Broadband Scattering of the Solar Spectrum by Spherical Metal Nanoparticles

RESEARCH ARTICLE

Broadband scattering of the solar spectrum by sphericalmetal nanoparticlesTristan L. Temple* and Darren M. Bagnall

Nano Group, Electronics and Computer Science, University of Southampton, Southampton, UK

ABSTRACT

Metal nanoparticles offer the possibility of improved light trapping in solar cells, but careful design is required to maximisescattering and minimise parasitic absorption across the wavelength range of interest. We present an analysis of the broad-band scattering and absorption characteristics of spherical metal nanoparticles, optimized for either crystalline silicon (c-Si)or amorphous silicon (a-Si:H) solar cells. A random two-dimensional array of optimally sized Ag spheres can scatter over97% of the AM1.5 spectrum from 400 to 1100 nm. Larger particles are required for c-Si devices than a-Si:H due to theincreased spectral range, with optimum particle sizes ranging from 60 nm for a-Si:H to 116 nm for c-Si. Positioning the par-ticles at the rear of the solar cell decreases absorption losses because these principally occur at short wavelengths. Increasingthe refractive index of the surrounding medium beyond the optimum value, which is 1.0 for a-Si:H and 1.6 for c-Si, shiftsabsorption to longer wavelengths and decreases scattering at short wavelengths. Ag nanoparticles scatter more of the solarspectrum than Au, Cu or Al nanoparticles. Of these other metals, Al can only be considered for a-Si:H applications due tohigh absorption in the near-infrared, whereas Au and Cu can only be considered for the rear of c-Si devices due to highabsorption in the ultraviolet (UV) and visible. In general, we demonstrate the importance of considering the broadbandoptical properties of metal nanoparticles for photovoltaic applications. Copyright © 2012 John Wiley & Sons, Ltd.

KEYWORDS

plasmonics; light trapping; metal nanoparticles

*Correspondence

Tristan Temple, Nano Group, ECS, University of Southampton, Southampton, SO17 1BJ, UK.Email: [email protected]

Received 8 August 2011; Revised 1 September 2011; Accepted 14 October 2011

1. INTRODUCTION

Metal nanoparticles strongly scatter and/or absorb light dueto the excitation and decay of localized surface plasmons(LSPs). The interaction is characterised by a resonant peak,which is highly sensitive to the size, shape and composi-tion of the nanoparticle, in addition to the properties ofthe surrounding medium. The sensitivity results in changesto the height, width and spectral position of the resonantpeak, and to the ratio of scattering to absorption. The sensi-tive and highly flexible optical properties of metal nano-particles have led to a wide range of interest in both thefundamental properties and applications including biosen-sing [1], cancer therapy [2] and photovoltaics [3].

Metal nanoparticles are of interest for silicon solar cellsbecause they can strongly scatter light despite having sub-wavelength dimensions. Therefore, metal nanoparticlescan provide light trapping in thin-film solar cells withoutintroducing substantial roughness into the device, which

can be detrimental to the electrical properties of the semi-conductor. However, the metal nanoparticles must be cor-rectly designed to maximise scattering across thewavelength range of interest and to minimise losses. Theoptical properties of metal nanoparticles are sensitive to awide range of parameters, and so design optimisation is acomplex task.

The optical cross-section of a metal nanoparticle is typ-ically larger than the geometric cross-section [4]. Whencomparing the optical properties of different sizes of nano-particles, it is helpful to normalise the scattering and ab-sorption cross-sections into dimensionless efficiencies,Qsca and Qabs, which are obtained by dividing the opticalcross-section by the geometric cross-section of the particle.This accounts for the fact that more particles can fit in thesame space if the radius is reduced, and offsets the asso-ciated reduction in optical cross-section. The radiative effi-ciency, Qrad, is equal to Qsca / (Qsca +Qabs). A particlethat only absorbs light has a Qrad of 0, whereas a particle

PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONSProg. Photovolt: Res. Appl. 2013; 21:600–611

Published online 8 January 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/pip.1237

Copyright © 2012 John Wiley & Sons, Ltd.600

that only scatters light has a Qrad of 1. One method to tunethe Qrad is to change the particle size because Qradincreases with the particle volume [5]. Increasing the particlesize or the refractive index of the surrounding medium alsoshifts the resonance position to longer wavelengths [5,6].LSPs are resonant excitations of conduction electrons andcan only be efficiently excited in metals that have no avail-able lower lying energy states across the wavelength rangeof interest. Suitable candidates include the alkali metals(K, Li, Na), the noble metals (Au, Cu, Ag) and Al. Giventhat we must integrate the nanoparticles into devices, werestrict our consideration to Au, Cu, Ag and Al.

The sum of absorption and scattering is known asextinction, Qext. The response of a randomly arrangedtwo-dimensional (2D) array of metal nanoparticles is equalto the superposition of the response of each constituentnanoparticle, provided that the surface coverage is lowenough that interparticle coupling can be neglected [7,8].Arranging the nanoparticles in a planar layer reduces thechance of photons scattered by one nanoparticle interactingwith another nanoparticle, which is undesirable for metalnanoparticles due to unavoidable absorption losses. For pe-riodic arrays of nanoparticles, far-field interaction of thescattered fields gives rise to diffraction, which changesthe angular distribution of scattering and modifies theextinction spectrum due to constructive and destructive in-terference [8,9]. The optimisation of diffraction by periodicarrays is a trade-off between maximising interaction withphotons (which requires a high surface coverage, i.e. asmall pitch) and maximising higher-order diffraction(which requires a large pitch) [10]. Two-dimensionalarrays of metal nanoparticles are typically deposited onthe front or rear of solar cells, with a general preferencefor the rear due to back-scattering losses for front-mountedparticles [11]. The optical properties of metal nanoparticleswill be modified by proximity to the complex multilayeredstructure of the solar cell. Proximity to a silicon waferdecreases the optical cross-section, red-shifts the extinctionpeak and increases the fraction of light scattered into the silicon(i.e. modifies the angular distribution of scattering) [12]. Theseproperties are also strongly sensitive to the particle shape [13].

