Voltage Tunable Spoof Surface Plasmon Polariton Waveguide Loaded with Ferroelectric Resonators

Abstract

A real-time tunable planar plasmonic waveguide based on a voltage-adjustable ferroelectric resonator is designed and investigated. The laminated ferroelectric compound resonator is composed of a ferroelectric Ba_0.85Ca_0.15Zr_0.9Ti_0.1O₃ (BCZT) layer, a PCB layer, as well as a localized spoof plasmonic metal layer, where the BCZT layer is beneficial for enhancing the voltage tunability in the spoof surface plasmon polariton (SSPP) waveguide. The simulated results show that the tuning range of the notch in the transmission curve, generated by the coupling between the ferroelectric compound resonator and the plasmonic waveguide, can achieve a variation of up to 8.8% thanks to the large tunability value in the BCZT ferroelectric layer. In addition, the notches consist of Fano resonant frequencies, the generation mechanism of which is elaborately discussed in terms of the temporal coupled mode theory.

1. Introduction

As is well-known, the spoof surface plasmon polariton (SSPP) concept proposed by J. B. Pendry is a kind of artificial EM wave propagating in corrugated metal, which mimics the natural inherent electron oscillation on the metal surface, named surface plasmon polaritons (SPPs) [1,2,3]. Since SSPPs can expand the SPP effect to GHz and THz bands through subwavelength engineering in metal, they have become more flexible and have generated increasing enthusiasm since their proposal. In 2012, Ander Pors et al. proposed the concept of a localized spoof surface plasmonic (LSSP), which refers to the 2D or 3D periodical plasmonic structures on the surface of metal particles, resonating in the excited standing wave mode [4]. The EM field of the resonating LSSP is highly confined near the metal interface, with a strong enhancement effect, and is highly sensitive to the geometry and local dielectric environment [5,6]. For example, in 2021, Zhang Xuanru proposed a ring plasma resonator to achieve both high efficiency and high Q capture mode excitation [7]. In particular, when utilizing coupled mode theory, multipolar plasmonic modes can be excited [8,9,10,11,12], which have sharper resonances than the basic dipole mode, and hence provide a higher Q value. Owing to these priorities, LSSPs have been widely used in optical antennas [13,14,15,16], chemical and biological sensors [17,18,19,20,21], and superlenses [22,23]. However, the previously designed LSSP was mostly static, which means the resonant properties had been solidified once the structure was fabricated. This unitary resonance is not conducive to modern complex communication environments. Thus, the real-time adjustability and experimental flexibility of multiple LSSPs become a major challenge.

Certain researchers started utilizing diverse measurement techniques to adjust LSSPs. For example, PIN diodes for the electric tuning of LSSPs have been employed [24,25,26]. Yuzan Xiong et al. utilized Yttrium Iron Garnet (YIG) ferrite for magnetically adjusting the ferromagnetic resonance field of the LSSP resonator [27]. Furthermore, a varactor diode and micro-electro-mechanical system (MEMS) have also been used as active elements for tuning LSSPs [28,29,30]. However, due to the complexity and variability of the communication environment, more regulatory measures need to be continuously expanded.

Ferroelectric materials can realize the exotic regulation of dielectric permittivity with an applied DC electric field, which has potential applications in tunable LSSP resonators [31,32,33,34,35]. The advanced features of ferroelectrics include their capability for continuous tunability, with high permittivity, low loss, and fast responses [36,37,38,39]. It is emphasized that the efficiency of the electrical tunability of ferroelectric resonators is larger than that of conventional liquid crystal devices [40,41,42,43]. The frequency stability, precision, durability, reliability, and vibration resistance of ferroelectric resonators are obviously superior to MEMSs and phase-change materials [44,45]. Also, ferroelectric resonators can be flexibly regulated by employing structure engineering as well as composition fluctuation [46,47,48,49,50,51]. However, until now, few works have comprehensively exploited the modulation mechanism of the ferroelectric resonator.

