Chapter 4. Photothermal Properties of Gold Nanoparticles Bruno Palpant
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4.2 Light Induced Heating of Gold Nanoparticles
4.2.1 Introduction: the key role of electron-phonon
scattering
It is more than likely that the optical response of metals confined at the
nanoscale shares many common characteristics with the bulk metal one.
Hence, as the dielectric function of gold in the visible range is dominated by the contributions of both interband and intraband transitions,1 the
optical properties of Au-NPs (provided their size is sufficiently large, see
Chapter 3, § 3.3) are ruled by the microscopic coupling and exchange mechanisms involved in such transitions. The plasmon resonance phenomenon —
which arises from the dipolar nature of the wave-driven collective electron
excitation allowed by the small NP size — is essentially a coherent set of inphase intraband transitions. It is then also closely dependent on the energy
dissipation processes involved in individual intraband transitions, which are
classically accounted for in the bulk metal response by the phenomenological scattering rate of the Drude model. If the Drude model is assumed to
depict the NP dielectric function, the spectral width of the SPR is directly
given by h¯ (see Chapter 3, §3.3).
It may certainly be interesting to remind ourselves of the origin of the
implication of collisions in metal intraband transitions. The conduction
electrons being quasi-free in noble metals,2 their dispersion law is quasiparabolic (apart from the gap opening at the edge point L of the Brillouin
zone stemming from the weak periodic ionic potential). Let us consider an
initial (final) state with energy Ei (Ef ) and wave vector ki (kf ). As the photon momentum is negligible against the electron one in the visible spectral
domain, an electron transition induced by simple photon absorption imposes
kf ≈ ki due to momentum conservation (the transition is said to be vertical).
This means that a final state in the conduction band with Ef = Ei cannot
be reached by such an inelastic electron-photon scattering. Thus, an optical
transition within the conduction band is impossible without the help of a
collision with a third particle or quasi-particle which provides the momentum difference between the initial and final states. Let us underline that such
an interband transition can occur through either emission or absorption
of a third particle. The significant contributions to the collision-assisted
intraband transitions at room temperature are the electron-electron and
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electron-phonon scattering ones.3 Indeed, electron scattering by defects and
impurities can be neglected, except at very low temperature. Furthermore,
the contribution of electron-electron collisions4 does not exceed 15% of
the total scattering rate.5 Even at higher electron temperatures reached with
ultrashort laser excitation (see below) the electron-phonon scattering contribution remains by far the dominant one. This reveals that the plasmon
resonance is tightly linked with the scattering of electrons with phonons.
Heat can be seen as a broadband incoherent statistical set of vibrations
and is then supposed to be supported, in a solid, by phonons. Heat transport
is ensured by carriers which are elementary particles, as electrons, phonons,
or even photons. In the former two cases the mechanism involved is conduction, whereas in the latter the transfer occurs by radiation. When exposed
to an incident light in (or in the vicinity of) the visible spectral domain, a
gold nanoparticle can gain energy by absorbing photons through electron
transitions. The main relaxation process is then the electron-phonon scattering, as stated above. Of course, an individual electron-phonon collision can
result either in an energy gain or an energy loss for the electron; but as the
initial energy supplied to the NP by photon absorption is input in the form
of electron excitations the subsequent overall energy transfer progresses
from electrons to lattice vibrations. This mechanism is then the cause of the
photo-induced heating of a metal NP.
To end this introduction, let us notice that before going deeper into any
calculation we can foresee the great interest of the plasmon resonance for
realizing nanosize heat sources. Indeed, the SPR absorption is an efficient
and fast way of inputting energy into a NP by macroscopic light excitation,
the inner exchange mechanisms then allowing this energy to be mainly
converted into heat at the nanoscale. This explains the spectacular recent
rise in studies and applications of Au-NPs for their photothermal ability
(see Chapter 10).
