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(M.eq.) Size dependence of the number, frequencies and Size dependence of the number, frequencies and radiative radiative decays of plasmon modes in a spherical free- decays of plasmon modes in a spherical free- electron cluster electron cluster K.Kolwas, A.Derkachova and S.Demianiuk Institute of Physics, Polish Acadamy of Sciences, Al. Lotników 32/46 02-668 Warsaw, Poland A B S T R A C T A B S T R A C T Nanoscale metal particles are well known for their ability to sustain collective electron plasma oscillations - plasmons. When we talk of plasmons, we have in mind the eigenmodes of the self-consistent Maxwell equations with appropriate boundary conditions. In [1-4] we solved exactly the eigenvalue problem for the sodium spherical particle. It resulted in dipole and higher polarity plasmon frequencies dependence l (R), l=1,2,...10 (as well as the plasmon radiative decays) as a function of the particle radius R for an arbitrarily large particle. We now re-examine the usual expectations for multipolar plasmon frequencies in the "low radius limit" of the classical picture: 0,l = p (l/(2l+1)) 1/2 , l=1,2,...10. We show, that 0,l are not the values of 0,l in the limit R -› 0 as usually assumed, but 0,l l (R= R min,l ) = ini,l (R min,l ). So ini,l are the frequencies of plasmon oscillation for the smallest particle radius R min,l 0 still possessing an eigenfrequency for given polarity l. R min,l can be e.g.: R min,l=4 = 6 nm, but it can be as large R min,l=10 = 87.2 nm. The confinement of free-electrons within the sphere restricts the number of modes l to the well defined number depending on sphere radius R and on free-electron concentration influencing the value of p . [1] K. Kolwas, S. Demianiuk, M. Kolwas, J. Phys. B 29 4761(1996). [2] K. Kolwas, S. Demianiuk, M. Kolwas, Appl. Phys. B 65 63 (1997). [3] K. Kolwas, Appl. Phys. B 66 467 (1998). [4] K. Kolwas, M. Kolwas, Opt. Appl. 29 515 (1999). [5] M.Born, E.Wolf. Principles of Optics. Pergamon Press, Oxford, 1975. Self-consistent Maxwell equations describing fields due to known currents and charges: No external sources: We are concerned with transverse solutions only (E = 0). For harmonic fields (M.eq.) reduces to the Helmholtz equation: Solution of the scalar equation in spherical coordinates: Continuity relations of tangential components of E and B + nontriviality of solutions for amplitudes A lm and B lm Dispersion relation for TM and TE field oscillations. Two independent solution of the vectorial equation: • TM mode (''transverse magnetic'': • TE mode (''transverse electric'': F O R M U L A T I O N OF T H E E I G E N V A L U E P R O B L E M: F O R M U L A T I O N OF T H E E I G E N V A L U E P R O B L E M: P L A S M O N F R E Q U E N C I E S A N D R A D I A T I V E D A M P I N G R A P L A S M O N F R E Q U E N C I E S A N D R A D I A T I V E D A M P I N G R A We allow the imaginary solutions for given R: - the eigenfrequencies of free- electron gas filling a spherical cavity of radius R (the frequencies of the filed oscillations), - the damping of oscillations. Let's define a function D l TM (z l ) of the complex arguments z l ( l ,R We are interested in zeros of D l TM (z l ) as a function of l and R: Dispersion relation for TM mode: If: l in given l is treated as a parameter to find, R is outside parameter with the successive values changed with the step R 2nm up to the final radius R=300nm. p, - plasma frequency and relaxation rate of the free electron gas accordingly. R E S U L T S R E S U L T S 0 50 100 