Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
The effect of the electron temperature on both the light absorption and the scattering by metal nanoparticles (MNs) with excitation of the surface plasmon electron vibrations is studied in the framework of the kinetic theory. The formulae for electroconductivity and polarizability tensors are derive...
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| Zitieren: | Temperature dependence of plasmon resonances in spheroidal metal nanoparticles / N.I. Grigorchuk // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С 33706:1-18. — Бібліогр.: 51 назв. — англ. |
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irk-123456789-1208202017-06-14T03:03:17Z Temperature dependence of plasmon resonances in spheroidal metal nanoparticles Grigorchuk, N.I. The effect of the electron temperature on both the light absorption and the scattering by metal nanoparticles (MNs) with excitation of the surface plasmon electron vibrations is studied in the framework of the kinetic theory. The formulae for electroconductivity and polarizability tensors are derived for finite temperatures of an electron gas. The electrical conductivity and the halfwidth of the surface plasmon resonance are studied in detail for a spherical MN. Depending on the size of MN, the efficiencies of light absorption and scattering with the temperature change are investigated. It is found, in particular, that the absorption efficiency can both increase and decrease with a temperature drop. The derived formulas make it possible to analytically calculate various optical and transport phenomena for MNs of any spheroidal shape embedded in any dielectric media. В рамках кiнетичної теорiї вивчається вплив температури електронiв на поглинання та розсiювання свiтла металевими наночастинками (МН) при збудженнi електронних коливань поверхневого плазмона. Одержанi формули для тензорiв електропровiдностi та поляризованостi для кiнцевих температур електронного газу. Детально вивченi електрична провiднiсть та напiвширина резонансу поверхневого плазмона для сферичної МН. Дослiджується ефективнiсть поглинання та розсiювання свiтла при змiнi температури в залежностi вiд розмiрiв МН. Зокрема, знайдено, що ефективнiсть поглинання може не тiльки збiльшуватися при зменшеннi температури, а також й зменшуватись. Одержанi формули дозволяють аналiтично обчислювати рiзноманiтнi оптичнi й транспортнi явища для МН довiльної сфероїдальної форми, що знаходяться в довiльному дiелектричному середовищi. 2013 Article Temperature dependence of plasmon resonances in spheroidal metal nanoparticles / N.I. Grigorchuk // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С 33706:1-18. — Бібліогр.: 51 назв. — англ. 1607-324X PACS: 78.67.-n, 65.80.-g, 73.23.-b, 68.49.Jk, 52.25.Os DOI:10.5488/CMP.16.33706 arXiv:1304.2160 http://dspace.nbuv.gov.ua/handle/123456789/120820 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The effect of the electron temperature on both the light absorption and the scattering by metal nanoparticles (MNs) with excitation of the surface plasmon electron vibrations is studied in the framework of the kinetic theory. The formulae for electroconductivity and polarizability tensors are derived for finite temperatures of an electron gas. The electrical conductivity and the halfwidth of the surface plasmon resonance are studied in detail for a spherical MN. Depending on the size of MN, the efficiencies of light absorption and scattering with the temperature change are investigated. It is found, in particular, that the absorption efficiency can both increase and decrease with a temperature drop. The derived formulas make it possible to analytically calculate various optical and transport phenomena for MNs of any spheroidal shape embedded in any dielectric media. |
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Article |
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Grigorchuk, N.I. |
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Grigorchuk, N.I. Temperature dependence of plasmon resonances in spheroidal metal nanoparticles Condensed Matter Physics |
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Grigorchuk, N.I. |
| author_sort |
Grigorchuk, N.I. |
| title |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
| title_short |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
| title_full |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
| title_fullStr |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
| title_full_unstemmed |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
| title_sort |
temperature dependence of plasmon resonances in spheroidal metal nanoparticles |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120820 |
| citation_txt |
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles / N.I. Grigorchuk // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С 33706:1-18. — Бібліогр.: 51 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT grigorchukni temperaturedependenceofplasmonresonancesinspheroidalmetalnanoparticles |
| first_indexed |
2025-07-08T18:40:28Z |
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2025-07-08T18:40:28Z |
| _version_ |
1837105211875786752 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 3, 33706: 1–18
DOI: 10.5488/CMP.16.33706
http://www.icmp.lviv.ua/journal
Temperature dependence of plasmon resonances in
spheroidal metal nanoparticles
N.I. Grigorchuk
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine,
14–b Metrologichna St., 03680 Kyiv, Ukraine
Received May 30, 2013, in final form July 1, 2013
The effect of the electron temperature on both the light absorption and the scattering by metal nanoparticles
(MNs) with excitation of the surface plasmon electron vibrations is studied in the framework of the kinetic
theory. The formulae for electroconductivity and polarizability tensors are derived for finite temperatures of
an electron gas. The electrical conductivity and the halfwidth of the surface plasmon resonance are studied
in detail for a spherical MN. Depending on the size of MN, the efficiencies of light absorption and scattering
with the temperature change are investigated. It is found, in particular, that the absorption efficiency can both
increase and decrease with a temperature drop. The derived formulas make it possible to analytically calculate
various optical and transport phenomena for MNs of any spheroidal shape embedded in any dielectric media.
Key words: electron temperature, metal nanoparticles, electroconductivity, polarizability tensor, surface
plasmon resonance
PACS: 78.67.-n, 65.80.-g, 73.23.-b, 68.49.Jk, 52.25.Os
1. Introduction
When the metal nanoparticle is illuminated with a laser beam by frequency which coincides with the
frequency of collective electron oscillations in the MN, the surface plasmon resonance (SPR) is excited.
The frequency and the width of SPR depends on the size, morphology, spatial orientation of MN and
on the dielectric environment [1, 2]. Resonance light scatterers are employed in various applications
ranging from surface enhanced Raman scattering [3], near-field scanning optical microscopes [4], to bio-
chemical imaging [5], surface enhanced fluorescence [6], subwavelength optical waveguides [7], cancer
therapy [8] etc.
The temperature effect on the optical and transport properties of the metal nanoparticles is very
important for pure and applied science of nanoparticles [9]. Since a MN absorbs laser energy in a thin-
surface layer, rapid local heating can occur at the surface. Thus, for a detailed analysis of the laser-light
absorption or reflection at MN surfaces, the properties of MNs at electron temperatures must be stud-
ied. The temperature dependence of the SP resonance is crucial due to a number of recent applications
of noble MNs in computer chips [10], thermally assisted magnetic recording [11], thermal cancer treat-
ment [12], catalysis and nanostructure growth [13]. The use of nano-objects as temperature sensors [14]
and thermometers [15] is quite promising due to their small sizes and short thermal relaxation time.
