Optical properties of noble-metallic nanoparticle chains
embedded in a graded-index host
K W Yu1,2,* and J J Xiao1
(1) The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
(2) Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin,
New Territories, Hong Kong, China
*kwyu@phy.cuhk.edu.hk
Abstract. We have studied plasmon resonance in chains of noble metal nanoparticles
immersed in a graded-index host. Within the nonretarded dipole approximation, we
calculate the effective linear and third-order nonlinear optical responses of silver and
gold nanoparticle chains. The absorption spectra exhibit a series of sharp peaks that
merge in a broadened enhancement band while the sharp peaks become a
continuous band for increasing chain length. These results suggest the existence of
coupled plasmon modes in the system. Moreover, these modes exhibit localizationdelocalization transition within the resonant plasmon band, which has been confirmed
numerically via the force oscillator method. We then examine the relationship between
the dipolar Mie resonance of isolated nanoparticles, the resonant band of an infinite
chain in a homogeneous host, and the resonant band of an infinite-size chain
embedded in a graded-index host with infinitesimal gradient.
There has been considerable interest in materials with sensitive and tunable optical responses. Noble
metals, typically gold, silver, and copper, have attracted significant attention as ingredients owing to
their unique optical properties. Many different microstructures have been exploited in an attempt to
access the intrinsically large and fast optical nonlinearity of metals, for example, in random
metallodielectric composites [1, 2], fractal films [2], and alternative bi-layers [3], etc. In these
structures, the enhanced local fields with respect to the applied field are generally associated with
surface plasmons (SPs), which are non-radiative evanescent electromagnetic eigenmodes, bounded at
metal-dielectric interfaces [4]. Appropriately designed metallic and metallodielectric nanostructures
can strongly localize and manipulate these plasmon modes and thus their interactions with light,
improving the efficiency of absorption and emission processes. In particular, interesting phenomena
occur in nanoscale structures at the plasmon frequency, at which optical responses are resonantly
enhanced. These have found increasing applications in integrated subwavelength optics, optical data
storage, solar cells, nanolithography, microscopy, and biosensing [4–6], as well as SP-mediated
fluorescence [7] and photoluminescence spectra [8, 9]. It is well known that SP resonances can be
dramatically altered by modifying the sample morphologies [10] such as shape, size and local
dielectric environment or surface chemical functionality of noble metal nanoparticles [11, 12], or by
organizing them in regular arrays such as dimers [13, 14] and chains [15], allowing controlled
coupling of SPs to other excitons [16, 17]. As improved nanofabrication methods now allow advanced
control of nanoparticle shape, arrangement of particle ensembles and patterning of metal layers,
substantial insights have been gained regarding the effects of these factors, much less is known,
however, about graded effects. That is, what are the consequences if there exists certain characteristic
gradient in plasmonic systems. In fact, many artificial optical metamaterials and elements with graded
features have been proposed and fabricated nowadays [18]. In this context, recently we have found a
giant enhanced optical resonant band in layered colloidal crystals within a graded-index host [19].
In this work, we examine the coupled plasmon modes of a chain of silver or gold nanoparticles
immersed in a host medium whose refractive index varies continuously along the chain. It turns out
that the presence of a graded host dramatically alters the SP modes and therefore their optical
properties, giving rise to tunable and controllable resonant spectra in these systems. Under certain
circumstances, there exist several sharp peaks in the resonant band, opening new avenue for
manipulating the SPs and improving SP-involved phenomena and applications, particularly in
metallic/organic hybrid system. This also offers exciting possibilities for controlling the optical
interaction between molecules in areas as diverse as photosynthesis and solid-state polymer lasers.
