Materials Transactions, Vol. 52, No. 3 (2011) pp. 547 to 553
#2011 The Japan Institute of Metals
EXPRESS REGULAR ARTICLE
Comparison of Theoretical Models of Electron-Phonon Coupling
in Thin Gold Films Irradiated by Femtosecond Pulse Lasers
Jae Bin Lee, Kwangu Kang and Seong Hyuk Lee*
School of Mechanical Engineering, Chung-Ang University, Seoul 156-756, Korea
This study reports on a comparison of theoretical models for electron-phonon coupling that is substantially associated with nonequilibrium energy transport in thin gold films irradiated by femtosecond pulse lasers. Three published electron-phonon coupling models were
analyzed with the use of a well-established two-temperature model to describe non-equilibrium energy transport between electrons and phonons.
Based on the numerical results, at lower fluence, all models showed nearly similar tendencies, whereas at higher fluence, constant electronphonon coupling forced unrealistically long electron-phonon equilibration times and spatially long diffusive regions as it failed to intrinsically
consider the effect of a high number density of excited d-band electrons. Even at higher fluence, however, both Lin’s and Chen’s models yielded
physically reasonable results, showing converging electron-phonon equilibration times and steep gradients in the spatial lattice temperature
profiles at higher laser fluence. In particular, Lin’s model predicts nonlinear characteristics of heat capacity and lattice temperature with respect
to laser fluence better than Chen’s model. Moreover, the electron-phonon relaxation time increased with laser fluence, whereas at laser fluence
greater than 0.05 J/cm2 , the thermal equilibrium time was nearly independent of the laser fluence. Thus, it was concluded that Lin’s model better
predicted the electron-phonon coupling phenomena in thin metal films irradiated by ultra-short pulse lasers.
[doi:10.2320/matertrans.M2010396]
(Received November 24, 2010; Accepted December 14, 2010; Published February 2, 2011)
Keywords: nonequilibrium energy transfer, femtosecond pulse laser, two-temperature model (TTM), electron-phonon coupling factor
(G-factor)
1.
Introduction
Ultra-short pulse lasers (<10 ps) have been regarded as
powerful tools for material modification (e.g., surface
cleaning and heating) and precise micro machining,1–3)
and their applications are rapidly growing. Unlike the
nanosecond pulse lasers which have been widely used in
industry, the ultrashort pulse lasers can minimize laserplume interaction and reduce the heat-affected zone because of the time scale difference. Such applications of
ultra-short pulse lasers require the fundamental investigation of non-equilibrium processes between electron
and phonons, fast phase transition, theoretical predictive
modeling and non-linear effects under the high fluence
laser pulses. In this study, we investigated non-equilibrium
energy transport in metallic films under ultra-fast laser
irradiations. For years, considerable effort has been devoted
to the theoretical and experimental study of non-equilibrium
energy transport in metallic thin films.4) Nevertheless,
the detailed physics behind opto-energy phenomena in thin
metal films under femtosecond pulse laser irradiation are
still debated and poorly understood because it is very
difficult to directly measure the carrier and lattice temperatures. For this reason, numerical models may be useful in
the analysis of energy transport in materials under pulsed
laser irradiations.2)
For the analysis of femtosecond pulse laser interactions
with metals, the two-temperature model (TTM) has been
widely used to describe electron-phonon interactions and
non-equilibrium heat transfer in metallic thin films.5–11)
Because the electron-electron interaction time (the order of
hundred femtoseconds) is shorter than the electron-phonon
interaction time (on the order of picoseconds), a strong nonequilibrium occurs between the electrons and phonons under
*Corresponding
author, E-mail: [email protected]
the irradiation of femtosecond laser pulses. The thermal
equilibrium among the excited electrons is reached within
a few hundred femtoseconds, depending on the excitation
energy above the Fermi level.12) Cooling of the electrons
occurs because of the electron-phonon energy transport and
the electronic diffusive motion. In TTM, the average energy
levels of hot electrons and cold lattices are described by two
separate temperatures, Te and Tl ,12) respectively. Energy
transport in time and space can be modeled by two coupled
diffusion equations describing the heat conduction of
electrons and lattices.13) For modeling the electron-phonon
energy transfer, both equations include an energy exchange
term that is proportional to an electron-phonon coupling
factor G and to the temperature differences between electrons
and lattices. In femtosecond laser material processing
applications, the temperature dependence of electron-phonon
coupling is very crucial in estimating an accurate ablation
depth. Nevertheless, less attention has been paid to investigate the electron-phonon coupling characteristics associated with the temperature dependency of the G-factor.
