Download Undamped collective surface plasmon oscillations along metallic

JOURNAL OF APPLIED PHYSICS 108, 084304 共2010兲
Undamped collective surface plasmon oscillations along metallic
nanosphere chains
W. Jacak,1,a兲 J. Krasnyj,1,2 J. Jacak,1 A. Chepok,2 L. Jacak,1 W. Donderowicz,1 D. Z. Hu,3
and D. M. Schaadt3
1
Institute of Physics, Wrocław University of Technology, Wyb. Wyspiańskiego 27, 50-370, Wrocław, Poland
Theor. Phys. Group, International University, Fontanskaya Doroga 33, Odessa 65-026, Ukraine
3
Institute of Applied Physics/DFG-Center for Functional Nanostructures, Karlsruhe Institute of Technology,
Weberstrasse 5, 76-133 Karlsruhe, Germany
2
共Received 14 July 2010; accepted 25 August 2010; published online 19 October 2010兲
The random-phase-approximation semiclassical scheme for describing plasmon excitations in large
metallic nanospheres 共with radius ⬃10– 60 nm兲 is developed for the case when a dynamical electric
field is present. The spectrum of plasmons in metallic nanospheres is determined including both
surface and volume-type excitations and their mutual connections. It is demonstrated that only
dipole-type surface plasmons can be excited by a homogeneous dynamical electric field. The
Lorentz friction due to irradiation of electromagnetic energy by plasmon oscillations is analyzed
with respect to sphere dimension. The resulting shift in resonance frequency due to plasmon
damping is compared with experimental data for various sphere radii. Wave-type collective
oscillations of surface plasmons in long chains of metallic nanospheres are described. The
undamped region for collective plasmon propagation along the metallic chain is found in agreement
with previous numerical simulations. © 2010 American Institute of Physics.
关doi:10.1063/1.3493263兴
I. INTRODUCTION
Experimental and theoretical invesitigations of plasmon
excitations in metallic nanocrystals rapidly grew mainly due
to possible applications in photovoltaics and microelectronics. A significant enhancement of absorption of incident light
in photodiode-systems with an active surface covered with
nanodimension metallic particles 共of Au, Ag, or Cu兲 with
planar density ⬃108 – 1010 / cm2 was observed.1–7 This is due
to a mediating role in light energy transport carried out by
surface plasmon oscillations in metallic nanocompounds on a
semiconductor surface. These findings are of practical importance for enhancement of solar cell efficiency, especially for
thin film cell technology. On the other hand, hybridized
states of surface plasmons and photons result in
plasmon-polaritons8 which are of high importance for applications in photonics and microelectronics,9,10 in particular,
for transportation of energy in metallic modified structures in
nanoscale.11,12
Surface plasmons in nanoparticles have been widely investigated since their classical description by Mie.13 Many
particular studies, including numerical modeling of multielectron clusters, have been carried out.14–18 They were
mostly developments of Kohn–Sham attitude in form of local density approximation 共LDA兲 or time dependent LDA
共TDLDA兲 共Refs. 14–17兲 for small metallic clusters only, up
to approximately 200 electrons 共as limited by numerical calculation constraints that grow rapidly with the number of
electrons兲. The random-phase-approximation 共RPA兲 was
formulated19 for description of volume plasmons in bulk
a兲
Electronic mail: [email protected].
0021-8979/2010/108共8兲/084304/12/$30.00
metals and utilized also for confined geometry mainly in a
numerical or seminumerical manner.16 Usually, in these
analyses the jellium model was assumed for description of
positive ion background in the metal and dynamics was addressed to the electron system only.16–18 Such an attitude is
preferable for clusters of simple metals, including noble metals 共also transition and alkali metals兲. It should be noted,
however, that the TDLDA approach was successfully applied
to nanoshells with radius above 10 nm and thickness up to a
few nanometers20 via acceleration of the numerical procedures. It was achieved by assuming screened effective Coulomb interaction, which enhanced convergence of iterations,
thus allowing for effective solution of Kohn–Sham type
equations for even more than 104 electrons within a jellium
model. For the nanoshell systems also numerical solution of
the RPA-type polarization self-consistent equation has been
found.21 These analyses showed that plasmon resonances in
nanoshells are similar to classical Mie-type ones, redshifted,
however, with respect to them, similarly as it was for small
clusters 共mostly due to spill-out of electrons beyond the jellium rim兲. The radiation effects were not included, corresponding to not severe contribution of radiation losses even
for relatively large radius of nanoshells, because the number
of electrons in a shell is much lower than in a full nanosphere of similar radius. One can thus expect that dimension
scaling of radiation losses, being of cubic type for
nanospheres,22 does not apply to shells, resulting in the overall shift in radiation effects in shells to larger radius region in
comparison to full spheres 共likely, above the radii considered
in Refs. 20 and 21兲.
In the present paper we generalize the bulk RPA
description,19 using a semiclassical approach for a large metallic nanosphere 共with radius of several tens of nanometers
108, 084304-1
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084304-2
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
and with 105 – 107 electrons兲 in an all-analytical calculus
version.22 The electron liquid oscillations of compressional
and translational type result in excitations inside the sphere
and on its surface, respectively. They are analyzed and referred to as volume and surface plasmons. Damping of plasmons due to electron scattering 共nonlocal processes兲 and due
to radiation losses 共retardation effects accounted for via the
Lorentz friction force兲 is included. The shift in the resonance
frequency of dipole-type surface plasmons 共only such plasmons are induced by homogeneous time-dependent electric
field兲, due to damping phenomena, is compared with the experimental data for various nanosphere radii.
Collective surface dipole-type plasmon oscillations in
the linear chain of metallic nanospheres are analyzed and
wave-type plasmon modes are described. A coupling in the
near-field regime between oscillating dipoles of surface plasmons, together with retardation effects for energy irradiation,
lead to the possibility of undamped propagation of plasmon
waves along the chain in the experimentally realistic region
of parameters 共separation of spheres in the chain and their
radii兲. These effects are of particular significance for plasmon arranged nondissipative transport of energy along metallic chains for application in nanoelectronics.
In Sec. II of this paper, the standard RPA theory in the
quasiclassical limit, is generalized for a confined system of
spherical shape. The resulting equations for volume and surface plasmons are solved in the following section 共with the
particularities of calculus shifted to the Appendix兲. Section
III contains a description of the Lorentz friction for surface
plasmons oscillations of the dipole-type. Analysis of collective wave-type surface plasmon oscillations in a chain of
metallic nanospheres is presented in Sec. V. Besides the theoretical model the comparison of the characteristic nanoscale
plasmon behavior with available experimental data, including our own measurements, is presented.