Previous calculations of the effect of the temperature on different properties of the MNs were per-
formed in the zero-temperature limit [16, 17] or in the interval 15÷ 1000 K only [9, 18–30]. While the
optical properties of both the bulk metals [31–35] and the MNs in the low-temperature regime are now
well studied, the understanding of the effects of electron temperature (typically of the order of 104 K)
on the plasmon modes has remained a challenge and is not so well understood as at the T = 0. Such an
out-of-equilibrium situation can easily be induced in metallic nanoparticles [36] using ultrashort optical
pulses. At time t = 0, only the electrons are excited by the laser pulse; then the excited electrons would
decay through light radiation, phonon excitation, and/or electron-electron scattering. In the most com-
monly used approach, the two-temperature model [34, 37, 38], assumes that both electronic and ionic
© N.I. Grigorchuk, 2013 33706-1
http://dx.doi.org/10.5488/CMP.16.33706
http://www.icmp.lviv.ua/journal
N.I. Grigorchuk
degrees of freedom are in thermal equilibrium conditions individually, but each degree of freedom has
its own temperature. The energy exchange between the electrons and the lattice in this model leads to a
time-dependent electronic temperature Te(t).
The effect of electron temperature was not studied in detail so far because a broad temperature inter-
val requires the use of materials with high thermal stability. In [22–25, 30], an improved Drudemodel was
used where the electron concentration and/or a plasma frequency are directly dependent on the electron
temperature. Only a few works have addressed this problem so far [39, 40].
In this work, we focus on optical properties of metal nanoparticles at relatively high electron tem-
peratures, whereas the ion temperature, Ti, is supposed to be much smaller [36]. Different behavior with
temperature of optical lines for MNs with different radii and shapes is a particular motivation for our
present study. The screening coming from the surrounding matrix is taken into account through a con-
stant dielectric function, ǫm.
To calculate the effect of the electron temperature on the optical and transport characteristics of
MNs, we use the kinetic theory method which accounts for the electron scattering on the MN boundary.
It is shown that the efficiency of absorption with temperature strongly depends on the size of MN. It is
predicted that the effect of temperature on the MNs with different radii can substantially modify their
optical properties. We present our theoretical results on the temperature dependence of the conductivity
and polarizability tensors related to the MNs having a spheroidal shape.
The rest of the paper is organized as follows. The Boltzmann equation approach to the problem is
presented in section 2. Section 3 contains the study of conductivity tensor in the spheroidal MNs at finite
temperatures. In section 4, we consider the effect of temperature on the polarizability of MN. Section 5 is
devoted to the study of light absorption and scattering crossection byMNs. The discussion of the obtained
results is available in section 6, while section 7 contains the conclusions.
2. Boltzmann equation
To account for the temperature effect on the conductivity and polarization properties of the MNs, we
use the Boltzmann equations approach. The advantage of this approach is that the obtained results can be
applied not only to MNswith a spherical shape, but to strongly anisotropic spheroidal (needle-like or disk-
shaped) MNs. Thus, it permits to study the effect of the particle shape on the physical values measured.
Second, the Boltzmann equation method enables us to investigate the MNs having sizes smaller than the
electron mean free path l . But in the case of sizes less than l , it provides the same results as known from
other approaches.
There exists a lower limit of applicability of Boltzmann method in a small radius limit when the par-
ticle size is comparable to the de Broglie wavelength of the electron, and the quantization of the electron
spectrum starts to play an essential role [41]. Practically, it is around a radius greater or less than 2 nm.
Let us consider a single ellipsoidal metal MNwith semiaxes a, b, c that is irradiated by an electromag-
netic (EM) wave whose electric field is given as
E = E0 exp[i(k ·r−ωt)]. (2.1)
Here, E0 is the amplitude of an electric field of a pump laser, ω is its frequency, k is the wave vector, and
r and t describe the spatial coordinates and time.
We restrict ourselves to the case of Rayleigh scattering, where the electromagnetic wavelength (from
a pump laser) λ∼ c/ω is much larger than the diameter of the nanoparticle d (=max{a,b,c}). Then, the
electromagnetic field around the MN can be considered as homogeneous. Placing the coordinate origin
in the center of the particle, the above-mentioned assumption is written as follows:
kr ≪ 1. (2.2)
The inequality (2.2) implies that the E field of the electromagnetic wave can be considered to be
spatially uniform on scales of the order of a particle size such that all the conduction electrons move
in-phase producing only dipole-type oscillations. The amplitude of such a field is linked to E0 by the
relation [42]
E
( j )
0 (0,ω)/E
( j )
in
(ω)= 1+L j [ǫ(ω)/ǫm−1], (2.3)
33706-2
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
where ǫ(ω) is the dielectric permittivity of the MN, ǫm is the dielectric constant of the adjacent medium
(i.e., solvent), and L j are depolarization factors in the j -th direction (in the principal axes of an ellipsoid).
The electric field Ein gives rise to high-frequency current inside the MN. To obtain the average density
of this current over the MN, it is necessary first of all to calculate the electron velocity distribution func-
tion. The field Ein has an effect on the equilibrium electron velocity distribution and thus determines the
appearance of a nonequilibrium addition f1(r,v, t) to the Fermi distribution function
f0(ε) = 1
exp
[
ε−µ
kBT
]
+1
.
Here, µ is the chemical potential, kB is Boltzmann’s constant, ε = mυ2/2 denotes the kinetic energy of
an electron, υ = |v| refers to the electron velocity, and m is the electron mass. As is well known [43,
44], the equilibrium function f0(ε) does not give any input to the current. Accounting for both the time
dependence of equation (2.1) and the inequality (2.2), the total distribution function of electrons can be
represented as follows:
f (r,v, t) = f0(ε)+ f1(r,v, t) ≡ f0(ε)+ f1(r,v)eiωt . (2.4)
We seek the function f1(r,v) as a solution to the linearized Boltzmann’s equation
(ν− iω) f1(r,v)+v
∂ f1(r,v)
∂r
+eEinv
∂ f0(ε)
∂ε
= 0, (2.5)
where e is the electron charge. In equation (2.5) we have assumed that the collision integral
(∂ f1/∂t)col =− f1/τ
is evaluated in the relaxation time approximation (τ = 1/ν, ν refers to electron collision frequency).
Strictly speaking, ν≡ ν(T ) is a temperature dependent value which can be presented as follows:
ν(T ) = nee2
m
K
Θ
(
T
Θ
)5
Θ/T
∫
0
z5ez
(ez −1)2
dz, (2.6)
where ne is the electron concentration, Θ is the Debye temperature, and K combines together the factors
depending on the details of the Fermi surface geometry and scattering matrix elements.