4.2.2 A series of energy exchanges
By simply depicting, in the preceding section, the link between the absorption of photons in gold NPs (at or off the plasmon resonance) and the energy
dissipation through electron-phonon collisions, we have implicitly introduced the notion of dynamics. The optical properties of such nano-objects
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Fig. 4.1. Schematic illustration of the series of energy exchanges involved in the optical impulse
response of a gold nanoparticle on a logarithmic time scale.
are indeed driven by a series of interlinked internal and external energy
exchanges, each of them characterized by a typical time scale. In order to
gain a deeper insight into these properties, it appears useful to study the NPs
impulse response, as is typical for dynamic systems in physics. Furthermore,
regarding the ability of Au-NPs to behave as thermal nanosources, such a
study may allow us (i) to understand the stationary regime of photo-induced
heating, and (ii) to propose new ways of controlling the heating.
Let us then imagine that a “very short”i light pulse is sent onto a gold
nanoparticle (see Fig. 4.1). Part of the incoming light is then absorbed
for inducing electron transitions. Let us stress that it is a priori possible to
induce, whether interband transitions, or intraband ones. In the former case,
photons generate individual electron-hole pairs, while in the latter they promote electrons up to higher levels within the conduction band. We have seen
in a previous section that in spherical Au-NPs both the plasmon resonance
and interband transitions can be excited in the same spectral range. In this
case the SPR is damped and broadened due to Landau damping. Beyond
this, for the sake of simplicity, we will disregard the possible excitation of
interband transitions in the description of the dynamics mechanisms. Provided the ultrashort wave packet spectrum matches the SPR energy, the
plasmon resonance is excited, that is, a resonant coupling with the electromagnetic wave induces a coherent set of in-phase electron excitations in
the conduction band. It has been shown through optical experiments using
second-harmonic generation autocorrelation, spectral hole burning or measurements of the SPR bandwidth of single NPs that the dephasing time (T2 )
i By very short, we mean the duration of which is about a few electromagnetic field oscillation cycles.
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of the SPR is a few femtoseconds.6 The NP is then left with an extra energy
stored in the electron gas.
As a matter of fact, a few electrons have gained photon energy by
absorption, the other ones remaining in the non-excited states. This puts
the electron distribution out of equilibrium. Energy is then redistributed
among the whole quasi-free electron gas by electron-electron collisions,
leading to the recovery of an internal thermal equilibrium within the conduction band. This process occurs on a time scale which ranges from a few
tens to several hundreds of femtoseconds, depending mainly on the initial
energy input (the higher the proportion of excited electrons in the gas the
faster the energy redistribution by collisions). At the same time, electrons
scatter with phonons. Actually, there are no real collisions with such quasiparticles; rather, a quantized vibration mode of the crystal lattice (i.e. a
phonon) induces a modification of the periodic potential experienced by
the electrons, and then a modification of the wave function of the latter. The
typical time scale of this process is about one picosecond.As for the electronelectron (e-e) scattering rate, the actual electron-phonon (e-ph) relaxation
time depends on several factors such as particle size and input excess energy.
Finally, as the NP is not isolated but is embedded in a medium, there is a
thermal energy exchange at the interface through phonon-phonon collisions,
leading to the cooling down of the NP. The dynamics of this process, as
will be seen later, are very sensitive to the heat transfer properties in the
host medium. It may range from a few picoseconds to nanoseconds. Let
us notice that, as Au-NPs are most of the time dispersed in a liquid or
solid insulating medium, the heat carriers in the latter are mainly phonons
(convection may be neglected at these small time scales). Of course, the NP
cooling goes together with the transient heating of its close environment,
which may be exploited to use gold NP as photo-induced heat nanosources.
If the neighbouring NPs are sufficiently close in the medium (that is, if
the NP density is high enough), then the thermal energy released by these
neighbours will affect the cooling dynamics. Let us stress that in the case of
very close NPs a thermal exchange through radiative transfer is possible.7
We won’t address this particular effect in this chapter.
As a consequence of this series of energy exchanges, the internal
energy of the electron gas subsequent to light pulse absorption undergoes (i) a sudden and strong rise, (ii) an inner redistribution within
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the electron gas (athermal regime), (iii) a fast decrease (e-ph scattering)
and (iv) a slow return back to equilibrium (thermal transfer to the host
medium).
The particle temperature is ruled by the balance between the gain of
energy from e-ph collisions and the heat release towards the host medium.
It then presents an increase on a picosecond time scale followed by a slow
decrease. In the subsequent sections, the different steps described above
will be examined in deeper detail.
