Optical and transport properties of spheroidal metal nanoparticles with account for the surface effect

Abstract

The kinetic approach is applied to develop the Drude-Sommerfeld model for
studying of the optical and electrical transport properties of spheroidal
metallic nanoparticles, when the free electron path is much greater than the
particle size. For the nanoparticles of an oblate or a prolate spheroidal shape
there has been found the dependence of the dielectric function and the electric
conductivity on a number of factors, including the frequency, the particle
radius, the spheroidal aspect ratio and the orientation of the electric field
with respect to the particle axes. The oscillations of the real and imaginary
parts of the dielectric permeability have been found with increasing of
particle size at some fixed frequencies or with frequency increasing at some
fixed radius of a nanoparticle. The results obtained in kinetic approach are
compared with the known from the classical model.

Full text

Page 1

arXiv:1107.2866v1 [physics.optics] 14 Jul 2011
Optical and transport properties of spheroidal metal nanoparticles with account for
the surface effect
Nicolas I. Grigorchuk
∗
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
14-b Metrologichna Str., Kyiv-143, Ukraine, 03680
Petro M. Tomchuk
†
Institute for Physics, National Academy of Sciences of Ukraine,
46, Nauky Ave., Kyiv-28, Ukraine, 03680
(Dated: July 15, 2011)
The kinetic approach is applied to develop the Drude-Sommerfeld model for studying of the optical
and electrical transport properties of spheroidal metallic nanoparticles, when the free electron path
is much greater than the particle size. For the nanoparticles of an oblate or a prolate spheroidal
shape there has been found the dependence of the dielectric function and the electric conductivity
on a number of factors, including the frequency, the particle radius, the spheroidal aspect ratio and
the orientation of the electric field with respect to the particle axes. The oscillations of the real
and imaginary parts of the dielectric permeability have been found with increasing of particle size
at some fixed frequencies or with frequency increasing at some fixed radius of a nanoparticle. The
results obtained in kinetic approach are compared with the known from the classical model.
PACS numbers: 78.67.Bf; 68.49.Jk; 73.23.-b; 78.67.-n; 52.25.Os
I. INTRODUCTION
Understanding how light interacts with matter at the
nanometre scale is a fundamental problem in nanopho-
tonics and optoelectronics. The variety of applications
are based on the surface-bound optical excitations in
metallic nanoparticles (MNs). They include plasmonic
nanomaterials, optical nanosensors, biomarkers, inte-
grated circuits and sub-wavelength waveguides, molec-
ular imaging, metamaterials with a negative refractive
index, optical antennas and so on (see, e.g., Refs.[1–7]).
The optical and transport properties of the MNs
have been investigated for a long time and the results
have been presented quite completely in numbers of
reviews8–11and monographs,12–16especially for MNs of
a spherical shape.
Dielectric function is the most important factor for
the design and optomization of plasmon nanometer-sized
structures. However, this function for MNs differs from
an ideal bulk metal. The difference depends on many fac-
tors, such as the size and shape of the MNs and the sur-
rounding media.18–22That is why the dielectric function
of MNs has been under comprehensive study for many
years.12–17The size effect becomes more significant when
the size d of MN is comparable with the electron mean
free path l. The ratio between d and l is a very impor-
tant physical characteristic, which eventually defines the
mechanism of an electron relaxation rate inside the MN.
As a rule, the cases d ≫ l and/or d ≪ l are studied. The
former case refers to the so called diffusive electron dy-
∗email: ngrigor@bitp.kiev.ua
†email: ptomchuk@iop.kiev.ua
namics and the latter one to the ballistic electron dynam-
ics. The diffusive case has been studied in details, since
it enables to use the Mie theory for a uniform media23or
the Maxwell-Garnett theory24for a composite media in
the calculations of the optical properties of the MN.
In particular, when d ≫ l, the following expression for
the dielectric permeability which results from the Drude-
Sommerfeld theory16,25has been often used:
ǫ(ω) = ǫ′(ω) + iǫ′′(ω) = 1 −
ω2
pl
ν2+ ω2+ iν
ω
ω2
pl
ν2+ ω2. (1)
Here ν is the collision frequency inside the particle bulk,
ωpl =
?4πne2/m is the frequency of plasma electrons
and mass, correspondingly, and n is the electron concen-
tration. The imaginary part of the dielectric permeabil-
ity ǫ′′(ω) is connected with the high-frequency (optical)
conductivity by the well known relation:
oscillations in the metal, e and m are the electron charge
σ(ω) =
ω
4πǫ′′(ω) =
ν
4π
ω2
pl
ν2+ ω2.(2)
For sufficiently low frequencies (ω → 0), one gets the
expression σ0= ne2/mν describing the statical conduc-
tivity.
In the case when the sizes of the MN are less than the
mean electronic free path in the particle, the mechanism
of electrons scattering is changed, and the surface of the
particle starts to play a dominate role. This effect will be
called further as the surface or boundary effect. Strictly
speaking, in this case neither Mie nor Drude-Sommerfeld
theory can be applied and an another approach ought to
be elaborated to present the optical properties of MNs.
As it was shown,26,27a clear size dependence of relaxation
dynamics is observed in experiments. This is a strong

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2
indication of an efficient electron-surface phonon interac-
tion in this regime. But the experimental results18,28–33
are often analyzed within the frame of the named theo-
ries, or to simplify the problem, the Eqs. (1), (2) have
been used with formal replacements15,33–35
ν →3
4
υF
R,
orν → AυF
S
4V,
(3)
where υF is the electron velocity at the Fermi surface, R
refers the particle radius, V and S refer the volume and
the surface area of the spherical particle, and A is a coef-
ficient obtained by fitting the calculations to the exper-
imental data. In other words, the collision frequency in
the particle volume is replaced by some effective collision
frequency. But such a replacement can be applied for the
MNs of a spherical shape only. If the geometry of MN
differs from the spherical one, and the condition d ≪ l
holds for at least one of the particle directions, then the
optical conductivity becomes the tensor quantity21and
the formal replacement similar to the one presented by
Eq. (3) can not be used any more. It is necessary to
look more closely at the effect of the particle boundaries
on the optical properties. For a theoretical study it is
convenient to choose the particle of a spheroidal shape.
That is because the results obtained for the particles of
such a shape can be easily extended to the particles of
other shapes by means of the formal transformation of
the spheroidal axes.
Thus, there arises a problem in the case d ≪ l, how to
calculate the values of ǫ or σ, when the mentioned above
classical models can not be used.
The present work is devoted to the elaboration of the
new approach, allowing to calculate the real and imag-
inary parts of the dielectric permeability for MNs of a
spheroidal shape in the case when the inequality d ≪ l
is hold.
Since both the real and the imaginary parts of ǫ(ω)
govern the numerous properties of the MNs, the neces-
sity of the detailed study of the boundary effect on the
dielectric permeability has become evident.
The rest of the paper is organized as follows.
kinetic approach to the problem is presented in Sec. II.
Section III contains the study of the conductivity in the
spheroidal MNs. In Sec. IV, we consider the limit cases
of the problem and the size effects. Sec. V is devoted
to the discussion of the obtained results, and Sec. VI
contains the conclusions.
The
II. KINETIC EQUATION METHOD
To account for the effect of MN boundaries on the op-
tical and electrical transport properties of the MN, we
will apply the kinetic equations approach. The advan-
tages of this approach is that the obtained results can be
applied to strongly anisotropic spheroidal (needle-like or
disk-shaped) MNs, but in the case of MNs of spherical
shape it transforms to the well known results16,25, like
the ones given by Eqs. (1), (2). Thus, it permits one to
study the effect of the particle shape on the measured
physical values. Secondly, the kinetic method enables to
investigate the MNs with sizes as more or less then the
electron mean-free path l. But, there exists the lower
limit of the applicability of this method in the small ra-
dius limit, when the particle size is comparable with the
de Broglie wavelength of the electron, and the quantiza-
tion of the electron spectrum start to play an essential
role.36–38Practically, it is around a radius of more or less
2 nm. Beyond the electronic Fermi distribution function,
the influence of quantum effects on the optical conduc-
tivity can manifest itself in the quantization of electron
pulses and angular momentums. It is important at low
temperatures, when the distance between successive en-
ergy levels is much higher than kBT.
Let us consider the single MN which is irradiated by
electromagnetic wave, whose electric field is given as
E = E0exp[i(kr − ωt)].(4)
Here E0 is the amplitude of an electric field, ω is its
frequency, k is the wave vector, and r and t describe the
spatial coordinates and time.
We will assume that the electromagnetic wave length
is far above the particle size. If one chooses the coordi-
nate origin in the center of the particle, then the above
mentioned assumption is written as
kr ≪ 1. (5)
The inequality (5) implies that the E-field of the elec-
tromagnetic wave can be considered as spatially uniform
on scales of the order of a particle size. This means that
the field represented by Eq. (4) induces inside of the MN
the electric field varying in time, but uniform in space.
The amplitude of such a field is connected with E0 by
the relation39
Ej
in(ω) =
Ej
0(0,ω)
1 + Lj[ǫ(ω) − 1], (6)
where Ljare depolarization factors in the j-th direction
(in the principal axes of an ellipsoid). The explicit ex-
pressions of Lj for a single MN of an ellipsoidal shape
can be found elsewhere (see, e.g., Refs. [40,41]).
The field Einhas an effect on the equilibrium electron
velocities distribution and, thus, determines the appear-
ance of a nonequilibrium addition f1(r,v,t) to the Fermi
distribution function f0(ε). Here ε = mυ2/2 is the ki-
netic energy of electron, and υ = |v| refers the electron
velocity. As is well known,42the equilibrium function
f0(ε) does not give any input to the current. With the
account for both the time dependence of Eq. (4) and
the inequality (5), the distribution function of electrons,
which generates the field Ein, can be written as
f(r,v,t) = f0(ε)+f1(r,v,t) ≡ f0(ε)+f1(r,v)eiωt. (7)

