New Physics on the Femtosecond Time Scale

Bernd Hüttner CphysFInstP DLR Stuttgart

Folie 1 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 2 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 3 Valencia Bernd Hüttner 3.-5.9.2008

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

1. Local thermal equilibrium vs. Nonequilibrium, T el  T ph vs. T el >> T ph 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/cm 2 , fs: F = 1-10mJ/cm 2

→

I 0 (fs)  10 3 ·I 0 (ns) Folie 4 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. New thermal and optical properties 4. Hyperbolic heat conduction equation (HHCE) 5. Summary Folie 5 Valencia Bernd Hüttner 3.-5.9.2008

2. Nonequilibrium of electron system Experimental result:  L =180fs, F abs =(300±90)  J/cm 2 , E L =1.84eV, d=30nm≈2·d opt Au

FD

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

Folie 6 Valencia Bernd Hüttner 3.-5.9.2008

Theoretical approach

Boltzmann equation



 

 v  t 

 

 e v E  r 

 

 E 



' 

  with the photon operator for Gaussian laser pulse         t 2  I e o n  2 L      s f o  E f o     E small parameter p  , T ph I .

 D T ph Ne .

f o  s development f  n m   0 n p f n  f o  p f 1  2 p f 2  ...

Folie 7 Valencia Bernd Hüttner 3.-5.9.2008

The first order reads and the 2 nd order p  f 1  t      f o  T    e v E  f o  E   pf 1    p 2  f  t 2   v    1 T    e v E  1  E    2   For the one photon distribution function we find  f o   o  f o  E f o    E f o  Folie 8 Valencia Bernd Hüttner 3.-5.9.2008

Theoretical electron energy distribution function vs energy with 300 µJ/cm 2 absorbed laser fluence at five time delays. The dashed line is the Fermi-Dirac function and the corresponding electron temperature T e is shown.

Folie 9 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 10 Valencia Bernd Hüttner 3.-5.9.2008

2. Enhanced importance of electron-electron scattering time

Fermi liquid theory: 

 1 eT  1 eE  1

 

4 2 e 2    E o 2 

 

20

Au

15  total  ph (300K)= 30fs  (fs) 10 5  e-e 0 0 2000 4000 T e (K) 6000 8000 1  10 4 Folie 11 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 12 Valencia Bernd Hüttner 3.-5.9.2008

     E       3. New thermal and optical properties  1 3.1 Thermal conductivity  1   k  E      2 T e v 2   , , ph       f o E   where the scattering time is given as   , , ph   1    , ph  ph    E   ( T e )  2  The integration yields ph    ph  eT     4    1 LTE ph   T e  e ph     G(T ) e  1 G(T ) e     2 2 k T B 24 E 2 F e 2   4 7 k T B e 4 480 E 4 F     e  ph  1   2 k T e 2 12  2 o  7  4 k T e 4 360  4 o .

Folie 13 Valencia Bernd Hüttner 3.-5.9.2008

6 5 4 3 2 1 0 0 10 9 8 7 15 14 13 12 11 λ e = λ 0 ·T e /T 0 Wiedemann-Franz λ 2 λ 1 + λ 2 1000 2000 3000 4000 5000 T (K) 6000 7000 8000 9000 1 10 4 Thermal conductivity of Au for the case of nonlocal thermal equilibrium at fixed Tph=300K: Solid upper curve  1 +  2 , dashed  ~T e , dashed-dotted curve  2 , and for the local thermal equilibrium T e =T ph =T: solid curve  1 , dotted curve  LTE , à experimental data taken from Weast Folie 14 Valencia Bernd Hüttner 3.-5.9.2008

But there is more

Time dependence of thermal conductivity

  

0

  1  3  2t  2  e

  t

t   2t e

 2t 

  2t    

 t 



2

     

0   t/  << 1:

  

0

  1 t 2   6

2

  

t/  >> 1:

  

0

ballistic behavior diffusive behavior

Folie 15 Valencia Bernd Hüttner 3.-5.9.2008

1,0 0,8 0,6 0,4 0,2 0,0 0,0 110 fs Al 0,2 570 fs Ag c e h ex 0,4 0,6 time (ps) 0,8 0.95

1,0 Solid: Al Dasded-dotted: Ag  =  -1 Vertical lines: Electron temperature relaxation time  T

Summary:   

0

T

ph

T

e

 

T e 2 ph Folie 16 Valencia Bernd Hüttner 3.-5.9.2008

Molecular dynamics and fluctuation-dissipation theorem Volz – Physical Review Letters

87

(2001) 74301 Folie 17 Valencia Bernd Hüttner 3.-5.9.2008



e

c

e 3.2 Thermal diffusivity with the specific heat of NFE c e   2 nk 2 E F 2 k B T e   e T e   e

T

ph    0 T e 2  e

f (T , T )

e ph Few examples: Folie 18 Valencia Bernd Hüttner 3.-5.9.2008

Electronic thermal diffusivity 120 100 80 60 40 20 Au F=300µJ/cm 2  L  L =180fs =1.84eV

d=375nm  e (T e ,T ph ,t)=const*f(t)/(T ph ·[1+b·T e 2 ])  e (T ph )=const*f(t)/T ph 0 0,0 0,2 0,4 0,6 0,8 t (ps) 1,0 1,2 1,4 1,6 1,8 Folie 19 Valencia Bernd Hüttner 3.-5.9.2008

Electronic thermal diffusivity 120 110 100 90 80 70 60 50 40 30 20 10 0 0,0 Au film 0,1 0,2  e (T e ,T ph ,t)=const*f(t)/(T ph ·[1+b·T e 2 ])  e (T ph )=const/T ph F=3mJ/cm 2  L =180fs  L =1.84eV

d=375nm 0,3 0,4 t (ps) 0,5 0,6 0,7 0,8 0,9 Folie 20 Valencia Bernd Hüttner 3.-5.9.2008

