Jpn. J. Appl. Phys. Vol. 38 (1999) pp. 2983–2987
Part 1, No. 5A, May 1999c
°1999
Publication Board, Japanese Journal of Applied Physics

Direct Measurements of Femtosecond Energy Dissipation Processes of Hot Electrons in a Gold Film
Akihide H IBARA, Tomohiro M ORISHITA, Isao T SUYUMOTO, Akira H ARATA1 ,Takehiko K ITAMORI and Tsuguo S AWADA ∗Department of Applied Chemistry, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
1 Department of Molecular Science and Technology, Graduate School of Engineering Science,
Kyushu University, 6-1 Kasugakoen, Kasuga, Fukuoka 816-8580, Japan(Received November 16, 1998; accepted for publication February 23, 1999)

Energy dissipation processes of hot electrons in a gold thin film were measured by a femtosecond time-resolved transientreflecting grating method. The processes were analyzed using the two-temperature model and the theory of the transient grating
method. It was found that the electron-phonon coupling and thermal diffusion processes could be investigated independently.
Temperature dependence of the electron-phonon coupling factor and the thermal conductivity was investigated quantitatively.
The results suggested that the nonthermal states of hot electrons still contribute to the dissipation processes on time scales ofseveral picoseconds.
KEYWORDS: hot electrons, transient reflecting grating method, electron-phonon coupling, transport phenomena

1. Introduction
The ultrafast dynamics of photoexcited electrons in metals
have been intensely investigated, using fs-laser techniques,in order to study nonequilibrium systems and particle-likebehavior of electrons in solids.1) The thermalization processes of the photoexcited electrons were directly measured
by a two-photon photoemission method, and ultrafast electron dynamics on metal surfaces was discussed.2, 3) These
processes have also been measured by a transient reflecting(TR) method. The TR method measures reflectivity changes(1R) by electron temperature rise (1Te ) after the thermalization. The TR measurements showed that the photoexcited electrons before thermalization have ballistic characteristics4, 5) and are thermalized within several hundred femtoseconds in gold films.6) Energy dissipation processes of hotelectrons after the thermalization have been analyzed by thetwo-temperature model (TTM), based on thermal diffusionand electron-phonon coupling.7–9) Intense interest has beenshown toward these two processes, not only in fundamental physics, but also in short-pulse laser-applications, suchas material processing. However, quantitative treatment has
been difficult, especially for the thermal conductivity κ andthe electron-phonon coupling factor G. When the pump pulse
is focused on the metal surface and the Te gradient is induced
in a spot size of several µm scale, the energy dissipation rateof hot electrons by thermal diffusion (1/τD ) becomes comparable to that by electron-phonon coupling (1/τe-p ). To quantify and separate these rates, it has been necessary to control
the initial Te gradient and to measure the energy dissipation
processes directly.
The transient reflecting grating (TRG) method is suitablefor investigating the energy dissipation process of hot electrons. In the TRG method, sample surfaces are excited, withperiodic patterns induced by interference between two pumppulses crossing on a sample surface. The periodic patterns
can be controlled by changing the optical setup (see §2). The
fundamental properties of the TRG method, such as diffraction efficiency and surface selectivity have been well investigated.10) In addition, the usefulness of the TRG method has∗ To

