Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters

Metamaterials have the potential to dramatically improve our ability to control interactions with electromagnetic (EM) radiation, broadly expanding the design space and enabling the implementation of unusual electromagnetic properties such as negative permittivity,¹ negative permeability,² and negative index of refraction.³ Moreover, metamaterials allow for the precise control of the electromagnetic constitutive properties of a structure over a volumetric region⁴; this latter property, akin to gradient index, has been shown to have unique benefits for the design of complex media such as the transformation optical media used to demonstrate electromagnetic cloaking.^4,5 The appreciation of controlling material properties through structure has led to a multitude of creative concepts, including beyond diffraction-limited imaging,⁶ gradient negative-index lenses,⁷ and perfect absorbers.⁸ 

Metamaterials provide tremendous flexibility by breaking prior design paradigms and allowing geometry—not just intrinsic material properties—to influence the electromagnetic response of a structure. However, since geometry so strongly influences the electromagnetic performance of metamaterials, it implies the potential for mechanical sensitivities not found in more traditional approaches. Specifically, mechanical strain, which by definition implies a change in the geometry of a structure, might be expected to produce a change in the electromagnetic response of a metamaterial. In order to successfully transition metamaterials onto operational systems, these relationships must be understood, so that the behavior of a metamaterial can be predicted in relevant environments.

Previous investigations have illustrated the connection between strain and deformation introduced into a metamaterial structure and its resulting electromagnetic scattering properties.^9–13 The present work expands upon those earlier laboratory validations, developing a fundamental understanding of the physical mechanisms that drive a metamaterial’s strain-dependent electromagnetic properties. That understanding then enables the accurate reproduction of a metamaterial’s full, complex permittivity and permeability curves, as a function of applied strain.

The unit cell studied here is an electric-LC resonator, or ELC, designed to operate at X-band, with a resonance expected in the electric permittivity at 9.33 GHz (Ref. 15). Figure 1 shows one element of the repeated cell of the artificial medium. The unit cell is assumed to be cubic, with lattice dimension d.

As in Ref. 10, the mechanical model of the metamaterial assumes the unit cell is part of a large (>10λ), load-bearing structure. As a result, the loading on the cell is symmetric at the boundaries and the copper contributes insignificantly to the overall stiffness of the structure, resulting in an approximately uniform in-plane strain profile throughout the unit cell.

A model of the unit cell was developed in ANSYS-HFSS,¹⁶ utilizing the process outlined in Ref. 10. Normal, in-plane strains, ranging from −5 to +5%, were applied independently in the X- (E_XX) and Y-directions (E_YY), and S-parameters were determined via full-wave simulations. Effective medium parameters (ɛ, μ) were retrieved utilizing a standard inversion algorithm.¹⁷ Of note, even though the dimensions of the unit cell necessarily change with the applied strain, the lattice constant d was kept constant in the parameter retrieval process for all configurations/strain states. Figure 2 shows the material’s permittivity at several different strain conditions.

The permittivity and permeability for this electrically resonant (nonmagnetic) metamaterial can be described analytically, via the following set of equations¹⁸ 

Equation (1) represents an approximate analytical form for the effective medium parameters of a thin layer of electrically resonant polarizable elements embedded within a cubic cell of dimension d. The effective permittivity of the layer is assumed to have the Drude-Lorentz form given by

where $fp$⁠, $f0$ and $Γe$ are the plasma frequency, resonance frequency and damping parameter of the ELC. These parameters are generally related to the geometry and equivalent circuit parameters associated with the element. The propagation constant, $α$⁠, has the form

The sine and cosine terms in Eq. (1) account for spatial dispersion—or the spatial inhomogeneity of the structure. While the ELC (as with other metamaterial structures) occupies a significant portion of the unit cell, the effective response can nevertheless be modeled as being situated in only a portion of the cell, the remainder being air (or whatever the host dielectric material is). The inhomogeneity introduces artifacts into the otherwise Lorentzianlike constitutive parameters that are well described by the additional terms in Eq. (1). In particular, utilizing the relationship for α, it is possible to remove the effects of spatial dispersion from the simulated constitutive parameters, and finally extract values for the Lorentzianlike ELC oscillator (⁠$ɛ−$⁠, shown in Fig. 4). Standard curve fitting procedures can then be applied to determine the values of ε_b, f₀, f_p, and Γ_e.

While Eq. (2) has been previously applied to evaluate electric metamaterial structures, it does not quite represent a full description of the ELC and must be modified through the use of a more detailed circuit model, depicted schematically in Fig. 3.

To obtain the appropriate expression for the effective permittivity, we first calculate the impedance for the circuit model shown in Fig. 3, which consists of a capacitance in series with a parallel LC circuit. The impedance of the structure can be written as $ztot=zCext+zL||zCint$⁠. Using $zC=1/(iωC)$ and $zL=iωL$⁠, we obtain

The expression for the impedance, Eq. (4), can be utilized in conjunction with a transmission line formulation to arrive at an approximate expression for the effective permittivity of the ELC. In the limit that the free-space wavelength is much larger than the unit cell dimension, the transmission line model gives¹⁹ 

Inserting the impedance expression Eq. (4) into Eq. (5), we obtain

where $ω02=(1/LCint)$⁠. The effect of resistive losses can be incorporated into the model by assuming the resistive paths are identical to the inductive paths, and letting $L→L+(R/iω)$⁠.

