JOURNAL OF APPLIED PHYSICS 107, 024907 共2010兲

A metal-to-insulator transition in cut-wire-grid metamaterials in theterahertz region
Keisuke Takano,1,a兲 Kyoji Shibuya,1 Koichi Akiyama,2 Takeshi Nagashima,1Fumiaki Miyamaru,3,4 and Masanori Hangyo1
1

Institute of Laser Engineering, Osaka University, 2-6 Yamadaoka, Suita, Osaka 565-0871, Japan
Advanced Technology R&D Center, Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-Honmachi,
Amagasaki, Hyogo 661-8661, Japan3
Department of Physics, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 3908621, Japan
4
PRESTO, Japan Science and Technology Agency, 2-1-1 Katahira, Aoba, Sendai 980-8577, Japan
2

共Received 13 October 2009; accepted 7 December 2009; published online 27 January 2010兲
Metamaterials with metallic wire-gird structures support extremely low frequency plasmons andtheir electromagnetic responses exhibit metallic characteristics described by the Drude model. In
this paper, we have investigated the electromagnetic responses of the wire-grid structures in theterahertz frequency range when periodic and random cuts are introduced in the wires. It is foundexperimentally that the wire-grid structures exhibit a transition from a metallic response of negative
permittivity to an insulators-like response of large positive permittivity at terahertz frequencies
when small cuts are introduced in the wires. It is also revealed from the finite-differencetime-domain simulation that such a transition is accompanied by an enhancement of the electricfields in the cuts, which is applicable to practical technologies. © 2010 American Institute ofPhysics. 关doi:10.1063/1.3284958兴

冉

I. INTRODUCTION␧ = ␧0 1 −The metamaterials of periodic arrays of metallic thinwires 共wire-gird structures兲 are the simplest and most popular structures to design the negative permittivity.1–6 The wiregrid structures behave like diluted metals for the electromagnetic waves polarized parallel to the wires. The effective
plasma frequency is depressed by the artificial structures tothe extremely low frequency region. The electromagnetic
waves polarized parallel to the wires are blocked below theeffective plasma frequency 共cutoff frequency兲 and transmitted above the effective plasma frequency. In the microwave
and terahertz regions, such properties of the metal wire-grids
have been used as polarizers and high-pass filters for a longtime.7–9
In the case of one-dimensional 共1D兲 wire-grid arrays关Fig. 1共a兲兴, the effective permittivity for TE waves 共incident
electric fields parallel to the wires兲 is described by冉␧ = ␧0 1 +冊

1,i␻␧0 pdZw

共1兲

Electronic mail: ktakano@ile.osaka-u.ac.jp.

0021-8979/2010/107共2兲/024907/6/$30.00

共2兲

where ␻p is the plasma frequency and ␶ is the relaxation
time, the effective plasma frequency and relaxation time are
obtained as␻p,eff =冑

1,
␧0 pdL␶eff =L.
R

共3兲

In Refs. 2–4, the authors described that if periodic cuts areinserted in the wires 关Fig. 1共b兲兴, the negative permittivity
metamaterials change into artificial dielectrics. Awai andco-workers10,11 have also proposed this structure as a highpermittivity artificial dielectric metamaterial. Recently, several similar structures are used to fabricate negative refractive index media and metamaterial absorbers.12,13 By introducing the cuts, capacitive impedance 1 / i␻C is added into
the impedance,
(a)

(b)

y

y

x

where Zw = i␻L + R is series of effective total inductive impedance i␻L and resistance R per unit length,2 and ␧0 is thepermittivity of vacuum. Here, the parallel capacitance between the adjacent wires are neglected, and the cross section
area of the unit cell is assumed to be pd, where p and d are
the period and thickness of the wires as shown in Fig. 1. The
effective permittivity is negative below an effective plasma
frequency so that the wire-grid structures exhibit plasmonic
responses. Comparing Eq. 共1兲 to the Drude model,
a兲

冊

␻2p
,
␻共␻ − i␶−1兲

x
w
R

R
L

x
z

L
l

C

d
p

a

x
z

FIG. 1. Schematic diagrams of 共a兲 wire-grid structure and 共b兲 cut-wire-grid
structure. Equivalent circuits are also shown.
107, 024907-1© 2010 American Institute of Physics

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024907-2

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Takano et al.

