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Supplementary Information
“Reconfigurable Terahertz Metamaterials”
H. Tao1, A. C. Strikwerda2, K. Fan1, W. J. Padilla3, X. Zhang1*, R. D. Averitt2*
1
Department of Mechanical Engineering, Boston University, Brookline, Massachusetts 02446,
USA; 2Department of Physics, Boston University, 590 Commonwealth Avenue, Boston,
Massachusetts 02215, USA; 3Department of Physics, Boston College, 140 Commonwealth
Avenue, Chestnut Hill, Massachusetts 02467, USA
*
e-mail: [email protected] or [email protected]
1. Fabrication and Dimensions of Cantilever Metamaterials
Supplementary Figure 1: Fabrication process flow.
The reconfigurable terahertz metamaterials were fabricated using a surface microfabrication
process, as shown in Supplementary Figure 1. The dimensions are shown in Supplementary
Figure 2. The process started with a 4’’ P type <100> 500 μm thick silicon wafer with LPCVD
deposited 400 nm thick SiNx films on both sides. The SiNx film on the front side serves as the
supporting plate for SRRs and the bottom layer of bimaterial legs, and the SiNx film on the back
side serves as the etching windows for structure release at the last step. The split ring resonators
(SRRs) and the top layer of bimaterial legs were patterned using standard photolithography
methods. A layer of 200-nm-thick Au/Cr film was e-beam evaporated followed by rinsing in
acetone for several minutes. The next step was to pattern the photoresist mask and it was
followed by the removal of unwanted SiNx layer employing reactive ion etching (RIE)
technique. Finally, the back side SiNx etching window was patterned by RIE with photoresist as
dry etching mask, and the Si substrate beneath the pop up metamaterial elements was etched in
KOH solution. The as-fabricated pop up structures were nearly flat with the bending angle under
5 degree. Rapid thermal annealing (RTA) was used to set the orientation of the SRRs at a
specific angle with respect to the substrate, as shown in Supplementary Figure 3.
Supplementary Figure 2: A single pop up SRR. Dimensions: unit cell a: 100μm; width of the supporting
silicon frame b: 12μm; length of the supporting SiNx plate c: 80μm; SRR side length d: 72μm; width of
the bimaterial leg e: 4μm; length of the bimaterial leg l: 80μm; SRR line width w: 8μm; SRR gap
distance g: 4μm.
Supplementary Figure 3: Scanned Electron Microscopy (SEM) photo of the as-fabricated pop up
metamaterials after RTA at 350°.
2. Simulation Details
The simulations were conducted using CST Microwave
Studios and the subsequent parameter extraction was
performed in accordance with [1]. Due to the highly nonCartesian orientation of these structures, the simulations were
performed using the tetrahedral mesh and unit cell boundary
conditions of the frequency solver. The value found in the
CST material library was used for gold (conductivity = 4.09e7
S/m) and the SiN was modeled using a constant permittivity of
7. To accurately simulate the bending of the bimaterial
cantilevers, they were modeled as thin curved strips of
constant length. For example, (see supplemental Figure 4) at a
bending angle of 30 degrees they were modeled as a 30 degree Supplemental Figure 4:
cylindrical arc of radius 432/pi. The SRR was then patterned Simulation unit cell.
in the plane of the bent cantilever, ensuring that it, too, will
have the correct angle. As a result of this method, it should be noted that the angle = 0
simulations were actually angle = .01 degrees. Since the physical thickness of these configurable
structures is poorly defined, a thickness of 100µm was used for parameter extraction regardless
of bending angle. While somewhat arbitrary, this thickness establishes a cubic unit cell which is
thicker than the structure even at large angles. This is particularly important when comparing the
strength of resonance at various bending angles, since the Lorentzian response is highly
dependant upon the thickness used in the extraction procedure.
While not displayed here, the permeability for the purely electric response and the
permittivity for the purely magnetic response display characteristic signatures of spatial
dispersion [2]. For example, spatial dispersion due to the higher frequency electric dipole can be
seen in the 0 degree permeability for the
magnetic resonator. This dipole response will
be covered further in section four.
3. Bianisotropic Response
Figure 2e of the main text presents the
experimentally measured data for the
configuration where the electric resonance is
driven for all angles of the SRRs, while the
magnetic coupling increases as the SRRs pop-up
out of plane. Surprisingly, there is no substantial
change in the transmission as a function of the
angle of the SRRs. The simulations shown in
Supplementary Figure 5 confirm this to be the
case. In essence, for this particular orientation of
the fields with respect to the SRR, an effective
isotropic response is obtained. This is because
of the bianisotropy of the individual SRRs.
Supplementary Figure 5:
Simulations of
bianisotropic configuration (Corresponding to
experimental data shown in Fig. 2e of main text).
