Effect of microscopic disorder on magnetic properties of metamaterials
Maxim V. Gorkunov,
1,2
Sergey A. Gredeskul,
1,3
Ilya V. Shadrivov,
1
and Yuri S. Kivshar
1
1
Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, Australian National University,
Canberra ACT 0200, Australia
2
Institute of Crystallography, Russian Academy of Science, Moscow 119333, Russia
3
Department of Physics, Ben-Gurion University, Beer-Sheva, Israel
共Received 10 October 2005; revised manuscript received 10 March 2006; published 16 May 2006
兲
We analyze the effect of microscopic disorder on the macroscopic properties of composite metamaterials and
study how weak statistically independent fluctuations of the parameters of the structure elements can modify
their collective magnetic response and left-handed properties. We demonstrate that even a weak microscopic
disorder may lead to a substantial modification of the metamaterial magnetic properties, and a 10% deviation
in the parameters of the microscopic resonant elements may lead to a substantial suppression of the wave
propagation in a wide frequency range. A noticeable suppression occurs also if more than 10% of the resonant
magnetic elements possess strongly different properties, and in the latter case the defects can create an addi-
tional weak resonant line. These results are of a key importance for characterizing and optimizing novel
composite metamaterials with the left-handed properties at terahertz and optical frequencies.
DOI: 10.1103/PhysRevE.73.056605 PACS number共s兲: 42.70.Qs, 41.20.Jb, 78.67.Pt
I. INTRODUCTION
Artificially fabricated composite conductive structures for
electromagnetic waves or metamaterials acquire a growing
attention of research during the last years due to their unique
properties of negative magnetic permeability and left-handed
wave propagation. Being composed of three-dimensional ar-
rays of identical conductive elements, the metamaterials
have much in common with conventional optical crystals
scaled to support the propagation of microwave or terahertz
radiation. In contrast to crystals, metamaterials allow tailor-
ing their macroscopic properties by adjusting the type and
geometry of their structural elements. In particular, it is pos-
sible to obtain negative permeability in magnetically reso-
nant metamaterials at the frequencies up to hundreds of tera-
hertz 关1,2兴. This appears to be especially useful for the
practical realization of the negative refraction phenomenon
关3兴.
The simplest method to create a magnetically resonant
metamaterial is to assemble a periodic lattice of the resonant
conductive elements 共RCEs兲, where each element is much
smaller than the wavelength of the propagating electromag-
netic waves, and can be well approximated by a linear LC
contour. A small slit provides the contour with a certain ca-
pacitance while its shape determines self-inductance. As a
result, a resonance of the induced currents and corresponding
magnetization resonance occurs. Remarkably, the resistive
losses in RCEs are low enough to provide the quality factor
of the resonance of the order of 10
3
关1兴.
During the recent years, a number of ideas have been
suggested for achieving better characteristics 关4兴, tunability
关5–7兴, nonlinear wave coupling 关8–11兴 in composite
metamaterials by inserting different types of active and pas-
sive electronic complements into the resonant circuits. One
of the common features of these seemingly different ap-
proaches is to modify the macroscopic properties by the
identical insertions into a microscopic structure of the
metamaterial, assuming technologically ideal fabrication
processes. It was always assumed that all RCEs are identical,
and no analysis of the effect of random deviations or disorder
was carried out. However, one should expect that even weak
fluctuations in the microscopic parameters may become criti-
cal near the magnetic resonance. On the other hand, after a
certain working time, a small amount of RCEs could expe-
rience a breakdown and will operate in a way different from
that of the majority of the elements. Thus, the problem of
disorder appears naturally in the physics of composite
metamaterials.
Recent experiments 关12兴 demonstrated a significant de-
crease of the wave transmission due to defect elements in
one-dimensional metamaterial structures. Low transmission
in a one-dimensional magnetoinductive waveguide with
weak deviations of the element properties was also found in
numerical simulations 关13兴. The first theoretical study of the
effects of disorder in three-dimensional metamaterials has
been made in Ref. 关14兴, where the magnetic susceptibility for
a spatially uniform system 关15兴 has been averaged with re-
spect to random variations of the RCE resonant frequency,
and the resulting change of the frequency dispersion of the
left-handed composite system has been found analytically.
