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In this paper, we present ultrafast measurements of the complex refractive index for copper up to a time delay of 20 ps with an accuracy <1% at laser fluences in the vicinity of the ablation threshold. The measured refractive index n and extinction coefficient k are supported by a simulation including the two - temperature model with an accurate description of thermal and optical properties and a thermomechanical model. Comparison of the measured time resolved optical properties with results of the simulation reveals underlying physical mechanisms in three distinct time delay regimes. It is found that in the early stage (-5 ps - 0 ps) the thermally excited d-band electrons make a major contribution to the laser pulse absorption and create a steep increase in transient optical properties n and k. In the second time regime (0 ps - 10 ps) the material expansion influences the plasma frequency, which is also reflected in the transient extinction coefficient. In contrast, the refractive index n follows the total collision frequency. Additionally, the electron-ion thermalization time can be attributed to a minimum of the extinction coefficient at ∼10 ps. In the third time regime (10 ps - 20 ps) the transient extinction coefficient k indicates the surface cooling-down process.
Content may be subject to copyright.
Applied
Surface
Science
417
(2017)
2–15
Contents
lists
available
at
ScienceDirect
Applied
Surface
Science
journal
h
om
epa
ge:
www.elsevier.com/locate/apsusc
Ultrafast
laser
processing
of
copper:
A
comparative
study
of
experimental
and
simulated
transient
optical
properties
Jan
Wintera,,
Stephan
Rappa,b,
Michael
Schmidtc,
Heinz
P.
Hubera
aDepartment
of
Applied
Sciences
and
Mechatronics,
Munich
University
of
Applied
Sciences,
Lothstrae
34,
80335
Munich,
Germany
bErlangen
Graduate
School
in
Advanced
Optical
Technologies
(SAOT),
Friedrich-Alexander-Universität
Erlangen-Nürnberg,
Paul-Gordan-Strae
6,
91052
Erlangen,
Germany
cLehrstuhl
für
Photonische
Technologien,
Friedrich-Alexander-Universität
Erlangen-Nürnberg,
Konrad-Zuse-Strae
3-5,
91052
Erlangen,
Germany
a
r
t
i
c
l
e
i
n
f
o
Article
history:
Received
11
November
2016
Received
in
revised
form
25
January
2017
Accepted
9
February
2017
Available
online
16
February
2017
Keywords:
Pump–probe
ellipsometry
Transport
properties
Optical
properties
Laser-metal
interaction
a
b
s
t
r
a
c
t
In
this
paper,
we
present
ultrafast
measurements
of
the
complex
refractive
index
for
copper
up
to
a
time
delay
of
20
ps
with
an
accuracy
<1%
at
laser
fluences
in
the
vicinity
of
the
ablation
threshold.
The
measured
refractive
index
n
and
extinction
coefficient
k
are
supported
by
a
simulation
including
the
two-temperature
model
with
an
accurate
description
of
thermal
and
optical
properties
and
a
thermo-
mechanical
model.
Comparison
of
the
measured
time
resolved
optical
properties
with
results
of
the
simulation
reveals
underlying
physical
mechanisms
in
three
distinct
time
delay
regimes.
It
is
found
that
in
the
early
stage
(5
ps
to
0
ps)
the
thermally
excited
d-band
electrons
make
a
major
contribution
to
the
laser
pulse
absorption
and
create
a
steep
increase
in
transient
optical
properties
n
and
k.
In
the
second
time
regime
(0–10
ps)
the
material
expansion
influences
the
plasma
frequency,
which
is
also
reflected
in
the
transient
extinction
coefficient.
In
contrast,
the
refractive
index
n
follows
the
total
collision
frequency.
Additionally,
the
electron-ion
thermalization
time
can
be
attributed
to
a
minimum
of
the
extinction
coef-
ficient
at
10
ps.
In
the
third
time
regime
(10–20
ps)
the
transient
extinction
coefficient
k
indicates
the
surface
cooling-down
process.
©
2017
The
Author(s).
Published
by
Elsevier
B.V.
This
is
an
open
access
article
under
the
CC
BY
license
(http://creativecommons.org/licenses/by/4.0/).
1.
Introduction
The
laser-matter
interactions
in
the
sub-ps
range
has
received
a
great
attention
over
the
last
two
decades
and
has
a
broad
spectrum
of
industrial
applications
in
material
processing
and
pre-
cise
micro-machining
[1–3].
A
deep
theoretical
understanding
of
the
fundamental
phenomena
of
the
non-equilibrium
dynamic
in
condensed
matter
is
needed
to
optimize
the
laser
process.
The
the-
oretical
description
of
the
laser-material
dynamics
also
requires
predictive
modeling
of
the
irradiated
metal
with
an
ultra-short
laser
pulse.
In
an
early
stage
of
the
non-equilibrium
process
the
laser
pulse
energy
is
absorbed
and
reflected,
respectively.
Trans-
mission
can
be
neglected,
as
the
samples
of
our
study
have
a
thickness
which
can
be
considered
large
compared
with
the
opti-
cal
penetration
depth.