Black – 0 degrees, red – 30 degrees, blue – 40
degrees, green – 60 degrees, magenta – 80
degrees.
To understand this somewhat counterintuitive result, we refer to Supplementary Figure
6 which shows a SRR with the axes labeled and two specific field configurations. Supplementary
Figure 6b is the field orientation at normal incidence with the SRRs in the plane of the substrate
while Supplementary Figure 6c shows, for
simplicity, the case where the angle of
incidence in 90 degrees and the SRRs is
remains in the plane of the substrate (this is
analogous to the case where the field remains
fixed and the SRRs are at 90 degrees with
respect to the substrate).
For propagation along the z direction
as shown in Supp. Fig. 6b, the dispersion Supplementary Figure 6: Schematic for effective
relation is [3-5]:
isotropic response from bianisotropy of SRR.
k   
2
2
z
0
(1)
yy
For propagation along the x direction as shown in Supp. Fig. 6c the dispersion relation is:
k   (  -    )
2
2
x
2
yy
zz
0
0
yz
(2)
In these equations, the yy is the electrical permittivity along y with the E oriented along y, zz is
the magnetic permittivity along z with H along z, and yz results from magnetoelectric coupling
and embodies the bianisotropy. Independence of the transmission on the orientation of the SRR
or, in the present discussion, on the angle of incidence will result if kx = kz. Equating (1) and (2)
and rewriting in terms of the associated susceptibilities yields:
 (1  )  
m,zz
e , yy
2
yz
(3)
Finally, in the limit where e,yy >> 1 and in the limit where the susceptibilities are written in
terms of the polarizabilities of the individual elements (i.e. coupling between unit cell is
neglected) our analysis suggests an independence on the orientation if the following expression
holds:
   
m
e
2
em
(4)
This equation just states that the magnetic susceptibility times the electric susceptibility is equal
to minus the square of the magnetoelectric susceptibility. This is known to hold explicitly for
split ring resonators [4] and hence the independent of the orientation of the SRRs that we
experimentally observe in Fig. 2e of the main text is expected and is a consequence of the
bianisotropy.
4. Higher Frequency Dipolar Response
In Figure 2f of the main text, there is no clear resonant response, but there is a change in the
transmission as a function of the cantilever bending. This results from two higher frequency
dipole resonances. These resonances arise from the vertical bars of the SRR structure and the
bending cantilever arms. These resonances can both be seen clearly in Sup. Fig. 7 which shows
the experimental data on the left and corresponding simulations on the right over a frequency
Supplementary Figure 7: The high frequency behavior of the samples (pictured in Sup. Fig. 1) from
measurement (left) and simulation (right). The lower frequency resonance is a result of the vertical bars
of the SRR structure, where as the higher resonance is a result of the cantilever arms.
range which includes these dipolar resonances. The lower and higher frequency resonances from
the figure are due to the SRR and the cantilever bars, respectively. As the bending angle
increases, the SRR dipole response weakens due and eventually disappears as the projection of
the electric field along the dipole decreases. This is in contrast to the dipolar response of the
cantilever arms which is present at every angle. These different responses are a result of the
physical orientation of the dipoles. The SRR bars, which are perfectly straight, rotate relative to
the electric field, decreasing their coupling as the bending angle increases. The cantilever arms,
however, have a fixed base which is always parallel to the electric field. Even at high bending
angles, these cantilevers will have a component parallel to the incident electric field, even though
a significant portion of the bar has bent out of plane. The other orientations not explicitly
mentioned here have similar high frequency behavior, which can be readily, and intuitively,
determined by inspecting the SRR and cantilever orientations in the unit cell with respect to the
electric field orientation and then applying the arguments described here.
5. Linewidth Considerations
The simulations in Fig. 3a of the main text suggest, and subsequent analysis confirms, that the
quality of the resonance increases as its strength decreases. Due to radiative damping between
elements in adjacent unit cells, this correlation between linewidth broadening and oscillator
strength is anticipated. However, the same correlation is not observed in the measured data. This
is because of inhomogeneities in the fabrication (i.e. unit cell to unit cell variations) and in the
bending angle introduced during RTA. A more accurate method of controlling the bending angle,
perhaps through the use of voltage control, may agree better with the simulations presented here.
Supplementary References:
[1] D. R. Smith, et al, Phys. Rev. E 71, 036617 (2005).
[2] R. Liu, et al, Phys. Rev E 76, 026606 (2007).
[3] R. Marques, et al, Phys. Rev B 65, 144440 (2002).
[4] R. Marques, et al, “Metamaterials with Negative Parameters”, Wiley, New Jersey (2008).
[5] X. Chen, et al, Phys. Rev. E, 71, 046610 (2005).