The method used in Ref. 关14兴 is based on a macroscopic
averaging performed prior to a statistical averaging. This is
possible only under the assumption that the resonant fre-
quency is a slowly varying random function of the coordi-
nates, and its correlation length r
c
satisfies the inequality
r
resp
Ⰶ r
c
Ⰶ, 共1兲
where r
resp
is the characteristic length of the local response
共which is usually of the order of a few lattice constants 关15兴兲,
and is the wavelength of the electromagnetic wave propa-
gating in the metamaterial. However, the opposite case,
r
c
Ⰶ r
resp
, 共2兲
appears to be more realistic because even the nearest neigh-
boring RCEs are statistically independent. Then the primary
PHYSICAL REVIEW E 73, 056605 共2006兲
1539-3755/2006/73共5兲/056605共8兲 ©2006 The American Physical Society056605-1
characteristic of the disordered structure is not the average
susceptibility itself but the current distribution, and the mag-
netic properties of the disordered system are determined by a
macroscopic average of the current. If the averaging length is
large with respect to the correlation length, i.e., if the in-
equality 共2兲 is fulfilled, then the macroscopic current is a
self-averaged quantity 关16兴, and it should coincide with its
ensemble average. In this case, the magnetic properties of the
system are described by a statistical mean current.
Below we study systematically the effect of disorder on
the averaged characteristics and permeability of metamateri-
als, and consider two practically important models of the
disordered composite metamaterials, assuming the capaci-
tances C
n
of different RCEs to be random quantities. In the
first model, we assume that the capacitances are completely
uncorrelated, but fluctuations are weak. The second model
corresponds to the small volume density n
˜
of defect RCEs
acting as impurities with substantially different capacitance
C
˜
. The difference can be very strong here covering two prac-
tically important cases of a casual RCE breakdown, C
˜
→ ⬁,
and the absence of some RCEs, C
˜
→ 0. The impurities make
the medium microscopically inhomogeneous. The volume in
which the local response is formed has to contain a large
number of impurities. In accordance with the condition 共2兲,
this implies an additional but not very restrictive condition:
the volume density of impurities should not be extremely
small, i.e., n
˜
Ⰷ
−3
.
The paper is organized as follows. In Sec. II we analyze
the magnetic response of an ideally ordered metamaterial.
We start with reminding briefly the method of permeability
calculation developed in 关15兴, and also describe a theoretical
approach based on the response function of the discrete com-
posite medium, which later allows us to describe the proper-
ties and response of rather general microscopically inhomo-
geneous media. Section III is devoted to the study of effects
of small deviations in each resonant element. In Sec. IV, we
deal with the case of strong but rarefied impurities. Our gen-
eral conclusions are accompanied by some specific examples
calculated numerically for the typical metamaterial param-
eters. Finally, in Sec. V we discuss the quality and reliability
requirements for the electronic components to be used as
RCE insertions in the composite magnetic metamaterials.
II. RESPONSE OF MAGNETIC METAMATERIALS
A. Effective permeability
First we consider a composite metamaterial created by an
ideal three-dimensional lattice of identical RCEs. The RCEs
are placed in the parallel planes normal to the z axis 共see Fig.
1兲. We denote N the total macroscopic number of the lattice
sites, and each site is characterized by the index n. We as-
sume that the wavelength of the electromagnetic radiation is
much larger than both the RCE size and lattice constant,
which allows us to describe the electromagnetic properties of
the metamaterial using the effective magnetic permeability
and effective dielectric permittivity.
To derive the permeability of the particular metamaterial
we apply the procedure developed earlier in Ref. 关15兴.In
brief, the main steps include the following.
共i兲 The magnetization of the metamaterial exposed to a
homogeneous oscillating magnetic field parallel to the z axis
is calculated, taking into account the mutual inductive
coupling of RCEs.
共ii兲 The volume averaging over a unit cell of the overall
microscopic magnetic field yields the macroscopic magnetic
induction.
共iii兲 Excluding the external field allows us to obtain the
relation between the magnetic induction and the macroscopic
magnetic field, that defines the permeability.
