linear and non-linear electron dynamics in finite systems claude guet cea, saclay 1.reminders on...
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Linear and non-linear electron dynamics in finite systemsClaude 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
FU-Berlin Colloquium17/12-04
Claude Guet
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
Quantum finite size effects on metallic particles
nmrNR SN 100 ~ 1 31
• discrete energy level spacing:
• surface effects :
• isomer effects: properties depend on cluster shape
• collision features are changed: electron mean-free path in bulkNR l
/F BN k T
1/3#surface atoms / 4N N
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
The optical properties are strongly affected by finite size effects associated with the coupling of the CM motion and intrinsic excitations
RENrEVa
a
..int
Erice, July 26-30, 2010 5
Jellium approximation to metallic nanoparticles
r RJ
Vio
n(r)
32
2
2
22
J
J
JJJ
Rrr
Ne
RrR
r
R
NerV
SJ rNR 31
a s
a
ba baa
a
r
re
rr
e
m
pH
3
22
,
22
22
1
2For all N electrons
inside
nrs
1
3
4 3
aa
ionba baa
a rVrr
e
m
pH
,
22
2
1
2
Erice, July 26-30, 2010
Separation of CM and intrinsic motions
RrUNm
PHH
,
2 '
2
aionn a
n
aaionaion
aa
rVRn
rVrVRrURrU
1 !
1
,,
L. Gerchikov, C. Guet, and A. Ipatov, Phys. Rev. A 66, 053202 (2002)
RrrrN
R aaa
a
' ;
1N
PpppP aaa
a
' ;
Erice, July 26-30, 2010
pNmR 232 16/5
0
2
NR
R
0th approximation. Model Separable Hamiltonian.
N interacting electrons in a confining HO potential and an external electric dipole field
3
22
2
2
1
2 '
sr
ReN
Nm
PHH
3
22
2,
saaionaion
aa r
eNRrVrVRrU
m
ne
mr
e
sMie 3
π4 2
3
22
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
resVHH 0
RURrUV effa
ares
,'
RUNm
PHH eff
2'
2
0
''0, anan rRr
Finite size effects and adiabatic approximation
separable Hamiltonian
Total eigenfunction of H0 is a product of wave functions
Erice, July 26-30, 2010
Averaging the exact potential over the electron density
Effective plasmon Hamiltonian
rdrVRrn
URU ionn
enn
neffeff
.!
1
11
Spherical symmetry => odd terms vanish
First non vanishing term rdNm
RNmRU iesp
speff
3
4 ;
22
22)2(
21
, nE spn
In this adiabatic approximation anharmonic terms originate from n=4,6,..
Energy spectrum
the external dipole field does not excite the intrinsic motion 0
At 2nd order the dipole excitation spectrum is purely harmonicspn nEE 0,00,
Erice, July 26-30, 2010
RUNm
PH effeff
2
2
Effective plasmon Hamiltonian. Jellium approximation
Js
i Rrr
er
34
3
First non vanishing term: rdNm
RNmRU iesp
speff
3
4 ;
22
22)2(
N
NMiesp
1
Spill-out electrons outside the ionic edgeN
m
ne
mr
e
sMie 3
π4 2
3
22
Erice, July 26-30, 2010
Coupling CM and intrinsic electron motions
RrWR
NNm
PHH sp
,
22
222'
a
aionres rVRRrVRrW '1 .,,
1 1 !
