Precise one-to-one equivalent nanocircuit models for layered metamaterials

A precise one-to-one equivalent nanocircuit model for layered metamaterials is presented in this work. The theoretical analysis establishes a precise link between the nanocircuit system and the optical film system by comparing between the optical transfer matrix of an optical film and the transmission matrix of the distributed-element model. Through dimensional analysis, the connection between the optical properties of the film and the distributed circuit components of the transmission line is revealed. Subsequently, the lumped-element model is simplified to the distributed-element model for nonmagnetic films with different optical features. Finally, the lumped-element model is further applied to multilayer metamaterials with different microstructures. All analysis is confirmed through the agreement between the S-parameters of the equivalent nanocircuit model and the reflection and transmission coefficients of the layered metamaterials.


Introduction
Electromagnetic metamaterials, or metamaterials for short, are artificial materials created to display electromagnetic characteristics not found in natural.They are made by arranging subwavelength-sized structures in predetermined patterns to manipulate the behavior of electromagnetic waves [1][2][3].In particular, layered metamaterials are a type of metamaterial made up of several subwavelength-thin layers stacked either periodically or non-periodically [4,5].The unique and exotic behaviors of the layered metamaterials are created by the electromagnetic properties that are specifically chosen for each layer.Like other metamaterials, layered metamaterials can also exhibit extraordinary electromagnetic properties at a specified operation frequency, such as negative refractive index [6,7], near-zero permittivity or permeability [8,9], anisotropic behavior [10,11], and other unusual responses [12], by carefully designing the geometry, composition, and arrangement of the layers.More importantly, the layered metamaterials can be designed with a sophisticated microstructure to operate across a broad range of frequencies, e.g. the broadband absorber [13,14], the broadband epsilon-near-zero (ENZ) metamaterials [15][16][17][18][19][20][21], and others [22].Inspired by metamaterials, metactronics [23], also named as metatronics [24], is promoted to develop new technologies with fantastical functions by fusing metamaterials with nanoelectronics and nanophotonics.For example, individual nanoparticles can be regarded as lumped circuit components such as nanoinductors, nanoresistors and nanocapacitors [25,26].Therefore, this enables the design of complicated metamaterials by applying tools developed for the design of electronic circuits.For instance, several different types of functional optical nanocircuits have been developed to assist in the design and optimization of metamaterials.These structures include absorbers [27], nanoantennas [28,29], filters [30,31], circuit boards [32,33], computational devices [34,35], and the mu-near-zero media [36], and they provide guidelines for the optimization of metamaterial structures to obtain the desired electromagnetic properties.In addition, the nanocircuit model aids in a better understanding of the anomalous properties of metamaterials [37].However, the explicit relationship between the nanocircuit and the metamaterial has not been definitively proposed.This encourages further comprehensive studies in this area.
In this work, a precise one-to-one equivalent nanocircuit model for the layered metamaterials is presented, theoretically establishing a precise connection between the nanocircuit system and the optical film system.Initially, the model is established from the comparison between the optical transfer matrix for an optical film and the transmission matrix (the ABCD matrix) of the distributed-element model (the transmission line model).This analysis reveals the relationship between the optical properties of the film and the distributed circuit components of the transmission line through dimensional analysis.Subsequently, the lumped-element model is simplified to the distributed-element model in light of a nonmagnetic film with different optical features.Specifically, the nonmagnetic film, with an electric permittivity respectively characterized by real constants, by the Drude model, and by the Lorentz model, is considered.Furthermore, the application of the lumped-element model is extended into the multilayer metamaterials with different microstructures.These include the periodic dielectric-metal multilayer metamaterial and the broadband ENZ metamaterials with a complicated multilayer microstructure.The agreement between the S-parameters of the equivalent nanocircuit model and the reflection and transmission coefficients of the layered metamaterials serves as clear validation for the accuracy of the lumped-element model.

