Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles
In this paper it is shown that the enhancement factor of the local electric field in metal covered ellipsoidal nanoparticles embedded in a dielectric host matrix has two maxima at two different frequencies. The second maximum for the metal covered inclusions with large dielectric core (small metal...
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nasplib_isofts_kiev_ua-123456789-1569842025-02-09T21:23:10Z Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles Пiдсилення локального електричного поля в текстурi кор-оболонка елiпсоїдних наночастинок метал/дiелектрик Ismail, A.A. Gholap, A.V. Abbo, Y.A. In this paper it is shown that the enhancement factor of the local electric field in metal covered ellipsoidal nanoparticles embedded in a dielectric host matrix has two maxima at two different frequencies. The second maximum for the metal covered inclusions with large dielectric core (small metal fraction p) is comparatively large. This maximum strongly depends on the depolarization factor of the core L⁽¹⁾z , keeping that of the shell L⁽²⁾z constant and is less than L⁽¹⁾z . If the frequency of the external radiation approaches the frequency of surface plasmons of a metal, the local field in the particle considerably increases. The importance of maximum value of enhancement factor |A|² of the ellipsoidal inclusion is emphasized in the case where the dielectric core exceeds metal fraction of the inclusion. The results of numerical computations for typical small silver particles are presented graphically. В цiй статтi показано, що коефiцiєнт пiдсилення локального електричного поля в елiпсоїдних частинках з металевим покриттям, вставлених в дiелектричну матрицю, має два максимуми при рiзних частотах. Другий максимум для включень з металiчним покриттям з великим дiелектричним кором (мала фракцiя металу p) є порiвняно великим. Цей максимум сильно залежить вiд коефiцiєнту деполяризацiї кору L⁽¹⁾z , коли L⁽²⁾z для оболонки залишається постiйним i меншим нiж L⁽¹⁾z . Якщо частота зовнiшнього випромiнювання наближається до частоти поверхневих плазмонiв металу, локальне поле в частинцi значно зростає. Наголошується, що максимальне значення коефiцiєнта пiдсилення |A|² елiпсоїдального включення стає особливо важливим у випадку, коли дiелектричний кор перевищує металеву фракцiю включення. Графiчно представленi результати числових обчислень для типових малих срiбних частинок. This work is dedicated to the late Professor V.N. Mal’nev who departed suddenly on January 22, 2015. We greatly acknowledge his invaluable contributions right from problem setting to almost its conclusion. Let his soul rest in peace 2017 Article Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles / A.A. Ismail, A.V. Gholap, Y.A. Abbo // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23401: 1–11. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 42.65.Pc, 42.79.Ta, 78.67.Sc, 78.67.-n DOI:10.5488/CMP.20.23401 arXiv:1706.07271 https://nasplib.isofts.kiev.ua/handle/123456789/156984 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
In this paper it is shown that the enhancement factor of the local electric field in metal covered ellipsoidal
nanoparticles embedded in a dielectric host matrix has two maxima at two different frequencies. The second
maximum for the metal covered inclusions with large dielectric core (small metal fraction p) is comparatively
large. This maximum strongly depends on the depolarization factor of the core L⁽¹⁾z
, keeping that of the shell
L⁽²⁾z
constant and is less than L⁽¹⁾z
. If the frequency of the external radiation approaches the frequency of
surface plasmons of a metal, the local field in the particle considerably increases. The importance of maximum
value of enhancement factor |A|²
of the ellipsoidal inclusion is emphasized in the case where the dielectric core
exceeds metal fraction of the inclusion. The results of numerical computations for typical small silver particles
are presented graphically. |
| format |
Article |
| author |
Ismail, A.A. Gholap, A.V. Abbo, Y.A. |
| spellingShingle |
Ismail, A.A. Gholap, A.V. Abbo, Y.A. Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles Condensed Matter Physics |
| author_facet |
Ismail, A.A. Gholap, A.V. Abbo, Y.A. |