Assuming a relatively weak non-equilibrium state, most
TTMs have previously used a constant G-factor that is
estimated using the electron-phonon relaxation time.14)
Otherwise, the G-factor has been taken as a parameter fitted
for agreement between calculated results and experimental
values for the ablation threshold in gold.13) For example, Gfactors acquired through fitting to experimental data showed
broad variation from 3:6 1017 to 10:5 1017 Wm3 K1
for Ni, and ð2:2 0:3Þ 1017 Wm3 K1 for Au.13,15–17)
However, there is growing experimental evidence that the
applicability of the constant G-factor is limited to a low
intensity laser pulses.18–20)
Kaganov et al.21) proposed a theoretical model of the
G-factor with electron relaxation time that is defined as
the inverse of the electron-phonon collision frequency and
usually assumed to be proportional to 1=Te , forcing the
548
J. B. Lee, K. Kang and S. H. Lee
G-factor to be constant. For consideration of temperature
dependency, Chen et al.22) modified the electron relaxation
time in Kaganov’s model by involving electron-phonon
scattering and electron-electron scattering.22) Recently, Lin
et al.12) performed computational analysis based on firstprinciples electronic structure calculations of the electron
density of states (DOS),12) showing large deviations from the
linear temperature dependence of the electron heat capacity
and a constant electron-phonon coupling. According to these
results, under irradiation with high power, thermal excitation
of the electrons below the Fermi level begins to contribute
to the energy transfer between electrons and phonons. The
main objective of this study is to compare three different
theoretical models such as Kaganov’s model, Chen’s model,
and Lin’s model to investigate the effects of each G-factor
on the non-equilibrium energy transport in thin gold films.
In this article, we analyze the transient evolution of electron
and phonon temperatures and examine detailed physics
behind energy relaxation process in thin films. Finally, the
performance and applicability of various electron-phonon
coupling models were discussed.
2.
Mathematical Representation and Computational
Details
This study used TTM to compare the effects of electronphonon coupling on the energy transfer between electrons
and the lattice, and the subsequent temperature changes
under irradiation from femtosecond pulse lasers. Since a
focused laser beam diameter is usually much larger than the
penetration depth of the metal, the energy transport can be
safely assumed to be a one-dimensional (1D) problem. The
spatial and temporal evolutions of the electron and lattice
temperatures are written as follows:11,23)
@Ce ðTe ÞTe
@
@Te
ð1Þ
ke ðTe Þ
¼
GðTe Tl Þ þ Sðz; tÞ
@z
@t
@z
@Cl ðTl ÞTl
ð2Þ
¼ GðTe Tl Þ
@t
0:939Fð1 RðtÞÞ
Sðz; tÞ ¼
tp
!
Zz
t2
exp ðz; tÞdz exp 2:773 2 ; ð3Þ
tp
0
where C is the heat capacity with respect to the temperature,
subscripts e and l are the electron and lattice, respectively,
ke ðTe Þ is the electron thermal conductivity, G is the electronphonon coupling factor, and Sðz; tÞ is the laser heating source
term.23) In eq. (3), RðtÞ, F, tp , and ðz; tÞ represent the
reflectivity at the surface, laser fluence, pulse duration, and
absorption coefficient, respectively.