150 200 10 15 20 25 30 35 40 l=6 l= 10 l= 5 l=4 l=3 l= 2 [fs] R [nm ] l=1 0 50 100 150 200 10 12 14 16 18 20 l= 10 l=6 l=5 l=4 l=3 l=2 [fs] R [nm ] l=1 a) b) Radiative decay of plasmon oscillations in sodium particle for different values of l and for relaxation rates of the free electron gas: a) = 0.5 eV; b) = 1 eV The smallest particle radii R min,l, still possessing an eigenfrequency of given polarity l as a function of l Frequencies of plasmon oscillation ini,l as a function of the smallest particle radius R min,l for different relaxation rates of free electron gas 0 20 40 60 80 100 3,0 3,2 3,4 3,6 3,8 4,0 eV eV eV ini , l [eV ] R mi n, l [nm ] l = 1,2,...,10 Comparison of plasmon frequencies and damping rates resulting from the exact and the approximated approach: Approximated (irrespective R value ): Exact: for: Conclusions: • If the sphere is too small, there is no related values of l (R) real nor complex. • For given multipolarity l the eigenfrequency l (R) can be attributed to the sphere of the radius R starting from R min,l 0. • Plasmon frequency l (R) in given l is weakly modified by the relaxation rate , while radiative damping rate ” (R) is strongly affected by in the rage of smaller sphere sizes. a) Resonance frequencies and b) radiative damping of plasmon oscillations as a function of the radius of sodium particle for different values of l =0). -0,6 -0,4 -0,2 0,0 0 50 100 150 200 250 300 l= 4 l=6 l= 3 l= 5 l= 2 l= 1 R [nm ] l '' (R) ,[eV ] b) 0 50 100 150 200 250 300 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 p / 2 p / 3 l= 12 l=3 l=2 l=1 l ( R )[eV ] R [nm ] a) -0 ,20 -0 ,15 -0 ,10 -0 ,05 0 ,00 0 50 100 150 200 250 300 l=7 l= 4 l=6 l= 8 l=3 l=5 l= 2 l=1 R [nm ] l '' (R) ,[eV ] 0 50 100 150 3,0 3,2 3,4 3,6 3,8 4,0 l =10 l =7 l =6 l =5 l =4 l =3 l =2 l =1 0, l = p [ l/2l +1] 1/2 p / 3 p / 2 l ( R ) [eV ] R [nm ] Legend : - Bessel, Hankel and Neuman cylindrical functions of the standard type defined according to the convention used e.g. in [5]. or where: Approximated Riccati-Bessel functions “for small arguments”: where: Using the approximated Riccati-Bessel functions in the dispersion relation, one gets: irrespective the value of the sphere radius R. R e ( ψ l ( z B ) ) I m ( ψ l ( z B ) ) Re(z B ) Im(z B ) Re(z B ) Im(z B ) Re(z B ) Im(z B ) Re(z B ) Im(z B ) l=1 l=8 R e ( l ( z B ) ) I m ( l ( z B ) ) Im(z H ) Re(z H ) Im(z H ) Re(z H ) Im(z H ) Re(z H ) Im(z H ) Re(z H ) l=1 l=8 Variation ranges of the functions l (z B (R)) and l (z H (R)) due to the dependence (R)=(R)+”(R) resulting from the dispersion relation; the example for l=1 and l=8. l and l (and their derivatives l ’ and l ’ in respect to the corresponding argument z B and z H ) were calculated exactly using the recurrence relation: with the two first terms of the series in the form: Exact Riccati-Bessel functions: Variation ranges of the arguments z B,l (R)=c -1 (R)R and z H,l (R)= c -1 ( ()) 1/2 (R)R of l (z B (R)) and l (z H (R)) functions due to the dependence (R)=(R)+”(R) resulting from the dispersion relation; the example for l=1 and l=8. 0 50 100 150 200 0,00 0,02 0,04 0,06 0,08 0,10 R e(z B ) R [nm ] 0 50 100 150 200 -6 -5 -4 -3 -2 -1 0 Im (z B ) R [nm ] 0 50 100 150 200 0 1 2 3 4 R e(z B ) x10 -5 R [nm ] 0 50 100 150 200 -5 -4 -3 -2 -1 0 Im (z B ) R [nm ] l = 1 l = 1 l = 8 l = 8 0 50 100 150 200 0,0 0,2 0,4 0,6 0,8 1,0 R e(z H ) R [nm ] 0 50 100 150 200 -0,4 -0,3 -0,2 -0,1 0,0 R [nm ] Im (z H ) 0 50 100 150 200 0 1 2 3 4 R e(z H ) R [nm ] 0 50 100 150 200 -4 -3 -2 -1 0 Im (z H ) x10 -5 R [nm ] l = 8 l = 8 l = 1 l = 1