II. RPA APPROACH TO ELECTRON EXCITATIONS IN
METALLIC NANOSPHERE
A. Derivation of RPA equation for local electron
density in a confined spherical geometry
冋
Ĥe = 兺 −
j=1
+
ប2ⵜ2j
− e2
2m
冕
ne共r0兲d3r0
+ e␸共R + r j,t兲
兩r j − r0兩
册
1
e2
+ ⌬E,
兺
2 j⫽j 兩r j − r j⬘兩
共2兲
⬘
where r j and m are the position 共with respect to the sphere
center兲 and the mass of the jth electron, respectively, R is the
position of the metallic sphere center, ⌬E represents electrostatic energy contribution from the ion “jellium,” and ␸共r , t兲
is the scalar potential of the external electric field. The corresponding electric field E共R + r j , t兲 = −grad j␸共R + r j , t兲. Assuming that the space-dependent variation in E is weak on
the scale of the sphere radius a, then E共R + r j , t兲 ⯝ E共R , t兲,
i.e., the electric field is homogeneous over the sphere. Then
␸共R + r j , t兲 ⯝ −E共R , t兲 · R + ␸1共R + r j , t兲, where ␸1共R + r j , t兲
= −r j · E共R , t兲. Hence, one can rewrite the Hamiltonian 共2兲 in
the form
Ĥe = Ĥe⬘ − eNE共R,t兲 · R,
where
Ne
冋
Ĥe⬘ = 兺 −
j=1
+
ប2ⵜ2j
− e2
2m
冕
共3兲
ne共r兲d3r0
+ e␸共r j,t兲
兩r j − r0兩
册
1
e2
+ ⌬E,
兺
2 j⫽j 兩r j − r j⬘兩
共4兲
⬘
and the corresponding 共to Ĥe兲 wave function can be represented as,
⌿共re,t兲 = ⌿⬘共re,t兲ei共eN/ប兲兰E共R,t兲·Rdt ,
共5兲
with iប ⳵ ⌿⬘ / ⳵t = Ĥe⬘⌿⬘, re = 共r1 , r2 , . . . , rN兲.
A local electron density can be written as follows:19
␳共r,t兲 = 具⌿共re,t兲兩 兺 ␦共r − r j兲兩⌿共re,t兲典 = 具⌿⬘共re,t兲兩 兺 ␦共r
j
j
− r j兲兩⌿⬘共re,t兲典,
共6兲
with the Fourier picture
Let us consider a metallic sphere with radius a located in
vacuum, ␧ = 1, ␮ = 1 and in the presence of dynamical electric
field 共magnetic field is assumed to be zero兲. We will consider
collective electrons in the metallic material. The model
jellium16–18 is assumed in order to account for the screening
background of positive ions in the form of static positive
charge uniformly distributed over the sphere
ne共r兲 = ne⌰共a − r兲,
Ne
共1兲
where ne = Ne / V and ne兩e兩 is the averaged positive charge
density, Ne is the number of collective electrons in the
sphere, V = 共4␲a3 / 3兲 is the sphere volume, and ⌰ is the
Heaviside step-function. Neglecting ion dynamics within the
jellium model, which is adopted in particular for description
of simple metals, e.g., noble, transition, and alkali metals, we
deal with the Hamiltonian for collective electrons,
˜␳共k,t兲 =
冕
␳共r,t兲e−ik·rd3r = 具⌿⬘共re,t兲兩␳ˆ 共k兲兩⌿⬘共re,t兲典, 共7兲
where the “operator” ␳ˆ 共k兲 = 兺 je−ik·r j.
Using the above notation one can rewrite Ĥe⬘ in the following form, in analogy to the bulk case23
Ne
冋 册 冕
Ĥe⬘ = 兺 −
j=1
ប2ⵜ2j
e2
−
2m
4␲2
冕
冕
+
e2
16␲3
+
e2
4␲2
d3kñe共k兲
1 ˆ+
关␳ 共k兲 + ␳ˆ 共k兲兴
k2
d3k˜␸1共k,t兲关␳ˆ+共k兲 + ␳ˆ 共k兲兴
d 3k
1 ˆ+
关␳ 共k兲␳ˆ 共k兲 − Ne兴 + ⌬E,
k2
where ñe共k兲 = 兰d3rne共r兲e−ik·r,
˜␸1共k , t兲 = 兰d3r␸共r , t兲e−ik·r.
4␲ / k2 = 兰d3r1 / re−ik·r,
共8兲
and
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084304-3
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
Utilizing this form of the electron Hamiltonian one can
write the second time-derivative of ␳ˆ 共k兲,
1
d2␳ˆ 共k,t兲
=
兵关␳ˆ 共k兲,Ĥe⬘兴,Ĥe⬘其,
dt2
共iប兲2
⫻具⌿⬘兩兺 je−ik·r j共ប2ⵜ2j / 2m兲兩⌿⬘典. Performing the inverse Fourier transform, Eq. 共11兲 finally attains the form,
共9兲
ប2ⵜ2j
2 2
⳵2␦␳共r,t兲
具⌿
兩
␦
共r
−
r
兲
ⵜ
兩⌿⬘典
=
−
⬘
兺
j
⳵ ,t2
3m
2m
j
再
which resolves itself into the equation
再
␻2p
ⵜ ⌰共a − r兲 ⵜ
+
4␲
d ␦␳ˆ 共k,t兲
ប
បk
= − 兺 e−ik·r j − 2 共k · ⵜ j兲2 + 2 ik · ⵜ j
2
dt
m
m
j
2
冎
2
ប 2k 4
e2
+
2 −
4m
m2␲2
e
−
m8␲3
−
e
m8␲3
e2
−
m2␲2
冕
冕
冕
2 2
冕
d qñe共k − q兲共k · q兲˜␸1共q,t兲
j
再
+
冎
ប 2k 4
e2
2 兩⌿⬘典 −
4m
m2␲2
− q兲
冕
2 2
⫻共k · q兲˜␸1共q,t兲 −
e
m8␲3
⫻共k · q兲˜␸1共q,t兲 −
e2
m2␲2
− q兲␦␳ˆ 共q兲兩⌿⬘典.
冋
⫻ 1+
冕
冕
冕
ប2ⵜ2j
兩⌿⬘典
2m
册
5 ␦␳共r,t兲
+¯ .
3 ne
共13兲
冋
册
⳵2␦␳共r兲
2 ⑀F 2
=
ⵜ ␦␳共r,t兲 − ␻2p␦␳共r,t兲 ⌰共a − r兲
2
3m
⳵t
d3qñe共k − q兲
−
d3q␦˜␳共k − q,t兲
−
d 3q
共12兲
Taking then into account the above approximation and that
ⵜ⌰共a − r兲 = − rr ␦共a − r兲 as well as that ␸1共R , r , t兲 = −r · E共R , t兲,
one can rewrite Eq. 共12兲 in the following manner:
d3qñe共k
k·q
e
˜
2 ␦␳共q,t兲 −
q
m8␲3
冎
3
3
ប2
ប2
⌰共a − r兲
⯝ 共3␲2兲2/3 ␳5/3共r,t兲 = 共3␲2兲2/3 n5/3
5
5
2m
2m e
共10兲
⳵ ␦˜␳共k,t兲
ប
បk
= 具⌿⬘兩 − 兺 e−ik·r j − 2 共k · ⵜ j兲2 + 2 ik · ⵜ j
2
⳵t
m
m
j
2
1
␦␳共r1,t兲
兩r − r1兩
ene
ⵜ 兵⌰共a − r兲 ⵜ ␸1共r1,t兲其.
