Estimation of electron temperature in heated metallic nanoparticle
A method is proposed for determining the temperature of hot electrons in a metallic nanoparticle embedded in a dielectric matrix under ultrashort laser pulses irradiation. The amplitude and power of the longitudinal spherical acoustic oscillations as functions of density and elastic properties of...
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irk-123456789-1570282019-06-20T01:28:22Z Estimation of electron temperature in heated metallic nanoparticle Grigorchuk, N.I. A method is proposed for determining the temperature of hot electrons in a metallic nanoparticle embedded in a dielectric matrix under ultrashort laser pulses irradiation. The amplitude and power of the longitudinal spherical acoustic oscillations as functions of density and elastic properties of the medium, the laser pulse duration, the electron temperature, radii of particles, and the electron-phonon coupling constant are obtained. The efficiency of the electron energy transfer from heated noble nanoparticles to a surrounding environment is estimated for different electron temperatures. Запропоновано метод визначення температури гарячих електронiв в металевiй наночастинцi, що знаходиться в середовищi пiд дiєю ультракоротких лазерних iмпульсiв. Одержанi амплiтуда i потужнiсть поздовжних сферичних акустичних коливань як функцiя густини i пружних властивостей середовища, тривалостi лазерного iмпульсу, радiусу частинки, електрон-фононної константи зв’язку та електронної температури. Зроблено оцiнку ефективностi передачi енергiї електронiв вiд пiдiгрiтих благородних наночастинок в оточуюче їх середовище за рiзних температур електронiв. 2017 Article Estimation of electron temperature in heated metallic nanoparticle / N.I. Grigorchuk // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43703: 1–7. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 78.67.Bf, 68.49.Jk, 73.23.-b, 52.25.Os, 36.40.Vz DOI:10.5488/CMP.20.43703 arXiv:1712.05355 http://dspace.nbuv.gov.ua/handle/123456789/157028 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A method is proposed for determining the temperature of hot electrons in a metallic nanoparticle embedded
in a dielectric matrix under ultrashort laser pulses irradiation. The amplitude and power of the longitudinal
spherical acoustic oscillations as functions of density and elastic properties of the medium, the laser pulse
duration, the electron temperature, radii of particles, and the electron-phonon coupling constant are obtained.
The efficiency of the electron energy transfer from heated noble nanoparticles to a surrounding environment is
estimated for different electron temperatures. |
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Grigorchuk, N.I. |
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Grigorchuk, N.I. Estimation of electron temperature in heated metallic nanoparticle Condensed Matter Physics |
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Grigorchuk, N.I. |
| author_sort |
Grigorchuk, N.I. |
| title |
Estimation of electron temperature in heated metallic nanoparticle |
| title_short |
Estimation of electron temperature in heated metallic nanoparticle |
| title_full |
Estimation of electron temperature in heated metallic nanoparticle |
| title_fullStr |
Estimation of electron temperature in heated metallic nanoparticle |
| title_full_unstemmed |
Estimation of electron temperature in heated metallic nanoparticle |
| title_sort |
estimation of electron temperature in heated metallic nanoparticle |
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Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157028 |
| citation_txt |
Estimation of electron temperature in heated metallic nanoparticle / N.I. Grigorchuk // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43703: 1–7. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT grigorchukni estimationofelectrontemperatureinheatedmetallicnanoparticle |
| first_indexed |
2025-07-14T09:22:16Z |
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2025-07-14T09:22:16Z |
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1837613645740113920 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 4, 43703: 1–7
DOI: 10.5488/CMP.20.43703
http://www.icmp.lviv.ua/journal
Estimation of electron temperature in heated
metallic nanoparticle
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 12, 2017, in final form June 19, 2017
A method is proposed for determining the temperature of hot electrons in a metallic nanoparticle embedded
in a dielectric matrix under ultrashort laser pulses irradiation. The amplitude and power of the longitudinal
spherical acoustic oscillations as functions of density and elastic properties of the medium, the laser pulse
duration, the electron temperature, radii of particles, and the electron-phonon coupling constant are obtained.
The efficiency of the electron energy transfer from heated noble nanoparticles to a surrounding environment is
estimated for different electron temperatures.
Key words: electron temperature, metal nanoparticle, ultrashort laser pulses
PACS: 78.67.Bf, 68.49.Jk, 73.23.-b, 52.25.Os, 36.40.Vz
1. Introduction
Metallic nanoparticles (MNs) are mainly studied due to their unique optical properties [1] and
extensive practical applications [2–5]. Over the last decade, the acoustic oscillations of MNs excited by
ultrashort laser pulses have been under intense study [1–9]. This is due to the availability of important
data relating to the elastic properties of these particles and their mechanical coupling to the surrounding
medium, which forms the foundation for the design of elasticity sensors in the nanometer range [10]. The
generation of sound waves byMNs was originally observed experimentally for noble nanoparticles [2–4].