The
pulse
absorption
and
reflection
within
the
material
is
described
by
the
optical
penetration
depth
and
reflection
coefficient.
The
penetration
depth
and
reflection
coef-
ficient
during
laser
impact
are
depicted
by
the
complex
refractive
index.
Whereby,
the
imaginary
part
describes
the
absorption
or
the
Corresponding
author.
E-mail
address:
jan.winter@hm.edu
(J.
Winter).
extinction
and
both,
real
and
imaginary
part,
describe
the
reflectiv-
ity
with
the
Fresnel
law.
In
metals,
the
absorption
measurements
have
been
focused
on
the
investigation
of
the
electron
dynamics
in
the
plasma
state
[4,5]
and
on
measurements
of
laser
energy
absorption
by
the
metal
sample
with
laser
calorimetry
technique
[6,7].
Theoretically
stud-
ies
of
femtosecond
laser
pulse
absorption
mechanisms
have
been
performed
by
a
comparison
of
the
experimental
founded
average
absorption
coefficient
with
the
simulated
absorption
[8–11].
While
these
studies
are
performed
at
very
high
laser
fluences,
the
reflec-
tivity
of
a
metal
surface
can
be
also
altered
significantly
in
a
very
low
fluence
regime
during
the
laser
pulses
owing
to
the
optical
electron
excitation.
The
change
in
reflectivity
has
often
been
used
to
study
the
relaxation
dynamics
of
non-equilibrium
electrons,
if
the
estimated
change
of
electron
temperature
does
not
exceed
100
K
and
ion
temperature
remains
at
room
temperature
[12–14].
Nevertheless,
these
ultra-fast
measurement
techniques
of
optical
properties
at
moderate
laser
fluences
for
a
wide
range
of
metals
have
mostly
been
limited
to
the
measurement
of
laser
induced
reflectivity
changes
[15,16].
However,
optical
values
as
refractive
index
and
extinction
coefficient
could
not
be
directly
resolved
in
the
measured
reflectivity.
Thus,
their
contribution
to
the
optical
penetration
depth
and
reflection
coefficient
cannot
be
attributed.
http://dx.doi.org/10.1016/j.apsusc.2017.02.070
0169-4332/©
2017
The
Author(s).
Published
by
Elsevier
B.V.
This
is
an
open
access
article
under
the
CC
BY
license
(http://creativecommons.org/licenses/by/4.0/).
J.
Winter
et
al.
/
Applied
Surface
Science
417
(2017)
2–15
3
One
way
to
measure
the
time-resolved
dielectric
function
of
a
material
with
a
femtosecond
time
resolution
was
proposed
by
Roeser
et
al.
by
using
a
dual-angle
reflectometry
technique.
This
method
is
based
on
measuring
the
absolute
reflectivity
at
two
angles
of
incidence.
The
optical
indices
are
calculated
from
the
measured
values
of
the
transient
reflectivity
change
by
numeri-
cal
inversion
of
the
Fresnel
formulas
at
the
predefined
and
fixed
angles
[17].
The
optical
dynamics
of
the
d-band
electrons
in
the
sur-
face
plasmonic
state
in
wolfram
and
the
ultrafast
destructuring
of
laser
irradiated
tungsten
around
the
ablation
threshold
were
exper-
imentally
mapped
by
using
a
dual-angle
one
color
time-resolved
pump–probe
ellipsometry
method
following
the
technique
from
Roeser
et
al.
[18,19].
The
alternative
way
to
determine
temporal
optical
changes
during
the
laser
irradiation
can
be
obtained
by
the
ultrafast
pump–probe
principle
connected
with
fundamental
ellipsometry
technique.
This
method
combines
the
advantages
of
ellipsometry
measurements
on
the
basis
of
the
analysis
of
the
polarization
state
from
reflected
light
and
time-resolved
pump–probe
principle
and
allows
to
determine
directly
the
transient
complex
refractive
index.
Recently,
such
a
pump–probe
ellipsometry
setup
was
developed
by
Rapp
et
al.
to
study
ultra-fast
changes
of
the
complex
refrac-
tive
index
during
and
after
the
material
irradiation
with
a
sub-ps
temporal
resolution
[20].
In
this
paper,
we
present
the
pump–probe
ellipsometric
mea-
surements
of
the
complex
refractive
index
for
copper
(Cu)
during
and
after
the
irradiation
with
ultra-short
laser
pulse
below
and
above
the
ablation
threshold.
The
transient
complex
refractive
index
is
split
into
the
real
and
imaginary
parts,
the
refractive
index
n
and
the
extinction
coefficient
k,
respectively.
Additionally,
a
numerical
simulation
for
the
optical
material
response
is
per-
formed
by
applying
the
framework
of
the
two-temperature
model
fully
coupled
with
the
thermoelasticity
theory.
The
combination
of
time-resolved
experiments
and
simulations
generates
a
profound
understanding
of
the
experimental
data
and
underlying
physical
mechanisms.
The
structure
of
the
paper
is
as
follows.
Firstly,
in
Section
2
the
pump–probe
ellipsometry
setup
and
the
data
processing
are
explained
briefly.