whom correspondence should be addressed.
sawada@hongo.ecc.u-tokyo.ac.jp

E-mail address:tbeen demonstrated in many applications, such as elucidation
carrier dynamics in semiconductors.11, 12) The thermal andacoustic properties of metal surfaces have been investigatedby ps-TRG measurements.13, 14) Recently, the TRG method
under surface plasmon resonance conditions has also beendeveloped.15) In fs-time-resolved measurements of metal surfaces, the TRG method is expected to be useful, especially inquantitative treatment of hot electron transport and electronphonon coupling in the surface plane.
Here, we report direct measurements of the energy dissipation processes of hot electrons in a gold film of 100 nm thickness using the fs-TRG method. The gold film was excitedby high laser fluence (∼ mJ·cm−2 ) and high photon energy(λpump = 400 nm). Since the thermalization process can be
regarded as instantaneous on highly excited surfaces,16) the
hot electrons with electron temperature Te were dynamicallyinvestigated in our fs-TRG measurements. The energy dissipation dynamics of the hot electrons was analyzed by TTM
and the theory of the transient grating method.17)
2. Experimental
A schematic figure of the TRG method is shown in Fig. 1.
Methodological descriptions for the TRG technique havebeen presented in refs. 10 and 17, so only a summaryis given here. TRG responses are measured by a pumpprobe technique. Two pump pulses of the same wavelength
λpump cross on a metal surface and form an optical interference pattern with the fringe spacing 3 which is tunable bychanging the crossing angle of the two pump pulses 2θ , as
3 = λpump /(2 sin θ ). Thus, the interference pattern along
the x-direction excited by p-polarized beams can be written as 2I + I cos(2π x/3), where I is the one-side intensity of the pump pulse. The temporally generated interference fringes act as a transient diffraction grating. A probe
pulse (λprobe = 800 nm) is incident normal to the grating, after passing through an optical delay line, and is diffracted by
the grating. Intensities of the diffracted light are recorded as
TRG signals.
TRG experiments were performed using a regeneratively
amplified Ti:sapphire laser system which had an autocorrelation of 200 fs, a wavelength of 800 nm and a repetition rateof 1 kHz. The pump light was frequency-doubled and split
in two parts. Each pump pulse had a spot size of 600 µm, a

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Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 5A

Fringe spacingΛ=A. H IBARA et al.

1∂= − [ln(Te − Tl )]
τe
∂t
1
1
+
=
τe-p
τD

λ pump2 sin θGold thin film

Pump pulse

z -axisx-axis2θ

Probe pulse
Diffracted light
Pump pulse=TRG signal

Fig. 1. A schematic figure of the TRG method.

fluence I of 0.38 to 2.25 mJ·cm−2 , and p-polarization. Initial
electron temperatures (Te0 ) were estimated as 1700 to 4200 K
from the absorbed energy and electronic heat capacity in thedetection volume. The fringe spacings were set to 1.1, 1.5,
2.1 and 2.5 µm by controlling the crossing angles. The polarizations of pump and probe beams were perpendicular toeach other. The diffracted light was detected by a positiveintrinsic-negative (PIN) photodiode. The output of the photodiode was fed into a fast-gated-integrator and its average signal was recorded as a TRG response. A polycrystalline gold
thin film deposited on a glass plate was used for the measurements. The reflectivity R of the gold film was measured as 0.4
at λpump = 400 nm. The penetration depths of the pump beam
(λpump = 400 nm) and the probe beam (λprobe = 800 nm) were
calculated as 16 and 12 nm, respectively. All the experiments
were performed in air at 296 K.
3. Theory
To analyze the energy dissipation processes, we used thetwo-temperature model (TTM).7) The TTM assumes that theelectron and lattice systems are in local equilibrium, that theelectron temperature Te is not in equilibrium with the lattice
temperature Tl on a time scale of several picoseconds, andthat Te is coupled with Tl by electron-phonon coupling. The
evolution of the systems can be written as a set of coupled
differential equations. Namely,∂ Te= ∇(κ∇Te ) − G(Te − Tl ) + Q,
(1a)
γ Te
∂t
∂ Tl
= G(Te − Tl ),
(1b)
Cl
∂t
where γ Te and C l are the electronic and lattice heat capacities, G is the electron-phonon coupling factor, κ is the thermal conductivity, and Q is the energy source term. In eq. (1b),diffusion and energy source terms are neglected on the basisof physical considerations. Here, we assume that the electron temperature decay is approximately equal to the temporal
evolution of Te −Tl and it is expressed by an exponential function. Under these assumptions, the decay time of the electron
temperature τe can be derived from eqs. (1a) and (1b).

(2)