Let us discuss the dielectric functions of typical noble metals. The Drude form is overwhelmingly
adopted in the literature because it offers invaluable insights into behavior of real metals in near and
far-infrared regime. By adding Lorentzian resonance terms, one can account for the interband
transitions and the validity covers the entire visible wavelength range, i.e., below 500 nm. For gold
and silver particles with radius a  10 nm, bulk dielectric constant  ( ) apply. We adopt the
following function for their dielectric dispersions over range of photon energy between 0.5 – 0.65 eV
 ( )  1    
3
 p2
aj
 2
, (1)
 (  i0 ) j 1  0 j   2  i j
where i   1 and  is in unit of eV;   ,  p , 0 , a1, 2 ,3 ,  ( 01, 02, 03) and (1, 2 ,3) are fitting
parameters taking the empirical optical constants of Johnson and Christy [20]. Figure 1 shows the real
part and the imaginary part of the dielectric functions (curves) given by Eq. (1) and the experimental
bulk data (symbols) for both silver [Fig. 1(a)] and gold [Fig. 1(b)]. The data actually covers the
frequencies from microwave, to the visible, and the ultraviolet portions in the spectrum, while in the
present work we mainly focus on the visible range of frequencies. By using Eq. (1), the electron
scattering loss is underestimated for these small particles.
Figure. 1. Dielectric constants over
photon energy 0:5 to 6:5 eV (wavelength
from around 190 nm to 2480 nm) of
Johnson and Christy [20] (symbols) and
from the analytical functions (curves) of
Eq. (1). The imaginary parts are
multiplied by a factor of 10. (a) silver, (b)
gold.
Wavelength (100 nm)
1510
5
100
250
Dielectric function
(a)
50
200
0
150
(b)
Fitted real part
Fitted imaginary
Johnson & Christy
Johnson & Christy
100
-50
50
-100
0
-50
-150
-100
-200
-150
Ag
-250
Au
-200
0
1
2
3
4
5
Photon Energy (eV)
6
7
0
1
2
3
4
5
6
7
Photon Energy (eV)
For silver and gold particles of a  50 nm, the electric dipole moment dominates the higher-order
multipolar moments, and the absorption exceeds scattering as for optical extinction. To highlight the
essential physics in the structure, there is no need to employ large-scale numerical computation
approaches such as the finite difference time domain method or the dyadic Green-function technique
[14]. We simply invoke the coupled dipole equation [5] which reads as En 

N
m n
Tmn  pm  En(0) ,
( n  1,2,..., N ), where E n( 0 ) represent external electric fields in the host, p m is the dipole moment
vector of the m-th particle, and Tmn ( m  n ) labels the near field susceptibility of vacuum. In
Cartesian coordinates ( x, y, z ), its electrostatic limit components (  ,  ) are given by the general
relation Tmn ( ,  )  (3dmn, dmn,  | dmn |2   , ) / | dmn |5 . Here d mn is the distance vector from the n-th
particle to m-th particle. Note that for convenience, we can denote Tnn  0 . For small spherical
particles, the coupled dipole equation can be reduced by associating just one dipole per particle with
the frequency-dependent dipole polarizability which relates p n of the nth particle and the local field
En around it, and in fact consists of isotropic linear and nonlinear contributions.
That is p n   n ( ) E n   n | E n2 |E n / 3 where  n ( )   2( n ) a 3 [ ( )   2( n ) ] /[ ( )  2 2( n ) ] . Here
 2( n ) is the dielectric constant of the host around the n-th particle. In the case of weak nonlinearity
 1 | E n | 2 | Re[ ( )] | in the D-E relationship Dn   ( ) E n  1 | E n | 2 E n [21], one has
2
2


3 2( n )
3 2( n )
n  
a 3 1 .
(2)
(n) 
(n)
  ( )  2 2   ( )  2 2
It is noteworthy that the linear local fields E n around the n-th particles are actually obtained by
assuming no intrinsic nonlinear response, i.e., we set 1  0 for solving the self-consistent equations,
which is appropriate provided that the nonlinear responses are much less than the linear ones. Next we
use the resultant linear local fields E n to extract the enhancement factor of the effective nonlinear
susceptibility [21, 22]
|E n | 2 En2  n
1
,

 f
1
3 E04 1
(3)
where f is the volume fraction of the nanoparticles. The average  is taken over the nanoparticles.
We consider nanoparticle chains with nearest-neighboring distance d  3a in a graded host of
 2( n)   left    (n  1) /( N  1), (n  1,2,..., N ) . (4)
Figure 2. Schematic of band construction for
the case of graded host from the bands of the
same chain in various homogeneous hosts. (a)
Extinction spectra and (b) third-order
nonlinear
enhancement
for
silver
nanoparticles chain ( N = 15 ) in a graded host
(solid curves) with  2 varying as Eq. (4), and
in homogeneous hosts with  2 = 2.0 (dashed
curves) and  2 = 5.0 (dotted curves). (c)-(e)
Mode patterns for typical excitations in the
chain with increased length ( N = 50 ) in the
graded host. The figures are for longitudinal
(L) polarization.