For simplicity, we also assume that the vortex electric field induced by magnetic component of the
external EM field gives a comparatively small input at the plasmon resonance frequencies and can be
neglected in equation (2.5).
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 electron reflection from the inner walls of the MN. We adopt, as is
usually done, the assumption of diffusive electron scattering by the boundary of MN. Then, the boundary
conditions can be presented in the form
f (r,v)|S = 0, vn < 0, (2.7)
where vn is the velocity of the component normal to the particle surface.
Along with diffusive scattering, the mirror boundary conditions at the nanoparticle surface were
examined in the literature for electron scattering (see, e.g., [43, 44]). 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. For the mirror mechanism to be dominant, the surface must
be perfectly smooth in the atomic scale because 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 was shown [45], the mirror boundary conditions give a small correction to the results
obtained with the account of only the diffusive electron reflections. Therefore, we choose more realistic
boundary conditions given by equation (2.7).
The boundary conditions (2.7) in the case of an ellipsoidal MN, broadly speaking, depend on the an-
gles, which complicates the solution of equation (2.5).
33706-3
N.I. Grigorchuk
It is rather easy to solve (2.5) and to satisfy the boundary conditions of (2.7) if one passes to a trans-
formed coordinate system, where an ellipsoid with semiaxes a,b,c (along the x, y , and z directions,
respectively) transforms into a sphere of radius R with the same volume:
x j =
d j
R
x′
j , R = (abc)1/3 , (2.8)
with j = 1, 2, 3, and d1 = a, d2 = b, d3 = c. A similar transformation should be made for the electron
velocities:
υ j =
d j
R
υ′j . (2.9)
Equation (2.5) and the boundary conditions (2.7) in transformed coordinate and velocity systems can
be rewritten as follows:
(ν− iω) f1(r′,v′)+v′
∂ f1(r′,v′)
∂r′
+eEinv′
∂ f0(ε)
∂ε
= 0, (2.10)
f (r′,v′)|r ′=R = 0, r′ ·v′ < 0. (2.11)
The first condition calls for zero equality of the electron distribution function at the nanoparticle surface,
and the second one, r′ · v′ < 0, means that an electron motion occurs only inside MN, and there is no
electron leakage through the NP surface.
Equation (2.10) presents the partial differential equation with boundary conditions (2.11). Using the
method of characteristics, one can obtain the solution to this equation in the form
f1(r′,v′, t) =−e
∂ f0
∂ε
v′Ein
1−exp[−(ν− iω) tc(r′,v′)]
ν− iω
, (2.12)
where the characteristic tc(r′,v′) can be presented as follows:
tc(r′,v′) = 1
v
′2
[
r′v′+
√
(R2 − r
′2)v
′2 + (r′v′)2
]
. (2.13)
The radius vector R determines the starting position of an electron at the moment tc = 0. The characteris-
tic curve of equation (2.13) depends only on the absolute value of R and does not depend on the direction
of R.
It is reasonable to point out that, in spite of the electric field which remains spatially uniform in-
side the MN, the distribution function (2.12) still depends on the coordinates due to the requirement to
obey the boundary conditions (2.11). Owing to this dependence, other physical parameters averaged with
f1(r′,v′) start to depend on coordinates too. Since the physical sensing involves only parameters averaged
over the whole MN volume V , it is necessary to fulfill the integration over all the coordinates inside the
MN. One can see this exemplified just below.
Using the solution (2.12), one can calculate the density of a high-frequency current induced by the EM
wave inside the MN. Performing the Fourier transformation of equation (2.12), we obtain
j(ω) = 2e
V
( m
2πħ
)3
∫∫
V
∫
d3r ′
∫∫∫
d3υ′ v′ f1(r′,v′,ω). (2.14)
3. Electric-conductivity tensor
Let us introduce the tensor of electric conductivity σαβ(ω) using the relationship
jα(ω)=
3
∑
β=1
σαβ(ω)E
(β)
in
(ω). (3.1)
Then, in accordance with both equations (2.12) and (2.14), the components of this tensor can be pre-
sented as follows:
σαβ(ω) =−2e
V
( m
2πħ
)3
∫∫
V
∫
d3r ′
∫∫∫
d3υ′ υα
[
eυβ
∂ f0
∂ε
1−e−(ν−iω)tc(r ′ ,υ′)
ν− iω
]
. (3.2)
33706-4
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
Electric-conductivity tensor σαβ = σβα is the symmetrical second-rank tensor. Since the integrand in
equation (3.2) is an odd function for nondiagonal tensor components α , β, and the integration is con-
ducted over the whole velocity space [−∞, ∞], only diagonal components in this equation have been
retained.
If we introduce a rectangular coordinate system and choose the fundamental directions x, y , z to be
coincident with the three principal axes of the ellipsoids, then we get
σ=
σxx 0 0
0 σy y 0
0 0 σzz
.
The tensor of electric conductivity, as can be seen from equation (3.2), is a complex value:
σαβ(ω)=σ′
αβ(ω)+ iσ′′
αβ(ω). (3.3)
The surface effect on the conducting phenomenon is described in equation (3.2) by means of a char-
acteristic tc(r′,v′). It accounts for the restrictions imposed on the electron movement by nanoparticle
surfaces. As one can see from equation (2.12), the value of tc is of the order of t ′ ∼ R/υF, where υF is
the Fermi velocity. This implies that the value reciprocal to tc will correspond to the vibration frequency
between the particle walls. Hence, the inequality νtc ≫ 1 indicates that the electron collision frequency
inside the MN bulk would significantly exceed the one for an electron collision with the surface of MN.
If this inequality is satisfied, one can direct tc → ∞. Then, the exponent in equation (3.2) vanishes and
finally we arrive at a standard expression for an electric-conductivity [1, 42]
σ(ω) = 1
4π
ω2
pl
ν− iω
, (3.4)
with
ω2
pl =
4e2m2υ3
F
3πħ3
. (3.5)
To be sure of that, it is necessary to take into account that the energy derivative of f0 in the zero
approximation in the small ratio of kBT /εF (εF is the Fermi energy) can be replaced by
∂ f0
∂ε
≈−δ(ε−εF), (3.6)
and passes in equation (3.2) to the integration over υ in the spherical coordinate system
∫
∞
∫
−∞
∫
d3 υ→
2π
∫
0
dϕ
π
∫
0
sinθdθ
∞
∫
0
υ2 dυ,
with the use of the formula ∞
∫
0
υ4 δ(υ2 −υ2
F)dυ=
υ3
F
2
. (3.7)
We denote with ϕ and θ the azimuthal and polar angles with respect to the ellipsoid rotation axis z, re-
spectively. Note here that only diagonal terms with υα = υβ = υ are retained after integration over all
angles. As one can see from equation (3.4), the conductivity becomes a scalar quantity in this approxima-
tion.
