# Effect of plasmonic Aluminum nanoparticles shapes on optical absorption
enhancement in silicon thin-film solar cells

Maedeh Rassekh1<EMAIL_ADDRESS>Reza Shirmohammadi2 Roghayeh
Ghasempour2 Fatemeh Razi Astaraei 2 Saber Farjami Shayesteh1 1Department of
Physics, University of Guilan, 41335-1914 Rasht, Iran.
2Faculty of New Sciences and Technologies, University of Tehran, Iran.

###### Abstract

Scattering from metal nanoparticles near their localized plasmon resonance;
especially, the resonances of noble metals which are mostly in the visible or
infrared part of the electromagnetic spectrum; is a way of improving light
absorption in thin-film solar cells. The surface plasmon resonance can be
affected by different factors such as the type, size, shape, and dielectric
properties of the surrounding medium. Here we investigate, using the Finite
Difference Time Domain (FDTD) method, how different shapes of aluminum
nanoparticles affect absorption enhancement in silicon thin-film solar cells.
Our results show that using these particles more than 30% conversion
efficiency for plasmonic solar cells can be achieved compared to a cell
without particles. We have also found that although the spherical particles
have the highest absorption peak, optimization of some parameters such as the
height of the cylinder or disk-shaped particles and their distance from the
substrate can increase the absorption. The results can provide more
information and insight to understand and optimize plasmonic particles for
solar cell applications.

solar cell, Localized surface plasmon, plasmonic particles, crystalline
silicon thin-film, light absorption.

††preprint: APS/123-QED

##  I Introduction

Over the past decades, using photovoltaic to produce electricity has been
known as one of the promising ways of solving or at least decreasing the
extreme speed of climate change which has become a defending issue in our time
Ranjbaran  _et al._ (2019). In practice, however, the higher cost of this
technology compared to fossil fuels has prevented their widespread use.
Scientists always have been looking for a way to reduce the price and increase
the efficiency of solar cells Shiraz  _et al._ (2018). An ideal solar cell
should have four main features: first, having a direct band-gap between 1.1 to
1.7 eV, second, consisting of non-toxic and available material in nature,
third, having a high photovoltaic conversion efficiency, and four, to be
produced easily for large areas Chopra  _et al._ (2004); Goetzberger  _et al._
(2002).

Among all the growing up kinds of solar cells Steele  _et al._ (2019); Wei
_et al._ (2019); Ananthakumar  _et al._ (2019); Darbe  _et al._ (2019);
Khatibi  _et al._ (2019a); Yao  _et al._ (2019); Khatibi  _et al._ (2019b),
thin-film solar cells Chopra  _et al._ (2004) always have been considered as
the low cost one. However, this cost reduction is due to the decrease in the
thickness of the active layer, which is mostly made of silicon because of its
cost-effective, earth abundance, non-toxicity, and processing availability Li
_et al._ (2010). Reduction in the thickness of the active layer of solar cells
can reduce efficiency by affecting their optical properties Pillai  _et al._
(2010). In general, the performance of a solar cell is determined by its
ability to absorb light and generate electron-hole pairs. Both features can be
limited by some factors such as light acceptance angle and spectral
sensitivity of the active layer. The second one is responsible for the loss of
a significant part of the solar spectrum Shi  _et al._ (2017). To increase
efficiency in this kind of solar cell, several techniques have been developed
for better light absorption Ghahremani and Fathy (2016); Liu  _et al._ (2019);
Eisenhauer  _et al._ (2019); Javadzadeh  _et al._ (2019); Akdemir  _et al._
(2020). Plasmonic, as a new technique; that makes use of the nanoscale
properties of metals; due to its potential to significantly enhance the cell’s
efficiency, has received great interest over the past decades Kumawat  _et
al._ (2019); Stuart and Hall (1998); Rockstuhl  _et al._ (2008); Beck  _et
al._ (2011a). Enhancements in photocurrent have been observed for a wide range
of semiconductors and solar cell configurations Pillai  _et al._ (2007);
Hägglund  _et al._ (2008); Derkacs  _et al._ (2006). To create a surface
plasmon, two materials with free electrons at their interface are needed. In
practice, this means that one of the two materials must be metal with free
conduction electrons Barnes  _et al._ (2003). The use of metal nanostructures
such as gold and silver nanoparticles on the front surface, back surface, or
inside the active layer can also excite the surface plasmon resonance Hutter
and Fendler (2004). Incident electromagnetic radiation interaction with
metallic nanoparticles results in collective electron oscillations, known as
Localized Surface Plasmon Resonances (LSPRs). The result of this interaction
is light field concentration and enhancement at visible and near-infrared
wavelengths to the subwavelength scale Rajeeva  _et al._ (2015). Surface
plasmons excitations can increase light trapping and absorption and reduce
reflection in thin substrates mainly by three methods: Scattering light into
the device from the surface of metal nanoparticles situated on the front
surface of the device that behave as bipolar (Far-field effect), near-field
increment, and direct creation of charge carriers in the semiconductor layer
Atwater and Polman (2011). The optical properties of metal nanoparticles are
closely related to the real and imaginary parts of their dielectric constants,
which depend on wavelength Chan  _et al._ (2008). This means that the spectral
response of the plasmonic nanoparticles is wavelength-dependent, and the
optical properties of these particles can be tuned by changing some of their
factors such as size, shape, and constituent material Kelly  _et al._ (2003).

