Linear and non-linear electron dynamics in finite systems
Claude Guet
CEA, Saclay
1. Reminders on surface plasmons in metallic nanoparticles
2. Red shifts and anharmonicities. Model based on separation
of CM and intrinsic excitations
3. Semi-classical TDDFT
4. Plasmon relaxation
5. Coupled dynamics of electrons and ions in nanoparticles
induced by short laser pulses
6. Finite size effects on the optical properties of denses plasmas
Erice, July 26-30, 2010
1
Collaborators
Jérôme Daligault
Theoretical Division, Los Alamos National Laboratory, Los Alamos,NM
Leonid Gerchikov, Andrei Ipatov
St Petersburg State Polytechnical University, St Petersburg, Russia
Walter Johnson
Dept of Physics, University of Notre-Dame, Notre-Dame, IN
George Bertsch
Institute of Nuclear Theory and Dept of Physics, Uni. of Washington, Seattle,WA
Claude Guet
FU-Berlin Colloquium
Quantum finite size effects on metallic particles
1
3
RN  N rS
1 ~ 100 nm
• collision features are changed: RN  electron mean-free path l in bulk

• discrete energy level spacing:
   F / N  kBT
• surface effects :
#surface atoms / N  4N 1/3
• isomer effects:
properties depend on cluster shape
Erice, July 26-30, 2010
3
Dipole surface plasmons in metallic nanoparticles
H. Haberland et al, PRL74, 1558 (1995)
The optical properties are strongly affected by
finite size effects
Erice, July 26-30, 2010
4
Dipole surface plasmons in jellium metal clusters
Vint

 
  E.ra   N E.R
a
The optical properties are strongly affected by finite size effects associated
with the coupling of the CM motion and intrinsic excitations
Erice, July 26-30, 2010
5
Jellium approximation to metallic nanoparticles
4
1
 rs3 
3
n
1
3
Vion(r)
RJ  N rS
Ne 2
VJ r   
2 RJ
r
2
a

r2 
3  2  r  RJ
 RJ 
Ne 2

r
RJ
r  RJ
2

p
1
e
H 
     V ion ra 
2 a ,b ra  rb
a 2m
a
For all N electrons
inside
2
pa2 1
r
e2
H 
      e2 a 3
2 a ,b ra  rb
2rs
a 2m
a
Erice, July 26-30, 2010
Separation of CM and intrinsic motions
L. Gerchikov, C. Guet, and A. Ipatov, Phys. Rev. A 66, 053202 (2002)
 1
R
N



'

P   pa ; pa  pa  P

'  
 ra ; ra  ra  R
a
a

 
P2
H  H'
 U r , R
2 Nm





 
 


U r , R  U ra , R   Vion ra   Vion ra
a
a



1  n
 
R Vion ra 
n 1 a n!
R
2
 3
R2
2 Nm p
R0
Erice, July 26-30, 2010
 N 5 / 6  1
N
0th approximation. Model Separable Hamiltonian.
N interacting electrons in a confining HO potential and an
external electric dipole field


2 2
 


NR
e
 U ra, R   Vion ra   Vion ra 
a
a
2rs3
2
P2
1
R
2
H  H' 
 Ne 3
2 Nm 2
rs
The CM motion decouples exactly from the intrinsic motion
Collective state (the whole dipole strength) at
frequency equal to the HO frequency, independently of
the interaction among particles and N.
Erice, July 26-30, 2010
2
Mie
e2
4πne2
 3
mrs
3m
Finite size effects and adiabatic approximation
H  H 0  Vres

 separable Hamiltonian
P2
H0  H ' 
 U eff R
2 Nm

' 
Vres  U ra , R  U eff R
 

a
Total eigenfunction of H0 is a product of wave functions

0
n ,
   

'
'
ra   n R   ra
 
Erice, July 26-30, 2010
Effective plasmon Hamiltonian
H eff


P2

 U eff R
2 Nm
Averaging the exact potential over the electron density
U eff
 


 
1
n
n 
R  U eff     e r R.  Vion r dr
n 1
n 1 n!
Spherical symmetry => odd terms vanish
First non vanishing term
( 2)
U eff
Energy spectrum