| author_sort |
Ismail, A.A. |
| title |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| title_short |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| title_full |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| title_fullStr |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| title_full_unstemmed |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| title_sort |
enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/156984 |
| citation_txt |
Enhancement of local electric field in core-shell orientation of ellipsoidal metal/dielectric nanoparticles / A.A. Ismail, A.V. Gholap, Y.A. Abbo // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23401: 1–11. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ismailaa enhancementoflocalelectricfieldincoreshellorientationofellipsoidalmetaldielectricnanoparticles AT gholapav enhancementoflocalelectricfieldincoreshellorientationofellipsoidalmetaldielectricnanoparticles AT abboya enhancementoflocalelectricfieldincoreshellorientationofellipsoidalmetaldielectricnanoparticles AT ismailaa pidsilennâlokalʹnogoelektričnogopolâvteksturikorobolonkaelipsoídnihnanočastinokmetaldielektrik AT gholapav pidsilennâlokalʹnogoelektričnogopolâvteksturikorobolonkaelipsoídnihnanočastinokmetaldielektrik AT abboya pidsilennâlokalʹnogoelektričnogopolâvteksturikorobolonkaelipsoídnihnanočastinokmetaldielektrik |
| first_indexed |
2025-11-30T23:12:49Z |
| last_indexed |
2025-11-30T23:12:49Z |
| _version_ |
1850258878686560256 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 2, 23401: 1–11
DOI: 10.5488/CMP.20.23401
http://www.icmp.lviv.ua/journal
Enhancement of local electric field in core-shell
orientation of ellipsoidal metal/dielectric
nanoparticles
∗
A.A. Ismail1, A.V. Gholap1, Y.A. Abbo2
1 Department of Physics, Addis Ababa University, P O Box 1176, Addis Ababa, Ethiopia
2 Department of Physics, Wollega University, P O Box 395, Nekemt, Ethiopia
Received October 20, 2016, in final form January 10, 2017
In this paper it is shown that the enhancement factor of the local electric field in metal covered ellipsoidal
nanoparticles embedded in a dielectric host matrix has two maxima at two different frequencies. The second
maximum for the metal covered inclusions with large dielectric core (small metal fraction p) is comparatively
large. This maximum strongly depends on the depolarization factor of the core L(1)z , keeping that of the shell
L(2)z constant and is less than L(1)z . If the frequency of the external radiation approaches the frequency of
surface plasmons of a metal, the local field in the particle considerably increases. The importance of maximum
value of enhancement factor |A|2 of the ellipsoidal inclusion is emphasized in the case where the dielectric core
exceeds metal fraction of the inclusion. The results of numerical computations for typical small silver particles
are presented graphically.
Key words: enhancement factor, ellipsoidal nanoinclusion, depolarization factor, local field, resonant
frequency
PACS: 42.65.Pc, 42.79.Ta, 78.67.Sc, 78.67.-n
1. Introduction
The enhancement of the local electric field of the incident electromagnetic radiation in the composites
of metal covered nanoparticles with dielectric core is of great importance due to different possible
applications such as surface enhanced Raman spectroscopy [1], metal enhanced florescence [2], quantum
electrodynamics [3, 4], nonlinear optical effect [5], quantum optomechanics [6], optical sensors [7] and
nano-optical tweezers [8]. It is known that the local electric field in the inclusions can be considerably
enhanced if a frequency of the incident radiation is close to the surface plasmon frequency [9]. This
problem was studied in connection with the optically induced bistability [10, 11] and it is accepted that
such an enhancement takes place only on one resonant frequency. It is clear that the nonlinear part of
the dielectric function (DF) is important only if the electric fields are comparable with the inner atomic
fields. At present, such fields may be achieved by laser radiation. Another interesting property of a pure
metal and metal-covered dielectric small particles is an abnormal enhancement of the local field, when
