View
215
Download
2
Category
Preview:
Citation preview
(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
Recommended