Below we present these calculations in more detail.
The external magnetic field H
z
共0兲
oscillating with the fre-
quency
gives rise to an electromotive force in each RCE,
E =i
0
SH
z
共0兲
, where S is the RCE area. On the other hand,
the relation between the external electromotive forces and
the induced currents I
n
is given by the mutual impedance
matrix 关17兴 Z
ˆ
with N
2
elements:
兺
m
Z
nm
I
m
= E
n
, 共3兲
where for a periodic structure Z
nm
=Z共n−m兲. The diagonal
elements of the matrix Z
ˆ
coincide with the RCE self-
impedance:
Z共0兲 =−i
L +
i
C
+ R, 共4兲
while the nondiagonal elements are determined by the mu-
tual inductance:
Z共n − m兲 =−i
M共n − m兲. 共5兲
In an ideally ordered composite, the currents I
n
are the
same, and they can be found as
I
0
=
E
Z
0
, Z
0
⬅
兺
n
Z共n兲. 共6兲
Accordingly, the magnetization of the metamaterial is found
as
M
z
= i
0
nS
2
H
z
共0兲
Z
0
−1
, 共7兲
where n is the volume density of RCEs.
FIG. 1. 共Color online兲 Schematic of two layers of the metama-
terial with random insertions of defect resonant elements 共shown in
green兲.
GORKUNOV et al. PHYSICAL REVIEW E 73, 056605 共2006兲
056605-2
Next, we perform the macroscopic averaging of the mag-
netic field 共external plus induced by RCEs兲. As shown in
关15兴, integrating the field over a unit cell yields a rather
general relation, which is independent of the details of the
material structure:
B
z
=
0
冉
H
z
共0兲
+
2
3
M
z
冊
. 共8兲
Combining Eqs. 共7兲 and 共8兲 with the relation B
z
=
0
共H
z
+ M
z
兲 yields the standard formula of macroscopic electrody-
namics B
z
=
0
zz
H
z
, with the effective permeability of the
form
zz
=
iZ
0
− 共2/3兲
0
nS
2
iZ
0
+ 共1/3兲
0
nS
2
. 共9兲
Note that the structure of this relation resembles the
Clausius-Mossotti formula obtained within the local field
theory approximation. As discussed in 关15兴, the permeability
共9兲 reduces to the Clausius-Mossotti relation in the limit of
infinitesimally small RCEs. However, for typical metamate-
rials, this limit is far from being realistic, and we rely on a
more general expression for permeability defined by Eq. 共9兲.
B. Metamaterial response function
Response function G appears to be another useful charac-
teristic of the metamaterial. Mathematically, it is an inverse
matrix of the mutual impedance matrix G
ˆ
=Z
ˆ
−1
, and it satis-
fies the equation
兺
m
Z共n − m兲G共m − l兲 =
␦
n,l
. 共10兲
This matrix has a clear physical meaning. Indeed, according
to Eq. 共10兲 it is the distribution of currents for the system
with only one RCE 共placed at the origin n=0兲 exposed to the
action of the unitary external electromotive force, i.e., E
n
=
␦
n,0
. Therefore, G共n兲 is nothing but the Green’s function of
the metamaterial.
To find G, we solve Eq. 共10兲 via the Fourier transform in
the reciprocal space. The latter consists of N wave vectors k
within the first Brillouin zone. Let us define the Fourier
transform f
˜
共k兲 of a discrete function f共n兲 as
f
˜
共k兲 =
1
N
兺
n
f共n兲e
−ik·n
so that the inverse transform has the form
f共n兲 =
兺
k
f
˜
共k兲e
ik·n
.
Then the inversion of the difference matrix can be easily
performed,
G共n兲 =
1
N
兺
k
e
ik·n
兺
m
Z共m兲e
−ik·m
. 共11兲
The poles of the Green’s function 关i.e., zeros of the de-
nominator in Eq. 共11兲兴 determine the spectrum of linear
waves that can be excited in the composite metamaterial. In
the magnetostatic approximation, only a part of the spectrum
can be revealed. Conventional relativistic “lightlike” electro-
magnetic waves remain beyond the validity of this approach.