Studies of the optical properties of metal nanoparticlestypically focus on the attributes of the extinction peak,but for broadband applications such as photovoltaics it isimportant to consider the optical properties across the en-tire wavelength range of interest. For photovoltaics, weuse the AM1.5 spectrum as a standard for solar irradiance.Cole and Halas used Mie theory simulations and downhillsimplex optimisation to find the sphere diameter requiredto maximise scattering of the AM1.5 spectrum, and foundthat a 30% surface coverage of 105-nm radius Ag spherescould scatter 98.8% of photons across the range of 380–820 nm [14]. Akimov and Koh used the finite-elementmethod to optimise absorption within an amorphous sili-con (a-Si:H) layer by a front-mounted periodic array ofAg nanospheres and found an optimum diameter of80 nm and a surface coverage of 11% [15]. Spinelli et al.used finite-different time-domain simulations to optimise

the anti-reflection effect of periodic Ag spheroid arrayson the front surface of silicon and found optimum para-meters of Ag spheroids with 200-nm width, 125-nmheight, 450-nm pitch and a 50-nm Si3N4 spacer layer[16]. However, there is still a need to explore the generaltrends of broadband scattering by nanoparticles and to ex-pand the parameter space to include random arrays, thesurroundingmedium, and other metals and semiconductors.

In this report, we use Mie theory simulations to investi-gate the broadband optical properties of random 2D arraysof spherical nanoparticles. Such arrays are readily fabri-cated by chemical synthesis [5,17] and have previouslybeen demonstrated to increase the efficiency of silicon so-lar cells [18]. Mie theory is limited to simulating the opticalproperties of spherical nanoparticles embedded in a non-absorbing homogeneous medium and so cannot capturethe effect of shape, near- and far-field interparticle cou-pling, and particle–substrate interactions (including thefraction of light coupled into the substrate). A numericalsimulation method is required to fully account for theseinteractions [13]. However, these approaches are manyorders of magnitudes slower than Mie theory [19], whichrestricts the number of simulations and hence the rangeand fidelity of the parameter space that can be explored.We aim to give an overview of the general trends of thebroadband optical properties of spherical and to highlightsome important issues in the design and optimisation ofhighly scattering nanoparticle arrays for photovoltaics. Itis likely that coupling effects will change the specific para-meters required for optimum scattering, but we do not ex-pect that the overall trends will be substantially modified.

In particular, we will investigate the role of the particlesize and constituent metal, the refractive index of the sur-rounding medium and the surface coverage. The use of abroadband analysis includes contributions from importantfeatures that occur away from the dipolar extinction peak,including higher-order modes and interband transitions.We consider optimisation for amorphous silicon (a-Si:H)and crystalline silicon (c-Si) solar cells, which are used todefine the input spectrum but are not directly included inthe simulation. For front-mounted nanoparticles, the band-gap of the semiconductor defines the spectral range of in-terest, whereas for rear-mounted nanoparticles, the effectof filtering of the solar spectrum by the semiconductorlayer must also be considered. In each case, the aim is tomaximise scattering and minimise absorption across theentire spectral range available to the solar cell.

2. SIMULATION METHOD

Simulations based on Mie theory were used to calculate theoptical properties of spherical metal nanoparticles. Mietheory is an exact solution to Maxwell’s equations for thecase of a spherical particle situated in a homogeneous,non-absorbing medium [20]. The solution is a summationof an infinite number of orders, L= 1, 2, 3 and so on. Eachmode order represents an excitation symmetry: dipolar,

Broadband scattering of the solar spectrumT. L. Temple and D. Bagnall

601Prog. Photovolt: Res. Appl. 2013; 21:600–611 © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/pip

quadrupolar, octupolar and so on. It is not possible to find asolution to an infinite sum, but in practice the series can betruncated as orders with large values of L contribute anegligible amount to the overall optical properties. In the spe-cial case where only the dipolar mode is required, the seriescan be truncated at L=1. An adapted version of the imple-mentation of Mie theory by Bohren and Huffman, BHMIE,was used in this work [20]. As an input, we require the wave-length, sphere radius, refractive index of the surrounding me-dium and the wavelength-dependent optical constants of thesphere itself. We have used the values tabulated by Johnsonand Christy for the case of the noble metals [21] and thevalues given by Palik for Al [22]. The use of the bulk opticalconstants is valid for particles sizes greater than approxi-mately 10 nm, as surface scattering effects can be neglected[23]. Scattering and absorption efficiency spectra were calcu-lated from 400 to 1100 nm, with a wavelength step of 1 nm.Parabolic interpolation was used to fit the optical constantsto the step size.

The scattering and absorption percentages wereobtained by multiplying the optical efficiencies by the per-centage surface coverage. For extinction values exceeding100%, scattering and absorption were truncated to 100% *Qrad and 100% * (1 – Qrad), respectively. The total scat-tering percentage was obtained by integrating the productof the input spectrum and the scattering spectrum and thendividing this by the total power in the input spectrum.Total absorption was calculated in the same way. Maxi-mum total scattering was calculated by finding the highesttotal scattering value for radii ranging from 5 to 150 nm.