Theoretical analysis
Consider an isotropic and homogeneous film that is irradiated by a normal incident plane electromagnetic wave propagating along the z-direction, as shown in figure 1(a).The film is of a thickness ∆z in the z-direction and infinitely extending in both the x-and y-directions, and possesses the electric permittivity ε = ε 0 ε r and the magnetic permeability µ = µ 0 µ r .According to the optical transfer matrix [38], the tangential components of the electromagnetic wave on both surfaces of the film can be related through the following equation in which n = √ ε r µ r denotes the refractive index of the film with respect to the relative permittivity r and the relative permeability µ r = µ ′ r − jµ ′ ′ r of the film, η = √ µ/ε denotes the characteristic impedance of the film, and k 0 = ω √ ε 0 µ 0 denotes the wave vector with respect to the angular frequency ω of the electromagnetic wave.It is important to mention that the time-harmonic term exp(jωt) is set as default with j 2 = −1 in this work.
On the other hand, the distributed-element model, i.e. the transmission-line model, exhibits a similar transmission process, wherein the series resistance per unit length, the series inductance per unit length, the shunt capacitance per unit length and the shunt conductance per unit length are individually denoted as R 0 , L 0 , C 0 and G 0 , as shown in figure 1(b).To facilitate analysis, the transmission matrix [39], i.e. the ABCD matrix, is introduced in to determine the relation between the total voltages V(z, t) = v(z) exp(jωt) and the total currents I(z, t) = i(z) exp(jωt) at the input-and output-port of the transmission line, in which To conduct a more thorough examination of the similarity between the two different systems, a novel parameter K = −jγ is introduced in, thus equation ( 2) can be transformed into the following format based on the identities cosh(a + jb) = cos(b − ja) and sinh(a + jb) = j sin(b − ja).According to equations ( 1) and (3), it is found that the parameter nk 0 in the optical transfer matrix The schematic profile of an isotropic and homogeneous film of the thickness ∆z along the z-direction and infinitely extending in the x-and y-direction.The film is of the electric permittivity ε = ε0εr and the magnetic permeability µ = µ0µr, with the background media (ε1, µ1) and (ε3, µ3) on its two sides, respectively.A plane electromagnetic wave interacts with the film with the zero angle of incidence along the z-direction.The propagation of the electromagnetic wave is represented by the incident, the reflected, and the transmitted waves at different interfaces between the film and the background media.(b) A small segment of the length ∆z in the distributed-element model, i.e. the transmission-line model, characterized by the series resistance per unit length R0, the series inductance per unit length L0, the shunt capacitance per unit length C0 and the shunt conductance per unit length G0.(c) The lumped-element model constructed by a series impedance Z = R + jωL in terms of a series resistance R and a series inductance L and a shunt admittance Y = G + jωC in terms of a shunt capacitance C and a shunt conductance G.
Table 1.The relationship among the optical properties of the film and the electronic components per unit length of the transmission line with the units in the square brackets.
and the parameter K in the ABCD matrix may stand for a similar physical quantity since both of them possess the same unit m −1 .Therefore, through the dimensional analysis of each term of the two parameters, the one-to-one relationship between the optical properties of the film and the electronic components of the transmission line can be obtained as shown in table 1.
As a consequence, the distributed-circuit model can be precisely mapped into the optical properties of the isotropic and homogeneous film and vice versa.
Although accuracy in mimicking the optical properties of the film, the equivalent distributed-element model is challenging to implement in applications.Therefore, it is worthwhile to develop a corresponding lumped-element model for the film.Simply, the corresponding lumped-element model is constructed by a series impedance Z = R + jωL in terms of a series resistance R and a series inductance L and a shunt admittance Y = G + jωC in terms of a shunt capacitance C and a shunt conductance G, as shown in figure 1(c).In the lumped-element model, the total voltages and the total currents at the input-and output-port of the lumped circuit follow the relationship [40] Table 2.The relationship among the optical properties of the film in the subwavelength scale and the electronic components of the lumped circuit with the units in the square brackets.
On the other hand, if the thickness of the film is in a deep subwavelength scale, i.e. nk 0 ≪ 1, equation ( 1) can also be approximated as Therefore, according to equations ( 7) and ( 8), the relationship between the lumped circuit and the film reads as and the detailed information is shown in table 2.
To verify the accuracy of the equivalent lumped-element model of the film, the S-parameters (S 11 and S 21 ) of the lumped-element model [39] (12) based on the ABCD matrix in equation ( 6) are introduced to compare with the reflection (r) and transmission (t) coefficients of the film [38] based on the optical transfer matrix in equation ( 1).Here, the parameters Z in and Z out denote the characteristic electrical impedance of the input-and output-port of the lumped-element model, while the parameters η 1 = √ µ 1 /ε 1 and η 3 = √ µ 3 /ε 3 denote the characteristic impedance of the background media on each side of the film.According to equations ( 11)-( 14), it is obvious that the characteristic electrical impedance Z in and Z out of the lumped-element model must be comparable to the characteristic impedance η 1 and η 3 of the background media.Without loss of generality, only the simplest case that Z in = Z out and

