Supplementary Material
Microwave Metamaterials with Fused Deposition 3D Printing of Highly Conductive
Filaments
Yangbo Xie1*, Shengrong Ye2*, Christopher Reyes2, Paeng Sithikong1, Bogdan Popa3, Benjamin
Wiley2††, and Steven A. Cummer1†
1
Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina
27708, USA
2
Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
3
Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109,
USA
* These authors contributed equally to this work
†
Corresponding authors:
[email protected]
[email protected]
1. The description of the simulation method used for extracting the effective material properties of
the three-dimensional unit cells
1.1 Simulation setup
The simulation is performed with the Radio Frequency (RF) module of a commercial Finite Element
Method (FEM)-based solver COMSOL Multiphysics. The setup of the simulation is shown in Fig. S1. In
order to generate the transverse electromagnetic (TEM) mode, the upper and lower boundaries are set as
perfect electric conductor (PEC) and the left and right boundaries are set as perfect magnetic conductor
(PMC). The front the back ports are set as scattering boundary condition with the front one being excited
with an electric field polarized along z-direction.
FIG. S1 Simulation setup for extracting the effective electromagnetic material properties of a metamaterial
unit cell.
S-parameters (S11 and S21) can be extracted with the following formulas:
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S11
 (E

S21
total
z
 Ezinc )dAf
E
inc
z
E

E
dAf
total
z
inc
z
dAb
dAf
Where E zinc is the defined incident electric field on the front port, E ztotal is the total electric field, Af is
the area of the front port and Ab is the area of the back port. The other parameters (refractive index,
impedance, permittivity, and permeability) can be retrieved using standard methods described in [20].
1.2 Comparison of the retrieved parameters of a single unit cell with those of 5 identical unit cells
In our experiment, 5 identical unit cells were placed inside a TEM waveguide to mimic the simulation setup
described in the previous section (1.1). We thus would like to verify numerically how close the retrieved
parameters with 5 unit cells represent those with a single unit cell. The setup of the simulation is shown in
Fig. S2, where the boundary conditions are the same as those in Fig. S1.
FIG. S2 Simulation setup with 5 identical metamaterial unit cells.
The retrieved permittivity and permeability between a single unit cell and those with 5 identical unit cells
are shown in Fig. S3. Their excellent agreement indicates that measurements with 5 identical unit cells in
a microstrip waveguide is a valid way to mimic the non-existing PEC-PMC waveguide used in an ideal
single unit cell retrieval simulation.
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FIG. S3 The comparison of retrieved permittivity and permeability between a single unit cell and those with 5
identical unit cells.
2. The numerical method of extracting the capacitances of the unit cells
The capacitance of a unit cell reflects the amount of charge a unit cell can hold under external excitation of
electric fields. In simple cases, such as for a unit cell consisting of only parallel plates with a wire in between
(so called “I-beam” metamaterial), the F-factor (or the away-from-resonance electric susceptibility) of the
Cd 2
unit cell is roughly proportional to its capacitance: F 
, where C is the capacitance of the two
 r  0Vtot
parallel plates, d is the separation between them,  r is the relative permittivity of the host medium of the
I-beam and Vtot is the total volume of the unit cell [1]. For more complicated 3D geometries, the relation
between capacitance and F-factor is less straightforward, but the capacitance value still can serve as a good
indicator of the strength of the electric responses of the unit cells to the excitation field and thus provides a
helpful guidance for designing the geometries of the 3D metamaterial particles.
FIG. S4 Simulation setup for extracting the capacitance of a metamaterial unit cell.
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The numerical study of the capacitance is performed with the ‘Electric Currents’ module in COMSOL
Multiphysics Finite-Element package. As shown in Fig. S4, the outer sphere boundary is set to be ‘Electric
Insolation’ and a sinusoidal time-varying voltage is applied to the upper and the lower boundaries of the
unit cell. By extracting the effective impedance between the upper and the lower boundaries, the
capacitance can be calculated as Ceffective 
im( Z 1 )