When MNs are excited by ultrashort pulses, the energy pulse is initially transferred to the gas of
free electrons which collide with one another as well as with the lattice vibrations, and redistribute this
energy (being thermalized) over a short time (of the order of tens of picoseconds in a 100 Å particle [3]).
The electron gas in the particle is immediately heated. Due to a low heat capacity of the electron gas
(compared to the lattice), there originates a short (but strong) electron pressure burst [11]. It is equivalent
to a short mechanical impact upon the particle surface over an infinitely small time interval and can
generate spherical sound waves in the medium surrounding the particle [12, 13].
In this article, we proposed a method for determining the temperature of hot electron gas in a metallic
nanoparticle embedded in a dielectric matrix under ultrashort laser pulses irradiation. To accomplish this,
the amplitude and power of the longitudinal sound wave generated by the excess pressure of an electron
gas inside the MN driven by ultrashort laser pulses are calculated. So far, this problem has been little
studied theoretically.
2. Initial principles and model
Let an ultrashort laser pulse of duration τ0 be incident on a spherical MN embedded in an infinite,
elastically isotropic, dielectric medium. Energy transfer from the laser to the lattice of a nanoparticle
depends much on the relationship between the particle sizes, the electron mean free path in the particle,
and the Debye length lD = πυF/ωD, which plays an important role in the energy exchange between the hot
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43703-1
https://doi.org/10.5488/CMP.20.43703
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
N.I. Grigorchuk
electrons and the lattice [14]. Its values for several metals are listed below in table 2. We shall consider
here the particles with radii R < lD/2.
The additional pressure that develops during heating of the electron gas by the laser pulse can
produce oscillations of the nanoparticle surface, which in turn, generate longitudinal acoustic waves in
the elastically isotropic space around the particle.
The equation for the propagation of longitudinal acoustic oscillations in this space is given by [15]
∇2uL(t) −
1
s2
L
∂2
∂t2 uL(t) = 0, (2.1)
where sL is the longitudinal sound speed in the medium. Elastic displacement waves uL are accompanied
by bulk compressions and expansions of the surrounding medium.
As boundary conditions for equation (2.1), we take a condition that all forces applied to the particle
surface (r = R) are equal to zero. This equality can be presented as [15][
σrr −
2Es
R2 ur (t)
] ����
r=R
= −δp(t). (2.2)
Here, ur = |ur | is the radial component of the displacement, R is the particle radius, Es is the surface
energy density, δp(t) is the time dependent additional pressure of the hot electrons, and σrr is the
stress tensor, whose components are expressed through the moduli K , µ of the uniform compression and
rigidity, which one can find elsewhere (e.g., in [15]). For spherical acoustic oscillations, one can assume
that uL(t) = ur (t).
The pressure of the electron gas on the surface of the nanoparticle can be written as the sum
p(t) = p0 + δp(t). (2.3)
Here, the first term represents the pressure of a degenerate Fermi gas at Te = 0 K, p0 =
2
5 nµ0, where
n = N/V is the electron density, V is the particle volume, and µ0 is the limiting value of the chemical
potential at Te = 0 K (µ0 = εF, where εF is the Fermi energy). The second term, δp(t), represents the
additional time dependent gas pressure owing to the electron mobility at temperatures Te > 0. It can be
written as [16]
δp(t) = αT2
e (t), α =
π2
6
n
k2
B
µ0
. (2.4)
3. Amplitude of the sound wave
To solve equation (2.1) with the boundary condition (2.2), we use the potential method [15] u(r, t) ≡
∇∇∇ϕ(r, t). Then, representing of function ϕ(r, t) in the form of spherical waves expanding from particle
center ϕ(r, t) = φ
(
t − (r − R)/sL
)
/r , where φ is an arbitrary, twice differentiable function and using
the boundary condition (2.2) at the nanoparticle surface, we found for φ an inhomogeneous differential
equation
∂2
∂t2 φ(t) + 2γ
∂
∂t
φ(t) + ω2
0 φ(t) = −δp(t)
R
ρ
, (3.1)
in which ρ is the mass density of the medium surrounding the nanoparticle, and
γ =
1
sLR
(
2s2
T +
Es
ρR
)
, ω2
0 =
2
R2
(
2s2
T +
Es
ρR
)
. (3.2)
Here, sT is a transverse sound speed [15]. The parametersω0 and γ are the oscillator eigenfrequency in the
absence of friction and the damping decrement, respectively. Note that when a solid medium surrounds
the nanoparticle, we must set Es = 0 in expressions (3.2). However, if the nanoparticle is embedded in
a liquid medium, then only the second terms in expressions (3.2) must be retained. The quantity 1/γ
specifies the time over which the oscillations are completely damped. It depends on the nanoparticle
radius [17].