the frequency of the incident electromagnetic wave approaches the surface plasmon frequency of the
metal [12]. The fact that the surface plasmon (SP) strongly depends on size, shape, distribution of metal
nanoparticles as well as on the surrounding dielectric matrix offers an opportunity for manufacturing new
promising nonlinear materials, nanodevices and optical elements. The ellipsoidal shape does represent
the most general geometry suitable for many practical applications: in particular, it allows us to analyze
two important limiting cases, namely the spherical and the cylindrical ones. The existence of a two-peak
∗E-mail: abdulhayi.abdella@aau.edu.et
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
23401-1
https://doi.org/10.5488/CMP.20.23401
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
A.A. Ismail, A.V. Gholap, Y.A. Abbo
value structure of the frequency dependence of the enhancement factor was first presented by Sisay and
Mal’nev [13] for a composite with metal coated spherical nanoinclusions. Under this context, the present
investigation provides a very general conceptual framework, including those specific cases previously
investigated. A detailed theoretical and numerical analysis of the local field enhancement in small metal
covered ellipsoidal inclusions in the electrostatic approximation is the aim of this study. In section 2
and 3, we analyze the distribution of electric potential in a coated ellipsoidal metal nanoparticle when the
incident electric field is parallel to one of the ellipsoid axes (z-axis), and the enhancement factor of local
field inside a metal covered ellipsoidal dielectric core embedded into a dielectric matrix, respectively.
Lastly, in section 4, the results of numerical calculations are illustrated graphically. In the conclusion, we
summarize the main results of the paper.
2. Electric potentials distribution in a coated ellipsoidal metal nanopar-
ticle
The most general smooth particle (the one without edges or corners) of regular shape of an ellipsoidal
coordinates [14], with semiaxes a > b > c [figure 1 (a)], can be obtained by considering the surface
which is specified by
x2
a2 +
y2
b2 +
z2
c2 = 1 (2.1)
without loss of generality, considering a family of curves defined by
f (q) ≡
x2
a2 + q
+
y2
b2 + q
+
z2
c2 + q
− 1 = 0
for q > −c2, f (q) = 0 defines an ellipsoid.
Consider a confocal core-shell ellipsoid shown in figure 1 (b), which can represent a wide range of
shapes from disks to rods. The principal semiaxes are a1, b1, and c1 for the core surface and a2, b2, and
c2 for the outer shell surface. Any confocal ellipsoidal surface can be expressed by
x2
a2
1 + q
+
y2
b2
1 + q
+
z2
c2
1 + q
= 1, (a1 > b1 > c1). (2.2)
This equation, a cubic in q, has three real roots ξ, η, and ζ that define the ellipsoidal coordinates. The
coordinate ξ is normal to the surface. The variables η and ζ are the parameters of confocal hyperboloids
Figure 1. (Color online) a) Ellipsoidal particle. b)Metal coated ellipsoidal particle with internal dielectric
core (c), external shell (s) embedded into a given matrix (m).
23401-2
Enhancement of local field in core-shell orientation of ellipsoidal nanoinclusions
and as such serve to measure the position on any ellipsoid ξ = constant. In other words, each ellipsoidal
surface is defined by a constant ξ. Therefore, ξ = 0 is the equation of the surface of inner ellipsoid and
ξ = t is that of the surface of the outer ellipsoid, where a2
1 + t = a2
2, b2
1 + t = b2
2, c2
1 + t = c2
2 . For a given
(x, y, z), if we assume x > 0, y > 0, z > 0, there is a one to one correspondence between (x, y, z) and
the three largest roots (ξ, η, ζ). This implies that the transformation to rectangular coordinates is obtained
by solving equation (2.2) simultaneously for x, y, z, which shows that an expression for the z will be
given as:
z =
[
(c2
1 + ξ)(c
2
1 + η)(c
2
1 + ζ)
(a2
1 − c2
1)(b
2
1 − c2
1)
]1/2
. (2.3)
We assume that the uniform electrostatic field E0 is directed along the z-axis. Then, the external field
potential can be written in the form
Φ0 = −E0z = −E0
[
(c2
1 + ξ)(c
2
1 + η)(c
2
1 + ζ)
(a2
1 − c2
1)(b
2
1 − c2
1)
]1/2
= −E0F1(ξ)G(η, ζ), (2.4)
where we substitute F1(ξ) = (c2
1 + ξ)
1/2 and G(η, ζ) = {(c2
1 + η)(c
2
1 + ζ)/[(a
2
1 − c2
1)(b
2
1 − c2
1)]}
1/2 from
equation (2.3).