1,,
n n a
neffaion
nnresres RUrVR
nRrVRrV
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 single-particle excitations. It couples unperturbed states and 0,n ,1n
Erice, July 26-30, 2010 12
Coupling CM and intrinsic electron motions
0, ,0,
2
20,
0,,
nn nnn EE
nWnE
Oscillator part spNnnRn 2/21211
022
2
20,2
02
2
1
v sp
aa
sp
nsp
rv
Ndn
dE
0 dz
rdVrv ion
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 larger than spThus the energy shift is negative as observed experimentally
Erice, July 26-30, 2010 13
• 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
20 40 60 80
2.6
2.8
3.0
3.2
3.4
RPAE
sp
Mie
(e
V)
N
Erice, July 26-30, 2010 14
Spill-out electrons are responsible
Jellium background potential does not contribute to the coupling in the interior
32
22
2
JMie
ion Rrm
rV
Adding to a linear term does not change the matrix elements
JJ
RrR
z
r
zNrv
33
rv
3
2
s
ion
r
ze
dz
rdV
32
sr
ze
Assuming all intrinsic excitations at one obtains
N
N
r spspssp
226
2
2
3
Erice, July 26-30, 2010 15
Spill-out parameter, plasmon frequency at 0th approximation and RPAE frequency
0.14 0.13 0.12 0.096 0.084 3.15 3.17 3.20 3.24 3.26 2.98 2.88 2.76 2.88 2.84
Erice, July 26-30, 2010
93Na
eVsp ,eVRPAE ,
59Na
41Na21Na
9Na
NN /
16
Many-body theory approach
motion intrinsic of states excited and are ,
plasmon torefers
)'()()'()(
0
0100
gs
nn
rRCrRC
th
ank
ank
k
Get the wave functions of the intrinsic excitations from RPAE
Some intrinsic levels close to unperturbed plasmon,1nE 0,nE
0,1,12
1000
k
vk
ksp CnWnCEn
kkk EH Erice, July 26-30, 2010 17
RPAE with projectors
0,
heehehheeh aPaYaPaX
Y
X
Y
X
P
P
AB
BA
P
P
0
0
0
0
rrz
zzrrP he
he
eheh
ehehhheehe
2
Erice, July 26-30, 2010 18
Recoupling CM and intrinsic motions
0,
heehehheeh aPaYaPaX
Y
X
Y
X
P
P
AB
BA
P
P
0
0
0
0
hhVeeB
ehVheA
ehhe
eehhheehhe
,
,
rdrd
rr
rrrrrdrd
rr
rrrrV hheehhee
hehe
Erice, July 26-30, 2010 19
Dipole excitation energies and strengthsRPAE and present model
1 2.438 3.3 2.482 2.453 5.22 2.978 89.4 2.963 84.43 4.536 3.4 4.485 4.567 3.64 4.771 2.3 4.743 4.802 4.05 5.515 0.6 5.503 5.526 0.9
Erice, July 26-30, 2010
kkpii feVeVfeV
9Na
20
Dipole excitation energies and strengthsRPAE and present model
1 1.020 0.04 1.038 1.036 0.08
2 1.193 0.004 1.194 1.194 0.0083 1.876 0.008 1.877 1.877 0.014 1.964 0.001 1.964 1.964 0.0025 2.841 40.6 2.798 44.6
6 3.036 11.8 2.972 3.033 8.57 3.175 20.1 3.021 3.178 15.38 3.390 7.1 3.105 3.397 6.19 3.439 0.5 3.353 3.440 0.610 3.553 1.6 3.525 3.549 2.6
Erice, July 26-30, 2010
kkpii feVeVfeV
93Na
21
Dipole excitation levels
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Firstappr.
Zeroappr.
RPAE
(eV
)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Firstappr.
Zeroappr.
RPAE
(e
V)
93Na
9Na
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
• 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
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
422
2
N RRmRU speff
rdrVRr
R ione34
4)().)((
!4
1 �
222
0,00, 2
3
soson N
nnEE
010!5
43/2
23Rr
e
Sione dr
d
r
Nerd
Separation of CM and intrinsic motions
The anharmonic frequency shift is negative but negligibly small
In agreement with Hagino’s result
The anharmonic frequency shift is negative but negligibly small
In agreement with Hagino’s result
For spherical jellium clusters
Using Bohr Sommerfeld quantization condition of orbits in the anharmonic potential
EqpHnpdqq
q ),(;
2
1
03
2 20,10,0,10 sp
nnnN
EEE
Erice, July 26-30, 2010
Anharmonicity due to coupling
0,, ,0,0,0,,0,
22221111
0,2
0,,
2
20,
40,
21,2,1 2211
0,,,0,0,,,0,
0,,
nnnn nnnnnn
nn nvn
nn
EEEEEE
nWnnWnnWnnWn
EE
nWnEE
Erice, July 26-30, 2010
22
2222 3
sp
spsp
N
N
r spspssp
226
2
2
3
0 sp
1,2,3 plasmon states in 93Na
41Na and
Line strength as fraction of
pure plasmon excitation
2
0kC
Erice, July 26-30, 2010 27
Anharmonicity at 0th approximation
02.6
2.8
3.0
3.2
3.4
3.6 Na+
93
3
2
1
sp
Mie
Ene
rgy
(eV
)
Excitation spectrum including 1,2, and 3-plasmons
Erice, July 26-30, 2010 28
Anharmonicity of plasmon mode
0.055 0.12 0.27 0.22 0.27 -0.023 -0.072 -0.0029 -0.0017 -0.0009
Erice, July 26-30, 2010
93Na
eV,eV,0
59Na
41Na21Na
9Na
1202 ppp EEE
Anharmonicity size comparable to the plasmon width
