1. 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

2. FU-Berlin Colloquium 17/12-04 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

3. Quantum finite size effects on metallic particles discrete energy level spacing: surface effects : isomer effects:properties depend on cluster shape collision features are changed:  Erice, July 26-30, 2010 3

4. 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

5. 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 Erice, July 26-30, 2010 5

6. Jellium approximation to metallic nanoparticles For all N electrons inside Erice, July 26-30, 2010

7. Separation of CM and intrinsic motions L. Gerchikov, C. Guet, and A. Ipatov, Phys. Rev. A 66, 053202 (2002) Erice, July 26-30, 2010

8. 0 th approximation. Model Separable Hamiltonian. N interacting electrons in a confining HO potential and an external electric dipole field 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

9. Finite size effects and adiabatic approximation separable Hamiltonian Total eigenfunction of H 0 is a product of wave functions Erice, July 26-30, 2010

10. Averaging the exact potential over the electron density Effective plasmon Hamiltonian Spherical symmetry => odd terms vanish First non vanishing term In this adiabatic approximation anharmonic terms originate from n=4,6,.. Energy spectrum the external dipole field does not excite the intrinsic motion At 2 nd order the dipole excitation spectrum is purely harmonic Erice, July 26-30, 2010

11. Effective plasmon Hamiltonian. Jellium approximation First non vanishing term: Spill-out electrons outside the ionic edge Erice, July 26-30, 2010

12. Coupling CM and intrinsic electron motions 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 Erice, July 26-30, 2010 12

13. Coupling CM and intrinsic electron motions Oscillator part 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 Thus the energy shift is negative as observed experimentally Erice, July 26-30, 2010 13

14. 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

15. Spill-out electrons are responsible Jellium background potential does not contribute to the coupling in the interior Adding to a linear term does not change the matrix elements Assuming all intrinsic excitations atone obtains Erice, July 26-30, 2010 15

16. Spill-out parameter, plasmon frequency at 0 th 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 16

17. Many-body theory approach Get the wave functions of the intrinsic excitations from RPAE Some intrinsic levels close to unperturbed plasmon Erice, July 26-30, 2010 17

18. RPAE with projectors Erice, July 26-30, 2010 18

19. Recoupling CM and intrinsic motions Erice, July 26-30, 2010 19

20. Dipole excitation energies and strengths RPAE and present model 12.438 3.3 2.482 2.453 5.2 2 2.978 89.4 2.963 84.4 3 4.536 3.4 4.485 4.567 3.6 4 4.771 2.3 4.743 4.802 4.0 5 5.515 0.6 5.503 5.526 0.9 Erice, July 26-30, 2010 20

21. Dipole excitation energies and strengths RPAE and present model 11.020 0.04 1.038 1.036 0.08 2 1.193 0.004 1.194 1.194 0.008 3 1.876 0.008 1.877 1.877 0.01 4 1.964 0.001 1.964 1.964 0.002 5 2.841 40.6 2.798 44.6 63.036 11.8 2.972 3.033 8.5 7 3.175 20.1 3.021 3.178 15.3 8 3.390 7.1 3.105 3.397 6.1 9 3.439 0.5 3.353 3.440 0.6 10 3.553 1.6 3.525 3.549 2.6 Erice, July 26-30, 2010 21

22. Dipole excitation levels Erice, July 26-30, 2010 22

23. 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

24. 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

25. Anharmonicity at 0th approximation 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 Erice, July 26-30, 2010

26. Anharmonicity due to coupling Erice, July 26-30, 2010

27. 1,2,3 plasmon states in and Line strength as fraction of pure plasmon excitation Erice, July 26-30, 2010 27

28. Anharmonicity at 0th approximation Excitation spectrum including 1,2, and 3-plasmons Erice, July 26-30, 2010 28

29. 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 Anharmonicity size comparable to the plasmon width  ~  <<  p Consequence: Nonlinear photoabsorption in metallic nanoparticles

30. Non-linear photoabsorption Model of anharmonic oscillator Photon transitions Relaxation 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

31. 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) 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)

32. 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

33. 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 approximation NO Frank-Kondon principle NO perturbative treatment

34. semi-classical TDDFT model N e electrons in an TD external potential In TDDFT, one works with the one-body density confinement by static ions external field From TD Kohn-Sham equations

35. semi-classical TDDFT model Wigner representation

36. Coupled dynamics of electrons and ions The only external potential is v ext (t) Two sets of motion equations for electrons and ions respectively Not the Born-Oppenheimer density Finally, our model is: for electrons for ions

37. Coupled dynamics of electrons and ions Approximations: Exchange-correlation potential from LDA Ionic potential 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)

38. Numerical integration. Pseudo-particles Gaussian Hamilton dynamics of pseudo-particles initial condition: phase-space volumes are conserved (Liouville theorem) over large time scales provided the number of pseudoparticles is large (N p ~10 6 )

39. Plasmon relaxation : ellipsoidal jellium models RxRx RzRz Plasmon lifetime: 90 fs => 0.015 eV Small distortions have sizeable effects

40. Plasmon relaxation : models with ions Na 55

41. 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 the self-consistent potential due to (amorphous and nonsymmetrical structure)  electronic dipole loses its coherence much faster

42. A typical laser experiment Icosahedral Na 147, laser I=10 11 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

43. Kinetic versus Coulombic effects Electron kinetic energyIon kinetic energy free ions fixed ions laser experiment free coulomb explosion of Compare simulations in which ions are either free to move or rigidly fixed Na 196,I=10 12 W.cm -2, w=w p, 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

44. Na 196 + Xe 25+ peripheral collision electron dipole (a.u.) 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 oscillations against the ions greatly enhance the explosion