Schmid et al. Nanoscale Research Letters 2014, 9:50http://www.nanoscalereslett.com/content/9/1/50
NANO EXPRESS Open Access
Plasmonic and photonic scattering and nearfields of nanoparticlesMartina Schmid*, Patrick Andrae and Phillip Manley
Abstract
We theoretically compare the scattering and near field of nanoparticles from different types of materials, eachcharacterized by specific optical properties that determine the interaction with light: metals with their free chargecarriers giving rise to plasmon resonances, dielectrics showing zero absorption in wide wavelength ranges, andsemiconductors combining the two beforehand mentioned properties plus a band gap. Our simulations are basedon Mie theory and on full 3D calculations of Maxwell’s equations with the finite element method. Scattering andabsorption cross sections, their division into the different order electric and magnetic modes, electromagnetic nearfield distributions around the nanoparticles at various wavelengths as well as angular distributions of the scatteredlight were investigated. The combined information from these calculations will give guidelines for choosingadequate nanoparticles when aiming at certain scattering properties. With a special focus on the integration intothin film solar cells, we will evaluate our results.
Keywords: Nanoparticles; Plasmonics; Photonics; Scattering; Near field; Mie theory; FEM simulations; Solar cellsPACS: 42.70.-a; 78.67.Bf; 73.20.Mf
BackgroundThe study of light scattering from small particles goes backfor more than a hundred years, as shown by the early the-ory by Mie in 1908 [1], but applications have been knownsince much longer, see for example the Lycurgus cup [2].Currently, nanoparticles find widespread applications inelaborate technologies - and they also require elaborate se-lection and tuning for each of the individual applications.The specific scattering of nanoparticles was shown to bebeneficial for enhanced outcoupling from LEDs [3], innano-waveguides [4] or nano-antennas [5]. The enhancednear fields are exploited, e.g., in Raman spectroscopy [6],near field optical microscopy [7], or biosensing [8].Another promising application for plasmonic and pho-
tonic nanoparticles is in photovoltaic devices for absorptionenhancement. Both metallic and dielectric nanoparticleshave been used for this purpose: Ag nanoparticles in Sisolar cell [9,10], Au and SiO2 on Si [11], SiO2 on Si [12],Ag on GaAs [13], Ag in organic solar cells [14], Ag in dye-sensitized solar cells [15], etc. There appears to havebeen a strong focus on Ag nanoparticles, yet also SiO2
* Correspondence: [email protected] Berlin für Materialien und Energie, Nanooptical Conceptsfor PV, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
nanoparticles are growing in interest. According to[16,17], several mechanisms have to be taken into accountwhen considering plasmonic nanoparticles for solar cellapplications: enhanced near fields, high (angle) scattering,and in the case of regular arrangements, coupling intoguided modes. Also, the dielectric nanoparticles comewith their specific promises for expected enhancement[18,19]. But which nanoparticle material will provide themost efficient light coupling?In a solar cell, the objectives for nanoparticle applica-
tion are as follows: in ultra-thin or low-absorbing photo-voltaic materials, plasmonic and photonic nanoparticlesare expected to enhance the absorption. This can beachieved by various mechanisms which ideally can becombined or for which the most promising one needs tobe identified. Firstly, nanoparticles may be able to locallyconcentrate light into their vicinity, i.e., generate a near-field enhancement, which then can lead to enhancedabsorption in a surrounding medium. Secondly, theyscatter light and therefore are able to redirect the ini-tially incident light for preferential scattering into thesolar cell, similar to traditional anti-reflection coatingsor back reflectors. Thirdly, the scattered light is ideallyscattered into modes that are otherwise subject to total
an Open Access article distributed under the terms of the Creative Commonsg/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionroperly cited.
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reflection (being related to a high angular scatteringdistribution) which leads to light trapping in a thinlayer. Finally, strong fields at interfaces can also lead toleaky modes enhancing the absorption in the vicinitysimilarly to the near fields.With the aim of judging which type of material is the
most promising one for the desired absorption enhance-ment, we compare the absorption and scattering behav-ior of different materials, each of which is characterizedby a particular refractive index. The task is to find howthe optical properties will influence the plasmonic/pho-tonic scattering behavior and how we need to tune ac-cording parameters. We compare metals and dielectricsbut will also address semiconductors, since for examplethe scattering of silicon nanoparticles has started to at-tract interest [20].
