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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

e−2intensity).

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

=

45◦with

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

180◦periodicity:

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)

−

B˛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

e−2intensity

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

m−3from

which

the

atomic

number

density

nat is

calculated

as

8.5

×

1028 m−3.

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

atom−1is

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