Let us assume an electrostatic approximation in which the wavelength of the incident electromagnetic
wave is much greater than the typical size of the inclusion. The distribution of electric potentials between
the interfaces of an ellipsoidal metal coated nanoparticle can be expressed as: (i) Φc in the dielectric
core, (ii) Φs in the metal cover shell and (iii) Φm in the embedded dielectric matrix. Under the action of
a constant external electric field E0, they can be described by the following expressions [15]
Φc = K1F1(ξ)G(η, ζ), −c2
1 < ξ < 0,
Φs = [K2F1(ξ) + K3F2(ξ)]G(η, ζ), 0 6 ξ < t,
Φm = Φ0 + Φp , t 6 ξ < ∞, (2.5)
which are the solutions of the Laplace’s equation in ellipsoidal coordinates stated as:
∇2
Φi = (η − ζ) f (ξ)
∂
∂ξ
[
f (ξ)
∂Φi
∂ξ
]
+ (ζ − ξ) f (η)
∂
∂η
[
f (η)
∂Φi
∂η
]
+ (ξ − η) f (ζ)
∂
∂ζ
[
f (ζ)
∂Φi
∂ζ
]
= 0, (2.6)
where f (q) = [(a2 + q)(b2 + q)(c2 + q)]1/2, here q stands for ξ, η, ζ in the function of f (ξ), f (η), f (ζ) of
the ellipsoidal coordinates in equation (2.6) above. The subscript “i” indicates the interface potentials in
the dielectric core i = c, metal shell i = s, and in the host matrix i = m, respectively. The potential Φm in
the surrounding medium of equation (2.5) is the sum ofΦ0 and the perturbing potentialΦp of the particle
which is given by
Φp = K4F2(ξ)G(η, ζ), (2.7)
where F2(ξ) = F1(ξ)
∫∞
ξ
dq/(c2
1 + q) f1(q), with the property limξ→∞ F2(ξ) = 0; f1(q) = [(a2
1 + q)(b2
1 +
q)(c2
1 + q)]1/2, and K1, K2, K3, K4 are unknown constants, to be determined by the boundary conditions
specified below. Therefore, the potential Φm, in the surrounding medium can be expressed by putting
equations (2.4) and (2.7) in (2.5) as:
Φm = [−E0F1(ξ) + K4F2(ξ)]G(η, ζ), t 6 ξ < ∞.
The boundary conditions for the potentials can be found from the continuity conditions of the potentials
themselves:
Φc = Φs at ξ = 0,
Φs = Φm at ξ = t, (2.8)
23401-3
A.A. Ismail, A.V. Gholap, Y.A. Abbo
and the normal components of the electric displacement vector:
ε1
∂Φc
∂ξ
= ε2
∂Φs
∂ξ
at ξ = 0,
ε2
∂Φs
∂ξ
= εm
∂Φm
∂ξ
at ξ = t. (2.9)
The unknown coefficients K1, K2, K3, K4 can be determined by substituting expressions of equation (2.5)
in the system of equations (2.8) and (2.9) at the boundaries of dielectric core-metal and metal-host matrix
interfaces, and solving simultaneously, we obtain a system of linear algebraic equations as listed below:
K1 = −
εmε2
Q
E0 , (2.10)
K2 = −
εm
[
(ε1 − ε2)L
(1)
z + ε2
]
Q
E0 , (2.11)
K3 =
a1b1c1εm(ε1 − ε2)
2Q
E0 , (2.12)
K4 = −
a2b2c2
2Q
{
f (ε1 − ε2)
[
L(2)z (ε2 − εm) − ε2
]
− (ε2 − εm)
[
L(1)z (ε1 − ε2) + ε2
]}
E0 , (2.13)
where Q = p∆, and
∆ = ε2
2
[
1 +
L(2)z − L(1)z
p
]
− qε2 + ε1εm . (2.14)
Here, q = ε1[1 − L(1)z /p] + εm{[L
(2)
z − 1]/p + 1}, and p = f L(2)z [L
(2)
z − 1] − L(1)z [L
(2)
z − 1] is a metal
fraction in the inclusion which is expressed by the volume fraction f = a1b1c1/(a2b2c2) of the core into
the whole inclusion, that is, the fraction of the total particle of the volume occupied by the inner ellipsoid.