In
Section
3
the
governing
equations
of
the
2T-
TE
model
are
given
with
an
accurate
description
of
modeling
from
transport
properties
in
Section
4
and
optical
model
in
Section
5.
In
Section
6
the
results
of
the
measured
real
refractive
index
and
extinction
coefficient
are
presented.
Finally,
the
experimental
data
are
compared
with
predictions
of
simulation
to
draw
conclusion
for
the
underlying
physical
mechanisms.
A
brief
summary
of
the
results
completes
the
paper
in
Section
7.
2.
Experimental
setup
For
the
time-resolved
ellipsometric
measurements,
sputtered
copper
(Cu)
films
with
a
thickness
of
430
nm
were
used.
The
abla-
tion
threshold
fluence
Fthr of
the
Cu
sample
was
determined
by
a
common
method
described
in
[21].
Assuming
an
ideal
Gaus-
sian
spatial
laser
beam
and
an
ideal
threshold
behavior
for
the
laser
ablation
process
Fthr was
determined
to
be
1.9
±
0.1
J/cm2
for
the
fundamental
infrared
wavelength
(
=
1056
nm)
at
normal
incidence
pulses
with
linear
polarization
and
680
fs
pulse
duration.
2.1.
Pump–probe
ellipsometry
setup
Fig.
1
shows
an
overview
of
the
experimental
pump–probe
ellip-
sometry
setup,
which
was
described
in
detail
in
[20].
Laser
pulses
at
a
center
wavelength
of
=
1056
nm
with
a
pulse
duration
of
=
680
fs
(FWHM)
emitted
by
a
Nd:glass
laser
source
are
divided
into
pump
and
probe
pulses
by
a
polarizing
beam
splitter
(ratio
of
Fig.
1.
Pump–probe
ellipsometry
setup:
laser
pulses
(
=
680
fs,
=
1056
nm)
are
divided
into
pump
and
probe
pulses.
The
pump
pulse
is
focused
at
the
sample
and
initiates
the
reaction.
The
probe
pulse
is
frequency
doubled
(SHG)
for
illumination.
It
is
coupled
in
the
ellipsometric
branch
of
the
setup
(incident
angle
=
70).
The
polarization
of
the
probe
beam
on
the
sample
is
adjusted
by
the
polarizer
(ϕ
=
45);
the
polarization
of
the
reflected
probe
light
is
analyzed
by
the
rotatable
analyzer.
The
locally
distributed
reflected
intensity
is
detected
by
a
CCD-camera.
To
temporally
delay
the
pump
against
the
probe
pulse
a
delay
line
(t
1
ns)
is
used.
90–10%).
A
mechanical
shutter
in
the
pump
path
separates
a
sin-
gle
pump
pulse,
used
for
initiating
the
ablation.
The
pump
pulse
is
focused
with
a
f
=
100
mm
lens
on
the
sample
(Gaussian
beam
radius
=
25
±
0.5
m
at
e2intensity).
The
probe
pulse
with
the
wavelength
of
probe =
528
nm
is
frequency
doubled,
probe =
540
fs
(FWHM))
and
weakly
focused
by
a
f
=
1000
mm
lens
for
illuminat-
ing
an
area
of
about
700
m
in
diameter
on
the
sample.
The
probe
pulse
is
coupled
into
the
ellipsometric
branch
of
the
setup.
The
incident
angle
is
fixed
at
=
70.
The
linear
polarization
angle
ϕ
of
the
probe
pulse
on
the
sample
is
adjusted
by
a
Glan-Laser-Prism
(“polarizer”)
(extinction
ratio
106:1)
to
ϕ
=
45with
respect
to
the
incident
plane.
The
process
area
is
imaged
by
a
20×
magnification
stress-free
and
thus
polarization
maintaining
microscope
objective
with
numerical
aperture
of
NA
=
0.42
according
the
optical
resolu-
tion
of
0.61/NA
=
0.8
m.
The
polarization
of
the
reflected
probe
pulse
after
transmission
through
the
objective
is
analyzed
by
a
rotating
thin
film
polarizer
(“analyzer”)
(extinction
ratio
2
×
106:1).
A
band-pass
filter
(530
±
10
nm)
in
front
of
the
CCD
detector
blocks
ambient
light
and
plasma
emission
or
scattered
pump
light
from
the
sample.
To
optically
delay
the
pump
against
the
probe
pulse,
the
pump
pulse
is
guided
over
a
variable
linear
translation
stage
(t
1
ns).
To
take
a
series
of
images
covering
the
ablation
pro-
cess,
the
sample
is
irradiated
by
a
pump–probe-pulse
combination
at
a
new
position
for
every
delay
time
t
and
for
every
analyzer
angle
.
For
every
delay
time,
the
intensity
I
on
the
CCD
is
detected
in
dependency
on
the
analyzer
angle
.
This
intensity
distribution
versus
the
analyzer
angle
can
be
fitted
by
Eq.
(1)
which
describes
a
harmonic
function
with
a
180periodicity:
I
=
I0[1
+
˛
cos(2)
+
ˇ
sin(2)],
(1)
The
Fourier-coefficients
˛
and
ˇ
are
determined
by
a
discrete
Fourier
transform.
