Photovoltaic absorption enhancement in thin-film solar cells by non-resonant beam collimation by submicron dielectric particlesConstantin R. Simovski, Alexander S. Shalin, Pavel M. Voroshilov, and Pavel A. Belov Citation: J. Appl. Phys. 114, 103104 (2013); doi: 10.1063/1.4820573 View online: http://dx.doi.org/10.1063/1.4820573 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i10 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Photovoltaic absorption enhancement in thin-film solar cellsby non-resonant beam collimation by submicron dielectric particles

Constantin R. Simovski,1 Alexander S. Shalin,2,3 Pavel M. Voroshilov,3 and Pavel A. Belov3

1School of Electrical Engineering, Department of Radio Science and Engineering, Aalto University,P.O. Box 13000, 00076 Aalto, Finland2Ulyanovsk Branch of Kotel’nikov Institute of Radio Engineering and Electronics, Goncharov Str. 48,432011 Ulyanovsk, Russia3National Research University of Information Technologies, Mechanics and Optics (ITMO),St. Petersburg 197101, Russia

(Received 11 June 2013; accepted 22 August 2013; published online 11 September 2013)

We propose the enhancement of the photovoltaic absorption in thin-film solar cells using densely

packed arrays (not obviously regular) of non-absorbing submicron or micron-sized dielectric spheres

located on top of the cell. The spheres can decrease reflection forming an effective blooming layer.

Simultaneously, they can suppress the transmission through the photovoltaic layer transforming the

incident radiation into a set of collimated beams. The focusing of the light inside the photovoltaic layer

allows enhanced absorption in it leading to the increase of the photovoltaic current. Every sphere

focuses the incident wave separately—this mechanism does not require collective effects or resonances

and therefore takes place in a wide spectral range. Since the fabrication of such the coating is easy, our

light-trapping structure may be cheaper than previously known light-trapping ones and perhaps even

than flat anti-reflecting coatings. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4820573]

I. INTRODUCTION

Photovoltaics (PV) is one of the most promising trends in

the modern energy industry. A large amount of toxic waste

generated by the high-purity semiconductor production forces

to pay attention to thin-film solar cells (TFSCs).1–4 The main

advantage of TFSCs is a very small amount of purified semi-

conductor required. The thickness of the PV layer can be as

small as 150-300 nm.1–4 The second important advantage is

the possibility of roll-to-roll processing on flexible substrates.4

The disadvantage of TFSCs is parasitic optical losses due to a

small thickness of the PV layer: the solar radiation transmit-

ting through the PV layer is then absorbed in the substrate. As

a result, the conventional anti-reflecting coating (ARC) can be

less efficient than so-called light-trapping structures (LTSs)

(see, e.g., in Refs. 5–19). LTSs reduce both the reflection

from a solar cell and the transmission through its PV layer.

They localize the incident electric field in the volume of the

PV semiconductor. Here, the transformation of the incident

wave field into near field is usually implied. Some authors

offer LTS performed as combinations of the PV medium with

photonic crystals;5–7 in some works, LTS are performed as

arrays of dielectric resonators,8,9 but most of the authors

develop plasmonic LTS.11–19 The common disadvantage of

plasmonic LTS and LTS based on dielectric resonators is their

resonant behavior that implies a rather narrow frequency

range of the light trapping effect poorly compatible with the

idea of so broadband structure as a solar cell. As to LTS per-

formed as photonic crystals, their stop-bands can be engi-

neered very broadband, however they are rather challenging

in fabrication and expensive.11–19

In this paper, we substantiate the possibility of non-

resonant, i.e., fundamentally broadband light trapping in the

PV layer of thickness 300–400 nm using analytical estimates

and accurate numerical simulations. The decrease of the

reflection of light and the suppression of the transmission

through the PV layer are carried out through two different

mechanisms. Our suggested LTS is so simple that can be even

cheaper for the mass production than the standard ARC.

II. THE MAIN MECHANISMS OF THE NON-RESONANTPHOTOVOLTAIC ABSORPTION ENHANCEMENT

A schematic view of our photovoltaic structure is shown

in Fig. 1. A layer of closely packed micron or submicron

dielectric (e.g., polystyrene, silica, etc.,) spheroidal particles is

placed on the surface of a PV layer of thickness d � 1000 nm.

