Broadband Electromagnetic Properties of Engineered Flexible Absorber Materials

Flexible and stretchable materials have attracted significant interest in wearable electronics and bioengineering fields. Recent developments also incorporate embedded microwave circuits and systems with engineered flexible materials that operate over a broad frequency range (≈1–100 GHz). Herein, a simple flip‐chip technique is used to evaluate frequency‐dependent electromagnetic properties of flexible materials and applied to evaluate engineered microwave absorbers based on self‐biased barium hexaferrite composites. On‐wafer error correction and de‐embedding techniques are applied to determine broadband electromagnetic properties of the material‐loaded transmission lines by placing the materials on top of coplanar waveguide transmission lines. Finite‐element simulations along with broadband measurements were employed to estimate the electromagnetic material properties. To demonstrate, flexible polydimethylsiloxane (PDMS) composites are fabricated with barium hexaferrite nanoparticles and complex permittivity and permeability of the composites are quantified under zero magnetic field bias up to 110 GHz. Frequency‐dependent composite permeability is fitted to models describing the ferromagnetic resonance of the self‐biased barium hexaferrite nanoparticles in polydimethylsiloxane, and constituent nanoparticle properties are estimated using the Maxwell–Garnett mixing model. This study paves the way to exploit a wide range of engineered materials in flexible, wearable, and biomedical electronics applications and presents a convenient methodology to extract important broadband electromagnetic properties of nanoparticles for customized electromagnetic applications.


Introduction
3] The applications of flexible materials and composites lie in wearable, implantable, and stretchable electronics due to their ability to conform to underlying shapes and seamlessly integrate with textiles, natural skin, or other biological tissue. [4,5]Flexibility and biocompatibility [6] in implanted electronics can provide alternative therapies for neurological disorders and also enable neural interfaces [7] that can have lifealtering applications in biomedical and bioengineering fields and personalized health monitoring systems. [8]Flexible materials are also widely used in microfluidic applications for chemical and biochemical analysis to test and evaluate properties of liquids, proteins, ions, and other essential ingredients in complex fluids, [9,10] and to create tunable metamaterials at microwave frequencies. [11] key requirement of flexible hybrid electronics is the ability to connect wirelessly to the outside world.This capability is more crucial now due to the increased machine-tomachine connectivity enabled by fifth-generation wireless communications (5G) and the internet of things (IoT).[14] Active components such as transistors, amplifiers, and sensors, as well as passive components such as filters and antennas [15,16] must be either flexible or fabricated on top of flexible engineered materials that are well characterized and optimized to function over broadband frequency domains. [17,18]Conventional semiconductors such as silicon and III-V semiconductors have been adapted to flexible hybrid approaches, and novel 1D/2D materials such as carbon nanofibers, [19] graphene, [20] and Cu/Ag nanoflakes [21] have been utilized to design soft and stretchable microwave electronics on flexible substrates such as paper, polyethylene terephthalate (PET), textile, polydimethylsiloxane (PDMS), Kapton tape, etc.The electromagnetic properties of these flexible materials can widely vary depending on fabrication methods but are critically important and can directly impact the achievable performance of flexible electronic devices and systems.
An important application of flexible materials at microwave frequencies is in the design of electromagnetic absorbers, [22][23][24][25][26][27] which have applications in electromagnetic interference, [28][29][30][31][32][33][34][35] antenna design, [36] IoT, [37] automotive radar, and for potential mitigation of eavesdropping via electromagnetic sidechannels, [38][39][40] with important consequences for hardware security assurance. [41]The electromagnetic absorbance of flexible materials can be engineered by varying composition to obtain desired performance and add increased functionality while minimizing deleterious effects such as loss and dispersion.Selfbiased materials such as barium hexaferrite offer the opportunity for tailoring absorption at specific frequencies, without the need for external magnetic field bias.[44][45] The mechanical and electromagnetic demands on flexible materials are increasing to enable a wide range of new applications in this rapidly growing field.Therefore, it is imperative to develop measurement techniques that can evaluate the broadband electromagnetic properties of flexible materials rapidly and accurately for the design and development of new materials, circuits, and systems for flexible wireless and RF applications.
Due to the intrinsic elastic and conformal nature of flexible materials, determination of electromagnetic properties can present challenges for non-rigid samples.The typical methods and standards used to extract permittivity and permeability values for such materials are the free-space method [46,47] or the transmission and reflection methods. [46,48,49]Implementation of free-space methods can be difficult since it requires samples of specific size and shape (typically >2 m per side for microwave bands) which can be difficult to obtain if the material is too expensive or non-rigid/semisolid in nature.In addition, these materials can easily compress or deform during setup, leading to inaccurate results.The transmission-reflection technique is the most promising method for these materials; [50] however, assess-ment of broadband properties is difficult due to frequency limitations and resonances of measurement systems along with the need for specific form factors to load waveguide and coaxial test cells.To achieve needed measurement accuracy, it is necessary to accurately determine the thickness and/or geometry of flexible materials, which is often a challenge, and coaxial and waveguide measurement cells can give rise to non-uniform airgaps and the material itself can vary spatially for non-uniform materials and composites.
We present an approach to accurately evaluate the permittivity and permeability of flexible composites over the frequency range 0.1-100 GHz based on "flip-chip" measurements exploiting broadband microfabricated transmission lines.In essence, we measure the signal propagation through a CPW transmission line, calibrate out the effects of the probe, and utilize the fringing fields penetrating the attached material to probe the complex permittivity of the composites.To accomplish this, we position the engineered flexible materials on top of a patterned set of coplanar waveguide (CPW) transmission lines (Figure 1), and analyze the frequency-dependent complex scattering (S-) parameters, similar to the approach used by, [51] under zero applied magnetic field bias.We then apply calibration and de-embedding approaches to determine the distributed circuit parameters of these materialloaded transmission lines.We perform simulations in ANSYS HFSS 2D Extractor to determine the frequency-dependent complex permeability and permittivity of the flexible composite samples from the loaded transmission line distributed circuit parameters.We apply this technique to characterize PDMS, in a manner similar to previous reports, [52] but without the need to fabricate metallic conductors directly on the material under test.We also measure barium hexaferrite (BaM)/PDMS composite samples for different BaM nanoparticle concentrations, and further analyze our results using mixing formulae [53,54] to determine the electromagnetic properties (permeability and permittivity) of the constituent nanoparticles (BaM) and the host material (PDMS) separately over the entire measurement frequency range.The broadband frequency measurements up to 110 GHz were carried out using an Anritsu VectorStar MS4647B Vector Network Analyzer (VNA) with Anritsu 3743A mmW Extenders while samples were probed using a Cascade Microtech RF Probe Station.
In Section 2 we describe the fabrication of transmission-line reference and test chips, as well as the BaM/PDMS composite samples.In Section 3 we present results of ANSYS HFSS 2D Extractor simulations that connect the electromagnetic material parameters to the experimentally determined distributed circuit parameters of the material-loaded transmission lines.Section 4 is a detailed description of microwave calibration and deembedding procedures used to determine the transmission line distributed circuit parameters from on-wafer scattering parameter measurements of our material-loaded transmission lines.In Section 5 we determine the frequency-dependent complex permittivity and permeability of the materials under test using the results from Sections 3 and 4 and analyze these composite material parameters using complex mixing formulae to determine the constituent material properties of the BaM nanoparticles along with the PDMS binder.We then apply these results to predict the free-space attenuation for composites with a range of BaM nanoparticle concentrations for electromagnetic shielding.