The
coefficients
˛
and
ˇ
describe
the
amplitude
and
phase
of
the
harmonic
function.
The
ellipsometric
angles
and
are
calculated
by
[22].
=
arctan 1
+
˛
1
˛tan(|ϕ|),
(2)
=
arccos ˇ
1
˛2,
(3)
4
J.
Winter
et
al.
/
Applied
Surface
Science
417
(2017)
2–15
The
ellipsometric
parameters
and
in
Eq.
(4)
describe
the
amplitude
ratio
and
the
phase
difference,
respectively,
of
the
p-
and
s-polarized
components
after
reflection:
tan
=|rp|
|rs|,
=
ırp
ırs.
(4)
where
ırp and
ırs represent
the
phases
of
the
p-
and
the
s-polarized
reflected
light.
The
ellipsometric
parameters
are
combined
in
the
fundamental
equation
of
ellipsometry
in
following
expression:
=
tan
e(i)=rp
rs
,(5)
The
complex
refractive
index
N
=
n
ik
of
an
optically
thick
sample
N1in
a
homogeneous
ambient
medium
N0can
be
then
calculated
by
Eq.
(6)
[23].
N1=
N0sin()1
+1
1
+
2
tan2().
(6)
In
this
work,
the
optically
thick
sample
is
represented
by
a
430
nm
thick
copper
film
on
a
glass
substrate
in
the
ambient
medium
air
(N0=
1
i·0).
It
has
to
be
emphasized
that
calculated
values
of
optical
indices
in
the
excited
matter
describe
the
state
of
matter
integrated
over
the
skin
depth
of
the
probe
pulse.
Varying
optical
properties
inside
the
material
within
the
penetration
depth
due
to
decreasing
energy
deposition
of
the
pump
pulse
with
increasing
depth
arise
in
gra-
dients
of
electron
and
ion
temperature
distributions.
Thus,
the
calculated
optical
indices
are
averaged
values
over
the
skin
depth
[20].
3.
Governing
equations
The
laser
radiation
at
the
surface
of
a
bulk
material
will
be
partially
reflected
or
transmitted
according
to
Fresnel’s
laws
and
absorbed
within
the
material
by
conduction
band
elec-
trons
depending
on
the
absorption
coefficient
at
the
irradiation
wavelength.
The
laser
pulse
energy
absorption
induces
a
non-
equilibrium
state
with
a
non-Fermi
distribution
in
the
electronic
system.
At
first,
within
a
few
tens
of
femtoseconds
the
energy
is
redistributed
among
the
electrons
through
electron–electron
inter-
actions
generating
a
hot
Fermi
distribution
with
temperature
Te
[12,24].
In
a
second
step,
the
electron
gas
loses
its
energy
through
energy
transfer
to
the
ion
due
to
electron-ion
coupling
within
the
electron-ion
scattering
process
until
the
local
equilibrium
is
reached.
Assuming
that
these
relaxation
times
are
small
compared
to
the
pulse
duration,
the
energy
transfer
between
the
electron
and
the
ion
systems
is
described
by
the
two-temperature
(2T)
model.
For
metals,
the
dynamic
of
non-equilibrium
state
associated
with
the
second
step
the
temporal
as
well
as
spatial
evolution
of
the
electron
and
ion
temperatures
in
the
target
material
induced
with
an
ultra-short
laser
pulse
are
generally
described
by
two
coupled
non-linear
differential
equations
[25,26].
The
heat
conduction
equation
for
electron
and
ion
system
from
the
irradiated
hot
surface
into
the
bulk
is
given
by
Eqs.
(7)
and
(8).
The
thermal
2T-model
is
supplemented
by
consideration
of
the
thermomechanical
effect,
which
is
fully
coupled
with
the
thermoe-
lasticity
theory
for
calculation
of
the
temporal
and
spatial
evolution
of
the
compressive
thermoelastic
pressure:
Ce(Te)Te
t=
·
[ke(Te,
Ti)Te]
G(Te)(Te
Ti)
+
S,
(7)
Ci(Ti)Ti
t=
G(Te)(Te
Ti)
3˛thBTi˙
ij,
(8)
where
C
is
the
heat
capacity
and
K
is
the
thermal
conductivity
with
respect
to
the
temperature
of
the
electron
and
ion
denoted
by
sub-
scripts
e
and
i,
G
is
the
electron-ion
coupling
factor,
and
S
is
the
laser
heating
source
term.
B
is
the
bulk
modulus,
˛th is
the
thermal
linear
expansion
coefficient
and ˙
ij is
the
material
induced
strain
rate.
The
corresponding
wave
equation
for
the
dynamic
of
the
dis-
placement
u
in
a
linear
elastically
isotropic
solid
is
expressed
in
Eq.
(9)
[27,28]:
2u
t2=E
2(1
+
)2u
E
2(1
+
)(1
2)(u)
th(Ti
T0)
+
Fe.
(9)
E
is
Young’s
modulus
and
is
the
Poisson
ratio.