In this paper, we use the promising environmentally friendly ferroelectric Ba_0.85Ca_0.15Zr_0.9Ti_0.1O₃ (BCZT) layer for fabricating an LSSP resonator. An exhaustive study on the structure engineering as well as the voltage tunability in the BCZT layer is conducted, which possesses high dielectric permittivity as well as a low loss at microwave frequencies, and will be beneficial for a compact microwave device with a low insertion loss [52,53,54,55]. Based on these characteristics, a compound ferroelectric LSSP resonator is designed and loaded onto a plasmonic waveguide to fulfill the real-time and accurate S-parameter regulation; S parameters, representing scattering parameters, are important parameters for describing the characteristics of microwave transmission and radio frequency components. Also, we demonstrate that multiple Fano modes can be aroused by the feeding of the ultracompact plasmonic structure, which can be tailored independently for filtering or sensing applications. A model based on temporal coupled mode theory is provided to describe the triple Fano behaviors, including dipole, quadrupole, and hexapole modes, which are all sensitive to the disk refractive index of the LSPR. Since the proposed LSSP is planar-textured, it provides significant flexibility in engineering the resonant properties, which possesses potential applications in compact filtering applications at microwave frequencies.

2. Experiments

The BCZT ferroelectrics were synthesized by the plasma-activated sintering (PAS) method. The calcined BCZT powders were plasma-activated for 30 s with the supplied pulsed current before sintering using a PAS device (ED-PAS-111, ELENIX, Japan). The axial pressure during PAS sintering is 50 MPa and the heating rate is 100 °C/min, with a holding time of 3 min at the sintering temperature of 1250 °C. The crystal structure of the BCZT ferroelectric resonator was examined by using a PANalytical Empyrean four-circle diffraction system. In order to test the dielectric properties of the BCZT ferroelectric resonator, a dielectric measurement equipment (ZY6173) loaded with a Keysight fixture 16197A, which can work in frequencies up to 3GHz, was utilized to test the permittivity and loss tangent of the ferroelectric resonator. Then, a compound resonator was designed employing a BCZT and PCB laminated structure, whose dielectric permittivity can be regulated flexibly in a large range by its thickness ratio (see Equations (1) and (2)). Finally, the compound resonator was loaded on the SSPP waveguide for fabricating a real-time tunable notched filter. The dispersion and Extinction Cross Section (ECS) properties of the BCZT compound resonator as well as the tunability of the notching properties were investigated using CST Studio Suite 2021 software simulation. The horizontal plane waves were used to excite the side of the resonator for the calculation of the ECS spectra. The boundary condition was set as an open boundary, and the time domain solver was selected with a grid resolution of 1000 nm in every direction of the mesh cell. A plane EM wave in the frequency range of 0 to 7 GHz, propagating in the x direction and with an electric field amplitude of 1 V/m, was employed as an incident source in the ECS investigation. A far-field/RCS monitor was selected to record the field information with the sampling of 700 frequency points.

3. Results and Discussion

3.1. Crystal Structure and Dielectric Properties of BCZT Ferroelectric

Figure 1 presents the X-ray diffraction (XRD) patterns of the BCZT ferroelectric resonator, sintered at 1250 °C. The indexed diffraction peaks indicate that the ferroelectric resonator displays a single phase without any impurity. The inset of Figure 1 depicts the cross section grain morphology of the as-sintered BCZT ferroelectric resonator. The well-densified grain structure and the even distributed grain size illustrate an optimized sintering condition at 1250 °C. Also, the BCZT ferroelectric resonator became crystallized in the trigonal–tetragonal morphotropic phase boundary (MPB) region at room temperature, with the small full width at half-maximum (FWHM), which suggests that the BCZT ferroelectric resonator has excellent crystallinity and will be beneficial for the large permittivity as well as tunability of the subsequent ferroelectric resonator.