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3
The function f1(r,v) can be found as a solution of the
linearized Boltzmann’s equation
(ν − iω)f1(r,v) + v∂f1
In Eq. (8) we have assumed that the collision integral
(∂f1/∂t)col= −f1/τ is evaluated in the relaxation time
approximation (τ = 1/ν). What is more, the function
f1(r,v) ought to satisfy the boundary conditions as well.
These conditions may be chosen from the character of
electrons reflection from the inner walls of the MN. We
will take, as is usually done, the assumption of a diffusive
electron scattering by the boundary of MN. Placing the
origin of the coordinates at the center of the particle, we
can present this boundary conditions in the form
∂r+ eEinv∂f0(ε)
∂ε
= 0. (8)
f(r,v)|S= 0,
vn< 0, (9)
where vnis the velocity of the component normal to the
particle surface.
Alongside with diffusive, the mirror boundary condi-
tions at the nanoparticle surface were examined in liter-
ature for electron scattering (see, e.g., Refs. [42, 43]). In
this case, each electron is reflected from the surface at
the same angle at which it falls to the surface. In diffuse
reflection, the electron is reflected from the surface at any
angle. In order that the mirror mechanism be dominant,
the surface must be perfectly smooth in the atomic scale,
since the degree of reflectivity of the boundary essentially
depends on its smoothness. Practically, for a nonplanar
border such smoothness is extremely difficult to achieve.
As it was shown43, the mirror boundary conditions give
a small corrections to the results obtained with the ac-
count of only the diffusive electron reflections. There-
fore, we chose more realistic boundary conditions given
by Eq. (9).
It is comparatively easy to solve Eq. (8) and to satisfy
the boundary conditions of Eq. (9), if one passes to the
transformed coordinate system, where an ellipsoid with
semiaxes R1,R2,R3transforms into a sphere of radius R
with the same total volume:
xj=Rj
Rx′
Similar transformation should be made for the electron
velocities as well: υj = υ′
one finds
f1(r,v,t) = −e∂f0
j,R = (R1R2R3)1/3. (10)
jRj/R. Then solving Eq. (8),
∂εvEin1 − exp[−(ν − iω) t′(r′,v′)]
ν − iω
,
(11)
where the characteristic of Eq. (8) can be presented as
t′=
1
v
′2
?
r′v′+
?
(R2− r
′2)v
′2+ (r′v′)2
?
.(12)
The characteristic curve of Eq. (12) depends only on the
absolute value of R and does not depend on the direc-
tion of R. The radius vector R determines the starting
position of an electron at the moment t′= 0.
Generally speaking, the presence of the surface changes
both the current and the field distributions. It is reason-
able to point out here that, though the electric field still
remains homogeneous inside of the MN (in accordance
with the Eq. (5)), the distribution function f1is the co-
ordinate dependent in any case due to the necessity to
meet the boundary condition given by Eq. (9).
Performing the Fourier transformation of Eq. (11), one
can calculate the density of a high-frequency current in-
duced by the electromagnetic wave of Eq. (4) inside the
MN, via the expression
j(r,ω) = 2e
?m
2π?
?3? ? ?
vf1(r,v,ω)d3υ. (13)
Let us introduce the tensor of the complex conductivity
αβ(r,ω) using the relationship
σc
jα(r,ω) =
3
?
β=1
σc
αβ(r,ω)Eβ
in. (14)
Then in accordance with the both Eq. (11) and Eq. (13),
the components of this tensor can be presented in the
form
σc
αβ(r,ω) = 2e
?m
1 − e−(ν−iω)t′(r′,υ′)
ν − iω
2π?
?3? ? ?
υα
?
??
−eυβ∂f0
∂ε
×
?
d3υ.(15)
Before the detailed study of the role of the particle sur-
face and the size effects, which will be displayed below,
it is worth to note here the following. The surface effect
on the conducting phenomenon is described in Eq. (15)
by means of the characteristic t′(r′,v′). It accounts for
the restrictions imposed on the electron movement by a
nanoparticle surfaces. As one can see from Eq. (12), the
value of t′is of the order of t′∼ R/υF, where υF is the
Fermi velocity. This implies that the value reciprocal to
t′will correspond to the vibration frequency between the
particle walls. Hence, the inequality νt′≫ 1 indicates
that the electron collision frequency inside the volume of
MN would significantly exceed the one for an electron
collision with the surface of MN. If pointed inequality is
satisfied, it is possible to direct t′→ ∞ and the expo-
nent in Eq. (15) can be neglected. Then, we obtain a
standard expression for the dielectric permeability, such
as given in Eq. (2). To ensure in that, it is necessary to
pass in Eq. (15) to the integration over υ in the spherical
coordinate system
?
∞
?
−∞
?
d3υ →
2π
?
0
dϕ
π
?
0
sinθdθ
∞
?
0
υ2dυ,
with the use of the following formulas
?∞
0
υ4δ(υ2− υ2
F)dυ =υ3
F
2, (16)