43703-2
Estimation of electron temperature in heated metallic nanoparticle
Equation (3.1) formally describes the motion of the oscillator under the action of an external driving
force of the form δp(t)R/ρ. We will consider that there are no oscillator eigenmodes without an external
force, and that they are generated only by an external force. If the driving force acting on the oscillator,
for example, is δ-shaped
δp(t) = pimδ(t − τ0), (3.3)
then the solution of equation (3.1) with the initial conditions φ = 0 and φ′ = −Rpim/ρ for t = τ0 is given
by
φ(t) = −e−γ(t−τ0) sin [ω(t − τ0)] Rpim/ρω, (3.4)
where ω =
√
ω2
0 − γ
2
0 , provided that ω0 > γ.
When the time dependence of the function Te(t) is known, the value of pim can be determined
explicitly. Comparing expressions (2.4) and (3.3), we find
α
∞∫
0
T2
e (t − τ0) dt = pim . (3.5)
The variation of the electron temperature with time can be modelled or determined analytically.
Let us suppose that initially (at t = τ0) the oscillator is not displaced, i.e., is at rest, and obeys the
initial conditions φ(τ0) = 0 and φ′(τ0) = 0. Then, the solution of equation (3.1) for an arbitrary form of
the driving force can be presented in an integral form [18]
φ(t) = −
Re−γt
ρω
t∫
τ0
δp(τ) eγτ sin [ω(t − τ)] dτ, (3.6)
with the conditions that δp(τ) , 0 and t > 0.
Equation (3.6) easily answers the question of how the oscillator behaves after an excess pressure
δp(τ) has acted on it over a short time interval (τ0, τ1). The upper limit in equation (3.6) for this case,
evidently, will be the quantity τ1. Then, using the mean-value theorem in equation (3.6), we obtain
φ(t) = −RIe−γteγξτ1 sin(ωt)/ρω, (3.7)
where τ0/τ1 < ξ < 1, while
∫τ1
τ0
δp(τ)dτ ≡ I presents the impulse of the excess pressure of the electron
gas. If ω0τ1ξ � 2π, then the explicit form of δp(τ) is not essential; an important point is only the value
of I. When ω0 � γ, the maximum displacement umax caused by the excess pressure “impact”, with using
potential method and equation (3.7), will be
umax = I/ρω0R. (3.8)
Taking into account here equation (3.2) for the case Es = 0, we obtain
umax, Es=0 = I/2
√
µρ, (3.9)
i.e., the magnitude of umax, Es=0 is independent of the particle radius (and is more independent since the
rigidity modulus and mass density of the nanoparticle are smaller). On the other hand, if the Es > 0, then
a dependence on the particle radius shows up, and the magnitude of umax is decreased as R is reduced.
The maximum displacement over the time τ1 − τ0 with account for the finite damping, has the form:
umax, Es=0 =
I(1 − Rγ/sL)
2√µρ
√
1 − (sT/sL)2
e−γ(τ1−τ0). (3.10)
Therefore, it is possible to determine the pressure impulse I by measuring the maximum displacement
of the physical object.
43703-3
N.I. Grigorchuk
4. Power radiated by nanoparticle surface
A certain part of the accumulated byMN’s energy is carried away from the particle into the surround-
ing matrix by sound waves. The instantaneous elastic energy flux from the surface of the sphere (for the
case of Es = 0) can be calculated using a formula [19]
W(t) = S δp(t) υr (R, t). (4.1)
Here, S is the area of the particle surface and υr is the radial or vibrational velocity.
In terms of time, conventionally, two stages in this process can be identified: one is an impetuous rise
in δp during an ultrashort time spell (t ∼ τ0) and another one is a slower decrease (under t > τ0) owing
to a transfer of the electron energy to the surrounding matrix. The dependence of the δp(t) within the
second time spell, can be well described by expression (2.4).