m
具⌿⬘兩 − 兺 ␦共r − r j兲
d3q␦␳ˆ 共k − q兲共k · q兲˜␸1共q,t兲
k·q
d3q␦␳ˆ 共k − q兲 2 ␦␳ˆ 共q兲,
q
d 3r 1
According to the Thomas–Fermi approximation19 the RPA
averaged kinetic energy can be represented as follows:
3
where ␦␳ˆ 共k兲 = ␳ˆ 共k兲 − ñe共k兲 is the operator of local electron
density fluctuations beyond the uniform distribution. Taking
into account that ␦˜␳共k , t兲 = 具⌿⬘共t兲兩␦␳ˆ 共k兲兩⌿⬘共t兲典 = ˜␳共k , t兲
− ñe共k兲, we find,
2
+
k·q
d3qñe共k − q兲 2 ␦␳ˆ 共q兲
q
冕
k·q
具⌿⬘兩␦␳ˆ 共k
q2
共11兲
One can simplify the above equation with the assumption
that ␦␳共r , t兲 = 1 / 8␲3兰eik·r␦˜␳共k , t兲d3k only weakly varies on
the interatomic scale, and, therefore, three components of the
first term on right-hand-side of Eq. 共11兲 can be estimated as:
k2vF2 ␦˜␳共k兲, k3vF / kT␦˜␳共k兲, and k4vF2 / kT2 ␦˜␳共k兲, respectively,
with 1 / kT the Thomas–Fermi radius,19 kT = 冑6␲nee2 / ⑀F, ⑀F is
the Fermi energy, and vF is the Fermi velocity. Thus, the
contribution of the second and third components of the first
term can be neglected in comparison to the first component.
Also small, and thus negligible, is the last term on the righthand-side of Eq. 共11兲, as it involves a product of two ␦˜␳
共which we assumed small ␦˜␳ / ne Ⰶ 1兲. This approach corresponds to RPA attitude formulated for bulk metal19,23 关note
that ␦␳ˆ 共0兲 = 0 and the coherent RPA contribution of interaction is contained in the second term in Eq. 共11兲兴. The last but
one term in Eq. 共11兲 can also be omitted if one confines it to
linear terms with respect to ␦˜␳ and ˜␸1. Next, due to spherical
symmetry,
具⌿⬘兩兺 je−ik·r j共ប2 / m2兲共k · ⵜ j兲2兩⌿⬘典 ⯝ 共2k2 / 3m兲
2
ⵜ
3m
冋
再冋
册
3
r
⑀Fne + ⑀F␦␳共r,t兲 ␦共a − r兲
5
r
冎
2 ⑀F r
ⵜ ␦␳共r,t兲
3mr
冕
+
␻2p r
ⵜ
4␲ r
+
ene r
· E共R,t兲 ␦共a − r兲.
m r
d 3r 1
1
␦␳共r1,t兲
兩r − r1兩
册
共14兲
In the above formula ␻ p is the bulk plasmon frequency, and
␻2p = 共4␲nee2 / m兲. The solution of Eq. 共14兲 can be decomposed into two parts that are related to the partial domains,
␦␳共r,t兲 =
再
␦␳1共r,t兲,
␦␳2共r,t兲,
for
r ⬍ a,
for
r ⱖ a,
共r → a+兲,
冎
共15兲
corresponding to the volume and surface excitations, respectively. These two parts of local electron density fluctuations
satisfy the equations,
⳵2␦␳1共r,t兲 2 ⑀F 2
ⵜ ␦␳1共r,t兲 − ␻2p␦␳1共r,t兲
=
3m
⳵ t2
共16兲
and
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084304-4
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
2
⳵2␦␳2共r,t兲
ⵜ
=−
2
⳵t
3m
−
+
再
再冋
册
3
r
⑀Fne + ⑀F␦␳2共r,t兲 ␦共a + ⑀ − r兲
5
r
冎
2 ⑀F r
ⵜ ␦␳2共r,t兲
3mr
␻2p r
ⵜ
4␲ r
冕
d 3r 1
1
关␦␳1共r1,t兲⌰共a − r1兲
兩r − r1兩
+ ␦␳2共r1,t兲⌰共r1 − a兲兴 +
ene r
· E共R,t兲
m r
冎
⫻␦共a + ⑀ − r兲.
共17兲
The ⑀ = 0+ shift in the delta argument, ␦共a + ⑀ − r兲 is introduced, as a formal character only, to avoid the singular point
position on the edge of integration 共for the delta distribution兲.
It is clear from Eq. 共16兲 that the volume plasmons are
independent of surface plasmons. However, surface plasmons can be excited by volume plasmons due to the last but
one term in Eq. 共17兲, which expresses a coupling between
surface and volume plasmons in the metallic nanosphere
within the RPA semiclassical picture. This is due to the possible influence of volume charges on surface fluctuations.
Oppositely, the surface charges cannot influence the interior
of the nanosphere.
When the metallic sphere is embedded in a dielectric
medium, the electrons on the surface interact with forces ␧
共dielectric susceptibility constant兲 times weaker compared to
vacuum. To account for this, one substitutes Eq. 共17兲 with the
following one:
2
⳵2␦␳2共r,t兲
ⵜ
=−
2
⳵t
3m
−
+
再
再冋
册
3
r
⑀Fne + ⑀F␦␳2共r,t兲 ␦共a + ⑀ − r兲
5
r
冕
d 3r 1
冎
冋
and
1
␦␳1共r1,t兲⌰共a − r1兲
兩r − r1兩
册
冎
1
ene r
· E共R,t兲 ␦共a
+ ␦␳2共r1,t兲⌰共r1 − a兲 +
␧
m r
+ ⑀ − r兲.
⳵2␦␳1共r,t兲 2 ⳵ ␦␳共r,t兲 2 ⑀F 2
=
ⵜ ␦˜␳1共r,t兲 − ␻2p␦␳1共r,t兲
+
3m
⳵ t2
␶0 ⳵ t
共19兲
2 ⑀F r
ⵜ ␦␳2共r,t兲
3mr
␻2p r
ⵜ
4␲ r
tuations, in opposition to a sharply defined domain of the
surface modes within the semiclassical approximation.
Moreover, within the semiclassical approach the inner material dielectric properties are not important 关as it follows from
Eq. 共18兲兴, what is no case, however, in TDLDA calculus,
with fuzzy surface.20,24 This leads to a certain overestimation
of the dielectric surroundings induced redshift in the semiclassical treatment in comparison to TDLDA when the domain of surface fluctuation was wider and, therefore, more
distant from screening dielectric medium polarization. This
manifest itself in stronger redshift for the sharp domain of
surface oscillations in semiclassical approximation, which
exceeds the Mie-type classical shift and also the TDLDA
redshift 共as it is presented in the next subsection for surface
plasmon frequencies兲. One can expect, however, that with
growing radius of the nanosphere, when spill-out is gradually
reducing, the limiting TDLDA would be convergent with the
semiclassical estimation. Note that the more thorough inclusion of dielectric missmatch 共as in the TDLDA method20,24兲
would have a particular significance for closely located
nanospheres 共in a chain or dimer兲 when multipolar modes
would contribute to mutual coupling of the spheres. In the
present paper we confine, however, our consideration to simplified dipole-type coupling only, what is justified for not too
small distance of nanospheres.