In the general case of an ellipsoidal-shaped MN, the electric conductivity is the tensor quantity (3.2).
Integrating it over all nanoparticle coordinates in a solid angle dΩ= sinθ dϕ dθ gives
1
V
∫∫
V
∫
d3r ′
[
1−e−(ν−iω)t ′(r′,υ′)
]
= 3
4
Ψ(ω,υ′), (3.8)
where we use
33706-5
N.I. Grigorchuk
∞
∫
0
dz z3 Erfc
(
z + a
z
)
= 1
4
(
a + 3
4
)
e−4a , (3.9)
provided that Re a > 0, and
∞
∫
0
dz z3 Erfc
(
z − a
z
)
= 1
2
(
a2 +a + 3
8
)
. (3.10)
The complex Ψ function entering the equation (3.8) has the form
Ψ(ω,υ′) =Φ(ω,υ′)−
4
q2
(
1+
1
q
)
e−q , (3.11)
with
Φ(ω,υ′) = 4
3
− 2
q
+ 4
q3
, q ≡ q(ω,υ′) = 2R
υ′
(ν− iω) , (3.12)
and υ′ (= ςυ) is a “deformed” electron velocity [17] with the coefficient of “deformation” ς j = R/d j . The
last summand in equation (3.11) represents the oscillation part of the Ψ function and the first one refers
to its smooth part.
Accounting for equation (3.8) and only diagonal components in equation (3.2), there remain the inte-
grals over all electron velocities. To calculate them, we pass to a spherical coordinate system with the z
axis directed along the rotation axis of the spheroid (as we have done it above). Then, equation (3.2) can
be rewritten as follows:
σ j j (ω) = 3e2m3
2(2πħ)3
1
ν− iω
2π
∫
0
dϕ
π
∫
0
sinθ dθ
∞
∫
0
υ2dυυ2
j
(
−∂ f0
∂ε
)
Ψ(ω,υ′), (3.13)
where υ j is the j -th component of the electron velocity, j = x, y, z, with
υ2
(x
y
) = υ2 sin2θ ·
(cos2ϕ
sin2 ϕ
)
,
υ2
z = υ2 cos2θ,
respectively.
It should be noted that the “deformed” electron velocity entering the Ψ function can be expressed
through the electron velocity in a Cartesian coordinate system as follows:
υ′ = υR
√
(
cos2ϕ
a2
+ sin2ϕ
b2
)
sin2θ+ cos2θ
c2
. (3.14)
Let us suppose that the particle in the matrix is modelled as a rotationally symmetric ellipsoid (a =
b ≡ R⊥, c ≡ R‖) with the symmetry axis along the z direction. The components of an electron velocity
parallel (υ‖) and perpendicular (υ⊥) to the spheroid revolution axis
υ‖ = υz = υcosθ, υ⊥ =
√
υ2
x +υ2
y = υsinθ (3.15)
play an important role in this case, and the υ′ ceases to depend on the angle ϕ.
Let us pass from an integration in equation (3.13) over electron velocities to the integration over
electron energies
υ4 dυ= 1
m
(
2ε
m
)3/2
dε (3.16)
and take into account the integrals
2π
∫
0
cos2ϕ dϕ=
2π
∫
0
sin2ϕ dϕ=π. (3.17)
33706-6
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
Then, with the use of equations (3.13), (3.16), and (3.17), we obtain for the main components of the com-
plex electric conductivity tensor, namely for
σxx =σy y ≡σ⊥ , σzz =σ‖ , (3.18)
the expression
σ(‖
⊥
)(ω) = 3e2
2π2
p
2m
ħ3
1
ν− iω
π/2
∫
0
(
cosθ′ sin2θ′
1
2 cos3θ′
)
dθ′
∞
∫
0
Ψ(ω,ε′θ′ )
(
−∂ f0
∂ε
)
ε3/2 dε
. (3.19)
We denote with θ′(= θ−π/2) the angle between the direction of an electron velocity and an axis perpen-
dicular to the spheroid rotation axes. Here, and below, the upper (lower) symbol in the parentheses on
the left-hand side of equation (3.19) corresponds to the upper (lower) expression in the parentheses on
the right-hand side of this equation.
TheΨ function in (3.19) depends now on both the energy ε′ and the angle θ′ because the parameter q
[see (3.12) and (3.14)] for a spheroidal particle becomes dependent on the angle θ and can be determined
as
q =
√
2m
ε
(ν− iω)
(
sin2θ
R2
⊥
+ cos2θ
R2
‖
)−1/2
≡ q(θ,ε), (3.20)
where R‖ and R⊥ are the semiaxes of the spheroid. The electron energy ε′ entering the Ψ function, be-
comes dependent on the angle θ too, and for a spheroid can be presented as follows:
ε′θ′ = εR2
(
cos2θ′
R2
⊥
+
sin2θ′
R2
‖
)
, (3.21)
where the equation (3.14) was used. Since the spheroid semiaxes R⊥ and R‖ can be easily expressed
through the radius of a sphere R of an equivalent volume
R⊥ = R
(
R⊥
R‖
)1/3
, R‖ = R
(
R⊥
R‖
)−2/3
, (3.22)
the energy ε′ does not depend on the particle radius R but depends only on the spheroid axes ratio.
If we restrict ourselves here to the case of low temperatures, then the equation (3.6) can be used, and
we find
∞
∫
0
Ψ(ω,ε′θ′ ) δ(ε−εF)ε3/2 dε=µ3/2
0 Ψ(ω,µ′
θ′ ) , (3.23)
where µ0 is the chemical potential at zero temperature. Substituting (3.23) into equation (3.19) and ac-
counting that the electron concentration at T = 0 can be presented as
n0 =
(2mµ0)3/2
3π2ħ3
, (3.24)
it is easy to check that equation (3.19) transforms to the form known from (see, e.g., [46]).