3. INCIDENT SPECTRUM

The distribution of power across the solar spectrum mustbe considered when designing optical systems for photo-voltaic applications. The solar spectrum reaching theEarth’s surface is attenuated by atmospheric absorptionand scattering, and so varies depending on the position ofthe sun and the relative thickness of the atmosphere at agiven position. The AM 1.5 spectrum is typically used asa standard for the design and optimisation of photovoltaicsystems and represents the spectrum at a solar zenith angleof 48.2o (1.5 = 1/cos(48.2)) [24]. The peak power output isin the visible range, but we must consider the real poweravailable to the solar cell. Electrons excited by photons withenergy above the bandgap will quickly drop down to theedge of the conduction band due to carrier thermalisation,that is, emission of phonons. Therefore, the maximumenergy that a photon can contribute to the solar cell is equalto the bandgap energy of the semiconductor, which is ap-proximately 1.1 and 1.6 eV for c-Si and a-Si:H, respectively.Accounting for thermalisation losses flattens the power spec-trum in the case of c-Si and shifts the maximum power pointto the bandgap energy in the case of a-Si:H (Figure 1).

For particles situated on the front of a solar cell, wemust consider absorption and scattering across the spectralrange of photons with energy greater than the bandgap of

the semiconductor. For c-Si solar cells, we will considerwavelengths from 400 to 1000 nm, with an energy of1.1 eV per photon. For a-Si:H solar cells, we will considerwavelengths from 400 to 700 nm, with an energy of 1.6 eVper photon. We note that the optical and electrical proper-ties of a-Si:H are highly dependent on the depositionconditions, but these represent typical values.

For particles situated on the rear of a solar cell, some ofthe spectrum will be absorbed by a single pass through thesemiconductor and as such need not be considered. Theamount of the spectrum absorbed by a single pass throughthe solar cell is dependent on the thickness of the layer andthe absorption coefficient of the semiconductor. We haveused Beer’s law to estimate the amount of the spectrumtransmitted through the semiconductor layer: T= exp(�4pkl / lo) where T is the fraction of light transmitted,k is the wavelength-dependent absorption coefficient ofthe semiconductor, l is the thickness of the semiconduc-tor and lo is the free-space wavelength. This simpletransmission analysis neglects reflection at the silicon-dielectric interfaces, but gives an approximation of thechange in the incident spectrum due to filtering by thesemiconductor layer. The values of k were obtainedfrom Green for c-Si [25] and by spectroscopic ellipso-metry for a-Si:H [26]. In this report, we will considerthree layers: 200 nm a-Si:H, 2 mm c-Si and 20 mm

Figure 1. Proportion of the AM 1.5 spectrum available at thefront and rear of (a) a-Si:H and (b) c-Si layers, accounting for car-

rier thermalisation losses.

Broadband scattering of the solar spectrum T. L. Temple and D. Bagnall

602 Prog. Photovolt: Res. Appl. 2013; 21:600–611 © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/pip

c-Si. These values of thickness correspond to very thindevices, where light trapping is of crucial importance. Itis important to note that these layers were only used tomodify the input spectrum; they were not directly in-cluded in the nanoparticle simulations.

The five input spectra used in this report are given inFigure 1. The spectra are considerably different in termsof the wavelength range of interest and so can be expectedto affect the optimum design of the metal nanoparticles.Almost all photons between 300 and 500 nm have beenabsorbed by a single pass through either a 200-nm a-Si:Hlayer or a 2-mm c-Si layer, and so need not be consideredfor nanoparticles situated at the rear of the solar cell. Therange of completely absorbed photons is extended to700 nm for a 20-mm c-Si layer. It is clear that a large pro-portion of light is not absorbed by a single pass throughany of the layers, thus demonstrating the need for lighttrapping in thin solar cells.

4. SURFACE COVERAGE

The minimum surface coverage of nanoparticles requiredto scatter and/or absorb all incident photons is dependenton the extinction efficiency, Qext, across the wavelengthrange of interest. In the absence of interparticle coupling,complete extinction of all incident photons will occurwhen the surface coverage (%) is equal to 100 / Qext. Foran array of identical nanoparticles, the maximum scatteringthat can be achieved is limited by the minimum Qext overthe wavelength range of interest. Therefore, it is the averageQext across the spectral range that must be maximised ratherthan the peak Qext. Increasing the average Qext enables alower surface coverage of nanoparticles to be used.

The optical properties of isolated spheres in a homoge-neous medium can be calculated using Mie theory. Theseresults can be used to approximate the far-field responseof a 2D array of identical nanoparticles, assuming thatthe particles are randomly positioned and the surface cov-erage is low enough that interparticle coupling can be ig-nored. Close proximity of two or more nanoparticlesresults in near-field coupling, which gives rise to a red-shiftof the resonant peak position [27,28] and a change in thenear-field intensity distribution [29]. For pairs of nanopar-ticles, the fractional shift in the wavelength positionincreases exponentially as the gap/diameter (g/d) ratio be-tween two particles decreases [30], with a peak shift ex-ceeding 10% for a gap size less than 0.25 d. The peakshift is fractional and so will be larger for longer peakwavelengths (i.e. larger nanoparticles) and also becomeslarger when the refractive index of the surrounding me-dium is increased [30]. Hence, interparticle coupling willmost strongly affect arrays with a high surface coverage(i.e. lower gap size), large spheres and a high refractiveindex surrounding medium.