In order to take an explicit account of the behavior of the pressure in the initial time spell, the function
δp(t) can be presented as the product of two functions,
δp(t) → θ(t − τ0) · αT2
e (t), (4.2)
where we use equation (2.4), while θ(t − τ0) is the Heaviside unit-step function that specifies the pressure
behavior when t → τ0. Let us also suppose that the electron temperature varies with time in accordance
with the rule
Te(t) = Te(τ0) e−β(t−τ0), β =
gR
Ce
, Ce = 3αTe , (4.3)
whereCe ≡ Ce
(
Te(τ0)
)
is the heat capacity of the electron gas, which depends on the electron temperature
taken at the instant of time t = τ0. The electron-phonon coupling constant in equation (4.3) can be
computed from the formula
gR =
27
16
n
m
α
kB
1
ρpR
(
π~
a
)3 (
U1
A
)2
. (4.4)
Here, a is the lattice constant for the MN, ρp denotes the mass density of the MN, U1 is the energy
required to detatch the first electron from the neutral unexcited atom, and A refers to the electron work
function of the metal. Formula (4.4) follows from equation (115) of the work [20] using the equation (2.4)
and the following relationships
ωD '
√
σ
ρp
(
π
a
)3
, υF =
~
m
(
3nπ2)1/3
, (4.5)
where ωD is the Debye frequency and σ is the surface energy density.
Taking into account equations (4.2) and (4.3), after integratingwith respect to τwithin limits 0 < τ < t
and some transformations, we finally obtain
W(t) =
4πR2
ρsL
δp(t) δpm(t)
ω2
0 + 4β(β − γ)
(
− 2β
( sL
R
− 2β
)
+ e(2β−γ)t0
{ [
ω2
0 + 2β
( sL
R
− 2γ
)]
cos (ωt0)
+
[
(ω2
0 − 2βγ)
( sL
R
− γ
)
− 2βω2
] sin (ωt0)
ω
})
, (4.6)
where t0 = t − τ0. Equation (4.6) gives the energy per unit time carried out by spherical sound waves
from the nanoparticle into the surrounding medium.
The analysis show that the power of the sonic signalW(t) is of the form of damped oscillations. The
number of oscillations depends much on the density of the matrix material. The amplitude of the sound
power is increased with an increase in the radius of MN, and it is fully damped over longer times. For
different metals (but with the same radius) the amplitude ofW(t) will be greater for MNs with a higher
coupling constant gR.
The maximum power of the acoustic signal is reached at the time instant t0 = 0. From equation (4.6),
it is equal to
Wmax
��
t0=0 = 4πR2[δp(τ0)]
2/ρsL. (4.7)
43703-4
Estimation of electron temperature in heated metallic nanoparticle
The temperature decrease of the electron gas is proportional to the product βTe, which characterizes
the cooling rate of the electron gas. Since the fall of the electron temperature occurs much more slowly
than the electron’s heating, we can assume that the influence of this mechanism on the sound power is
negligible in the first approximation. Formally, this allows us to direct β → 0 in equation (4.6). In this
case, the sound signal will correspond to exponentially damped oscillations.
The time dependence of the excess pressure in the case of β → 0, can also be estimated using the
equation
δp(t)
��
β→0 ≈ −2gR t Te(τ0)/3, (4.8)
which follows from the energy balance equation for the electrons. The maximum power of the signal
[equation (4.7)], with the account for approximation (4.8), is
Wmax
��
t=τ0, β→0 =
16
9
π
ρsL
[gR τ0 R Te(τ0)]
2, (4.9)
where the dependence on the laser pulse duration appears explicitly.
5. Total energy of the oscillations of the metal nanoparticle
We now determine the total energy transferred from the pulsating spherical surface to the surrounding
matrix. We will consider the case β , 0, when the powerW(t) is given by equation (4.6). Integrating
equation (4.6) with respect to time over the interval 0 < t < ∞, in view of expressions (2.4) and (3.2) at
Es = 0, we obtain
E =
∞∫
τ0
W(t) dt =
2πR
ρsL
sL + 2βR
ω2
0 + 4β(β + γ)
α2T4
e . (5.1)
Equation (5.1) shows that the total energy is proportional to the fourth power of the electron temperature.
In the case β = 0, it is easy to reveal that this energy depends on the volume of the particle and is
reciprocal to the rigidity modulus of the medium.
On the other hand, the work necessary to shift the spherical surface at the maximal distance umax
[defined by equation (3.10)], is
E = 8πRu2
maxαT2
e . (5.2)
Comparing it with equation (5.1), one finds
Te = 2umax
√
ρsL
α
ω2
0 + 4β(β + γ)
sL + 2βR
. (5.3)
Another way to determine Te is as follows. The total energy obtained by the electron gas in the MN is
Etot = 3VαT2
e /2. (5.4)
The efficiency of the energy transfer η from the pulsating spherical MN to the sound oscillations can be
estimated taking the ratio of equations (5.1) and (5.4). It has the simplest form in the case β→ 0
η(T) ≡ E |β→0/Etot = αT2
e /4µ. (5.5)
The temperature dependence of η(T) for noble nanoparticles embedded in a plexiglass matrix is plotted in
figure 1. The calculations were done using equations (5.1) and (5.4), with account for the characteristics
both of the matrix and the MNs given in tables 1 and 2. The efficiency increases as the square of the
temperature. Actually, it increases if the rigidity modulus of the medium in which the nanoparticle is
embedded gets smaller. So, knowing from experiment the efficiency η, it is easy to estimate the electron
temperature in MN. The highest possible ηT for noble metal nanoparticles with R = 100 Å embedded in
a plexiglass matrix are presented in table 2 at Te = 104.