The variables L(1)z and L(2)z are the geometrical factors for the inner and outer confocal ellipsoids, and, ε1,
ε2, and εm are the dielectric functions (DFs) of the core, metal shell, and the host matrix (the surrounding
medium), respectively. Since we assume that a uniform, parallel electric field E0 is directed along the
z-axis, and is thus along the major semi-axis of the ellipsoid (a), the local field Eloc in the dielectric core
of the inclusion can be obtained with the help of the relation, Eloc = −∇Φ. The corresponding local fields
in each portion between the interfaces are; Ec
loc of the core, Es
loc of the shell and Em
loc of the surrounding
medium of metal coated inclusion. They can be presented with the relation
Ec
loc = −∇Φc = AE0 , (2.15)
Es
loc = −∇Φs = BE0 + CE0 , (2.16)
Em
loc = −∇Φm = E0 + DE0 , (2.17)
the factors that relate the local fields with the external incident electric field between the interfaces are
given as below:
A =
ε2εm
Q
, (2.18)
B =
εm
[
(ε1 − ε2)L
(1)
z + ε2
]
Q
, (2.19)
C = −
a1b1c1εm(ε1 − ε2)
2Q
∞∫
ξ
dq
(c2
1 + q) f1(q)
, (2.20)
D =
a2b2c2
2Q
[
(ε2 − εm)
{
ε2 + (ε1 − ε2)
[
L(1)z − f L(2)z
]}
+ f ε2(ε1 − ε2)
] ∞∫
ξ
dq
(c2
1 + q) f1(q)
. (2.21)
23401-4
Enhancement of local field in core-shell orientation of ellipsoidal nanoinclusions
It is important to remark that the external field is completely controlled by the coefficient K4 (or/and D).
At sufficiently large distances from the particle, the perturbing potential in equation (2.7) is negligible
i.e., when ξ � a2
2, therefore, we require that limξ→∞Φp = 0. We note that at distances r from the origin
which are much greater than the largest semi-axis of the shell a2 to any point on the ellipsoid ξ, then
x2 + y2 + z2 = ξ ' r2; the integral in equation (2.7), that can enter the constant D of expression in
equation (2.21) is approximately:
∞∫
ξ
dξ
(c2
1 + ξ) f1(ξ)
'
∞∫
ξ
dξ
ξ5/2 =
2
3
ξ−3/2,
(
ξ ' r2 � a2
2
)
, (2.22)
and, therefore, the potential Φp is given
Φp ∼
E0 cos θ
r2
a2b2c2
3Q
[
(ε2 − εm)
{
ε2 + (ε1 − ε2)
[
L(1)z − f L(2)z
]}
+ f (ε1 − ε2)ε2
]
, (r � a2), (2.23)
since the potential of ideal dipole is given byΦ = P cos θ/(4πεmr2), we can recognize the equation (2.23)
as the potential of a dipole with the moment
P = 4πεm
a2b2c2
3Q
[
(ε2 − εm)
{
ε2 + (ε1 − ε2)
[
L(1)z − f L(2)z
]}
+ f (ε1 − ε2)ε2
]
. (2.24)
Therefore, this yields the polarizability
αz =
υ
[
(ε2 − εm)
{
ε2 + (ε1 − ε2)
[
L(1)z − f L(2)z
]}
+ f (ε1 − ε2)ε2
]
Q
, (2.25)
where υ = 4πa2b2c2/3 is the volume of the particle, and f = a1b1c1/(a2b2c2) is the volume fraction.