Here τe-p and τD are decay times by electron-phonon coupling
and thermal diffusion, respectively, and are written as
G
G
1
=
+ ,
(2a)
τe-p
γ Te
Cl
∇(κ∇Te )
1
=
.
(2b)
τD
γ Te (Te − Tl )From eqs. (2), (2a) and (2b), the decay time τe can be estimated by averaging eqs. (2a) and (2b) temporally and spatially. To our knowledge, there have been no reports on the estimation hot electron energy loss rate including the diffusion
term. In most conventional investigations by TR measurements,6) eq. (2b) has been neglected. This approximation isvalid only when film thickness is comparable to light penetration depth and the spot size of the pump beam is sufficiently
large to neglect diffusion in the surface plane.
To clarify the thermal diffusion of hot electrons, the diffusion term was divided into two parts; thermal diffusion in the
direction of the excitation fringe (parallel to the surface, xaxis in Fig. 1) and that in the thickness direction (perpendicular to the surface, z-axis in Fig. 1). The contributions of thesetwo diffusion parts to TRG responses are very different. Under our experimental setup of excitation fringe spacings (1.1
to 2.5 µm) and the 100-nm-thick gold target, decay rates forthe diffusion part parallel to the surface plane are estimatedto be nearly comparable to the rates of electron-phonon coupling. In contrast, decay rates for the diffusion part perpendicular to the surface are much faster because of its higher
temperature gradient.
The calculated temperature distributions along x- and zdirections are shown in Figs. 2(a) and 2(b), respectively. The
full scale of the transverse axis of Fig. 2(a) corresponds to
one period of the excitation fringe. The distributions are numerically calculated by eqs. (1a) and (1b) under the excitation conditions of Q = 2.25 mJ·cm−1 , 3 = 1.1 µm and thepulse width of 200 fs using the values of γ , C l , G and κ for
Au18, 19) (γ = 67.6 J·m−3 ·K−2 , C l = 2.5 × 106 J·m−3 ·K−1 ,G = 3.5 × 1016 W·m−3 ·K−1 and κ = 317 W·m−1 ·K−1 ).
As shown in Figs. 2(a) and 2(b), the temperature distribution
along the z-axis becomes nearly homogeneous on a time scaleof 1 ps while that along the x-axis is still clearly inhomogeneous at the delay time of 2 ps. This means that the diffusion
occurring perpendicular to the surface is very quickly and has
little effect on the TRG responses. Therefore, the diffusion
term (eq. (2b)) may be regarded as one-dimensional in the direction of the excitation fringe (x-axis). Under this condition,eq. (2) can be transformed by using the theory of the transient
grating method:17)
µ ¶2
1
2π
1
=
+D
,
(3)
τe
τe-p
3
where D is the thermal diffusivity of the hot electrons. Equation (3) means that τe-p and D can be calculated from the relationship between τe and 3. To obtain thermal conductivities κand electron-phonon coupling factors G from eqs. (2) and (3),a characteristic electron-temperature should be defined. The

A. H IBARA et al.

Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 5A

(a)

4000
Te (K)

0 ps
3000
2000
1000

2 ps
Λ=1.1 µm

1.0
TRG signal (arb. units)

5000

Te (K)

4000
3000

-2

e

0.75 mJcm(τ =2.2ps)

-2

e

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

Time (ps)Fig. 3. TRG responses on a 100-nm-thick gold film with pump fluence of0.75 and 2.25 mJ·cm−2 and an excitation fringe spacing of 1.1 µm. Solidlines: best-fit results by an exponential function which convoluted the temporal shape of the pump and probe pulses.

(b)

0 ps

2.25 mJcm(τ =4.4ps)

0.5

0.0

0
0.0 0.20 0.40 0.60 0.80 1.0
x (µm)
5000

2985

10.0

2ps

Λ=1.1 µm
Λ=2.5 µm

2000

0
0

e

Thickness = 100 nm

τ (ps)

1000
20

40
60
z (nm)

80

100

5.0

Fig. 2. (a)Temperature distributions for delay times of 0 and 2 ps in the direction of the excitation fringe at the surface. (b) Temperature distributionsin the thickness direction at x = 0 in (a). The distributions are calculated
numerically by eqs. (1a) and (1b).

mean initial electron-temperature in the probe volume Te0 waschosen as the characteristic temperature.6, 16)
4. Results and Discussion
Typical TRG responses with a fringe spacing 3 of 1.1 µm
and pump fluence I of 0.75 and 2.25 mJ·cm−2 are shown in
Fig. 3. The decay curves in Fig. 3 denote only electronic energy dissipation processes because the generation of longitudinal and surface acoustic waves observed in ps-TRG measurements13) should occur on a time scale of several tens ofpicoseconds. The solid lines in Fig. 3 are best-fit resultsof an exponential function. The exponential function corresponds to the decay of the intensity of the diffracted light,
which is proportional to (1²)2 .17) For an ambient temperature of ∼300 K, 1² is related to the electron temperature riseas 1² ∝ 1Te . Therefore, the decay time of the fitted exponential function corresponds to half the decay time of electron
temperature τe . As shown in Fig. 3, τe depends strongly on
pump fluence I .
Dependence of the decay times τe on pump fluence I for

0.0
0.0

1.0
2.0
-2
Pump fluence (mJcm )Fig. 4. Dependence of the decay times of electron temperature τe on pumpfluences for 3 of 1.1 and 2.5 µm. Solid line: theoretical values given byeq. (2a).