Reference and Test Substrates
Reference and test dies were fabricated on the same 75 mm diameter fused silica substrates (500 μm thickness) in the same fabrication process to minimize die-to-die variation of device electromagnetic response.The coplanar waveguide (CPW) structures were defined with optical photolithography and fabricated with a Ti adhesion layer 10 nm in thickness and an Au electrode layer 600 nm in thickness by use of electron-beam evaporation.A liftoff process was used to obtain well-defined conductor edges and device cross-sectional structure.Both reference and test die have CPW transmission lines with 50 μm wide center conductors, and 200 μm wide ground planes, with a gap between the signal line and the ground planes of 5 μm.Reference and test chips are shown in Figure 1, along with schematic diagrams of the test chip configuration with the material under test placed on top.
The reference and test chips have different device layouts while keeping the same cross-sectional CPW device geometry.The reference die is used to isolate the parameters of the unloaded CPW transmission lines and perform on-wafer calibrations with all the artifacts necessary for multiline-through-reflect-line (mTRL) calibration [55] : a symmetric short circuit reflect standard, series resistor standard (lumped resistance ≈ 50Ω) and seven other CPW transmission lines with lengths 0.420, 1.000, 1.735, 3.135, 4.595, 7.615, and 9.97 mm.The series resistor is analyzed to define the reference impedance of the TRL calibrations, and subsequently modeled as a lumped circuit element to perform broadband calibrations that are more accurate over a wider frequency range (compared to mTRL calibrations alone) as explained in previous work. [56]he test chips (Figure 1) have eight identical CPW transmission lines with length of 11 mm and identical cross-sectional structure to the CPW devices on the reference wafer.During mea-surements our flexible composite was placed to cover all CPW lines on the test chips from top to bottom while sections of the left and right were open for probe landing (Figure 1).Reported results for PDMS and composite samples were obtained by averaging the distributed circuit parameters over multiple loaded transmission lines.Once a set of calibration chips and test chips were fabricated, they could be re-used multiple times to obtain the material properties of any number of samples, which makes our method fast and cost-effective.The approach eliminates the need to have samples fabricated to specific shapes or sizes with tight dimensions, although rectangular samples or samples with linear edges simplify the analysis.The samples should only be thick enough to contain all electromagnetic fields within the material.Therefore, the uncertainty for the extracted properties would likely increase if the samples were <100 μm in thickness.The size of our material samples was quite small (≈10 mm × 10 mm, with a thickness of ≈1 mm) and smaller or irregularly shaped samples can be measured, provided they lie flat and fit on top of CPW transmission lines on our test chips.