sp2 R 2
4

2
R  Nm
; sp 
e i dr

2
3Nm
 
En,    sp n  12 
the external dipole field does not excite the intrinsic motion
At 2nd order the dipole excitation spectrum is purely harmonic
 0
En,0  E0,0  nsp
In this adiabatic approximation anharmonic terms originate from n=4,6,..
Erice, July 26-30, 2010
Effective plasmon Hamiltonian. Jellium approximation
3e
i r  
 r  RJ 
3
4 rs
First non vanishing term:
( 2)
U eff

sp2 R 2
4

2
R  Nm
; sp 
e i dr

2
3Nm
 
sp  Mie 1 
N
2
Mie
N
N
Spill-out electrons outside the ionic edge
e2
4πne2
 3
mrs
3m
Erice, July 26-30, 2010
Coupling CM and intrinsic electron motions
Vres


 





 

 1   n 
n  
n
r , R  Vres r , R    R   Vion ra   U eff R 
n 1
n 1  n!
a


 
 
'
1 
W r , R  Vres r , R   R. Vion ra






 
a
This potential is associated with an extra time-dependent EM field arising in the
CM system due to the plasmon oscillation.
At 1st order it is a separable interaction between dipole plasmon and singleparticle excitations. It couples unperturbed states n,0 and n1,
2
2
2




R

P
sp
'
  W r , R
H  H 
N
 2 Nm

2


 
Erice, July 26-30, 2010
12
Coupling CM and intrinsic electron motions
2 
En , 0 
Oscillator part
2 
 sp 

n  n ,  0
n 1 R n 
dEn 2, 0
1

dn
2 N sp
     0
n, W n,0
2
En , 0  En,
n  1 2  1 2 / 2 Nsp


2   vra  0
2
a
2
2



v0
sp


 dVion r 
vr  
dz
Creation/annihilation of one dipole plasmon generates a dipole excitation of the intrinsic motion
In small Na cationic clusters almost all dipole excitations have energies

sp
larger than
Thus the energy shift is negative as observed experimentally
Erice, July 26-30, 2010
13
3.4
 (eV)
3.2
3.0
2.8
Mie
sp
RPAE
2.6
20
40
60
80
N
•
The main contribution to the observed red shift is due to
the repulsion interaction between the dipole plasmon
and the intrinsic excitations of higher energies.
RPAE accounts properly for this process
In addition: partial transfer of strength into states of higher
energies preserving the TRK sum rule
Erice, July 26-30, 2010
14
Spill-out electrons are responsible
Jellium background potential does not contribute to the coupling in the interior

mMie 2
2
Vion r  
r  3RJ
2
2


Adding to v r  a linear term
dVion r 
2 z
e 3
dz
rs

z
e 3
rs
2
does not change the matrix elements
 z

z 
vr   N  3  3  r  RJ 
RJ 
r
Assuming all intrinsic excitations at
2 

sp 
one obtains

3
2
2rs6sp sp
  2
Erice, July 26-30, 2010

N
N
15
Spill-out parameter, plasmon frequency at 0th approximation
and RPAE frequency
Na9

Na21
N / N
0.14
0.13
 sp , eV
3.15
2.98
3.17
2.88
 RPAE , eV

Na59

Na93
0.12
0.096
0.084
3.20
2.76
3.24
2.88
3.26
2.84
Na

41
Erice, July 26-30, 2010
16
Many-body theory approach
Get the wave functions of the intrinsic excitations from RPAE
Some intrinsic levels
close to unperturbed plasmon En , 0
En1,




k
 k  C  n ( R)0 (ra ' )   C  n 1 ( R) (ra ' )
k
0
 0
n refers t o n th plasmon
0 ,  are gs and excit edst at esof int rinsicmot ion

 k
1

k




n




E
C

n



1
,

W
n



1
,

C



sp
0
k 
0
0 
v  0

2




H k  Ek k
Erice, July 26-30, 2010
17
RPAE with projectors


  
zeh z he
P e r  h r      ee hh 
2
zeh
e h  

eh



  
  e r  h r 



   X eh
Pae ah  Yeh Pah ae  0
e,h
 P 0  A

 
 0 P  B
B  P 0  X  
 X 

       
 