MethodsMie theoryWe calculate the elastic interaction of an electromag-netic wave with a homogenous spherical particle usingthe Mie solution to Maxwell’s equations. The Mie theorygives the scattered external (scattering, extinction) andinternal field of the particle (absorption, field penetrationinside the sphere). The matrix form can be used to showthe relation between incident (subscript I) and scattered(subscript S) fields:
EjjSE⊥S
� �¼ ei
2πλ res
−i 2πλ resS2 S3S4 S1
� �EjjIE⊥I
� �ð1Þ
Where res is the resulting vector of the far field, S is theamplitude scattering matrix, and λ is the wavelength of theincident light with the electromagnetic wave componentsE∥ I and E⊥ I. The scattering amplitudes can be solved for asphere with S3 = S4 = 0. However, the result of the scatter-ing amplitudes S1 and S2 will still depend on the scatteringangle and azimuthal angle. For the calculation in the Miesimulation of nanoparticles with variable radius, we con-centrate on calculating the cross section with the Mie coef-ficients, which will no longer depend on the scatteringangles. First, we calculate the Mie coefficients for the exter-nal field in an infinite and homogenous medium [21]:
al ¼ n~ λð Þψl n~ λð Þxð Þψ0l xð Þ−ψl xð Þψ0
l n~ λð Þxð Þn~ λð Þψl n~ λð Þxð Þξ 0
l xð Þ−ξ l xð Þψ0l n~ λð Þxð Þ ð2Þ
bl ¼ ψl n~ λð Þxð Þψ0l xð Þ−n~ λð Þψl xð Þψ0
l n~ λð Þxð Þψl n~ λð Þxð Þξ 0
l xð Þ−n~ λð Þξ l xð Þψ 0l n~ λð Þxð Þ ð3Þ
Where l is the mode, x is the size parameter x ¼ 2πλ r,
and ñ is the complex reflective index. To simplify theformulas for calculation, the Riccati-Bessel functions ψl(p)and ξl(p) are used. We can calculate the scattered
field by using the boundary conditions and adding upthe resulting wave vectors of the particle scattering lead-ing to the scattering cross section Csca and the extinctioncross section Cext:
Csca ¼ λ2
2π
X∞l¼1
2l þ 1ð Þ alj j2 þ blj j2� � ð4Þ
Cext ¼ λ2
2π
X∞l¼1
2l þ 1ð ÞRe al þ blf g ð5Þ
The absorption cross section Cabs results as
Cabs ¼ Cext−Csca ð6ÞThe normalized cross sections Q - which we will show
in the following - are calculated by dividing C throughthe particle area πr2. The different modes and the separ-ation of the electric and magnetic field is done by the in-dividual calculation of al and bl with l for any relevantnumber (e.g., 1, 2, 3, 4,…).The scattering efficiency is defined as
Qeff ¼Qsca
Qsca þ Qabsð7Þ
3D FEM calculationsWe solve Maxwell’s equations in full 3D with the finiteelement method (FEM) using the software packageJCMwave, Berlin, Germany [22]. The FEM is a vari-ational method whereby a partial differential equation issolved by dividing up the entire simulation domain intosmall elements. Each element provides local solutionswhich, when added together, form a complete solutionover the entire domain. Due to the inherently localizednature of the method, different regions of space can bemodeled with different accuracy. This allows demandingregions like metallic interfaces to be calculated with ahigh accuracy without compromising on total computa-tion time.The time harmonic ansatz along with the assumptions
of linear, isotropic media and no free charges or currentsallows Maxwell’s equations to be written as a curlequation:
1ε∇� 1
μ∇� E − ω2E ¼ 0 ð8Þ
Where ε and μ are the permittivity and the permeabil-ity of the medium respectively, E is the electric field vec-tor, and ω is the frequency of the electromagneticradiation. This equation can be solved numerically bydiscretization of the curl operator (∇×) using the finiteelement method. After the discretization, a linear systemof equations needs to be solved to calculate the field
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scattered by the geometry in question. During our calcu-lations, the finite element degree and grid discretizationwere refined to ensure a convergence in the scatteringand absorption cross sections to the 0.01 level.For the calculation of normalized scattering and ab-
sorption cross sections, the Poynting flux of the scat-tered field through the exterior domain and the net totalflux into the absorbing medium were used. The normal-ized cross section is then:
Q ¼ Φ
ΦI=CN:P:
CC:D:ð9Þ
Where Φ is the scattered or absorbed flux, ΦI is theincident flux, and CN.P. and CC.D. are the cross-sectionalarea of the nanoparticle and computational domain, re-spectively. The calculation of the angular far field spectrumis achieved by an evaluation of the Rayleigh-Sommerfelddiffraction integral.In our calculations, Mie theory was mainly used for
extraction of scattering and absorption cross sections,their division into the different order electric and mag-netic modes and their representation as maps of wave-length and nanoparticle radius. The electromagneticnear fields and the angular distributions of scatteredlight were preferentially calculated with 3D FEM simula-tions. Whereas Mie theory is a fast calculation method,it cannot handle nanoparticles at an interface which wewill address in our last chapter. The comparison of thetwo calculation approaches for the simple case of a nano-particle in vacuum (air) gives us confidence about the con-formity of the two methods where possible. If not statedotherwise, a spherical nanoparticle in air is investigatedand cross sections are always the normalized values.
