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Active metamaterials with negative static dielectric susceptibility
F. Castles and P. S. Grant
arXiv:1704.06575v1 [] 20 Apr 2017
Department of Materials, University of Oxford,
Parks Road, Oxford OX1 3PH, United Kingdom
It is proposed that negative static dielectric susceptibility values may be obtained in materials
that consume energy. Preliminary experimental evidence is presented for a single unit-cell of an
active metamaterial.
It is generally believed that the static dielectric susceptibility of all materials must be
positive; that is, for an isotropic material the scalar static susceptibility must be positive,
and for an anisotropic material each principal value of the tensor static susceptibility must
be positive. This belief is based on both theoretical arguments (e.g., Ref. [1] p. 58), and the
empirical fact that when the static susceptibility is interpreted conventionally and measured
reliably, a positive value is always experimentally obtained [2]. Here we investigate whether,
despite such strong indications to the contrary, it may nevertheless be possible to develop
new materials that exhibit negative static susceptibility.
One may ask whether the concept of metamaterials can be exploited to generate negative
static susceptibility, given previous successes in making metamaterials with unusual properties such as negative refractive index [3]. The answer appears to be no, however, since there
is no reason to suppose that the general thermodynamic arguments presented in Ref. [1]
do not apply also to metamaterials; the arguments concern the macroscopic properties of
a material treated as a homogeneous medium and it is irrelevant whether the material is
inhomogeneous on an intermediate length scale, as is the case for metamaterials, in addition to the always-present inhomogeneities on the atomic length scale. The question of
the electromagnetic properties of metamaterials at zero frequency has also been specifically
addressed in Ref. [4] where it is concluded, using an alternative theoretical argument, that
a negative static dielectric susceptibility is “impossible to realize” [4]. Thus, the concept of
metamaterials does not, in itself, appear to permit negative static susceptibility.
We suggest that the key to achieving negative static susceptibility is to consider active
materials, by which we mean materials that consume energy. Theoretical arguments such
as those presented in Refs. [1, 4] implicitly assume that the medium is passive, and do not
necessarily apply to active materials. It has been well established that active metamaterials
may exhibit wave behavior not possible in their passive counterparts (e.g., Refs. [5–7]).
Here, we apply the concept of active metamaterials to the problem of electrostatic material
The dispersion of the constitutive parameters of metamaterials, both passive and active,
has been investigated in Ref. [8]; when applied to the question of the static electric properties
of a material, such arguments appear to rigorously rule out the possibility of negative static
dielectric permittivity, i.e., of values of the static susceptibility that are less than minus
one. This applies to all materials, passive or active. The question of active materials
with a susceptibility greater than minus one but less than zero is not addressed by such
arguments, and we hypothesize that a negative static susceptibility in this range is possible
in active materials. Our methodology is to attempt to realize such a material experimentally,
exploiting the design freedom afforded by metamaterials.
The challenge is to create a material that polarizes in essentially the opposite direction
to normal under the action of a static electric field. For the present, we focus on creating an
anisotropic material for which one principal component χz of the static susceptibility tensor
is negative, i.e., for which the induced polarization is in the opposite direction to an electric
field applied along the ±z-axis. The general design concept is as follows: each meta-atom
contains (1) a mechanism by which to detect the local electric field, (2) two conductors that
may be charged to create an artificial dipole in the ±z-direction, (3) a mechanism by which
the conductors may be charged by a given amount in response to the detected electric field,
and (4) a means by which to supply the energy necessary to charge the conductors. If the
required components are ‘electrically small’ compared to the artificial dipole, the net dipole
moment of the meta-atom can be dominated by the artificial dipole, which may be chosen
to orient in the opposite direction to normal. On a sufficiently large length scale, many such
unit cells may be considered a homogeneous material that should exhibit a negative χz .