PDMS and Barium Hexaferrites
Barium hexaferrite nanoparticles were commercially obtained from Nanostructures and Amorphous Materials Inc.The PDMS matrix was fabricated by mixing a base and curing agent (SYL-GARD 184 Silicone Elastomer from Sigma Aldrich).The PDMS monomer base was mixed with the curing agent with 10:1 ratio, and the barium hexaferrite samples were prepared by adding 30% and 60% weight ratios of powder to the PDMS mixture and mixing with a spatula prior to being transferred to tape molds.In order to check the homogeneity of the mixing process and resulting composite materials we made two PDMS samples, three 30% (w/w) BaM samples, and one 60% (w/w) BaM sample for measurement, characterization, and comparison.The mold and the composites were placed in a vacuum desiccator for 20 min to outgas any air bubbles.The samples were cured in an oven at 75 °C for 24 h before they were carefully removed from the molds using razor blades.This process allowed us to obtain samples ≈10 mm × 10 mm in area by 1 mm thickness and subsequently placed on top of our test chips for broadband evaluation.

Modeling and Simulation
Material-loaded transmission lines were modeled and analyzed with ANSYS 2D Extractor using the distributed circuit model shown in Figure 2, consisting of the distributed resistance (R), inductance (L), capacitance (C), and conductance (G), all per unit length.These distributed circuit parameters are determined by the conductor cross-sectional geometry, conductivity of the metallic electrodes, substrate material properties, and superstrate material properties.Using a quasi-static analysis, we can separate the contributions of the conductor, substate, and superstrate layers to the total shunt admittance (Y = G + iC) and series impedance (Z = R + iL) of the distributed circuit model of the device under test.
The distributed capacitance and conductance of the materialloaded transmission lines (referred to as the device under test or DUT) depend on the waveguide cross-sectional geometry as well as the dielectric properties of the substrate layer (C sub , G sub ) and the dielectric response of the superstrate layer (C super , G super ), which can be written as follows under quasi-static assumptions: If we assume that the CPW transmission lines are fabricated on a non-magnetic substrate (such as fused silica in our case), then the substrate layer does not contribute to the resistance (R) and inductance (L) per unit length.The total resistance and inductance per unit length of the loaded transmission lines are then determined by the cross-sectional geometry, along with the properties of the metallic conductor layer (L cond , R cond ) and the magnetic response of the superstrate layer (L super , R super ), and under the same quasi-static assumptions can be written as: In the following sections, we describe the details of our 2D finiteelement simulations that link the measured and de-embedded distributed circuit parameters (C, G, L, and R) to the substrate and superstrate material properties (complex permittivity and permeability).We apply these 2D quasi-static simulations to describe the contributions of the different circuit layers shown in Figure 1 to the circuit parameters of the loaded transmission line DUT.The quasi-static analysis assumes single-mode propagation along the transmission line and has been validated up to 110 GHz for the transmission line geometries used here, fabricated on low-loss fused silica dielectric substrates.

Finite Element Models -Conductor Layer
Any current-carrying set of conductors has an associated resistance and inductance per unit length that depend on frequency, the dc conductivity of the metallic electrodes, and the crosssectional geometry of the CPW transmission lines.We used measurements with an optical microscope and profilometer to determine the conductor dimensions (cross-sectional conductor dimensions and conductor thickness) for our CPW transmission lines.These dimensions were used in 2D finite-element simulations to calculate the inductance and resistance per unit length for our CPW transmission lines, once the direct current (dc) conductivity of the metallic conductors was known.The dependencies of the conductor layer inductance and resistance per unit length are given by: where  is the angular frequency,  dc is the dc conductivity, and the cross-sectional dimensions are described in Figure 1.
The dc conductivity was determined through measurements of the dc resistance of the center conductor on the reference chip for lines of different lengths (Section 2.1).The dc resistance per unit length (R l ) was defined by the slope of a linear fit of the dc resistance versus line length data.This dc resistance per unit length was divided by the center conductor cross-sectional area to obtain the dc conductivity: where w c is the center conductor width and t c is the center conductor thickness.We obtained a value of ≈3.64 × 10 7 1/Ωm for  DC for our Au conductors, which is consistent with literature values for the conductivity of gold at 25 °C. [57]Since we co-fabricated our reference and test chips on the same wafer, we assumed that the conductivity and dimensions of the test chip were identical to the reference chip and used the same  DC in our subsequent simulations.We directly validate our simulation results by comparing the calculated frequency-dependent R cond , L cond with the experimentally determined R(), L() for the reference chips, [56] and with the circuit parameters obtained for the bare test chips without any material under test (see Section 4.4).

