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Linear and non-linear electron dynamics in finite systems

Linear and non-linear electron dynamics in finite systems. Claude Guet CEA, Saclay. Reminders on surface plasmons in metallic nanoparticles Red shifts and anharmonicities. Model based on separation of CM and intrinsic excitations Semi-classical TDDFT Plasmon relaxation

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Linear and non-linear electron dynamics in finite systems

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  1. Linear and non-linear electron dynamics in finite systems Claude Guet CEA, Saclay • Reminders on surface plasmons in metallic nanoparticles • Red shifts and anharmonicities. Model based on separation of CM and intrinsic excitations • Semi-classical TDDFT • Plasmon relaxation • Coupled dynamics of electrons and ions in nanoparticles induced by short laser pulses • Finite size effects on the optical properties of denses plasmas Erice, July 26-30, 2010

  2. 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

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

  4. Dipole surface plasmons in metallic nanoparticles collective oscillation of electrons The optical properties are strongly affected by finite size effects H. Haberland et al, PRL74, 1558 (1995) Erice, July 26-30, 2010

  5. Dipole surface plasmons in jellium metal clusters collective oscillation of electrons 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

  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. 0th approximation. Model Separable Hamiltonian. N interacting electrons in a confining HO potential and an external electric dipole field The CM motion decouplesexactlyfrom the intrinsic motion Collective state (the wholedipolestrength) atfrequencyequal to the HO frequency, independently of the interaction amongparticles and N. Erice, July 26-30, 2010

  9. Finite size effects and adiabatic approximation separable Hamiltonian Total eigenfunction of H0 is a product of wave functions Erice, July 26-30, 2010

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

  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

  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

  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 at one obtains Erice, July 26-30, 2010

  16. Spill-out parameter, plasmon frequency at 0th approximation and RPAE frequency 0.14 0.13 0.12 0.096 0.084 3.153.173.20 3.24 3.26 2.98 2.88 2.76 2.88 2.84 Erice, July 26-30, 2010

  17. Many-body theory approach Get the wavefunctions of the intrinsic excitations from RPAE Someintrinsiclevels close to unperturbedplasmon Erice, July 26-30, 2010

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

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

  20. Dipole excitation energies and strengthsRPAE and present model 1 2.4383.32.4822.453 5.2 2 2.97889.42.963 84.4 3 4.536 3.44.4854.567 3.6 4 4.7712.34.7434.802 4.0 5 5.5150.65.5035.526 0.9 Erice, July 26-30, 2010

  21. Dipole excitation energies and strengthsRPAE and present model 1 1.0200.041.0381.036 0.08 2 1.1930.0041.1941.194 0.008 3 1.876 0.008 1.8771.877 0.01 4 1.9640.0011.9641.964 0.002 5 2.84140.6 2.798 44.6 6 3.03611.82.9723.033 8.5 7 3.17520.13.0213.178 15.3 8 3.390 7.1 3.1053.397 6.1 9 3.439 0.53.3533.440 0.6 10 3.5531.6 3.5253.549 2.6 Erice, July 26-30, 2010

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

  23. Beyond the linear regime In linearregimewhereonly one electron-hole pair canbeexcitedat a moment of time, the excitation spectrumcalculatedwithinour approximation coincideswith the results of standard lineartheory (RPAE). We have a clearunderstanding of the plasmonfrequency: the red shift resultsfrom the repulsion interaction between the collective mode and intrinsicelectronic excitations Advantage of the method: itallows one to go beyond the linearresponse and to calculate the excitation of severalplasmons. We’llseethatthereis an anharmonicblueshift whichresultsfrom the couplinginteraction Erice, July 26-30, 2010

  24. Anharmonicity of collective excitations in metallic clusters F. Catara, Ph. Chomaz, N. Van Giai, Phys. Rev. B 48, 18207 (1993) Boson Expansion Method=> stronganharmoniceffects in contrastwith the nuclearGR F. Calvayrac, P.G. Reinhard and E. Suraud, Phys. Rev. B52 R17056 (1995) Real time TDLDA=> smallanharmonicity K. Hagino, Phys. Rev. B60 R2197 (1999) TD variationalprinciple=>highlyharmonicbehavior of dipoleplasmon LG Gerchikov, C. Guet, and A. Ipatov, Phys. Rev. A 66, 53202 (2002) Sizeableanharmonicity Erice, July 26-30, 2010

  25. Anharmonicity at 0th approximation Separation of CM and intrinsic motions For spherical jellium clusters Using Bohr Sommerfeld quantization condition of orbits in the anharmonic potential The anharmonicfrequency shift isnegative but negligiblysmall In agreement withHagino’sresult 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

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

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

  30. Photon transitions Non-linear photoabsorption Relaxation Na+41 Model of anharmonic oscillator • 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

  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) 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)

  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 confinement by static ions external field Ne electrons in an TD external potential In TDDFT, one works with the one-body density From TD Kohn-Sham equations

  35. semi-classical TDDFT model Wigner representation

  36. Coupled dynamics of electrons and ions The only external potential is vext (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. Gaussian • initial condition: Numerical integration. Pseudo-particles Hamilton dynamics of pseudo-particles • phase-space volumes are conserved (Liouville theorem) over large time scales • provided the number of pseudoparticles is large (Np~106)

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

  40. Plasmon relaxation : models with ions Na55

  41. 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

  42. 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 kinetic energy (a.u.) Electronic dipole (a.u.) Ionic kinetic energy (a.u.) Residual cluster charge Ionic radial distribution

  43. Kinetic versus Coulombic effects 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 free ions free coulomb explosion of Compare simulations in which ions are either free to move or rigidly fixed Na196 ,I=1012 W.cm-2 , w=wp , T=100 fs  The electronic kinetic pressure plays a major role in the cluster explosion

  44. Na196 + Xe25+ peripheral collision electron dipole (a.u.) Q(t) time (fs) time (fs) Ion kinetic energy (eV) time (fs) The envelopes of electric fields and the final cluster charges are similar the strong electron oscillations against the ions greatly enhance the explosion

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