Resonance modes of a metal-semiconductor-metal multilayer mediated by electric charge

Electromagnetic fields around metal–semiconductor–metal (MSM) multilayers with square island top layers were numerically simulated to elucidate the difference in physics between the circuit resonance and Fabry–Pérot interference mediated by the surface plasmon polaritons (SPP). In the current study, the top and bottom metal layers were made of gold, and the intermediate semiconductor layer was a gallium antimony (GaSb). The lumped-element and Fabry–Pérot interference models showed less accuracy when the island width of the MSM multilayer was comparatively smaller. Since the capacitor and SPP could not be supported between the top and bottom gold layers, the anti-reflection mode of the gold–GaSb bilayer mainly affected the absorptance. However, when the width of the island was sufficiently large, the time-lapse development of the electromagnetic fields at resonant wavelengths showed strong electric and magnetic responses relating to the circuit resonance. Simultaneously, the electric fields depicted the movement of the electric charge, which coupled to the short-range surface plasmon polariton (SRSP) existing at the thin GaSb layer sandwiched by two gold layers. The wavelength of the SRSP approximately corresponded to that of the Fabry–Pérot interference. It was revealed that the lumped-element and Fabry–Pérot interference models indicated the same resonant mode from two different perspectives in physics.


Introduction
Electromagnetic metamaterials have attracted the attention of many scientists investigating applications of negative refraction [1], optical cloaking [2], perfect blackbodies [3], and wavelength-selective emitters and absorbers [4][5][6][7][8], among others. These metamaterials consist of various nano/micro-structures that manipulate the propagation behavior of electromagnetic waves at wavelengths longer than their structures. As candidate metamaterials, a one-dimensional grating [5,9], split-ring resonators [10], three-dimensional porous materials [11], and an aperiodic multilayer [12] were proposed over the past several years. To reveal the physics establishing the metamaterials, the electromagnetic properties of these metamaterials were often evaluated using numerical analysis methods, such as finite difference time domain (FDTD) or rigorous coupled wave analysis (RCWA). Concurrently, the resonant frequencies of the metamaterials were formulated using equivalent physical models. A representative model is a lumped-element model that expresses a group oscillation of free electrons and electric current inside a nano/micro-structure as an electric circuit [13][14][15]. An electromagnetic wave at the resonant frequency is almost perfectly absorbed because the electric impedance of the circuit is zero at such a frequency. Researchers in the fields of engineering and optics studied the magnetic properties of such circuits and identified metamaterials with negative magnetic permeabilities at resonant frequencies [4,[16][17][18]. Several researchers coined a term for the resonance at a frequency with negative permeability, a magnetic plasmon polariton (MPP) [19,20] or simply a magnetic polariton (MP) [4,5,9,[21][22][23]. This is an analogy for the surface plasmon polariton (SPP) at a frequency with negative permittivity. The zero-impedance frequencies of nanostructures have been theoretically and experimentally studied. For example, the electromagnetic properties of metal-insulator-metal (MIM) [21,24], metal-dielectric-metal (MDM) [25], and metal-semiconductormetal (MSM) [26,27] multilayers have been studied for over 20 years. Here, the top metal layer can have a rectangular [21,24,28], cross [21,29], or split-ring shape [10,18,30], among others. Since their significant absorptive and emissive characteristics are useful to improve the efficiency of thermophotovoltaic (TPV) power generation, MSM multilayers using a TPV semiconductor, e.g., gallium antimony (GaSb), were proposed to selectively absorb thermal radiation exceeding the bandgap energy level [26,27]. Moreover, a MIM multilayer used as a frequency-tunable emitter is also a convenient way to provide such a radiation to a TPV cell [4].
