Valencia Bernd Hüttner 3.-5.9.2008 Folie 1 New Physics on the Femtosecond Time Scale Bernd Hüttner...

Folie 1

Valencia Bernd Hüttner 3.-5.9.2008

New Physics on the Femtosecond Time Scale

Bernd Hüttner CphysFInstPDLR Stuttgart


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Overview

1. What are the distinctions between ns and fs laser pulse interaction?

2. Nonequilibrium of electron system

4. New thermal and optical properties

5. Hyperbolic heat conduction equation (HHCE)

6. Summary

3. Enhanced importance of electron-electron scattering time


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Overview





6. Summary



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1. Local thermal equilibrium vs. Nonequilibrium, Tel Tph vs. Tel >> Tph


2. Electron-electron scattering time smaller than electron-phonon one

3. Changing of optical and thermal properties, e.g. time dependent

4. Relaxation time is in the order or above the laser pulse duration, PHCE HHCE or diffusive ballistic behavior

5. Intensity, ns: F = 1-10J/cm2, fs: F = 1-10mJ/cm2 → I0(fs) 103·I0(ns)


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Overview





5. Summary


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Experimental result: L=180fs, Fabs=(300±90)J/cm2, EL=1.84eV, d=30nm≈2·dopt

Figure 1: Experimental electron energy distribution function taken from Fann et al.

FD

Au


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Theoretical approach

Boltzmann equation

f k, t f k, t f k, tv e v E P(k,k ', t) f k ', t f k, t d k G f k

rt'

t,

E

2

2L

st

so

o o o os 1

I eG f f E f E H E f E f E

n

with the photon operator for Gaussian laser pulse

small parameter p , T ph

.I D T ph..Ne

development f p f f p f p fnn o

n

m

12

2

0

...


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pf

tv T T

f

Te v E

f

E

pfG fo o

o

1 1 The first order reads

and the 2nd order pf

tv T T

p f

Te v E

p f

E

p fG p f2 2 1 1

22

1

o o o o o of E f E f E f E H E f E f E

For the one photon distribution function we find


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Theoretical electron energy distribution function vs energy with 300 µJ/cm2 absorbed laser fluence at five time delays. The dashed line is the Fermi-Dirac function and the corresponding electron temperature Te is shown.


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Overview





6. Summary



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21 1 1 2 2e e eT eE e o4 T eV E

Fermi liquid theory:

0 2000 4000 6000 8000 1 104

0

5

10

15

20

total

e-e

Te (K)

(fs)

ph (300K)= 30fs

Au


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Overview





6. Summary



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3.1 Thermal conductivity


2 2 2 4 4

e B e B e1 LTE e 2 4

e F Fph e ph

T k T 7 k T1G(T )

G(T ) 24 E 480ET 1 z T ,T

E T T

T

z T T E Te ph

ph ph

e ph e

, ,, ( )

1

2

ph ph 2 2e ph e ph ph

eT e

Tz T ,T 4 T (eV) T

T

where the scattering time is given as

The integration yields

E T v f T 1

G Tk T k T

eB e

o

B e

o

112

7

360

2 2 2

2

4 4 4

4

.

1

22

E T

Tv E T T

f

Ee

ee ph

o

k

, ,


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0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1 1040

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

T (K)

Therm

al

co

nducti

vit

y (

W/c

mK

)

Thermal conductivity of Au for the case of nonlocal thermal equilibrium at fixed Tph=300K: Solid upper curve 1+2,

dashed ~Te, dashed-dotted curve 2, and for the local thermal equilibrium Te=Tph=T: solid curve 1, dotted

curve LTE, à experimental data taken from Weast

λ1+ λ2

λ2

λe= λ0·Te/T0

Wiedemann-Franz


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Time dependence of thermal conductivity

2t 2t t

0 0

3 21 e e 1 e f t,

2t t 2t 2t

t/ << 1:2

0

1 t 1 t... .

2 6

ballistic behavior

t/ >> 1: 0 diffusive behavior

But there is more


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Summary: e2e

0 e ph

ph

Tf (T ,T )

T 1 const·t

T

0,0 0,2 0,4 0,6 0,8 1,0

0,0

0,2

0,4

0,6

0,8

1,00.95

570 fs110 fs

f(t,

)

time (ps)

Solid: AlDasded-dotted: Ag= -1

Vertical lines:Electron temperaturerelaxation time T

eT

ex

c

h

AgAl


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Volz – Physical Review Letters 87 (2001) 74301

Molecular dynamics and fluctuation-dissipation theorem


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3.2 Thermal diffusivity

ee

ec

B

22k

e e e eF

nkc T T

2 E

with the specific heat of NFE

2e

0e e ph

ph e

t

Tf (T ,T )

T 1 const·

Few examples:


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0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,80

20

40

60

80

100

120

Au

F=300µJ/cm2

L=180fs

L=1.84eV

d=375nm

Electronic thermal diffusivity

e(T

e,T

ph,t)=const*f(t)/(T

ph·[1+b·T

e

2])

e(T

ph)=const*f(t)/T

ph

e (c

m2 /s

)

t (ps)


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0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,90

10

20

30

40

50

60

70

80

90

100

110

120

Au film

F=3mJ/cm2

L=180fs

L=1.84eV

d=375nm

e(T

e,T

ph,t)=const*f(t)/(T

ph·[1+b·T

e

2])

e(T

ph)=const/T

ph

Electronic thermal diffusivity e (

cm2 /s

)

t (ps)


Folie 21

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

1000

2000

3000

4000

5000

6000full lines:

e(T

e,T

ph,t)=const*f(t)/(T

ph·[1+b·T

e

2])

dashed lines: e(T

ph)=const/T

ph

Au film

F=3mJ/cm2

L=180fs

L=1.84eV

d=375nm

Electron temperature

z=0.0nm z=1.5nm z=3.0nm z=4.5nm z=6.0nm

Te (

K)

t (ps)


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What is with ballistic behavior?