Finite Element Models -Substrate Layer
The contribution of the substrate layer to the DUT capacitance per unit length was determined using 2D finiteelement simulations with varying substrate permittivity  ′ r,sub ( r,sub =  ′ r,sub − j ′′ r,sub ).For this case, the capacitance per unit length depends only on the real part of the relative permittivity of the substrate and a proportionality constant that depends on cross-sectional geometry: Figure 3a shows the dependence of C sub on the real part of the permittivity  ′ r,sub , plotted as (C DUT -C 0 ) versus ( ′ r,sub − 1), where C 0 is the capacitance per unit length of the entire structure with air as the dielectric.These results were calculated without dielectric loss (  ′′ r,sub = 0); we assume that the same proportionality constant can be used in order to relate any experimentallydetermined G sub to  ′′ r,sub Although not required for the subsequent analysis, the substrate permittivity  r,sub can be determined from measurements of the reference and air-loaded test chips, using the above analysis for the resistance and inductance per unit length along with calculated dependence of the substrate layer capacitance on  ′ r,sub in Figure 3a.From this analysis we obtain values in the absolute range 3.82-3.86for the permittivity of the fused silica substrate from data on both the reference chip and the air-loaded test chips, consistent with literature values. [58]This provides some validation of the use of quasi-static simulations to 110 GHz for the geometries analyzed here.

Finite Element Models -Superstrate Layer (Dielectric Properties)
Once we have calculated the distributed circuit parameters of the conductor and substrate layers, we fix these values in finiteelement simulations and in an analogous manner determine the dependence of the superstrate layer capacitance and conductance (C super ) on the complex superstrate permittivity ( ′ r,super ): Figure 3b shows C super = (C DUT − C sub ) versus ( ′ r,super − 1) from which we can determine the proportionality constant k super .When C super is experimentally determined from measurements, this allows us to determine the real part of the relative permittivity  ′ r,super of the superstrate material at each measured frequency point.The imaginary contribution to the permittivity is calculated from G super using the same proportionality constant: In Section 4, we use this analysis to calculate both the real and the imaginary part of the superstrate permittivity from the experimentally determined C super (), G super ().
We can then analyze the resulting frequency-dependent complex permittivity function using the Cole-Cole function: [59] The real parameters that describe the material response include  r,∞ that represents the permittivity at the high-frequency limit, and three parameters for each discrete relaxation including Δ j , which represents the dielectric increment,  j , which represents the relaxation time of the electromagnetic response, and ∝ j , which represents the broadening parameter.For the results presented here, we use only a single relaxation term (j = 1 in Equation ( 12)).The Cole-Cole function reduces to the Debye equation for ∝ = 1.

Finite Element Models -Superstrate Layer (Magnetic Properties)
An analogous method to that described in the previous section can be applied to calculate the contribution of the superstrate to the overall transmission line inductance per unit length, as a function of the superstrate permeability.We use lossless models to determine the dependence of the inductance on the real part of the superstrate permittivity, shown in Figure 3c, where the contribution to the inductance of the (lossless) conductor layer has been subtracted out.This allows us to determine the dependence of the superstrate contribution to the inductance on the real part of the superstrate permittivity: For this case, we see that the superstrate inductance does not depend linearly on  ′ r,super as was the case for the superstrate capacitance dependence on  ′ r,super .Nonetheless, we can still use Equation (13) with the simulation data in Figure 3c to interpolate a value for the real permeability from the measured superstrate inductance at each frequency point.We then relate the resistance per unit length (divided by the angular frequency ) to the imaginary part of the superstrate permeability using the same proportionality constant: The interpolation of the permeability from the measured R(), L() per unit length is analogous to the process explained in Section 3.4 for the determination of the complex permittivity from the measured C(), G().
Once the complex permeability function  * r () =  ′ r − j ′′ r has been determined from the measured L super , R super , we note that the resulting frequency-dependent complex permeability func-tion for many materials can be accurately described by a sum of terms describing ferromagnetic resonance: [60,61] where the real model parameters are the dc susceptibility  DC,j the ferromagnetic resonance frequency  r,j and the ferromagnetic resonance linewidth Δ r,j .