A  0 P  Y 
Y 
Erice, July 26-30, 2010
18
Recoupling CM and intrinsic motions

   X eh
Pae ah  Yeh Pah ae  0
e,h
 P 0  A

 
 0 P  B
B  P 0  X  
 X 

       
 
A  0 P  Y 
Y 
Aeh,eh   e   h  hh ee  eh V he
Beh,eh  ee V hh
 e h V  e h








 e r  e r  h r  h r     e r  e r  h r  h r   

dr dr   
dr dr 
 
 
r  r
r  r
Erice, July 26-30, 2010
19
Dipole excitation energies and strengths
RPAE and present model

9
Na
i eV 
fi
1
2.438
3.3
2
3
4
5
2.978
4.536
4.771
5.515
89.4
3.4
2.3
0.6
 p eV   k eV  f k
2.482
2.453
4.485
4.743
5.503
2.963 84.4
4.567 3.6
4.802 4.0
5.526 0.9
Erice, July 26-30, 2010
5.2
20
Dipole excitation energies and strengths
RPAE and present model

93
Na
i eV 
fi
 p eV   k eV  f k
1
1.020
0.04
1.038
1.036
0.08
2
3
4
5
6
7
8
9
10
1.193
1.876
1.964
2.841
3.036
3.175
3.390
3.439
3.553
0.004
0.008
0.001
40.6
11.8
20.1
7.1
0.5
1.6
1.194
1.877
1.964
1.194
1.877
1.964
2.798
3.033
3.178
3.397
3.440
3.549
0.008
0.01
0.002
44.6
8.5
15.3
6.1
0.6
2.6
2.972
3.021
3.105
3.353
3.525
Erice, July 26-30, 2010
21
Dipole excitation levels

9

93
Na
Na
5.0
4.5
3.5
4.0
3.0
2.5
3.0
 (eV)
 (eV)
3.5
2.5
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
Zero First RPAE
appr. appr.
0.5
Zero First RPAE
appr. appr.
0.0
Erice, July 26-30, 2010
22
Beyond the linear regime
•
In linear regime where only one electron-hole pair can be excited at a moment of time, the
excitation spectrum calculated within our approximation coincides with the results of
standard linear theory (RPAE).
•
We have a clear understanding of the plasmon frequency: the red shift results from the
repulsion interaction between the collective mode and intrinsic electronic excitations
•
Advantage of the method: it allows one to go beyond the linear response and to calculate the
excitation of several plasmons. We’ll see that there is an anharmonic blue shift which results
from the coupling interaction
Erice, July 26-30, 2010
23
Anharmonicity of collective excitations in metallic clusters
F. Catara, Ph. Chomaz, N. Van Giai, Phys. Rev. B 48, 18207 (1993)
Boson Expansion Method => strong anharmonic effects in contrast with the nuclear GR
F. Calvayrac, P.G. Reinhard and E. Suraud, Phys. Rev. B52 R17056 (1995)
Real time TDLDA=> small anharmonicity
K. Hagino, Phys. Rev. B60 R2197 (1999)
TD variational principle=>highly harmonic behavior of dipole plasmon
LG Gerchikov, C. Guet, and A. Ipatov, Phys. Rev. A 66, 53202 (2002)
Sizeable anharmonicity
Erice, July 26-30, 2010
24
Anharmonicity at 0th approximation
Separation of CM and intrinsic motions
U eff

