossRef]

31. B. J. Yang and N. Nagaosa, “Classification of stable three-dimensional Dirac semimetals with nontrivial topology,” Nat. Commun. 5(1), 4898 (2014). [CrossRef]

32. S. Huang, S. Xu, I. Belopolski, C. C. Lee, G. Chang, T. R. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, D. Sanchez, H. Zheng, H. T. Jeng, A. Bansil, T. Neupert, H. Lin, and M. Z. Hasan, “New type of Weyl semimetal with quadratic double Weyl fermions,” Proc. Natl. Acad. Sci. 113(5), 1180–1185 (2016). [CrossRef]

33. Q. Liu and A. Zunger, “Predicted realization of cubic Dirac fermion in quasi-one-dimensional transition-metal monochalcogenides,” Phys. Rev. X 7(2), 021019 (2017). [CrossRef]

34. J. Wu, L. Jiang, X. Dai, and Y. Xiang, “Tunable and enhanced Faraday rotation induced by the epsilon-near-zero response of a Weyl semimetal,” Phys. Rev. A 105(4), 043519 (2022). [CrossRef]

35. T. Li, C. Yin, and F. Wu, “Giant enhancement of Faraday rotation in Weyl semimetal assisted by optical Tamm state,” Phys. Lett. A 437, 128103 (2022). [CrossRef]

36. K. Sonowal, A. Singh, and A. Agarwal, “Giant optical activity and Kerr effect in type-I and type-II Weyl semimetals,” Phys. Rev. B 100(8), 085436 (2019). [CrossRef]

37. J. Wu, R. Zeng, J. Liang, D. Huang, Y. Xiang, and X. Dai, “Faraday rotation enhancement and characteristic of the Weyl node separation and tilt degree by resonant tunneling,” J. Appl. Phys. 133(23), 233102 (2023). [CrossRef]

38. C. Yang, B. Zhao, W. Cai, and Z M. Zhang, “Mid-infrared broadband circular polarizer based on Weyl semimetals,” Opt. Express 30(2), 3035–3046 (2022). [CrossRef]

39. Y. Wang, C. Khandekar, X. Gao, T. Li, D. Jiao, and Z. Jacob, “Broadband circularly polarized thermal radiation from magnetic Weyl semimetals,” Opt. Mater. Express 11(11), 3880–3895 (2021). [CrossRef]

40. V. S. Asadchy, C. Guo, B. Zhao, and S. Fan, “Sub-Wavelength Passive Optical Isolators Using Photonic Structures Based on Weyl Semimetals,” Adv. Opt. Mater. 8(16), 2000100 (2020). [CrossRef]

41. J. Wu, Y. Xiang, and X. Dai, “Tunable broadband compact optical isolator based on Weyl semimetal,” Results Phys. 46(10), 106290 (2023). [CrossRef]

42. H. Zhu, H. Luo, Q. Li, D. Zhao, L. Cai, K. Du, Z. Xu, P. Ghosh, and M. Qiu, “Tunable narrowband mid-infrared thermal emitter with a bilayer cavity enhanced Tamm Plasmon,” Opt. Lett. 43(21), 5230–5233 (2018). [CrossRef]

43. Š. Višňovský, K. Postava, and T. Yamaguchi, “Magneto-optic polar Kerr and Faraday effects in magnetic superlattices,” Czechoslov. J. Phys. 51(9), 917–949 (2001) [CrossRef]

44. J. Wu, Y. Liang, J. Guo, L. Jiang, X. Dai, and Y. Xiang, “Tunable and multichannel terahertz perfect absorber due to Tamm plasmons with topological insulators,” Plasmonics 15(1), 83–91 (2020). [CrossRef]

45. S. Jena, R. B. Tokas, P. Sarkar, J. S. Misal, S. M. Haque, K. D. Rao, S. Thakur, and N. K. Sahoo, “Omnidirectional photonic band gap in magnetron sputtered TiO₂/SiO₂ one dimensional photonic crystal,” Thin Solid Films 599, 138–144 (2016). [CrossRef]

46. M. Deb, E. Popova, and N. Keller, “Different magneto-optical response of magnetic sublattices as a function of temperature in ferrimagnetic bismuth iron garnet films,” Phys. Rev. B 100(22), 224410 (2019). [CrossRef]

Spin-dependent and tunable perfect absorption in a Fabry-Perot cavity containing a multi-Weyl semimetal

Abstract

1. Introduction

2. Theoretical model and method

3. Results and discussions

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (21)

Optics Express