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4
n =8π
3
?mυF
2π?
?3
, (17)
and take into account that the energy derivative of f0in
zero approximation in the small ratio of kBT/εF, can be
replaced by
∂f0
∂ε
≈ −δ(ε − εF),(18)
as well as the fact that only the diagonal terms with
υα= υβ= υ are retained after integration over all angles.
At the end of this section, we would like to pay atten-
tion on the next two important circumstances. i) Though
the inner field Einis spatially uniform, the distribution
function f1(r,v,t), and, consequently, the density of the
current j(r,ω) are depended on the coordinates. This
dependence is imposed by the boundary conditions of
Eq. (8). ii) The density of the current and the compo-
nents of the conductivity tensor have the physical sense
only if they are averaged over the particle volume. For
instance, it is easy to ensure that the energy absorbed by
a single MN is determined either by an averaged density
of the current ?j? or by an averaged tensor of the com-
plex conductivity ?σc
Eq. (14)).
αβ(r,ω)? (with the account of the
III. CONDUCTIVITY OF SPHEROIDAL
METALLIC NANOPARTICLE
Let us average over coordinates the components of
the conductivity complex tensor represented by Eq. (15).
The necessity of such an averaging arises from the fact
that the power absorbed by a single MN is caused by the
conductivity averaged over the volume of MN. Then, in
view of Eq. (16), one gets the expression
?σc
αβ(r,ω)? =4e2
×?d3υ υαυβδ(υ2− υ2
where the angle brackets denotes the averaging. Firstly,
we can fulfil the integration in Eq. (19) over all electron
coordinates. In accordance with Ref. [39], one can find
that
m
?m
2π?
?3
?
1
ν−iω
1 − e(ν−iω)t′?
?
d3r
V
F)
, (19)
1
V
?
d3r′?
1 − e−(ν−iω)t′(r′,υ′)?
=3
4Ψ(q), (20)
where the following notations have been used
Ψ(q) =4
3−2
q+4
q3−4
q2
?
1 +1
q
?
e−q, (21)
q =2R
υ′(ν − iω) ≡ q1− iq2. (22)
Further, we will take into account only the diagonal com-
ponents of the conductivity tensor. Based on Eq. (20),
we can rewrite Eq. (19) as
?σc
αα(r,ω)? =
3e2m2
(2π?)3
?
1
ν − iω
αδ(υ2− υ2
×d3υ υ2
F)Ψ(q). (23)
Let us restrict ourselves only with the single MN of
spheroidal shape. To calculate the residual integral, we
pass to the spherical coordinate system with a z-axis di-
rected along the rotation axis of the spheroid (as we have
done it in the previous section), and take into account
that the components of an electron velocity parallel (υ?)
and perpendicular (υ⊥) to this axis are defined as
υ?= υz= υ cosθ,υ⊥=
?
υ2
x+ υ2
y= υ sinθ, (24)
correspondingly, where
υ2
(x
y)= υ2sin2θ ·
?cos2ϕ
sin2ϕ
?
.
After integration in Eq. (23) over the azimuthal angle ϕ
and over all electron velocities, in view of Eqs. (16), (19),
we obtain for the ?- and ⊥-components of the conductiv-
ity tensor the expressions
σc
?≡ ?σc
zz(r,ω)? =9
π/2
?
0
4
ne2
m
1
ν − iω
×
sinθcos2θ Ψ(θ) dθ|υ=υF,(25)
and
σc
⊥≡ ?σc
xx(r,ω)? = ?σc
yy(r,ω)?
π/2
?
0
=
9
8
ne2
m
1
ν − iω
sin3θ Ψ(θ) dθ|υ=υF. (26)
The subscript υ = υF means that the electron velocity
in the final expressions should be taken on the Fermi
surface. The Ψ-function in Eqs. (25), (26) varies now
with the angle θ, because q (see Eq. (22)) for a spheroidal
particle becomes dependent on the angle θ, and can be
determined as
q =
2
υF
ν − iω
?cos2θ
R2
?
+sin2θ
R2
⊥
≡ q(θ), (27)
where R?and R⊥are the semiaxes of the spheroid. Such
form for q follows from the form of ”deformed” electron
velocity, which enters into Eq. (22), and for a spheroid is
υ′= υR
??sinθ
R⊥
?2
+
?cosθ
R?
?2
≡ υ′(θ),(28)
where the velocity components υ?and υ⊥, presented by
Eq. (24), were used. The velocity υ′does not depend on

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5
the particle radius R, but only on the spheroid aspect
ratio. In the case of a spherical particle R?= R⊥≡ R,
and υ′= υ.
There is a well known common relation42between the
tensor components of the complex dielectric permeability
and the components of a complex conductivity tensor
?ǫαβ(r,ω)? = δαβ+ i4π
ω?σc
αβ(r,ω)?. (29)
If one separates the real and the imaginary parts in the
expressions for both the complex tensors of the dielectric
permeability and the conductivity, i.e., presents
?ǫc
αβ(r,ω)? = ǫ′
αβ(ω) + iǫ′′
αβ(ω), (30)
and
?σc
αβ(r,ω)? = σ′
αβ(ω) + iσ′′
αβ(ω),(31)
then in a correspondence with Eq. (29), one obtains the
next two relations
ǫ′
αβ(ω) = δαβ(ω) −4π
ωσ′′
αβ(ω),(32)
and
ǫ′′
αβ(ω) =4π
ωσ′
αβ(ω). (33)
Finally, using Eqs. (25), (26) and Eq. (31), one gets for
a spheroidal particle
σ′
(?
⊥)(ω) =9
4
ne2
mℜ



1
ν − iω
π/2
?
0
?sinθ cos2θ
1
2sin3θ
?
Ψ(θ) dθ


(34)

υ=υF
and
σ′′
(?
⊥)(ω) =9
4
ne2
mℑ



1
ν − iω
π/2
?
0
?sinθcos2θ
1
2sin3θ
?
Ψ(θ) dθ


(35)

υ=υF
.
The upper (lower) symbols in the parentheses of the left
hand sides of Eqs. (34), (35) correspond to the upper
(lower) symbols in the parentheses of the right hand sides
of these equations. Formulas (34), (35) are the funda-
mental equations for calculations of the dielectric func-
tion and, therefore, for studying the optical properties of
MNs.
For illustration, we present in Fig. 1 the frequency
dependence of an imaginary part of the dielectric per-
meability ratio components for a spheroidal Au particle,
which is obtained by numerical evaluating the integrals
in Eqs. (33), (34). It is worth to note that the radius R?
is directed along the revolution axis of the spheroid, and
R⊥– transverse to it. These spheroidal radiuses can be
expressed through the radius of a sphere (of an equivava-
lent volume) as
R = (R?R2
⊥)1/3.
The calculations were carried out using such parameters
for Au: ν = 3.39×1013at 00C [39], n = 5.9×1022cm−3,
υF= 1.39 × 108cm/s [44].
0 204060 80100
0.5
1.0
1.5
2.0
2.5
Ω?Νs
Im?Ε???Im?Ε??
FIG. 1. The dependence of the ratio ǫ′′
0.1, upper curve) and oblate (R⊥/R? = 10, lower curve) Au
particle (with R = 50˚ A) versus frequencies ratio ω/νs, where
νs= υF/(2R).
⊥/ǫ′′
?for prolate (R⊥/R?=
As one can see, the ratio of ǫ′′
creasing frequency both for the prolate and the oblate Au
nanoparticles. These oscillations have a damping char-
acter both for prolate and oblate particles, but differ in
the period of oscillations. For a given prolate nanopar-
ticle the oscillations occur around the constant value
ǫ′′
?≃ 4/3 and for an oblate ones – in the vicinity of the
another constant ǫ′′
?≃ 1/2. The period of oscillations
depend only slightly on the particle volume, however, the
oscillation amplitude is more pronounced for particles of
smaller radii.
It is also worth noting that the intensity of the surface
mode is determined by the magnitude of the imaginary
component of the material dielectric constant. Materials
with a small ǫ′′have a large, narrow absorption peak,
whereas materials with a large ǫ′′have a small broader
absorption peak.31
The ratio of the real parts of ǫ′
with frequency, therewith at the frequency of ω ≃ ωpl
exhibits the singularity. Thus, we demonstrate below the
plots for ǫ′
?(ω) separately.
Figure 2 shows the real part of the dielectric perme-
ability components for Au nanoparticle as a function of
normalized frequency, obtained by using Eqs. (32), (35)
in numerical calculations. The magnitude of ǫ′reaches
minimum value at ω → 0, and ǫ′→ 0 at ω → ωpl. Among
the two components of ǫ′, the frequency dependence is
more pronounced for the longitudinal one: ǫ′
smallest negative value at ω → 0 in the case of the pro-
late Au nanoparticle, and the maximal negative value –
for the oblate one. The absolute magnitude of the both
components of ǫ′is essentially enhanced as the radius of
the particle is increased (especially, at ω → 0).
⊥/ǫ′′
?oscillates with in-
⊥/ǫ′′
⊥/ǫ′′
⊥/ǫ′
?does not oscillate
⊥(ω) and ǫ′
?attains the