Zw = i␻L +

1
+ R.
i␻C

共4兲

Equations 共1兲 and 共4兲 predict that the electromagnetic responses of the wire-grid structures exhibit a transition frommetallic one to insulating one below the LC resonance frequency 1 / 2␲冑LC by introducing cuts. Additionally, the transition happens abruptly even if the inserted cut width is verysmall because the inductive impedance decreases with decreasing ␻ and the capacitive impedance increases with ␻−1in the frequency region much lower than the LC resonancefrequency. The permittivity becomes a positive constantvalue at very low frequencies,冉␧ ⬇ ␧0 1 +

冊

C
.
␧0 pd

共5兲

The drastic change of the electromagnetic responses is desirable for an electromagnetic modulation or sensing using
metamaterials.14–16
In this paper, we demonstrate the drastic transitions fromthe metals to insulators in the metamaterials with the cutwire-grid structures experimentally in the terahertz frequency
range. Additionally, the case of random cuts is discussed
from the standpoint of the comparison of the cut-wire-gridmetamaterials with the real materials such as conducting
polymers. The finite-difference time-domain 共FDTD兲 simulations reveal that the electric fields concentrate in the small
cuts at low frequencies. This electric field confinement isuseful for improving the sensitivity of terahertz sensors andthe enhancement of nonlinear optical effects.
II. FABRICATION OF SAMPLES AND TERAHERTZ
TIME-DOMAIN SPECTROSCOPY

Printing using a general-purpose printer 共ALPS MD5500兲 with spot-color ink imitating a gold color 共ALPS
MDC-FMG兲 is a simple and useful method for fabricating
two-dimensional 共2D兲 metallic structures.17 The spot-color
ink exhibits metallic responses in the terahertz frequency region. It is possible to print arbitrary 2D patterns on papers
with the maximum resolution of about 100 ␮m. The ink
thickness is about 2.5 ␮m.
The transmission spectra of the printed 2D patterns were
measured by the terahertz time-domain spectroscopy
共terahertz-TDS兲 system. The transmittance T and phase shift␾ at each frequency are obtained simultaneously by theterahertz-TDS measurements. Assuming the permeability ␮= 1, the optical constants are calculated from the following
relationship between the complex amplitude transmission coefficient 冑Tei␾ and the optical constants in a two-layer system 共ink and paper in our case, see Fig. 2兲,18,19

Air

Layer 1n0n1Layer 2Airn2n3 = n0k3 = k0

k0

k1 = n1k 0

k 2 = n2 k 0

E0+ exp(ik 0 z)

E1+ exp(ik1 z)

E2+ exp(ik 2 z)

E0+ exp(−ik 0 z) E1+ exp(−ik1 z) E2+ exp(−ik 2 z)

E3+ exp(ik 3 z)

r01 t01

r12 t12

r23 t23z=0z = d1z = d1 + d2

FIG. 2. Electromagnetic wave propagation in a two-layer system. E+i , E−i , ni,
and ki 共i = 0, 1, 2, and 3兲 are amplitudes of the electric fields, refractive
index, and wavenumber in each layer. The superscripts + and – denote thepositive and negative directions. tij and rij are the Fresnel transmission andreflection coefficients. The subscript ij denotes the interfaces between thelayers i and j. Equation 共6兲 is derived from the boundary conditions at z
= 0, d1, and d1 + d2 共Ref. 19兲.

conductivity of the ink daubed all over the substrate paper.
The thickness of the paper d2 is 0.07 mm and the refractive
index n2 is 1.5+ i0.05, which is obtained by the terahertzTDS measurement. The responses of the metallic ink are
fitted well to the Drude model 关Eq. 共2兲兴 with ␯p = ␻p / 2␲= 9.1 THz and ␶ = 0.1 ps. The ink behaves like the Drude
metals in the terahertz region. All structures examined in thispaper were fabricated by this method.
III. A TRANSITION FROM METALS TO INSULATORSIN CUT-WIRE-GRID METAMATERIALSThe cut-wire-grid structures were printed on the substrate papers and their transmittance and phase shift spectra
were measured by the terahertz-TDS system. A schematicand photograph are shown in Fig. 1 and the inset of Fig. 4,
respectively. The dimensions of the structures are indicatedin the caption of Fig. 4. The transmittance and phase shift+冑Tei␾ = E+3
E0=

t01t12t23ei共k1d1+k2d2兲−ik3共d1+d2兲
.
1 + r01r12e2ik1d1 + r01r23e2i共k1d1+k2d2兲 + r12r23e2ik2d2
共6兲

Figure 3 shows the real parts of the relative permittivity and

FIG. 3. 共Color online兲 Real parts of the relative permittivity and conductivity of the metallic ink on the paper. Open circles are obtained by theterahertz-TDS measurements. Solid lines are fitted to the Drude model
whose plasma frequency is ␯p = ␻p / 2␲ = 9.1 THz and the relaxation time is␶ = 0.1 ps.