In order to construct a tunable microwave device, the dielectric properties as well as the turnability of the BCZT resonator should be declared firstly. Therefore, we begin to investigate the dielectric permittivity as well as the unloaded quality factor of the BCZT ferroelectric ceramic at the 0.1GHz~1 GHz frequency range; the results are exhibited in Figure 2.

As can be viewed, the BCZT ceramic has a stable permittivity in a wide frequency range from 0.1 GHz to 1 GHz, and the permittivity at the 1GHz frequency band is around 190. The other important issue for ferroelectric devices is the loss tangent of the BCZT dielectric resonator in GHz band ranges from 4.0 × 10⁻⁴ to 6.3 × 10⁻⁴. The voltage tunability of the BCZT ferroelectric can be altered according to the data reported by D. Mercier et al. [56,57], where a 75% capacitance tuning is obtained in the Pt/BCZT/Pt heterostructure system under the biased voltage of 20 V. Our previous work on the dielectric tunability of the BCZT heterostructure system also shows a larger than 60% capacitance tunability [55]. The large permittivity as well as the high tunability of the BCZT ferroelectric resonator will be beneficial for the following design of the large tunability in the ferroelectric compound resonator.

3.2. Design of LSSP Resonator with BCZT Ferroelectric Compound Substrate

The following design of the LSSP resonator starts with the design of a compound tunable ferroelectric substrate. As shown in Figure 3, a bilayer compound substrate was constructed using a laminated BCZT and PCB substrate. Then, the LSSP metal structure was printed on the surface of the compound substrate. Therefore, the substrate permittivity of the LSSP resonator can be flexibly adjusted by the applied voltage. According to the superposition principle of EM theory, the final effective permittivity of the compound substrate can be designed by controlling the relative thickness ratio of each layer, as depicted in Equations (1) and (2):

$${{\displaystyle \epsilon }}_i={{\displaystyle \sum }}_i({{\displaystyle \epsilon }}_i{{\displaystyle h}}_i{{\displaystyle P}}_{e,i})$$

(1)

where ε_i, h_i, and P_e,i (i = 1, 2, 3) refer to the permittivity, laminated thickness, and filling coefficient of the ith layer in the LSSP resonator, respectively. As can be seen from Figure 3b, since the value of the dielectric permittivity of the whole substrate employed in the simulation is actually an equivalent average value, the equivalent dielectric permittivity can be adjusted by the P_e,i value, which ranges between 0 and 1 [58].

$$P_{e,i}={\displaystyle \frac{(1/2){\displaystyle \underset{V}{\int }{\epsilon }_{r,i}}\overrightarrow{E}\times {\overrightarrow{E}}^{*}dV}{{\displaystyle \sum _i((1/2){\displaystyle \underset{V}{\int }{\epsilon }_{r,i}}\overrightarrow{E}\times {\overrightarrow{E}}^{*}dV)}}}={\displaystyle \frac{{\displaystyle \underset{S}{\int }{\epsilon }_{r,i}}{E}_{\theta }^2\times rdS}{{\displaystyle \sum _i({\displaystyle \underset{S}{\int }{\epsilon }_{r,i}}{E}_{\theta }^2\times rdS)}}}$$

(2)

where ε_r,i (i = 1, 2, 3) denotes the permittivity of the ith layer in the compound resonator, E* represents the complex conjugation of the electric field E, while E_θ depicts the component of the electric field oriented along the θ direction. From the above equation, it can be revealed that the precise adjustment of the equivalent dielectric permittivity of the resonator can be achieved by judiciously designing the thickness and filling coefficient of each layer in the composite LSSP. For example, when the compound substrate is composed of two layers, where the lower layer is a PCB dielectric layer with a permittivity of 2.2 and thickness of h₁ = 0.5 mm, and the upper layer is a BCZT ferroelectric layer with a permittivity of 190 in the working frequency range and with a thickness of h₂ = 5.4 μm, then the permittivity of the compound substrate can be calculated from Equations (1) and (2), and the result exhibits an equivalent permittivity value of 3.5. Then, as the applied voltage is loaded on BCZT layer, the permittivity will experience a 60% reduction, and the equivalent permittivity of the compound substrate will become 2.9. We numerically compared the ECS properties of the LSSP resonator with the compound substrate and with a single substrate of the equivalent permittivity, as illustrated in Figure 3b. The result reveals that the two LSSP resonators with the above two different substrates actually possess similar ECS properties, which indicates the equivalence of the two substrates and verifies the validity of the design strategy of the compound tunable ferroelectric substrate. Also, it is worth pointing out that although the regulation range employed in the LSSP ferroelectric resonator is from 3.5 to 2.9, which is similar to that of the nematic liquid crystals [38], the former actually possesses a larger regulation range than that of the latter, since the permittivity variation range of the former can be adjusted by the thickness ratio in the laminated composition.