The discussions about the lumped-element model relating to MPs have been developed; however, they sometimes evaded several fundamental questions. At first, the suffix '-on' in 'polariton' indicates a quantum or elementary particle. While an SPP originates a plasmon, the quantized group oscillation of the electric charge, past studies of MPPs and MPs have not revealed the existence of an equivalent quantum. If it corresponds to a magnon, the quantized spin wave related to the magnetism [31,32], the resonant phenomena should be called magnon polariton [31]. However, the magnon frequency is usually a few GHz, which is ordinary difficult to exhibit a magnetic response at the near-infrared regime [31]. At second, the magnetic response at the resonant frequency is often investigated independently from an electric or a plasmonic response of the nanostructures. Several studies showed a relation between the resonant mode and classical Fabry-Pérot interference, which is an electrical response originated by a propagating wave or SPP [26,27,33]. Wang and Zhang declared that the MP and Fabry-Pérot interference have individual physics and are distinguishable [23]. However, few perspectives for both the magnetic response at the Fabry-Pérot interference frequency and the behavior of SPP at the MP frequency might result in misunderstandings of the nature of resonance. Because the electric and magnetic responses are not independent considering Maxwell's equations, the SPP affects both the electric and magnetic fields around the nanostructures. In the current study, time-step developments of the electromagnetic fields around MSM multilayers of various dimensions are numerically analyzed to simultaneously display the magnetic field and the electric charge, the origin of SPP. Moreover, each resonant mode is theoretically compared with the lumped-element and Fabry-Pérot interference models. The distinguishability of the MP and Fabry-Pérot interference are discussed through numerical and theoretical approaches.

Numerical simulation
In the current study, Maxwell's equations for electromagnetic fields were numerically solved using a three dimensional FDTD approach. This was done to obtain the spectral absorptance of the MSM multilayer and to show the electromagnetic field around them [34]. The simulation was conducted using a Fortran program developed by the authors. The simulation results for the representative case were validated using data obtained employing other open-source software such as MEEP [35]. Figure 1 shows a schematic representation of the MSM multilayer. The bottom and top metal layers are made of gold, while the semiconductor, a GaSb layer, is sandwiched between the gold layers. The relative magnetic permeabilities of GaSb and gold were set to be 1.0. The complex permittivity of gold [36] was fitted according to the monopole Drude model: while that of GaSb [37] was fitted using the monopole Lorentz model: Here, ε ∞ is the respective permittivity at an infinite angular frequency, ω Drude is the plasma frequency, Γ Drude is the carrier relaxation rate, f Lorentz represents the strength of the Lorentz oscillator, ω Lorentz is the central frequency of the Lorentz oscillator, and Γ Lorentz is the damping factor. Table 1 shows each of these parameters for the two materials used to construct the MSM multilayer in the current study. These optical properties were introduced into the simulation using a piecewise linear recursive convolution method [38].
The configuration of the MSM multilayer, referred to as the island geometry, is shown in figure 1. The characteristic parameters are defined as follows: w and l respectively show the x-and y-directional island widths, h (=100 nm) is the height of the top gold layer, Λ is the pitch length of each unit cell, and d (=100 nm) is the thickness of the semiconductor. A square configuration was assumed with w=l in this study to focus on the simplified resonant mode excited around an isotropic geometry. Since this study aims to analyze electromagnetic fields, especially at a thin semiconductor layer, the island height, h, was fixed at a constant value to simplify the discussion. The thickness of the gold bottom layer, 200 nm, is sufficient to eliminate transmission of near-infrared rays. So, the spectral absorptance of the multilayer can be calculated from the intensity ratio of the difference between the incident and reflected rays to the incident ray. One period of each multilayer is set in the computational area, as depicted by the green lines in figure 1. The side boundaries of computational area are a simple periodic boundary condition, while both the top and bottom boundaries are set as a second-order perfectly matched layer (PML) [39,40]. The spectral absorptance of the MSM multilayer was evaluated using an E x -polarized plane wave that was generated at the top boundary for the incident rays and vertically irradiated the multilayer. The spatial resolution of computational grid was 5.0 nm and the time resolution was 5.0×10 −18 s.
In this study, simulations were conducted using the TSUBAME 3.0 supercomputer of the Global Scientific Information and Computing Center at the Tokyo Institute of Technology and a personal workstation with a multi-core processor (Ryzen Threadripper 3970X; AMD, Santa Clara, California, United States).

