Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative index behavior

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by  E. V. Ponizovskaya

Ensembles of plasmonic nanospheres at optical frequencies and a problem of negative
index behavior
E.V. Ponizovskaya and A.M. Bratkovsky
Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, California 94304
(Dated: February 6, 2008)
7
Arraysofmetallicnanoparticlessupportindividualandcollectiveplasmonicexcitationsthatcon-
0
tribute to unusual phenomena like surface enhanced Raman scattering, anomalous transparency,
0
negativeindex,andsubwavelengthresolutioninvariousmetamaterials. Wehaveexaminedtheelec-
2
tromagneticresponseof dualKron’slattice andfilmscontaining uptothreemonolayers ofmetallic

n nanospheres. It appears that open cubicKron’s lattice exhibits ‘soft’ electromagnetic response but
a nonegativeindexbehavior. Theclose-packedarraysbehavesimilarly: thereareplasmonresonances
J
andveryhightransmissionatcertainwavelengthsthataremuchlargerthantheseparationbetween

9 theparticles,anda‘soft’magneticresponse,withsmallbutpositiveeffectiveindexofrefraction. It
would beinteresting to check those predictions experimentally.
]
l
l PACSnumbers: 78.20.Ci,42.30.Wb, 73.20.Mf,42.25.Bs
a
h
-
s I. INTRODUCTION wavelength resolution (perfect lens) [9]. The effect ap-
e
pears because ideal NIM slab supports surface plasmon
m
Media with strong dispersion of the refractive index modes that are in resonance with incident radiation at
t. may supportbackwardwaves,as is obvious fromthe fol- anyanglesofincidence[10, 11]. The incidentfieldpumps
a lowing relation established by Lord Rayleigh in 1877: those plasmon modes up and this lead to enhancement
m
of evanescent waves reaching the surface on the image
- dn side of the slab further from the source. The induced
d ng =n λ (1)
− dλ displacementcurrentsre-emitthe lightthat reconstructs
n
o where n is the group index for waves with wavelength the image of the source without loss of resolution for
g
c λ, phase velocity is v = c/n , group velocity v = c/n, features smaller than λ. Similar effect involving surface
[ g g plasmons is also responsible for extraordinary transmis-
where c is the light speed in vacuo. Hence, the group
sion in thin metallic hole arrays[12]. This behavior is
1 velocitymaybecomenegativeinasystemwithlargepos-
v itive dispersion. The structures that support backward quite fragile, however, and limited by losses and spatial
6 waveshavebeenknownsinceearly1900sandwidelyused dispersion (metamaterials granularity)[10, 13, 14]. Nev-
8 ertheless, it has indeed been demonstrated experimen-
in antenna and electronics technologies since 1950s. Me-
1 tally by Lagarkov and Kissel that the NIM slab built
diawherethedirectionsofphaseandgroupvelocitiesare
1
with SRR interspersed with the wire mesh is able to
0 opposite are known to produce negative refraction[1]. It
resolve features λ/6 in the source separated by for
7 was noted in Ref. [1] that e.g. dispersion of light close ∼
microwave radiation (f = 1.7GHz)[15]. Later on, the
0 to excitonic frequencies in solids can be negative. It is
/ also true of artificial metamaterials with strong spatial sub-wavelengthresolution ( λ/6) was demonstrated by
t ∼
a dispersion, like photonic crystals [2, 3]. Pafomov [4] and N. Fang et al. in the visible range using silver slab as
m a plasmonic medium[16]. The silver slab is not a ma-
Veselago [5] showed that backward waves can propagate
terial with both permittivity and permeability negative
- in isotropic medium with simultaneously negative per-
d (double negative), it only has ǫ < 0 as any metal at
mittivity ǫ and permeability µ, exhibiting negative re-
n frequencies below plasmon frequency, and µ > 0, but
fraction, inverse Doppler and Vavilov-Cherenkov effects.
o
the effect is still possible because the system is in qua-
c Currently, such metamaterials are called negative index
sistatic limit where the sign of permeability µ drops out
: metamaterials (NIM). Veselago has noticed that a slab
v of the result for image intensity[9]. Various papers de-
of NIM with thickness d would work as a flat lens. The
i
X lenscouldnotproduceanimageofadistantobject,since scribe metamaterials that show NIM-like response at far
infrared frequencies[17] and, most interestingly, in near-
r it cannot focus a parallel beam of light, but produces a
a optical and optical interval[18, 19, 20, 21, 22].
replica of the object if it is placed less than distance d
away from the nearest surface of the slab.