Bearing the above substrate equivalence in mind, the LSSP can be constructed on a single equivalent substrate instead of the compound substrate for convenience in the following calculation model. As depicted in panels a and b of Figure 4, the configuration of the LSSP resonator consists of three parts; the upper layer is composed of two disjoint electrodes, the middle layer is the equivalent substrate with a permittivity of ε₁ = 3.5 and thickness of t = 0.65 mm, and the bottom layer is composed of a metal disk with radius r₁ and six evenly distributed metal spokes. Figure 4c shows the unit structure diagram of the LSSP resonator, where the geometric parameters are listed in

Table 1. The denotation and physical dimensions of each part of the designed SSPP waveguide loaded with the LSSP resonator.

Structure	Symbol	Size (mm)
the lower surface metal strip length	l1	5.25
the upper metal strip length	l2	4.71
the gap between upper center circle and metal strip	w1	0.1
the upper surface metal strip width	w2	0.08
the upper and lower center metal circle radius	r1	1
the upper metal ring width	w	0.44
the lower surface metal strip width	a	0.7
the periodic of the circle slots	p	6.545
the thickness of the dielectric substrate	t	0.65

. The top electrode is composed of two parallel and nonintersecting rectangular metals, in which each metal electrode has a width of 0.06 mm and a gap of 0.7 mm. The permittivity of ε₁ can be regulated by applying voltage at both ends of the electrode. The metal electrode is annealed copper with a thickness of 0.018 mm.

The resonant mode of the LSSP can be essentially understood as a standing wave mode propagating in a waveguide with circular arranged periodic units. When the disk circumference L of the LSSP resonator is equal to an integer n multiple the effective wavelength of the propagating mode, the standing wave resonance will be aroused [53]. Thus, the dipole mode, quadrupole mode, and hexapole mode can be excited, where the three modes represent n = 1, 2, and 3, respectively, and the wave vector k_LSSP of the LSSP resonator can be described as follows:

$$k_{LSSP}=2n\pi /L$$

(3)

Then, the dispersion characteristics of the LSSP resonator as a function of the permittivity is investigated. The simulated results illustrate that when ε₁ = 3.5, the asymptotic frequency is slightly lower than that of ε₁ = 2.9, deviating further from the light line, as shown in Figure 4d. The distance between the dispersion curve and the light line indicates the degree of constraint on the field; this means that as ε₁ increases, the constrained field gradually increases, resulting in less radiation loss of the LSSP resonator. Moreover, according to the dispersion relationship of the LSSP unit structure and its circumference, it can be seen that when the permittivity is 3.5, the dipole frequency of the resonator should appear at 3.6 GHz, while the quadrupole and the hexapole resonance appear at 5.68 GHz and 6.35 GHz, respectively, in terms of Equation (3). It can be seen that the calculated resonances according to the dispersion curve correspond quite well with the resonant peaks in the ECS spectra, as exhibited in Figure 3a.