Dielectric function of materialsFor the above mentioned calculation methods alongwith the particular geometry, the optical constants ofthe materials, i.e., the dielectric functions, are thefundamental input parameters. Therefore, we nowbring together the essentials of describing the dielec-tric function of a material which we will use in thefollowing.The dielectric function ∈ = ∈ 1 + i ∈ 2 relates to the
refractive index ñ = n + ik as
�¼ ne2 ð10ÞThe dielectric function of a material strongly depends
on its electronic states: metals are dominated by freeelectrons whereas dielectrics have no free movablecharges and semiconductors are characterized by a bandgap plus possibly free charge carriers. The correspondingdielectric functions are often times described by modelsof which the most common ones are summarized below:
Metals - Drude formula
�¼ 1−ω2p
ω2 þ iωγð11Þ
With the damping γ and the plasma frequency ωP re-lated to the free charge carrier concentration ne and theeffective mass m* by ℏ
ωp ¼ffiffiffiffiffiffiffiffiffiffiffinee2
m��0¼ Ep=ℏ
sð12Þ
Whereas the plasma frequency relates to a property of a bulkmaterial, for a spherical nanoparticle with radius r madefrom a material that can be described by the Drude formula,the resonance conditions for particle plasmons given by∈ = −2 may be fulfilled. This condition results from the po-larizability α which is derived for small particles [21] as
α ¼ 4π�0r3 �−1�þ2
ð13Þ
Metals may also show significant interband transitionsand related absorption which can be described by aLorentz oscillator compare also the semiconductors.
Dielectrics - Cauchy equations
n2 ¼ 1þ B1λ2
λ2−C1þ B2λ
2
λ2−C2þ B3λ
2
λ2−C3ð14Þ
With the Sellmeier coefficients B1, 2, 3 and C1, 2, 3. TheCauchy equation can be approximated by a constant re-fractive index value for longer wavelengths.
Semiconductors - Tauc-Lorentz modelCombine the Tauc joint density of states with the Lorentzoscillator model for ∈ 2:
�2 ¼AE0C E−Eg
� �2E2−E2
0
� �2 þ C2E2⋅1E
; E > Eg
0 ; E < Eg
8><>: ð15Þ
and ∈ 1 is defined according to the Kramers-Kronig relation
�1 Eð Þ ¼ �1;∞ þ 2πPZ∞
Eg
ζ�2 Eð Þζ2−E2
dζ ð16Þ
For the presence of significant free charge carriers inthe semiconductor, the Tauc-Lorentz model can be com-bined with the Drude formula.For the materials used in the following calculations,
we give the parameters used to fit the dielectric functionwith the models above in Table 1 and show the refract-ive index (n,k) in Figure 1.
Table 1 Fitting paramaters for the materials used in the calculations
A (eV) C (eV) E0 (eV) Eg (eV) ∈ 1,∞ Ep (eV) γ (eV)
Fitting parameters according to Equations 15 and 16 (A, C, E0, Eg, ∈ 1,∞) and Equations 11 and 12 (Ep, γ) for the materials used in the calculations.
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Results and discussionWe start with investigating the scattering and near fieldsof metallic nanoparticles and later contrast them tothose from dielectric particles. These considerations willfurther lead us to address nanoparticles made fromsemiconducting materials. To finally evaluate the effi-ciency of the nanoparticles’ scattering for light trappingpurposes, we will address the angular distribution of thescattered light including the consideration of a substrate.