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6
0510 15
?1000
?800
?600
?400
?200
0
Ω?Νs
Re?Ε??, Re?Ε??
FIG. 2. The dependence of ǫ′
0.1, solid lines) and the oblate (R⊥/R?= 10, dashed lines) Au par-
ticle (with R = 50˚ A) versus frequencies ratio ω/νs. Thick curves
are for ?-components, and thin curves – for ⊥-components of ǫ′.
⊥,ǫ′
?for both the prolate (R⊥/R?=
Below, we will consider several approaches, enable us
to derive the explicit analytical expressions for σ′and σ′′
from Eqs. (34), (35).
IV. LIMIT CASES. SIZE EFFECTS
A. Frequency approach
The straightforward evaluations of the dielectric per-
meability of a single MN or it conductance can be made,
when Ψ-function entering in Eq. (23) or in Eqs. (34), (35)
takes the simplest form. Let us introduce the value
νs=
υ′
2R,
(36)
which will characterize the frequency of electron collisions
with the spherical MN surfaces, and R is the sphere ra-
dius. This allow us to rewrite Eq. (22) as
q =ν
νs
− iω
νs.(37)
i) First, we consider, for example, the case |q| ≫ 1.
In the frequency scale, this implies that both inequalities
ν ≫ νs and ω ≫ νs must be executed. Then, Eq. (21)
reduces to the form
Ψ(q)|q≫1≈4
3−2
q+4
q3− ··· .(38)
Accounting for Eqs. (38) and (22), we can calculate ap-
proximately the real and imaginary parts of the ratio
ℜ
?Ψ(θ)
ν − iω
?
|q|≫1
≈4
3
ν
ν2+ ω2−υ′(θ)
R
ν2− ω2
(ν2+ ω2)2+ ··· ,
(39)
ℑ
?Ψ(θ)
ν − iω
?
|q|≫1
≈4
3
ω
ν2+ ω2−2υ′(θ)
R
ν ω
(ν2+ ω2)2+··· ,
(40)
which enters into Eqs. (34), (35). Here, the velocity υ′(θ)
is given by Eq. (28). The formula (39) at ν → 0 agrees
with an earlier estimation given in Ref. [45].
Substituting Eq. (39) into Eq. (34), and Eq. (40) into
Eq. (35), correspondingly, and using the values of the in-
tegrals I?and I⊥given in the Appendix A, one obtains
for the real and imaginary parts of the conductivity ten-
sor components the expressions
σ′
(?
⊥)(ω) ≈ne2
m
?
ν
ν2+ ω2−9
2νs
ν2− ω2
(ν2+ ω2)2I(?
⊥)+ ···
?
,
(41)
σ′′
(?
⊥)(ω) ≈ne2
m
?
ω
ν2+ ω2− 9νs
ν ω
(ν2+ ω2)2I(?
⊥)+ ···
?
(42)
,
provided that the conditions ν ≫ νs, ω ≫ νsand υ = υF
are satisfied. To meet the requirement of ν ≫ νs, e.g.,
for Au particle at 0oC, it is necessary that its radius
should be R ≫ 400˚ A. The first term in the parentheses
of Eqs. (41), (42) results from the integration over angle
π/2
?
0
?sinθ cos2θ
1
2sin3θ
?
dθ =1
3. (43)
The second term in the parentheses of Eqs. (41), (42)
contains the integrals I which accounts for the particle
nonsphericity and doesn’t depend on the frequency. The
simplest result for σ can be obtained for a spherical MN,
when the integrals I for different polarizations coincide
with each other, and are equal to 1/3 (see Eqs. (A5) in
Appendix A).
As one can see, the first term in the parentheses of
Eqs. (41), (42) describes the Drude-Sommerfeld results
for a spherical particle, and the second one gives the
first correction of the kinetic theory to the volume elec-
tron scattering, allowing to account an electron scatter-
ing from the surfaces of the nanoparticle as well. The
calculations of the frequency dependencies of the ra-
tio ǫ′′
?with employing of Eqs. (33) and (41) (and
the conditions pointed therein) give the results qualita-
tively similar to our previous numerical calculations (us-
ing Eqs. (33), (34)), depicted in Fig. 1, though without
any oscillations. But quantitatively, the results obtained
from Eq. (34) are of one or even two orders of magnitude
lower (depending on the radius of MN) than those fol-
lowing else from the Drude-Sommerfeld formula or from
Eqs. (41).
ii) For the opposite limit case, when |q| ≪ 1 (or ν ≪ νs,
and ω ≪ νs), we can take advantage of the expansion
e−q≃ 1 − q +q2
⊥/ǫ′′
2!−q3
3!+q4
4!−q5
5!+q6
6!− ··· ,
and then from Eq. (21), we immediately obtain
Ψ(q)||q|≪1≈1
2q −
2
15q2+
1
36q3− ··· .(44)

Page 7

7
In this case, we can find for real and imaginary parts of
the mentioned above ratio the relations
ℜ
?Ψ(θ)
ν − iω
?
|q|≪1
≈
R
υ′(θ)−
8
15ν
?
R
υ′(θ)
?2
+ ··· , (45)
ℑ
?Ψ(θ)
ν − iω
?
|q|≪1
≈
8
15ω
?
R
υ′(θ)
?2
+ ··· . (46)
This permits us to get for σ components the following
expressions:
σ′
(?
⊥)≈9
4
ne2
m
?R
υI<
(?
⊥)|υ=υF−
2
15
ν
ν2
s
J(?
⊥)+ ···
?
, (47)
σ′′
(?
⊥)(ω) ≈
9
30
ne2
m
ω
ν2
s
?
J(?
⊥)+ ···
?
, (48)
provided that ν ≪ νs, ω ≪ νs, and υ = υF. In other
words, the first condition means that the radius of MN
should obey the condition R ≪ υF/(2ν). The first term
in the parentheses of Eqs. (47) is the result of the inte-
gration over angle θ with υ′(θ) in the denominator (see
Appendix A, Eqs. (A6), (A7)), and the second one can
be given by
J(?
⊥)=
1
R2
R2
?R2
⊥− R2
⊥
R2
?
?
±
?1
1/2
?
R2
∓
?
1
1
2(R⊥/R?)2
?
×
R?
?
R2
⊥− R2
?
arctan
?
⊥− R2
R?
?

. (49)
The upper (lower) signs and upper (lower) symbols in the
parentheses of the right hand side of Eq. (49) correspond
to those in the left hand side of this equation. In the case
of prolate spheroidal particles (R?> R⊥), one should
made in Eqs. (49) only the following replacement
arctan
?
R2
⊥− R2
R?
?
→1
2iln
??????
R?−
R?+
?
?
R2
?− R2
R2
⊥
?− R2
⊥
??????
, (50)
and in the case of a spherical symmetric MN, R?= R⊥≡
R, and J?= J⊥≡ 1/3.
As one can see from Eq. (47), the real part of σ(?
doesn’t depend on the frequency, but only on the particle
geometry, which is defined here by the parameters I<and
J. For imaginary part of σ(?
linear enhancement with the frequency. This implies that
⊥)
⊥), we have from Eq. (48) the
ǫ′
(?
⊥)≈ 1 −9
30
?ωpl
νs
?2
J(?
⊥), (51)
doesn’t depend on the frequency, provided that the con-
ditions ω ≪ νsand ν ≪ νsare fulfilled.
B.Size approach
It remains to examine the cases of different relations
between the particle sizes and the conduction electron
mean-free path inside a particle. Using Eq. (28), we can
rewrite Eq. (22) in somewhat another form
q = q1(θ) − iq2(θ), (52)
with
q1(θ) =
1
??
l
2R?
?2
cos2θ +
?
l
2R⊥
?2
sin2θ
, (53)
where 2R?is the length of the MN along the z axis, which
is directed along the principal spheroid axis, 2R⊥is the
MN size along the x or y directions, and
l =υF
ν
(54)
has the sense of the length of an electron mean-free path
(for Au at 00C, e.g., l ≃ 410˚ A), and
q2(θ) =
1
??νs⊥
ω
?2sin2θ +?νs?
ω
?2cos2θ
.(55)
Here
νs?=
υF
2R?,νs⊥=
υF
2R⊥
(56)
are the frequencies of the electron collision with the par-
ticle surfaces along and across the z-axis of a spheroid,
correspondingly.
Let us consider the possible relations between l and
particle sizes 2R?, 2R⊥.
i) The conduction electron mean-free path is much
less than the sizes of the particle along particular direc-
tions: l ≪ 2R⊥,
q1(θ)|υ=υF≫ 1. In this case, an electron is scattered
predominately inside the volume of MN. If, moreover,
q2→ 0, i.e., νs,?,νs,⊥≫ ω, then from Eq. (21), one gets
Ψ ≃4
l ≪ 2R?. As follows from Eq. (53),
3.(57)
Substituting Eq. (57) into Eqs. (34), (35), and using
Eq. (43), we obtain the Drude-Sommerfeld formulas for
real and imaginary parts of σ, presented above by the
first term in the parenthesis of Eqs. (41), (42).
ii) The mean-free path of a conduction electron is much
greater than the particle size along particular directions:
l ≫ 2R⊥,
occurs mainly from the inner surface of the MN. The
electrons oscillate between the walls of the particle with
different frequencies, excluding νs,?, νs,⊥. In accordance
with Eq. (53), the inequality q1(θ) ≪ 1 holds only for
q1. The parameter q2 remains arbitrary. Formally, we
l ≫ 2R?. In this case, an electron scattering