Numerical results
In detail, regarding a general isotropic and homogeneous film, the series resistance R = ωµ 0 µ ′ ′ r ∆z can be set as zero and the series inductance L = µ 0 µ ′ r ∆z can be set as µ 0 ∆z because the permeability of most natural materials is close to the permeability of free space, i.e. µ ′ r = 1 and µ ′ ′ r = 0.In consideration of simplicity, this condition is taken into account in this work.Additionally, it should be noted that because no electrical components truly operate as conductance in a real scenario, a more in-depth examination of the shunt conductance in the lumped-circuit model is necessary.Given that the shunt conductance depends on the permittivity of the film, i.e.G = ωε 0 ε ′ ′ r ∆z, its corresponding lumped circuit should be inspected in relation to three scenarios related to the permittivity of three common materials.The first case is the simplest, that the permittivity of the film can be regarded as a real constant in the frequency range of the probing electromagnetic wave, e.g. the permittivity of some dielectric films in the optical frequency range.Therefore, the equivalent lumped circuit of the film can be constructed with a series inductance L = µ 0 ∆z and a shunt capacitance C = ε 0 ε ′ r ∆z, as shown in figure 2(a).Take a SiC film of the permittivity ε ′ r = 2.65 2 [41] and the thickness ∆z = 5 nm as an example.In such a case, the corresponding series inductance is L = 6.283 pH and the corresponding shunt capacitance is C = 0.3109 aF.In addition, figure 2 It is clear that the results of the equivalent lumped circuit are well coincident with that of the film, which confirms the theoretical analysis.There is only one slight deviation between the argument of the S 11 parameter and that of the r coefficient.This is probably because the element A ̸ = D in the ABCD matrix of equation ( 6), while the element m 11 = m 22 in the optical transfer matrix of equation (1).As a result, the numerator of the S 11 parameter in equation (11) will be greater than that of the r coefficient in equation ( 13) when the all characteristic impedances are equal, and this leads to a greater argument in the S 11 parameter.This deviation can be observed in all of the following results.However, this slight deviation is acceptable because it decreases as the thickness of the film decreases.
The second case is the Drude model, which is generally applied to characterize the permittivity of metals in terms of the dielectric constant ε ∞ , the plasma frequency ω p and the damping factor γ. Here, the electrical conductivity reads σ (ω) = σ 0 1 + jωτ (16) in terms of the relaxation time τ = 1/γ and the direct current conductivity σ 0 = ω 2 p τ ε 0 .According to the result in table 2, there is Therefore, a resistance R ′ and an inductance L ′ of the values can be introduced into the shunt branch standing for the shunt conductance as G = R ′ + jωL ′ .Meanwhile, the capacitance in the other shunt branch should be of the value Therefore, the resistance R ′ , the inductance L ′ and the capacity C form the shunt admittance Y of the lumped circuit.Finally, the equivalent lumped circuit for the Drude model takes the configuration of figure 2 The final case is the Lorentz model that generally characterizes the permittivity of dielectric.For simplicity, only the simple Lorentz model is considered, which describes the permittivity of dielectric as with only one resonance frequency ω 0 and the damping factor γ. According to the result in table 2, the conductance can be determined as As a result, the shunt conductance G can be constructed by a resistance R ′ = (σ 0 ∆z) −1 , an inductance L ′ = τ (σ 0 ∆z) −1 and an additional capacitor as