However, as we demonstrate below, small wave vectors and
high group velocities of the “lightlike” waves make their
contribution negligible. The excitations that determine the
Green’s function on a microscopic scale can be well explored
within the magnetostatic approximation. Being first predicted
in Ref. 关18兴, the magnetostatic excitations were observed in
ab initio simulations of a metamaterial sample exposed to an
external magnetic field 关15兴 and soon after they were ex-
plored theoretically in one-, two-, and three-dimensional
metamaterial structures in Refs. 关13,19,20兴. Although the
waves are similar in many aspects to quasistatic excitations
in magnetic substances 共magnetostatic waves and magnons兲,
in metamaterials they are called magnetoinductive waves. In
the following we will use this established terminology.
Until now, the spectra of magnetoinductive waves in
three-dimensional metamaterials were analyzed only in the
nearest and next-nearest neighbor approximations 关13,20兴,
which provide only qualitative information. As will be dis-
cussed below in Sec. III B, in order to obtain quantitatively
reliable results we should take into account hundreds of
neighbors.
After changing the summation over the macroscopic num-
ber of k vectors by the integration over the first Brillouin
zone, B
1
, we obtain
G共n兲 =
1
共2
兲
3
n
冕
B
1
e
ik·n
d
3
k
兺
m
Z共m兲e
−ik·m
. 共12兲
With the help of Eqs. 共4兲 and 共5兲 we can rewrite the denomi-
nator to obtain
G共n兲 =
i
C
共2
兲
3
n
冕
B
1
⍀
2
共k兲e
ik·n
d
3
k
2
− ⍀
2
共k兲 + i
␥
共k兲
. 共13兲
Here the spectrum of magnetoinductive waves, ⍀共k兲,isin
agreement with the results of Ref. 关13兴,
⍀
2
共k兲 =
0
2
1+L
−1
兺
n
M共n兲e
ik·n
, 共14兲
and the damping coefficient is determined as
␥
共k兲 =
⍀
2
共k兲
Q
0
, 共15兲
where
0
=共LC兲
−1/2
and Q =
0
L/R are the RCE resonant fre-
quency and quality factor, respectively.
To isolate the contribution of singular poles, we split the
integral in Eq. 共13兲 into the real and imaginary parts,
G共n兲 =
C
共2
兲
3
n
关A共n兲 + iB共n兲兴, 共16兲
where the imaginary part allows a straightforward numerical
integration,
EFFECT OF MICROSCOPIC DISORDER ON MAGNETIC¼ PHYSICAL REVIEW E 73, 056605 共2006兲
056605-3
B共n兲 =
冕
B
1
⍀
2
共k兲关
2
− ⍀
2
共k兲兴e
ik·n
关
2
− ⍀
2
共k兲兴
2
+
2
␥
2
共k兲
d
3
k, 共17兲
whereas the real part
A共n兲 =
冕
B
1
⍀
2
共k兲
␥
共k兲e
ik·n
关
2
− ⍀
2
共k兲兴
2
+
2
␥
2
共k兲
d
3
k 共18兲
acquires a contribution at the surface S
K
built by the magne-
toinductive wave vectors K共
兲, obeying the relation
⍀关K共
兲兴=
. Since the RCE quality factor is high, the sin-
gular part dominates in Eq. 共18兲. In the vicinity of the surface
S
K
, we can write ⍀共k兲⯝
+k·v共K兲, where v共K兲 stands for
the group velocity. Next, we integrate along the surface nor-
mal reducing the volume integration in Eq. 共18兲 to the sur-
face integral over S
K
,
A共n兲 =
冕
S
K
e
iK·n
d
2
K
v
共K兲
, 共19兲
which is more suitable for numerical calculations.
Analyzing Eqs. 共17兲 and 共19兲, it is easy to conclude that
accounting for the “lightlike” modes would not lead to any
noticeable corrections. These modes are located in the center
of the Brillouin zone near the point k = 0, and their contribu-
tion to the integral B is apparently small. On the other hand,
the light group velocity is about two orders of magnitude
higher than that of the magnetoinductive waves. Therefore,
the “lightlike” part in A is also negligible.