Electron heat capacity is commonly expressed as a linear
function of the electron temperature, Ce ðTe Þ ¼ Te , where
¼ 2 ne kB2 =2"F .24) Because this Sommerfeld expansion is
no longer valid for high electron temperatures, the electron
heat capacity needs to be calculated from the following
equation;12)
Z1
@ f ð"; ; Te Þ
Ce ðTe Þ ¼
gð"Þ"d";
ð4Þ
@Te
1
where gð"Þ is the electron DOS, is the chemical potential
at Te , and f ð"; ; Te Þ is the Fermi distribution function. The
electron thermal conductivity in eq. (1) based on Drude
theory can be expressed as follows:25)
ke ðTe Þ ¼
1
Ce ðTe ÞVe2 ðTe Þe ðTe Þ;
3
ð5Þ
where Ve denotes the electron velocity, and e means the
electron-electron relaxation time with respect to the electron
temperature.25) It is noted that for e related to the collision
frequency, there are three different regimes; the cold solid
regime, hot plasma regime, and intermediate regime, depending on the melting temperature and Fermi temperature as
follows:26)
1=2
me
Ji Tl
ðiÞ v ¼ ks
ðTe < Tmelt Þ
ð6Þ
M
h TD
Zne ln ðiiÞ v ¼ ks
ðTe TF Þ
ð7Þ
Te3=2
Ve
ðiiiÞ v ¼ ks
ðTmelt < Te < TF Þ;
ð8Þ
r0
where Z is the ionization degree and ne is the electron number
density, ln is the Coulomb logarithm, M is the ion mass, me
is the electron mass, Ji the is ionization potential, h is the
reduced Planck constant, TD is the Debye temperature, r0 is
the ion sphere radius, and Ve is the electron velocity. Here,
the empirical constant ks can be obtained from experimental
data using the interfacial condition.26)
To precisely calculate the reflectivity and absorption
coefficient, the dielectric function and complex refractive
index are determined by the Drude model.26,27) The dielectric
function of metal consists of a real part "1 and an imaginary
part "2 . The two parts are expressed by "1 ¼ 1 ð!P2 =ð!P2 þ
2 Þ and "2 ¼ !P2 =ð!L3 þ 2 !L Þ, respectively.26,27) Here, !P
is the plasma frequency, !L is the laser frequency, and is
the collision frequency suggested by Eidmann et al.26) The
complex refractive index can be expressed by n ¼ ðð"1 þ
ð"21 þ "22 Þ1=2 Þ=2Þ1=2 and k ¼ ðð"1 þ ð"21 þ "22 Þ1=2 Þ=2Þ1=2 ,
where n and k represent the refractive index and the
extinction coefficient, respectively.27) The reflectivity and
the absorption coefficient are determined as follows:27)
ðn 1Þ2 þ k2
ð9Þ
ðn þ 1Þ2 þ k2
4k
;
ð10Þ
¼
where means the wavelength of the incident laser. Kaganov
et al.21) suggested that the electron-phonon coupling factor
was calculated from the approximation of a free-electron
metal with a Debye phonon spectrum.21) For Te and Tl much
greater than the Debye temperature, the electron-phonon
coupling factor can be expressed as follows:21,22)
R¼
2 me Cs2 ne
ðTl TD ; Te Tl Tl Þ
ð11Þ
6 ðTl ÞTl
2
me Cs2 ne
G¼
ðTe TD ; Tl Te Þ; ð12Þ
6 ðTe ÞðTe Tl Þ
where me is the effective electron mass, Cs is the speed of
sound, ne is the number density of electrons, and ðTe Þ is the
electron relaxation time. Under conditions for which the
G¼
Comparison of Theoretical Models of Electron-Phonon Coupling in Thin Gold Films Irradiated by Femtosecond Pulse Lasers
electron relaxation time is proportional to the inverse of
the lattice temperature, Tl and the electron temperature and
lattice temperature are equal, ðTe Þ 1=Te , Te ¼ Tl , the
coupling factor given by eqs. (11) and (12) are independent
of temperature, i.e., constant.21) The constant model using a
constant electron-phonon coupling has been widely used in
TTM, despite a large temperature difference between the
electron and lattice.
Chen et al.22) showed that electron thermal relaxation was
closely associated with electron-electron and electron-phonon scattering, and presented the electron relaxation time as
e ¼ ðe-e þ e-p Þ1 where e-e and e-p are electron-electron
and electron-phonon collision rates, respectively.22) According to Chen’s model, when Tl > TD and Te TF , the
electron-electron and electron-phonon collision rates are
expressed by e-e ¼ eo ðTF =Te Þ2 and e-p ¼ po ðTF =Tl Þ,
respectively, where eo and po are empirically determined
for metals.22) Thus, the electron-phonon coupling of Chen’s
model can be written as follows:22)
Table 1
549
Physical properties of gold used in this study.