3 of 1.1 and 2.5 µm is shown in Fig. 4. The solid line shows
the theoretical values of τe-p estimated by eq. (2a). The decay times for 3 = 1.1 µm are clearly shorter than those for
3 = 2.5 µm. This means that the diffusion term in eq. (2)affects τe . To obtain the thermal diffusivity D in eq. (3),the dependence of the decay times τe on the excitation fringe
spacings 3 is shown in Fig. 5. The solid lines in Fig. 5 are
the least-squares-fitted data of eq. (3). Obtained values of Dand τe-p for various pump fluences I are summarized in Table
I. Initial electron temperatures Te0 are calculated from deposited energies and temperature-dependent heat capacities.
Since two time-coincident pump pulses were crossed on thesample surface in the TRG measurements, 2I × (1 − R) was2986

A. H IBARA et al.

Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 5A

10.0

12.0

8.0

10.0

G=2.5 x1016(Wm-3K-1)
G=3.5 x1016

8.0

I=2.25 mJcm

τe-p (ps)6.0eτ (ps)

-2

4.0

6.0
G=4.5 x1016

4.0
2.0
0.0
0.0

-2

I=0.75 mJcm

0.5 1.0 1.5 2.0 2.5
Fringe spacing Λ (µm)

2.0
3.0

Fig. 5. Dependence of the decay times τe on the excitation fringe spacings
3. Solid lines: least-squares-fitted data of eq. (3).

Table I. Obtained values of D and τe-p for various pump fluences I . Initial
electron temperatures Te0 are calculated from deposited energies and heat
capacities of the detection volume.
I(mJ·cm−2 )D(103 m2 ·s−1 )

τe-p
(ps)

Te0
(K)

0.38
0.75
1.13
1.50
1.88
2.25

7.2
7.4
4.9
3.3
4.0
3.2

3.7
4.2
4.1
5.1
6.8
8.0

1700
2400
2900
3400
3900
4200

chosen as the spatially averaged deposited energy. This simple assumption was justified by comparison of the estimated
Te0 and the numerically calculated one.
Our TRG measurements demonstrated that the electronphonon coupling process could be distinguished from thethermal diffusion on the metal surfaces, as shown in Table I.
This means that the electron-phonon coupling process andthermal diffusion can be investigated independently and quantitatively.
First, we discuss the electron-phonon coupling process.
Dependence of the decay times τe-p on the initial electron temperature Te0 is shown in Fig. 6. The error bars in the plot
indicate the standard deviations of the fitted values. The theoretical values calculated by eq. (2a) are also shown in Fig. 6(solid lines) by using three temperature-independent electronphonon coupling factors. The values calculated with G =3.5 × 1016 W·m−3 ·K−1 , which was previously reported,19, 20)
seem to agree with the experimental ones. However, our results seem to include the systematic errors relative to the theoretical values. This may correspond to the temperature dependence of the electron-phonon coupling. Although the temperature dependence of G has been investigated experimentallyand theoretically by many investigators,21, 22) the exact dependence has not been clarified. The temperature dependence has
been expressed by G ∝ Ten with 1 < n < 3 in one previous

0.0
0

2000

4000
Te0 (K)

6000

Fig. 6. Dependence of the decay times τe-p on the initial electron temperature Te0 . Solid lines: theoretical values calculated by eq. (2a) for
G = 2.5 × 1016 , 3.5 × 1016 , 4.5 × 1016 W·m−3 ·K−1 .