6000 Au film 5000 4000 F=3mJ/cm 2  L =180fs  L =1.84eV

d=375nm 3000 Electron temperature full 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 z=0.0nm

z=1.5nm

z=3.0nm

z=4.5nm

z=6.0nm

2000 1000 0,0 0,1 0,2 0,3 0,4 t (ps) 0,5 0,6 0,7 0,8 0,9 Folie 21 Valencia Bernd Hüttner 3.-5.9.2008

Einstein relation: What is with ballistic behavior?

x 2   e t ~    t t 2 if t /   1 ballistic 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 Folie 22 Valencia Bernd Hüttner 3.-5.9.2008

Dielectric function Optical properties  , T , T e ph     0 i 4    , T , T e ph  We find the electrical current by multiplying the BE with –e·v    e  v



 

 t d k

 e  v 2



 

 r d k 

e 2

d k ' d k





v 2

E

e







 

 E

v 

d k 



d k

 e   j e  t   j e  n e 2  e m  E  e 2 T  e  v 2  T

 

e



B ph



tr

 

D Folie 23 Valencia Bernd Hüttner 3.-5.9.2008

The integration reads for the first order contribution ph    D D  d E E   E, T , T e ph      ph       f o E    , T , T e ph             i D  z 12   2 2 k T B e 2 24  o 2    1   z      2 2 k T B e 2 24  2 o z  z  6 2 D     2  D      with the abbreviations   , T T ph    1   e ph 1   2    D z   ph  T Folie 24 Valencia Bernd Hüttner 3.-5.9.2008

Relations between the optical functions Complex refractive index n   ,T T ph   Re  ph   , k   ph   Im  Optical penetration depth and absorption  ,T ,T e ph    c ph  ,T ,T e ph  ph    ,T ,T e ph ph    1 2  1 An example: hat-top profile with  =1eV,  L =500fs, I abs =10GW/cm 2 , I abs =20GW/cm 2 Folie 25 Valencia Bernd Hüttner 3.-5.9.2008

Surface temperature distributions of gold 10000 8000  L  L = 500 fs = 1 eV Surface temperature of the electrons 6000 T e I abs = 20 GW/cm 2 T e I abs = 10 GW/cm 2 4000 2000 0,0 Laser pulse profile 0,2 0,4 t (ps) 0,6 0,8 1,0 300 0,0 Surface temperature of the phonons 450 400 T ph I abs = 20 GW/cm 2 T ph I abs = 10 GW/cm 2  L  L = 500 fs = 1 eV 350 0,2 0,4 t (ps) 0,6 0,8 1,0 Folie 26 Valencia Bernd Hüttner 3.-5.9.2008

24,0 23,5 23,0 22,5 22,0 21,5 0,0 0,2 Optical penetration depth Absorption depth theory I=20 GW/cm 2 theory I=10 GW/cm 2 Drude I=20 GW/cm 2  L = 500 fs 0,4 t (ps) 0,6 0,8 1,0 Folie 27 Valencia Bernd Hüttner 3.-5.9.2008

Absorption Absorption 10 4 2 8 6 0 0,0 0,2 theory I=20 GW/cm 2 theory I=10 GW/cm 2 Drude I=20 GW/cm 2  L = 500 fs 0,4 t (ps) 0,6 0,8 1,0 Folie 28 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 29 Valencia Bernd Hüttner 3.-5.9.2008

4. Hyperbolic heat conduction equation (HHCE) Multiply BE by the product of the energy difference (E  ) times the velocity      

 

 t dk  v  E     v 2  E    

 

 r dk  f k ', t  e  v 2  E    E dk 'dk    v  E     

 

 E  d k d k .

 Solving the integrals leads to Cattaneo’s equation  Q   j Q  t   j Q T with 3  k B  e T e e  e     T h ex Folie 30 Valencia Bernd Hüttner 3.-5.9.2008

  q  t e  q e q   e  u e  t   e    e  T e  h ex  T e  T ph  Cattaneo equation Energy conservation Extended two temperature model  1  2 u e   t 2    t c ph  T ph  t   u e  t   h ex   1  e    x   t  T e  x   T e   e  T ph   h ex  T e  T ph  with  2  x T e 2    c e h ex Folie 31 Valencia Bernd Hüttner 3.-5.9.2008

Electron temperature distribution 1400 1200 1000 800 600 400 F abs =1.77µJ/cm 2  L =180fs d=30nm 0·L opt 0.1·L opt 0.2·L opt 0.3·L opt 0.4·L opt 0.5·L opt L opt =13.5nm

200 0,2 0,4 0,6 0,8 time (ps) Electron temperature as a function of time for a Au-film with thickness of d=30nm ..\..\..\Mathematics\FlexPDE5\Files\Archiv\Different laser profiles.pg5

Folie 32 Valencia Bernd Hüttner 3.-5.9.2008

Overview

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

2. Nonequilibrium of electron system 3. Enhanced importance of electron-electron scattering time 4. New thermal and optical properties 5. Hyperbolic heat conduction equation (HHCE) 6. Summary Folie 33 Valencia Bernd Hüttner 3.-5.9.2008

5. Summary

The essential new points on the femtosecond time scale

• Nonequilibrium distribution of electrons – deviations from FD distribution • Nonequilibrium between electrons and phonons – T e >> T ph • 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 x 2 t 2 Folie 34 Valencia Bernd Hüttner 3.-5.9.2008

Folie 35 Valencia Bernd Hüttner 3.-5.9.2008