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024907-3

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Takano et al.

(a)

(a)

(b)

(b)

1 mm

FIG. 4. 共Color online兲 Transmittance and phase shift of the cut-wire-girdstructures printed on the substrate papers. Their characteristic dimensions
are the wire period p = 0.4 mm, wire width d = 0.2 mm, cut period l= 1 mm, and cut with w = 0, 0.1, 0.2, and 0.3 mm. The substrate papers havethe thickness of 0.07 mm and the refractive index of 1.5+ i0.05. The ink
thickness is about 2.5 ␮m. The inset photograph shows a typical picture of
the sample with the cut width of 0.1 mm

spectra for the TE waves are shown in Fig. 4. Without thecuts in the wires, it shows a typical wire-grid response.6,17
The dips at around 0.75 THz in the transmission spectra
correspond to the frequency of the first order diffraction fromthe wire period ␯d = c / p, where c is the speed of light invacuum. By introducing the cuts in the wires, stop bands
attributed to the LC resonance appear around 0.2 THz.3 We
observe the drastic increase in the transmittance and thechange from negative values to positive ones in the phaseshift at the frequencies below the stop bands; they are more
remarkable for larger cut widths. Because the capacitance isdominant in the impedance of the series of L, C, and R below
the LC resonance, the electromagnetic waves can transmit
through the open circuit consisted of the capacitance of thecuts.7
Assuming the printed pattern as a uniform layer with the
thickness of 2.5 ␮m in the lateral directions 共layer 1 in Fig.
2兲, the effective permittivity and conductivity are deduced
using Eq. 共6兲. Ikonen et al.4 discussed that how the effective
thickness of the wire-grid structures should be chosen. For
the case of the single-layered wire-grid, the effective thickness is chosen to be the wire diameter 共corresponding to therealistic thickness of the single wire-grid兲 so that a homogeneous medium having the effective permittivity and thickness almost reproduces the transmittance and reflectance ofthe single-wire grid structure. Following this, we use thethickness of ink 2.5 ␮m as layer 1. However, it is noted thatthe effective permittivity is not meaningful for a multilayered
wire-grid.
Figure 5 indicates the real parts of the effective relative
permittivity and conductivity of the cut-wire-grid structures.
It must be noted that these effective values are meaningful

FIG. 5. 共Color online兲 Effective conductivity and relative permittivity deduced from the transmittance and phase shift in Fig. 4. Open circles are theexperimental data. Solid curves in the permittivity spectra are the fitted
curves to Eqs. 共1兲 and 共4兲. The fitting parameters are listed in Table I.

below 0.3 THz where the diffraction waves due to the periodic cuts do not exist. For the cut width w = 0 mm 共no cuts兲,the responses 共␴1 and ␧1兲 are fitted to the Drude model with
␯p = ␻p / 2␲ = 0.99 THz and ␶ = 2 ps. When the cuts are introduced in the wires, the permittivity and conductivity exhibit
the Lorentz-type dispersions described by the following
equation:冉␧ = ␧0 1 +冊

1/␧0 pdL,
1/LC − ␻2 + i␻R/L

共7兲

which is derived by substituting Eq. 共4兲 into Eq. 共1兲. In thevicinity of the LC resonance the permittivity exhibits large
positive and negative values.
Below the LC resonance frequency, the introduction of
the cuts changes the effective permittivity from large negative values to large positive values. This means that the cutsin the wires change the cut-wire structures from metallic toinsulating metamaterials. The conductivity in the low frequency limit ␴共0兲 becomes 0 by the introduction of the cuts关Fig. 5共a兲兴. It reflects that the dc current in the wires isblocked by the cuts, even though the cut width is muchsmaller than the incident wavelength. By fitting the effective
permittivity to Eq. 共7兲 in the frequency range upto 0.4 THz,the effective LCR parameters are estimated 关Fig. 5共b兲兴. The
fitting parameters are listed in Table I. The responses without
TABLE I. Fitting parameters in Fig. 5共b兲.
Cut width
共mm兲

L
共10−7 H / m兲C
共10−18 Fm兲

R
共104 ⍀ m兲

0
0.1
0.2
0.3

0.78
1.13
1.92
2.44

⬎1000
7.67
2.55
1.70

3.66
8.20
6.05
6.38

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024907-4

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Takano et al.