Page 8

8
can put q1(θ) → 0, and for a real and imaginary parts of
Ψ-function one finds
ℜ Ψ(q)|q1→0=4
3+4q2
2
?
cosq2−1
q2sinq2
?
, (58)
and
ℑ Ψ(q)|q1→0= −2
q2−4
q3
2
+4
q2
2
?1
q2cosq2+ sinq2
?
. (59)
The equations (58), (59) allow to fulfil the calculation of
the real and the imaginary parts of the ratio Ψ/(ν −iω),
as we have done it before. Then, we obtain
ℜ
?Ψ(q)
ν − iω
?
q1→0
=1
ω
?2
q2
−4
q2
2
sinq2+4
q3
2
(1 − cosq2)
?
,
(60)
ℑ
?Ψ(q)
ν − iω
?
q1→0
=4
ω
?1
3+1q2
2
?
cosq2−1
q2sinq2
??
.
(61)
The parameter q2, defined by Eq. (55), is governed by
the frequency. Depending on the ratio between the inci-
dent frequency and the frequencies νs,?, νs,⊥, the value
of q2can be greater or less than 1. This makes it difficult
to perform subsequent analytical calculations of the in-
tegrals involved into Eqs. (34), (35). Below, we dwell on
some particular cases for which the calculations of σ are
the most simple. It should be noted also that the cor-
responding expressions for the components of ǫ can be
easily obtained by substituting of σ into Eqs. (32), (33).
1. Conductivity of a spherical MN
In the case of a spherical MN there are three char-
acteristic frequencies which are considered usually: the
frequency of an incident electromagnetic field ω, the col-
lision frequency of electrons in the particle volume ν, and
the vibration frequency between the particle walls νs(if
the particle size is less than the electron mean free path).
When ν > νs, the mechanism of an electron scattering in
the bulk is dominated, and an electron scattering from
the particle surface gives only small corrections of the or-
der of νs/ν. But we are interested in the case, when the
mechanism of the surface electron scattering dominates,
which corresponds to ν < νs.
For particles of a spherical shape, the electric con-
ductivity becomes a scalar quantity, and one can put
R?= R⊥ ≡ R in Eq. (27), then q and Ψ-function in
Eqs. (25), (26) are not depended on the angle θ. With
an account for Eq. (43), this makes it possible to obtain
for σ the simple expression
σc
sph= σc
?= σc
⊥=3
4
ne2
m
Ψ(q)
ν − iω|υ=υF,
sph, one can use
(62)
with q = 2R(ν − iω)/υF. To calculate σc
either Eqs. (39), (40), or Eqs. (45), (46), or Eqs. (60), (61)
for different limit cases considered above. For example,
we restrict ourselves here only to the case, when ν ≪ νs.
In this limit case, to a first approximation, one can put
q1 → 0. Then Eqs. (60), (61) with q2 = ω/νs can be
used in Eq. (62). As a result, one obtains for real and
imaginary parts of σ the following expressions
σ′
sph≃
3
8πνs
ω2
ω2
pl
?
1 −2νs
ω
sinω
νs
+2ν2
s
ω2
?
1 − cosω
νs
??
(63)
,
σ′′
sph≃
ω2
pl
4πω
?
1 + 3
?νs
ω
?2?
cosω
νs
−νs
ωsinω
νs
??
, (64)
provided that ν ≪ νs, where νs = υF/(2R). The last
expression in the Drude-Sommerfeld approximation looks
like14
σ′′
sph=
ω
4π
ω2
pl
ν2+ ω2. (65)
If one puts the oscillation terms in Eqs. (63), (64) equal
to zero, and uses Eqs. (32), (33), then one obtains the
expression for real part of the dielectric function which
coincides with Eq. (1) at ν → 0; but for imaginary part
of the dielectric function, one gets Eq. (2) only if the
replacement ν → 3νs/2 will be done. This replacement
is the same one as presented by Eq. (3), which has been
often used in a phenomenological approximation.
02468 10
0.0
0.2
0.4
0.6
0.8
1.0
Ω?Νs
Re?Σsph??Σ0;
Im?Σsph??Σ0
FIG. 3. The real σ′(solid lines) and imaginary σ′′(dashed lines)
parts of the ratio of an electric conductivity to the statical con-
ductivity σ0 vs frequencies ratio, for a spherical Au particle with
R = 200˚ A. The thin lines correspond to the results obtained us-
ing the kinetic method and the bold lines – the Drude-Sommerfeld
formulas.
The calculated results of the real and imaginary parts
of an electric conductivity across the normalized to νs
frequency are shown in Fig. 3 for spherical Au nanopar-
ticle, obtained with the use of both the kinetic method
and the Drude-Sommerfeld formulas.
ity is measured on the scale of the statical conductiv-
ity σ0 = ne2/(mν). For calculations of σ, Eqs. (34),
The conductiv-

Page 9

9
(35) have been used in the kinetic case, and Eqs. (2)
and (65) – in the Drude-Zommerfeld case.
tration, the numerical parameters for Au particle44and
ωpl = 1.37 × 1016s−1were chosen. As it can be seen
in Fig. 3, the kinetic method appreciably changes the
frequency dependence of σsph at low frequencies, and
at high frequencies (ω ≫ νs) gives the same result for
σsph as that which follows from the Drude-Zommerfeld
formula. The difference between kinetic and Drude-
Zommerfeld results is enhanced markedly as the parti-
cle radius is decreased. The real part of σ is peaked at
ω/νs→ 0 in both cases, whereas the imaginary part of
σ – at ω = νs in the Drude-Zommerfeld case, and at
ω ≈ 4νs– using the kinetic method.
For an extremely low ω ≪ νs or an extremely high
ω ≫ νs frequencies, one can find from Eqs. (63), (64)
after some algebra, the next simple approximation for σ′
For illus-
σ′
sph=
3
16πω2
pl·
?
R
υF,
υF
Rω2,
ω ≪ νs,
ω ≫ νs
, (66)
and for σ′′:
σ′′
sph=
ω2
4π·
pl



ω
2
?
R
υF
1
ω,
?2
,ω ≪ νs,
ω ≫ νs
. (67)
The results of Eqs. (66), (67) at ω ≫ νs correspond to
those presented above by first terms in the square brack-
ets of Eq. (63), (64), accordingly.
The calculations with employing of Eqs. (63), (64) for
Au particle with R = 200˚ A give the results similar qual-
itatively to the ones above presented, but quantitatively
they are at the σ maximum of approximately 30% higher.
In spite of the oscillation terms in square brackets of
Eqs. (63), (64), the ratio of both σ′
does’t oscillate with the frequency owing to the cut-off
factors before these brackets.
sph/σ0 and σ′′
sph/σ0
5.?10?7
1.?10?6
1.5?10?6
2.?10?6
2.5?10?6
3.?10?6
?550
?500
?450
Nanoparticle Radius ?cm?
Re?Ε?
FIG. 4. The dependence of ǫ′for the spherical Au particle versus
a radius R at the frequencies of ω =: 5.7 × 1014s−1(dashed line),
6 × 1014s−1(solid line), and 6.3 × 1014s−1(doted line).
Now, let us discuss shortly the dependence of the di-
electric permeability on the size of MN. In Fig. 4, we
present the results of our numerical calculations of ǫ(R)
using Eqs. (32), (35) at the fixed ω. For illustration,
we choose such frequencies from the frequency scale for
which the above dependencies are the most pronounced
for Au particle.
5.?10?7
1.?10?61.5?10?62.?10?62.5?10?63.?10?6
100
200
300
400
500
Nanoparticle Radius ?cm?
Im?Ε?
FIG. 5. The dependence of ǫ′′for the spherical Au particle across
the radius R at the frequencies of ω =: 5 × 1014s−1(dashed line),
6 × 1014s−1(solid line), and 7 × 1014s−1(doted line).
The real ǫ′as well as the imaginary ǫ′′parts of the di-
electric permeability oscillate, when the particle radius is
increased. These oscillations have a damping character
and practically are vanished for MNs of high radiuses.
The real part of ǫ tends to +1, and the imaginary part
tends to 0 at ω → 0. The real part has a first main
minimum and the imaginary part has a first main max-
imum at small values of R. Both the minimum and the
maximum of ǫ are slightly shifted towards the greater
R with the frequency decreasing. In other words, this
means that the resonance energy peak shifts toward the
red-side with increasing sizes of Au particle. The oscilla-
tion period in accordance with Eqs. (63), (64) is defined
by
T = πυF
Rω
(68)
and, as one can see, essentially depends on the product
of Rω. It becomes shorter at Rω ≪ υF, and at Rω ≫ υF
– extends. Two types of oscillations may occur: at fixed
ω with varying of R, or at fixed R with changing of ω.
For every fixed frequency there is the constant of ǫ′or
ǫ′′around which those quantities oscillate with altering
of R. At high frequencies, the period of oscillations is
sharply decreased and their amplitude are considerably
lowered proportionally to the factors (ωpl/ω)2or ω2
before square brackets in Eqs. (63), (64), correspondingly.
If we assume that the dielectric function of a surround-
ing media ǫm is close to unity, then for a spherical Au
particle at a plasmon frequency ω = ωpl/√1 + 2ǫmwe get
the closely packed oscillations of ǫ′(R) within the ampli-
tude interval −2 ÷ 1. The behavior of σ′(R) and σ′′(R)
at the frequencies ω ≪ νsand ω ≫ νscan be seen from
Eqs. (66), (67) as well.
pl/ω