Figure 2(a) shows the maximum total scattering ofthe five input spectra by Ag spheres, for a surface coverageranging from 1% to 50% and a surrounding medium

with refractive index N = 1.5. The maximum total scatter-ing value was obtained by choosing the highest total scat-tering value for radii varying from 5 to 150nm. Thescattering increases approximately linearly for low valuesof surface coverage, until it becomes limited by absorptionand reaches a maximum value. For a-Si:H devices, the maxi-mum scattering saturates at approximately 26% surfacecoverage, whereas for c-Si a surface coverage of 31% isrequired to saturate the scattering response. These valuesare in agreement with the rough figure of 30% reportedby Cole and Halas [14]. It is interesting to note that thesame surface coverage is required for particles positionedon either the front or rear of a given device. Rear-mountednanoparticles can attain a higher maximum scatteringvalue than front-mounted particles because of the reducedspectral range and the importance of absorption at shortwavelengths, which will be discussed in later sections.The sharp discontinuity in the front a-Si:H absorption spec-trum between 24% and 25% is due to a change in optimumparticle radius from 73 to 127 nm, resulting in a 0.5% dropin absorption. The optimisation routine selects the particlesize with the highest total scattering, but it is clear thatimproved results could be obtained by also accounting forabsorption. For example, allowing for a small reduction in

Figure 2. (a) Maximum total scattering and (b) the correspondingtotal absorption, for five different incident spectra. The Ag sphereradius was varied from 5 to 150 nm, with a surrounding medium

refractive index N=1.5. Note the different scales.



the already high value of total scattering may result in amuch larger relative reduction of absorption.

For the remainder of this report, we will consider a sur-face coverage of 30%, as this provides high total scatteringfor both a-Si:H and c-Si spectra and is readily achievedexperimentally.

5. PARTICLE RADIUS AND HIGHER-ORDER MODES

The optical properties of metal nanoparticles are highlysensitive to the particle size. Small nanoparticles featurestrong, narrow resonances that are predominantly absorbing,whereas large nanoparticles feature weak, broad resonancesthat are predominantly scattering [5]. Increasing the particlesize also shifts the resonant peak to longer wavelengths.Figure 3 shows the scattering and absorption spectra ofspherical nanoparticles with radii varying from 25 to100 nm. Increasing the sphere radius broadens and attenuatesthe scattering peak and moves it to longer wavelengths. The25nm radius sphere has the highest peak scattering effi-ciency, but only scatters a narrow range of wavelengthsand also has the highest absorption efficiency. The 100-nmradius sphere has a weaker peak scattering efficiency, butcan scatter a considerably broader range of wavelengthsand hence has a higher average scattering efficiency acrossthe spectral range. Comparing absorption efficiency values

in isolation can be misleading, as it is the radiative efficiencythat dictates the minimum overall absorption. For example,consider two metal nanoparticles: particle A with Qsca= 10and Qabs=5, and particle B with Qsca= 6 and Qabs= 4.Particle A has a higher absorption efficiency but also ahigher Qrad (0.66 c.f. 0.60), and so for a suitably highsurface coverage, an array of particle A will absorb lessthan an array of particle B.

The primary resonance peak is known as the dipolar(L= 1) mode, but for sufficiently large nanoparticles,higher-order modes can also be excited, resulting in spectrawith multiple peaks. The peak position shifts to shorterwavelengths and becomes narrower and weaker as themode order is increased. Figure 3 shows that the quadrupo-lar (L= 2) mode of the 25-nm sphere only contributes toabsorption, whereas the higher-order modes of the largerparticles contribute to both the scattering and the absorp-tion. The radiative efficiency decreases with increasingmode order, but higher-order modes can still achievelarge radiative efficiencies [11]. The radiative efficiencyof all modes increases with particle size and is close tounity for the dipolar mode of large nanoparticles. Eachmode is only absorbing at short wavelengths: once it isshifted to long wavelength, it has a high Qrad. It is clearthat there is a threshold where absorption becomes negli-gible, which occurs at a wavelength of around 700 nm forspherical Ag nanoparticles in a medium with N= 1.5(Figure 3(b)).

For the remainder of this section, we will investigate thescattering and absorption properties of an array of Agspheres with an overall surface coverage of 30% and asurrounding medium with refractive index N = 1.5, forexample glass or SiO2. We will first consider scatteringand absorption of the amorphous silicon spectra. Morethan 95% of the a-Si:H spectra are scattered by Ag sphereswith a radius of 54 nm or larger, with peak values of 97.3%and 98.5% for front- and rear-mounted spheres, respec-tively (Figure 4(a)). The total scattering value is almostconstant for radii between 60 and 150 nm, but the relativecontribution from higher-order modes increases for largerparticles. Hence, it is possible to alter the scattering contri-bution of higher-order modes from less than 1% of totalscattering for a 60 nm radius to more than 60% for a150 nm radius. Although the total scattering stays constant,the effect on the photocurrent of a solar cell will change, asthe angular distribution of scattering and the interactionwith substrates is different for each mode order. The ab-sorption initially increases as the radius increases, but thenreaches a peak and decreases with increasing radius (Fig-ure 4(b)). The peak absorption is due entirely to the dipolarmode, whereas higher-order modes are the dominant causeof absorption in larger (r> 100 nm) spheres. Absorptionby higher-order modes results in less than 3% and 2%absorption of the front and rear a-Si:H spectra, respectively(Figure 4(b)).