Reference Wafer Calibration
Multiline TRL and resistor calibrations were carried out using the devices on the reference chip as described in one of our previous reports. [56]This calibration process allowed us to perform a probe-tip calibration for S-parameter measurements of the test chip that establishes a reference plane with a real impedance of 50 ohms at the location of the probe tips (see Figure 4).The multiline TRL calibration also returns the complex propagation constant () for the transmission lines on the reference chip.
Since the reference and test chips are co-fabricated, we can assume the propagation constant of the reference wafer is identical to the propagation constant of the unloaded (in-air) part of the transmission lines on the test wafer.The propagation constant is thus used to de-embed the offset length L for each transmission line, for each port, which allows us to translate the 50-ohm reference plane to the location of the material under test (Figure 4).The offset length L is measured with an optical microscope using the scales alongside CPW transmission lines in the test wafers as shown in Figure 4.The image processing software is initially calibrated using the scale, and then the distance from port to the edge of the sample (L) is measured.This process provides L with a standard deviation of 1.7 μm, which is negligible when compared to the total length of the line (11 mm).The reference chip calibration also allows us to determine the contributions to the inductance and resistance per unit length due to the metallic conductor layer, shown in Figure 5 (labeled as "Reference").We compare the experimentally determined R(), L() with the values calculated based on the analysis of the previous section using the dc conductivity (data labeled as "Calc" in Figure 5), demonstrating excellent agreement over the entire frequency range.

Test Wafer Analysis
Swept-frequency scattering parameter (S-parameters) measurements can be easily converted to transmission parameter matrices (T-parameters) via algebraic transformations that are welldocumented in literature. [62]This formalism of T-parameters is used since a measurement can be easily represented by a cascade of T-matrices that represent different sections of a transmission line.With the calibration procedure outlined in Section 4.1, the scattering parameters of the material-loaded transmission line can be described using the transfer parameter matrices (T-matrices) as follows: [62] where M  represents the measured (2 × 2) transmission matrix for a material-loaded transmission line of length , T  represents the actual transmission matrix for the same line, and Q Zn Zm represents the impedance transformation matrix, describing the change in characteristic impedance at the interface from impedance Z n (of the bare CPW lines) to impedance Z m (of the material-loaded lines), for example.
The transmission matrix for a transmission line of length  can be calculated if the propagation constant () is known for that waveguide segment Here () is the propagation constant defined in terms of the RLCG distributed circuit parameters by The impedance mismatch between two segments is accounted by the impedance transformation matrix that depends on the characteristic impedances of each section (Z m and Z n , one of which can be 50 ohms), and is defined as Here Z n () is the characteristic impedance of the transmission line defined in terms of the RLCG distributed circuit parameters of that section as

Distributed Circuit Parameter Extraction
The reference wafer calibration and test wafer de-embedding process allow us to correct for experimental effects that distort the measurements and determine the scattering parameters of a uniform section of transmission line loaded by a material-undertest at each frequency point.Once the scattering parameters are calibrated and de-embedded, the distributed circuit parameters R, L, C, G are extracted for each frequency point by using a leastsquares-optimization method.This method minimizes the following error function: [63] S where S ij calc is determined at each frequency point using the distributed circuit parameters and the equations in Section 4.2.The error function can be optimized for all four distributed circuit parameters at each frequency point by allowing all four parameters to vary, or if one or two parameters are known they can be held fixed while we allow the remainder to vary to obtain a better estimate for the unknown circuit parameters.The approach described here was applied to test wafer measurements in air and loaded with PDMS and BaM composites (shown in Figure 1) to extract the distributed circuit parameters and estimate the corresponding material properties.

Test Chip Results in Air
The bare test chip is the simplest case to analyze since the superstrate relative permeability and permittivity are known to be 1 + 0i.The inductance and resistance per unit length for the test chip are assumed to be identical to those we determined from the reference chip, and thus during parameter fitting, R and L were fixed to calculated values (shown as "calc" in Figure 5) and capacitance (C) and conductance (G) per unit length were optimized using Equation ( 21) above.The results for the capacitance and conductance are shown in Figure 6, labeled as "air".This value of the capacitance is approximately constant with frequency, and the conductance is approximately zero.This value of capacitance per unit length in air can be related to the permittivity of the substrate material through the simulation data in Figure 3a and gives a substrate permittivity value of 3.85 +/− 0.02 when averaged over the frequency range 0.1-110 GHz, where the uncertainty is estimated from the standard deviation over frequency.This is consistent with literature values for fused silica. [58]

PDMS-Loaded Test Chip Results
The same approach is applied to the test chip loaded with PDMS.In this case, the superstrate material also has a dielectric permittivity greater than 1 that increases the overall device capacitance (C DUT ).Since pure PDMS is assumed to be non-magnetic, it does not contribute to the inductance and resistance per unit length of the PDMS-loaded transmission line.Thus, we can again assume R, L of the loaded test chip are given by those of the reference wafer (shown as "calculated" in Figure 5) and fix them in our optimization process to determine capacitance and conductance per unit length for the PDMS-loaded sample.The resulting capacitance and conductance are shown in Figure 6, labeled as "PDMS".The two PDMS samples (marked as PDMS -1 and PDMS -2 in Figure 6) show similar capacitance and extracted permittivity.The capacitance shows some dispersion with frequency and displays a non-zero value for the conductance.(This capacitance per unit length is used to calculate the permittivity of the PDMS material through the simulations shown in Figure 3b, as discussed in Section 5).