In the case of longitudinal ( L ) polarization (i.e., the external fields parallel to the chain axis) or
transverse ( T ) polarization, the N coupled dipole equations are being able to be transformed into a
~
matrix form as E = T AE  E (0) . More precisely, in the longitudinal and the transverse cases,
E = {En : n = 1,2,, N} and E (0) = {E n(0) : n = 1,2, , N } are simply N-dimensional vectors and
A is N  N diagonal matrix of the polarizability  n ( ) with  2( n ) given by Eq. (4). In the non~
~
retardation dipolar model, Tmn = Tmn ( z, z)/ 2( n) and Tmn ( x, x)/ 2( n ) for the L and T polarization,
respectively, which approximately takes into account the screening effect due to the host and basically
depends on the periodicity d of the chain and the polarization (L or T).
Figures 2(a) and 2(b) show in logarithmic scale the extinction of the system and the nonlinear
enhancement factor  as defined by Eq. (3), respectively. The wavelength in these figures is for light
in free space. It is noteworthy that in the calculations, we assume that the dielectric constant of the
host around the n-th nanoparticle is approximately homogeneous, e.g., denoted by  2( n ) . The results in
Fig. 2 are for the longitudinal polarization. It is seen that the presence of a gradient in the host leads to
a broadened and giant enhanced resonant band (thick solid curves) in the spectra that falls between the
resonant peaks of the same chain immersed in homogenous host with  2 =  L = 2.0 (dashed curves)
or with  2 =  left    = 5.0 (dotted curves), indicating that the broadened resonant band for chain in
graded host in some sense stems from the hybridization of the non-graded cases. Notice that an
increased host dielectric constant leads to strong redshifts of the coupled plasmon resonant peaks due
to both reduced coupling between the particles and decreased dipolar Mie resonance  0 of isolated
particles, which occurs at  = 396 (  0 = 3.13 eV) nm and  = 509 nm (  0 = 2.43 eV) for
 2 = 2.0 and  2 = 5.0 , respectively. These two resonant frequencies are indicated by the two solid
vertical lines in Figs. 2(a) and 2(b). It is well known that the strong coupling inside an infinite chain
within a homogenous host of dielectric constant  2 not only redshifts the extinction peaks with
respect to that of isolated particles, but also expands the resonance at  0 to a band [23]
 2 =  02  2 ( L,T )12 cos(kd ) cosh (d ),
(5)
where k is the wave number of the plasmon wave,  is the attenuation coefficient, and  ( L ) = 2
and  ( T ) = 1 for the longitudinal and the transverse polarizations, respectively. Here the nearest
coupling strength 1  0.93/  2 eV and the second-order correction due to dissipations is
cosh (d )  1.0013 . These are very typical and realistic values for silver nanoparticles of diameter
a = 25 nm and periodicity d = 3a = 75 nm [23]. We therefore display the bands (shaded regions)
given by Eq. (5) in Figs. 2(a) and 2(b). We have specifically chosen the two extremities of the host
dielectric constant. That is, for (i)  2 =  left = 2.0 , the band is at around 365  437 nm
(  = 2.84  3.40 eV) and for (ii)  2 =  left    = 5.0 at around 482  541 nm (  = 2.57  2.29
eV). It is quite interesting that although the band expression [Eq. (5)] is for infinite-sized chain in
homogeneous host, the two vertical dashed lines in Figs. 2(a) and 2(b) which represent the upper
bound of (i) and the lower bound of (ii) approximately give the plasmon mode band boundaries of the
chain in (iii) a graded host characterized by Eq. (4). As a matter of fact, if the chain length (particle
number N ) increases, the gradient in the host becomes infinitesimal, and we can break up the graded
chain into many segments, each of which still represents an infinite chain but in a homogeneous host.