The PV film is deposited on the aluminum-doped zinc oxide

(AZO) substrate. This system can be either obtained using the

atomic layer deposition (ALD), grown using the self-

organization of colloidal nanoparticles or simply mechanically

applied using the centrifuge (micron-sized silica and polysty-

rene particles are available on the market). It is necessary to

mention that the self-organization is much cheaper than

ALD-methods. In our simulations, the PV layer was doped

crystalline silicon with density of minor carriers in both

p-doped (150 nm-thick) and n-doped (150-nm thick) parts

equal to 3 � 1018 cm�3. Polystyrene spheres were chosen from

the reasons of low cost, absence of dispersion of the optical

constant in the visible range, and simplicity of the fabrication

process of both ALD and colloid self-assembly. The optical

constants of materials were taken from Refs. 20 and 21.

Obviously, two conditions must be satisfied to achieve

high values of the PV absorption using the LTS of spheres.

The first condition is minimization of the reflection from the

structure. For that, a layer of nanoparticles should operate as

an ARC. The second one is a concentration of electromag-

netic energy in the volume of the semiconductor. For this,

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the LTS should prevent the transmission through the PV

layer.

The suppression of reflection occurs through the interac-

tion of the incident wave with the nanostructure mostly due

to the dipole component scattered by the nanoparticles.22–24

The localization of the radiation arises due to the high-

intensity decaying harmonics that occur through excitation

of multipole moments of high order.25 The last effect can be

simply described as the collimation of the transmitted light

impinging every sphere. The balance of these two mecha-

nisms should give an optimal effect. Here, we do not use any

of resonance mechanisms as well as no photonic crystal

regime—the strict regularity of the array of closely packed

spheres is not a crucial condition. Thus, we believe that the

total light absorption after the correct design of all parame-

ters should be enhanced significantly compared to the open

interface of the solar cell.

III. FIELD CONCENTRATION

First, let us consider a monolayer of nanospheres

located on the surface of a semi-infinite crystalline silicon.

We use this approach to determine the approximate range

of the nanoparticles radii that allows to achieve the optimal

absorption. The resulting spectra will be calculated further

using numerical methods without these approximations.

Hereinafter, we will assume the normally incident wave,

which is polarized along the x axis.

The transmission of radiation into the substrate plays

an important role due to the small thickness of PV layer in

the structure. Therefore, let us explore the possibility of

light collimation through the excitation of high-order multi-

poles in the nanospheres. The stronger is the focusing of the

incident wave inside the PV layer, the larger is the ratio of

the local field to the incident one jE=Eij, and hence the

absorption of light in the PV layer. Respectively, as the

absorption increases, the parasitic transmittance through the

PV layer decreases. Of course, this focusing is diffraction-

limited, but the question is, how onerous restriction is the dif-

fraction limit for realistic TFSCs? Elementary estimations show

that this restriction has no practical significance for

d > 200–300 nm, although in the literature, it is often stated

otherwise.6,10,19,26,27 For example, in amorphous silicon, the

wavelength is shortened about 4 times as compared with the

wavelength in free space (being averaged over the visible

range). So, the wavelength in the material is as small as

100–200 nm. However, the thickness of the PV layer should not

be smaller than 100 nm, otherwise it will be thinner than the

width of the depletion region and the efficiency of the TFSC

will drop unacceptably. For an a-Si TFSC that does not yield in

its efficiency to a wafer solar cell, the PV layer should have

thickness d � 300 nm, i.e., be larger than the wavelength in the

medium. Therefore, for LTS, there is no any obvious need in

huge local field enhancement and nanometer scaled hot spots.

The concentration of the field by such purely wave process as

the wave beam collimation (achievable with dielectric micro- or

nanospheres) can be sufficient to get an optimal absorption of

the incident wave. Our calculations show that the beam collima-

tion inside the silicon layer of thickness d ¼ 300–400 nm is

enough to substantially reduce the transmission through the PV

layer if the effective beam width is nearly equal to d. The con-

centration of the field in the z-direction (along the wave inci-

dence) is also desirable; but under the condition of the sufficient

collimation, it is not critical.