The
volume
force
Fe=
·
Peis
the
hot
electron
blast
force
induced
by
the
interaction
among
electron
system
in
a
non-equilibrium
state,
where
Peis
the
electronic
pressure
with
Pe=
2/3*Ce(Te)*Te[29].
In
the
present
work
Eq.
(9)
of
mechanical
motion
for
a
contin-
uum
is
solved
for
a
2D-axisymmetric
geometry
in
the
framework
of
finite
element
method
as
the
Cauchy–Green
equation
using
the
software
“Comsol
Multiphysics”.
A
similar
model
for
the
mechani-
cal
motion
was
discussed
in
detail
in
a
previous
paper
[30].
The
phase
transition
from
solid
to
liquid
in
2T-model
simu-
lation
occurs,
when
the
equilibrium
temperature
for
melting
is
reached
and
takes
away
the
latent
heat
of
melting
known
as
the
fusion
enthalpy
until
the
phase
transition
is
completely
finished.
The
implementation
in
finite
difference
model
is
performed
with
the
ion
temperature
dependence
of
the
heat
capacity
in
the
form
of
a
Gaussian
distribution
function
of
a
finite
width
T
centered
around
the
melting
temperature
Tm[30,31].
The
finite
width
T
in
the
Gaussian
function
is
smearing
the
delta
function
in
order
to
improve
numerical
solvability.
For
a
solid
and
liquid
state
the
spe-
cific
heat
capacity
was
taken
from
literature
[32,33].
In
the
liquid
state
above
Tmno
shear
waves
can
be
supported.
In
modeling
of
liquid
state
in
Eq.
(9)
the
Young’s
modulus
is
decreasing,
while
the
Poisson
ratio
is
increasing
with
temperature
up
to
the
value
of
0.5
in
such
way
that
the
bulk
modulus
remain
nearly
constant
[30].
In
this
work,
a
free-standing
rectangular
copper
(Cu)
film
with
a
thickness
of
2
m
and
50
m
radial
length
within
the
2D-
axisymmetric
geometry
is
assumed
to
be
irradiated
on
the
front
surface
along
the
axial
symmetry
z-axis
by
a
temporal
and
spa-
tial
Gaussian
intensity
pulse
shape.
The
intensity
decrease
in
axial
direction
in
the
case
of
a
temporal
and
spatial
variation
of
the
absorption
coefficient,
˛(r,
z,
t)
is
expressed
mathematically
in
a
general
form
and
the
volumetric
laser
heat
source
with
a
finite
spot
size
is
given
by
the
following
equation,
S(r,
z,
t)
=
(1
R(r,
0,
t))
·2F0
p·ln
2
·
exp 2r2
w2
0·
˛(r,
z,
t)
·
exp 0
z
˛(r,
z,
t)dz
4
ln
2t
tp2,
(10)
where
R
is
the
reflectivity,
(1
R)
the
absorbed
fraction
of
the
laser
intensity,
F0is
the
peak
fluence,
r
is
the
radial
spatial
coordinate,
w0is
the
beam
radius
at
e2intensity
level,
t
is
the
time
variable,
pis
the
pulse
duration
(FWHM)
and
˛
is
the
absorption
coefficient
and
z
is
the
axial
spatial
coordinate.
The
optical
properties
R
and
˛
can
be
calculated
from
Fresnel
equation
[34]
according
to
R
=(n
1)2+
k2
(n
+
1)2+
k2,
˛
=2ωLk
c0
.
(11)
where
n
is
the
refractive
index,
k
is
the
extinction
coefficient
and
c0is
the
speed
of
light
in
vacuum.
J.
Winter
et
al.
/
Applied
Surface
Science
417
(2017)
2–15
5
The
constants
refractive
index
n
and
the
extinction
coefficient
k
are
expressed
in
relation
to
real
and
imaginary
part
of
the
dielectric
function
rand
i:
n
=2
r+
2
i+
r
2,
k
=2
r+
2
i
r
2.
(12)
4.
Transport
properties
Most
properties
of
metals
like
electrical
and
thermal
conductiv-
ity
can
be
described
in
a
good
approximation
with
the
free
electron
gas
(FEG)
theory.
In
the
FEG
theory
of
metals
proposed
by
Sommer-
feld
the
valence
electrons
in
a
crystal
structure
of
a
metallic
solid
are
considered
as
an
ideal
gas
is
attraction
completely
neglected
and
hence
the
valence
electrons
can
be
considered
as
free
and
independent
particles
[35,36].
In
the
case
of
copper,
the
electron
energy
in
the
half
filled
s-band
can
be
approximated
by
a
parabolic
dispersion
ε(k)
=
2k2/2m
as
function
of
wave
vector
k.
Within
the
FEG
theory
the
density
of
states
(DOS)
in
the
parabolic
valence
band
can
be
determined
by
gs(ε)
=3
2
ns
εFε
εF1/2
,
(13)
where
nsis
the
number
of
electrons
per
unit
volume
in
the
s-band
and
εFis
the
Fermi
energy
defined
by
the
zero
temperature
[35].