Page 10

10
2. Conductivity of an oblate MN
For oblate particles
l > 2R⊥> 2R?. (69)
We shall consider, for convenience, the frequency inter-
vals ω ≪ νs,⊥ and ω ≫ νs,?. According to Eq. (55),
the former interval corresponds to q2|υ=υF< 1, and the
latter one – to q2|υ=υF> 1, respectively. It is easy to
show that Eqs. (60), (61) for these frequency intervals
transform to
ℜ
?Ψ(q)
ν − iω
?
q1→0
≈1
ω·
?q2
2,
2
q2,
ω ≪ νs,⊥,
ω ≫ νs,?
, (70)
ℑ
?Ψ(q)
ν − iω
?
q1→0
≈1
ω·
?q2
2
6,
4
3,
ω ≪ νs,⊥,
ω ≫ νs,?
. (71)
Then, the integrals in Eqs. (34), (35) can be calculated
exactly in the approximations of (70), (71), and for an
arbitrary spheroid aspect ratio between R?and R⊥ we
obtain the following expressions for the components of
an electric conductivity
σ′
(?
⊥)(ω) ≃
9
16π
ω2
pl
ω
·




?

Rω
υFI<
(?
⊥),ω ≪ νs,⊥
ω ≫ νs,?
υF
RωI>
(?
⊥),
, (72)
σ′′
(?
⊥)(ω) ≃
1
4π
ω2
pl
ω
·



3
2
Rω
υF
?2
1,
J(?
ω ≫ νs,?
⊥),ω ≪ νs,⊥
. (73)
In the case of ω ≪ νs,⊥, Eq. (72) coincides with the first
term found previously in Eq. (47), and in the case of
ω ≫ νs,?, it coincides with the second term of Eq. (41)
at ν = 0. Similarly, Eq. (73) in the case of ω ≪ νs,⊥
agrees with the first term found previously in Eq. (48)
with an accuracy of a numerical coefficient, and in the
case of ω ≫ νs,?coincides with the first term of Eq. (42)
at ν = 0.
The above expressions can be transformed to their sim-
plest forms for strongly deformed particles. Thus, for
strongly oblate MNs, when R⊥≫ R?, using Eqs. (A3),
(A9) from Appendix A, it can be easily found that at low
frequencies
σ′
?=
9
32πω2
?
pl
R?
υF,
σ′
⊥= σ′
?
ln2R⊥
R?−1
2
?


,ω ≪ νs,⊥, (74)
and at high frequencies:
σ′
?=
9
64π
σ′
?ωpl
ω
?2υF
R?,
⊥= σ′
?/2
?
,ω ≫ νs,?. (75)
Similarly, for σ′′in the case of strongly oblate MNs at
low frequencies, one gets
σ′′
?=
3
8πω
?
?ωplR?
R?arctanR⊥
υF
?2
R?− 1
,
σ′′
⊥=1
2σ′′
?
R⊥
?





,ω ≪ νs,⊥, (76)
and at high frequencies:
σ′′
⊥≡ σ′′
?=
ω2
4πω,
pl
ω ≫ νs,?.(77)
It remains to consider, how the optical properties of
MNs evolve at νs,⊥ ≤ ω ≤ νs,?between the low- and
high-frequency interval.This interval is just that for
which the parameter q2 can be greater or less than 1.
In this case, only the numerical evaluations of the inte-
grals entered into Eqs. (34), (35) can be performed. The
obtained results for real part of σ can be found in Ref.
[45].
3. Conductivity of a prolate MN
For prolate particles
l > 2R?> 2R⊥. (78)
It is also worth noting that the approximations (70), (71)
still remain to be valid for the case of inequalities (78), if
the transposition νs,⊥⇆ νs,?is carried out in Eqs. (70),
(71). As a result, the integrals in Eqs. (34), (35) can
be easily calculated, and for prolate particles with an
arbitrary aspect ratio of R?/R⊥, one obtains the results
similar to ones given by Eqs. (72), (73), in which the
both replacements (50) and (A8) have been made for J
and the integrals I,I<. Below, we write down only the
results for a strongly prolate particles (R?≫ R⊥) at low
and high frequencies. Using Eqs. (A4), (A10), it is easy
to find that
σ′
?=
σ′
9
64ω2
⊥= σ′
pl
?/2
R⊥
υF,
?
,ω ≪ νs,?,(79)
σ′
?=
9
256
⊥= 3σ′
?ωpl
ω
?2υF
R⊥,
σ′
?/2
?
,ω ≫ νs,⊥.(80)
For σ′′, just as for σ′, in the case of strongly prolate
MNs, one finds
σ′′
⊥=
3
16πω
?ωplR⊥
?
υF
?2
2R?
,
σ′′
?= −2σ′′
⊥
1 + lnR⊥
?





,ω ≪ νs,?, (81)
and at high frequencies ω ≫ νs,⊥, we have found σ′′
σ′′
pl/(4πω), which coincides exactly with Eq. (77)
for oblate particles.
?≃
⊥= ω2