Changing to the crystalline silicon spectrum results in ashift of the total scattering and absorption trends, as shownin Figure 5. Larger spheres are required to scatter the c-Si

Figure 3. Calculated (a) scattering and (b) absorption spectra ofAg spheres with radii ranging from 25 to 100 nm and N=1.5.



spectrum because it is a much broader wavelength rangethan the a-Si:H spectrum. More than 95% of incidentphotons are scattered by spheres with radii between 100and 135 nm for all c-Si spectra. Unlike a-Si:H, there is arelatively narrow range of optimum particle size for maxi-mum scattering of the c-Si spectra. This is because the totalscattering decreases when the particle size is increasedenough for the dipolar mode to be shifted out of the spec-tral range of interest. Additionally, some contribution fromhigher-order modes is required to reach the maximum scat-tering values for the front spectrum and the rear 20-mmspectrum. The dipolar mode alone can only scatter a max-imum of 85.8% of the spectrum reaching front-mountednanoparticles. Including higher-order modes increases themaximum total scattering to 98.2% because of increasedscattering at short wavelengths. Higher-order modes con-tribute less to scattering by rear-mounted particles, withscattering of the rear 20-mm spectrum being almost entirelydue to the dipolar mode. Peak scattering by the dipolarmodes occur for sphere sizes of 89, 99 and 112 nm forthe front c-Si, rear 2 mm and rear 20-mm spectra, respec-tively. However, with the inclusion of higher-order modes,the peak value is reached for radii between 112 and 116 nmfor all three spectra. The absorption trends are similar tothe a-Si:H case, but with lower overall values because ab-sorption at short wavelengths is averaged over a broaderspectral range. At the peak scattering radii, the total

absorption is 1.61%, 0.81% and 0.45% for the front, rear2mm and rear 20mm spectra, respectively. The particularlylow value of absorption for nanoparticles at the rear of a20mm device is due to the fact that most of the photons be-low 700 nm have been absorbed in the c-Si (Figure 1(b)),and absorption at wavelengths above 700 nm is minimalfor N= 1.5 (Figure 3(b)).

6. REFRACTIVE INDEX OFSURROUNDING MEDIUM

Solar cells are composed of numerous layers, each with adifferent refractive index. The optical properties of metalnanoparticles are highly sensitive to the surroundingmedium, and so we must consider the refractive indexof the layers surrounding them. The lowest refractiveindex in a commercial solar cell is approximately 1.5,corresponding to SiO2, glass or index-matched encapsula-tion polymers. In a laboratory cell, the nanoparticles maybe situated between a substrate and air, and as such, the ef-fective refractive index of the surrounding medium will belower. The layer with the highest refractive index in a solarcell is usually the semiconductor itself: the real part of therefractive index of crystalline silicon varies from 5.57 at400 nm to 3.52 at 1200 nm at 1200 nm [25]. Therefore,

Figure 4. (a) Total scattering and (b) total absorption of Ag spheresmounted at the front and rear of a-Si:H solar cells, with N=1.5.Dotted lines denote the dipolar-only case, and solid lines denote thecase including contributions from dipolar and higher-order modes.

Figure 5. (a) Total scattering and (b) total absorption of Ag spheresmounted at the front and rear of c-Si solar cells, with N=1.5. Dot-ted lines denote the dipolar-only case, and solid lines denote thecase including contributions from dipolar and higher-order modes.



the refractive index of the medium surrounding the metalnanoparticles will change according to where in the solarcell the particles are positioned and which layers arepresent in the solar cell structure. In this section, we willinvestigate the effect of a homogeneous non-absorbingsurrounding medium with N varying from 1.0 to 3.5.Values between 1.0 and 2.5 represent nanoparticles em-bedded in a dielectric medium next to the semiconductorlayer, whereas larger values represent the nanoparticle par-tially or completely embedded within the semiconductor.We note that damping of the resonance due to absorptionwithin the semiconductor has not been accounted for.

Increasing the refractive index of the surrounding me-dium broadens the scattering peak and shifts it to longerwavelengths, as shown in Figure 6(a). An increase ofrefractive index from 1.5 to 3.5 shifts the dipolar scatteringpeak from 525 to 1124 nm. For N= 3.5, the dipolar scatter-ing peak occurs outside the wavelength range of interest,but can still scatter a large proportion of the solar spectrumdue to the very large peak width. Tuning the peak positionto longer wavelengths by increasing the refractive index ofthe medium results in less attenuation of the scatteringpeak than tuning the peak position by increasing the spheresize. However, the absorption peaks are not as stronglyattenuated as they are when the sphere size is increased,and—crucially—they are shifted to longer wavelengths(Figure 6(b)). Absorption by spheres in a medium with

N= 1.5 is largely restricted to wavelengths lower than700 nm, but increasing the refractive index will push thethreshold for low absorption to longer wavelengths.