A two-wire capacitor is used to verify this numerical method. The analytic estimation of the capacitor is
Ccal  2.5 1013 F , calculated using the formula Ctwo  wire 
 0
l , where the wire separation
d
ln( )
r
d  28 mm , wire radius r  1 mm , and the wire length l  30 mm (a computer model is shown in Fig.
S5). The simulated capacitance is Csimu  2.72 10
13
F , agreeing well with the analytic estimation.
FIG. S5 Simulation setup for extracting the capacitance of a two-wire capacitor.
3. The comparison of the influence of S11 and S21 on the retrieved impedance and permittivity
In this numerical study, we investigate the influences from S11 and S21 on the retrieved impedance and
permittivity. We will demonstrate that S11 has much large influence in the parameter retrieval process, and
the deviation of retrieved refractive index/impedance and permittivity/permeability is likely to be caused
by the small measurement error in S11.
We design our numerical tests as follows:
First, we create a new hybrid ‘experimental’ S-parameter dataset that consists of the simulated S11 and the
measured S21. Fig. S7 shows the results of the retrieved refractive index/impedance and
permittivity/permeability when this new hybrid S-parameter dataset is used for the retrieval. Obvious
improvement in agreement is found with the hybrid ‘experimental’ dataset, compared to the original results
(Fig. S6, which is the same as Fig. 4(e) in the main text).
Secondly, we create another new hybrid ‘experimental’ S-parameter dataset that consists of the simulated
S21 and the measured S11. Figure S8 shows the results of the retrieved refractive index/impedance and
permittivity/permeability when this new hybrid S-parameter dataset is used for the retrieval. No significant
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improvement in agreement is found with the hybrid ‘experimental’ dataset, compared to the original results
in Fig. S6.
The above study demonstrates that S11 has much large influence in the parameter retrieval process, and the
deviation of retrieved refractive index/impedance and permittivity/permeability is likely caused by the
small measurement error in S11.
FIG. S6 The comparison of the parameters between the simulations and measurements. Both S11 and S21 are
directly obtained from experiments. This plot is the same as that in Fig. 4(e) in the main text and is shown here
for the convenience of comparison.
FIG. S7 The comparison of the parameters between the simulations and the results from a new hybrid
‘experimental’ dataset that consists of the simulated S11 and the measured S21. Significantly improved
agreement over the results in Fig. S6 is found between the retrieved refractive index/impedance and
permittivity/permeability.
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FIG. S8 The comparison of the parameters between the simulations and the results from a new hybrid
‘experimental’ dataset that consists of the simulated S21 and the measured S11. The improvement in agreement
over the results in Fig. S6 is not as significant as the case shown in Fig. S7.
4. The description of the employed highly conductive filament and the setting of the 3D printer
Electrifi Filament is a non-hazardous, proprietary metal-polymer composite that consists primarily of a
biodegradable polyester and copper. It has a density of 2.4-2.5 g/cm3 and melts at around 60 °C. The 1.75
mm Electrifi filament was used to fabricate the microwave metamaterials on a Prusa i3 desktop FDM 3D
printer (HICTOP Technology, China). The print temperature was set at 140 °C and the build plate set at
room temperature. The microwave metamaterials were printed with a 0.4 mm nozzle on a polystyrene
substrate at a print speed of 15 mm/s with a 110% infill. The Cura LulzBot Edition was used as the slicing
software, in which Z-offset may be varied in the setting according to the thickness of the polystyrene
substrate.
5. Effects of small voids in the metamaterial unit cells
Because of the resolution limits of the employed 3D printer, several small voids were found in the sample
shown in Fig. 4b in the main text. These sub-millimeter size voids do not affect the properties of the
metamaterial performance. We performed the following simulations to verify that.
A high-permittivity metamaterial unit cell with 5 voids (1 mm cube) on each side is simulated and compared
with a unit cell without any void. The permittivity and permeability retrieved of the unit cell with voids and
the unit cell without voids are plotted in Fig. S9. Their overlapping data proves that their properties are the
same.
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FIG. S9 Comparison of the simulated results of a high-permittivity metamaterial unit cell with voids and one
without voids.
6. The effects of the height-to-thickness ratio on the effective material properties of the highpermittivity unit cell
We simulated the effects of the height-to-thickness ratio (HTR) on the effective material properties of the
high-permittivity unit cell. The unit cell demonstrated in the main text is a 3-cm cube with a HTR = 1.0. In
the simulation, we fixed the thickness of the unit cell to be 3 cm while sweeping the height of the unit cell
from 2.4 cm to 3.6 cm (corresponding to a HTR ranging from 0.8 to 1.2) and retrieved their permittivity
and permeability.
The results are shown in Fig. S10. While the real parts of the permeability of all five unit cells are closely
distributed in a small range between 0.6 to 0.75 (below 1 GHz), the real parts of the permittivity have span
a wider range of values. The lowest HTR (= 0.8) gives a low permittivity value around 11.5 (below 1 GHz)
and the highest HTR (= 1.2) gives a high permittivity value around 20.8 (below 1 GHz). These results may
indicate a way to tune the permittivity of the unit cell by changing the HTR value.
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FIG. S10 Comparison of the simulated parameters of five high-permittivity metamaterial unit cells with
different height-to-thickness ratio (HTR). (A) Geometries of these five unit cells. (B) The real parts of
permittivity (left) and the real parts of the permeability (right). The imaginary parts for both the permittivity
and the permeability are close to zero and are not shown in the plots.
Supplementary References
[1] Popa, B. I. Simplified Design Techniques for Physically Realizable Metamaterials and Applications (Doctoral
dissertation, Duke University, 2007).
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