Here, it may be mentioned that, letting q = ξ + t to solve the integral that will be substituted by the
geometrical factors of the depolarization which is given by
Lk
z =
akbkck
2
∞∫
0
dq
(c2
k
+ q) fk(q)
, (k = 1, 2), (2.26)
the expression in equation (2.25) is equivalent to equation (2.21). Therefore, equation (2.21) coincides
with the corresponding result shown in [15] for the polarizability of a coated ellipsoid. The coefficients
A, B, C, D are consistent with the coated sphere that can be verified by putting L(1)z = L(2)z = 1/3,
Q = 2p∆/9, as shown by Sisay and Mal’nev [13].
3. Resonant frequencies and enhancement factor of local field in metal
covered ellipsoidal inclusion
Among equations (2.18)–(2.21), we need only the coefficients A and D that enter the potential of the
local field in the inclusion “core” and the induced dipole moment of the inclusion. Let us consider that
the dielectric function of metal ε2 is chosen to be in Drude form [16],
ε2 = ε∞ −
1
z(z + iγ)
. (3.1)
Here, we introduced dimensionless frequencies z = ω
ωp
, and γ = ν
ωp
(ω and ωp are the frequency of the
incident radiation and the plasma frequency of the metal shell, respectively; ν is the electron collision
frequency). ε∞ is a constant that can be a function of the frequency and depends on the type of a metal.
The real ε′2 and imaginary ε′′2 parts of dielectric function are given by
ε′2 = ε
′
∞ −
1
z2 + γ2 and ε′′2 = ε
′′
∞ +
γ
z(z2 + γ2)
. (3.2)
23401-5
A.A. Ismail, A.V. Gholap, Y.A. Abbo
The dielectric function of the inclusion core ε1, in general case, includes a non-linear part with respect
to the local field.
ε1 = ε10 + χ |E|2, (3.3)
where ε10 is the linear part of DF, χ is the nonlinear Kerr coefficient, |E| is the local field in the core.
For week incident fields |E| � ε10, the local field is presented as in equation (2.15) Ec
loc = AE0. We call
A the enhancement factor, which is in general a complex function. It would be convenient to deal with
the real quantity |A|2, which can be presented as follows
|A|2 =
ε2
m
p2
ε′22 + ε
′′2
2{(
ε′22 − ε
′′2
2
) [
1 + L
(2)
z −L
(1)
z
p
]
− qε′2 + ε1εm
}2
+ ε′′22
{
2ε′2
[
1 + L
(2)
z −L
(1)
z
p
]
− q
}2 . (3.4)
For the sake of simplicity, we ignore the imaginary parts of ε1 and εm.
For an analytic analysis, let us consider an ideal case when a decay of the plasma vibrations is
extremely small γ � 1. In this case, the second term in the denominator of equation (3.4) is proportional
to ε′′22 ∼ γ2 which is very small. Therefore, maximum of the enhancement factor |A|2 corresponds to
zero of the first term in the denominator of (3.4). This condition gives the quadratic equation with respect
to ε′2,
ε′22
[
1 +
L(2)z − L(1)z
p
]
− qε′2 + ε1εm = 0. (3.5)
It has two roots, and we obtain two different resonant frequencies zr. In the real inclusions, γ is not
extremely small but finite. The behavior of |A|2 as a function z in this case can be analyzed only
numerically. The results of this study are presented in the following section.
4. Numerical results and discussion
We start our numerical calculations with the enhancement factor of a pure metal particle |Am |
2.