Recently,unusualpropertiesofNIMsystemsattracted
II. ELECTROMAGNETIC RESPONSE OF
considerable attention followed first theoretical [6, 7]
METALLIC NANOPARTICLE ASSEMBLIES
and then experimental [8] demonstration of feasibility
ofnegativerefractionby periodic metallic metamaterials
like periodic wire meshes producing negative ǫ [6] and The metallic periodic systems like ‘fishnet’ structure
split-ring resonators (SRR) giving negative µ [7] in mi- [18] support an infinite set (bands) of electromagnetic
crowaveregionof incident radiation. Pendryhas showed (EM) waves, ω = ωn, where k is the wavevector (quasi-
k
theoretically that ideal Veselago lens can produce sub- momentum), n = 1,2,... the number of the band (Flo-
2
quet mode). At small wavevectors k such a crystal can spheres embedded in an organic matrix. We see from
be characterized by effective permittivity ǫ and perme- Fig.1dthattherealisticcaseofAgspheresinthe dielec-
ability µ[23]. Because of strong dispersion, it is easy to tricmatrix the electromagneticresponseofthe systemis
find crystals where some of the higher bands (second or ‘soft’(Re(n) is small)butpositive. Correspondingly,the
third) support backward waves corresponding to nega- dual Kron’s lattice of Ag nanoparticles in vacuo shows
tive group velocity, see Refs.[2, 3]. If such a band exists only a weak negative index behavior, Re(n) 0.3 but
≈ −
alone in a particular (usually narrow) frequency range, it is overwhelmed by losses, since Im(n) Re(n). It is
≫| |
the crystalwouldoperate as a NIM at those frequencies. likely, therefore, that the weak negative index behavior
Eleftheriadeset al. haveconsideredrecentlyapossibility of Ag spheres in dual Kron’s lattice won’t be observable
of an isotropic 3D NIM crystals (3D transmission lines) in sub-wavelengthfocusing experiments.
that might support backward waves in the first band if To get more insight into electromagnetic (EM) re-
madeoflumpedcapacitorsandinductances,Ref.[24],see sponse of nanoparticle arrays, we have performed
Fig.1a. Thisisdirectlyfollowsfromthe3Dtransmission extensive Finite Difference Time Domain (FDTD)
line(TL)modelsuggestedbyKronforMaxwellequations modeling[27] of dual Kron lattices (Fig. 1) and close
in isotropic space with positive permittivity and perme- packed layers of spherical Ag nanoparticles (Figs. 2-4).
ability, ǫ,µ>0[25]. In the case of ‘positive’ medium the The dielectric constant of Ag was assumed to have a
dispersion at small k-vectors is positive for the lowest Drude form
energy band, where eigenfrequency ω = ωk is a growing ω2
functionofk. Itshouldbecomeadecreasingfunctionofk ǫ(ω)=1 p , (2)
aftera dualtransformationofthe TL.Indeed,this is the − ω(ω+iγ)
standard passband-to-stopband transformation, where
with ω = 9.04 eV, γ = 0.02 eV [28]. The bare Mie res-
the dispersion remains exactly the same after a substi- p
tution C ⇋ L with obvious replacement ω ⇋ ω′ = ω−1, onance of Ag nanoparticles would be at ωM =ωp/√3=
5.22 eV or wavelength λ = 238 nm[29]. The trans-
which means that the group velocity for the dual band M
′ ′ mission characteristics of the slabs have been calculated
changes sign: v = dω/dk > 0 v (ω ) < 0. The dual
g → g by FDTD and then used to estimate the effective index
transformation does indeed result in negative dispersion
of refraction n, permittivity ǫ and permeability µ from
in the doubly degenerate first band around the Γ point
complex scattering coefficients according to a standard
[24]. However, the Kron TL lattice contains a few ele-
procedure[30].