Equivalent lumped-element model for layered metamaterials
In addition to being applied in the single film, the above outcome can also be utilized in layered metamaterials.Take the simple periodic dielectric-metal multilayer metamaterial as an example.The multilayer is constructed with SiC and Au of the permittivity, ε d and ε m , keeping previous values.Figure 3(a) shows one unit of the periodic SiC-Au multilayer metamaterial with respect to a plane electromagnetic wave that is incident normal to the interface between the layers, as an insert figure, and the corresponding equivalent lumped circuit.In this circuit, the series inductance L d1 and the shunt capacitance On the other hand, figure 3(d) depicts another configuration that the plane electromagnetic wave is incident parallel to the interface between the layers of the SiC-Au multilayer metamaterial unit cell as the insert figure, and the corresponding equivalent lumped circuit.To construct the equivalent lumped circuit in such a case, the effective medium theory (EMT) is applied.According to the EMT, the effective permittivity ε e of the unit cell with respect to the probing electromagnetic wave is with the filling ratio defined as As the unit cell is regarded as a homogeneous medium, the series inductance can be determined as L l = µ d l, where the parameter l denotes the length of the unit cell.
Concerning the remain part of the equivalent lumped circuit, recall the relation between the total admittance Y e and two series admittances Y m and Y d , and compare it with the EMT in equation (24).As a consequence, there are for the SiC layer and

Equivalent lumped-element model for broadband ENZ metamaterials
Besides the simple layered metamaterials, the lumped-element model can also be applied in more complicated layered microstructures.According to the authors' previous work [21], a unit cell of the broadband ENZ metamaterials is constructed as shown in figure 4    the plasma frequency ω p = 2π × 265 × 10 12 rad s −1 , and the damping factor γ = 2π × 20 × 10 12 rad s −1 [44].
The unit cell possesses 6 layered-sublayers forming a symmetric microstructure.The length and the height of the unit cell is l = 180 nm and d = 5 nm, respectively.The filling ratio of the ITO in the nth sublayer f 2 /d and that of the nth sublayer in the unit cell l n /l follows the values in table 3.In accordance with this, the equivalent lumped circuit can be built using the results of the previous study and is shown in figure 4(b).Meanwhile, the electric components in the equivalent lumped circuit have values in table 4. Similarly, the accurateness of the equivalent lumped circuit can be verified by comparing the absolute value and the argument of the S 11 and S 21 parameters [solid curves] of the equivalent lumped circuit and that of the r and t coefficients [empty circles] of the unit cell as shown in figures 4(c) and (d), respectively.The S 11 and S 21 parameters are theoretically calculated from the ABCD matrix of the equivalent lumped circuit, while the r and t coefficients are numerically calculated by the finite element method.It is clear that the absolute value and the argument obtained from the equivalent lumped circuit and that obtained from the unit cell are coincident with each other, respectively.The existence of the acceptable deviations is because, instead of the thickness d m is considered to determine the values of the shunt electric components in the equivalent lumped circuit for every sublayer.In general, the equivalent lumped circuit does exhibit similar characteristics as the broadband ENZ unit cell.

Conclusions
In conclusion, a precise one-to-one equivalent nanocircuit model for layered metamaterials has been successfully developed and demonstrated in this work.The theoretical analysis has established a precise and reliable connection between the nanocircuit system and the optical film system.By comparing the optical transfer matrix of an optical film with the transmission matrix of the distributed-element model, the clear relationship between the optical properties of the film and the distributed circuit components of the transmission line has been revealed through dimensional analysis.The subsequent simplification of the distributed-element model to a lumped-element model has been proven effective for nonmagnetic films exhibiting typical optical properties, which possess the electric permittivity individually characterized by real constants, by the Drude model, and by the Lorentz model.Importantly, the applicability of the lumped-element model has been extended to encompass diverse multilayer metamaterials, ranging from periodic dielectric-metal multilayer structures to broadband ENZ metamaterials characterized by complex multilayer microstructures.The robustness and accuracy of the lumped-element model have been theoretically validated by the agreement observed between the S-parameters obtained from the equivalent nanocircuit model and the reflection and transmission coefficients measured from the layered metamaterials.Furthermore, it is worth mentioning that, even though it is called the nanocircuit model, the lumped-element model is generally a collective effect of the lumped components under the macroscale.Therefore, it shows a possibility to demonstrate the electromagnetic properties of metamaterials in a much easier way by applying the equivalent nanocircuit model.Consequently, this work not only establishes a solid foundation for understanding the behavior of layered metamaterials but also contributes to the advancement of nanocircuit modeling techniques in optical systems.Future studies can further explore the application of this model to other complex optical systems, paving the way for more comprehensive investigations and technological advancements in this field.