III. WEAK FLUCTUATIONS
A. Magnetic permeability of disordered structures
In order to model the effect of a weak disorder in the
composite media, we assume that the values of the RCE
self-impedances experience random uncorrelated deviations
due to the capacitance fluctuations. This may correspond to
the results of a real fabrication process when the capaci-
tances of different resonators are not identical, and they can
be treated as independent quantities. In this case, the corre-
lation radius coincides with the lattice constant, r
c
=a, and
the inequality 共2兲 is satisfied. Obviously, strong uncertainty
should totally destroy the macroscopic metamaterial re-
sponse. We assume here that the fluctuations are weak and
study their effect on the permeability in the second order of
the perturbation theory with respect to the capacitance fluc-
tuations, using the methods known in the solid state physics
关21兴.
We define the local fluctuations of the RCE impedance as
␦
n
=
1
冉
1
C
n
−
1
C
冊
, 共20兲
where
1
C
⬅
冓
1
C
n
冔
, 共21兲
and the angle brackets stand for the statistical averaging.
To calculate the magnetic permeability of a disordered
metamaterial, we turn again to the composite medium ex-
posed to a homogeneous external radiation. Equation 共3兲
now takes the form
兺
m
Z共n − m兲I
m
+ i
␦
n
I
n
= E. 共22兲
Applying an iterating procedure to Eq. 共22兲, we obtain
I
n
= I
0
− i
兺
m
G共n − m兲
␦
m
I
0
−
兺
m,p
G共n − m兲G共m − p兲
␦
m
␦
p
I
p
.
共23兲
The assumption of weak fluctuations allows us to substitute
I
0
instead of I
p
into the last term of Eq. 共23兲.
The important step in our subsequent analysis is the mac-
roscopic averaging of this equation. The size of the 共macro-
scopic兲 volume of averaging should be small with respect to
the wavelength of the external field. On the other hand, this
size is much larger than the radius of the capacitance corre-
lations. Therefore, from the statistical point of view, the vol-
ume of averaging can be considered infinite. As a result,
instead of the volume averaging the statistical averaging can
be performed 关16兴. Taking into account that 具
␦
n
典=0, and the
capacitances of different RCEs are statistically independent,
具
␦
m
␦
p
典⬀
␦
m,p
we obtain that in the second order of the per-
turbation theory the average current induced in RCEs can be
presented as 关cf. Eq. 共6兲兴
具I
n
典 =
E
Z
eff
, 共24兲
where the effective impedance
Z
eff
= Z
0
+
␦
2
G共0兲共25兲
involves the square of the standard deviation,
␦
2
= 具
␦
n
2
典. 共26兲
Accordingly, the permeability of the weakly disordered
metamaterial takes the form 共9兲 with the impedance Z
0
being
replaced by the effective impedance Z
eff
.
We note that the fluctuations contribute to both imaginary
and real parts of Z
eff
. The imaginary part of the correction is
determined by B共0兲 and it affects mainly the real part of the
permeability. The real part of the correction leads to an in-
crease of Im共
zz
兲, and it enhances the effective dissipation.
Remarkably, this occurs even when RCE losses are absent,
and the energy is dissipated without conversion into heating,
i.e., this dissipation mechanism is analogous to the Landau
damping. The nature of this nonheating dissipation becomes
clear if we note that it is determined by the term A共0兲 that
combines the contributions from all magnetostatic waves ex-
cited at the given frequency
. This suggests that the inci-
dent radiation experiences scattering on microscopic fluctua-
tions. The similar effect arising in electromagnetic media
without positional order 共see, e.g., Ref. 关22兴 and references
therein兲 is known as scattering losses or Raleigh scattering.
In our case, a very small correlation radius suppresses the
scattering into “lightlike” modes, but the scattering into mag-
netoinductive waves remains strong.