3.3. Tunability of SSPP Waveguide Loaded with BCZT LSSP Resonator

According to the above, this investigation is conducive to the higher tunability of ε_r in the BCZT compound resonator, which, in turn, will be beneficial for the following tunability of the plasmonic waveguide. Figure 5 presents the plasmonic waveguide with the BCZT compound resonator loaded symmetrically along the center of SSPP waveguide. The transmitted power distribution was numerically monitored in the x-y plane 0.5 mm above the coupled LSSP resonator at the resonant frequencies. The plasmonic waveguide is settled on a Rogers RT5880 substrate, with a substrate thickness of 0.5 mm, a permittivity of 2.2, and a loss tangent of 0.0009.

After loading the LSSP resonator, we conducted a numerical simulation on the S parameter of the filter, and the results are shown in Figure 5. The power flow distribution forms counterclockwise closed loops in the LSSP and the first three excited resonant modes are dipole, quadrupole, and hexapole resonant modes, as depicted in Figure 5c. Also, when the dielectric permittivity of the resonator varies from 3.5 to 2.9, the entire resonance peak position undergoes a blue shift, where the resonant frequency of the dipole mode shifts from 3.44 GHz to 3.65 GHz, with a blue shift of 210 MHz, or a 6.1% controllable range. Meanwhile, the resonant frequency of the quadrupole mode shifts from 5.39 GHz to 5.71 GHz, with a blue shift of 320 MHz, or a regulation range of 5.9%, and the hexapole mode shifts from 6.07 GHz to 6.44 GHz, with a tuning range of 6.1%, which is consistent with the variation pattern in the ECS spectra of the LSSP resonator illustrated in Figure 3a. Also, the transmission properties show that the excited dipole, the quadrupoles, as well as the hexapole of the loaded LSSP resonator are all Fano resonance frequencies because of the asymmetrically distributed resonant line shape. Thus, in the spectral domain, the interference between these plasmonic modes can be expressed with an analytical Fano interference model to fit the spectral lines as follows:

$$T(\omega )=T_0+A_0{\displaystyle \frac{{ [q+2(\omega -{\omega }_0)/\Gamma ]}^2}{1+{ \left[2\right(\omega -{\omega }_0)/\Gamma ]}^2}}$$

(4)

where ω₀ represents the Fano resonant frequencies, ω is the working angular frequency, Γ is the resonance line width, T₀ characterizes the background scattering parameter, A₀ is the coupling parameter between the background and the Fano resonances, and q depicts the Breit–Wigner–Fano parameter, which determines the asymmetry of the resonance spectral line.

The fitting spectral line utilizing the analytical Fano format as well as the simulated spectra are shown in the red and black line, respectively, in Figure 6. As can be seen, a well-matched spectra within the three oscillations (n = 1, 2, 3) can be obtained for the LSSP structure. The resonant frequencies of the dipole, quadrupole, and the hexapole modes center at 3.43 GHz, 5.39 GHz, and 6.06 GHz, respectively, and the corresponding line widths of the modes are 0.0125 GHz, 0.078 GHz, and 0.045 GHz, respectively. Thus, the Q value of the resonator can be calculated using the following formula:

$$Q={\displaystyle \frac{{\omega }_0}{\Gamma }}$$

(5)

The results show that the Q values of the dipole, quadrupole, and hexapole modes of the LSSP resonator can be calculated as 275, 80, and 135, respectively.

3.4. Coupling Mechanism Between BCZT LSSP Resonator and SSPP Waveguide

To reveal the formation of the Fano response of the LSSP resonator, the coupling mechanism of the near-field EM between the three Fano modes and the SSPP modes in plasmonic waveguide is investigated in terms of temporal coupled mode theory (TCMT). The excited Fano modes should stem from the interference of the two transmission paths, as illustrated in Figure 7. An equivalent network is therefore employed to describe the EM field coupling characteristics between the two plasmonic structures, as illustrated in the inset of Figure 7.

