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