43703-5
N.I. Grigorchuk
0 2000 4000 6000 8000 10 000
0
10
20
30
40
Te
Η
H%
L
Figure 1. (Color online) Dependence of the sound signal power efficiency on electron temperature for Cu
nanoparticle (solid curve), Ag (dashed curve), and Au (dotted curve), embedded in plexiglass.
Table 1. Constants for the plexiglass [21].
Medium K µ ρ sL sT
(dyn/cm2) (dyn/cm2) (g/cm3) (cm/s) (cm/s)
Plexiglass 5.83·1010 1.48·1010 1.18 2.57 ·105 1.12·105
Table 2. Physical parameters of the noble MNs.
Metal lD (Å) α
(
erg
cm3 K2
)
n
(
cm−3) [22] η104 (%)
Cu 1197 235.5 8.45 · 1022 39.5
Ag 1552 208.35 5.85 · 1022 35.15
Au 1968 208.9 5.90 · 1022 35.3
6. Conclusions
Themethod has been proposed for the estimation of hot electron temperature in metallic nanoparticles
embedded in a dielectric matrix under irradiation of ultrashort laser pulses. Analytic expressions have
been derived for the amplitude and power of longitudinal spherical sound oscillations as function of the
density and elastic properties of the medium, the laser pulse duration, electron temperature, nanoparticle
radius, and electron-phonon coupling constants.
The maximal displacement of any oscillator in the MN’s material is due to the “impact” of the excess
pressure of the electron gas. The magnitude of the sound signal power at the moment of the laser pulse
ending can be used to estimate the maximal electron temperature in the MN. The latter is determined by
the cooling rate of the electron gas and the rate of its pressure change.
The efficiency with which the energy of the hot electrons is carried away by sound oscillations
has been examined for the noble metals. It has been shown that the sound energy transfer efficiency is
considerably higher in the medium with smaller rigidity moduli.
Acknowledgements
Author is indebted to DFFD of Ukraine for single financial support of this work.
43703-6
Estimation of electron temperature in heated metallic nanoparticle
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Визначення температури електронiв у пiдiгрiтiй металевiй
наночастинцi
М.I. Григорчук
Iнститут теоретичної фiзики iм.М.М. Боголюбова НАН України,
вул.Метрологiчна, 14-б, 03680 Київ, Україна
Запропоновано метод визначення температури гарячих електронiв в металевiй наночастинцi, що зна-
ходиться в середовищi пiд дiєю ультракоротких лазерних iмпульсiв. Одержанi амплiтуда i потужнiсть
поздовжних сферичних акустичних коливань як функцiя густини i пружних властивостей середовища,
тривалостi лазерного iмпульсу, радiусу частинки, електрон-фононної константи зв’язку та електронної
температури. Зроблено оцiнку ефективностi передачi енергiї електронiв вiд пiдiгрiтих благородних нано-
частинок в оточуюче їх середовище за рiзних температур електронiв.
Ключовi слова: електронна температура, металева наночастинка, ультракороткi лазернi iмпульси
43703-7
https://doi.org/10.1039/C3CS60367A
https://doi.org/10.1063/1.476374
https://doi.org/10.1021/jp504467j
https://doi.org/10.3390/ma6114879
https://doi.org/10.1016/j.ijheatmasstransfer.2007.01.049
https://doi.org/10.1103/PhysRevB.86.075463
https://doi.org/10.1103/PhysRevB.80.155456
https://doi.org/10.5488/CMP.16.33706
https://doi.org/10.1103/PhysRevLett.85.792
https://doi.org/10.1103/PhysRevB.69.195416
https://doi.org/10.1016/j.cplett.2004.11.072
https://doi.org/10.1146/annurev.physchem.57.032905.104533
https://doi.org/10.1063/1.125460
https://doi.org/10.1016/S0370-1573(99)00094-0
Introduction
Initial principles and model
Amplitude of the sound wave
Power radiated by nanoparticle surface
Total energy of the oscillations of the metal nanoparticle
Conclusions
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