Page 11

11
Comparing Eqs. (79)–(81) for a strongly prolate par-
ticle to the corresponding ones (72)–(76) for a strongly
oblate particle shows that the character of the frequency
dependence of the corresponding components of the elec-
tric conductivity tensor remains practically the same.
The behavior of this dependence reminds the asymptotic
behavior of the Drude-Zommerfeld frequency dependence
for an electron scattering in the volume of MN; this can
be easily verified from the first summand in the right-
hand sides of Eqs. (41), (42) by setting ω ≪ ν or ω ≫ ν.
The difference consists only in the fact that for an elec-
tron scattering in the volume, the high-frequency con-
ductivity close to the frequency ω ≈ ν goes smoothly
to the saturation. However, for the strongly asymmetric
MNs there exists an entire transitional region between
the minimum and the maximum transit rates, where the
frequency dependence of σ appreciably differs from the
volume case.
In the case of the frequency interval νs,?≤ ω ≤ νs,⊥,
the numerical evaluations of the integrals involved into
Eqs. (34), (35) can be performed and the obtained results
for σ have been presented in Ref. [45].
V.RESULTS AND DISCUSSION
One of the most important cases takes place for the fre-
quencies ω ≫ νs, with νs= υF/(2R) ≫ ν. In terms of q-
language this corresponds to the situation when |q1| ≪ 1
and |q2| ≫ 1. In this case, one may use the expressions
(58), (59) at q1→ 0 to obtain the results for an asymp-
totically high value of q2. As a result, we get for a real
and imaginary parts of an electric conductivity of a single
spheroidal MN with an arbitrary aspect ratio of R⊥/R?
the following expressions
σ′
(?
⊥)(ω) ≈
1
4π
?ωpl
ω
?2?
ν +9
4
υF
RI(?
⊥)+ ···
?
, (82)
σ′′
(?
⊥)(ω) ≈
1
4π
?ωpl
ω
?2?
ω −9
4
υF
RI(?
⊥)+ ···
?
, (83)
provided that ω ≫ νs≫ ν. The parameters I?, I⊥are
given in Appendix A by Eqs. (A1), (A2).
For a MN of a spherical shape these expressions take
the simplest form
σ′
(q1≪1
q2≫1)(ω) ≃
3
8π
?ωpl
ω
?2
νs+ ··· ,(84)
σ′′
(q1≪1
q2≫1)(ω) ≃
1
4π
?ωpl
ω
?2?
ω −3
2νs+ ···
?
, (85)
provided that ω ≫ νs ≫ ν. The last formula demon-
strates simply that the electron interaction with the sur-
face of a spherical MN can shift the imaginary part of σ
towards the red side of the frequency scale. The smaller
is the radius of the MN, the greater is the correction to
the frequency shift.
Below, we will discuss the size dependence of optical
properties of MNs in more details and will illustrate some
results obtained in above sections.
0
1.?1015
2.?1015
3.?1015
4.?1015
1
2
3
4
5
6
7
Ω, s?1
Re?Ε???Re?Ε??
FIG. 6. The frequency dependence of the reduced dielectric con-
stant of Au particles with different R,˚ A: 50 (thick lines) and 150
(thin lines). Full lines are for a prolate particle with R⊥/R?= 0.1,
and dashed lines are for an oblate one with R⊥/R?= 10.
In Fig. 6, the results of numerical calculations of the
ratio between transversal and longitudinal components
of the dielectric permeability versus the frequency are
given for Au particles of different radiuses. The calcu-
lations were performed with the use of Eqs. (32), (35)
and the same numerical parameters as above used. At
high frequencies (ω ≫ νs), the ratio of ǫ′
different radiuses of Au particle. This result follows as
well from Eqs. (32), (73) at ω ≫ νs for particles of an
oblate shape, and from Eqs. (32), (77) – for particles of a
prolate shape. At low frequencies (ω ≪ νs), the particle
size is strongly effected on the ratio of Re(ǫ⊥)/Re(ǫ?).
As it can be seen in Fig. 6, the curves for MNs of prolate
and oblate shapes are merged into one with frequency
grows, but for particles of a greater radiuses it becomes
at smaller frequencies.
The behavior of ǫ′as a function of the ellipsoid aspect
ratio R⊥/R?at the frequency of a plasmon resonance in
Au nanoparticle, embedded in the dielectric media with
ǫm= 1, is plotted in Fig. 7. The numerical calculations
with employing of Eqs. (32), (35) were performed for two
radiuses of a nanoparticle and for two light polarizations.
As one can see in this figure, the longitudinal component
of ǫ′in an oblate MN depends more stronger on the ratio
R⊥/R?than the transverse one. It becomes especially
pronounced for Au nanoparticle (with R = 50˚ A) of small
oblateness (e.g., R⊥/R?< 15), when keeping the laser
frequency fixed at ω = ωpl/√1 + 2ǫm: the oscillation
magnitude of ǫ′
much larger than ones for ǫ′
oscillations is enhanced and the period is extended with
increasing of the ellipsoid aspect ratio R⊥/R?. For MNs
⊥/ǫ′
?→ 1 for
⊥(round the constant of ǫ′= −2) becomes
?. The amplitude of these

Page 12

12
of greater radiuses, the number of oscillations is decreased
and their period is extended (compare the thin and thick
curves in the Fig. 7 for Au particles with R = 50 and
150˚ A, for example). The both components of ǫ′no longer
oscillate at high oblateness (R⊥/R?> 80) of Au particle
with R = 50˚ A.
0 100 200300400500
?2.0
?1.5
?1.0
?0.5
0.0
0.5
R??R?
Re?Ε?,??
FIG. 7. The dependence of the real part of the dielectric constant of
Au particles with different R,˚ A: 50 (thick lines) and 150 (thin lines)
on the ellipsoid aspect ratio, at a frequency of plasmon resonance
ω = ωpl/√3 ≃ 7.91 × 1015s−1. Solid line is for ?- and dashed line
for ⊥-component.
When the size of the particle is large enough, the ?- and
⊥-components of ǫ′start to come together (see Fig. 2).
For instance, at the frequency ω ≈ 6 × 1014s−1it takes
place for Au particle with R = 130˚ A. At smaller frequen-
cies it occurs for greater radiuses.
Presented dependencies display mainly the behavior of
ǫ′for an oblate nanoparticle (R⊥/R?> 1). In the case
of a prolate nanoparticle (0 < R⊥/R?< 1), the numeri-
cal calculations with employing the same Eqs. (32), (35)
for Au nanoparticle of a small size (∼ 50˚ A) at the fre-
quency of the plasmon resonance give the oscillations of
a transversal component of ǫ′, the amplitude of which is
enhanced and the period is reduced as soon as the pro-
lateness of the MN is increased. The longitudinal com-
ponent of ǫ′
?oscillates in the prolate MN as well, but its
amplitude is considerably smaller and the period is much
larger than the proper ones for ǫ′
The imaginary part of the dielectric function as a func-
tion of ellipsoid aspect ratio R⊥/R?is shown in Fig. 8.
The numerical calculations were performed with the use
of Eqs. (33), (34) at the plasmon frequency for two ra-
diuses of nanoparticle and for two light polarizations.
Since for prolate MN, the value of ǫ′′practically doesn’t
depend on the spheroid aspect ratio R⊥/R?(except for
the case of a very high prolateness), in Fig. 8 we present
only the results for an oblate MN. In contrast to the size
behavior of ǫ′, described above, one can see that the weak
oscillations of ǫ′′hold together with linear increasing of
ǫ′′just at small values of the aspect ratio R⊥/R?. For Au
particles with R = 50˚ A it takes place until R⊥/R?< 150,
⊥.
and is more sensible for the longitudinal component of
ǫ′′than for the transversal one. The both components
of ǫ′′reach the same maximum at some aspect ratios of
R⊥/R?, which value depends on the radius of particle.
In the example depicting in Fig. 8 (for Au particle with
R = 50˚ A), the values of ǫ′′
R⊥/R? > 140 for ?-component and at R⊥/R? ≃ 700
for ⊥-component of ǫ′′. The another interesting feature
of the dependence of ǫ′′on R⊥/R?is the intersection of
curves for ?- and ⊥-components for an oblate particle, as
it takes place usually for the particle of a spherical shape.
?= ǫ′′
⊥≃ 1.8 have peaks at
0 200 400600800 10001200 1400
0.0
0.5
1.0
1.5
R??R?
Im?Ε?,??
FIG. 8. The dependence of the imaginary part of the dielectric
constant of Au particles with different R,˚ A: 50 (thick lines) and
150 (thin lines) on the ellipsoid aspect ratio, at the frequency of a
plasmon resonance ω ≃ 7.91 × 1015s−1. Solid line is for ?- and
dashed line for ⊥-component.
Finally, we illustrate, how one may estimate the life-
time of surface plasmon excitation (or any others) in a
MN using the above derived formulas for σ. As follows
from our previous calculations21,39, the linewidth can be
represented as
Γβ(ω) = 4πLασαβ(ω), (86)
where Lα is defined above after Eq. (6) and σαβ is the
real part of the conductivity tensor, given by Eq. (23) for
most general situations or by Eq. (34) for the spheroidal
MN. For other particular cases, one may use for Reσ
Eqs. (41), (47), (63), (66), (72), (74), (75), (79), (80), and
(82), presented above. In particular, considering only
the nanoparticles of a spherical shape (L = 1/3) and
restricting ourselves to the case ν ≪ νs, we can choose
Eq. (63) for illustration. Substituting it into Eq. (86), we
obtain
Γ(ω) ≃υF
4R
?ωpl
ω
?2?
1 −2νs
ω
sinω
νs
+2ν2
s
ω2
?
1 − cosω
νs
??
(87)
,
Taking into account only the first term in (87), we recover
the well-known17,36,371/R dependence of Γ.