Figure 1 .
Figure 1.(a)The schematic profile of an isotropic and homogeneous film of the thickness ∆z along the z-direction and infinitely extending in the x-and y-direction.The film is of the electric permittivity ε = ε0εr and the magnetic permeability µ = µ0µr, with the background media (ε1, µ1) and (ε3, µ3) on its two sides, respectively.A plane electromagnetic wave interacts with the film with the zero angle of incidence along the z-direction.The propagation of the electromagnetic wave is represented by the incident, the reflected, and the transmitted waves at different interfaces between the film and the background media.(b) A small segment of the length ∆z in the distributed-element model, i.e. the transmission-line model, characterized by the series resistance per unit length R0, the series inductance per unit length L0, the shunt capacitance per unit length C0 and the shunt conductance per unit length G0.(c) The lumped-element model constructed by a series impedance Z = R + jωL in terms of a series resistance R and a series inductance L and a shunt admittance Y = G + jωC in terms of a shunt capacitance C and a shunt conductance G.

Figure 2 .
Figure 2. The equivalent lumped circuit for an isotropic and homogenous non-magnetic film with the relative permittivity characterized by (a) a constant, (d) the Drude model and (g) the Lorentz model.The absolute value of the S11 and S21 parameters [solid curves] of the corresponding equivalent lumped circuit and the absolute value of the r and t coefficients [empty circles] of the film constructed by (b) SiC [constant permittivity], (e) Au [Drude model] and (h) α-Si [Lorentz model].The argument of the S11 and S21 parameters [solid curves] of the corresponding equivalent lumped circuit and the argument of the r and t coefficients [empty circles] of the film constructed by (c) SiC [constant permittivity], (f) Au [Drude model] and (i) α-Si [Lorentz model].
figure 2(c) demonstrates the argument of the S 11 and S 21 parameters [solid curves] of the equivalent lumped circuit compared with the argument of the r and t coefficients [empty circles] of the film.It is clear that the results of the equivalent lumped circuit are well coincident with that of the film, which confirms the theoretical analysis.There is only one slight deviation between the argument of the S 11 parameter and that of the r coefficient.This is probably because the element A ̸ = D in the ABCD matrix of equation (6), while the element m 11 = m 22 in the optical transfer matrix of equation(1).As a result, the numerator of the S 11 parameter in equation(11) will be greater than that of the r coefficient in equation (13) when the all characteristic impedances are equal, and this leads to a greater argument in the S 11 parameter.This deviation can be observed in all of the following results.However, this slight deviation is acceptable because it decreases as the thickness of the film decreases.The second case is the Drude model, which is generally applied to characterize the permittivity of metals (d).To verify the analysis, an Au film of the thickness ∆z = 5 nm is considered.The Drude model for Au possesses the dielectric constant ε ∞ = 9, the plasma frequency ω p = 13.8 × 10 15 rad s −1 and the damping factor γ = 0.11 × 10 15 rad s −1[42].As a result, the values of each electric component in the equivalent lumped circuit read L = 6.283 fH, C = 0.3984 aF, L ′ = 0.1186 pH and R ′ = 13.05Ω.As the same, both the absolute value and the argument of the S 11 and S 21 parameters [solid curves] and that of the r and t coefficients [empty circles] are individually compared in figures 2(e) and (f), and it is clear all results are well coincident.
the simple Lorentz model can also be represented by a capacitor C = ε 0 ε ∞ ∆z in the other shunt branch similar to the Drude model.Finally, the equivalent lumped circuit for the simple Lorentz model is depicted in figure 2(g).Take an α-Si film of the thickness ∆z = 5 nm to verify the analysis.Here, the permittivity of α-Si follows the simple Lorentz model with the dielectric constant ε ∞ = 3.109, the plasma frequency ω p = 2π × 3.627 × 10 15 rad s −1 , the resonance frequency ω 0 = 2π × 0.9503 × 10 15 rad s −1 , and the damping factor γ = 2π × 0.4643 × 10 15 rad s −1 [43].Therefore, the values of electric components in the equivalent lumped circuit read L = 6.283 fH, C = 0.1377 aF, L ′ = 43.48pH, R ′ = 126.8Ω and C ′ = 0.6451 aF.It is clear that both the absolute value and the argument of the S 11 and S 21 parameters [solid curves] are coincident with that of the r and t coefficients as shown in figures 2(h) and (i), respectively.Meanwhile, because of the Lorentz resonance at the resonance frequency, figure 2(i) also illustrates that the argument of the S 11 parameter and that of the r coefficient flips around the resonance frequency, i.e. the phase flipping.
C d represent the SiC layer of the thickness d 1 , while the series inductance L d2 , the shunt capacitance C m , the shunt inductance L m and the shunt resistance R m represent the Au layer of the thickness d 2 .Regarding a general case, that is d 1 = 6 nm and d 2 = 4 nm, the values of the electric components in the equivalent lumped circuit read L d1 = 7.540 fH, C d = 0.3731 aF, L d2 = 5.027 fH, C m = 0.3188 aF, L m = 0.1483 pH, and R m = 16.31Ω. Correspondingly, figures 3(b) and (c) individually display the absolute value and the arguments of the S 11 and S 21 parameters [solid curves] of the equivalent lumped circuit and that of the r and t coefficients [empty circles] of the multilayer.All result are coincident well.