Let us estimate the dependence of the quadrupole

moment of nanospheres (or spheroid) on wavelength and

size of particles. We used the method proposed in Ref. 28

for qualitative evaluations for the case of isolated sphere tak-

ing into account its reflection in the substrate (a semi-infinite

crystalline silicon). Details of this approach are presented in

the paper.28 Taking into account the field inhomogeneity

inside the particle, its internal polarization can be approxi-

mately represented as follows:28

PpðrÞ ¼ Ppðr0Þ þ ðDrrÞPpjr0: (1)

Here, r0 is the coordinate of the center of the particle, r is

the radius vector of a point inside the particle, Dr ¼ r� r0.

Thus, we consider inhomogeneity of the field that character-

izes the quadrupole moment taking into account the gradient

of polarization,28 which is proportional to the gradient of the

electric field because PpðrÞ ¼ e0ðe� 1ÞEðrÞ. We can write

an expression for the vertical gradient of the field inside a

nanosphere

@Ex

@z� 120apb expð�iak0Þð1þ R̂12 expð2iak0ÞÞz4

3a2apbðapb� 32z3Þ þ 80z5ð�apbþ 8z3Þ ;(2)

where k0 is the wave number in vacuum, a is the radius of a

sphere, and it is denoted

ap ¼ a3 e� 1

eþ 2; b ¼ em � 1

eþ 2:

Here, em is the dielectric permittivity of the substrate (PV

medium), and z is the distance from the point inside the parti-

cle to the substrate. In the further estimations, one uses the

FIG. 1. A schematic view of thin film solar cell with nanosphere coating.

Polystyrene particles (e ¼ 2:53) are packed in a square array on the surface

of the doped crystalline silicon film of thickness d¼ 300 nm. The density of

carriers in the silicon layer is assumed to be 3 � 1018 cm�3. Substrate—AZO

(es ¼ 3:53þ i0:004 at the wavelength of 600 nm).

103104-2 Simovski et al. J. Appl. Phys. 114, 103104 (2013)

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averaged value of the field gradient (averaging is done

over the particle cross section). Then, z in formula (2) is

taken z¼ a. Expression (2) after multiplication by ð3=4pÞðe� 1Þ=ðeþ 2Þ gives the averaged gradient of polarization

of the nanosphere and therefore determines the vertical (zz)

component of the quadrupole moment.28 Fig. 2 shows the

distribution of j@Exða; �=@zj according to which the position

of maxima and minima of the quadrupole moment on the

wavelength axis is practically a linear function of a, and is

significant at a ¼ 400� 500 nm. For such values of a,

according to the results of our numerical calculation, multi-

pole moments of high order play an important role in the

formation of the scattering spectrum. The presence of high-

order multipoles results in the formation of the collimated

beam below the sphere illuminated by the plane wave inci-

dent from top. This effect is known and under the name

of photonic nanojet has been considered in Refs. 29–31.

This effect in our structure is illustrated by Fig. 3.

Here, we depict electric field amplitude, magnetic

field amplitude, and local energy density in a cross section

of the PV structure at a frequency 575 THz for the case

a ¼ 450 nm. Simulations were done using the package

Comsol Multiphysics. It should be noted that the effective

width of the collimated beam in the PV layer at this fre-

quency is about 370 nm. This collimation is sufficient for

effective light absorption in the PV layer. Less than 10% of

the incident power is transmitted into the substrate at this fre-

quency. About 35% of the energy incident on the sphere is

reflected in this case (which is not worse than for the open

surface of the PV layer).

We made analytical estimations of the quadrupole

moment based on formula (2). They are in good agreement

with the results of exact numerical simulations. A difference

is caused by the influence of higher-order multipole

moments (also, notice that the results presented in Fig. 2

were obtained in the approximation of semi-infinite silicon

substrate, whereas the numerical simulations were done for

the case d ¼ 300 nm and the AZO substrate). It should be

noted that larger spheres can further enhance the effect of

collimation but the increase of a also leads to the increase of

the reflection and scattering losses.

IV. SUPPRESSION OF REFLECTION

The next step is the suppression of reflection. It can be

achieved through a dense arrangement of spheres, whereby

the upper surface of the array emulates a textured interface.

Let us consider this system qualitatively in the frame of

dipole approximation. According to our studies,22–24 the

array of nanospheres (even closely packed) under this

approximation can be replaced by a flat reflective surface

(often called metasurface) referred to the plane crossing

centres of spheres. Then, the reflection can be described by

the Airy-like formula for two interfaces24

FIG. 2. The electric field gradient inside the particle along the z-axis

depending on the radius of spheres and frequency. A single spherical particle

is placed on the surface of a semi-infinite substrate of doped crystalline sili-

con with a carrier concentration of 3 � 1018 cm�3. The external wave is nor-

mally incident and x-polarized.