3.1. Conductivity of a spherical MN
For particles having a spherical shape, the electric conductivity becomes a scalar quantity, and one
can put R‖ = R⊥ ≡ R, ε′ = ε in equations (3.20) and (3.16); then, q and the Ψ function cease to dependent
on the angle θ′,
π/2
∫
0
(
cosθ′ sin2θ′
1
2 cos3θ′
)
dθ′ =
1
3
, (3.25)
33706-7
N.I. Grigorchuk
and equation (3.19) reduces to the form
σsph(ω) =
e2
p
2m
2π2ħ3(ν− iω)
∞
∫
0
Ψ(ω,ε)
(
−
∂ f0
∂ε
)
ε3/2 dε. (3.26)
Let us make the following change of variables in equation (3.26)
η= ε−µ
kBT
, dη= dε
kBT
. (3.27)
Then
− ∂ f0
∂ε
dε= e−η
(1+e−η)2
dη , (3.28)
and the integral in (3.26) can be presented as
∞
∫
−µ/(kBT )
Ψ(ω,η) (µ+ηkBT )3/2 e−η
(1+e−η)2
dη≡ I , (3.29)
where
Ψ(ω,η) = 4
3
−
√
2
m
(µ+ηkT )
R(ν− iω)
+
[
2
m
(µ+ηkT )
]3/2
2R3(ν− iω)3
−
2
m
(µ+ηkT )
R2(ν− iω)2
1+
√
2
m
(µ+ηkT )
2R(ν− iω)
exp
− 2R(ν− iω)
√
2
m
(µ+ηkT )
,
(3.30)
with [47]
µ≡µ(T ) ≃µ0
[
1− π2
12
(
kBT
µ0
)2]
. (3.31)
Since the input of the integrand tends to zero as η→−∞, the lower limit in this integral can be extended
to the −∞. The derivative (−∂ f0/∂ε) has a maximum at the point ε = µ. So, it is conveniently to expand
the product of Ψ · (µ+ηkBT )3/2 ≡Ξ(η) in the powers of η
Ξ(η) ≃ µ3/2
Ψ(0)+
[
3
2
kBT
p
µΨ(0)+µ3/2
Ψ
′(0)
]
η
+ 1
8
p
µ
[
3(kBT )2
Ψ(0)+12kBTµΨ′(0)+4µ2
Ψ
′′(0)
]
η2 +0[η]3 , (3.32)
with
Ψ(0) =Ψ(ω,η)
∣
∣
∣
η→0
, Ψ
′(0) = ∂
∂η
Ψ(ω,η)
∣
∣
∣
η→0
, Ψ
′′(0) = ∂2
∂η2
Ψ(ω,η)
∣
∣
∣
η→0
. (3.33)
Note, that Ψ(0) at υ′ = υF coincides with the Ψ(ω) defined by equation (3.11).
Then, the integral (3.29) can be presented as the sum of integrals
I ≃ µ3/2
Ψ(0) I1 +
[
3
2
kBT
p
µΨ(0)+µ3/2
Ψ
′(0)
]
I2
+1
2
[
3(kBT )2
4
p
µ
Ψ(0)+3kBT
p
µΨ′(0)+µ3/2
Ψ
′′(0)
]
I3 , (3.34)
where the integrals I j =
∫∞
−∞ dηη j−1 exp(−η)
/[
1+exp(−η)
]2
are calculated to be: I1 = 1, I2 = 0 and I3 =
π2/6.
33706-8
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
Using I j and (3.27), equation (3.26) transforms into
σsph(ω) =
e2
√
2mµ
2π2ħ3(ν− iω)
{
µ
[
1+
π2
8
(
kBT
µ
)2]
Ψ(0)+
π2
6
[
3kBT Ψ
′(0)+µΨ′′(0)
]
+0[η]3
}
. (3.35)
Here,
Ψ
′(0) = kBT
µqµ
[
−1+ 6
q2
µ
− 6
qµ
e−qµ
(
1+ 1
qµ
+
qµ
3
)
]
, (3.36)
Ψ
′′(0) = (kBT )2
2µ2qµ
[
1+ 6
q2
µ
−4e−qµ
(
1+
qµ
2
+ 3
2qµ
+ 3
2q2
µ
)]
, (3.37)
with
qµ = 2R(ν− iω)
√
2µ/m
. (3.38)
Substituting (3.36)–(3.37) into equation (3.35), we obtain
σsph(ω,T ) = me2µR
π2ħ3qµ
[Ψ(0)+αT ΨT (ω)] , (3.39)
with
αT = π2
6
(
kBT
µ
)2
, (3.40)
ΨT (ω)=ΦT (ω)−
[
1+ 8
qµ
+ 24
q2
µ
(
1+ 1
qµ
)
]
e−qµ , (3.41)
ΦT (ω)= 1− 4
qµ
+ 24
q3
µ
. (3.42)
When the temperature T → 0 in equation (3.40), then (3.39) is reduced to the form
σsph(ω)=
e2µ3/2
0
p
2m
2π2ħ3
Ψ(0)
ν− iω
, (3.43)
known from other calculations (see, e.g., [48]).
The conductivity of MN given by (3.39) is a complex value. Practically, we need to know both the real
and the imaginary parts of it.
Further analytical calculations are possible for some particular cases. In the case of a spherical MN,
there are three actual frequencies that are usually considered: the frequency of an incident electromag-
netic field ω, the collision frequency of electrons in the particle volume ν, and the vibration frequency
between the particle walls νs = υF/(2R) (provided the particle size is less than the electron mean free
path). When ν> νs, the mechanism of an electron scattering in the bulk dominates, and an electron scat-
tering from the particle surface gives only small corrections of the order of νs/ν. However, a much more
interesting case is when the mechanism of the surface electron scattering dominates, which corresponds
to the inequality ν< νs.