MetalsThe dielectric function of a metal being characterized bythe free electrons can, in wide ranges, be described bythe Drude formula (see Equation 11). As a metal, Agwas chosen, which is the most popular material for plas-monic application since it has a low absorption in thevisible region. A fit to the Drude equation with plasmafrequency as given in Table 1 results in a good approxi-mation of Ag data from Palik [23] in the wavelengthrange above 300 nm; below interband transitions existwhich cannot be reproduced with this model (compareFigure 1a). In Figure 2, the scattering cross section Qsca
and the scattering efficiency Qeff are shown in subfiguresa and b, respectively, for a Drude-fitted Ag sphericalnanoparticle in air. These maps of scattering efficiencyas a function of wavelength and particle radius canquickly be calculated based on Mie theory. They allowthe estimation of the required particle size for most ef-fectively exploiting the scattering while having a lowparasitic absorption and for tuning the resonance fre-quency to the desired wavelength range. From Figure 2,we can see that nanoparticles with a radius of <50 nm
Figure 1 Refractive index (n,k) of the materials used in the calculationTauc-Lorentz fit, and (d) GZO with combined Tauc-Lorentz and Drude fit; f
are subject to strong absorption, whereas nanoparticleswith r = 50 nm are already dominated by scattering. Therelated resonance wavelengths however appear at λ <500 nm. In terms of the application to devices whichmainly work in the visible range of light, a shift of themain resonance to λ approximately 700 nm is desirableand can be achieved by choosing bigger nanoparticles -r = 120 nm appears a good choice judging from the mapsin Figure 2.Figure 3a shows the according scattering cross section
of a 120-nm radius nanoparticle from Ag with dielectricfunction fitted according to the Drude model. The sumas well as the division into the individual order modes isgiven. The main resonance at λ approximately 700 nmcan be attributed to the dipole electric mode, the domin-ant peaks at shorter wavelengths related to the quadru-pole, the hexapole, and the octopole electric mode. Wewant to note that for the metallic nanoparticles, the res-onance peaks result from maxima of the electric modes.Magnetic modes only appear at shorter wavelengths andare much less pronounced. Comparing the scatteringto the absorption cross section (see Additional file 1:Figure S1), the lower order modes, i.e., especially thedipole mode, are more favorable for efficient scatter-ing. The near field distributions of the electromagneticfield around the nanoparticle are given in Figure 3b atthe peak wavelengths of the dominant electric modes.Since the nanoparticle investigated is of metallic na-ture, we find no strong electromagnetic field inside theparticle where the free charge carriers can compensatelocal fields. However, the metal fulfills the particleplasmon resonance condition (see Equation 13), and
s. (a) Ag with Drude fit, (b) a-Si with Tauc-Lorentz fit, (c) AZO withitting parameters according to Table 1.
Figure 2 Scattering maps for metallic nanoparticles. (a) Scattering cross section and (b) scattering efficiency of a spherical Ag nanoparticlewith refractive index data obtained from a Drude fit to data from Palik.
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the related plasmonic collective oscillations of the elec-trons cause strong electromagnetic fields to build uparound the surface of the nanoparticle which are charac-terized by knots according to the respective order. Aslightly stronger electromagnetic field in the forward dir-ection is the result of interference with the incident light.
DielectricsDielectrics show an imaginary part of the refractive indexwhich is zero, i.e., no absorption, which makes them favor-able to be used as the material for scattering nanoparticles.The main question is whether these dielectric nanoparticlescan give scattering cross sections comparable to the ones ofmetallic nanoparticles. The refractive index of a typicaldielectric is often times described with a Cauchy model,yet since it is constant over a wide wavelength range, we ap-proximate it with n = const (=2 here) and k = 0. We choosen = 2 since the value is a compromise for the most popularoxides SiO2 (n approximately 1.5) and TiO2 (n approxi-mately 2.5) or also Al2O3 (n approximately 1.7) andZrO2 (n approximately 2.2). Si3N4 would be the examplewith the direct value n approximately 2 in the wavelengthrange above 400 nm.Scattering cross section maps (the absorption cross
sections always being zero) again give guidelines for anadequate radius in order to obtain the main scatteringresonance at λ approximately 700 nm (see Additionalfile 2: Figure S2). This requirement is fulfilled for the di-electric nanoparticle (in air) with n = 2, k = 0 for a radiusof 170 nm which is distinctly larger than in the case ofmetallic nanoparticles (r = 120 nm). Figure 4a representsthe total scattering cross section with the main reson-ance around 700 nm together with the division into thedifferent order electromagnetic modes which are mani-fold for this medium-sized nanoparticle. As Figure 4ashows, the magnetic modes dominate the peaks ofthe scattering cross section and the electric modes
contribute in the form of a broader background. Themaximum scattering cross section reaches a value ofnearly 6 which is the same as for the 120-nm radiusDrude-fitted Ag nanoparticle. From this point of view,the dielectric nanoparticles appear to perform equallywell or, considering the zero absorption, even better thanthe metallic ones. Looking at the near fields of the dom-inant resonance modes (Figure 4b), however reveals dis-tinct differences: the magnetic modes of the dielectricnanoparticles appear to localize the electromagnetic fieldinside the particle and the direction of light extractionseems to be preferential to the direct forward direction,i.e., the dielectric nanoparticle appears like a lens. There isa strong near field in this direction in contrast to theremaining surface of the nanoparticle. We will come backto a detailed comparison of the angular distributions ofthe scattered light in a later section. Here, we only recordthat dielectric nanoparticles are characterized by a strongscattering, yet not by a pronounced near field enhance-ment around the particle.