Property
Value
Reference
23
Boltzmann constant
1:3806503 10
[m2 kg s2 K1 ]
25)
Mass of electron
9:1 1031 [kg]
25)
Mass of ion
196.97 [a.m.u]
25)
Debye temperature
170 [K]
25)
Melting temperature
1337 [K]
25)
Fermi temperature
6:39 104 [K]
11)
First ionization potential
9.22 [eV]
11)
Laser wavelength
1053 [nm]
Speed of sound
1:9 103 [ms1 ]
23)
2 me Cs2 ne
ð13Þ
½Ae ðTe þ Tl Þ þ Bl ;
6
where Ae ¼ ðeo TF2 Þ1 and Bl ¼ ðeo TF2 Þ1 . For gold films,
constants Ae and Bl are 1:2 107 and 1:23 1011 K1 s1 ,
respectively.22)
Lin et al.12) proposed a new theoretical model of electronphonon coupling to consider the scattering of electrons away
from the Fermi surface at high electron temperatures. The
rigorous form of temperature-dependent electron-phonon
coupling can be expressed as:12)
Z
h kB h!2 i 1 2
@f
d";
ð14Þ
GðTe Þ ¼
g ð"Þ gð"F Þ
@"
1
G¼
where h is the reduced Planck constant, kB is the Boltzmann
constant, and gð"Þ is the electron DOS at the energy level ".
The above equation is considered a major variation of the Gfactor at high electron temperatures, whereas it requires some
difficult procedures for the electronic structure calculation of
the electron DOS.
Numerically, the finite difference method (FDM) was used
with a fully-implicit scheme for one-dimensional transient
diffusion equations of electron and lattice temperatures (see
eqs. (1) and (2)). Von Neumann boundary conditions were
used at the top and bottom surfaces of a thin gold film. With a
constant electron number density of 5:9 1028 m3 ,8,28) it
was assumed that the electron and lattice temperatures
(300 K) were initially in thermal equilibrium. The film
thickness was 300 nm and the laser pulse was 100 fs for the
calculation at a fixed wavelength of 1053 nm. The physical
properties of thin gold film are summarized in Table 1.
3.
Results and Discussion
Figure 1 compares the estimated electron-phonon coupling G-factors from the three different models assuming
constant electron-phonon coupling values. At room temperature, the G-factors predicted from all models are in the range
of 1:9{3:9 1016 Wm3 K1 , which shows good agreement
with experimentally reported values of 0:8{4:0 1016
Wm3 K1 .29–32) It should be noted that the G-factors
Fig. 1 The G-factor with respect to the electron temperature for different
G-factor models.
estimated by Kaganov’s model became nearly constant in
the range of the electron temperature, indicating that this
model fails to describe the temperature dependency of the
electron-phonon coupling. Hence, Kaganov’s model is
excluded from further discussion. Chen’s model shows a
linear increase in the G-factor with the increase of Te ,
because electron-electron scattering strengthens the G-factor
with the increase of Te found in Chen’s model. However,
the role of electron-electron scattering on the G-factor is not
yet fully understood. In Lin’s model, the G-factor remains
constant below 3000 K, while it starts to increase significantly above 3000 K. The strong increase of the G-factor in
Lin’s model is attributed to the large number of excited d
band electrons while only s electrons were excited below
3000 K.12)
Numerical predictions from each model are compared with
experimental measurements13) in Fig. 2. The irradiated
sample is a 300 nm gold thin film and the incident laser
fluence was F ¼ 0:001 J/cm2 (low laser fluence). The change
in normalized electron temperature was defined as ðTe Teq Þ=ðTe Teq Þmax , which is analogous to the change in
normalized reflectivity.14) Because each model shows a small
difference in the G-factor at low temperature in Fig. 1, the
reflectivity values predicted from the models were similar as
550
J. B. Lee, K. Kang and S. H. Lee
Fig. 2 Comparisons of the predicted electron temperatures with experimental data at low fluence (F ¼ 0:001 J/cm2 , tP ¼ 100 fs, df ¼ 300 nm).
in Fig. 2 and all showed fairly good agreement with the
experimental data. This agreement assures that the constant
G-factor that is commonly used in TTM is available at low
fluences.
