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Chapter 4. Photothermal Properties of Gold Nanoparticles Bruno Palpant

Chapter 4. Photothermal Properties of Gold Nanoparticles Bruno Palpant

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November 15, 2012



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Gold Nanoparticles for Physics, Biology and Chemistry



B. Palpant



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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