By the argument of graded coupled oscillators [24], the two vertical dashed lines give exactly the
lower and the upper bounds of the resonant band in case (iii). These are the most significant results in
this paper. In fact the L- or T-polarized case, dispersion of dipolar plasmon normal modes in the chain
can simply be determined by vanishing nonlinearity.
det{M ( )} = 0,
(6)
~
where M nn ( ) = 1/ n ( ), (n = 1,, N ) and M mn ( ) = Tmn , ( m  n ). In presence of loss and/or
~ , one can alternatively use the forced
radiation, instead of solving Eq. (6) for allowable complex 
oscillator method [25]. The mode pattern is represented by P = { pn : n = 1,, N} , illustrated in Figs.
2(c), 2(d), and 2(e), where the real part of P is plotted as a function of site index n . The frequencies
are chosen as  = 2.3 eV (  = 539 nm), 2.5 eV (  = 496 nm), 2.7 eV (  = 459 nm), 2.9 eV
(  = 428 nm), and 3.2 eV (  = 387 nm) according to the six vertical lines in Figs. 2(a) and 2(b), in
an attempt to exhibit their distinguished characteristics. Indeed, we clearly see that the modes with low
frequency [Fig. 2(c)] is spatially confined at the host side with large  2 , the high frequency mode [Fig.
2(e)] tends to residue at the host side with relatively small  2 , while the modes with intermediate
frequencies that fall fairly inside the middle part of the band are basically extended, as seen in Fig.
2(d). We emphasis that for full retarded dipolar model, Bloch theorem can be applied to extract
analytical dispersion relations for periodic lattices in both the longitudinal case and the transverse case
of one-dimensional infinite chains [26, 27]. However, it is hard to handle in similar way for the graded
case because of lack of translational symmetry. In this regard, a perturbation approach [28] may be
helpful for infinitesimal gradient. It is also possible to include retardation effect on the polarizability
 n ( ) for larger particles, for example, simply as 1/ n ( )  1/ n ( )  i 2 2 /3c 2 [26].
Notice that for the transverse case, the resonant band for homogeneous host and for graded host
are blueshifted with respect to their counterparts in the longitudinal case, respectively, as seen in Figs.
3(a) and 3(b), where the resonant band for the case (i) and the case (ii) is at around 379  414 nm
(  = 2.99  3.27 eV) and 495  524 nm (  = 2.36  2.50 eV), respectively. Although this
blueshift is well recognized in the case of homogeneous host, we easily prove and show that the
blueshift persists in the case of graded host. It is natural expected that a decrease in the interparticle
distance d (i.e., an increased 1 ) will cause further blueshifting and expansion of the spectra for the
transverse polarization, while opposite redshifting and expansion for the longitudinal polarization,
provided that the other conditions keep unchanged. In Figs. 3(a) and 3(b), the shaded regions represent
the bands predicted by Eq. (5) for homogeneous host of cases (i) and (ii), and our above argument on
the band for graded host applies to this case as well, which is confirmed by a careful examination on
transverse plasmon modes for silver nanoparticles in graded host.
Figure. 3. Same as Figs. 2(a) and 2(b), but (a) and (b)
for the transverse (T) polarization. (c) and (d) are for
gold nanoparticle chain in the case of longitudinal (L)
polarization.
Unlike silver, in which plasmon peak dominates the optical properties in the whole visible regime,
interband transitions also contribute largely for gold at short wavelengths. We repeat the calculations
for gold nanoparticles with the same parameters as in Fig. 2. The results are shown in Figs. 3(c) and
3(d) for the longitudinal polarization only. Although similarly broadened resonant bands are also
observed, they are substantially reduced, not as remarkable as those exhibited by silver nanoparticles
(see Fig. 2). This is perhaps due to the intrinsic dielectric properties of gold that interband transitions
lead to a large imaginary part in the dielectric function as seen in Fig. 1(b), which suppress the
resonances in both the cases of homogeneous host and graded host. Therefore, it appears that silver is
more favorable for the useful behaviors we reported. Retardation effect will be noticeable for a system
with increased dimensions so that the spectrum reported in this paper could be altered. Further work is
thus necessary to explore this regime. However, the results obtained in this paper using the
nonretarded approximation provide a first reference frame to initiate experimental works.
This work was supported by RGC Earmarked Grant of Hong Kong SAR Government.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
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