Figure 3 shows the transient evolution of the electron and
lattice temperatures at the top surface as predicted by Chen’s
model and Lin’s model. For calculation, the Au film depth
is 300 nm and the laser fluence used was 0.2 J/cm2 (high
fluence case). The peak of the electron temperature in all
cases reached 10000 K, regardless of G-factor model. The
optical constants can be indirectly changed by different Gfactor models because the optical constants are functions of
electron and lattice temperatures, as shown in eqs. (6)–(8).
However, the effect of differences in the optical constants
of G-factors on electron and lattice temperature can be
neglected because the laser pulse duration is too short to yield
any significant changes in the electron and lattice temperature. The peak values of the electron temperature from each
model were estimated to be similar, but the values for
electron-phonon thermal equilibration time differ. When
using a constant value for electron-phonon coupling, the
electron-phonon equilibration time was approximately 25 ps,
which is much longer than the equilibration times predicted
by Lin’s and Chen’ models as shown in Fig. 3. As a result,
electron-phonon coupling is considered a very important
factor affecting thermal equilibrium.
As previously noted, most research has used a constant
value of 2:2 1016 Wm3 K1 , suggesting a very low
electron-phonon energy exchange rate, even at a maximum
electron temperature of 10000 K, which results in very slow
increase in the lattice temperature. Alternatively, at a
maximum electron temperature, Lin’s model and Chen’s
models predicted G-factors values much greater than the
constant, indicating that the lattice temperatures for both
models reached steady state conditions within a few picoseconds due to faster energy exchange rates between
electrons and phonons. The electron energy was excited by
laser pulse decay through electron-electron and electronphonon scattering. The electron-electron collision time was
Fig. 3 Predicted electron and phonon temperatures at high fluence (F ¼
0:2 J/cm2 , tP ¼ 100 fs, df ¼ 300 nm): (a) constant model; (b) Chen’s
model; (c) Lin’s model.
approximately 1 fs and electron-phonon collision time was
approximately 100 fs.24) Since a thermal equilibrium requires
5–20 collisions between energy carriers, the electron-phonon
relaxation time is 1012 s.24,33) Thus, at a high laser fluence,
the use of a constant electron-phonon coupling is no longer
valid, but Chen’s and Lin’s models are appropriate for
predicting energy transport between electrons and phonons.
Comparison of Theoretical Models of Electron-Phonon Coupling in Thin Gold Films Irradiated by Femtosecond Pulse Lasers
551
Fig. 4 The lattice temperature rises with respect to the laser fluence
(0.001–0.2 J/cm2 ) at t ¼ 1 ps for different G-factor models.
Fig. 5 The predicted electron-phonon relaxation time, which is defined as
the time required for the lattice temperature to reach 95% of the
equilibration temperature for different G-factor models.
Figure 4 represents the predicted lattice temperature with
respect to laser fluence at 1 ps after irradiation. When using a
constant value, very little change in the lattice temperature
was observed, even if the laser fluence increased. Alternatively, in Lin’s and Chen’s models, the lattice temperature
increased with the increase in laser fluence. One picosecond
was sufficient for building up electron-phonon relaxation,
and within the electron-phonon relaxation time, the electrons
transferred their energy to phonons. Thus, it is important to
examine the effects of laser fluence on electron-phonon
relaxation time.
Figure 5 illustrates electron-phonon thermal relaxation
time with respect to laser fluence. The thermal electronphonon relaxation time was numerically determined using
the time required for 95% steady-state lattice temperature.
When a constant G value was used, the predicted electronphonon relaxation time increased continuously with laser
fluence. Alternatively, Lin’s and Chen’s models predicted
that the electron-phonon relaxation time converges to 5 ps
approximately. Considering that average electron-phonon
relaxation time was a few picoseconds,24,33) both models
showed good agreement in predicting the relaxation time,
and the use of constant electron-phonon coupling was no
longer appropriate.