(a)

(b)

1 mm

FIG. 6. 共Color online兲 Effective conductivity and relative permittivity of therandom cut-wire-grid metamaterials. The inset photo is the case of w
= 0.1 mm.

the cuts mean that the capacitance value is assumed to be
infinity. The effective capacitance decreases by two orders ofmagnitude when the small cuts are introduced. The introduction of the capacitive impedance is obviously a dominantfactor for the change of the electromagnetic responses. In
contrast, owing to the small volume of the cuts, effective
inductance and resistance are less sensitive to the cuts. Thus,
the introduction of the cuts can change the electromagnetic
responses below the LC resonance frequency without changing the responses above the LC resonance frequency as discussed in Ref. 3.
IV. RANDOM CUT-WIRE-GRID METAMATERIALS

It is interesting to compare the responses of the cut-wiregird structures to those of real materials. Recently, Lee etal.20 reported an achievement of truly metallic conducting
polymers. The conductivity ␴共␻兲 of their high-quality polyaniline samples is typical of conventional metals, which fits to
the Drude model. However, the most conducting polymers
exhibit a decrease in the conductivity as the frequency decreases in the low frequency region owing to the disorderinduced localization of the charge carries. These polymers
behave like disordered metals near the metal-insulatortransition.21–24 The disorders exist randomly in the materials.
Figure 6 shows the effective conductivity and relative permittivity of the cut-wire-grid structures in which the cuts are
introduced randomly in the wires. The average period of thecuts is 1 mm. The randomness broadens the LC resonanceand the conductivity shows a broad peak. The introducing of
the cuts causes a transition from the metallic to insulating
responses as well as the periodic cases.
A number of reports successfully explained the behaviorof ␴共␻兲 of the conducting polymers using inhomogeneousmodel. One of the disorders in the metallic polymers arises
from the inhomogeneities in the mesoscopic structures. In

the inhomogeneous disorder model, the conductivity is modeled as series of the conductivity in the metallic region anddisordered region. The Drude model describes the conductivity in the metallic regions ␴m共␻兲 in the polymer. In contrast, the conductivity in the inhomogeneous disorders is
modeled as ␴b共␻兲 = ␳−1 + i␻␧b where, ␳ is the resistivity andi␻␧b is the admittance in the inhomogeneous barrier 共insulating兲 regions with the permittivity ␧b. The macroscopic
conductivity of the polymers is represented by ␴−1共␻兲= 关gm␴m共␻兲兴−1 + 关gb␴b共␻兲兴−1, where gm and gb are the geometrical factors.21,22 The conducting polymers are metallic in
the high frequency region because the Drude responses are
dominant. However, in the low frequency region, the polymers become insulators because the admittance of the disorder regions is dominant. Above perspective on the conducting polymers is almost the same as the treatment of the cutwire-grid structures in this paper. Actually, Fig. 6 is very
similar to the optical responses of the conducting polymers
in the references. We believe that the metal-insulator transition observed for the conducting polymer samples is explained partly by the structure change shown here.
V. FDTD SIMULATIONTo characterize the electromagnetic wave propagation in
the cut-wire-grid structures, we performed the FDTD simulations using a commercial software package.25 The simulation model and calculated transmission spectra are shown in
Fig. 7. The ink thickness of our simulation model is thicker
than the experimental samples owing to the limit of our computer power. However, the transmittance shows good agreements with the experiments. For the cut width of w
= 0.1 mm, the electric field distribution on the x-y plane,
which is in the paper and 10 ␮m away from the metallic ink
was calculated 共Fig. 8兲. As the frequency decreases, the electric fields are concentrated in the cuts. Although the wavelength is about 100 times larger than the cut width at 0.03
THz, the transmittance is larger than 0.5 and the electric field
distribution is highly concentrated in the cuts. The field intensity in the cuts is more than 40 times larger than that of
the incident waves. It is inferred that at the frequencies much
lower than the LC resonance, the transmittance and the electric field enhancements become much higher.
The electric field concentration in the low frequency region can be understood by regarding the cuts as parallel-platewaveguides. Because the cutoff frequency does not exist in
the metallic parallel-plate waveguide for the incident waves
polarized perpendicular to the plate, the electromagnetic
waves whose wavelength is much longer than the width ofthe metallic parallel-plates can exist in the waveguide as the
TEM mode.26,27 Thus, the long wavelength terahertz waves
can be squeezed in the cuts as seen in Fig. 8共a兲. Amazingly,
the terahertz waves can be squeezed into the slit even if theslit width is several tens of nanometer.28,29 In Ref. 29, theterahertz waves are focused and strongly enhanced in the␭ / 30 000 width gold slit. Their experiments imply that themetallic response of the wire-grid structures become the insulating one by the introducing the cuts with the width of
only several tens of nanometers.