SemiconductorsAfter having seen both the benefits of the metallic as well asof the dielectric nanoparticles, we move on to consideringnanoparticles of semiconductor material which might com-bine the two particular properties of free charge carriersand an area of approximately zero absorption. In the case ofa semiconductor, furthermore, its band gap needs to beconsidered which can be achieved using the Tauc-Lorentzcombined density of states and an oscillator model.In our investigations, we address three different semi-
conductors: amorphous silicon (a-Si), Al-doped ZnO(AZO), and Ga-doped ZnO (GZO). The refractive indexdata was fitted using parameters from [24,25] for a-Si,from [26] for AZO, and from [27] for GZO, see Table 1.Only the latter one has a significant free charge carrierconcentration according to the parameters used here,
Figure 3 Scattering and near fields of a metallic nanoparticle. (a) Scattering cross section of a 120-nm radius Ag nanoparticle with dielectricfunction according to a Drude fit; sum and allocation to different order and electromagnetic (E/M) modes. (b) Near field distribution of theelectromagnetic field around the nanoparticle for the dipole, the quadrupole, the hexapole, and the octopole electric mode at wavelengthsof 688, 426, 340, and 298 nm, respectively, which correspond to the maxima in scattering (incident light from the top).
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which leads to a pronounced plasmon resonance; the di-electric function of a-Si and AZO is simply characterizedby the band gap and the constant refractive index at lon-ger wavelengths, see also Figure 1b,c,d.Figure 5 compares the scattering efficiencies for spher-
ical nanoparticles (in air) from the three semiconductorswhich are characterized by a band gap around 800 nm(for a-Si) and 400 nm (for AZO and GZO). For wave-lengths below the band gap (i.e., in terms of energyabove), the absorption is dominant, and thus scatteringcan only be exploited for wavelengths well beyond theband gap. Since this is the case above 1,000 nm only for
Figure 4 Scattering and near fields of a dielectric nanoparticle. (a) Scaindex n = 2 and k = 0; sum and allocation to different order and electromagfield around the nanoparticle for the dipole, the quadrupole, the hexapole,and 322 nm, respectively, which correspond to the maxima in scattering (i
the a-Si nanoparticles, they cannot be expected to per-form well in a device operating in the visible wavelengthrange. The band gap has to be chosen as low (in wave-lengths, but high in energy) as possible. For AZO, thescattering efficiency is 1 for wavelengths larger than theband gap at around 400 nm making it comparable to adielectric. This is not surprising since low-doped semi-conducting materials far away from a specific resonancewill show dielectric-like behavior. Comparing a dielectricnanoparticle to one made of a low-doped semiconductor,the latter loses in terms of scattering efficiency since itshows parasitic absorption below the band gap.
ttering cross section of a 170-nm radius nanoparticle with refractivenetic (E/M) modes. (b) Near field distribution of the electromagneticand the octopole magnetic mode at wavelengths of 700, 502, 392,ncident light from the top).
Figure 5 Maps of scattering efficiency for semiconductor nanoparticles. Spherical particle made from (a) a-Si, (b) AZO, and (c) GZO withrefractive indices fitted with parameters from [24,25], [26], and [27], respectively (note the different wavelength range in (c)).
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For the highly doped semiconductor, the situation isslightly different. Also here, parasitic absorption domi-nates for wavelengths below the band gap. But additionally,the free charge carriers of the highly doped semiconductorlead to further parasitic absorption in the wavelength rangewhere they become dominant, compare Figure 5c (and alsosee the Additional file 3: Figure S3 for the individual ab-sorption and scattering cross sections). Yet, they also giverise to a plasmonic resonance since the according require-ments for the refractive index (∈ 1 = −2) can be fulfilled.For GZO, the conditions are met at λ approximately2,000 nm so that a further resonance occurs here. Thispeak can be attributed to the dipole electric mode asshown in Figure 6 where the sum of the scattering crosssection for an r = 170 nm GZO nanoparticle is depicted to-gether with the different order electric and magneticmodes. Going from short to long wavelengths, we first seethat the band gap and related absorption dominates; then,the constant refractive index gives rise to a dielectric-like
Figure 6 Scattering and near fields of a semiconductornanoparticle. Scattering cross section of a 170 nm radiusnanoparticle from GZO (refractive index data fitted with parametersfrom [27]) and near field distribution of the electromagnetic fieldaround the nanoparticle for the quadrupole magnetic mode at468 nm and the dipole electric mode at 1,978 nm as insets (incidentlight from the top.
scattering behavior with dominant magnetic modes and fi-nally the free charge carriers lead to a plasmon resonancein the infrared. The near field pictures in the inset revealthe typical electromagnetic field distribution of a dielectricnanoparticle for wavelengths up to 600 nm and one com-monly seen in metallic nanoparticles at λ approximately2,000 nm. The dielectric modes are virtually identical tothe ones shown in Figure 4b; the metal-like mode howeverno longer occurs as pronounced as in Figure 3b.The finding for the GZO nanoparticle of low pro-
nounced plasmonic near field modes together with thefact that a plasmon resonance at λ = 2,000 nm cannot beexploited when working in the visible regime suggeststhat we should tune the plasma frequency of the semi-conductor such that we obtain a plasmon resonance inthe visible. Yet, this would lead us back to the case of ametal described by the Drude formula, so that we onceagain end up with a trade-off between metallic and di-electric scattering properties.