Let us also assume that both volume and surface plasmon oscillations are damped with the time ratio ␶0 which can
be phenomenologically accounted for via the additional
term, −共2 / ␶0兲关⳵␦␳共r , t兲 / ⳵t兴, to the right-hand-side of above
equations. They attain the form,
共18兲
Simultaneously, Eq. 共16兲 does not change, which reveals the
fact that the dielectric surroundings does not influence volume plasmon modes, oppositely to surface modes. One can
expect that influence of surrounding dielectric medium
would cause a redshift in surface plasmon resonance, what is
clear as the polarization of nearby dielectric medium induced
by a fluctuating surface charge, reduces this charge due to
screening, which results in turn in lowering of oscillation
energy. This would be accounted for within even the classical Mie-type approach. It should be noted, however, that in
Ref. 20 the stronger redshift induced by dielectric missmatch
is found in comparison to the Mie-type classical estimation,
for the case of nanoshells. It agrees with similar analyses for
small metallic clusters.24 In both these cases spill-out is important, resulting in a widened fuzzy domain of surface fluc-
⳵2␦␳2共r,t兲 2 ⳵ ␦␳共r,t兲
+
⳵ t2
␶0 ⳵ t
再冋
2
ⵜ
3m
−
2 ⑀F r
ⵜ ␦˜␳2共r,t兲
3mr
+
再
␻2p r
ⵜ
4␲ r
冕
册
3
r
⑀Fne + ⑀F␦␳2共r,t兲 ␦共a + ⑀ − r兲
5
r
=−
d 3r 1
冋
冎
1
␦␳1共r1,t兲⌰共a − r1兲
兩r − r1兩
册
冎
1
ene r
· E共R,t兲 ␦共a + ⑀ − r兲.
+ ␦␳2共r1,t兲⌰共r1 − a兲 +
␧
m r
共20兲
From Eqs. 共19兲 and 共20兲 it is noticeable that the electric field
homogeneous over the nanosphere does not excite volumetype plasmon oscillations but only contributes to surface
plasmons.
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B. Solution of RPA equations: volume and surface
plasmons frequencies
Equations 共19兲 and 共20兲 can be solved upon imposing
boundary and symmetry conditions—see Appendix. Let us
write both parts of the electron fluctuation in the following
manner:
␦␳1共r,t兲 = ne关f 1共r兲 + F共r,t兲兴,
for
r ⬍ a,
␦␳2共r,t兲 = ne f 2共r兲 + ␴共⍀,t兲␦共r + ⑀ − a兲,
r ⱖ a,
for
共r → a+兲,
共21兲
and let us choose convenient initial conditions F共r , t兲 兩t=0 = 0,
␴共⍀ , t兲 兩t=0 = 0, 关⍀ = 共␪ , ␺兲—spherical angles兲, and 关1
+ f 1共r兲兴 兩r=a = f 2共r兲 兩r=a 共continuity condition兲, F共r , t兲 兩r→a = 0,
兰␳共r , t兲d3r = Ne 共neutrality condition兲.
We thus arrive at an explicit form of the solutions of
Eqs. 共19兲 and 共20兲 共as it is described in the Appendix兲
f 1共r兲 = −
kTa + 1 −k 共a−r兲 1 − e−2kTr
e T
,
2
k Tr
冋
for
册
for
=
冑
4␲ ene
兵Ez共R,t兲␦m0 + 冑2关Ex共R,t兲␦m1 + Ey共R,t兲␦m−1兴其,
3 m
共25兲
where ␻1 = ␻01 = 共␻ p / 冑3␧兲 共it is a dipole-type surface plasmon Mie frequency13兲. Only this function contributes to the
dynamical response to the homogeneous electric field 共for
the assumed initial conditions兲. From the above it follows
thus that the local electron density 共within the RPA attitude兲
has the form,
␳共r,t兲 = ␳0共r兲 + ␳1共r,t兲,
r ⱖ a,
⬁
⬘ t兲e−t/␶
Almn jl共knlr兲Y lm共⍀兲sin共␻nl
兺
兺
l=1 m=−l n=1
and
⬁
l
兺 Y lm共⍀兲
l=1 m=−l
␴共⍀,t兲 = 兺
册
冋
⫻
⬁
l␻2p − 共2l + 1兲␻2nl
a
0
dr1
共23兲
l
ne关1 + f 1共r兲兴,
for
ne f 2共r兲,
r ⱖ a, r → a+
冦
0,
兺 兺 Almn
rl+2
1
⬘ t兲e−t/␶0 ,
jl共knlr1兲sin共␻nl
al+2
共24兲
where jl共␰兲 = 冑共␲ / 2␰兲Il+1/2共␰兲 is the spherical Bessel funcis
the
spherical
function,
␻nl
tion,
Y lm共⍀兲
= ␻ p冑1 + 共x2nl / kT2 a2兲 are the frequencies of electron volume
free self-oscillations 共volume plasmon frequencies兲, xnl are
nodes of the Bessel function jl共␰兲, ␻0l = ␻ p冑l / ␧共2l + 1兲 are
the frequencies of electron surface free self-oscillations 共surface plasmon frequencies兲, and knl = xnl / a; ␻⬘ = 冑␻2 − 共1 / ␶20兲
are the shifted frequencies for all modes due to damping. The
coefficients Blm and Almn are determined by the initial conditions. We assumed that ␦␳共r , t = 0兲 = 0, so Blm = 0 and Almn
= 0, except for l = 1 in the former case 共of Blm兲, which corresponds to homogeneous electric field excitation. This is described by the function Q1m共t兲 in the general solution 共24兲.
The function Q1m共t兲 satisfies the equation,
冎
共27兲
for
r ⬍ a,
兺
Q1m共t兲Y 1m共⍀兲
for
r ⱖ a,
r→a+.
m=−1
冧
共28兲
In general, F共r , t兲 共volume plasmons兲 and ␴共⍀ , t兲 共surface
plasmons兲 contribute to plasmon oscillations. However, in
the case of homogeneous perturbation, only the surface l = 1
mode is excited.
For plasmon oscillations given by Eq. 共28兲 one can calculate the corresponding dipole,
D共R,t兲 = e
Y lm共⍀兲ne
for
r ⬍ a,
1
⬁
l=1 m=−l n=1
共l + 1兲␻2p
冕
0
Blm
⬘ t兲e−t/␶0共1 − ␦1l兲
sin共␻0l
a2
+ Q1m共t兲␦1l + 兺
⫻
␳1共r,t兲 =
⬁
l
F共r,t兲 = 兺
再
and the nonequilibrium, of surface plasmon oscillation type
for the homogeneous forcing field,
共22兲
where kT = 冑共6␲nee2 / ⑀F兲 = 冑共3␻2p / vF2 兲. For time-dependent
parts of electron fluctuations, we find,
共26兲
where the RPA equilibrium electron distribution 共correcting
the uniform distribution ne兲,
␳0共r兲 =
r ⬍ a,
k Ta + 1
e−kT共r−a兲
,
共1 − e−2kTa兲
2
k Tr
f 2共r兲 = kTa −
⳵2Q1m共t兲 2 ⳵ Q1m共t兲
+ ␻21Q1m共t兲
+
⳵ t2
␶0 ⳵ t
冕
d3rr␳共r,t兲 =
4␲
eq共R,t兲a3 ,
3
共29兲
Q1−1共R , t兲
where
Q11共R , t兲 = 冑共8␲ / 3兲qx共R , t兲,
= 冑共8␲ / 3兲qy共R , t兲, Q10共R , t兲 = 冑共4␲ / 3兲qx共R , t兲, and q共R , t兲
satisfies the equation 关see Eq. 共25兲兴,
冋
册
⳵2 2 ⳵
ene
+ ␻21 q共R,t兲 =
E共R,t兲.