mentsperunitcelland,asaconsequence,thereisalsoan-
The calculated effective permittivity and permeabil-
other ‘spurious’ band present in the first Brillouin zone.
ity of the dual Kron’s lattice of Ag particles with radius
The3DTLcrystalcanbefairlywellimpedancematched
R=100nm show a series of sharp resonances, Re(ǫ)<0
to free space in GHz range andit supports a backward
∼ at λ>0.8µm, and Re(µ) becomes negative at λ<1µm.
wave. The ‘spurious’ band can couple to highly atten-
InthesameregionRe(ǫ)isalsonegative,andweseethat
uated evanescent waves, but losses may be the limiting
the material has a small negative index at λ 0.9µm,
factor in subwavelengthresolution experiment with dual ≈
where Re(n) 0.3. However, the losses are very large
Kron’s TL lattice rather than the coupling to the ‘spuri- ≈ −
there, Im(n) 1.7, and this will likely preclude sub-
ous’ band[24]. ≈
wavelengthresolutionwiththelensmadewiththismeta-
It would be interesting to find an implementation of material. Large losses seem to be quite general for sys-
the dual Kron 3D transmission line model for NIM in temsofmetallicnanoparticles[26]andmayseverelylimit
visible range. With this in mind, Engheta et al. have their use for the purpose of sub-wavelength resolution.
speculated that Ag nanoparticles can play a role similar It is easily understoodthat the nanospheresshould be
to lumped inductance at optical frequencies. One can almost touching to increase capacitive coupling between
try, forinstance, substituting lumped inductances by Ag them. To see this, we considered Kron’s lattice with Ag
nanoparticlesinthedualKronlattice,hopingthattheca- nanoparticles with radius R = 300nm with the gap of
pacitivecouplingbetweentheparticleswouldbringabout 300nmbetweenthen,embeddedinamaterialwithdielec-
thesamebandstructureasin3DTL[26],seeFig.1b. The tric constant ǫ = 6.5. In this case the index remained
m
nanoparticle system in Fig. 1b is a poor representation positive at all wavelengths of interest, Re(n) 2.5 at
≈
of 3D TL system shown in Fig. 1a: although one does λ=1.2 3.2µm, since the displacementcurrentwerein-
−
have both electric and magnetic response from metal- effective in producing Re(µ)<0, and Re(ǫ) only dipped
licnanoparticlesinacfield,theycannotbeconsideredas below zero at a couple of very narrow intervals around
lumpedcircuitelements. Indeed,theinteractionbetween resonant wavelengths. Since the volume fraction of the
particles beyond nearest neighbors is likely important in metalwasquite small,the losseswere minimal, typically
open structures like Kron’s cubic lattice, in addition to Im(n)<0.3. To getNIM behavior,one needs to sharply
their non-negligibleself-capacitance. We havecalculated increase the metal fraction, but this immediately leads
thetwocases: (i)latticeofAgspheresinvacuo(dielectric to prohibitively large losses, as described above.
constant of the matrix is unity, n = 1, and (ii) dielec- We have also studied arrays of Ag nanospheres with
m
tric constantof the matrix is n =1.4. The secondcase radius R = 30nm close packed into one, two, and three
m
correspondsto experimentally accessible case of metallic monolayers,Figs.2-4. Suchsystemscanbeprepared,for
3
instance,byLangmuir-Blodgetttechnique. Theparticles
havebeenstackedinatriangularclosepackedmonolayer
(A), close packed bilayer with AB packing (B is a trian-
gularlayerofnanospheresplacedontopoftheAlayerso
that the top sphere touches three underlying spheres in
the layer A). Note that there are ‘see-through’ channels
ofdeepsub-wavelengthsizeineitherAmonolayerorAB
bilayerthatfacilitate hightransmissionofatλ 1.1µm,
∼
Figs. 2-4. The effect is apparently similar to extraordi-
nary transmission of light through sub-wavelength hole
arrays in metallic films[12]. It is worth comparing this
with Kron’s system, Fig. 1b, which is an open cubic lat-
tice and has high transparency even when a slab with 3
unit cells is considered. The transmission remains high
4 evenfortouchingorslightlyoverlappingnanoparticles,so
(d) Im(n) the assembly is transparent and conductive at the same
3 time. Quiteexpectedly,thetransmissionofABbilayeris
reduced in comparison with a monolayer,from T 80%
≈
2 to about 60%, but it still remains very large.