Barium Hexaferrite Composite-Loaded Test Chip Results
For BaM composite-loaded test chips, the analysis is somewhat more complicated than for the bare PDMS samples, since the superstrate will, in general, contribute to all four distributed circuit parameters RLCG.After de-embedding the scattering parameters for the loaded line, we allow all four distributed circuit parameters (RLCG) to vary and optimize Equation ( 21  This procedure yields good estimates for the distributed circuit parameters at lower frequencies but leads to inaccuracies at higher frequencies where the length of the loaded transmission line becomes comparable to an integer number of half wavelengths.To reduce the uncertainty at higher frequencies, we model the extracted CG values with the same frequency dependence we observed with the PDMS sample and use this estimate in Equation ( 21) to optimize RL estimates at each frequency point.We then scale the frequency-dependent estimate for CG as necessary (typically <1%) to ensure that the RL determined from the optimization procedure matches the results for the unloaded transmission line in Figure 5 at the highest measurement frequency, where we expect the effective permeability to approach 1. Fully optimized RL values for different engineered BaM compositions are shown in Figure 5 for the frequency range 100 MHz-110 GHz.The R, L values for BaM composites clearly show the ferromagnetic resonance at ≈45 GHz. [51]Ferromagnetic resonance occurs in BaM samples under zero applied magnetic field due to the presence of uniaxial anisotropy in the crystal structure of these materials.The frequency dependence of the DUT total R, L can be described using contributions from the conductor layer (shown as "calculated" in Figure 5) and results for the superstrate BaM composite, and thus we can adopt a model that accounts for the intrinsic conductor parameters and the FMR response using Equation (15a,b).Once this final RL model is obtained for composites, we hold these fit R, L values fixed, and vary CG to optimize Equation (15).The extracted CG values for various BaM concentrations are shown in Figure 6.The apparent peak in G/ at ≈50 GHz for the higher concentration material is due to inaccuracies in modeling and fitting the ferromagnetic resonance peaks.These FMR fitting errors in R and L manifest as deviations in C and G around the same frequencies, and this behavior is therefore an artifact of the optimization process.

Effective Permittivity from C and G
We next calculate the effective permittivity of the superstrate layer using the experimentally determined capacitance and conductance per unit length.We employ the finite-element simulation results from Section 3 to determine the effective permittivity from the measured quantities.As outlined above, the total capacitance per unit length (C DUT ) is determined and capacitance of the empty test chip (C sub ) is subtracted to determine contribution to the capacitance of the superstrate (Equation ( 1)).The results are then used to interpolate values for the real part of the superstrate permittivity using finite-element simulation data in Figure 3b along with Equation (10).The imaginary part of the superstrate permittivity was also determined using Equation (11) with the assumption that the entire contribution to the measured conductance comes from the superstrate (this corresponds to the assumption that the substrate conductance is negligible).Effective permittivity results for superstrate layers with different BaM compositions are summarized in Figure 7.For pure PDMS material, this frequency-dependent complex permittivity can be fit into a single-term Cole-Cole function outlined in Equation (12).These fits to the PDMS data are plotted in Figure 7, and the corresponding fit parameters are reported in Table 1.These results are consistent with the dielectric permittivity determined in literature, [52] especially considering that the detailed values and frequency dependence can depend on the specific formulation and curing conditions of the PDMS material.The fits to the frequency-dependent permittivity of the composite materials are described in Section 5.4 below.
Table 1.Fit parameters for the medium (PDMS), and particle (BaM) contributions to the effective permittivity and permeability in Figures 7 and 8.The static permittivity for the medium is obtained by combining the parameters in Equation ( 12) for the dielectric increment Δϵ (ϵ r,sϵ r,∞ ), relaxation time ( cc ), broadening parameter ( cc) and the high-frequency permittivity (ϵ ,∞ ).

Effective Permeability from L and R
A similar approach was used to extract the effective permeability of BaM-PDMS composites from L, R values by use of the finite-element simulation results outlined in Section 3. Inductance of the superstrate layer is calculated by taking the difference between measured L DUT (with superstrate) and L cond (in air for empty structure) values as outlined in Equation ( 3).We then use the results of the finite-element simulations in Figure 3c to interpolate a value for the real part of the superstrate permeability ( ′ r,super ) from the differential inductance ΔL = L DUT − L cond at each frequency point according to Equation (13).The imaginary part of the superstrate permeability ( ′′ r,super ) was calculated using Equation (14).Effective permeability for the superstrate layer with different BaM compositions are summarized in Figure 8.