Einstein relation:2

2e

t if t / 1 ballisticx t ~

t if t / 1 diffusive

Sample thickness vs time of flight for various Au films 50, 100, 150, 200, and 300nm thick.

Brorson et al. –

Phys. Rev. Lett. 59 (1987) 1962


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Optical properties

e ph 0 e ph

4,T ,T i ,T ,T

We find the electrical current by multiplying the BE with –e·v

Dielectric function

2 2 2f k, t f k, t f k, t

d k d k E d kt r E

P k,k ', t f k ', t f k, t d k '

e v e

d k G f k

v e v

e v e v , t d k

ee

ee e

j

tj

n e

mE

e

Tv T

2 2

2e D

B ph tr2 k T


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2 2 2 2 2B e D

2o

1 e ph D

2 2 2 2 2B e D

D 2o

k T zz1 z 1 z

12 24 6

,T ,T

k T zzi 1

6 1 z 24 6 1 z

, ,,

T Tz T T

e ph

e ph D

1

12 2 2

with the abbreviationsph

T

z

The integration reads for the first order contribution

e ph oDe ph

o D e ph

E,T ,T f,T ,T , t d E E

E1 i E,T ,T


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Relations between the optical functions

e, ph e, ph e, ph e, phn ,T T Re ,T T , k ,T T Im ,T T

2

e ph

e ph e ph

e ph e ph

,T ,T 1c,T ,T , A ,T ,T 1

k ,T ,T ,T ,T 1

An example: hat-top profile with =1eV, L=500fs, Iabs=10GW/cm2, Iabs=20GW/cm2

Complex refractive index

Optical penetration depth and absorption


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0,0 0,2 0,4 0,6 0,8 1,0

2000

4000

6000

8000

10000

Laser pulse profile

L = 500 fs

L = 1 eV

Surface temperature of the electrons

Te I

abs = 20 GW/cm2

Te I

abs = 10 GW/cm2

Te(

K)

t (ps)

0,0 0,2 0,4 0,6 0,8 1,0300

350

400

450

Surface temperature of the phonons

Tph

Iabs

= 20 GW/cm2

Tph

Iabs

= 10 GW/cm2

L = 500 fs

L = 1 eV

Tph

(K)

t (ps)

Surface temperature distributions of gold


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0,0 0,2 0,4 0,6 0,8 1,021,5

22,0

22,5

23,0

23,5

24,0

theory I=20 GW/cm2

theory I=10 GW/cm2

Drude I=20 GW/cm2

L = 500 fs

Absorption depth (

nm

)

t (ps)

Optical penetration depth


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0,0 0,2 0,4 0,6 0,8 1,0

0

2

4

6

8

10

Absorption

theory I=20 GW/cm2

theory I=10 GW/cm2

Drude I=20 GW/cm2

L = 500 fs

A (

%)

t (ps)

Absorption


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Overview





6. Summary



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Multiply BE by the product of the energy difference (E - ) times the velocity

2 2f k, t f k, t f k, t

dk dk e E d kt r E

P k,k ', t f k ', t f k, t dk 'dk G f k,

v E v E v E

v kE .v dE t

Q

QQ

j

tj T

Solving the integrals leads to Cattaneo’s equation

e eB e eQ T2

exe

c Tk T

h3

with


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ee e e e

e ee ex e ph

qq T T

tu T

q S x, t h T Tt

Cattaneo equation

Energy conservation

e2

e e e e ee 2

ex e ph

phph ex e

2e e

2 e

ph

u T T T

t x x x

1 S x, t h 1 T T

Tc

T

h T T

T

t

u T

t

t t

Extended two temperature model

e e

ex

c T

hwith


Folie 32

Electron temperature as a function of time for a Au-film with thickness of d=30nm

0,2 0,4 0,6 0,8200

400

600

800

1000

1200

1400

Fabs

=1.77µJ/cm2

L=180fs

d=30nm

Electron temperature distribution

0·Lopt

0.1·Lopt

0.2·Lopt

0.3·Lopt

0.4·Lopt

0.5·Lopt

Lopt

=13.5nm

T (

K)

time (ps)

..\..\..\Mathematics\FlexPDE5\Files\Archiv\Different laser profiles.pg5


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Overview





6. Summary



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5. Summary

• Nonequilibrium distribution of electrons – deviations from FD distribution• Nonequilibrium between electrons and phonons – Te >> Tph

• Changed dependence of temperature of the thermal and electrical conductivity due to electron-electron scattering time • Both conductivities become implicit and explicit time dependent• Change of optical properties (partly drastic)• Extended two temperature model (HHCE) must be used for the determination of the electron temperature leading to temperature waves• Ballistic electron transport -

The essential new points on the femtosecond time scale

2 2x t


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Valencia Bernd Hüttner 3.-5.9.2008 Folie 1 New Physics on the Femtosecond Time Scale Bernd Hüttner...

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Transcript of Valencia Bernd Hüttner 3.-5.9.2008 Folie 1 New Physics on the Femtosecond Time Scale Bernd Hüttner...