Figure 3 .
Figure 3.The equivalent lumped circuit for a unit of the periodic dielectric-metal multilayer metamaterial with respect to the probing electromagnetic wave propagating (a) normal to the interface [normal-incidence configuration] and (d) parallel to the interface [parallel-incidence configuration] between the layers.For a unit of the periodic SiC-Au multilayer metamaterial, the absolute value of the S11 and S21 parameters [solid curves] of the corresponding equivalent lumped circuit and the absolute value of the r and t coefficients [empty circles] of the unit cell with respect to (b) the normal-incidence configuration and (e) the parallel-incidence configuration.Furthermore, the argument of the S11 and S21 parameters [solid curves] of the corresponding equivalent lumped circuit and the argument of the r and t coefficients [empty circles] of the SiC-Au multilayer unit cell with respect to (c) the normal-incidence configuration and (f) the parallel-incidence configuration.
(a).The broadband ENZ unit cell is composed by two different materials, an ideal dielectric component and a kind of indium tin oxide (ITO) as the metallic component.Simply, the relative permittivity of the ideal dielectric is set as ε d = 1.Meanwhile, the relative permittivity of the ITO ε m is characterized by the Drude model with the dielectric constant ε ∞ = 1,

Figure 4 .
Figure 4. (a) The schematic profile for a unit cell of the broadband ENZ metamaterial with respect to the probing electromagnetic wave.The unit cell is a two-phase composite, the dielectric component and the metallic component, and possesses 6 layered-sublayers forming a symmetric microstructure.(b) The equivalent lumped circuit for the unit cell of the broadband ENZ metamaterial.(c) The absolute value of the S11 and S21 parameters [solid curves] of the equivalent lumped circuit and the absolute value of the r and t coefficients [empty circles] of the broadband ENZ unit cell.(d) The argument of the S11 and S21 parameters [solid curves] of the equivalent lumped circuit and the argument of the r and t coefficients [empty circles] of the broadband ENZ unit cell.

Table 3 .
The filling ratio of the ITO in each sublayer and that of each sublayer in the unit cell.

Table 4 .
The values of each electric component in the equivalent lumped circuit.