Maxwell-Garnet Mixing Formulae
The Maxwell-Garnet equation provides the effective dielectric permittivity of a mixture of two or more component materials.The derivation uses static and quasi-static arguments and assumes homogeneous inclusions of one or more materials in another homogenous medium. [64]The simplest mixing formula provides the effective permittivity ( eff ) of a composite where spherical inclusions of dielectric permittivity ( i ) are dispersed in a homogenous medium ( e ).
In a parallel form, an equation for effective permeability can also be derived using the same principles.
where f represents the volume fraction of inclusions in the homogeneous medium.We assumed our BaM nanoparticles are spherical homogenous inclusions in a PDMS binder and used Equations ( 22) and ( 23) to estimate the permittivity and permeability of the nanoparticles in the subsequent sections.

Particle Permittivity and Permeability
The effective permeabilities of BaM/PDMS composites determined experimentally can be used to estimate the constituent barium hexaferrite nanoparticle properties using Maxwell-Garnet equation above.Since PDMS and BaM nanoparticles are mixed in 30% and 60% weight ratios, the volume fractions (f) in the composite matrix can be calculated using material density, Complex effective permittivity increases with increasing BaM weight fraction, and the three 30% (w/w) BaM composites and 60% (w/w) BaM composite show good agreement and consistency among samples.Data for the 30% and 60% (by weight) composites were used to estimate the real part of the permittivity for BaM nanoparticles, with an average of 16.65 + 0i over frequency range of 100 MHz-110 GHz, which is reported in Table 1. [65,66]Using this extracted BaM particle permittivity and Cole-Cole model fit for PDMS medium we calculated the effective composite permittivity (ϵ eff, calc ) and compared it against the experimental permittivity data for each composite concentration as outlined in Figure 7a.The calculated BaM permittivity values for composites (dotted lines in Figure 7) show excellent agreement with measured values, which attests to the homogeneity and reproducibility of the composite materials.
The same approach was used to calculate the BaM particle permeability (shown in Figure S1, Supporting Information) from effective permeabilities in Figure 8.We used a permeability value of 1 + 0i for PDMS for calculation and fit the resulting frequencydependent BaM particle permeability to a three-term FMR response (Equation (15a,b)).Our experimental data reveal two types of magnetic resonance related to the domain wall (DW) vibration (at lower frequency) and gyro-magnetic spin or magnetization precession (at higher frequency) that can explain the frequency dispersion of permeability in agreement with prior work. [67]The parameters used to fit the FMR response for BaM nanoparticles in Figure S1 (Supporting Information) are also reported in Table 1.The fit values obtained for particle permeability can then be used to calculate the effective permeability for both composite concentrations (shown as solid lines in Figure 8) and compared against the measured values.These fit values for the composite permeability (μ eff, calc ) for both 30% and 60% composites show excellent agreement with measured values.The tabulated permittivity and permeability values for BaM can be used as initial estimates for further modeling and calculations, demonstrating the value of the approach.

Estimate of Free-Space Attenuation
The result of the above analysis was used to estimate the attenuation that can be achieved by designing PDMS composites with different BaM volume concentrations.Using PDMS and BaM particle parameters listed in Table 1, we calculated the effective permittivity (ϵ eff ) and permeability (μ eff ) for composites with volume fractions of 10%, 20%, 30%, and 40% using Equation (16a,b).These data for ϵ eff and μ eff can be used to estimate the free space propagation constant as follows: [68] k free = i √  eff ⋅  eff (24)   Attenuation constants for the composites were given by the real part of k free , which was converted to units of dB cm −1 .Figure 9 shows the free space attenuation constant for PDMS composites with different BaM volume fractions, which demonstrates the potential for attenuation greater than 10 dB cm −1 for higher volume Estimates for the free-space attenuation constant for engineered BaM/PDMS composites for volume fractions of 10%, 20%, 30%, and 40%.The materials show attenuation >10 dB cm −1 and for 30% or higher volume fractions at ≈45 GHz that makes them excellent candidates for microwave absorber applications at mm-wave frequencies.
fractions at ≈45 GHz.Such composite materials can thus be engineered and applied in microwave absorber designs once their properties are known.As noted, the frequency-dependent material properties for nanoparticles and flexible composites are difficult to measure using conventional methods; this example illustrates the importance of the current material property extraction technique and demonstrates its utility for the design, modeling, and engineering of flexible materials for broadband applications.

Conclusion
We have introduced an experimental de-embedding technique to determine the distributed circuit parameters of material-loaded transmission lines from 100 MHz to 100 GHz.We have applied this technique to determine the broadband permittivity and permeability of flexible engineered materials that display electromagnetic absorption in the 40-60 GHz range.Our measurements of materials with varying concentrations of barium hexaferrite (BaM) nanoparticles in a polydimethylsoloxine (PDMS) matrix also allowed us to determine the broadband permeability and permittivity of the individual constituent particles through the application of Maxwell-Garnett mixing formulae.We applied this approach to demonstrate the design of flexible absorbers with estimates of free space attenuation greater than 10 dB cm −1 that can be achieved using BaM/PDMS composites.We expect this approach to find widespread application for accurately determining the broadband electromagnetic properties of nanoparticles and nanoparticle-based flexible composites over a wide range of frequencies up to 100 GHz.Such engineered materials will be valuable for the wide range of microwave frequency applications envisioned for flexible hybrid electronics.