4.2.3 Athermal regime
Monovalent bulk metals like alkali and noble metals exhibit quasi-free electron behaviour in the conduction band.ii Indeed, their Fermi surface in the
reciprocal space is very close to a sphere, which reveals a parabolic dispersion law:
h¯ 2 k2
E(k) =
,
(1)
2m∗
where E, k and m∗ are the electron energy, wave vector and effective mass,
respectively, and h¯ denotes the reduced Planck constant.2 This property
bestows on us the right to use the quasi-free electron model for describing
the conduction electron gas. Hence, at thermal equilibrium, the latter obeys
a Fermi–Dirac distribution:
E − EF −1
f (E) = 1 + exp
.
(2)
kB T e
Te is the electron temperature, defined as a means to characterize the electron internal energy at equilibrium, EF is the Fermi energy and kB the Boltzmann constant. Before light excitation (Fig. 4.2, left), Te equals the lattice
temperature Tl , and Tl = T0 (initial temperature). When sending the light
pulse, part of it is absorbed to induce electron transitions the nature of
which is, either intraband only if the photon energy hω
¯ is lower than the
interband transition threshold, Eib , or otherwise both intra- and interband.
For the sake of simplicity, we will restrict ourselves to the first case only.
The electron distribution is then dug just below EF and those electrons
ii For noble metals, nevertheless, the lattice periodic potential opens a gap at the point L of the edge
of the Brillouin zone.
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Fig. 4.2. Evolution of the conduction electron occupation rate as a function of electron energy subsequent to a light pulse absorption. Left: distribution at initial room temperature; middle: just after photon
absorption; right: after internal thermalization.
which have gained energy by photon absorption are promoted to energy
levels just above EF (Fig. 4.2, middle). One is then left with a distribution
out of thermal equilibrium; f does not follow a Fermi–Dirac distribution
anymore and no electron temperature can be defined. This is the athermal regime. Note that this regime can be neglected when the excitation
pulse width is long relative to the typical e-ph scattering time, as in this
case the electron and the phonon gas are at every instant at quasi-thermal
equilibrium.
By electron-electron collisions the energy is internally redistributed
within the electron gas. This process is as efficient as the number of excited
electrons is high, which explains that the duration of the athermal regime
decreases with increasing laser power. The distribution then recovers a
Fermi–Dirac statistics at a temperature Te > T0 . This is known as the
Fermi smearing.
To describe the electron properties in the athermal regime the relevant
parameter then appears to be the electron distribution f (E, t) which depends
on energy and time, the dynamics of which is governed by the Boltzmann
equation8−10 :
∂f (E, t)
∂f (E, t)
=
∂t
∂t
+
source
∂f (E, t)
∂t
+
e−e
∂f (E, t)
∂t
.
(3)
e−ph
Here we neglect electron diffusion (assuming that the particle size is
smaller than the wave penetration depth, the excitation can be considered
as homogeneous) as well as the environment (the influence of which will
be significant at longer times). The source term refers to the instantaneous
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modification of f under photon absorption. Its value at a given electron
energy then depends on photon energy hω
¯ and is proportional to the number
of photons absorbed per time unit, this number being proportional to the
instantaneous power absorbed from the laser, Pabs (t), as illustrated on
Fig. 4.2 (middle graph). Pabs (t) can be evaluated from the particle absorption cross section (or absorption coefficient of the medium knowing the
NP density) and the laser pulse intensity and time profile. The second
and third terms in Eq. 2 denote the contributions of electron-electron and
electron-phonon scattering, respectively, to the variation rate of f . Several approaches have been proposed to treat these contributions.8−10 The
a priori most rigorous one consists in including explicitly all the detailed
scattering processes.10 For the e-e term, it amounts to integrate over all
the wave vectors of the 2nd electron and all the momentums exchanged in
the elastic collision. For the e-ph term, both the absorption and emission
of phonons have to be accounted for and the integration runs over all the
wave vectors of the phonons exchanged. A significant simplification can be
introduced by considering the weak perturbation regime, which validates the
use of the relaxation time approximation for the two scattering contributions
separately:
∂f (E, t)
f (E, t) − f0 (E)
.