report21) and 0.5 < n < 1 in another report.22) When thesefindings are considered, the relationship between Te and τe-p
(eq. (2a)) is expressed by τe-p ∝ Tem with m < 1. However,
our results appear to show the relationship τe-p ∝ Tem withm > 1, which corresponds to G ∝ Ten with n < 0. Thissuggests that the coupling strength between the electron andphonon systems becomes weak in the high Te region. In addition, nonthermal states, which have been reported to exist
before thermalization processes,4) are suggested to affect theenergy dissipation processes even on time scales of picoseconds. While nonthermal behaviors of the hot electrons withina picosecond have been reported by several groups,2, 4, 5) thoseon a longer time scale have not, to our knowkedge, been reported. Nonthermal processes on time scales of picoseconds
are suggested in this paper.
Next, the thermal diffusion process is discussed. The thermal conductivities κ can be calculated with the thermal diffusivities shown in Table I using the relationship κ = Dγ Te0 .
Figure 7 shows the dependence of κ on initial electron temperature. The error bars in the plot indicate the standard deviations of the fitted values. When κ is assumed to depend on
the temperature, its tendency can be explained by the relaxation time approximation (RTA). In a previous report,19) thetemperature dependence of κ was expressed as
γ Te
1
(4)
κ = vF2
3 ATe2 + BTl
by deviating from 1/τ = ATe2 + BTl and a general expressionof κ = (1/3)vF2 τ γ Te , where vF is the electron Fermi velocity, A and B are constants and τ is the electron relaxation
time. Using the experimental value of the electron velocity
1.02 × 106 m·s−1 for vF 5) and the temperature coefficient of
the thermal resistivity,23) A is calculated as 6.5 × 106 K−2 ·s−1 .
Since κ is assumed to be 317 W·m−1 ·K−1 at Te = 300 K,18) B
is chosen as 6.9 × 1010 K−1 ·s−1 . Theoretical values of κ areshown in Fig. 7. For the equilibrium state of Tl = Te in eq. (4),the calculated values of κ agree well with the values found in

A. H IBARA et al.

Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 5A

κ =DγTe0 (Wm-1K-1)

1500

1000

Tl=300 K

500

Tl=Te

0
0

2000
4000
Te0 (K)

6000

Fig. 7. Comparison between experimental and theoretical values of κ,
where ◦ and ¦ indicate experimental results for hot electrons and literature
values for equilibrium state (ref. 18), respectively. Solid lines: theoretical predictions for equilibrium state (Tl = Te ) and nonequilibrium state(Tl = 300 K).

the literature (diamond18) ). The predictions for nonequilibrium states calculated by eq. (4) are shown in Fig. 7, where
Tl is fixed at 300 K (ambient temperature) in eq. (4). Our results seem to be explained well by eq. (4) without adjustableparameters. However, there is some question as to whether or
not the expression of the relaxation time in eq. (4) is valid.
The calculation by eq. (4) predicts that κ has a maximum
value at Te = 1700 K, which means that the electronic resistivity of a metallic sample has a minimum value at a hightemperature of 1700 K. The prediction that κ increases withrising Te for Te < 1700 K seems to require careful consideration.
On the other hand, when κ is assumed to be independent of
the temperature, our results may not be explained by the RTA.
In this case, one possibile explanation of our results is that
hot electrons act as ballistic particles, even on a time scale ofseveral picoseconds after thermalization. When the mean-free
path of the electrons is assumed to be on the same order of theexcitation fringe spacing 3, the scattering time is estimatedas several picoseconds. Although the ballistic behaviors ofphotoexcited electrons in metals before their thermalization
have been reported,5) ballistic behaviors of hot electrons afterthermalization have not been pointed out, to our knowledge.
This nonthermal property of the hot electrons on time scalesof picoseconds may be similar to that in discussions of theelectron-phonon coupling process.
5. Conclusions
The direct measurements of electron-phonon coupling andtransport processes of hot electrons in the surface plane havebeen performed on a gold thin film. The possibility that
fs-TRG measurements can be a powerful tool for studying
ultrafast dynamics in metals was demonstrated. Our TRG

2987

measurements and analysis, based on TTM, showed that theelectron-phonon coupling process and thermal diffusion canbe investigated independently and quantitatively. From thediscussion regarding the electron-phonon coupling factor G
and the thermal conductivity κ, nonthermal processes on timescales of picoseconds were suggested. However, some interesting problems as to the hot electron properties, such astemperature dependences of G and κ, remain. These energy
dissipation processes of hot electrons are expected to affectthe surface displacement dynamics such as the generation of
longitudinal acoustic waves and surface acoustic waves. The
applicability of the TRG method in the investigation of thesurface displacement dynamics has been demonstrated by our
research.12–15) The results of the surface displacement dynamics monitored by the fs-TRG will be discussed elsewhere.
Since the photothermal methods, including the TGR method,
have useful properties such as interface selectivity, dynamic
investigations can be applied to more complicated systems.
The fs-TRG method is also expected to be a powerful tool forinvestigating femtochemistry at solid/liquid interfaces.24)
Acknowledgement
This work is supported by a Grant-in-Aid for Specially Promoted Research (No. 07102004) from the Ministry of Education, Science, Sports and Culture.

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