Angular scattering distribution and substrateTo further judge whether metallic or dielectric nanopar-ticles are performing better for light trapping purpose,we now address, in addition to the scattering cross sec-tions and the electromagnetic near field distributions,the angular distribution of the scattered light.Figure 7a compares the angular distribution of scattered
light for a metallic (Ag Drude fit) to that of a dielectric(n = 2, k = 0) nanoparticle (in air) at the respective res-onance wavelength of the quadrupole electric or mag-netic mode: λ = 426 nm for the metallic nanoparticlewith 120 nm radius and λ = 502 nm for the dielectricone with r = 170 nm. For the dielectric nanoparticle,the forward scattering dominates whereas for the me-tallic nanoparticle, additional lobes emerge, which forthe higher order modes, are additionally directed sidewards.Up to now, we were investigating the nanoparticles in
a homogeneous surrounding of n = 1 (i.e., in vacuum/air). With respect to the application in a device, placingthe nanoparticles at an interface is a more realistic
Figure 7 Angular scattering distributions. Of (a) the quadrupole (magnetic) mode at λ = 502 nm of a dielectric nanoparticle (n = 2, k = 0,r = 170 nm, in blue) and the quadrupole (electric) mode at λ = 426 nm of a metallic nanoparticle (Ag fitted with Drude model, r = 120 nm, inred) in air; (b) dipole, (c) quadrupole, and (d) hexapole electric mode of the above mentioned metallic particle in air (red) and on a substratewith n = 1.5 (green) at the resonance wavelengths of 688/914 nm (b), 426/524 nm (c), and 340/420 nm (d) (incident light from the top).
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configuration. This also plays an important role whenjudging about scattering efficiencies. In the following, wewill consider the case of a spherical nanoparticle embed-ded 50 % into a substrate. This symmetric configurationis readily comparable to the situation of a nanoparticlein a homogeneous medium, and there is a comparableexperimental configuration where the nanoparticle isembedded into a rough front side layer of the device.The following simulations of nanoparticles at interfacesrely on full 3D simulations as they are performed withthe finite element method because Mie theory is notcapable of taking substrates into account.Firstly, the integration of the nanoparticle into a sub-
strate leads to a well-known redshift of the plasmonicresonances. For the Ag nanoparticle with the dielectricfunction fitted to the Drude model and a radius of120 nm, the dipole resonance shifts from 688 to 914 nm
Figure 8 Angular scattering distribution and scattering cross sectionof light scattered from an r = 170 nm, n = 2, k = 0 dielectric nanoparticle inair/n = 3 interface (magenta) (incident light from the top); (b) shows thethe quadrupole resonance were chosen for the representation of the an
when embedding it into a substrate with refractive indexn = 1.5. But secondly, and here most importantly, the an-gular distribution of the scattered light experiences astronger orientation to the forward direction andadditional sidewards pointing lobes become more pro-nounced. Figure 7b,c,d highlights this observation bycomparing the scattering distribution of the dipole, thequadrupole, and the hexapole mode in air and on thesubstrate at the respective resonance wavelengths.Thus, in the case of metallic nanoparticles, the embed-
ding into a substrate helps to broaden the angular distri-bution of the scattered light and to potentially trap itin a thin layer. But how about the dielectric nanoparti-cles with their initial preferential scattering to the for-ward direction? Figure 8 represents in subfigure a the3D angular distribution of the light scattered from anr = 170 nm, n = 2, k = 0 nanoparticle at the resonance
for a dielectric nanoparticle at an interface. (a) Angular distributionair, i.e., n = 1 (blue), at an air/n = 1.5 interface (turquoise) and at anaccording scattering cross sections from which the wavelengths ofgular distributions in (a), i.e., 502, 490, and 502 nm.