FIG. 3. Electric field amplitude (a), magnetic field amplitude (b), and energy density (c) in the cross section of the PV structure at frequency of 575 THz.

Radius of spheres is equal to 450 nm, the external wave is normally incident, the other parameters are the same as in Fig. 1. A horizontal line at the sphere cen-

ter shows the location of the metasurface modeling the array.

103104-3 Simovski et al. J. Appl. Phys. 114, 103104 (2013)

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r̂ ¼ Eref lð0; tÞEIð0; tÞ

¼ R̂l þ R̂12 expf2iðk0DÞg1� R̂21R̂l expf2iðk0DÞg

;

t̂ ¼ EtransðD; tÞEIð0; tÞ

¼ T̂ l � T̂12 expfiðk0DÞg1� R̂21R̂l expf2iðk0DÞg

;

(3)

where k0 is the wave vector of the incident field in vacuum,

r̂; t̂ are reflection and transmission coefficients, Eref lð0; tÞ is

the strength of the reflected field (on a plane through the

centers of nanoparticles—the locations of point dipoles,

which model spherical particles (i.e., radius-vectors of the

spheres centers). EtransðD; tÞ is the strength of the transmit-

ted field, T̂ and R̂ are tensors of the Fresnel reflection and

transmission coefficients (12—incident wave from the

vacuum to the medium and 21—from the medium to the

vacuum). Tensors T̂ l and R̂l characterize the reflection and

the transmission of light by the metasurface emulating the

layer of nanoparticles. According to Refs. 22–24, they can

be approximately written as

jR̂lðk0Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

Fðk0Þ2 þ 1

s;

Fðk0Þ �k0

4p2n1

�jd1 � d2j

ap� 4:29

jd1 � d2j1=2

�

þ 1

3p

�1:461þ 1:797

ffiffiffiffiL0p

k0

þ 1:776L0

k20

�; (4)

qlðk0Þ � arctg

"� 0:155þ

�0:683� jd1 � d2j3=2

2app

�k0ffiffiffiffiL0p

� 0:191ffiffiffiffiL0p

k0

� 0:188

k20

L0

#þ sign½n� n1�p; (5)

T̂ l ¼ 1þ R̂l; (6)

where n1 is the refractive index of the medium in which par-

ticles are arranged (in our case n1 ¼ 1), R̂l ¼ jR̂ljexpðiqlÞ;L0 ¼ ð2pn1Þ2jd1 � d2j; k0 is the wavelength in vacuum,

and d1; d2 are the lattice constants of the layer. Radiation

damping of nanospheres is taken into account in the expres-

sions (4) and (5). Note that the dipole approximation can not

be used for the quantitative calculation for the particles of

size of a few hundred nanometers. However, it is quite appli-

cable for the qualitative assessment of the frequency aver-

aged effect.

Reflection spectrum of the system depending on fre-

quency � ¼ x=2p and the radius a is depicted in Fig. 4. The

regular arrangement of spheres in the square grid and slightly

aperiodic arrangement give practically the same result. As fol-

lows from Fig. 4, the broadband reflection minimum occurs

within the range a ¼ 400� 600 nm. The structure operates as

an ARC, since the averaged reflectance is lower than that of

the open surface of the doped silicon. For lower values of a,the reflection is rapidly growing. A similar increasing of the

reflection also occurs at larger values of a. Moreover, the scat-

tering maxima arise in the upper half-space and result in sig-

nificant scattering losses for a > 600 nm.

Fig. 5 shows the results of an exact numerical calcula-

tion of the reflection from the structure depicted in Fig. 1 in

the operating frequency range for the open surface of the so-

lar cell (dashed line) and in the presence of the nanospheres

(solid line). The radius of nanospheres is chosen to be

a ¼ 450 nm. The power reflectance averaged over the visible

range was calculated for two cases (open surface and nano-

spheres), and the decrease of the reflection is as follows:

GR ¼

ðRspheres dxðRopen dx

¼ 0:66: (7)

Thus, our structure operates as anti-reflecting coating reduc-

ing the integral reflection by 34%.