Theoretical models
3.1. Lumped-element model Figure 2(a) shows a schematic of the representative lumped-element or circuit model proposed by several researchers over the past 20 years [4,28,41]. Here, the electric current in the metal layer originates from a group oscillation of free electrons oscillated by an incident ray. When its oscillation frequency is higher than the reciprocal of the Drude relaxation time of the metal, the group oscillation of the free electrons contributes to forming a capacitor and inductor that modulate the multilayer absorptance. For gold, the relaxation time is 3.0×10 −14 s and its reciprocal is 3.3×10 13 Hz [42]. Thus, radiation with wavelengths shorter than 9.0 μm contributes to the group oscillation. In this study, mutual inductance between the top and bottom gold layers and kinetic inductance inside the gold slab, L m and L k , are defined as follows: is the penetration depth of the electromagnetic wave inside gold, which is the length that the intensity of incident wave attenuates at a rate of e −1 , and 2 ) is the extinction coefficient of gold. Additionally, the capacitance values between the top and bottom metal layers, C m , and between the two islands, C g , are defined as follows: where, ε′ GaSb is the real permittivity of GaSb. The imaginary part, ε′G aSb , does not affect the capacitance. The parameter, c′, is a factor to account for the fringe effects of an electric field around the island layer. Generally, c′, has a value in the range of 0.2 to 0.3 [21,23,27,41,43]. The non-dimensional constant, A=8.0, is an empirical parameter introduced in this study to amplify C g under conditions with narrow channels. Finally, the total impedance of the circuit shown in figure 2(a) is described as: When Z total (ω r ) becomes zero, the free electrons inside an island oscillate without resistance. The resonant angular frequency, ω r , can be mathematically calculated as follows: corresponds to an ultraviolet ray. Thus, a negative value of C C g 2 m 2 + is the only meaningful solution. It is notable that Z total (ω r ) can be zero at more than two frequencies because permittivities have an angular frequency dependency. The resonance at the circuit model was correlated with the absorption of electromagnetic waves at the semiconductor layer. Oscillation of free electrons produces a current loop between the island and bottom gold layer. As a result, a strong magnetic field is excited at the semiconductor or insulator layer, as reported in an earlier study and referred to as MP [4,5,23]. Moreover, several resonant modes at higher-order modes, originated by an LC circuit with multiple capacitors and inductors, have been called MP2, MP3, and so on. Figure 2(b) shows a predicted circuit model for the second resonant mode corresponding to MP2. Divided capacitors, C′, and inductors, L′, are correlated to C m , L m , and L k as follows: In figure 2(b), the red circled area works as a bridge circuit. Since the capacitors and inductors stand symmetrically, the capacitor C′ at the center of the island can be neglected. Therefore, the total impedance of the circuit is described as follows: Figure 2(c) shows another predicted model that ignores the contribution of the capacitor, C g . The total impedance of the circuit and resonant angular frequency can be written as follows: ) corresponds to the eigenmode frequency for a limited part of the circuit. Thus, only )is the frequency related to the absorption of an electromagnetic wave.

