Plan of the lectures 1. Introductory remarks on metallic nanostructures Relevant quantities and typical physical parameters Applications. Linear electron response: Mie theory and generalizations 3. Nonlinear response Survey of various models from N-body to macroscopic Mean-field approximation (Hartree and Vlasov equations) 4. Beyond the mean-field approximation Hartree-Fock equations Time-dependent density functional theory (DFT) and local-density approximation (LDA) 5. Macroscopic models: quantum hydrodynamics Linear theory and comparison of various models 6. Spin dynamics: experimental results and recent theoretical advances 7. Illustration: the nonlinear electron dynamics in thin metal films Master Lecture 1
Electron dynamics qualitative aspects We have seen that the typical timescale of collective phenomena is πω 1 p 1 fs We need short laser pulses to probe this timescale. Pump-probe experiments Master Lecture
Linear response: driven-damped harmonic oscillator β damping oscillations forcing Steady state solution (t ): Resonance becomes broader with increasing damping Master Lecture 3
The response is directly proportional to the excitation For instance, variation of the electron density is proportional to laser field When the frequency ω of the excitation is close to the natural frequency of the system, we have resonance enhanced absorption For electron gas, natural frequency ~ ω p In the presence of damping, the resonance becomes «broad» Damping rate Resonant energy = ħω Main purpose of linear theory: determine resonant frequency and damping rate Master Lecture 4
Mie theory 1D model Def. of electric dipole n 0 n i Ehrenfest theorem n 0 (x) Poisson s equation Initial shift of the electron density by a distance d(0): Induced change in electric field (from Poisson s equation) δn We obtain: Master Lecture 5
ω Finally the dipole obeys the harmonic oscillator equation: If n 0 = const. = n i it can be taken out of the integral: We obtain oscillations at the plasma frequency. This is the fundamental result of Mie theory No dependence on size, temperature, The metal species (Au, Ag, ) appears only in the plasma frequency, through the electron density. No damping: purely oscillatory mode at a single frequency Limitations Derivation in 1D and, of course, purely linear response Master Lecture 6
Spill-out effect: illustration Numerical computation of the ground state at zero temperature Sodium nanoparticle n e (r) / n 0 n i Spill-out r / a G. Weick, PhD thesis, IPCMS, Strasbourg, (006) Master Lecture 7
Spill-out effect: qualitative picture The electron density at equilibrium is not equal to the ion density n 0 The electrons spill out of a length δ n 0 / This leads to a reduction of the oscillation frequency -a a a+δ x Master Lecture 8
The correction to the frequency goes as 1/a This is the first correction we have found to the simple plasmon frequency It is true in any number of dimensions It s just the surface to volume ratio For instance, in 3D: Lithium clusters ħω (ev) Assuming δr does not depend on R N 1/3 Data : C. Brechignac et al, PRL 70, 036 (1993) Graph : G. Weick, PhD thesis, IPCMS, Strasbourg, (006) Master Lecture 9
Thomas-Fermi theory of spill-out Valid at Te = 0 Kinetic energy electrostatic energy = We obtain the density as a function of the electrostatic potential V(x) Poisson s equation: Linearize Poisson s equation, assuming ev << µ λ ( µ E F at T = 0) = e The spill-out is a screening effect λdoes not depend on particle s size Master Lecture 10