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024907-5

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Takano et al.

(a)
y z
x

m

1m

l-w

p
Paper

a

l

2

0.

Incident polarization|Ex|=1m
m Metallic ink

FIG. 7. 共Color online兲 共a兲 Unit cell of the FDTD simulation model. We used
the periodic boundary conditions in the x and y directions. The incident
waves are linearly polarized along the x direction and have the amplitude ofunity. The period of the cuts l is 1 mm, wire period p is 0.4 mm, and wire
width a is 0.2 mm. The substrate paper has the refractive index of 1.5+ i0.05 and thickness of 0.07 mm. The metallic ink is defined as a Drude
metal whose plasma frequency and collision time are obtained from Fig. 3.
The thickness of the ink is 0.01 mm. 共b兲 Change of the transmission spectrum of the cut-wire-grid structures when the cut width w is changed from 0
to 0.3 mm.

It is noted that this kind of electric field enhancements is
effective over a wide frequency range from microwave to
optical regions because of the scaling law for Maxwell equations. Miyazaki and Kurokawa30–32 succeeded in squeezing
visible light into the metal-insulator-metal nanocavities
whose thickness and length is several nanometer and several
tens of nanometer, respectively. The fundamental principle oftheir field confinement for the optical frequency range issame as that for the cut-wire-gird structures mentioned here.
(a)

(b)

Metal ink
0.03 THz

0|Ex|

0.22 THz

6.6

(c)

0

1.6

(d)

0.33 THz

0

|Ex|

|Ex|

1.6

0.67 THz

0

|Ex|

1.7

FIG. 8. 共Color online兲 Electric field distributions in the cut wire-gridstructures. The observation areas are placed on the x-y plane in the paperand 10 ␮m away from the metallic ink.

If the cut-wire-grids are scaled down to exhibit the LC resonances at the few hundred nanometers incident wavelengths,
the visible light is concentrated in the few nanometer cuts.
Such electric field confinement in the cut-wire-grid structures
is also important for the spatially selective enhancement of
nonlinear optical effects33,34 or sensing applications for small
amount samples.
VI. CONCLUSION

We demonstrated the sharp transition from the plasmonic
to insulating responses in the metamaterials having the cutwire-grid structures. The wire-grid structures behave as
Drude-like metals for the electromagnetic waves polarized
parallel to the metal wires. By introducing the cuts in thewires, the capacitive impedance drastically changes the electromagnetic responses. Especially in the low frequency region, the effective conductivity and permittivity exhibit atransition from the metallic to insulating responses owing to
the capacitive impedance of the cuts. We believe that thetransitions of the cut-wire-grid metamaterial responses imitate partly those of the real materials such as conducting
polymers.
The drastic spectral changes by the small structural differences are attractive for the terahertz applications such as
the intensity modulation devices. If the cut-wire structures
are made on semiconductor substrates, the terahertz transmittance can be controlled by irradiating the gap with light orapplying voltage.14,15 Additionally, the FDTD simulations reveal that the electric fields are concentrated in the cuts whose
widths are much smaller than the wavelength in the low
frequency region. Recently, some high power terahertz radiation sources are becoming popular.35–37 The field concentration by metallic structures presented in this paper is effective
for further enhancement of the fields to induce nonlinear effects in the terahertz region.
ACKNOWLEDGMENTS

This work has been partly supported by a Grant-in-Aid
for Scientific Research for Priority Area “Strong PhotonMolecule Coupling Fields for Chemical Reactions” 共Grant
No. 20043019兲 from The Ministry of Education, Culture,
Science, and Technology, Japan and a Grant-in-Aid for Scientific Research 共A兲 共Grant No. 20246022兲 from Japan Society of Promotion of Science. F.M. is also supported by the
Murata Science Foundation and the Konica Minolta Imaging
Science Foundation.
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