2 +
⳵t
␶0 ⳵ t
m
共30兲
III. LORENTZ FRICTION FOR NANOSPHERE
PLASMONS
Considering the nanosphere plasmons induced by the
homogeneous electric field, as described in the above paragraph, one can note that these plasmons are themselves a
source of electromagnetic 共e-m兲 radiation. This radiation
takes away the energy of plasmons resulting in their damping, which can be described as the Lorentz friction.25 This
e-m wave emission causes electron friction which can be
described as an additional electric field,25
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EL =
2 ⳵3D共t兲
,
3␧v2 ⳵ t3
共31兲
where v = 共c / 冑␧兲 is the light velocity in the dielectric medium, and D共t兲 the dipole of the nanosphere. According to
Eq. 共29兲 we arrive at the following:
EL =
2e 4␲ 3 ⳵3q共t兲
.
a
3␧v2 3
⳵ t3
册
冉 冊
2 ␻ 1a
ene
E共R,t兲 +
m
3␻1 v
⳵ q共R,t兲
.
⳵ t3
3 3
共33兲
If one rewrites the above equation 共for E = 0兲 in the form,
冋
册
冋
⳵2
+ ␻21 q共R,t兲
⳵ t2
冉 冊
2
2 ␻ 1a
⳵
− q共R,t兲 +
=
⳵t
␶0
3␻1 v
册
⳵ q共R,t兲
,
⳵ t2
3 2
共34兲
thus, one notes that the zeroth order approximation 共neglecting attenuation兲 corresponds to the equation,
冋
册
⳵2
+ ␻21 q共R,t兲 = 0.
⳵ t2
共35兲
In order to solve Eq. 共34兲 in the next step of perturbation,
one can substitute, in the right-hand-side of this equation,
⳵2q共t兲 / ⳵t2 by −␻21q共t兲 关according to Eq. 共35兲兴.
Therefore, if one assumes the above estimation,
共⳵3q共t兲 / ⳵t3兲 ⯝ −␻21共⳵q共t兲 / ⳵t兲, one can include the Lorentz
friction in a renormalized damping term,
冋
册
⳵2 2 ⳵
ene
+ ␻21 q共R,t兲 =
E共R,t兲,
+
⳵ t2 ␶ ⳵ t
m
where
共Air兲 1
共Water兲 1.4
共Colloidal solution兲 2
冉 冊
1 1 ␻ 1 ␻ 1a
= +
␶ ␶0 3 v
⳵2 2 ⳵
+
+ ␻21 q共R,t兲
⳵ t2 ␶0 ⳵ t
=
Refraction rate of the
surrounding medium, n0
共32兲
Substituting this into Eq. 共30兲, we get,
冋
TABLE I. a0—nanosphere radius corresponding to minimal damping.
共36兲
3
⯝
Au, a0
共nm兲
Ag, a0
共nm兲
Cu, a0
共nm兲
8.8
9.14
9.99
8.44
9.18
10.04
8.46
9.20
10.04
冉 冊
v F C v F ␻ 1 ␻ 1a
+
+
3
2␭B 2a
v
3
,
共37兲
where we used for 共1 / ␶0兲 ⯝ 共vF / 2␭B兲 + 共CvF / 2a兲, 共␭B is the
free path in bulk, vF is the Fermi velocity, and C ⯝ 1 is a
constant兲,26–28 which corresponds to inclusion of plasmon
damping due to electron scattering on other electrons and on
the nanoparticle boundary. Renormalized damping causes a
change in the shift in self-frequencies of free surface plasmons, ␻1⬘ = 冑␻21 − 共1 / ␶2兲.
Using Eq. 共37兲 one can determine the radius a0 that corresponds to minimal damping,
a0 =
冑3
␻p
共vFc3冑␧/2兲1/4 .
共38兲
For nanoparticles of gold, silver and copper in air, in water,
and in a colloidal solution, one can find a0 ⱕ 10 nm 共see
Table I兲, which corresponds to the experimental data.29,30 For
a ⬎ a0 damping increases due to Lorentz friction 共proportional to a3兲 but for a ⬍ a0, the damping due to electron scattering dominates and grows with lowering a 关as ⬃共1 / a兲, see
Fig. 1兴, which agrees with experimental observations.26,29
Surface plasmon oscillations cause attenuation of the incident e-m radiation, with maximum of attenuation at the
resonant frequency22 ␻1⬘ = 冑␻21 − 共1 / ␶2兲. This frequency diminishes with the rise of a, for a ⬎ a0 according to Eq. 共37兲,
which agrees with experimental observations for Au and Ag
presented in Fig. 2, and Table II 共Au兲 and Table III 共Ag兲.
Note that the Lorentz friction, being of the third order
with respect to time-retardation shift,25 includes the retardation effects via plasmon damping induced by irradiation.31
The other channel of plasmon damping corresponds to non-
FIG. 1. 共Color兲 Effective damping ratios for surface plasmon oscillations for Ag in air 共left and middle兲 and Au colloidal solutions 共right兲 from Eq. 共38兲,
respectively. The red curve represents the sum of both terms, the green is the term ⬃共1 / a兲, and the blue is the term ⬃a3; the minimum corresponds to the
minimal damping for radius a0 from Eq. 共39兲.
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084304-7
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FIG. 2. 共Color兲 Extinction spectra for nanospheres of Au 共a兲 and Ag 共b兲 in colloidal water solution for various sphere radii; Lorentzian shapes correspond to
light attenuation due to surface plasmons exciting in nanospheres; for large radii 共⬃75 nm for Au and ⬃30– 40 nm for Ag兲 the second attenuation peak
appears corresponding possibly to volume mode excited at nonexact dipole approximation regime.
local effects including scattering on the nanoparticle
surface.32 We have demonstrated that the retardation and the
nonlocal effects lead to opposite, redshift and blueshift in the
resonance frequency, similarly as it was indicated in Ref. 31.
above formula, R0 ⬍ ␭, n = R0 / R0, and v = c / 冑␧ = c / n0.
When both vectors R and R0 are along the z-axis 共the
linear chain兲 the above equation can be resolved as,
E␣共R,R0,t兲 =
IV. UNDAMPED SURFACE PLASMON COLLECTIVE
EXCITATION ALONG A CHAIN OF METALLIC
NANOSPHERES
Let us consider a linear chain of metallic nanospheres
with radii a in a dielectric medium with dielectric constant ␧.
