(m.eq.) size dependence of the number, frequencies and radiative decays of plasmon modes in a...
TRANSCRIPT
(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