Interference of surface plasmon polaritons
Fabry-Pérot interferences are possible physics relating to the electric response for an MSM multilayer. Ni et al described a Fabry-Pérot interference in a perpendicular direction to the vacuum-semiconductor interface that exhibited a resonant peak independent of an island or grating width, w [27]. In this case, an electromagnetic wave exhibited a Fabry-Pérot interference as a propagating mode. However, Liu and Takahara proposed another Fabry-Pérot interference model in a parallel direction to the interface and described that its resonant peak exhibits an island width dependency [33]. The resonant mode satisfies following relation: where, k, f, and m respectively denote the wavenumber of the resonant mode, phase retardation, and the order of the resonant mode (m=1, 2, 3K). Figure 3(a) shows a schematic of the interference inside the semiconductor layer. Since the semiconductor thickness, d, is much shorter than the wavelength of the incident infrared ray, the incident wave cannot penetrate into the area sandwiched by the top and bottom metal layers as a propagating mode. Instead, the incident wave produces a polarization of the electric charges at the lower surface of the island and the upper surface of the bottom layer. Therefore, the incident wave couples with a transverse magnetic (TM) mode of an SPP supported by the longitudinal oscillation of electric charge and is permitted to enter the semiconductor layer. It is notable that the wavelength of the SPP is not the same as that of the propagation mode at vacuum. A rigorous relation between the angular frequency of the incident ray and the where k x and k z1 are wavenumbers of the SP in the x-and z-directions, r ij is the reflection coefficient between media i and j. The green dotted line in figure 3(c) corresponds to dispersion relations of the SPP oscillated inside the GaSb layer [45]. In this figure, the black and blue solid lines indicate light lines in a vacuum and in GaSb, respectively, depending on their refractive indices. The dispersion relation of the SPP exists at a higher wavenumber than the light line in the GaSb layer. Then, the wavelength of the SPP is smaller than that of a propagating wave in the GaSb layer. The wavenumber involving the Fabry-Pérot interference can be derived using the dispersion relation. According to equation (14), the wavenumber of the interfered wave becomes (mπf)/w with a unit of 2π/m. Dotted white and light blue vertical lines in figure 3(c) respectively show the wavenumbers, k 1,300 and k 2,300 , corresponding to the first and second interference modes with w=300 nm, assuming the phase retardation, f, is zero. Then, the Fabry-Pérot interference model estimates the first and second peak wavelengths, λ SPP−1,300 =2.8 μm and λ SPP−2,300 =1.49 μm. Note that the phase retardation is usually larger than zero; thus, the peak slightly redshifts from the prediction [33]. Figure 4(a) shows a contour plot of the spectral absorptance for an MSM multilayer with various channel widths between islands, Λ − w, while the island width was fixed at 300 nm. Two absorptance peaks asymptotically approach the wavelengths of 3.3 and 1.84 μm, respectively, with increased channel width, Λ − w. The blue band, calculated using the circuit model in equation (7) employing a fringe factor, c′, from 0.2 to 0.3, well superposes on the peak at 3.3 μm. The lumped-element model describes the beginning of the peak for any channel width. Using no amplification constant, A, the circuit model indicated a shorter wavelength range, from 3.42 to 4.07 μm, for a channel width of 40 nm. Although the constant, A, was not rigorously based on physics, it improved the accuracy of the lumped-element model for channels narrower than the island width. However, the first interference wavelength of the SPP (SPP-1, 300) slightly deviates from the peak wavelength compared to the lumped-element model. It is because the phase retardation of the Fabry-Pérot interference was assumed to be zero in the current calculation. In order to reduce the discrepancy, the phase retardation should be 0.17π, which is consistent with the past study [33]. Contrary to the first peak, the lumped-element model shows diminished accuracy for the second mode at 1.84 μm. The white and light blue bands respectively depict the range of the second resonant wavelengths calculated using equations (11) and (12). The estimated model including C g exhibits a wavelength close to the first rather than the second mode. Another model with no C g indicates a range from 1.33 to 1.62 μm, which is much closer to the peak. These results imply that equation (12) is adequate to express the impedance related to the second mode of circuit resonance, and the second mode is independent of C g . Here, the second interference wavelength of the SPP (SPP-2, 300) is the same as the peak wavelength with a channel width, Λ − w, of 40 nm. The second peak slightly redshifts with increasing channel width and approaches 1.84 μm, which corresponds to the anti-reflection mode or the Fabry-Pérot interference of the gold-GaSb bilayer for the propagation wave. The spectral absorptance of the bilayer is simply calculated using the following formula [34]:

Simulation results
is the phase shift in the GaSb layer, N 1 (=n i is the complex refractive index of GaSb, while θ 1 is a refraction angle. For the normal incidence with θ 1 =0, the approximate wavelength of the first peak is 4n 1 d while κ 1 affects a slight peak shift. Figure 4(b) shows a contour plot of spectral absorptance for an MSM multilayer with various island widths, w=l, and a fixed channel width between two islands, Λ − w=40 nm. A series of wide band peaks, which redshift proportional to the island width, w=l, is observed in the region from 2.5 to 9.0 μm for the range of island widths from 60 to 600 nm. The lumped-element model for the first mode also exhibits a redshift trend and similar values. Moreover, the wavelengths indicated by the two models, i.e., the lumped-element model for the second mode with no C g and the interference model, superpose on the secondary peak in the range of 1.0 to 2.5 μm. Although these two models have different backgrounds in physics, the lumped-element model may essentially indicate the resonant wavelength at which the SPP oscillates interfered by the island width. Figure 5(a) shows the electromagnetic field distributions around the MSM multilayer at the second peak wavelength, 1.49 μm. The island and channel widths, w and Λ − w, were respectively 300 and 40 nm. At t=0, the lower left and right corners of the island were respectively charged positively and negatively. As a counterpart, opposite electric charges were observed at the surface of the bottom gold layer. As progressing time, these electric charges gradually shifted to the center and formed a longitudinal wave at the lower surface of the island and the upper surface of the bottom layer. These longitudinal waves are the odd mode of the SPP, called the short-range surface plasmon polariton (SRSP) [46]. The SPP exhibits a strong electric field in the z-direction, E z , observed at the quarter length of the island width from the lower corners of the island at t=T/4 s, where T is one wave period. The electric charges shifted to the center until t=5T/12 s. However, the charges from the left and right sides canceled each other in the next T/12 s. At t=T/2 s, the lower left and right corners of the island have opposite charges to those at t=0 s. These electric charges alternately switch at an interval of a half period, T/2 s. E z below the lower island corners and center, as shown by the dotted red lines in figure 5(a), were kept at almost zero through one cycle. This is because the longitudinal wave and SPP generated at the lower left and right island corners have opposite phases and interfered with each other. Here, the distances between each node are almost half an island width. Thus, the wavenumber of the SPP satisfies the relation in equation (14) with m=2. Moreover, the y-directional magnetic field, H y , exhibits high intensities at the center and lower corners of the island at t=0 s. The intensity of H y drastically diminishes at t=T/4 s and recovers at t=T/2 s. Through one cycle, H y is kept at zero at the quarter length of the island from the bottom corners of the island, as depicted by dotted green lines in figure 5(a). H y shows a maximum amplitude T/4 s earlier than E z , and there are differences in the node position by a quarter wavelength. These gaps in time and space for the electromagnetic field are a classical feature of stationary waves observed at the interferometer. Figure 5(b) shows the distribution of the electric field for Λ =900 nm and w=300 nm at the second peak wavelength, 1.84 μm. When the phase retardation in equation (14) is zero, the left and right nodes for E z should be fixed at the two geometric edges described by the dotted yellow lines in this figure. However, the electric field did not clearly exhibit two nodes due to expansion of the electric field in the GaSb layer to a region outside the geometric edge lines. Similarly, the node for H y shifted closer to the geometric edge, and it shows an expansion of the SPP wavelength. This is because electric charges at the lower surface are less affected by those of nearby islands compared to the case where Λ =340 nm. Therefore, the wavelength of the longitudinal wave increased by approximately 80 nm with expanding distance between the positive and negative charges, and it causes a phase retardation. The phase retardation caused a redshift of the second resonant peak, and the peak superposes on the anti-reflection mode of the gold-GaSb bilayer. With smaller island widths compared to the pitch, Λ, the SPP mode less affected the resonant wavelength while a part of the propagation wave coupled with the SPP.