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of the quadrupole magnetic mode when situated in air(blue legend) and half in air, half in an n = 1.5 substrate(turquoise legend). The shape appears almost un-changed, rather reduced to a smaller range of angleswhen considering that normally, the propagation an-gles of light will increase inside a substrate due toSnell’s law. Thus, the strong forward scattering re-mains for this substrate which however has a lower re-fractive index than the nanoparticle itself. Also, thescattering cross section becomes narrowed and theresonance peaks even blueshifted, see Figure 8b. Incontrast, the substrate refractive index was set to n = 3for the third angular scattering distribution shown inFigure 8a (magenta legend). Now that the substrate re-fractive index is larger than the particle refractiveindex, a strongly pronounced scattering into higherangle modes is observed. Therefore, it appears thatalso dielectric nanoparticles can profit from an en-hanced angular distribution of scattered light whenembedded into a high refractive index substrate. Yet, thenormalized scattering cross section is not just redshiftedbut also subject to a strong damping, see Figure 8b. Thisdamping is significantly more pronounced than for metal-lic nanoparticles - more than 60 % here compared toapproximately 20 % in the corresponding case ofmetals (see also Additional file 4: Figure S4).Finally, with the integration of a substrate, leaky
modes may emerge for the dielectric nanoparticles that,like enhanced near fields, can promote absorption in theunderlying layer. Figure 9 shows the electromagneticnear field distribution around the dielectric nanoparticlewith n = 2, k = 0, and r = 170 nm when embedded half inair and half in the substrate with (subfigure a) n = 1.5and (subfigure b) n = 3. For the case of the low-indexsubstrate, we find stronger forward scattering, which isin agreement with the angular scattering distributions,
Figure 9 Near field distributions of a dielectric nanoparticle at an intek = 0, and r = 170 nm, embedded half in air, half in a substrate with refraand the hexapole modes are shown for the wavelengths of 680/816 nmthe maxima in scattering, see Figure 8b (incident light from the top).
and the local field in the direct forward direction is en-hanced and appears more pronounced than for thenanoparticle in air, compare Figure 4b. However, for thehigh-index substrate, the local electromagnetic field ismore concentrated inside the nanoparticle or directedsidewards which can be correlated to the angular scat-tering distribution as well. Seeing these two cases to-gether, we can conclude that leaky modes from dielectricnanoparticles occur if the substrate refractive index islower than the one of the nanoparticles and that thelocal fields are more pronounced in the material withthe lower refractive index (which also may be the nano-particle if the substrate has a higher refractive index).A high angular scattering distribution is present for
metallic nanoparticles in vacuum and can easily be rein-forced by the integration of a substrate without showingsignificant losses in overall scattering efficiency. Yet, forthe dielectric nanoparticles, a high angular scatteringdistribution can only be achieved for high-index sub-strates which comes along with significant losses in scat-tering efficiency; these dielectric nanoparticles mayrather benefit from leaky modes appearing with low-index substrates that can lead to enhanced absorptionsimilar to the enhanced near fields around metallicnanoparticles.
ConclusionsEvaluating scattering and near field properties of metallicand dielectric nanoparticles, we firstly found that thescattering cross sections can, in both cases, reach a valueof several times the geometrical cross sections. For thedielectric nanoparticles, no parasitic absorption exists,whereas for the metallic ones, non-zero absorption crosssections are present, which however can be reduced byincreasing the particle radius. The nanoparticle radiuscan be used to tune the resonance position to the
rface. Electromagnetic field around a dielectric nanoparticle n = 2,ctive index (a) n = 1.5 and (b) n = 3. The dipole, the quadrupole,, 490/502 nm, and 396/346 nm, respectively, which correspond to
Schmid et al. Nanoscale Research Letters 2014, 9:50 Page 10 of 11http://www.nanoscalereslett.com/content/9/1/50
desired wavelengths. Scattering cross section maps, cal-culated here with Mie theory, give a fast overview ofthe parameter field and quickly show that dielectricnanoparticles with a refractive index around 2 requiresignificantly larger radii (approximately 1.5 times) thanmetallic ones from, e.g., Ag in order to obtain similarresonance wavelengths. The electromagnetic near fieldsaround the two different nanoparticle types also signifi-cantly differ; whereas for the metallic nanoparticles, thefield vanishes inside and builds up a strong localized fieldaround the surface, the dielectric nanoparticles havestrong fields inside, which however are not absorbed butpreferentially scattered to the forward direction. These ob-servations of both typical dielectric and metallic near-fields are found for semiconducting materials. On the onehand, they have a region of constant refractive index andzero absorption and thus a dielectric-like scattering behavior,but on the other hand, they can also show significant chargecarriers and thus metallic plasmon resonances. However,since the semiconductor also has a band gap and accordinghigh absorption for wavelengths below, it may only be ofinterest when the band to band absorption is outside thewavelength range in focus. Although semiconductors showthe scattering properties of both dielectrics and metals, itwas not possible to combine the two effects constructively.Depending on the application, one or the other type of ma-terial by itself may be preferred to a combination of both.Aside from the scattering ability and the near field dis-