=
τ(E)
∂t
(4)
f0 is the initial equilibrium distribution and τ the typical collision time.
In the case of the e-ph term, one has to distinguish the contribution of
the spontaneous emission of phonons from the ones of phonon stimulated
emission and absorption, the two latter depending on the number of states
available in the phonon bath.9 For the e-e scattering term, the Landau theory
of Fermi liquids allows us to express the e-e mean collision time: τe−e (E) =
τ0 EF2 /(E − EF )2 .11 This accounts for the fact that the scattering probability
decreases as E gets closer to the Fermi level, which stems from the Pauli
principle. τ0 is a few tenths of a femtosecond. In the limit of very weak
perturbations, some authors have split the distribution into a major part still
at thermal equilibrium and a small athermal part: f = fthermal + fathermal ,
with
+∞
−∞
+∞
fathermal (E)dE <<
82
−∞
fthermal (E)dE.10
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The model can be further refined, for instance by including the competition
between transitions within the conduction band and the creation of electronhole pairs in the valence and conduction bands.12
Let us notice that the only difference between the approaches for a bulk
metal and a NP has up to now rested on the disappearance of the electron
diffusion term in the second case of Eq. 2 (provided that the excitation
can be considered as homogeneous in the NP). In fact, some authors have
shown that in the relaxation time approximation the e-e and e-ph mean
collision times exhibit a size dependence.13 Indeed, they both decrease with
decreasing particle radius R due to the decrease of the Coulomb interaction
screening. Moreover, the appearance of low-frequency acoustic vibration
modes in finite-size NP induces a new relaxation channel for the electrons,
which increases the e-ph scattering rate.14
4.2.4 Thermal regime
Once the thermal equilibrium is recovered within the conduction electron gas, the energetic couplings within the nanoparticles and with their
environment can be treated by more classical thermodynamics approaches
as the electron temperature can be defined. Let us stress that, while the
model described in the preceding section enables us to account for the offequilibrium situation for the conduction electrons, it is restricted to short
times after excitation as the heat exchange with the host medium is not
accounted for. Moreover, in many cases the athermal regime can be disregarded, as in pump-probe time-resolved experiments carried out with a high
pump pulse intensity which shortens the athermal regime duration, or for
which the time scale under consideration is much larger than this duration.
It can of course be also neglected for long pulse excitation (pulse width
larger than the e-ph collision time) as in this case the electron gas and the
crystal lattice are permanently at equilibrium.
4.2.4.1 Two-temperature model
For describing the coupling between electrons and phonons, the twotemperature model (TTM) developed for bulk metals has been adapted
to NPs. It consists in writing the two coupled differential equations ruling
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the time evolution of the electron and lattice internal energies:
∂Te
Ce
(5)
= −G(Te − Tl ) + Pabs (t),
∂t
∂Tl
Cl
(6)
= G(Te − Tl ).
∂t
Here Ce and Cl denote the electron gas and lattice specific heats, respectively.
As in noble metals the conduction electrons exhibit a quasi-free electron
behaviour, as stated in §4.2.3, Ce can be deduced from free electron quantum
statistics: Ce = γe Te , where γe is a constant the value of which depends on
the metal (for gold, γe = 66 J m−3 K−2 ). G is the e-ph coupling constant
(for gold, G = 3 × 1016 W m−3 K−1 ). As for the athermal regime, Pabs (t)
represents the instantaneous power absorbed per metal volume unit (source
term) and has the profile of the incident light pulse. Again, as the NP size
is assumed to be smaller than the light penetration depth, the excitation is
homogeneous and electron diffusion can be neglected.