Second resonant mode
The abovementioned electromagnetic field also can be described using the lumped-element model for the second resonant mode. Figure 5(c) shows a schematic diagram of the electric charge adapted to the lumped-element model. The electric charges are localized anti-symmetrically inside the island layer when the incident wave vertically irradiates the MSM multilayer. These electric charges form two capacitors with opposite charges at the left and right sides of the circuit. Here, two adjacent capacitors should be oppositely charged at the resonant mode. Consequently, the electric charge at the central capacitor is canceled and kept at zero. Despite no electric charge, the capacitor still affects the resonant frequency. Thus, the lumped-element model for the second mode, corresponds to the MP2 mode in previous studies, approximately exhibits the peak wavelength. However, the electromagnetic field distribution in figure 5 is different from the MP2 mode and similar to the MP3 mode because a previous study observed symmetrical electric charges inside the island layer using an inclined incident wave [4]. These results reveal that simulations with different incident angles cause different distributions in the magnetic fields for each resonant mode. Regardless of the incident angle and magnetic field, the wavelength of the SPP and the lumped-element model can clearly show that the second peak is the second resonant mode. Figure 6(a) depicts a time-lapse development of the electromagnetic field around the MSM multilayer at the wavelength of the first peak, 5.45 μm for Λ − w=40 nm and w=l=300 nm. At t=0 s, H y in the negative direction of the y-axis was strongly confined inside the GaSb layer, and the electric flux lines showed a circular current loop in the anticlockwise direction. The current loop disappeared at t=T/36 s, and the electric charge started to localize at the left and right sides of the island over 8T/36 s. At t=T/4 s, the localized charges formed a strong electric field at the vacuum gap between two islands. The electric field expanded into the vacuum region above the island. Simultaneously, the electric field was attenuated along the interface between GaSb and gold. These results show that the effective wavelength of the SPP extended to w+2h, keeping the phase retardation at 0.17π. This irregular expansion of the SPP caused the discrepancy from the first peak. The expansion also affected the lumped-element model. The electric field corresponding to the capacitor C g was stronger than C m , although C m is numerically larger than C g . The constant, A, in equation (6) reduces deviation between the lumped-element model and the exact electromagnetic field around the multilayer with a narrow channel, and improves the accuracy of the lumped-element model.

First resonant mode
At t=T/4 s, H y at the GaSb layer exhibited the same magnitude as the incident wave since there is a gap in time when the electric and magnetic fields each reach maximum amplitudes. The magnetic field was confined and amplified in the positive direction of the y-axis over the next T/4 s as the localized electric charge dissipated. A current loop in the clockwise direction was observed at t=T/2 s. The current loop was formed at a time after t=35T/36 s, when the magnetic field had finished recovering. Although previous research indicated that the electrical loop significantly induced a magnetic field and MP [4,21,23], the loop was observed during only a little moment in the time-lapse development of the electric field. The current loop seems to be a temporary state existing during the switching of the electric charge. Since the SRSP mode controls the oscillation of the electric charge, the SPP should be the origin of both the electric and magnetic responses observed around the MSM multilayer. These results supports that the Fabry-Pérot interference model for the SRSP mode indicated the same resonant mode as the lumped-element model, which shows the resonant frequency of the MP. Figure 7 shows a contour plot of spectral absorptance with a fixed island width, w=100 nm. The resonant wavelength of the lumped-element model deviates from the absorptance peak when the channel width is larger than 100 nm. Moreover, the peak wavelength almost perfectly superposes the anti-reflection mode of the gold-GaSb bilayer. Thus, the island layer and circuit resonance less affect the resonant wavelength with increasing channel width. Furthermore, the spectral absorptance of the MSM multilayer asymptotically approaches that of the gold-GaSb bilayer. Both the Fabry-Pérot interference and lumped-element models lose accuracy when the island width is comparable to the semiconductor thickness. Figure 8(a) depicts the electric field vector around an MSM multilayer with a structure of Λ − w=600 nm and w=l=300 nm, which is a geometry where the lumped-element model accurately exhibits the peak wavelength. The electric field was oscillated by an incident ray with a wavelength of 3.33 μm corresponding to the absorptance peak. The E z , have opposite directions at the upper and lower surfaces of the island layer. It can be clearly seen that the lower left and the right corners of the rectangular island are, respectively, positively and negatively charged. The substrate directly below the island is charged negatively and positively as counterparts of the electric charges in the island. These electric charges are the SRSP inside the gold layer, and promote two capacitors, C m , at the left and right halves of the island layer. Figure 8(b) shows the electromagnetic field distribution around the MSM multilayer with a structure such that Λ − w=800 nm and w=l=100 nm. The wavelength of the incident wave corresponds to the first peak, 1.89 μm. Similar to the case for w=300 nm, the strong electric field around the rectangular island exhibits polarized electric charges at the lower left and the right corners. However, the electric field around the island spreads along the interface between the vacuum and GaSb layers, and the capacitor, C m , is not observed at the GaSb layer. The electric field distribution resembles the anti-reflection mode of the gold-GaSb bilayer shown in figure 8(c). In the case of the bilayer, the electric field is kept at zero along the interface between the gold and GaSb layers, while the electric field reaches maximum intensity at the GaSb-vacuum interface. A comparison between figures 8(b) and (c) implies that the electric charge in the polarized island layer cancels the electric field inside the island against external electric fields. Because neither the circuit nor the SRSP was formed between the top and bottom gold layers when the island width was comparatively smaller, the lumped-element and Fabry-Pérot interference models lost accuracy.