The numerical values of the dielectric functions of the composite used in this section are taken from
[12, 13, 16]. |Am |
2 can be obtained from equation (2.18) by setting p = 1 and making substitution
ε1 → ε2, and L(1)z = L(2)z = L as well
|Am |
2 =
ε2
m
[L(ε′2 − εm) + εm]
2 + L2ε′′22
, (4.1)
where the subscript “m” indicates pure metal inclusion. In figure 2, we present this quantity versus z,
and one can see that the enhancement factor |Am |
2 in the physically interesting range of parameters
sharply depends on the frequency of an incident electromagnetic wave ω and weakly depends on the
depolarization factor decreasing with L. It can easily be seen from equation (4.1) that |Am |
2 = 1 as L → 0.
The maximum value of |Am |
2 at the constant values used for numerical calculations of the particle and
the host matrix [12] is around 250.
As it is shown in figure 3, by changing εm and L, one can obtain even larger |Am |
2 which is around
500. This means that, at comparatively large applied fields E0 in the vicinity of the corresponding plasma
resonance, it is necessary to consider the nonlinear terms in the dielectric function of equation (3.3).
Further, we will compare it with |A|2 of metal covered inclusions with different dielectric cores. This
quantity is calculated with the help of the enhancement factor of equation (3.4) with typical numerical
values of the dielectric functions of a composite.
Figures 4–7 show the dependence of |A|2 on the depolarization factor of the shell and the core at
different metal fraction (p), respectively. Keeping the minimum value of the depolarization factor of
the core constant, it is observed that the enhancement factor |A|2 weakly depends on the depolarization
factor of the shell decreasing with L(2)z , and the second maximum will appear as shown in figure 4. At
23401-6
Enhancement of local field in core-shell orientation of ellipsoidal nanoinclusions
L1
L2
L3
L4
L5
L6
L7
0.0 0.1 0.2 0.3 0.4 0.5
Z
50
100
150
200
250
¤Am ¤
2
Figure 2. (Color online) The enhancement factor |Am |2 for a small silver particle versus a function of
the dimensionless frequency z at different L (L1 = 0.02; L2 = 0.05; L3 = 0.1, L4 = 0.2, L5 = 0.33,
L6 = 0.5, L7 = 0.8) with parameters of the particle εm = 2.25, ε′∞ = 4.5, ε′′∞ = 0.16, ωp = 1.46 · 1016,
ν = 1.68 · 1014, γ = 1.15 · 10−2.
L=0.8
L=0.5
L=0.33
L=0.2
L=0.1
L=0.05
L=0.02
0.0 0.1 0.2 0.3 0.4 0.5
Z
100
200
300
400
500
600
¤Am ¤
2
Figure 3. (Color online) The enhancement factor |Am |2 for a small silver particle versus a function of the
dimensionless frequency z at different L similar to figure 2 but with εm = 6.
this instant, the extent of |A|2 value for the small core (large p) becomes comparable with the large core
(small p) of the inclusion. When the maximum value of the core depolarization factor is kept at constant,
as seen from figure 5, the second maximum of the |A|2 is dominant but weakly depends on depolarization
factor of the shell, decreasing with L(2)z which in turn leaves the appearance on the first maxima for the
thin metal fraction of the inclusions.
Inspecting graphs in figure 6, one can find that the second maximum of the enhancement factor |A|2
strongly depends on L(1)z suppressing the result of the first maximum that corresponds to a small core
(large p) which leads to its disappearance at L(1)z = 0.8 resulting in the |A|2 value around 650 for a small
metal fraction.
When L(2)z = 0.8 is kept constant, the dependance of the secondmaximum on the depolarization factor
of the core is not important, while the first maximum increases with L(1)z as shown in figure 7. Here, it
should be emphasized that, unlike spherical case [13], ellipsoidal inclusions having large dielectric core
23401-7
A.A. Ismail, A.V. Gholap, Y.A. Abbo
Figure 4. (Color online) The enhancement factor |Ac |2 for a silver ellipsoidal coated nanoparticle versus z
for L(1)z (constant but minimum) < L(2)z (variable), at p = 0.2, 0.5, 0.8. Here and further we use the
following parameters: ε∞ = 4.5, ε1 = 6, εm = 2.25.