The
Fermi
energy
defines
the
highest
occupied
state
in
electron
DOS.
In
the
DOS
of
the
free
electron
gas
only
valence
electrons
in
the
s-band
are
considered.
Thus,
at
low
temperatures
a
relatively
small
electronic
excitation
induced
by
s-band
intraband
transition
around
the
Fermi
energy
can
be
expected.
At
higher
temperatures
the
Fermi
energy
is
smearing
about
kBTe
and
the
electrons
from
high
density
of
states
in
the
low-lying
d-
band
can
be
excited
to
higher
energies
in
the
conduction
s-band.
Thus,
the
thermal
excitation
of
d-band
electrons
with
a
high
density
of
states
from
copper
cannot
be
neglected
and
must
be
considered
at
higher
temperatures.
Thus,
at
large
thermal
excitation
Eq.
(13)
must
be
recast
as
the
sum
of
the
separated
electronic
DOS
for
s-
and
d-band
according
to
gtot(ε)
=
gs+
gd.
The
electronic
DOS
of
the
conduction
s-band
within
the
FEG
theory
is
described
by
Eq.
(13)
for
Cu
with
a
Fermi
energy
of
εF=
10.02
eV.
The
electron
configuration
corresponding
to
a
full
d-
band
and
a
half
filled
s-band
is
3d104s1.
The
mass
density
of
the
FCC
solid
crystal
structure
of
Cu
is
8960
kg
m3from
which
the
atomic
number
density
nat is
calculated
as
8.5
×
1028 m3.
The
high
density
of
states
of
the
d-band
at
zero
electron
temperature
in
Cu
can
again
be
approximated
by
a
root
function
similar
to
the
idea
of
Petrov
et
al.
[37,38]
within
two-parabolic
approximation
(2P)
of
the
cold
electron
DOS.
The
DOS
in
the
d-band
of
copper
can
be
simply
approximated
by:
gd(ε)
=3
2
(8.5
×
1029)
εF,d ε
3.93
εF,d 1/2
·
H(ε
7.87).
(14)
where
H
is
the
Heaviside
step
function
and
εF,d=
3.94
eV
[38]
is
the
energy
between
the
lower
and
upper
edge
of
the
d-band.
The
interband
transition
energy
of
2.15
eV
is
taken
from
experimental
investigations
of
the
transient
thermoreflectance
measurements
[39].
The
full
cold
electron
DOS
performed
with
Eqs.
(13)
and
(14)
is
shown
in
Fig.
2.
The
electronic
contribution
of
the
electron
gas
to
the
constant
volume
specific
heat
of
a
metal
can
be
calculated
in
dependence
on
the
electron
temperature
as
following:
Ce(Te)
=+∞
−∞
(ε
εF)f
(ε,
,
Te)
Te
g(ε)dε,
(15)
Fig.
2.
s–d
band
approximation
of
the
electron
DOS
for
Cu
with
root
function
for
s-
and
d-band.
Electron
densities
in
the
s-
and
d-bands
are
ns=
1.75nat and
nd=
9.75nat
at
Te=
0
K,
respectively
[38].
Additionally
the
Fermi
function
at
various
Teis
shown.
where
g(ε)
is
the
total
electron
density
of
states
(DOS),
involving
the
s-
and
d-band,
at
energy
level
ε,
f(ε,
,
Te)
is
the
Fermi–Dirac
distribution
function
f(ε,
,
Te)
=
1/[exp[(ε
)/kBTe]
+
1]1obeying
the
Pauli
exclusion
principle,
εFis
the
Fermi
energy
at
zero
temper-
ature,
and
the
chemical
potential.
The
calculation
of
the
electron
heat
capacity
from
Eq.
(15)
requires
the
determination
of
the
total
electron
DOS
and
the
derivative
of
the
Fermi
function
with
respect
to
the
electron
temperature
f/Te.
For
metals
the
Fermi
energy
Fis
equivalent
to
the
chemical
potential
at
0
K
and
f/Teis
only
non-
zero
near
F.
If
the
electron
temperature
is
essentially
lower
than
the
Fermi
temperature,
the
electron
heat
capacity
can
be
derived
from
the
commonly
used
Sommerfeld
expansion
of
the
FEG
model
in
metals.
From
Sommerfeld
theory
the
linear
temperature
depen-
dence
of
the
electron
heat
capacity
can
be
obtained
[36].
However,
at
higher
electron
temperatures
the
Sommerfeld
theory
of
metals
is
not
valid
and
the
electron
heat
capacity
can
be
obtained
for
ther-
mal
excited
electrons
from
Eq.
(15)
by
using
a
description
of
the
electronic
properties
of
the
total
electronic
DOS
and
f/Te.
In
this
work
the
theory
of
the
electron-ion
coupling
in
depen-
dence
on
electron
temperature
is
based
on
a
general
description
of
the
electron-ion
energy
exchange
rate
for
electron
and
ion
temper-
atures
[40–42].
An
alternative
theory
of
the
electron-ion
coupling
was
developed
by
Petrov
et
al.