Critical point for the circuit resonance
When the resonant wavelength of the lumped-element model is shorter than that of the anti-reflection mode, the capacitor, C m , is not observed, as shown in figure 8(b). Therefore, circuit resonance is independently observed only when the lumped-element model shows a much longer wavelength than the anti-reflection mode. With increasing Λ − w, the first resonant angular frequency of the lumped-element model in equation (8)    and λ r also has an inverse correlation to d, while it cannot be simply written. The wavelength of the antireflection mode is approximately 4n GaSb d. Thus, the island width and the semiconductor thickness need to be respectively large and small to observe circuit resonance. Empirically, the critical length of w is twice or three times the semiconductor thickness. The discussions focusing on the identical electric charge inside the island showed deep relationships between the lumped-element and Fabry-Pérot interference models for the MSM multilayer with a sufficiently large island. The electric response at the circuit resonance frequency resembles the Fabry-Pérot interference for the SRSP mode, and the magnetic response at the Fabry-Pérot interference frequency is similar to the MP. As a result, the two physical models for the MSM multilayer indicate the same resonant mode at the near-infrared wavelength. Moreover, since the magnetic permeabilities of gold and GaSb are 1.0, the origin of the magnetic resonance called MP is a magnetic field excited by the SPP.

Conclusions
An MSM multilayer with an island geometry for the top metal layer and a thin GaSb as a middle layer were analyzed to clarify the distinguishability between the circuit resonance and Fabry-Pérot interference mediated by the SPP. The electromagnetic fields around the MSM multilayer were numerically determined using an FDTD method to obtain the spectral absorptance and electromagnetic field distribution. For the second resonant mode, the peak wavelength agreed to the lumped-element model, and the electromagnetic field distribution showed three pairs of electric charges corresponding to three capacitors used in the lumpedelement model. Simultaneously, the electric charges formed longitudinal waves supporting the SRSP inside the GaSb layer. The SRSP interfered with itself at the GaSb region sandwiched between the two gold layers. As a result, the resonant wavelength also agreed with the Fabry-Pérot interference model considering the wavelength and phase retardation of the SPP. For the first mode, because the SPP expanded to the sidewall of the island, the effective capacitance of the circuit and phase retardation of the SPP increased slightly. Therefore, several modifications are required for both lumped-element and Fabry-Pérot interference models to improve the accuracy. The electric and magnetic fields respectively depicted the current loop and enhanced magnetic field described in earlier studies. However, the time-lapse visualization showed little contribution to the current loop for exciting the magnetic field. It also showed that both the lumped-element and Fabry-Pérot interference models indicated the same resonant mode, and the magnetic response at the GaSb layer originates a development of the SPP rather than the magnetic resonance or MP. Here, both the lumped-element and Fabry-Pérot interference models became invalid due to the anti-reflection mode of the gold-GaSb bilayer when the semiconductor thickness was greater than one-third of the island width. It is because the MSM multilayer with a small island could mediate neither the circuit nor the SRSP between the top and bottom gold layers.