V. PHOTOVOLTAIC ABSORPTION

For an accurate calculation of the reflection and PV

absorption spectra for the system (see in Fig. 1), we used the

FIG. 4. The dependence of the reflection coefficient on the radius of spheres

and frequency. The monolayer of mutually touching spheres is placed on a

semi-infinite doped silicon substrate. The external wave is normally incident.

FIG. 5. Reflectance from the PV structure described in Fig. 1 for the pure

surface (dashed line) and at the presence of a coating of nanospheres (solid

line). Radius of spheres is equal to 450 nm. The external wave is normally

incident. Numerical calculations are performed in Comsol Multiphysics and

CST Microwave Studio.

103104-4 Simovski et al. J. Appl. Phys. 114, 103104 (2013)

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Finite Element Method implemented in the commercial

product Comsol Multiphysics and verified these simulations

using another commercial package—CST Microwave Studio

implementing the Finite-Difference-Time-Domain method.

The results obtained using these software packages are

in good agreement with each other and with the fields

calculated analytically for the open surface of the PV layer

(without nanospheres). Mutual agreements prove the reliabil-

ity of our results.

According to preliminary estimates made in the previous

sections of this paper, the best balance of the two mechanisms

for increasing the efficiency of the PV conversion occurs in the

range of a ¼ 400� 550 nm. Further, we optimized the struc-

ture by the brute force method varying the value of the radius

with the step Da ¼ 25 nm. The results show that the greatest

PV absorption occurs when a ¼ 450 nm. Fig. 6 shows the

absorption spectrum for the two cases: open solar cell (dashed

line) and in the presence of nanospheres (solid line). As follows

from Fig. 6, the proposed easily fabricated nanostructure pro-

vides a significant increase of the PV absorption. Approximate

estimates of the quadrupole moment, shown in Fig. 2, are in

agreement with the result: it has a wide maximum in the red

region of the spectrum for the specified radius of spheres, cen-

tered at frequency 550 THz. Thus, the maximal PV conversion

is really achieved via the deal between two mechanisms—max-

imal field concentration and minimal reflection, as it was pre-

dicted in the previous sections. The most pronounced spectral

peaks of absorption in Fig. 6 correspond to the regime of a

standing wave in the PV layer. To see it, we have analyzed the

observed Fabry-Perot resonances of reflection, transmission,

and absorption. The dependence of the optical path difference

of two waves reflected from two boundaries of the PV layer are

expressed in effective wavelengths keff ¼ k=nSi, where nSi is

the refraction index of silicon and unified with the correspond-

ing frequencies in Table I. When the path difference is equal to

an integer number of wavelengths (the standing-wave regime),

at the corresponding frequency, we observe the absorption

maximum linked to the total reflection minimum. Accordingly,

the minimal absorption occurs when the path difference is a

half-integer number. This regime corresponds to maximal

transmittance into the substrate. These speculations can be also

confirmed by the comparison of Figs. 5 and 6. So, the best re-

gime corresponds to the maximal reflection from the interface

between the PV layer and the substrate that corresponds to the

weak total reflection. Definitely, this unusual result origins

from the additional local field enhancement in the maxima of

the standing wave, which increases the PV absorption. The

worst regime is the low reflection from the substrate.

A shift of Fabry-Perot resonances of our structure with

respect to those for the open surface (see Fig. 5) is caused by

the presence of nanospheres. The monolayer of nanospheres

can be qualitatively emulated also using the effective me-

dium model. The Maxwell-Garnett mixing rule gives for the

effective refractive index of the array of spheres the estima-

tion neff � 1:28. The resonant path differences of the waves

reflected from two boundaries of this layer expressed in

effective wavelengths keff ¼ k=neff and the corresponding

frequencies are shown in Table II.

In this case, damping of the reflection and growing of

the PV absorption occurs at half-integer path difference as

the reflection from both the lower and upper interfaces of the

film occurs with a phase shift of p. This is the usual regime

of the blooming layer. This type of absorption is seen from

comparison of Figs. 5 and 6.

The PV absorption in presence of the LTS averaged

over the visible range as compared to that calculated for the

open surface is as follows:

GA ¼

ðAspheres d�ðAopen d�

¼ 1:44: (8)

As follows from (8), the proposed system with the layer

of nanospheres allows to achieve 44% of PV absorption

enhancement in comparison with the solar cell without such

a coating. This is because of the decreasing of light transmis-

sion through the PV-layer, which occurs due to the localiza-

tion of radiation in the volume of the PV layer (see Fig. 3).