A dielectric loss as well as weak radiative loss of the LSSP resonator are considered azimuthally as a resistance R, where the amplitude damping factor in the LSSP is denoted as γ. Regarding the distinction between γ and Γ, Γ represents the resonance linewidth, which is a measure of the energy dissipation and directly determines the quality factor (Q) of the resonance. In contrast, γ describes the system’s intrinsic damping characteristics, which contribute to the total loss, but are treated as a separate parameter in our model. The normalized EM power due to the interference of the two transmission paths can thus be depicted as follows:

$$E_2=t{E}_1+jr{E}_4$$

(6)

$$E_3=t{E}_4+jr{E}_2$$

(7)

$$E_4=(1-\gamma )\mathrm{e}\mathrm{x}\mathrm{p}\left(j\theta \right)E_3$$

(8)

Then, the transmission coefficient E₂/E₁ can be deduced as follows:

$${\displaystyle \frac{E_2}{E_1}}={\displaystyle \frac{t+(1-\gamma )\mathrm{e}\mathrm{x}\mathrm{p}\left(j\theta \right)}{1+t(1-\gamma )\mathrm{e}\mathrm{x}\mathrm{p}\left(j\theta \right)}}$$

(9)

where t represents the coupling through nonradiative damping channels, r describes the coupling process from the LSSP resonator back to the plasmonic waveguide, j stands for imaginary unit, and E stands for electric field strength. For the lossless coupling process between the SSPP waveguide and LSSP resonator, we assume t² + r² = 1. The rotated phase delay θ distributed along the azimuthal direction of the LSSP resonator is in the range of 0 to 2π for the dipole Fano mode. In terms of the low loss of the BCZT substrate in the LSSP resonator, we consider the weak value of R in the equivalent circuit. This parameter represents both the dielectric and radiative losses of the LSSP resonator, thereby influencing the system’s damping characteristics. We set $\gamma =0.05(\omega -{\omega }_0)/\omega$ [54], where ${\omega }_0$ represents the dipole resonant frequency, and t = 0.996 to fit the normalized transmission coefficient.

This evidently indicates that the coupling effect between the SSPP waveguide and the LSSP resonator can deeply trap the power flow, appearing with a strong band notch in the transmission spectrum, as exhibited in Figure 7. Also, the theoretical calculated real part ratio |E₂/E₁| in Equation (9), standing for transmission intensity ratio, matches quite well with the numerically simulated one. It can thus be validated that the asymmetric field fed by the SSPP waveguide could lead to a unidirectional EM coupling into the LSSP resonator, and therefore create an asymmetric resonant line shape, forming the Fano resonant frequencies.

4. Conclusions

In summary, a LSSP resonator with a BCZT compound ferroelectric substrate is designed and employed for the tunability of the SSPP waveguide. The results show that high dielectric tunability can be achieved in the BCZT laminated structure, which is beneficial for the deep tunability of the notch in the plasmonic waveguide. When a pair of mirror-symmetrical LSSP resonators are coupled to the SSPP WG, the central frequency of the notch can be dynamically adjusted in real-time by about 8.8% due to the high dielectric tunability of the BCZT substrate. In addition, the notch generated by the plasmonic coupling enables the excitation of triple Fano resonances, which stem from a unidirectional EM coupling between the LSSP resonator and the SSPP waveguide. The proposal of this study achieves the real-time dynamic adjustment of waveguide performance, which is a significant innovation in the field of waveguide design. Meanwhile, the BCZT layer, as a ferroelectric material, exhibits excellent voltage response characteristics. By applying a voltage to it, the dielectric constant can be significantly changed, thereby enabling the real-time control of the SSPP waveguide performance. This enhancement of voltage tunability is a major highlight of this study.

The method provides a new idea for a compact physical size, minimal insertion loss, large tunability, and real-time flexible regulation in notch filter scenarios. The proposed novel waveguide structure provides new ideas and methods for the development of waveguide technology. By adjusting the waveguide performance in real-time, it can meet the needs of different application scenarios and promote the application of waveguide technology in fields such as communication, sensing, and imaging. Also, since the proposed LSSP is a planar-textured disk, it is more flexible in engineering a Fano line shape, which can be of practical interest for a variety of applications including enhanced biosensing, on-chip communication, and optical information processing.