Figure 1 .
Figure 1.a) Reference chip, air and composite loaded test chips subject to measurement.PDMS composites with 30% and 60% (weight ratio) barium hexaferrite were directly placed on test chips with CPWs for flip chip measurements.The reference chip is used for on-wafer calibration with necessary devices to perform multiline through-reflect-line (mTRL) and series resistor calibration.The test chip has eight identical transmission lines of ≈11 mm.b) Schematic diagrams of the top-view and c) cross-section of the composite (superstrate) loaded transmission line are shown on the right-hand side and bottom of the image.

Figure 2 .
Figure 2. Distributed circuit transmission line model that is used for simulations and calculations, showing frequency-dependent circuit elements corresponding to inductance L, resistance R, capacitance C, and conductance G, all per unit length.The image on the right-hand side depicts the electric field vectors for gold transmission lines on fused silica substrate that are exaggerated for clarity (image not to scale).The fields originate from the signal line and terminate on the two ground planes or on the grounded metallic chuck.

Figure 3 .
Figure 3. ANSYS HFSS 2D Extractor simulation data for a) differential substrate capacitance versus substrate permittivity b) differential superstrate capacitance versus superstrate permittivity c) Differential inductance versus superstrate permeability.The calculated values are normalized as described in the text.This simulation results are used to interpolate properties of materials under test from experimentally determined distributed circuit parameters.

Figure 4 .
Figure 4. Material loaded transmission lines and de-embedding in test wafer a) left side ports b) right side ports.The cables and probes from the vector network analyzer (VNA) are connected to the reference plane during measurement.The TRL calibration procedure translates 50 Ohm reference plane to probe pads.The air-loaded distance (L) of the transmission line is de-embedded to obtain distributed circuit parameters for BaM material loaded CPW regions.

Figure 5 .
Figure 5. Calculated and measured distributed circuit parameters determined for a) inductance per unit length, and b) resistance per unit length from reference and test chips with three 30% BaM samples (marked as BaM 30%-1, -2, and -3), and BaM 60% sample.The conductivity of  = 3.64 × 10 7 1/Ωm is determined from measurements of the dc resistance for reference wafer transmission lines of different lengths.Ferromagnetic resonance behavior is observed around ≈45 GHz in both the inductance and resistance per unit length.

Figure 6 .
Figure 6.Frequency dependence of the a) capacitance per unit length (C DUT ), and b) conductance per unit length (L DUT ) divided by the angular frequency, for the bare test chip (air), two PDMS samples, three 30% BaM samples (marked as BaM 30%-1, -2, and -3), and the BaM 60% sample.These values are obtained by fixing RL and optimizing CG to fit the measured, de-embedded data using Equation (15) as explained in Sections 4.3-4.6.
) at each frequency point.The optimization algorithm needs an initial guess for these values, and we utilize the RLCG values of bare PDMS samples.Adv.Mater.Technol.2023, 8, 2300887 © 2023 The Authors.Advanced Materials Technologies published by Wiley-VCH GmbH.This article has been contributed to by U.S. Government employees and their work is in the public domain in the USA

Figure 7 .
Figure 7. Calculated effective permittivity (ϵ eff ) versus frequency for two PDMS samples, three 30% BaM samples (marked as BaM 30%-1, -2, and -3), and the BaM 60% sample.a) Real part of effective permittivity ( ′ r,eff ) and b) imaginary part of effective permittivity ( ′′ r,eff ) are calculated via interpolation using simulations results, effective capacitance, and conductance from Figure 6.The Cole-Cole model for PDMS and effective permittivity for BaM (determined via Maxwell-Garnet formulae) are used to calculate fits for composites, shown as dotted lines.

Figure 8 .
Figure 8. Calculated effective permeability (μ eff ) versus frequency for three 30% BaM samples, and the BaM 60% sample.a) Real part of effective permittivity ( ′ r,eff ) and b) imaginary part of effective permittivity ( ′′ r,eff ) are calculated via interpolation using ANSYS HFSS 2D Extractor simulations, effective inductance, and resistance.This technique allows us to observe and extract ferromagnetic resonance (FMR) behavior of BaM materials.Fits to the ferromagnetic resonance response are shown in dotted lines for BaM composites.

Figure 9 .
Figure 9. Estimates for the free-space attenuation constant for engineered BaM/PDMS composites for volume fractions of 10%, 20%, 30%, and 40%.The materials show attenuation >10 dB cm −1 and for 30% or higher volume fractions at ≈45 GHz that makes them excellent candidates for microwave absorber applications at mm-wave frequencies.