Plots of total scattering and absorption of the front c-Sispectrum as a function of sphere size and surrounding me-dium N are given in Figure 7. In each case, the scatteringincreases with radius before reaching a maximum value,which occurs at lower radii for increasing values of N.With the exception of N= 1, the maximum total scatteringvalue decreases for higher values of N, dropping from98.2% at N= 1.5 to 89.7% at N = 3.5. Absorption increaseswith N, but the differences are negligible for N≥ 2 at themaximum total scattering radii (i.e. r> 75 nm). Absorptiononly increases from 1.6% at N= 1.5 to 2.9% at N= 3.5, andso the reduction in total scattering is primarily due toincomplete scattering of the spectrum. Increasing the re-fractive index shifts all of the peaks to longer wavelengths,such that only the ‘tail’ of the dipolar peak contributes toscattering at short wavelengths. For example, Figure 6(a)shows that a 50-nm sphere in a medium N= 3.5 has a scat-tering efficiency less than 3 for wavelengths between 300and 630 nm. This is in contrast to tuning by size, wherethe higher-order modes are not so strongly red-shifted,and so the scattering efficiency at short wavelengthsremains high (Figure 3(a)). The low-maximum total scat-tering value for N = 1.0 is due to the fact that the dipolarmode of the largest nanoparticle used in this study

Figure 6. Calculated (a) scattering and (b) absorption spectra of50 nm radius Ag spheres embedded in media with N=1.5, 2.5

and 3.5.

Figure 7. (a) Total scattering and (b) total absorption of the c-Sispectrum by Ag spheres with a 30% surface coverage, embed-

ded in a medium with N=1.5



(r = 150 nm) does not have a peak sufficiently far into thenear-infrared (NIR) to scatter all photons in that range.

Figure 8 shows the variation of the maximum total scat-tering value and the corresponding absorption for each ofthe input spectra. The discontinuities in the absorption plotsare due to large changes in optimum particle radius requiredto maximise total scattering. For all three c-Si spectra, themaximum scattering value increases rapidly until a peak atN=1.6 and then decreases gradually with increasing N. Forthe front spectrum and the rear 2mm spectrum, the reductionis due to both increased absorption at long wavelengths and areduction of scattering at short wavelengths, as discussedpreviously. The modest reduction in scattering for the rear20mm spectrum, from 99.5% to 98.5%, is entirely due to in-creased absorption in the near infrared.

The wavelength range of interest for the front and reara-Si:H spectra are narrow and restricted to wavelengthsbelow 700 nm. As such, there is no benefit to shifting thescattering peak to longer wavelengths by increasing the re-fractive index, resulting in an optimum value of N= 1.0 fora-Si:H devices. Increasing the refractive index increasesthe spectral range of absorption, and so reduces total scat-tering. The total scattering is limited by absorption (i.e.scattering + absorption = 100%) until N = 1.7 for the frontspectrum, and until N= 2.6 for the rear spectrum. Beyondthese values, losses due to incomplete scattering becomeimportant because the scattering peaks are shifted into the

near infrared. In practice, a surrounding medium withN= 1 (i.e. air or vacuum) cannot be achieved for metalnanoparticles integrated into an encapsulated solar cell,and so the optimum scattering condition for a-Si:H cannotbe achieved. Increasing the refractive index to a realisticvalue of N= 1.5 reduces total scattering (and increases totalabsorption) by 1.2% and 0.6% absolute for the front spec-trum and rear spectrum, respectively.

The choice of surrounding medium is constrained bythe design of the solar cell. From the above results, it isclear that the nanoparticles should not be situated withinthe active layer of the device for optical reasons, which isin addition to the fact that the nanoparticles are likely tonegatively affect the electrical properties of the active layer[31]. Therefore, the nanoparticles should be situated in thedielectric layers outside the active layer, which will rangein refractive index from around 1.5 (glass, SiO2) to 2.0(TCOs, Si3N4). The absolute reduction in maximum scat-tering across this range of N is less than 0.3% for rear c-Si, less than 1.1% for front c-Si and rear a-Si:H, and2.3% for front a-Si:H.

7. METAL

Ag, Au, Cu and Al can all support LSPs in the visible andNIR with a high radiative efficiency [8]. Ag is usually thepreferred metal for plasmonic applications, as it offerslower absorption losses and generally higher opticalcross-sections than the other metals. However, when con-sidering the choice of metal, we must also take into ac-count other factors, including surface termination,physical abundance and fabrication issues. Ag is a raremetal and can form an absorbing oxide layer, which maycause damping and increased absorption [32,33]. By con-trast, Au does not form an oxide, and the oxide layer thatforms on Al is non-absorbing and so does not negativelyaffect the optical properties [8,34]. The oxide layer thatforms on Cu damps the resonance [35], but the influenceon the radiative efficiency has not been studied. The choiceof metal also restricts which fabrication methods can beused. Hence, there is still motivation to investigate metalsother than Ag.

The optical properties of metals can be separated intofree-electron and interband regions. LSPs can only be ex-cited efficiently at wavelengths where the optical proper-ties of the metal are predominantly due to the behaviourof electrons in the conduction band (i.e. free electrons).Interband transitions act as additional non-radiative decaychannels and so either damp or prohibit the excitation ofLSPs. The noble metals (Ag, Au and Cu) are characterisedby a threshold below which the optical properties aredominated by interband transitions, which occurs at wave-lengths of approximately 327, 517 and 590 nm for Ag, Auand Cu, respectively [21]. By contrast, Al has a weak inter-band region centred near 827 nm [22].