[37,38]
using
an
accurate
con-
struction
of
electron
DOS
function
within
the
2P
approximation.
This
approach
is
not
directly
related
to
one
proposed
by
Lin
et
al.
[42].
The
general
temperature
dependency
of
electron-ion
coupling
within
this
approach
can
be
expressed
as
G(Te)
=kBω2
g(εF)+∞
−∞
g2(ε)f
εdε,
(16)
where
is
the
dimensionless
electron–phonon
mass
enhancement
parameter
[43],
ω2
is
the
second
moment
of
the
phonon
spec-
trum
defined
by
McMillan
[44]
and
g(εF)
is
the
electron
DOS
at
the
Fermi
level
εF.
In
the
current
work,
this
value
is
assumed
to
be
ω2=29
meV2[45].
At
low
electron
temperature,
f/E
reduces
to
a
delta
function
centered
at
Fand
Eq.
(16)
induces
the
expression
of
the
constant
electron-ion
coupling
factor
[40].
At
higher
electron
temperatures
the
delta
function
f/Tein
Eq.
(15)
and
f/ε
in
Eq.
(16)
at
energy
ε
is
shifting
away
from
εF.
This
shift
becomes
non-negligible
far
away
from
εFand
induces
retroactively
a
temperature
dependence
of
the
electron
heat
capacity
Ce(Te)
and
an
electron-ion
coupling
factor
G(Te).
For
the
thermal
excitation
of
a
d-band
the
computa-
tional
evaluation
of
Ce(Te)
and
G(Te)
requires
the
knowledge
of
the
chemical
potential
in
dependence
on
electron
temperature
(Te).
The
chemical
potential
provides
the
characteristic
energy,
which
represent
the
internal
energy
change
of
the
system,
when
6
J.
Winter
et
al.
/
Applied
Surface
Science
417
(2017)
2–15
one
more
particle
is
added
under
the
condition
of
constant
entropy
and
volume
=
U/N|S,V[46,47].
At
0
K
all
states
below
the
Fermi
energy
εFare
occupied
up
to
εFand
above
εFall
states
in
the
Fermi
gas
are
empty.
Thus,
at
temperature
0
K
the
system
must
be
in
a
state
with
the
minimum
possible
energy.
The
entropy
is
related
to
the
number
of
possible
states
and
for
a
system
with
a
certain
number
of
particles
only
one
exists,
the
so-called
ground
state
with
minimum
energy,
where
the
entropy
is
equal
to
zero
accordingly
to
the
third
law
of
thermodynamics
[48].
In
a
s-band
with
a
constant
electron
number
density,
all
low-lying
states
at
0
K
are
filled
with
the
electrons
up
to
the
Fermi
energy
corresponding
to
Pauli’s
exclu-
sion
principle
of
fermions.
The
entropy
is
therefore
equal
to
zero.
At
rising
electron
temperature
the
electrons
of
the
s-band
begin
to
occupy
excited
states
around
the
Fermi
energy
and
left
behind
unallocated
states
below
εF.
This
change
of
the
micro
states
num-
ber
accordingly
leads
to
an
internal
energy
increase
and
thus
to
the
entropy
increase.
The
chemical
potential
begin
to
move
towards
lower
energies
in
order
to
keep
the
number
of
electrons.
The
chemi-
cal
potential
according
to
the
Sommerfeld
expansion
theory
for
the
FEG
model
is
proportional
to
T2
e.
A
more
accurate
description
of
the
chemical
potential
shift
can
be
derived
by
considering
the
appropriate
thermodynamic
poten-
tials.
According
to
Wentzcovitch
et
al.
an
asymmetric
electronic
occupation
density
around
Fermi
energy
is
thermodynamically
related
to
the
Mermin’s
formulation
[49].
The
Mermin
theorem
defines
an
thermodynamic
potential
by
minimizing
the
free
energy
=
U
TeS
instead
of
the
internal
energy
U.
This
minimum
value
of
free
energy
is
corresponding
to
a
stationary
stable
state
with
an
density
of
the
electron
gas
in
thermal
equilibrium
at
a
fixed
electronic
temperature
for
Te>
0.
The
minimum
of
the
Mermin
free
energy
is
a
more
exact
thermodynamic
description
for
reaching
self-consistency
in
the
presence
of
a
small
occupation
gap
after
excitation
of
localized
low-lying
electrons.
The
Mermin
functional
for
the
free
energy
includes
also
fluctuations
in
the
electron
num-
ber
by
allowing
variations
in
the
chemical
potential
in
addition
to
an
almost
constant
density
at
a
fixed
temperature
[49,50].
Thus,
the
chemical
potential
in
Fig.
3(a)
varies
with
the
electron
temper-
ature
to
keep
the
total
number
of
electrons
in
the
system
constant
by
balancing
the
total
number
of
empty
states
below
and
the
num-
ber
of
occupied
states
above
the
level
of
the
chemical
potential.
This
corresponds
to
a
minimization
of
the
Mermin
free
energy.
The
chemical
potential
(Te)
in
dependence
on
the
electron
temperature
can
be
found
directly
by
evaluating
the
implicit
Eq.