General Mie theory of the surface plasmon (1908) Spherical nanoparticle immersed in external field E 0 Electrostatic response: free charges tend to shield the external field Inside the sphere the electric field is* E int = E 0 3ε m ε + ε m < E ε > ε We make the following assumptions: No magnetic field effect (E/B ~ c) Electric field wavelength >> R = radius of nanoparticle For visible-light lasers λ 400-800 nm Thus, we consider only the time variation of the field E = E 0 exp( iωt) The dielectric constant depends on the frequency 0 if m * J. D. Jackson, Classical Electrodynamics Master Lecture 11
The dielectric constant has a real and an imaginary part ε ( ω) = ε1( ω) + iε( ω) E int = E 0 3ε m ε + ε m When ε << ε 1, the resonance condition is ε 1 = ε m The dielectric constant also determines the photo-absorption cross-section It remains to be determined the frequency dependence of ε 1 (ω). Master Lecture 1
Frequency-dependent dielectric constant Drude theory for time-dependent electric field Equation of motion j = electric current Fourier transform: p p exp( iωt) microscopic Ohm s law σ (ω) = conductivity Master Lecture 13
Maxwell s equations Use: J = σ E iω t ε b ( ω) = ε ε 0 Bound electrons Free electrons Master Lecture 14
Mie resonance E int = E 0 3ε m ε + ε m < E 0 if ε > ε m When ε << ε 1, the resonance condition is ε 1 = ε m Remember the frequency-dependent dielectric constant If Γ << ω, the real part of ε is: = ε m Finally, we obtain the Mie frequency: if ε m ε 1 b Surface plasmon Master Lecture 15
Mie resonance in 1D, D, and 3D Dimensionality Geometry Resonant frequency 1D Thin film ω p D Planar surface ω p / 3D Sphere ω p / 3 Master Lecture 16
Does Mie theory work? 1. Pure Mie: ω = ω p / 3 ω Mie spill-out. Spill-out correction ω ω Mie 3 ( 1 K N 1/ ) ω / ω Mie spill-out Numerics (TDLDA) TDLDA 3. Self-consistent calculation using TDLDA (time-dependent local density approximation) Na 83 N 1/3 Spill-out ω ω Mie ( 3) 1 N 1/ K ; K > K TDLDA Mie ħω (ev) G. Weick, PhD thesis, IPCMS, Strasbourg (006) Master Lecture 17
Experiment vs. Mie theory and TDLDA simulations Experiment Recover Mie value for large N Simulation C. Yannouleas et al., Phys. Rev. B 47, 9849 (1993) Master Lecture 18
Linewidth of the Mie resonance damping Sources of damping 1. Electron-electron collisions (e-e). Electron-phonons collisions (e-ph): interactions with lattice 3. Coupling between collective modes (plasmon) and singleparticle modes: Landau damping 4. Radiation damping 5. Master Lecture 19
Linewidth of the Mie resonance damping Collision rate in the bulk material Γ = V F / L L is the bulk mean free path: L (Na) = 34 nm; L (Ag) = 5 nm (at T=73K) When L > R (size of the particle) then one should replace L with R Collisions with the particle s surfaces This picture is not quite correct quantum-mechanically The boundaries determine the shape of the wave functions everywhere Kawabata and Kubo (1966) computed the quantum damping rate Still obtain 1/R behavior. Physically: coupling of the collective plasmon excitation to single-particle modes ( Landau damping ) Γ = Γ V + A F R Master Lecture 0
Linewidth experimental and numerical results Num. C. Yannouleas et al., Phys. Rev. B 47, 9849 (1993) Exp. R. H. Doremus, J. Chem. Phys. 4, 414 (1965). Numerics ~ Γ 1 Silver ~ R 1/R theory Master Lecture 1
Radiation damping An oscillating electric dipole radiates electromagnetic energy This is a source of damping of the electronic energy W: W = 1 Nmv = The total radiated power is (see Jackson, Classical Electrodynamics) dw dt 1 = 4πε 0 I ( kd ) 1 c 1 Nm I k = p e ω ω = Qω sp sp sp : / c current : wave v = dω vector sp = p e ω sp Electric dipole: p=ed d + dw dt Q e 4 sp 0 4 sp ω p N ω p 3 3 4πε c 4πε c 0 W = Γ rad This yields Γ rad e 0 4πε mc 3 ω sp N R 3 Proportional to the volume: significant only for large nanoparticles Γ e hω rad sp 1 3 ev N N N sp hc 10 8 (4 ) ω mc ε 0 137 511 kev Master Lecture