Let us suppose that the mean free path of a conduction electron l is much greater than the particle
size. In this case, an electron scattering occurs mainly from the inner surface of the MN. The electrons
oscillate between the walls of the particle with the frequency νs. In the case of ν≪ νs, to a first approxi-
mation, one can formally put ν→ 0. If one introduces the designation
qi =
ω
νs
, (3.44)
then for real and imaginary parts of the ratiosΨ/(ν− iω) andΨT /(ν− iω) we obtain
Re
[
Ψ
ν− iω
]
ν→0
= 1
ω
[
2
qi
− 4
q2
i
sin qi +
4
q3
i
(1−cos qi )
]
, (3.45)
Im
[
Ψ
ν− iω
]
ν→0
= 4
ω
[
1
3
+ 1
q2
i
(
cos qi −
sin qi
qi
)
]
, (3.46)
33706-9
N.I. Grigorchuk
and
Re
[
ΨT
ν− iω
]
ν→0
= 1
ω
[(
1− 24
q2
i
)
sin qi +
4
qi
(
1+ 6
q2
i
)
+ 8
qi
(
1− 3
q2
i
)
cos qi
]
, (3.47)
Im
[
ΨT
ν− iω
]
ν→0
= 1
ω
[
1−
(
1− 24
q2
i
)
cos qi +
8
qi
(
1− 3
q2
i
)
sin qi
]
. (3.48)
Substituting (3.45)–(3.48) into equation (3.39), one obtains for the real and imaginary parts of σ the fol-
lowing expressions:
σ′
sph(ω,T ) = 3
4
e2ne(T )
mω
{
2
qi
− 4
q2
i
sin qi +
4
q3
i
(1−cos qi )
+ αT
[
4
qi
+ 24
q3
i
+
(
1− 24
q2
i
)
sin qi +
8
qi
(
1− 3
q2
i
)
cos qi
]}
, (3.49)
σ′′
sph(ω,T ) = 3
4
e2ne(T )
mω
{
4
3
+ 4
q2
i
(
cos qi −
sin qi
qi
)
+ αT
[
1+ 8
qi
(
1− 3
q2
i
)
sin qi −
(
1− 24
q2
i
)
cos qi
]}
, (3.50)
provided that ν≪ νs. In the presented equations, the electron concentration depends on the temperature
as [47]
ne(T ) = n0
{
1+ π2
8
[
kBT
µ
]2}
. (3.51)
If one neglects the oscillation terms in the equation (3.50), then we obtain
σ′′
sph(ω,T ) ≃
ω2
pl
(T )
4πω
(
1+ 3
4
αT
)
, (3.52)
with ω2
pl
(T ) = 4πe2ne(T )/m.
At T → 0, equations (3.49) and (3.50) agree with the expressions known from earlier calculations
[46, 48].
3.2. Linewidth and “figure of merit”
The linewidth of the particle-plasmon resonance is controlled by lifetime broadening due to various
decay processes. Knowing the expressions for conductivity tensor (3.19) and (3.49), the halfwidth of the
plasmon line absorption can be calculated simply by means of a relation
Γ j (ω,T ) = 4π
ǫ∞+n2(1/L j −1)
Reσ j j (ω,T ), (3.53)
where n(= p
ǫm) is the refractive index of the surrounding medium and ε∞ ≡ 1+ ǫinter is the high fre-
quency dielectric constant due to interband and core transitions of the inner electrons in aMN’s material.
The “figure of merit” (FOM) can be expressed through the σ j j as well. For MNs having a spheroidal
shape we found [48]
FOM(‖
⊥
) =
nωpl
[
1/L( ‖
⊥
) −1
]
4π
√
ε∞+
[
1/L( ‖
⊥
)−1
]
n2
σ′−1
(‖
⊥
) (ω,T ), (3.54)
with geometrical factors [49]
L‖ =
1
(1+R‖/R⊥)1.6
, L⊥ = 1
2
(1−L‖) . (3.55)
33706-10
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
Finally, it should be reminded that the real and imaginary parts of a dielectric constant tensor ǫ j j can
be easily obtained with the help of tensor σ j j , as [47]
ǫ′j j (ω,T ) = ǫ∞− 4π
ω
σ′′
j j (ω,T ), (3.56)
ǫ′′j j (ω,T ) = 4π
ω
σ′
j j (ω,T ). (3.57)
The sign in the last equation and before the second term in equation (3.56) corresponds to the exponent
sign in time dependence of an external electric field E(t) = E0e−iωt . The imaginary part of the dielectric
constant provides basically the damping and broadening of SPR.
The linewidth of SPR and FOM are strongly dependent on the particle radius. We will illustrate this
dependence below, having built the temperature behavior of SPR linewidth for different R as an example.
The SPR linewidth is plotted in figure 1 against temperature for spherical Au nanoparticles with two
different radii embedded in the water (n = 1.33). Calculations weremade according to the formulas (3.53)
and (3.49) for the following parameters of Au nanoparticle [50]: υF = 1.394×108 cm/s, TF = 6.41×104 K,
ωpl = 1.37×1016 s−1, and ǫ∞ = 9.84. The electron concentration was estimated from equation (3.51) with
n0 given above by equation (3.24). For Au, n0 ≃ 5.9×1022 cm−3. The behavior of Γ(T ) is governed by the
temperature dependence of the ǫ′′(T ) in accordance with equations (3.53) and (3.57).
Figure 1. (Color online) SPR linewidth vs temperature for spherical Au particles with different R (Å): 50
(1) and 54.7 (2), embedded in water.
High temperature of an electron gas can be achieved inside the MN by using laser pumping at SPR
frequency when hot electrons are produced. As the temperature of an electron gas decreases after an
initial excitation, the SPR linewidth can increase (see curve 1, figure 1) or decrease (curve 2) depending
on the particle radius. Such different behavior is due to the linewidth oscillations with the particle radius
changing [48]. Earlier [30, 33], the results for Ag and Au nanoparticles were reported where the linewidth
is only reduced linearly with the temperature lowering over a range of crystal temperatures.
Let us calculate the polarizability tensor for MNs at finite temperatures.
4. Polarizability tensor
The dipole electric moment for a spherical particle embedded in the media with dielectric constant
ǫm, can be written as follows:
d j (ω)= ǫm
3
∑
k=1
α j k (ω)Ek (ω), (4.1)
where Ek is the k component of an external electric field and α j k is the polarizability tensor of the MN.
The calculation of d is especially simple for the case when the particle sizes are small compared to some
33706-11
N.I. Grigorchuk
“wavelength” δ∼ c/(
p
|ǫ|ω), which corresponds to the frequency ω in the particle bulk. In this case, one
can calculate the polarizability of MN using the formulae obtained for an external uniform statical field.
In general terms, this tensor can be treated as being a complex value
α j k (ω)=α′
j k (ω)+ iα′′
j k (ω). (4.2)
Tensor of electric conductivity was introduced by equation (3.1). The component of the inner elec-
tric field in equation (3.1) is linked to the component of an external electric filed in equation (4.1) by
expression (2.3). Since the density of electric current is related to the dipole moment by a simple relation
j (t)=
1
V
∂
∂t
d(t), (4.3)
it is easy to establish the connection between polarizability and conductivity of MN. Performing the
Fourier transformation of equation (4.3), we obtain
j (ω)= iω
V
d(ω). (4.4)
Then, using equations (4.1), (4.4), and (3.1), the polarizability tensor can be expressed through the con-
ductivity tensor α j j by means of
α j j (ω,T ) =−
V
ǫmω
iσ j j (ω,T )
1+L j [ǫ j j (ω)/ǫm−1]
. (4.5)
In equation (4.5) for MN with a spheroidal shape, parameter V represents the volume of spheroid (=
4
3πR2
⊥R‖).