20 40 60 800.00
0.05
0.10
0.15
0.20
0.25
0.30
(eV
)
N
~<< p
Consequence: Nonlinear photoabsorption in metallic nanoparticles
Non-linear photoabsorptionModel of anharmonic oscillator
Photon transitions
Relaxation
0 2 4 6 8 100
2
4
6
8
10n=2
n=1
n=0
2.6 2.8 3.0 3.2 3.40
20
40
60
80
100
Ph
oto
ab
sorp
tion
, %
, eV
I=2*108 W/cm2
I=2*109 W/cm2
I=2*1010 W/cm2
I=4.5*1010 W/cm2
I=8*1010 W/cm2
Na+41
Non-linear effects:
• Blue shift of resonance maximum
• Decrease of resonance maximum amplitude due to the break of resonance condition
Erice, July 26-30, 2010 30
semi-classical TDDFT modelJ. 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)
M. Gross and C. Guet, Z. Phys. D 33, 289 (1995)
Phys. Rev. A54, R2547 (1996)
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)
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
• Needs for theoretical description– Take the coupled electron-ion dynamics into account – Describe interaction processes on a fs time scale– Go beyond the linear response regime
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 electrons– Classical evolution of the ions
As such: NO Born-Oppenheimer approximationNO Frank-Kondon principle NO perturbative treatment
semi-classical TDDFT modelNe electrons in an TD external potential
In TDDFT, one works with the one-body density ˆ( , ) | ( ) |e KSn r t r n t r
2ˆˆ ˆ ˆ( ),
2KS KS KS
d pi n v t n
dt m
( ) ( )ext conf lasv t v v t
confinement bystatic ions
externalfield
From TD Kohn-Sham equations
•
trntrv eext ,,
ee nOOn
trne ,
rnvrnvtrvtrnv eXCeHexteKS
;;,,;
semi-classical TDDFT model
0
ˆ ( , , ) ( , , )Wigner
KS KSn f r p t f r p t
0lim ( , ) ( , ) ( , , )en r t n r t f r p t dp
•
2
0lim( ) ( ; , ),
2 KS
f pKS eqs v n r t f
t m
•
0
Wigner representation
Coupled dynamics of electrons and ions
The only external potential is vext (t)
Two sets of motion equations for electrons and ions respectively
2 2
2( ) ( , ) ( , ) ( )
II las I e ie IR
J I I J
d ZM R t v R t n r t v R r dr
dt R R
Not the Born-Oppenheimer density
Finally, our model is:2
( ; , ),2 KS
f pv n r t f
t m
2
2ˆ( ) ,I I I
dM R t F R n
dt
for electrons
for ions
Coupled dynamics of electrons and ions
( )iev r
Approximations:
Exchange-correlation potential from LDA
Ionic potential
trnvnv xcxc ,
ii
N
I
N
iPSie RrvH
i e �
1 1
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)
Numerical integration. Pseudo-particles
1
( , , ) ( ) ( )pN
er i i
ip
Nf r p t g r r t p p t
N
Gaussian
( , )
i i
ir KS i
dr p
dt mdp
g v r tdt
Hamilton dynamics of pseudo-particles
• initial condition: 3
2( , , ) ( , )
(2 ) Ff r p t h r p
• phase-space volumes are conserved (Liouville theorem) over large time scales provided the number of pseudoparticles is large (Np~106)
tprfRnvtprfm
p
t
tprfpJKSrr ,,,,,
,,
Plasmon relaxation : ellipsoidal jellium models
15
cl clx y Mie
21
5clz Mie
Rx
Rz
1
3
2
3
2
2
2
2
x y N
z N
R R R
R R
55Na
Plasmon lifetime: 90 fs => 0.015 eV
Small distortions have sizeable effects
Plasmon relaxation : models with ions
Na55
pseudo-particles trajectories
Spherical jellium
Hard-core pseudopotential
Trajectories are stable, planar, scattered on edges of the self-consistent potential
Trajectories are “chaotic”, three-dimensional,scattered on the anharmonicites of theself-consistent potential due to (amorphous and nonsymmetrical structure)
electronic dipole loses its coherence much faster
A typical laser experimentIcosahedral 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.)
Residualcluster charge
Electronickinetic energy
(a.u.)
Ionickinetic energy
(a.u.)
Ionic radialdistribution
Kinetic versus Coulombic effects
Electron kinetic energy Ion kinetic energy
freeions
fixedions
laserexperiment
free coulombexplosion of
46196Na
elecKE
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
The electronic kinetic pressure plays a major role in the cluster explosion
Na196 + Xe25+ peripheral collision
electron dipole(a.u.)
time (fs) time (fs)
Q(t)
The envelopes of electric fields
and the final cluster charges
are similar
time (fs)
Ion kinetic energy(eV)
the strong electron oscillationsagainst the ions greatly enhancethe explosion