n Re(n), nm=1.4 System with three layers of nanospheres packed in
1 ABC sequence characteristic of face-centered cubic lat-
tice (fcc) is interesting, because there is no ‘see-through’
0 channelsinABCstack. However,eveninthiscasetrans-
Re(n), Ag spheres in vacuo
mission exceeds 40% at the resonance, Fig. 4. We see
-1 from Figs. 2-4 that one to three monolayers of close
0.5 1.0 1.5 2.0 2.5 packed metallic Ag nanospheres produce an extraordi-
Wavelength ( m) nary transmission in the vicinity of λ = 1.1µm, which
is much larger than the radius of the spheres R and the
lattice spacinga.Itis alsomuchlargerthanthe Mie res-
5
onant wavelength λ =238 nm. Since in ABC trilayers
(c) nm=1.4 there is no open “chMannels” for light to squeeze through
Im( )
the film, the explanation of unusual ‘transparent metal’
behavior lies in the fact that the incident light strongly
couplestoanarray. Strongcouplingofthe incidentlight
, 0 Re( ) is obviously facilitated by periodic ‘roughness’ of the ar-
rays, since the (quasi)momentum conservation is easier
Im( ) to meet. It would be interesting to see how surface plas-
monpolaritonsaresupportedbyarraysofnanoparticles,
Re( )
in other words, what kind of surface plasmon waveguid-
-5 ing is possible with arraysof nanoparticles consideredin
0.5 1.0 1.5 2.0 2.5
the present paper[32].
Wavelength ( m)
Importantly, electric field concentrates in the region
where the spheres touch, especially when the centers of
FIG. 1: Schematic of dual Kron lattice [24] (a) approxi-
the spheres are oriented along a polarization of incident
mated by the cubic lattice of Ag nanoparticles with radius
electric field, which looks analogous to corresponding
R = 100 nm and lattice period a = 300nm (b). Particles
electrostatic problem of two close metallic spheres. In
themselves show some inductance (marked by the coil sign)
fact,thelocalelectricfieldenhancementexceedsafactor
and are capacitively coupled to each other, as schematically
indicated in the graph (b). There is some resemblance to of η 30, which would facilitate strong Raman signal if
∼
the lattice of lumped elements[24] (a). (c) The effective per- thespecieswerepositionedontheparticlesnearthefield
mittivity ǫ(ω) and permeability µ(ω) of a slab (2 u.c.) of Ag peaks (Raman enhancement factor η4 106).
∼
nanoparticles(b). (d)Realandimaginarypartsofindexn,for In terms of electromagnetic response, there is a clear
Kron’slatticeinvacuoandinmatrixmaterialwithnm =1.4. topological difference between a monolayer and a few-
Ag lattice in the matrix shows some ‘softness’ at λ ≈0.9µm
layer systems. Indeed, at normal incidence the magnetic
[smallRe(n)],whereassysteminvacuoexhibitsaweaknega-
fieldintheincidentwavecannotexciteanydisplacement
tiveindexbehaviorwithRe(n)≈−0.3 butIm(n)≫ |Re(n)|.