We assume that spheres are located along the z-axis direction
equidistantly with the separation of the sphere centers d
⬎ 2a.26,33 At time t = 0 we assume the excitation of plasmon
oscillations via a Dirac delta, ⬃␦共t兲, shaped signal of electric
field. Taking into account the mutual interaction of the induced surface plasmons on the spheres via the radiation of
dipole oscillations, we aim to determine the stationary state
of the whole infinite chain. For a separation d much shorter
than the wavelength ␭ of the e-m wave corresponding to
surface plasmon self-frequency in a single nanosphere, the
dipole-type plasmon radiation can be treated within the nearfield regime, at least for nearest neighboring spheres. In the
near-field region a ⬍ R0 ⬍ ␭, the radiation of the dipole D共t兲
is not a planar wave 共as it is for far-field region, R0 Ⰷ ␭兲 but
is of only a retarded electric field 共without a magnetic field兲25
E共R,R0,t兲 =
1
再 冋 冉 冊册
冉 冊冎
␧R30
3n n · D R,t −
− D R,t −
R0
v
R0
v
共39兲
,
R is the position of the sphere 共center兲 irradiating e-m energy
due to its dipole surface plasmon oscillations, R0 is the position of another sphere 共center兲, with respect to the center of
the former one, where the field E共R , R0 , t兲 is given by the
冉
冊
␴␣
R0
,
3 D␣ R,t −
v
␧R0
共40兲
where ␣ = 共x , y , z兲, ␴x = ␴y = −1, and ␴z = 2. Assuming that the
z-axis origin coincides with the center of one sphere in the
chain, for the lth sphere located in the point Rl = 共0 , 0 , ld兲, an
electric field caused by neighboring spheres, E共Rm , Rml , t兲,
and the Lorentz friction force caused by self-radiation,
EL共Rl , t兲, have to be accounted for. By virtue of Eq. 共30兲 the
equation for the surface plasmon oscillation of the lth sphere
is,
冋
册
m=⬁
⳵2 2 ⳵
ene
+ ␻21 q共Rl,t兲 =
+
E共Rm,Rml,t兲
兺
⳵ t2 ␶0 ⳵ t
m m=−⬁,m⫽l,Rml⬍␭
+
ene
EL共Rl,t兲,
m
共41兲
provided that the dipole field of the mth sphere can be treated
as homogeneous over the lth sphere and the sum over m is
confined by the distance of the mth sphere from the lth
sphere not exceeding the near-field range 共⬃␭兲. In the case
of the equidistant chain, Rl = ld and Rml = 兩l − m兩d, and using
Eqs. 共40兲, 共29兲, and 共32兲, one can rewrite Eq. 共41兲 in the
form,
冋
冉 冊
2 ␻ 1a
⳵2 2 ⳵
+
−
⳵ t2 ␶0 ⳵ t 3␻1 v
m=⬁
3
册
⳵3
+ ␻21 q␣共ld,t兲
⳵ t3
a3
= ␴␣␻21 3
兺
d m=−⬁,m⫽l,兩l−m兩d⬍␭
冉
冊
d
q␣ md,t − 兩l − m兩
v
,
兩l − m兩3
共42兲
where ␣ = x , y, which describe the transversal plasmon modes
TABLE II. Resonant frequency for e-m wave attenuation in Au nanospheres.
Radius of nanospheres 共nm兲
ប␻⬘1 共experiment兲 共eV兲
ប␻⬘1 共theory兲 共eV兲, n0 = 1.4
ប␻⬘1 共theory兲 共eV兲, n0 = 2
10
2.371
3.721
2.604
15
2.362
3.720
2.603
20
2.357
3.716
2.600
25
2.340
2.702
2.590
30
2.316
3.666
2.565
40
2.248
3.415
2.388
50
2.172
2.374
1.656
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084304-8
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Jacak et al.
冉 冊
1
1 ␻ 2a ␻ ␣a
= + 1
␶ ␣共 ␻ ␣兲 ␶ 0 3 v
v
TABLE III. Resonant frequency for e-m wave attenuation in Ag nanospheres.
Radius of nanospheres 共nm兲
ប␻⬘1 共experiment兲 共eV兲
ប␻⬘1 共theory兲 共eV兲, n0 = 1.4
ប␻⬘1 共theory兲 共eV兲, n0 = 2
10
3.024
3.707
2.595
20
2.911
3.702
2.591
30
2.633
3.654
2.557
40
2.385
3.410
2.384
2
⬁
cos共mkd兲
a3
+ ␴␣␻21 3 兺
m3
d m=1,md⬍␭
冉 冊
sin
␻␣md
v
.
␻␣
共47兲
and ␣ = z, which describes the longitudinal mode;
= 共␻2p / 3␧兲 = 共4␲nee2 / 3␧m兲 and v = 共c / 冑␧兲. The above equation coincides with the appropriate one from Refs. 26 and 33,
if one assumes that 共4␲ / 3兲a3ne = N = 1 and neglects the retardation of the field.
Taking into account the periodicity of the infinite chain,
one can consider the solution of the above equation in the
form,
␻21
q␣共ld,t兲 = q̃␣共k,t兲e
−ikld
If we confine the sum in Eq. 共46兲 to m = 1 共the nearest neighbor approximation兲, we get,
冋
˜ ␣2 共␻␣兲 ⯝ ␻21 1 −
␻
and from Eq. 共47兲,
1 ␻ 2a 3
1
= + 1 2
␶ ␣共 ␻ ␣兲 ␶ 0 4 v d
共43兲
.
The right-hand-side term in Eq. 共42兲 attains the form,
m=⬁
兺
m=−⬁,m⫽l
l−1
兺
=
冉
d
q␣ md,t − 兩l − m兩
v
兩l − m兩3
m=−⬁
冉
d
q␣ md,t − 兩l − m兩
v
兩l − m兩3
冉
m=⬁
+
兺
m=l+1
= 2e−ikld 兺
m=1
1 ␻ 2a 3
1
= + 1 2
␶ z共 ␻ z兲 ␶ 0 2 v d
for
冊
冉 冊
3
˜ ␣2 共␻␣兲
␻
=
册
⳵3
+ ␻21 q̃␣共k,t兲
⳵ t3
2i␻␣
˜ ␣2 共␻␣兲 = 0,
+␻
␶ ␣共 ␻ ␣兲
=
␻21
冋
⬁
共44兲
共45兲
冉 冊册
cos共mkd兲
2 ␴ ␣a 3
␻␣md
1−
cos
兺
m3
d3 m=1,md⬍␭
v
共46兲
and
␻ zd
v
2
+ 共kd − ␲兲2 −
册
␲2
,
3
␣ = z.
共50兲
⬁
This equation is linear and, therefore, we look for the solutions of the shape: q̃␣共k , t兲 = Q̃␣共k兲ei␻␣t, and we arrive at the
condition,
where,
冋冉 冊
共49兲
1
1
=
关sin共kmd + ␻␣md/v兲 − sin共kmd − ␻␣md/v兲兴
兺
2␻␣ m=1 m3
cos共mkd兲
a3
2 兺
q̃␣共k,t − md/v兲.
m3
d3 m=1,md⬍␭
− ␻␣2 +
册
␲2
,
3
冉 冊
⬁
⬁
= ␴␣␻21
− 共kd − ␲兲2 +
cos共mkd兲
1
␻␣md
sin
兺
3
m
␻␣ m=1
v
cos共mkd兲
q̃␣共k,t − md/v兲.
m3
2 ␻ 1a
⳵2 2 ⳵
−
2 +
⳵t
␶0 ⳵ t 3␻1 v
2
In the derivation of the two above formulae the following
summation was performed:34
Thus, the Eq. 共42兲 can be written as follows:
冋
冋冉 冊
␻ ␣d
v
共48兲
␣ = x,y
and
冊
d
q␣ md,t − 兩l − m兩
v
兩l − m兩3
⬁
for
冊
冉 冊册
2 ␴ ␣a 3
␻ ␣d
3 cos共kd兲cos
d
v
冋
册
k2d2 ␻␣2 d2
d ␲2 ␲
− kd +
+
.
4
2
2v2
v 6
Because the terms in the sum drop quickly to zero, the above
formula approximates well the sum with limitation md ⬍ ␭.