GORKUNOV et al. PHYSICAL REVIEW E 73, 056605 共2006兲
056605-4
B. Weak disorder in a typical metamaterial
As a specific example of the application of our theory, we
calculate the averaged magnetic permittivity numerically for
typical metamaterial parameters with a weak disorder. We
assume circular RCEs with radius r
0
=2 mm, wire thickness
l=0.1 mm, which leads to the self-inductance L = 8.36 nHn
共see Ref. 关17兴兲. To obtain RCEs with the resonant frequency
0
=6
⫻10
9
rad/ s 共
0
=3 GHz兲, we take C =0.34 pF. The
lattice constants are a = 2.1r
0
in the plane and b = r
0
in the z
direction. The RCE quality factor Q can reach the values of
10
3
关1兴. However, we expect that the insertion of diodes or
other electronic components can lower this value to Q=10
2
.
First, we calculate the linear spectrum of magnetoinduc-
tive waves from Eq. 共14兲 and find a strong evidence of the
long-range interaction effects: the lattice sum converges
rather slow. In particular, in order to get an accuracy of a few
percent, we have to expand the summation radius to at least
ten lattice constants. A further increase of the summation
limits would be unjustified, since the sum should be calcu-
lated over the distances much smaller than the radiation
wavelength. We believe that the problem of the exact calcu-
lation of the spectrum of magnetoinductive waves in the
three-dimensional case needs a separate detailed consider-
ation. On the other hand, we observe that the maximum un-
certainty in the spectrum takes place along specific directions
perpendicular to the edges of the Brillouin zone. Therefore,
the resulting error in the integrals determining the discrete
Green’s function is extremely small.
Evaluating numerically the integrals A共0兲 and B共0兲 ac-
cording to Eqs. 共17兲 and 共19兲, we obtain the effective imped-
ance and corresponding magnetic permeability. The fre-
quency dispersion of the permeability
zz
is presented in Fig.
2 for the unperturbed metamaterial and for several values of
the standard deviation. Apparently, already 10% uncertainty
changes dramatically the permeability frequency dispersion
near the resonance. The effect is most pronounced for the
frequencies below the resonance. In a wide range, the imagi-
nary part of
becomes comparable with the real part, and a
weakly disordered nondissipative medium acts as strongly
dissipative. In the range of negative
above the resonance,
the losses are also considerably higher than those in a perfect
metamaterial. As a result, the frequency range appropriate for
the negative refraction shrinks.
IV. RAREFIED STRONG DEFECTS
A. Concentration expansion
In this section we consider a metamaterial with a small
amount of randomly distributed defective RCEs that differ
strongly from the regular RCEs, and therefore can be treated
as impurities. We assume that the dimensionless concentra-
tion of such impurities c= n
˜
/n is low, i.e., c Ⰶ 1. As before,
we assume that the deviations from the structure parameters
appear due to a difference in the RCE capacitances, and the
capacitance of the impurity, C
˜
, can even become very large
corresponding to a casual breakdown. We neglect any corre-
lation between the impurities, but assume that two impurities
cannot be placed on the same site. The applicability condi-
tion, n
˜
Ⰷ
−3
, defines the lower limit for the impurity concen-
tration, c
min
⬃10
−3
for a typical metamaterial. Above this
limit, we can calculate statistically averaged current and con-
struct its concentration expansion using standard techniques
关16,23兴.
In the system with impurities, the microscopic current dis-
tribution is substantially inhomogeneous. We are interested
in the normalized averaged value, defined as
具I典 =
1
N
兺
n
I
n
, 共27兲
where the summation is performed over a volume that in-
cludes a macroscopic number of impurities. The concentra-
tion expansion of the averaged current can be written in the
following form 关16,23兴:
具I典 = I
0
+ c
兺
p
1
关I
1
共p
1
兲 − I
0
兴
+
c
2
2
兺
p
1
⫽p
2
关I
2
共p
1
,p
2
兲 − I
1
共p
1
兲 − I
1
共p
2
兲 + I
0
兴 + ... ,
共28兲
where
FIG. 2. Real 共a兲 and imaginary 共b兲 parts of the magnetic perme-
ability of metamaterial without fluctuations and with weak capaci-
tance fluctuations,
␦
0
C, marked on the plots.
EFFECT OF MICROSCOPIC DISORDER ON MAGNETIC¼ PHYSICAL REVIEW E 73, 056605 共2006兲
056605-5