 N
R  m sp2 R 2  R 4
2
  4
 3
1

 (r )( R.) Vion (r )d r
4  e
4! R
N de
4
3
2

e iond r  e

5!
10rS dr
2/3
For spherical jellium clusters
Using Bohr Sommerfeld quantization condition
of orbits in the anharmonic potential

q2
q1
pdq  n ; H ( p, q)  E
En,0  E0,0
 0  En1,0  2En,0  En1,0 
3
 nso  n
2 N 2so2
2
3
0
2
Nsp 
The anharmonic frequency shift is negative but negligibly small
In agreement with Hagino’s result
Erice, July 26-30, 2010
r  R0
Anharmonicity due to coupling
4 
n, W n,0
 E
2 
En , 0  En , 0
 
n1, n2 , n  n , 1 ,
2
n  n ,  0
 En , 0 
2
n , v

n,0 W n1 , 1 n1 , 1 W n,0 n,0 W n2 , 2 n2 , 2 W n,0
E
2 0
n,0

    3  
 
     
2 
2
2
sp
sp
2
2
2
sp
  sp


 En1 , 1 En , 0  En, 0  En , 0  En2 , 2
2 
sp 
  0
Erice, July 26-30, 2010

3


N
2rs6sp sp2  2 N
1,2,3 plasmon states in
Na

and
41

93
Na
Line strength as fraction of
pure plasmon excitation
C
k 2
0
Erice, July 26-30, 2010
27
Anharmonicity at 0th approximation
Excitation spectrum including 1,2, and 3-plasmons
Na
Energy (eV)
3.6
Mie
3.4
sp
3.0
3
1
2.8
Erice, July 26-30, 2010
93
2
3.2
2.6
+
0
28
Anharmonicity of plasmon mode
  E p 2  E p 0  2E p 1
 , eV
 0 , eV
Na9

Na21
0.055
-0.023
0.12
-0.072
0.30
Na

41
0.27
-0.0029

Na59

Na93
0.22
-0.0017
0.27
-0.0009
Anharmonicity size comparable to the plasmon width
 (eV)
0.25
~<< p
0.20
0.15
0.10
0.05
0.00
20
40
N
60
80
Consequence: Nonlinear photoabsorption
in metallic nanoparticles
Erice, July 26-30, 2010
Non-linear photoabsorption
Model of anharmonic oscillator
n=2
10
8
Photon transitions
6
n=1
4
Relaxation
2
0
n=0
0
2
4
6
8
10
Photoabsorption, %
Na+41
100
8
2
I=2*10 W/cm
9
2
I=2*10 W/cm
10
2
I=2*10 W/cm
10
2
I=4.5*10 W/cm
10
2
I=8*10 W/cm
80
60
Non-linear effects:
40
• Blue shift of resonance maximum
20
• Decrease of resonance maximum amplitude
due to the break of resonance condition
0
2.6
2.8
3.0
, eV
3.2
3.4
Erice, July 26-30, 2010
30
semi-classical TDDFT model
J. Daligault and C Guet, Phys. Rev A 64, 043203 (2001)
J. Daligault and C Guet, J. Phys. A: Math Gen. 36, 5847 (2003)
J. Daligault, PhD thesis, Grenoble Université (2001)
L. Plagne and C. Guet, Phys. Rev A 59, 4461 (1999)
L. Plagne, PhD thesis, Grenoble Université (2001)
L. Plagne, J. Daligault, K. Yabana, T. Tazawa, Y. Abe, and C. Guet, Phys. Rev A 61, 0332001 (2000)
J. Daligault, F. Chandezon, C. Guet, B. Huber and S. Tomita, Phys. Rev A 66, 0332005 (2002)
M. Gross and C. Guet, Z. Phys. D 33, 289 (1995)
Phys. Rev. A54, R2547 (1996)
Femtosecond electron dynamics in
metal clusters
• Interaction with intense laser pulses
• Interaction with HCI
• Time-resolved femltosecond techniques
– Time evolution of e-e and e-ion energy exchange
– Impact of e-ion interactions on the plasmon relaxation
Present work
• Model: Real-time dynamics of ions and electrons
in 3D Na clusters
– N ions and N electrons with N : 10 to 1000
– Time scale: several hundreds of fs
– Non-linear regime
• Approximation: limit h 0 of the TDDFT
equations
– « semi-classical » Vlasov equation for the delocalized
semi-classical TDDFT model
Ne electrons in an TD external potential
vext (t )  vconf  vlas (t )
confinement by
static ions
In TDDFT, one works with the one-body density