Assuming now, ␻␣ = ␻␣⬘ + i␻␣⬙ , the Eq. 共45兲 gives the dependence of ␻␣⬘ and ␻␣⬙ on k. The general solution of Eq. 共42兲
attains the form,
Ns
q␣共ld,t兲 = 兺 Q̃␣共kn兲ei共␻␣⬘ 共kn兲t−knld兲−␻␣⬙ 共kn兲t ,
共51兲
n=1
where kn = 共2␲n / Nsd兲, L = Nsd is the assumed length of the
chain with Ns spheres, when periodic 共Born–Karman-type兲
boundary conditions are imposed. The components of Eq.
共51兲 describe monochromatic waves with wavelength ␭n
= 共2␲ / kn兲 = 共L / n兲, which are analogous to planar waves in
crystals, when damping is not big, i.e., when ␻␣⬙ Ⰶ ␻␣⬘ . Provided with this inequality, one can approximate: for the
transversal modes 共␣ = x , y兲,
冋
˜ ␣2 = ␻21 1 +
共␻␣⬘ 兲2 = ␻
册
2a3
cos共kd兲cos共␻␣⬘ d/v兲 ,
d3
共52兲
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084304-9
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
FIG. 3. 共Color兲 Wave vector kd for undamped surface plasmon propagation along the metallic nanosphere chain vs sphere separation d at 共top兲 constant
d / a = 3, with a the sphere radius and 共bottom兲 for a = 20 nm for Ag sphere chains, respectively; transversal modes are shown in red while longitudinal modes
are shown in blue.
␻␣⬙ =
1
1 ␻ 2a 3
= + 1 2
␶␣ ␶0 4vd
冋冉 冊
␻␣⬘ d
v
− 共kd − ␲兲2 +
and for the longitudinal mode 共␣ = z兲,
冋
˜ z2 = ␻21 1 −
共␻z⬘兲2 = ␻
␻z⬙ =
␻21a3
2
1 1
= +
␶z ␶0 2vd
册
␲2
,
3
2
册
4a3
cos共kd兲cos共␻z⬘d/v兲 ,
d3
冋冉 冊
␻␣⬘ d
v
2
共54兲
␲
3
2
+ 共kd − ␲兲2 −
共53兲
册
.
共55兲
From Eqs. 共53兲 and 共55兲 it follows that ␻␣⬙ can change its
sign. In the case of ␻␣⬙ ⬍ 0 the oscillations are destabilized,
which could be avoided by inclusions of some nonlinear
terms neglected in the expression for the Lorentz friction,
which in more accurate form25 includes also a small nonlinear term with respect to D, aside from the term with ⳵3D / ⳵t3.
Including this will result in damping of too highly rising
oscillations and will lead to stable oscillation amplitude. Due
to this stabilization caused by nonlinear effects, undamped
wave modes of dipole oscillations will propagate in the chain
in the region of parameters where ␻␣⬙ ⱕ 0 共and with fixed
amplitude accommodated by the nonlinear term兲. The condi-
tion ␻␣⬙ = 共1 / ␶␣兲 = 0, for critical parameters, resolves into,
冉 冊
␻ ␣d
v
2
= 共kd − ␲兲2 −
4vd2
␲2
,
−
3 ␶0␻21a3
共56兲
for ␣ = 共x , y兲 and for ␣ = z,
冉 冊
␻ zd
v
2
= − 共kd − ␲兲2 +
2vd2
␲2
.
−
3 ␶0␻21a3
共57兲
共␻␣d / v兲 obtained from the above equations leads to determination of the dependence of wave vector k with respect to
parameters d and a, via Eqs. 共52兲–共55兲. Solutions for these
equations, found numerically for the chain of Ag nanospheres, are depicted in Fig. 3.
Undamped plasmon waves in the chain appear if d
⬍ dmax and have k = ␲ / d, dmax = 98.5共68.8兲 nm for transversal共longitudinal兲 modes. For example, for Ag spheres with
the radius a = 20 nm and the separation d = 60 nm, undamped transversal modes appear for 0 ⱕ kd ⱕ ␲ / 4 or
3␲ / 4 ⱕ kd ⱕ 2␲ and an undamped longitudal mode for
3␲ / 4 ⱕ kd ⱕ 5␲ / 4.
Let us underline that the determined undamped plasmon
oscillation wave modes would explain similar, numerically
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084304-10
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
observed behavior.35–37 Within that numerical analysis two
types of collective surface plasmons with distinct propagation along the nanosphere chain were identified, and called
as quasistatic 共ordinary兲 and nonquasistatic 共extraordinary兲
surface plasmon modes. The quasistatic modes were damped
while nonquasistatic ones were undamped 共and additionally,
the latter turn out to be relatively robust against the disorder
in the chain兲.35 One can associate these observations with
undamped collective plasmon modes with the negative
imaginary part of the frequency, accessible in the described
above regions of the chain parameters and the wave vector
values 共stabilized by nonlinear terms of Lorentz friction at a
certain amplitude兲, which are, however, accompanied by ordinary damped modes with the positive imaginary frequency
part.
f 1共r兲 = ␣
e−kTa −k r k r
共e T − e T 兲,
k Tr
共A3兲
where ␣ is a constant, kT = 冑6␲nee2 / ⑀F = 冑3␻2p / vF2 共kT is the
inverse Thomas–Fermi radius兲, ␻ p = 冑4␲nee2 / m; 共bulk plasmon frequency兲.
Since we assume F共r , 0兲 = 0, then for function F共r , t兲 the
solution can be taken as,
F共r,t兲 = F␻共r兲sin共␻⬘t兲e−␶0t ,
共A4兲
where ␻⬘ = 冑␻2 + 1 / ␶20. F␻共r兲 satisfies the equation 共Helmholtz equation兲,
ⵜ2F␻共r兲 + k2F␻共r兲 = 0,
共A5兲
with k2 = 共␻2 − ␻2p / vF2 / 3兲. A solution of the above equation,
nonsingular at r = 0, is as follows,
V. CONCLUSIONS
In the present paper we analyzed plasmons in large metallic nanospheres induced by a homogeneous timedependent electric field. Within an all-analytical RPA quasiclassical approach, the volume and surface plasmons are
described. It was verified that only dipole-type of surface
plasmons can be induced by a homogeneous field 共while
none of volume modes兲. An irradiation of energy by plasmon
oscillations is described within the Lorentz friction effect. Its
scaling with the nanosphere dimension leads to a sphere radius dependent shift in resonant frequency, similarly as observed in experiments, for nonsmall nanospheres 共with radii
10–80 nm兲. This description of surface dipole-type plasmon
oscillations in a single nanosphere is applied to the analysis
of collective oscillations in linear chains of metallic nanospheres. The wave-type collective plasmon oscillations in the
chain are also considered. The undamped region of wave
propagation through the chain is found for a certain sphere
separation in the chain with the corresponding wavelength of
plasmon waves. This phenomenon confirms similar behavior
observed by numerical simulations.35
F␻共r兲 = Ajl共kr兲Y lm共⍀兲,
共A6兲
where A is a constant, jl共␰兲 = 冑␲ / 共2␰兲Il+1/2共␰兲 is the spherical
Bessel function 关In共␰兲 is the Bessel function of the first order兴, Y lm共⍀兲 is the spherical function 共⍀ is the spherical
angle兲. Owing to the quasiclassical boundary condition,
F共r , t兲 兩r→a = 0, one has to demand jl共ka兲 = 0, which leads to
the discrete values of k = knl = xnl / a, 共where xnl, n
= 1 , 2 , 3 , . . ., are nodes of jl兲, and next to the discretisation of
self-frequencies ␻,
冉
2
␻nl
= ␻2p 1 +
x2nl
kT2 a2
冊
共A7兲
.