currentsthat wouldfor a close loops,since the field does
not ‘see’ any such loops. In AB and ABC layers there
arethree-memberedloopsinthe fcc lattice thatcanpro-
ducesuchamagneticresponse. Asfollowsfromthe data
4
1.0 1.0
1 layer of spheres
A
0.8 0.8
R
T
R, A 0.6 R R, A 0.6 T
T, T, 2 layers of spheres
0.4 0.4
0.2
0.2
A
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength ( m)
Wavelength ( m)
3
10
Im( )
0 (a) 2 Im(n)
-10 Re( ) n
1
2
(b)
1 Re( )
Re(n)
0
0
-1 Im( )
-2 0.5 1.0 1.5 2.0 2.5 3.0
Wavelength ( m)
2 Im(n)
(c)
n 1
FIG. 3: (Top panel) Transmission T, reflection R, and ab-
0 Re(n) sorption A of a two (AB packing sequence) layers of closely
packednanosphereswith radiusa=30nm. Noteaverylarge
0.5 1.0 1.5 2.0 2.5 3.0 transmission atλ=1.1um,muchlargerthantheperiodofthe
wavelength ( m) monolayer, λ ≫ a. (Bottom panel) The effective index n of
the bi-layer of Ag nanoparticles. We observe a transmission
in excess of 50% at λ=1.1µm.
FIG. 2: (Top panel) Transmission T, reflection R, and ab-
sorptionAofamonolayerofcloselypackednanosphereswith
radius a = 30nm. Note a very large transmission T ≈ 0.75
bly of metallic nanoparticles can exhibit effective nega-
at λ=1.1um, much larger than the period of the monolayer,
tive index behavior[26]. We have foundthat dual Kron’s
λ≫a. (Bottompanel)Permittivityǫ(a),permeabilityµ,and
latticeofcloselyspacedAgnanoparticlesdoesnotexhibit
theeffective index n of the monolayer of Ag nanoparticles.
negative index behavior while embedded in a matrix, as
we have shown for a matrix with n = 1.4. Kron’s lat-
m
tice in vacuo may show a weak negative index behav-
in Figs. 3,4, we do see some ‘softness’ in magnetic re-
ior. However, it is overwhelmed by losses that would
sponseoftheABandABCarraysinRe(µ)atλ.0.8µm,
preclude sub-wavelength resolution with a slab of this
Fig.2b. Atthesametime,theindexforthemonolayeris
metamaterial. We also made an extensive FDTD study
positive and does not show any particular ‘softness’, cf.
of close-packed fcc-like arrays of Ag nanospheres. Ap-
Fig. 2.
parently, they do support collective plasmon excitations
andextraordinarytransparency. Evenintheopaquefcc-
like trilayer of Ag nanospheres, where there are no see-
III. CONCLUSIONS
through channels for light, the plasmons do transfer the
excitation from the front to back surface and re-emit it,
We have analyzed various arrays of metallic nanopar- providingforhightransparencyinexcessof40-50%. Ina
ticles in search for an isotropic negative index be- monolayerofAgparticlesthetransparencyisinexcessof
havior that was envisaged for dual Kron’s cubic lat- 80%. Interestingly,the monolayerdoesnotshowelectro-
tice of lumped circuit elements (3D transmission line magnetic ‘softness’ in response, whereas in both bi- and
model)[24]. In particular,we wanted to see if the assem- tri-layers(ABandABCpackedfilms)wehavefoundthat
5
Re(n) 0.3 at λ 0.7µm. Although the arrays do not
≈ ≈
behaveas negativeindex media, we see anintriguingbe-
1.0
havior characteristic of a ‘transparent metal’. All those
predictions would be very interesting to test experimen-
0.8 A R
tally.
A
R, 0.6
T, T
0.4 3 layers of spheres
0.2
0.0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength ( m)
3
2 Im(n)
n
1
Re(n)
0
0.5 1.0 1.5 2.0 2.5 3.0
Wavelength ( m)
FIG. 4: (Top panel) Transmission T, reflection R, and ab-
sorptionAofathree(ABCpackingsequence)layersofclosely
packed nanospheres with radius a = 30nm. (Bottom panel)
The effective index n of thebi-layer of Ag nanoparticles. Al-
thoughthereisno‘see-through’channelsinthisstructure,we
still observea large transmission at λ=1.1µm.
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