Finite Size Effects in Conducting Nanoparticles: Classical and Quantum Theories

1. George Y. Panasyuk Wright-Patterson Air Force Base, OH, February 8, 2011 Finite Size Effects in Conducting Nanoparticles: Classical and Quantum Theories

2. Outline of the Talk • Theory of classical confinement of electrons in conducting nanoparticles • Quantum theory of electric polarization in metal nanofilms

3. Part 1 • Theory of classical confinement of electrons in conducting nanoparticles and derivation of nonlinear polarizabilities in: (1)1D (slab), (2) 3D (sphere) geometries. • Nonlinear refraction

4. Classical Solution z _ z _ _ _ _ _ E _ E θ a ρ → ∞ + + + + + + , ρ + +

5. If nanoparticle size a ≤ 10 ℓ , where ℓ is atomic scale size: -Finite size corrections - Quantum corrections U. Kreibig and L. Genzel, Surf. Sci. (1985) Discrete electron states in a nanoparticle: F. Hache, D. Ricard, and C. Flyzanis, J. Opt. Soc. Am. (1986) S.G. Rautian, J. Exp. Theor. Phys. (1997) Additional, pure classical mechanism that leads to small-size effects: classical confinement effect G.Y. Panasyuk, J.C. Schotland, V.A. Markel, PRL (2008)

6. Zero conductivity region h + + + L Eext Conducting region σ Negative surface charge density _ _ _ z

7. Nonlinearity: Dipole moment per unit area Two small parameters:

8. Nonlinearity in optical response: 3D case z Negative surface charge density, σ(θ) E _ _ _ _ Conductivity region O' δ O Zero conductivity region, ρ + + + +

9. Resulting Equation: Solution: where τ = ωt – φandΛ(ω) same as in 1D case

10. Dipole moment: 3D case - accurate if |δ| < a -Consistent with the classical solution

11. Result Introducing:

12. Harmonic generation - No second harmonic generation despite of the second order nonlinearity over the electric field - Zero order (n = 0) → nonlinear refraction • First order (n = 1) → Third harmonic generation

13. Nonlinear refraction

14. Classical confinement: Magnitude of nonlinearity Nonlinear polarizability ~ I = the power of incident beam Silver:

16. Part 2. Quantum theory of electron confinement in metal nanofilms • Classical arguments based on macroscopic Maxwell equations gives only qualitative understanding of the finite size effects • Quantum mechanical treatment brings quantitatively accurate theory

17. Rautian’s theory (1997) • Most advanced theory of optical nonlinearities Shortcomings: - No e-e interaction beyond Paoli principle; - Electrons are driven by a uniform field; - Infinite potential barrier at the surface Our approach: Density functional theory (DFT) G.Y. Panasyuk, J.C. Schotland, V.A. Markel, arXiv:1101.1908v1 10 Jan 2011

18. Starting Point

19. Hohenberg and Kohn Theorems (1964) T1. T2. For as given

20. Kohn – Sham ansatz

21. Kohn-Sham Equations, 1D case -h/2 h/2 0 z -zm/2 Zm/2 h = ma = slab thickness ~ several nm a = atomic cell size: fcc for silver, a = 0.41 nm

22. 1D case (nanofilm) - Rigid BC - Free BC

23. Continuation of 1D case

24. Determination of the Factor W n=n max+1 . n=n max . . n=3 n=2 n=1

25. Solving the KS equations Exit condition ? No Yes Output

26. Dipole moment 1. Numerical 2. Perturbation: Pure classical arguments gives: and

27. Normalized Dipole Moment

28. Nonlinearities in the Dipole Moment Possible experiment: C. Torres-Torres, A.V. Khomenko at al., Optics Express 15, 9248 (2007)

29. Comparison with the Classical Confinement Theory

30. Electron density and electric field distribution m = 8 atomic layers

31. Electron density m = 8 atomic layers

32. Electron density and electric field distribution m = 32 atomic layers

33. Conclusions ● Pure classical mechanism leading to finite-size effects and nonlinearity of optical response is found and described. It is non-perturbative and fully accounts on electron-electron interaction ● Nonlinear response appears in the second order in electromagnetic field and is distinguishable from other optical nonlinearities by its dependence on the power of incident beam.

34. ● Quantum theory of electron confinement for metal nanofilms was developed and used to compute the nonlinear response of the nanofilm to external electric field ● Emergence of macroscopic behavior and correspondence to our classical theory of electron confinement for thick films was demonstrated ● As was shown, the sign and overall magnitude of the nonlinear corrections depends on type of boundary conditions