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

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

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

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

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)

h E ext Zero conductivity region Conducting region σ z L Negative surface charge density +++ _ __

Nonlinearity: Dipole moment per unit area Two small parameters:

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

Solution: where τ = ωt – φ and Λ(ω) same as in 1D case Resulting Equation:

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

Introducing: Result

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

Nonlinear refraction

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

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

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: v1 10 Jan 2011

Starting Point

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

Kohn – Sham ansatz

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

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

Continuation of 1D case

n=1 n=2 n=3 n=n max n=n max Determination of the Factor W

Solving the KS equations Exit condition ? No Yes Output

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

Normalized Dipole Moment

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

Comparison with the Classical Confinement Theory

m = 8 atomic layers Electron density and electric field distribution

Electron density m = 8 atomic layers

Electron density and electric field distribution m = 32 atomic layers

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.

● 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