Figure 5. (Color online) The enhancement factor |Ac |2 for a silver ellipsoidal coated nanoparticle versus z
for L(1)z (constant but maximum) > L(2)z (variable), at p = 0.2, 0.5, 0.8.
23401-8
Enhancement of local field in core-shell orientation of ellipsoidal nanoinclusions
Figure 6. (Color online) The enhancement factor |Ac |2 for a silver ellipsoidal coated nanoparticle versus z
for L(2)z (constant but minimum) < L(1)z (variable).
Figure 7. (Color online) The enhancement factor |Ac |2 for a silver ellipsoidal coated nanoparticle versus z
for L(2)z (constant but maximum) > L(1)z (variable), at p = 0.2, 0.5, 0.8.
23401-9
A.A. Ismail, A.V. Gholap, Y.A. Abbo
that exceeds the fraction of metal (small p), the maximum value of |A|2 is important depending on the
suitable change of the depolarization factor of the shell and the core. This leads us to speak further about
the composites of metal covered inclusions but not about the composites of dielectric inclusions having
a metal core. Thus, we may presumably say that composites of metal covered inclusions behave in a
manner different from the dielectric inclusions having a metal core.
5. Conclusion
In this paper, we have shown that depolarization factors L(1)z of the core and L(2)z of the shell are
the only factor that determines the magnitude of the enhancement factor. The enhancement factor of
the local field in a metal covered ellipsoidal inclusion with dielectric core in a linear host matrix has
two maxima at two different frequencies that depends on the value of the core L(1)z and the shell L(2)z
depolarization factor. It may be noted that in sphere the maxima are important in inclusions having large
fraction of metal (p = 0.9) that exceeds the fraction of the dielectric core, while in our case, the maxima
are important when the metal fraction (p = 0.2) of ellipsoidal particle is very small.
6. Acknowledgements
This work is dedicated to the late Professor V.N.Mal’nev who departed suddenly on January 22, 2015.
We greatly acknowledge his invaluable contributions right from problem setting to almost its conclusion.
Let his soul rest in peace.
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Enhancement of local field in core-shell orientation of ellipsoidal nanoinclusions
Пiдсилення локального електричного поля в текстурi
кор-оболонка елiпсоїдних наночастинок метал/дiелектрик
A.A. Iзмаiл1, А.В. Голап1, Й.А. Аббо2
1 Фiзичний факультет, унiверситет Аддiс-Абеби, м. Аддiс-Абеба, Ефiопiя
2 Фiзичний факультет, унiверситет Уоллега, P O Box 395, Некемте, Ефiопiя
В цiй статтi показано, що коефiцiєнт пiдсилення локального електричного поля в елiпсоїдних частинках
з металевим покриттям, вставлених в дiелектричну матрицю, має два максимуми при рiзних частотах.
Другий максимум для включень з металiчним покриттям з великим дiелектричним кором (мала фракцiя
металу p) є порiвняно великим. Цей максимум сильно залежить вiд коефiцiєнту деполяризацiї кору L(1)z ,
коли L(2)z для оболонки залишається постiйним i меншим нiж L(1)z . Якщо частота зовнiшнього випромi-
нювання наближається до частоти поверхневих плазмонiв металу, локальне поле в частинцi значно зро-
стає. Наголошується,що максимальне значення коефiцiєнта пiдсилення |A|2 елiпсоїдального включення
стає особливо важливим у випадку, коли дiелектричний кор перевищує металеву фракцiю включення.
Графiчно представленi результати числових обчислень для типових малих срiбних частинок.
Ключовi слова: коефiцiєнт пiдсилення, елiпсоїдальне включення, коефiцiєнт деполяризацiї, локальне
поле, резонансна частота
23401-11
Introduction
Electric potentials distribution in a coated ellipsoidal metal nanoparticle
Resonant frequencies and enhancement factor of local field in metal covered ellipsoidal inclusion
Numerical results and discussion
Conclusion
Acknowledgements
|