Using equations (3.56) and (3.57) for an isolated MN in host, we come to the Clausius-Mossotti dipole
polarizability [1]
α j j (ω,T ) = V
4πL j
ǫ j j (ω,T )−ǫm
ǫ j j (ω,T )+ (1/L j −1)ǫm
, (4.6)
where tensor ǫ j j corresponds to a given frequency ω, and in the case of frequencies close to the plasma
oscillations of electrons in metal, it can be presented as follows:
ǫ j j (ω,T ) ≃ ǫ∞−
(ωpl
ω
)2
+ 4π
ω
iσ j j (ω,T ). (4.7)
Optical absorption generally measures the Imǫ(ω,T ).
5. Light absorption and scattering
In classical electrodynamics it is supposed that the light scattering results from the polarization of the
scattered particle when it is illuminated with a beam of light. Since the spatially uniform field causes only
a dipole polarization, the nanoparticle scatters the light like a vibration dipole [1]. In the case of particles
of a spheroidal shape, the dipole moment of the MN can be presented in the form [51]
d =α⊥ǫmE+ (α‖−α⊥)ǫm(mE)m , (5.1)
where m is the unit vector directed along the spheroid axis of revolution. Accounting for m ·E = E cosφ,
where φ is the angle between the spheroid axis of revolution and the direction of the electric field E, the
equation (5.1) can be rewritten as follows:
d =
√
α2
⊥ sin2φ+α2
‖ cos2φ ǫmE . (5.2)
Using our forerunning formulae for d and α, it is easy to calculate the temperature dependence of
light absorption and scattering. The overall absorption and scattering cross-sections for a spheroidal MN
33706-12
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
are given by
Cabs = 4π
ω
c
p
ǫm
[
(Imα‖)cos2φ+ (Imα⊥)sin2φ
]
, (5.3)
Csca =
8
3
π
(ω
c
p
ǫm
)4
(
|α‖|2 cos2φ+|α⊥|2 sin2φ
)
. (5.4)
The square of diagonal components of the polarizability tensor can be presented in the Lorenzian form
as follows:
|α(‖
⊥
)|2 =
[
V
4πL( ‖
⊥
)
]2
[
(1−ξm)ω2 −ω2
(‖
⊥
)
]2
+
[
2ωγ(‖
⊥
)
]2
[
ω2 −ω2
(‖
⊥
)
]2
+
[
2ωγ(‖
⊥
)
]2
, (5.5)
and the imaginary part of the polarizability tensor is
Imα(‖
⊥
) =
[
V
4πL(‖
⊥
)
]
2ω3ξmγ(‖
⊥
)
[
ω2 −ω2
(‖
⊥
)
]2
+
[
2ωγ(‖
⊥
)
]2
. (5.6)
Here,
ξm = ǫm
ǫm+L(‖
⊥
)(1−ǫm)
, (5.7)
ω2
(‖
⊥
) ≡ω2
(‖
⊥
)(T ) =
L( ‖
⊥
)
ǫm+L( ‖
⊥
)(1−ǫm)
ω2
pl(T ) (5.8)
are the ‖ and ⊥ surface plasmon frequencies, and
γ(‖
⊥
) ≡ γ(‖
⊥
)(ω,T ) =
2πL( ‖
⊥
)
ǫm+L( ‖
⊥
)(1−ǫm)
Reσ(‖
⊥
)(ω,T ) (5.9)
is the half-width of the resonance for light polarized along (‖) or across (⊥) the rotation axis of the
spheroid, σ‖, σ⊥ are the corresponding components of the conductivity tensor, and L‖, L⊥ are the ge-
ometrical factors given above by equations (3.55).
On resonance, the polarizability tensor looks as follows:
α(‖
⊥
)(ω,T ) ≃ V
8π
ξmω(‖
⊥
)(T )
L(‖
⊥
)γ(‖
⊥
)(ω,T )
, (5.10)
with γ(‖
⊥
) taken at ω=ω(‖
⊥
). The contribution of nonresonant frequencies is so small that their input into
the scattering cross section can be neglected.
In the case of MN having a spherical shape L‖ = L⊥ = 1/3, equations (5.7)–(5.9) become as follows:
ξm = 3ǫm
2ǫm+1
, (5.11)
ω2
(‖
⊥
)(T ) ≡ωsph =
ω2
pl
(T )
2ǫm+1
, (5.12)
γ(‖
⊥
)(ω,T ) ≡ γsph(ω,T ) = 2π
2ǫm+1
σ(ω,T ) . (5.13)
Then, the expressions (5.3) and (5.4) can be easily transformed to the forms
Cabs(ω,T ) = 9V ε3/2
m
ω
c
ǫ′′(ω,T )
[ǫ′(ω,T )+2ǫm]2 + [ǫ′′(ω,T )]2
, (5.14)
Csca(ω,T ) = 3V 2
2π
ǫ2
m
(ω
c
)4 [ǫ′(ω,T )−ǫm]2 + [ǫ′′(ω,T )]2
[ǫ′(ω,T )+2ǫm]2 + [ǫ′′(ω,T )]2
, (5.15)
33706-13
N.I. Grigorchuk
where we have combined equations (5.11)–(5.13) with the real and imaginary parts of equation (4.7). The
most intense cross-sections are observed at the frequency which corresponds to the plasmon resonance
of a spherical nanoparticle in the vacuum. Formulae (5.15) and (5.14) yield a resonance when ǫ′(ω,T )
= −2ǫm. The absorption and the scattering cross-sections defined by equations (5.3) and (5.4) reach the
maximal values at the frequencies of the SPRsω=ω(‖
⊥
) as well. Near the surface, plasmon resonance light
may interact with the particle (at low temperatures) over a cross-sectional area usually larger than the
geometric cross section of the particle because the polarizability of the particle becomes very high in this
frequency range.
We calculate the scattering efficiency of the MN, which is defined as the ratio
Sef =
C
Cgeom
, Cgeom =π
(
c
ωpl
)2
, (5.16)
where Cgeom is the geometrical cross-section of an individual particle and C is defined by equations (5.3)
and (5.4), or by (5.14) and (5.15).
Figure 2. (Color online) The absorption crossection vs temperature for spherical Au particles with differ-
ent R (Å): 78 (1), 100 (2), 125 (3), and 150 (4), embedded in water.
Figure 2 shows the variation of an absorption efficiency as a function of temperature for gold
nanoparticles having a spherical shape embedded in water (n = 1.33). The calculations were performed
in accordance with equations (5.16) and (5.14), with the use of the same parameters as above for figure 1.