Figure 6 presents electron and phonon temperatures at
different times (0.1, 1, 5, 10, and 15 ps after irradiation). In
the simulation, the Au film depth was 300 nm and the laser
fluence was fixed at 0.2 J/cm2 . As shown in Fig. 6(a), even
if all the electron-phonon coupling models predicted nearly
the same electron temperature, the prediction of the lattice
temperature depends on electron-phonon coupling. This
demonstrates traditional non-equilibrium characteristics resulting from scale differences between the energy relaxation
time and pulse duration. With increasing time, electron
energy is transferred to the lattice phonon, showing the
increase in lattice temperature and highly dependency on the
electron-phonon coupling. The higher G-factors in Lin’s and
Chen’s models impel larger lattice temperature excursions.
Because the energy diffusion of electrons at t ¼ 1{5 ps was
much greater than the energy transferred to the lattice
phonon, as shown in Fig. 6(e), the electron temperatures
estimated by different models at t ¼ 5 ps differ less than the
lattice temperatures. It should be noted that the diffusion time
of the phonon 0:1 ns, which is a time scale one order of
magnitude greater than our calculation time. Therefore,
lattice thermal conduction is safely ignored in eq. (2). At
t ¼ 15 ps, the electron temperatures and lattice temperatures
in Lin’s and Chen’s models almost reached the equilibrium
state as time progressed.
4.
Conclusions
The present study numerically investigated electronphonon coupling characteristics that substantially affected
non-equilibrium energy transport in thin gold films heated by
femtosecond pulse lasers. Theoretical models were compared
and discussed regarding the physical interpretation of the
cooling rates of hot electrons in thin gold films. Based on the
numerical results, at lower laser fluence, Lin’s and Chen’s
models yielded nearly the same predictions, compared to
the results of a model that assumed a constant G factor
(ð2:2 0:3Þ 1016 Wm3 K1 ) because only s-band electrons were excited for a relatively low electron temperature
(below 3000 K), and consequently, G-factor did not show
much influence. Alternatively, at high laser fluence, constant
electron-phonon coupling cannot consider the effects of the
increase in excited d-band electrons on the energy transfer,
whereas Lin’s and Chen’s models accurately predicted the
energy transport between energy carriers in metallic thin
films irradiated by a femtosecond pulse laser. Thus, the Gfactor should be carefully determined when describing nonequilibrium energy transport. Moreover, the electron-phonon
relaxation time increases with laser fluence up to 0.05 J/cm2 .
However, when the laser fluence was greater than 0.05
J/cm2 , the equilibration time was not influenced much by
laser fluence.
Acknowledgements
This work was supported by the Korea Research Founda-
552
J. B. Lee, K. Kang and S. H. Lee
Fig. 6
The spatial electron and the lattice temperature distribution (F ¼ 0:2 J/cm2 , tP ¼ 100 fs, df ¼ 300 nm) at 100 fs, 1, 5, 10, and 15 ps.
tion Grant funded by the Korean Government (MOEHRD,
Basic Research Promotion Fund) (KRF-2008-331-D00064),
and it was also sponsored by the Human Resources Development of the Korea Institute of Energy Technology Evaluation
and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge and Economy (2008-P-EP-HME-05-0000).
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Nomenclature
Ce electron heat capacity
Cl lattice heat capacity
Cs speed of sound
F laser fluence
G electron-phonon coupling constant
h reduced Planck constant
Ji first ionization potential
k extinction coefficient
ke electron thermal conductivity
ks numerical correction constant
M ion mass
me electron mass
ni ion number density
ne electron number density
n refractive index
r0 ion sphere radius
R reflectivity
S laser heating source term
t time
tp pulse duration time
TD Debye temperature
TF Fermi temperature
Te electron temperature
Tl lattice temperature
Tmelt melting temperature of gold
Ve electron velocity
Z ionization degree
Greek symbols
absorption coefficient
optical penetration depth
electron heat capacity constant
laser wavelength
ln Coulomb logarithm
electron-phonon relaxation time
collision frequency
!L laser frequency
!p plasma frequency
Subscripts
e electron
l lattice
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