tribution, also the angular distribution of the scatteredlight plays a crucial role for applications. Considering inparticular the application to ultra-thin solar cells, bothan enhanced near field and a particular scattering of thenanoparticle may contribute to enhance the absorption.In a homogeneous medium, the near field is strongeraround the metallic nanoparticle, the scattering effi-ciency (scattering over scattering plus absorption) isstronger for non-absorbing dielectric nanoparticles, sothat up to that point, no decision about the ideal choice
Table 2 Aims and according requirements for efficient nanopbackground of application in solar cells
Aims Metallic nanoparticles Dielectri
High scattering efficiency(low absorption)
For big particles
High near-field enhancement ✓ Improved atmo
High scattering into the solar cell ✓ ✓Scattering
High scattering into large angles ✓ ✓On high
High scattering efficiency at interface ✓ Drop in sc
High local leaky modes ✓ ✓On low
of material can be made. However, when looking at theangular distribution of the scattered light, we saw thatthe dielectric nanoparticles show a strong scattering injust the forward direction, but the metallic nanoparticlesimmediately give rise to scattering lobes directed to highangles. This scattering into high angles allows light trap-ping in thin layers and therefore absorption enhance-ment. Scattering lobes in directions which are otherwisesubject to total reflection work most efficiently and arepreferentially characteristic for metallic nanoparticles.Bringing the nanoparticles at an interface of two mate-rials as it is the typical configuration in solar cells, fur-ther improves the high angle scattering of metallicnanoparticles. For dielectric nanoparticles, an interfacecan also give rise to additional lobes directed to high an-gles, yet this is only the case when the refractive indexof the substrate is larger than the one of the nanoparti-cle. In the end, this means a high-index substrate, whichhowever causes significant damping to the scatteringefficiency for dielectric nanoparticles. The low-indexsubstrate in combination with dielectric nanoparticlesmay be beneficial for generating leaky modes with astrong near field directed towards the substrate thatcan lead to absorption enhancement. So after thetrade-off between high scattering efficiency and pro-nounced scattering into high angles, between enhancednear fields and leaky modes, there is no ultimate pref-erential choice between metallic and dielectric nano-particles. Yet, Table 2 gives an overview on the mainaims for efficient scattering and near fields and howthey may be fulfilled, having in mind the example ofabsorption enhancement in solar cells. For the finalevaluation of performance of a certain nanoparticle in-tegrated in e.g., a solar cell, the complete device struc-ture including the nanoparticles and the specific geometrythen has to be calculated. It will also depend on the par-ticular solar cell concept whether near field enhancementor scattering turns out more beneficial [28,29]. Finally,
article scattering and near fields with the special
c nanoparticles Semiconductor nanoparticle
✓ Dielectric or metal like behavior depending onthe according properties dominating the dispersion
interface (see leakydes below)
Dielectric or metal like behavior depending onthe according properties dominating the dispersion
towards higher n Dielectric or metal like behavior depending onthe according properties dominating the dispersion
-index substrate Dielectric or metal like behavior depending onthe according properties dominating the dispersion
attering efficiency Dielectric or metal like behavior depending onthe according properties dominating the dispersion
-index substrate Dielectric or metal like behavior depending onthe according properties dominating the dispersion
Schmid et al. Nanoscale Research Letters 2014, 9:50 Page 11 of 11http://www.nanoscalereslett.com/content/9/1/50
aside the theoretical optimization, experimental boundaryconditions will define the configurations that are feasiblein the end.
Additional files
Additional file 1: Figure S1. Absorption cross section of a 120-nmradius Ag nanoparticle with dielectric function according to a Drudefit: sum and allocation to different modes.
Additional file 2: Figure S2. Map of scattering cross section for aspherical dielectric nanoparticle with n = 2 and k = 0.
Additional file 3: Figure S3. Maps of (a) scattering cross section and(b) scattering efficiency for a spherical nanoparticle from GZOsemiconductor (refractive index data fitted with parameters from [27]).
Additional file 4: Figure S4. Scattering cross section of a Agnanoparticle (fitted with Drude model) of r =120 nm in vacuum andwhen placed onto a substrate with n = 1.5.
Competing interestsThe authors declare that they have no competing interests.
Authors’ contributionsMS developed the idea of comparing optical scattering and near fieldproperties of nanoparticles made from different materials. She drafted themanuscript and ran the simulations. PA provided and adapted the code forthe Mie simulations and PM set up the FEM calculations. All authorscontributed to the preparation and revision of the manuscript. All authorsread and approved the manuscript.
Authors’ informationMS is the leader of the Young Investigator Group ‘Nanooptical concepts forChalcopyrite solar cells’ at the Helmholtz-Zentrum Berlin. PA and PM are PhDstudents in the group.
AcknowledgementsRegarding simulations with the finite element method, the collaborationwith Frank Schmidt’s group from the Zuse-Institut Berlin is acknowledged.Funding from the Helmholtz-Association for Young Investigator groupswithin the Initiative and Networking fund (VH-NG-928) is greatlyacknowledged.
Received: 12 November 2013 Accepted: 8 January 2014Published: 29 January 2014
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doi:10.1186/1556-276X-9-50Cite this article as: Schmid et al.: Plasmonic and photonic scattering andnear fields of nanoparticles. Nanoscale Research Letters 2014 9:50.
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