vext r , t   ne r , t 
  ne 

ne r , t 
ne (r , t )  r | nˆKS (t ) | r

 O   One 
From TD Kohn-Sham equations
 pˆ 2

d
i
nˆ KS  
 vˆKS (t ), nˆ KS 
dt
 2m

•
external
field




vKS ne ; r , t   vext r , t   vH ne ; r   vXC ne ; r 
semi-classical TDDFT model
0
Wigner representation
nˆ KS
0
Wigner

f KS ( r , p, t )

f ( r , p, t )
lim ne ( r , t )  n( r , t )   f ( r , p, t )dp
0
•
lim(
KS eqs)
•
0

 p2

f

 vKS ( n; r , t ), f 
t
 2m

Coupled dynamics of electrons and ions
The only external potential is vext (t)
Two sets of motion equations for electrons and ions respectively

d2
Z2
M 2 RI (t )  RI vlas ( RI , t )   ne (r , t )vie ( RI  r )dr  
dt

J  I RI  RJ
Not the Born-Oppenheimer density
Finally, our model is:

f  p 2

 vKS (n; r , t ), f 
t  2m

d2
M 2 RI (t )  FˆI
dt
R , n 
I
for electrons
for ions



Coupled dynamics of electrons and ions
Approximations:
Exchange-correlation potential from LDA
Ionic potential
Ni
Ne
H ie   vPS
I 1 i 1

 
ri  Ri

The « hard-core » potential gives a maximum degree of
transferability in the sense that it can reproduce the
important physical properties of a system irrespective
of its number of atoms or arrangement
Kümmel, Brack, Reinhard PRB 62, 7602 (2000)

vxc n  vxc nr , t 
vie ( r )
Numerical integration. Pseudo-particles
 


 
 
f r , p, t 
p





   r f r , p, t   r vKS n, RJ    p f r , p, t 
t
m
Ne
f ( r , p, t ) 
Np
Np
 g  r  r ( t )    p  p (t ) 
i 1
r
i
i
Gaussian
dri pi

dt m
dpi
  g r vKS ( ri , t )
dt
Hamilton dynamics of pseudo-particles
• initial condition:
f ( r , p, t ) 
2
(2 )
3
   F  h( r , p ) 
• phase-space volumes are conserved (Liouville theorem) over large time scales
provided the number of pseudoparticles is large (Np~106)
Plasmon relaxation : ellipsoidal jellium models
1
3
 2  
Rx  R y  
 RN
 2 
Rx
Rz
2
3
 2 
Rz  
 RN
 2  


    Mie 1  
5

cl
x
cl
y

Na 55
zcl  Mie  1 

2 

5 
Plasmon lifetime: 90 fs => 0.015 eV
Small distortions have sizeable effects
Plasmon relaxation : models with ions
Na55
pseudo-particles trajectories
Spherical jellium
Trajectories are stable, planar, scattered on edges of
the self-consistent potential
Hard-core pseudopotential
Trajectories are “chaotic”, three-dimensional,
scattered on the anharmonicites of the
self-consistent potential due to (amorphous and
nonsymmetrical structure)
 electronic dipole loses its coherence much faster
A typical laser experiment
Icosahedral Na147 , laser I=1011 W.cm-2 , las= p=3.1 eV, duration 200 fs
Laser field
E(t)
(a.u.)
Electronic dipole
(a.u.)
Residual
cluster charge
Electronic
kinetic energy
(a.u.)
Ionic
kinetic energy
(a.u.)
Ionic radial
distribution
Kinetic versus Coulombic effects
Compare simulations in which ions are either free to move or rigidly fixed
Na196 ,I=1012 W.cm-2 , w=wp , T=100 fs
Results:
the cluster charge at t=T is the same Q=46
BUT the energy transfers are very different
Electron kinetic energy
Ion kinetic energy
laser
experiment
fixed
ions
EKelec
free
ions
free coulomb
explosion of
46
Na196
 The electronic kinetic pressure plays a major role in the cluster explosion
Na196 + Xe25+ peripheral collision
electron dipole
(a.u.)
The envelopes of
electric fields
and the final cluster
charges
are similar
Q(t)
time (fs)
time (fs)
the strong electron oscillations
against the ions greatly enhance
the explosion
Ion kinetic energy
(eV)
time (fs)