We have checked that

GT ¼

ðTspheres d�ðTopen d�

< 1; (9)

i.e., the obtained reflection suppression does not lead to the

increase of the transmission. The achieved gain in the PV

FIG. 6. Absorption coefficient for the pure surface solar cell (dashed line)

and at the presence of the nanospheres (solid line). All parameters are the

same as in Fig. 5.

TABLE I. The optical path difference (expressed in effective wavelengths) of two waves reflected from the boundaries of the PV layer at the corresponding

frequencies. Near these frequencies, the absorption A has minima and maxima.

Path difference (in keff ) 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Frequency (THz) 445.23 493.89 539.36 578.70 613.91 646.60 674.70 698.51 718.55 736.53 751.43 764.58 775.50 790.64

103104-5 Simovski et al. J. Appl. Phys. 114, 103104 (2013)

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absorption is not huge (the total PV absorption would corre-

spond to the twice higher gain), however this result is only

the first step in the direction we pointed out and we hope to

improve it.

VI. CONCLUSION

In this paper, we proposed an array of non-resonant sub-

micron dielectric particles (e.g., polystyrene spheres) with

negligible optical losses for a significant increase of the PV

absorption in thin-film solar cells. Instead of the utilization

of a resonator regime of light-trapping, we suggest a broad-

band regime, which is the deal between the reduction of the

transmittance through the thin PV layer and the reduction of

the overall reflectance. Each sphere (or spheroid) transforms

the incident light into a collimated light beam propagating in

the lossy PV layer with a strong attenuation. Stronger colli-

mation of the beam leads to its greater attenuation. Thus, it is

possible to prevent an increase of the light transmission into

the substrate even for the silicon layer as thin as d ¼ 300 nm.

The decrease of the reflection is the collective effect. It can

be described as the effect of an effective layer formed by

closely packed spheres. For larger spheres, the collimation of

the beam in the PV layer is higher but the reflection also

increases, so that there is an optimal radius of the spheres.

The best balance has been achieved in our numerical exam-

ple. This example corresponds to the gain in the PV absorp-

tion GA ¼ 1:44. We believe that a design of the TFSC can be

found for which this gain will be higher than we have

obtained in the present paper.

It should be noted that the use of the array of polysty-

rene or silica spheres instead of a flat anti-reflecting coating

will hardly increase the cost of the fabrication. Really, in

Ref. 32, it was especially proposed to use arrays of polysty-

rene spheres with a diameter d¼ 1 micron in order to reduce

the cost of production of an ARC for the solar cells of the

wafer-type based on gallium arsenide. In that paper, authors

analyzed only the possibility of reducing the reflection of

light from the PV layer by the use of a structured surface.

The problem of preventing the transmission of light into the

substrate in Ref. 32 was not studied because it was not rele-

vant for the GaAs PV layer (which is more absorptive than

the silicon one) with the thickness larger than d¼ 5 microns

Respectively, in Ref. 32, the field distribution inside the

spheres and under them was not considered. The message of

Ref. 32 that the application of the densely packed array of

microspheres, which are produced commercially on the sur-

face of the semiconductor, is simpler and cheaper than grow-

ing textured or planar blooming layer, whereas the

suppression of the reflection is nearly the same.

We hope that physical mechanisms of the interaction of

light with dielectric particles we have revealed in this paper

will help to realize a significant increase of the PV absorp-

tion in thin-film solar cells in a way more attractive for

industrial adaptation than LTS based on photonic crystals,

arrays of plasmonic or dielectric nanoantennas, and other

resonant nanostructures.

ACKNOWLEDGMENTS

This work has been partially supported by the grants of

Russian Fund for Basic Research (number 12-02-31423-mol a,

number 13-02-90765-mol rf nr, 14-08-31730-mol a), of the

Ministry of Education and Science of Russian Federation (num-

bers 14.B37.21.1634, 14.B37.21.0307, and 11.G34.31.0020)

and of the Dynasty Foundation.

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TABLE II. The dependence of the optical path difference (in effective

wavelengths) of two waves reflected from the effective boundaries of the

layer emulating the array of nanospheres at the corresponding frequency.

Path difference 3.5 4 4.5 5 5.5 6

Frequency (THz) 459.00 524.57 590.14 655.71 721.29 786.86

103104-6 Simovski et al. J. Appl. Phys. 114, 103104 (2013)

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