A preliminary implementation of this concept is outlined in Fig. 1. When subject to a
vertical electric field produced by the external test electrodes the two copper plates in the
‘sense block’ are raised to different electric potentials. This potential difference provides
the input to an instrumentation amplifier which, in response, applies an amplified potential
difference across the copper plates in the ‘drive block’. Blocks were implemented using
copper clad epoxy laminate (FR4 with 0.4 mm dielectric and 18 µm copper on each side)
with Kapton tape applied to the top and bottom. The total thickness of the blocks was
measured using electronic calipers to be 0.56 mm.
The effective susceptibility of the meta-atom was determined via capacitance measurements on external test electrodes. Static capacitance values were obtained as follows: with
the bottom external test electrode grounded throughout, the output lead from a dc power
supply was touched to the top external test electrode, raising it to the known potential V (as
indicated on the power supply display) and depositing on it charge Q. With the lead from
the dc power supply no longer touching the top external test electrode, Q was measured
using an electrometer. The capacitance could then be calculated via C ≡ Q/V or, as was
FIG. 1. Schematic of the meta-atom. a, Blocks are composed of two parallel conductive layers,
separated and capped by insulator. b, The unit cell is composed of a sense block and a drive block.
c, Top view of unit cell showing lateral dimensions. d, Side view of unit cell between external test
electrodes showing connections to external amplifier and power supply. Relative vertical dimensions
in a, b, and d are shown ≈ × 100 for clarity.
done to check the linearity of the response and to reduce the random uncertainty of the
measurement, from the gradient of a plot of Q(V ).
Two 200 mm × 200 mm external parallel-plate copper test electrodes were set to a separation of ≈ 0.6 mm and, with no material between the plates, a series of measurements of
Q(V ) was taken, as shown in Fig. 2. The value of the empty capacitance determined from
the gradient of the plot is C0 = (608 ± 9) pF. The drive and sense blocks were then inserted
between the external test electrodes and, with the gain of the instrumentation amplifier
tuned to a suitable value, another series of measurements were taken, as shown in Fig. 2,
producing a capacitance value C = (446 ± 15) pF. The effective susceptibility of the unit
cell is determined via χz ≡ C/C0 − 1, giving a value χz = −0.27 ± 0.03.
meta-atom inserted
V (V)
FIG. 2. The charge Q on the external test electrodes as a function of the potential difference V
applied across them. The gradients of the linear fits provide the capacitance of the empty test
electrodes (air as dielectric), and the capacitance with the meta-atom inserted. The gradient,
and therefore the capacitance, with the meta-atom inserted is less than that for air, indicating a
negative dielectric susceptibility.
While this procedure results in an experimentally determined value of χz that is clearly
negative, the result must be considered preliminary in the respect that it concerns only a single unit cell for which the necessary circuitry and power supply are implemented externally.
Further experiments are being undertaken on multiple unit cells with internal circuitry and
power, as well as those for which more than one component of the static susceptibility tensor
is negative.
Materials with negative χ(0) should exhibit a number of novel properties, such as the
ability to be stably levitated in an electrostatic field.
This work was funded by the Engineering and Physical Sciences Research Council UK
(Grant No. EP/I034548/1).
[1] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media
(Course of Theoretical Physics, Vol. 8), 2nd ed. (Butterworth-Heinemann, Oxford, 1984).
[2] By a conventional interpretation, we mean the susceptibility of a homogeneous and macroscopic
sample of material under the action of a static electric field created by external test electrodes.
[3] R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).
[4] B. Wood and J. B. Pendry, J. Phys.: Condens. Matter 19, 076208 (2007).
[5] F. Auzanneau and R. W. Ziolkowski, IEEE Trans. Antennas Propag. 47, 1330 (1999).
[6] S. A. Tretyakov, Microw. Opt. Technol. Lett. 31, 163 (2001).
[7] S. A. Cummer, in Nonlinear, Tunable and Active Metamaterials, edited by I. V. Shadrivov,
M. Lapine, and Y. S. Kivshar (Springer, 2015) p. 21.
[8] S. A. Tretyakov and S. I. Maslovski, IEEE Antennas Propag. Mag. 49, 37 (2007).