The parameters involved in the TTM are usually taken as the bulk phase
ones, but it may be easy to replace some of them in a phenomenological
manner to account for finite size effects in NP. This has been done, for
instance, to deduce the size-dependent G value from pump-probe experiments using Eqs. 5 and 6.15 Some other authors have shown that G may
vary with temperature which itself depends on laser power.16
4.2.4.2 Three-temperature model
The TTM completely neglects the thermal influence of the surrounding host
medium. This might be valid as long as the heat exchange at the interface
remains negligible, that is, when photo-heating with an ultrashort pulse and
considering the electron temperature only during the first few picoseconds
(see Fig. 4.3). Of course, if the contact between the particle and the matrix
is poor, if the thermal resistance at the interface is high, or if the matrix has
a low thermal conductivity, its influence can be neglected over a larger time
scale. In the general case, it has to be taken into account. This is the purpose
of the three-temperature model (3TM). For this, Eq. 6 has to be modified
as to add the contribution of the instantaneous heat released through the
interface, H(t):
H(t)
∂Tl
Cl
= G(Te − Tl ) −
,
(7)
∂t
V
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Fig. 4.3. Time evolution of the electron (Te , thick lines) and lattice (Tl , thin lines) temperatures of a
Au-NP in silica after light pulse absorption [Pabs (t), grey line] calculated without (TTM, dashed line)
and with (3TM, solid line) considering the heat release to the host medium and a purely diffusive heat
transfer in the latter. Pulse duration and peak power absorbed are worth 110 fs and 1.4 × 1021 W m−3 ,
respectively. After Ref. 16.
where V is the NP volume. The time evolution of the factor H (and subsequently of Te and Tl ) then strongly depends on the characteristics of the
thermal transport in the surrounding medium, namely, the ability of the
latter to evacuate heat and then to cool down the particle. This point will
be given a special attention in the next section. Let us underline beforehand the role that the quality of the contact may play at the interface. It has
been recently shown that an interface thermal resistance (known as Kapitza
resistance) may modify the NP cooling down dynamics.17 The value of this
resistance has been extracted from time-resolved pump-probe experiments.
The authors ascribe this phenomenon to the acoustic impedance mismatch
between the NP and its surrounding medium. It is also likely in some other
situations, depending on the synthesis technique, that the medium close to
the interface presents a partial porosity or that the mismatch induces lattice
defects, which may also affect the thermal resistance.
4.2.5 Heat transfer to the host medium
4.2.5.1 Different approaches depending on the heat transfer
characteristics
As we have seen, the cooling down of a Au-NP after ultrashort light
pulse absorption or, by extension, the topography of the temperature field
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around a Au-NP under light irradiation — whatever the time profile of the
latter — strongly depends on the characteristics of the heat transfer in the NP
host medium. It is easily understandable that, for instance, the higher the
thermal conductivity of this medium, the faster the relaxation and then the
lower the temperature at a given distance from the NP. This has been experimentally shown to influence the dynamics even at short times,18 especially
by comparing the optical relaxation of silver or gold NPs in silica (or glass)
and alumina, the conductivities of which are roughly in the ratio of one to
thirty.19,20 Beyond the only conductivity, the detailed mechanisms involved
in the heat transport through the surrounding medium play a crucial role in
the Au-NP photo-induced thermal response. This involvement depends on
the excitation conditions as well as on the observation ones as we will now
show. The different approaches which are used to describe heat transport in a
medium can be split into two categories. In the first one, known as molecular
dynamics (MD), matter is described by its constituents (atoms) and the modelling consists in determining the motion of each atom by (usually) solving
the classical second law of Newton once the suited analytical description
of the interatomic forces is chosen.21 Once all the motions are calculated,
statistical physics allows us to determine relevant quantities of the system
thermodynamics as their mean values and fluctuations. Whereas, as we have
seen above, molecular dynamics has allowed us to address the problem of
photo-induced phase transform and partial melting of Au-NPs, this powerful
method has up to now been used very little to model the thermal transport
in the medium surrounding a metal NP subsequent to the photo-induced
heating of the latter.22 Rather, continuous-media approaches are employed.
They consider that all the media can be described by continuously varying
quantities such as phonon density, energy and flux. In this category, the
most general theory appropriate for this problem is the Boltzmann transport equation (BTE). It allows us to describe the time evolution of the local
phonon density and is particularly suited for off-equilibrium situations. If
the spectral composition of the heat transport is disregarded (namely, if
a spectrally-integrated phonon mean free path, ph , and lifetime, τph , can
be defined in a phenomenological manner) then they may be used as relevant parameters to validate successive simplifications of the BTE. First,
the definition of a characteristic phonon lifetime itself may result in the use
of the time relaxation approximation that we have already addressed in the
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