The general solution for F共r , t兲 thus has the form,
⬁
l
⬁
⬘ t兲e−␶ t .
Almn jl共knlr兲Y lm共⍀兲sin共␻nl
兺
兺
l=0 m=−l n=1
F共r,t兲 = 兺
0
共A8兲
ACKNOWLEDGMENTS
Supported by the Polish KBN Project No. N N202
260734 and the FNP Fellowship Start 共W.J.兲, as well as DFG
under Grant No. SCHA 1576/1-1 共D. Z. Hu and D.S.兲.
APPENDIX: ANALYTICAL SOLUTION OF PLASMON
EQUATIONS FOR THE NANOSPHERE
Let us solve first the Eq. 共19兲, assuming a solution in the
form,
␦␳1共r,t兲 = ne关f 1共r兲 + F共r,t兲兴,
for
r ⬍ a.
共A1兲
Equation 共19兲 resolves thus into,
ⵜ2 f 1共r兲 − kT2 f 1共r兲 = 0,
⳵2F共r,t兲 2 ⳵ F共r,t兲 vF2 2
= ⵜ F共r,t兲 − ␻2pF共r,t兲.
+
3
⳵ t2
␶0 ⳵ t
共A2兲
The solution for function f 1共r兲 共nonsingular at r = 0兲 thus has
the form,
A solution of Eq. 共20兲 we represent as,
␦␳2共r,t兲 = ne f 2共r兲 + ␴共⍀,t兲␦共r + ⑀ − a兲,
for
r ⱖ a,
共r → a + ,i . e . , ⑀ = 0+兲.
共A9兲
The neutrality condition, 兰␳共r , t兲d3r = Ne, with ␦␳2共r , t兲
= ␴共␻ , t兲␦共a + ⑀ − r兲 + ne f 2共r兲, 共⑀ → 0兲, can be rewritten as follows: −兰a0drr2 f 1共r兲 = 兰⬁a drr2 f 2共r兲, 兰a0d3rF共r , t兲 = 0, and
兰d⍀␴共⍀ , t兲 = 0. Taking into account also the continuity condition on the spherical particle surface, 1 + f 1共a兲 = f 2共a兲, one
can obtain: f 2共r兲 = ␤e−kT共r−a兲 / 共kTr兲. It is possible to fit ␣ 关see
Eq. 共A3兲兴 and ␤ constants: ␣ = 共kTa + 1 / 2兲 and ␤ = kTa − 共kTa
+ 1 / 2兲共1 − e−2kTa兲—which gives Eqs. 共22兲.
From the condition 兰a0d3rF共r , t兲 = 0 and from Eq. 共A8兲 it
follows that A00n = 0, 共because of 兰d⍀Y lm共␻兲 = 4␲␦l0␦m0兲.
To remove the Dirac delta functions we integrate both
sides of the Eq. 共20兲 with respect to the radius 共兰⬁0 r2dr. . .兲
and then we take the limit to the sphere surface, ⑀ → 0. This
results in the following equation for surface plasmons:
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084304-11
J. Appl. Phys. 108, 084304 共2010兲
Jacak et al.
⬁
l
⳵2␴共⍀,t兲 2 ⳵␴共⍀,t兲
2
+
Y lm共⍀兲
= − 兺 兺 ␻0l
⳵ t2
␶0 ⳵ t
l=0 m=−l
⬁
+ ␻2pne 兺
l
兺
冕
⬁
兺 Almn
l=0 m=−l n=1
ⴱ
d⍀1␴共⍀1,t兲Y lm
共⍀1兲
l+1
Y lm共⍀兲
2l + 1
冕
a
dr1
0
rl+2
ene
1
冑4␲/3关Ez共R,t兲Y 10共⍀兲
l+2 j l共knlr1兲sin共␻nlt兲 +
a
m
+ 冑2Ex共R,t兲Y 11共⍀兲 + 冑2Ey共R,t兲Y 1−1共⍀兲兴,
where ␻20l = ␻2p共l / 2l + 1兲. In derivation of the above equation
the following formulae were exploited, 共for a ⬍ r1兲,
⳵2Q1m共t兲 2 ⳵ Q1m共t兲
2
+ ␻01
+
Q1m共t兲
⳵ t2
␶0 ⳵ t
⬁
=
1
⳵
⳵
al
=
Pl共cos ␥兲
兺
⳵ a 冑a2 + r21 − 2ar1 cos ␥ ⳵ a l=0 rl+1
1
⬁
=兺
l=0
la
Pl共cos ␥兲,
ene
冑4␲/3关Ez共R,t兲␦m0 + 冑2Ex共R,t兲␦m1
m
+ 冑2Ey共R,t兲␦m−1兴.
l−1
rl+1
1
共A10兲
共A11兲
共A15兲
Thus ␴共␻ , t兲 attains the form,
⬁
l
B
lm
⬘ t兲e−t/␶
Y lm共⍀兲 2 sin共␻0l
兺
a
l=2 m=−l
␴共⍀,t兲 = 兺
where Pl共cos ␥兲 is the Legendre polynomial 关Pl共cos ␥兲
ⴱ
l
Y lm共⍀兲Y lm
共⍀1兲兴, ␥ is an angle between
= 4␲ / 共2l + 1兲兺m=−l
vectors a = ar̂ and r1, and 共for a ⬎ r1兲,
0
1
+
兺
Q1m共t兲Y 1m共⍀兲
m=−1
1
⳵
2
2
冑
⳵ a a + r1 − 2ar1 cos ␥
⬁
⬁
⬁
l
兺
4␲
l=0 m=−l
⫻
l + 1 rl1
ⴱ
Y lm共⍀兲Y lm
共⍀1兲.
2l + 1 al+2
共A12兲
l
兺 qlm共t兲Y lm共⍀兲.
␴共⍀,t兲 = 兺
共A13兲
l=0 m=−l
From the condition 兰␴共␻t兲d⍀ = 0 it follows that q00 = 0. Taking into account the initial condition ␴共␻ , 0兲 = 0 we get 共for
l ⱖ 1兲,
qlm共t兲 =
Blm
⬘ t兲e−t/␶0共1 − ␦l1兲 + Q1m共t兲␦l1
sin共␻0l
a2
⬁
+ 兺 Almn
n=1
⫻
冕
a
0
dr1
共l + 1兲␻2p
l␻2p − 共2l + 1兲␻2nl
冕
a
0
Taking into account the spherical symmetry, one can assume
the solution of the Eq. 共A10兲 in the form,
⬁
⬁
兺兺
l=1 m=−l n=1
+兺
rl1
⳵
= 兺 l+1 Pl共cos ␥兲
⳵ a l=0 a
=−兺
l
ne
rl+2
1
⬘ t兲e−t/␶0 ,
jl共knlr1兲sin共␻nl
al+2
共A14兲
2
where ␻0l
− 1 / ␶20 and Q1m共t兲 satisfies the equation,
⬘ = 冑␻0l
dr1
Anlm
共l + 1兲␻2p
l␻2p − 共2l + 1兲␻2nl
rl+2
1
⬘ t兲e−t/␶0 .
jl共knlr1兲sin共␻nl
al+2
Y lm共⍀兲ne
共A16兲
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