Following a process similar to that outlined in Ref. 10, we utilize analytic expressions for the equivalent circuit parameters required in Eq. (6). Since these expressions reveal the explicit dependence of the circuit parameters on geometry, a complete analytical description of the strain-dependent permittivity and permeability is facilitated. The capacitance (C) for a coplanar capacitor can be approximated as²⁰ 

W, s and H are the geometrical parameters of the ELC, as depicted in Fig. 1. $ɛS$ is the dielectric constant of the substrate, h_s is thickness of the substrate, and $α=5/2$⁠. The self-inductance (L) of a thin, conducting strip has the form²¹ 

where l is length and b is the width of the conducting strip. The resistance of any length of conductor can be simply approximated by

where σ is the conductivity, and l and A the length and cross-sectional area of the conductor

Inserting the analytic circuit parameter equations into Eq. (6) results in an expression for the effective permittivity that can be compared with that retrieved from a full wave simulation. We begin by considering an unstrained unit cell to obtain a baseline response. While Eqs. (7)–(9) are only approximate (the unit cell geometry lies outside some of the assumptions used for their derivation), they are nevertheless accurate enough to enable curve-fitting between the two functions. A good curve-fit is achieved by using the following values: C_int = 2.94 × 10⁻¹⁴ F, C_ext = 4.84 × 10⁻¹⁴ F, L = 3.74 × 10⁻⁹ H, and R = 0.74 Ω. Figure 4 shows the $ɛ−$ extracted from the simulation, compared to the analytic expression. For the resistance calculation, it is well known that the harmonic input causes the induced currents to reside on the surface of the conductor. Good agreement with the curve fit results was achieved by using one skin-depth at resonance to calculate the cross-sectional area of the conductor in Eq. (9).

A curve fit was also performed on the original form of $ɛ−$ to check the appropriateness of the values used for the circuit model. The following relationships demonstrate good agreement between the alternative forms of the analytic expression (⁠$ɛ−$⁠) (Ref. 22):

With $ɛ−$ successfully defined, the effects of spatial dispersion are then included via the relationship:¹⁸ 

This last piece of information is crucial, because it implies that full knowledge of $ɛ−$ allows full reconstruction of the permittivity and permeability curves. So understanding the strain-dependent behavior of the metamaterial is distilled to understanding the strain-dependent behavior of four critical parameters: C_int, C_ext, L, and R. Furthermore, each of these parameters is a readily described function of geometry. Thus, mechanical strain (a change in geometry) can be readily integrated into Eqs. (7)–(9) to ascertain the strain-dependent electromagnetic behavior of the metamaterial.

We utilize Eq. (11) to determine the phase advance across the unit cell, subsequently applying the determined values of $ɛ−$ and αd to Eq. (1). In this way, the frequency dependent permittivity and permeability functions can be determined. Figure 5 shows the analytically-constructed permittivity compared to the permittivity retrieved from the full-wave simulation for the baseline/unstrained unit cell.

Equations (7)–(9) are then used to describe the strain-dependence of C_int, C_ext, L, and R. These trends are linearized, with minor loss of accuracy, providing the following strain-dependent descriptions for the circuit elements of Eq. (6):

where the primes denote the strained configuration.

As a general rule, a metamaterial’s sensitivity to strain is configuration dependent. Relationships like Eqs. (7)–(9) can be used to quickly and efficiently determine the strain–dependence (i.e., the slopes in the above expressions) of many unit cell variations that share the same circuit model.

Using the relationships of Eq. (12), the values for the capacitances, inductance and resistance were determined for a given strain state. These values were input into Eq. (2), and the aforementioned process repeated to produce the permittivity curves for the deformed/strained unit cell. In Figs. 6 and 7, the strain-dependent analytic expression for the permittivity is compared to that found by retrieving the relevant parameters extracted from full-wave simulations. The curves shown are from the extreme values of the strain envelope modeled for this effort; E_XX = E_YY = −5% and E_XX = E_YY = +5%.

As Figs. 6 and 7 clearly demonstrate, these relatively simple expressions accurately reproduce the quite disparate frequency responses at the different strain states.

The agreement found for the permittivity of the ELC as derived from the analytic model and the full wave simulations demonstrates that a metamaterial’s strain-dependent permittivity and permeability curves can be accurately reproduced though the use of an equivalent circuit model, and explicit expressions for those equivalent circuit elements (as verified using full-wave simulations). The expressions for capacitance, inductance, and resistance are readily described functions of geometry, and are thus easily modified to include small changes to that geometry (i.e., mechanical strain). Similar expressions and processes could be used to describe the strain-dependent electromagnetic behavior of magnetic metamaterials, owing to similar analytic expressions for their constitutive properties and equivalent circuit elements.

The use of analytical approximations to the circuit parameters as a means to introduce the effect of strain might seem unnecessary, since numerical simulations can readily account for these effects. Still, the set of analytical formulas enables a library to be developed for a given structure, and its performance rapidly assessed without resorting to full wave simulations. Moreover, as other physical effects are included, such as temperature dependence, quasianalytical approaches become increasingly efficient and advantageous.

B.J.A. would like to acknowledge support from the U.S. Air Force Office of Scientific Research.