For zero temperature, we get the classical result [1, 2], when the SPR intensity depends on the parti-
cle radius: larger particles have a larger scattering or absorption cross sections. The situation drastically
changes at finite electron temperatures. As we can see from figure 2, the absorption efficiency not only in-
creases (right hand sides of curves 1–4) with the temperature drop, but can also decrease (left hand sides
of curves 1–4). We have an interesting situation when the absorption crossection with the temperature
lowering increases at first, reaches its peak, then decreases, and starts to increase again. These observa-
tions are explained by the conductivity of MN, which oscillate with the particle radius changing [46].
In the case of light scattering (see figure 3), the effect of MN size is more pronounced because the
scattering cross-section is proportional to the square of the particle volume.
In figure 3, one can see that the efficiency of light scattering at a resonance frequency does not only
increase (curves 1–3) with the growth of the MN radius, but can also decrease (curve 4). The latter means
that the efficiency of scattering at a resonance frequency can be suppressed with the temperature rise for
some particle radii. Depending on the electron temperature, the scattering efficiency reaches the maxi-
mum value, whose peak position depends on the nanoparticle radius.
The peak positions of the resonance plasmon absorption (or scattering) by Au nanoparticle embed-
ded in water can be shifted toward shorter wavelengths as the temperature of electron gas is lowered
33706-14
Temperature dependence of plasmon resonances in spheroidal metal nanoparticles
Figure 3. (Color online) The same as in figure 2, but for scattering crossection.
after an initial excitation. This behavior is consistent with the known temperature-dependent shift of the
differential transmission peak in an isolated sodium nanoparticle [26].
6. Discussion of results
Here, we shortly discuss the effects caused by deviations from the spherical particle shape. When the
symmetry of the particle decreases, the number of resonance peaks increases. If the shape of a particle
deviates from the spherical one, the 3-fold degeneracy dipole mode splits into two (for spheroid) or three
(for ellipsoid) modes giving rise to corresponding scattering peaks. In this case, the plasmon response
strongly depends on the particle position relative to the illumination direction. To evaluate the electron
temperature effect on the absorption or scattering crossections for various illumination directions, one
can perform numerical calculations using equations (5.3)–(5.9) and (3.19).
The particle absorption or scattering effects depend on the size and shape of individual particles.
Metallic particles that are much smaller than the wavelength of light tend to absorb still more and hence
the absorption dominates. The scattered light intensity from small MNs is extremely low. Indeed, Rayleigh
scattering theory predicts that the scattering cross section of a spherical cluster should drop as the sixth
power of the diameter.
The quantity that is measured experimentally is frequency-resolved time-dependent transmis-
sion [26] Υon after the excitation of the MN by a pump laser pulse. Then, the static equilibrium trans-
mission Υoff of the sample measured in the absence of the pump is subtracted to obtain the relative
differential transmission [19] δΥ/Υ=(Υon−Υoff)/Υoff. In the thermal equilibrium, the δΥ can be related
to the temperature dependent absorption cross section by [37]
δΥ
Υ
=− 3
2πR2
[Cabs(ω,T )−Cabs(ω,T0)] , (6.1)
where T0 is the temperature of the MN before excitation. This relation holds for very small sizes of MN
when the reflectivity of the MN is very small.
A direct comparison of theoretical results with most of the available experimental measurements of
the optical properties of MNs is still a matter of debate because inhomogeneities in nanoparticle size,
shape, and local environment hide the homogeneous width of the surface plasmon resonance.
33706-15
N.I. Grigorchuk
7. Conclusions
We have applied a kinetic theory to the calculation of the optical properties of a metal nanoparti-
cle embedded in a dielectric media at different temperatures. It permits to determine the cross-section
of light absorption or scattering for various polarizations of the incident electromagnetic wave. The ob-
tained analytical formulas provide the evaluation of the dynamics of light absorption or scattering inten-
sities at the plasmon frequencies for nanoparticles with different radii and shapes, when the temperature
of electron gas becomes settled after the action of a laser pulse.
The analytical expressions for electroconductivity and polarizability tensors have been obtained for
the case of MNs with a spherical shape. They permit to calculate a number of other physical quantities
(e.g., resonance linewidth, figure of merit, etc.) for a particular temperature. We theoretically studied the
variation of both the linewidth of SPR and the efficiency of the light absorption and scattering byMNwith
the change of the electron temperature. The case where the frequency of a laser beam is close to surface
plasmon frequencies of a spheroidal MN is studied in detail.
The high sensitivity of the temperature dependence of SPR linewidth to the radius of a particle was es-
tablished. Even a small variation in the particle radius can drastically change the run of the SPR linewidth
curve vs temperature. This is due to the linewidth oscillations when the radius of MN is varied. The ef-
ficiencies of light absorption and scattering for various temperatures strongly depend on the particle
radius as well. The Au nanoparticles with different radii are used for illustration.
Acknowledgement
The author is indebted to Prof. P.M. Tomchuk for helpful discussions and useful comments.
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N.I. Grigorchuk
Температурна залежнiсть плазмонних резонансiв у
сфероїдальних металевих наночастинках
М.I. Григорчук
Iнститут теоретичної фiзики iм. М.М. Боголюбова НАН України,
вул. Метрологiчна, 14–б, 03680 Київ, Україна
В рамках кiнетичної теорiї вивчається вплив температури електронiв на поглинання та розсiювання
свiтла металевими наночастинками (МН) при збудженнi електронних коливань поверхневого плазмо-
на. Одержанi формули для тензорiв електропровiдностi та поляризованостi для кiнцевих температур
електронного газу. Детально вивченi електрична провiднiсть та напiвширина резонансу поверхневого
плазмона для сферичної МН. Дослiджується ефективнiсть поглинання та розсiювання свiтла при змiнi
температури в залежностi вiд розмiрiв МН. Зокрема, знайдено, що ефективнiсть поглинання може не
тiльки збiльшуватися при зменшеннi температури, а також й зменшуватись. Одержанi формули дозволя-
ють аналiтично обчислювати рiзноманiтнi оптичнi й транспортнi явища для МН довiльної сфероїдальної
форми, що знаходяться в довiльному дiелектричному середовищi.
Ключовi слова: електронна температура, металевi наночастинки, електропровiднiсть, тензор
поляризованостi, резонанс поверхневого плазмона
33706-18
Introduction
Boltzmann equation
Electric-conductivity tensor
Conductivity of a spherical MN
Linewidth and ``figure of merit''
Polarizability tensor
Light absorption and scattering
Discussion of results
Conclusions
|