(17)
at
various
electron
temperatures
Te.
The
result
after
integra-
tion
Eq.
(17)
at
distinct
and
Temust
be
equal
to
the
constant
value
of
the
total
number
of
the
electronic
density,
ntot [35]:
ntot =
ns+
nd=+∞
−∞
f
(ε,
(Te),
Te)
·[gs(ε)
+
gd(ε)]dε.
(17)
whereby
g
is
the
d-band
and
conduction
s-band
with
correspond-
ing
constant
values
of
the
electron
number
density
n
denoted
by
subscripts
with
s
and
d.
The
prediction
of
the
thermophysical
properties
in
dependence
on
electron
temperature
calculated
from
total
cold
electronic
spec-
trum
are
demonstrated
in
Fig.
3.
For
Cu
shown
in
Fig.
3(a)
the
calculation
of
the
chemical
potential
(Te)
in
dependence
on
elec-
tron
temperature
below
electron
temperature
4
kK
using
s–d
band
approximation
follows
the
FEG
model.
As
the
electron
tem-
perature
exceeds
4
kK,
the
electrons
at
the
energy
levels
below
the
Fermi
energy
of
the
d-band
edge
are
thermally
excited
and
the
chemical
potential
is
moving
toward
higher
energies
in
order
to
minimize
the
free
energy
and
ensure
the
self-consistency
with
the
excited
d-band
electrons.
Now,
by
using
the
normalization
condition
in
Eq.
(17)
and
the
known
(Te)
the
number
of
free
electrons
in
the
s-band
for
a
given
Tecan
be
expressed
similarly
Fig.
3.
Evolution
of
(a)
chemical
potentials,
(b)
number
of
free
electrons,
electronic
heat
capacity
and
(d)
electron–ion
coupling
factor
as
a
function
of
the
electronic
temperature.
Data
obtained
from
s–d
band
approximation
is
in
black
lines
or
dashes.
Data
for
comparison
from
previously
reported
works
in
different
colors
as
indicated
in
labelling.
to
Ref.
[51]
as
Ne=
Ntot
Nd,
where
Ntot =
11
atom1is
the
sum
of
electrons
in
the
s-
and
d-band
and
Ndis
the
number
of
d-band
electrons
per
atom,
as
shown
in
Fig.
3(b).
Fig.
3(c)
shows
the
total
heat
capacity
of
electrons.
At
low
electron
temperature
the
electron
heat
capacity
dependency
of
Cu
follows
the
linear
depen-
dence,
Ce(Te)
=
·
Tewith
an
experimental
constant
of
96.8
J/m3/K2
[32].
Whereas,
at
higher
electron
temperature
above
4
kK
a
significant
number
of
d-band
electrons
are
thermally
excited
in
the
conductive
s-band
and
the
electronic
heat
capacity
begins
to
increase
significantly.
The
strength
of
the
electron-ion
coupling
of
4.22
×
1016 W/m3/K
in
Fig.
3(d)
remains
approximately
constant
at
low
electron
temperature
below
4
kK.
This
observation
is
con-
sistent
with
the
reported
value
of
the
electron-ion
coupling
factor
from
ultra-fast
pump–probe
experiments
performed
under
con-
dition,
if
the
estimated
maximum
electron
temperature
change
does
not
exceed
40
K
[52].
Fig.
3(d)
displays
a
high
increase
with
further
temperature
rise
by
thermal
excitation
of
d-band
elec-
trons.
The
electron-ion
coupling
and
electron
heat
capacity
at
up
to
20
kK
electron
temperatures
was
experimentally
investigated
by
Cho
et
al.
with
time-resolved
X-ray
absorption
spectroscopy
[53,54].
Overall,
it
was
found
in
the
experiment
that
the
electron-
ion
coupling
was
in
the
range
of
4–6
×
1017 W/m3/K
on
electron
temperatures
10–20
kK,
Fig.
3(d).
In
compliance
with
Cho
et
al.
these
values
of
the
electron-ion
phonon
coupling
are
also
well
reproduced
in
the
calculation
by
using
electronic
liquid
DOS
of
the
warm
dense
copper
in
the
liquid
phase.
The
temperature
J.
Winter
et
al.
/
Applied
Surface
Science
417
(2017)
2–15
7
dependent
heat
capacity
was
as
well
experimentally
estimated
by
using
a
linear
fit
in
the
temperature
range
of
15–23
kK
and
determined
electron
heat
capacity
of
(380
±
80)
J/kg/K
at
20
kK
corresponding
to
(3.5
±
0.7)
×
106J/m3/K.
The
results
of
the
electron-ion
coupling
using
the
exact
electron
DOS
in
the
frame
of
first-principle
calculation
shown
in
Fig.
3(d)
[38,42]
are
in
agree-
ment
with
the
electron-ion
strength
calculation
presented
in
this
work.
Finally,
the
chemical
potential,
the
number
of
free
electrons
and
the
electron
heat
capacity
for
copper
displayed
in
Fig.
3(a)–(c)
are
compared
with
previously
reported
works
from
the
first
prin-
ciple