(M.eq.)

Size dependence of the number, frequencies and radiative Size dependence of the number, frequencies and radiative decays of plasmon modes in a spherical free-electron clusterdecays of plasmon modes in a spherical free-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 TA 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= Rmin,l) =

ini,l(Rmin,l). So ini,l are the frequencies of plasmon oscillation for the

smallest particle radius Rmin,l 0 still possessing an eigenfrequency for

given polarity l. Rmin,l can be e.g.: Rmin,l=4 = 6 nm, but it can be as large

Rmin,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 equationsdescribing 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 Alm and Blm

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 T E SP 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 T E S

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 DlTM(zl) of the complex arguments zl(l,R):

We are interested in zeros of DlTM(zl) 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 SR E S U L T S

0 50 100 150 20010

15

20

25

30

35

40

l=6

l=10

l=5l=4

l=3

l=2

[fs

]

R [nm]

l=1

0 50 100 150 20010

12

14

16

18

20

l=10

l=6

l=5l=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 Rmin,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 Rmin,l for different relaxation rates of free electron gas

0 20 40 60 80 1003,0

3,2

3,4

3,6

3,8

4,0

eVeV

eV

ini,l

[eV

]

Rmin,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 Rmin,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

0,00 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 3000,0

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=2l=1

R [nm]

l'' (R),

[eV

]

0 50 100 1503,0

3,2

3,4

3,6

3,8

4,0

l=10

l=7

l=6

l=5

l=4l=3l=2l=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.

Re(

ψl(z

B))

Im(ψ

l(zB))

Re(zB ) Im

(z B)

Re(zB ) Im

(z B)

Re(zB ) Im

(z B)

Re(zB ) Im

(z B)

l=1 l=8

Re(

l(z

B))

Im(

l(zB))

Im(z H

)Re(zH )

Im(z H

)Re(zH )

Im(z H

)Re(zH )

Im(z H

)Re(zH )

l=1 l=8

Variation ranges of the functions l (zB(R)) and l (zH(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 zB and zH) 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 zB,l(R)=c-1 (R)R and zH,l(R)= c-1 ( ())1/2 (R)R of l (zB(R)) and l (zH(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 2000,00

0,02

0,04

0,06

0,08

0,10

Re(

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 2000

1

2

3

4

Re(

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 2000,0

0,2

0,4

0,6

0,8

1,0

Re(

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 2000

1

2

3

4

Re(

z H)

R [nm]

0 50 100 150 200

-4

-3

-2

-1

0

Im(z

H)

x10-5

R [nm]

l = 8

l = 8l = 1

l = 1