Page 13

13
Γ0(ω,R) =1
4
υF
R
?ωpl
ω
?2
.(88)
As seen from Eqs. (87), (88), the lifetime (1/Γ) of an
excitation in the MN depends not only on the nanopar-
ticle radius, but also on the frequency (at which a given
excitement is reasonable). For the frequency, which cor-
responds to the excitation of surface plasmon in MN in
a vacuum, ω = ωpl/√3, the following relation can be
obtained from Eq. (88) in energy units:
ΓSP
0 (R) =3
4?υF
R.
(89)
The oscillating terms in Eq. (87) give rise to the oscil-
lation of Γ around of Γ0as a function of both the particle
radius and the frequency. They can be represented at the
frequency of surface plasmon as follows
ΓSP
osc(R) ≃3√3
4
?
ωpl
√3 υF
2Rωpl
?υF
R
?2
?
×
?
−sin2Rωpl
√3 υF
+
1 − cos2Rωpl
√3 υF
??
.(90)
The amplitude and period of oscillations can be esti-
mated by the values
Γmax
osc ≃3√3?υ2
F
4ωplR2,T =
√3πυF
Rωpl
, (91)
accordingly.
It is important to note that in kinetic method this os-
cillatory behavior of Γ follows solely from the conditions
of an electron scattering on the nanoparticle surface.
Figure 9 shows the full linewidth Γ = ΓSP
is obtained by numerical evaluating of Eqs. (89), (90) for
Na nanoparticles with parameters44: ωpl = 9.18 × 1015
s−1and υF= 1.07×108cm/s. One can see that oscillat-
ing terms represent an important correction to ΓSP
at small particle radii. This our result for Na nanopar-
ticles only qualitative agrees with the similar results ob-
tained in Refs. [37,38]), since we try to apply the kinetic
method to the range of R, where the quantum effects
(like, e.g., the Landau damping) play an important role.
But, mostly, for the kinetics, the next inequality could
be met
0 +ΓSP
osc, which
0 (R)
R ≫
2π?
mυF.(92)
In order to study the significance of the oscillatory be-
havior in more general situations, it is necessary to per-
form the calculations of Eq. (86), using the real parts of
Eq. (23) or Eq. (34). This will be done separately.
10203040
0.5
1.0
1.5
2.0
2.5
R?aB
? ?eV?
FIG. 9. Linewidth Γ(R) of the surface plasmon resonance, as a
function of radius for Na nanoparticles in units of Bohr radius aB≃
0.53˚ A. The smooth term Γ0(R) is given by Eq. (89) (dashed) and
the solid line corresponds to the sum of Eqs. (89) and (90).
There are several experimental data for a dielec-
tric constant of powder of Ag and Al,46and for Ag
nanoparticles.47For a single Au particle we have found
only the experimental data for an optical response.48A
direct comparison of theoretical results with most of the
available experimental measurements of the optical prop-
erties of MNs are still a matter of debate because inho-
mogeneous in nanoparticle size, shape and local environ-
ment hide the homogeneous width of the surface plasmon
resonance. There are very different data even for bulk
permittivity of Au,49,50especially for its imaginary part.
VI.CONCLUSIONS
The kinetic equation method is used to study the pecu-
liarities of the electron interactions with the surface of a
spheroidal metal nanoparticle, when the electron scatter-
ing from the particle surface becomes a dominant effect.
The special attention was paid to study the modification
of the Drude-Zommerfeld model applying to the optical
properties of MN. The real and imaginary parts of the
dielectric permeability at the frequencies above and be-
low the characteristic frequency of a free electron passage
between the walls of the particle were calculated for a sin-
gle oblate and a prolate MN whose dimensions are much
smaller than the wavelength of an electromagnetic wave.
It was established for spherical MNs that the kinetic
method appreciably changes the frequency dependence
of electrical conductivity at low frequencies, and at high
frequencies (ω ≫ νs) gives the same result as one ob-
tained from the Drude-Zommerfeld formula. Quantita-
tively, the results obtained by kinetic method are of one
or even two orders of magnitude lower (depending on the
radius of MN) than those following from classical formu-
las. The difference between results is enhanced markedly
as the particle radius is decreased and the nanoparticle
surface starts to play the more pronounced role.

Page 14

14
The frequency dependencies of the components of the
electric conductivity tensor σ were found and their de-
pendence on the spheroidal aspect ratio was investigated.
Simple analytical expressions were found for this tensor
in a strongly oblate or prolate MNs at low and high fre-
quencies.
The electron interaction with the surface of a spherical
MN can shift the imaginary part of σ towards the red side
of the frequency scale. The smaller is the radius of the
MN, the greater is the correction to the frequency shift.
Two types of oscillations were established for small
enough MNs: at fixed ω with varying of R, or at fixed
R with changing of ω. These oscillations have a damp-
ing character and practically are vanished else at high
frequencies or for MNs of high radiuses.
The ratio of Im(ǫ⊥)/Im(ǫ?) oscillates with increasing
the frequency both for the prolate and the oblate MNs.
In contrast, the ratio of the real parts of Re(ǫ⊥)/Re(ǫ?)
does not oscillate with frequency. Together with that, the
real ǫ as well as the imaginary ǫ part of the dielectric per-
meability oscillate, when the particle radius is increased.
It was found that the particle size strongly effects the
ratio of Re(ǫ⊥)/Re(ǫ?) at low frequencies (ω ≪ νs).
Acknowledgments
Authors would like to thank Doctor E.A. Ponezha for
her valuable comments and useful remarks.
Appendix A
Below, we present the values for integrals introduced
in the Sec. IV.
I? =
1
υ
π/2
?
0
sinθ cos2θυ′(θ)dθ =
R
8R?
2R2
R2
⊥− R2
⊥− R2
?
?
−
R
8R⊥
R3
?
(R2
⊥− R2
?)3/2ln
?????
R⊥
R?
+
?
R2
R2
⊥
?
− 1
?????.(A1)
I⊥ =
1
υ
π/2
?
0
1
2sin3θυ′(θ)dθ =
3R
16R?
−
R
16R?
R2
⊥− R2
⊥
R2
?
+
R
4R⊥
R?
?
R2
⊥− R2
?
R2
?
?
1 +
R2
?
4(R2
⊥− R2
?)
?
× ln
?????
R⊥
R?
+
R2
⊥
?
− 1
?????. (A2)
For strongly oblate or prolate MNs, using the Eqs. (84),
(85), it is easy to find that
I(?
⊥),R⊥≫R?≃1
4
R
R?
·
?
1
1/2,(A3)
I(?
⊥),R⊥≪R?≃
π
16
R
R⊥
·
?
1
3/2,(A4)
correspondingly, and in the limit case of R⊥= R?≡ R,
one gets
lim
R?→R⊥I?= lim
R?→R⊥I⊥=1
3.(A5)
We advance here as well the another typical integrals,
which one meets under calculation of σ.
I<
?
= υ
π/2
?
0
sinθ cos2θ
υ′(θ)
dθ =
R?R⊥
⊥− R2
2R(R2
?)
×

R⊥−
R2
?
?
R2
⊥− R2
?
ln
?????
R⊥
R?
+
?
R2
R2
⊥
?
− 1
?????

, (A6)
I<
⊥= υ
π/2
?
0
sin3θ
2υ′(θ)dθ = −
R?R2
⊥− R2
⊥
4R(R2
?)
+
R?R⊥(2R2
4R(R2
⊥− R2
?)3/2ln
?)
⊥− R2
?????
R⊥
R?
+
?
R2
R2
⊥
?
− 1
?????.(A7)
In the case of prolate particles (R?> R⊥), one should
perform in Eqs. (85), (A1) and in Eqs. (A5), (A6) the
next replacement
1
iln
?????
R⊥
R?
+ i
?
1 −R2
⊥
R2
?
?????→ arcsin
?
1 −R2
⊥
R2
?
.(A8)
For strongly oblate or prolate MNs (R⊥≫ R?), using
Eqs. (A6), (A7), it is easy to obtain that
I<
(?
⊥),R⊥≫R?≃1
2
R?
R·
?
1
ln2R⊥
R?− 1/2, (A9)
I<
(?
⊥),R⊥≪R?≃π
4
R⊥
R
·
?
1
1/2,(A10)
respectively, and in the limit case of R⊥= R?≡ R, one
finds from Eqs. (A5), (A6)
lim
R?→R⊥I<
?=lim
R?→R⊥I<
⊥=1
3.(A11)

Page 15

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