Figure 9 shows the scattering and absorption spectra forspheres of Ag, Au, Cu and Al with the same radius. The

Figure 8. (a) Maximum achievable total scattering and (b) the as-sociated total absorption for Ag spheres as a function of incidentspectrum and refractive index, with a 30% surface coverage.



scattering spectra of the noble metal spheres are nearlyidentical beyond ~650 nm, that is, beyond the interbandtransition thresholds for all three metals. The thresholdfor both Cu and Au can be seen as a large increase in ab-sorption at wavelengths below 700 nm. Absorption at shortwavelengths is primarily due to higher-order modes for Agand interband transitions for Au and Cu. Within the inter-band region, the absorption and scattering efficiencies areapproximately equal. Beyond the interband region, Auand Cu have low absorption but still considerably higherthan Ag in the same wavelength range. The scatteringefficiency of Al is lower than the noble metals for wave-lengths above 570 nm. The Al dipolar peak cannot beclearly resolved, as it is weak and has a non-Lorentzianlineshape. The reason for the distorted peak is the Alinterband region, which is clearly seen as an increase inabsorption at long wavelengths. Absorption within the Alinterband region is considerably lower than the Au or Cuinterband regions, but it occurs in the NIR, which is themost important region of the spectrum for light trappingin c-Si solar cells. The radiative efficiency of the Al sphereis higher than the Au and Cu spheres for wavelengths be-low 628 nm but is lower at longer wavelengths, reachinga minimum of 0.84 at 823 nm. Al has higher absorptionthan Ag across most of the spectral range, but importantly,it also has a lower scattering, resulting in a considerablypoorer radiative efficiency.

The total scattering and absorption trends for nanoparti-cles on the front surface of a c-Si cell are given inFigure 10. The maximum total scattering is 98.2%, 85.9%,84.5% and 80.9% for Ag, Au, Cu and Al spheres, respec-tively, and occurs at radii between 110 and 122 nm for allfour metals. The corresponding absorption at the scatteringmaximum is 1.6%, 12.6%, 15.3% and 10.1% for Ag, Au,Cu and Al, respectively. Therefore, the maximum total scat-tering is limited by absorption for the noble metals and by acombination of absorption and a low scattering efficiency forAl. Cu exhibits the highest total absorption, peaking at26.5% for a radius of 31 nm. The absorption of Al spheresreaches a maximum at a larger radius than the other metals,but is still lower than Au and Cu for all radii. Ag has thehighest scattering and lowest absorption of all the metalsfor all sphere sizes, except for very small radii where Alhas the lowest absorption.

Figure 11 gives an overview of the maximum achiev-able total scattering and the corresponding absorption asa function of metal type for the front and rear of c-Si anda-Si:H solar cells. Ag has the highest scattering and lowestabsorption for all spectra. Absorption due to interbandtransitions in Au and Cu mainly affects short wavelengthsand so is highly detrimental to nanoparticles situated on thefront surface of the device. This is particularly importantfor nanoparticles on the front of a-Si:H solar cells, wherethe spectrum is mainly in the visible, and results inFigure 9. Calculated (a) scattering efficiency and (b) absorption

efficiency of spheres with 100 nm radius as a function of constit-uent metal, embedded in a medium with N=1.5.

Figure 10. (a) Total scattering and (b) total absorption of of thec-Si spectrum by metal spheres with a 30% surface coverage,

embedded in a medium with N=1.5



absorption losses of 22.4% and 26.4% for Au and Cuspheres, respectively. Al absorbs less and scatters more thanCu andAu for the front a-Si:H spectrum, but is slightly lowerin both respects compared with Auwhen considering the reara-Si:H spectrum. Cu has the highest absorption for both a-Si:H spectra. Absorption by Al is lower than Au and Cu on thefront of a c-Si device, but it also scatters around 5% fewer ofthe photons in this spectrum. Al has the highest absorptionand lowest scattering at the rear of c-Si devices due to thepresence of the interband region in the NIR.

8. SUMMARY AND DISCUSSION

The size, constituent metal and position of spherical nano-particles strongly affect their broadband scattering andabsorption properties. Incorrect design will lead to highabsorption losses and/or incomplete scattering of the inci-dent spectrum. Filtering of the incident spectrum by thesemiconductor layer mitigates absorption by higher-ordermodes and interband transitions, and hence, it is preferableto position nanoparticles at the rear of devices where pos-sible. Interband transitions contribute considerably moreto total absorption than higher-order modes, which are infact required to maximise scattering of the spectrum avail-able to thin c-Si devices.

Monodisperse arrays of Ag nanospheres can scattermore than 97% of the spectrum available at the front or rearof c-Si and a-Si:H solar cells. A surface coverage of around30% is sufficient to saturate the scattering response for botha-Si:H and c-Si. Maximum scattering of the solar spectrumwas found for sphere radii within the 110–116 nm range forc-Si and for radii larger than 60 nm for a-Si:H. Total scatter-ing reaches a peak at smaller sphere sizes when therefractive index of the surrounding medium is increased,but the maximum attainable total scattering percentage isdecreased. It is preferable to tune the peak position using

the particle radius rather than by increasing the refractiveindex of the surrounding medium, which should be keptlow to restrict absorption losses to short wavelengths. Opti-mum values of N= 1.0 and N = 1.6 were found for a-Si:Hand c-Si, respectively.

The optical properties of Ag nanoparticles clearly out-perform those of Au, Cu and Al in all respects. However,it is still useful to consider the other metals for scenarioswhere Ag cannot be used, or in the case that experimentalAg nanoparticles cannot reach the performance predictedby theory. The choice of metal depends on the incidentspectrum: Au and Cu nanoparticles have poor radiative ef-ficiency in the visible, whereas Al nanoparticles have poorradiative efficiency in the NIR. Therefore, Al offers betterperformance than Au or Cu for the front of a-Si:H solarcells, but should not be used at the rear of c-Si devices.Al and Au offer similar performance at the rear of a-Si:Hand can also be considered for the front of c-Si, with Aloffering lower absorption but also lower scattering. In gen-eral, Au is preferable to Cu for all device configurations,but the difference becomes small for particles situated atthe rear of wafer-based c-Si devices.

Our simple analysis neglected particle–particle andparticle–semiconductor coupling. Particle–particle interac-tions will most strongly affect the surface coverage resultsand will possibly define upper and/or lower limits to thesurface coverage. Further study is required on the opticalproperties of dense arrays of nanoparticles, particularlyconcerning changes to radiative efficiency and the angulardistribution of scattering. Proximity to a substrate reducesthe polarizability (and hence the extinction efficiency) ofmetal nanoparticles, and light reflected from the substratealters the total field driving the resonance [36]. The prox-imity of the substrate will also increase the average refrac-tive index of the medium surrounding the nanoparticleresulting in a red-shift of the resonance, but this effect willbe relatively minor for nanoparticles a few tens of nano-metres or more away from the semiconductor layer [15].Hence, the principle effect of the substrate interaction willbe to alter the magnitude of the optical cross-sections,which will in turn alter the required surface coverage.The angular distribution of scattering has not been consid-ered in this work, but it is an important consideration fornanoparticles on the front surface of solar cells, or on eithersurface of bifacial devices. In these cases, photons scat-tered away from the substrate are lost [11], and so scatter-ing into the substrate must be maximised. This can beachieved by situating the nanoparticle close to the sub-strate, at the cost of reducing the optical cross-section forfront-mounted particles [12]. The height of large sphericalparticles may also limit scattering into the substrate due tothe large distance between the ‘effective dipole’ and thesubstrate [37]. The angular distribution of scattering is oflesser importance for nanoparticles situated between thesemiconductor and the rear contact/reflector.

In this study, the optimum parameters were found bysimply maximising the total scattering of the incident spec-trum. For a saturated scattering response, this will yield the

Figure 11. Maximum total scattering and the associated totalabsorption by spherical metal nanoparticles as a function of inci-dent spectrum and metal type, for a 30% surface coverage and

surrounding medium N=1.5.



lowest absorption, but for the case of incomplete scattering(for example due to a low surface coverage, or a high re-fractive index medium), it is possible that a different sizesphere will have considerably lower absorption but onlyslightly lower scattering, and so will be preferable. The op-timisation will be improved by considering the relativeimportance of scattering and absorption at specific wave-lengths within the spectrum, rather than treating all wave-lengths equally in the optimisation. Absorption is moredetrimental at wavelengths where the semiconductor ispoorly absorbing, as poorly absorbed photons will havemore passes through the solar cell, and hence more interac-tions with the nanoparticles.

We have restricted our analysis to spherical nanoparti-cles, but changing the shape can be used to further tunethe optical properties. The radiative efficiency and thewidth, height and position of the scattering peak are allsensitive to particle shape [8,38]. Tuning the peak positionby changing the particle size results in broad, attenuated,scattering peaks due to radiation damping and the excita-tion of higher-order modes. In contrast, tuning the peakposition by changing the particle shape can preserve oreven enhance the quality factor of the resonance, resultingin large optical efficiencies in the NIR. For example,increasing the aspect ratio of a nanorod results in a strongred-shifting of the peak position and an increase of theextinction efficiency [8,39]. Large, narrow resonances inthe NIR will lower the required surface coverage, whichwill in turn lower the overall losses within the interbandregions of Au and Cu. Broadband interaction can beachieved by combining multiple particles types, with reso-nances at distinct parts of the spectrum, into a single array.However, this will add considerable complexity to arraydesign and fabrication, and tuning by shape can lead to areduction of the radiative efficiency [8]. Further investiga-tion is required to optimise the nanoparticle shape forbroadband scattering applications.

9. CONCLUSIONS

For a-Si:H solar cells, the optimum conditions are a 26%coverage of Ag spheres with radius of 55 nm or higher,with a surrounding medium assumed to be N = 1.5 for prac-tical reasons. For c-Si solar cells, the optimum conditionsare a 32% coverage of Ag spheres with radius between110 and 116 nm, embedded in a medium with N = 1.6.The same parameters provide the optimum results for bothfront and rear-mounted nanoparticles, but the maximumscattering that can be achieved increases for particles situ-ated at the rear of devices. Parasitic absorption by higher-order modes only adds between 1% and 3% absolute tothe total absorption, provided that the refractive index ofthe surrounding medium is 1.6 or lower. Interband transi-tions in Cu result in an absolute increase in absorption ofbetween 3.6% and 23.7% compared with Ag, dependingon the device type and particle position. Ag provides thehighest scattering and lowest absorption for both the front

and rear of a-Si:H and c-Si solar cells. The choice betweenthe other metals—Au, Cu and Al—is dependent on the par-ticle position and the spectral range of interest. Al is suitablefor the front and rear of a-Si:H devices and the front of c-Sidevices; Au is suitable for the rear of a-Si:H and the frontor rear of c-Si; Cu can only be considered for the rear of thickc-Si devices. Under ideal conditions, over 97% of the AM1.5spectrum available to a-Si:H and c-Si solar cells can be scat-tered by a random 2D array of spherical Ag nanoparticles.Metal nanoparticles have great potential for light-trappingapplications, but they must be carefully designed to maxi-mise scattering and minimise parasitic absorption.

ACKNOWLEDGEMENTS

The authors thank R. S.A. Sesuraj and R. Santbergen forhelpful comments and suggestions.

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Broadband Scattering of the Solar Spectrum by Spherical Metal Nanoparticles