It has been shown by S. H. Lim et al. that gold sphere nanoparticles with 100
nm in diameter positioned on top of a Si p-n junction photodiode can enhance
light absorption as a result of the increment in electromagnetic field
amplitude within the semiconductor Lim  _et al._ (2007). Finite-difference
time-domain calculations for a single Ag or Au particle separated from a semi-
infinite Si substrate by a 10 nm silicon dioxide layer by Catchpole et al.
have shown that silver particles enhance the optical path length much higher
than gold particles Catchpole and Polman (2008). T.L. Temple et al. have found
that spherical silver nanoparticles with the same diameter suppress light
absorption below the surface plasmon resonance frequency Temple  _et al._
(2009). Akimov and Koh have demonstrated that aluminum nanoparticles result in
higher optical absorption in the photoactive region than silver nanoparticles.
They have found that Al is the best candidate among the commonly used metals
due to its potential for non-resonant coupling and can double the enhancement
compared to the resonant enhancement by small Ag nanoparticles for a wide
range of surface coverage Akimov and Koh (2010). After that many experiments
showed that Al nanoparticles are good candidates for frontside plasmonic
scatterers since they give a broadband increase in the coupling of light into
silicon Villesen  _et al._ (2012); Uhrenfeldt  _et al._ (2013a, b); Zhang  _et
al._ (2013); Uhrenfeldt  _et al._ (2015).

The resonance of light inside the nanoparticles is also important because it
can result in absorption of light by the particle, which will be lost as heat,
and leads to a decrease in cell efficiency. In summary for nanoparticles
positioned on the front surface, scattering the light mostly forward to the
substrate is required. Absorbing by them and/or back-scattering will reduce
efficiency. Thus, absorption enhancement in the substrate has to compensate
these losses prior to cause a positive net gain. In the case of silver, its
plasmonic resonance is in the solar spectrum. As a result, a significant
absorption occurs by the nanoparticles themselves. In the case of aluminum,
the resonance is outside the important part of the solar spectrum. In
addition, aluminum particles are well oxidized, and their properties change
slightly with shape and size. More importantly, their dispersion properties
are stronger than that of silver nanoparticles Akimov and Koh (2011). However,
it should be noted that aluminum and aluminum oxide particles exhibit toxic
effects only in excessive concentrations, and hence, in general, their
toxicity is considered low Ambardar  _et al._ (2020); Krewski  _et al._
(2007).

Figure 1: Comparison of the dielectric function of Au, Ag, and Al: real part
(a), and imaginary part (b).

In general, to support plasmon excitation, the material needs to exhibit a
large negative real and small positive imaginary dielectric function Willets
and Van Duyne (2007). (Fig. 1) shows the real and imaginary parts of the
dielectric functions of Au, Ag, and Al, as obtained from Palik Edward and
Palik (1985). Au and Ag can only show plasmon excitations at wavelengths
longer than about 450 nm and 350 nm, respectively. The real part of their
dielectric function is positive below these wavelengths (Fig. 1-a). While Al
should be plasmonically active from 200 nm to just below 800 nm. The Al peak
at 800 nm in (Fig. 1-b) is due to the transition between two different bands
associated with the conduction electrons (i.e., an interband transition) Brust
(1970), which is accompanied by the rise in the imaginary part of the
dielectric function. This makes Al less attractive at long wavelengths rather
than Au and Ag. Nevertheless, Al has a remarkable potential over the visible
and short wavelengths of the electromagnetic spectrum. In addition, the
natural abundance of aluminum together with its low cost makes it attractive
for practical applications Kuiri (2020).

Recently, Mukti et al. have shown that spherical Al nanoparticles provide
better light trapping compared with silver nanoparticles Mukti  _et al._
(2019). In practice, we have nanoparticles in different sizes, shapes, and
distances from the substrate. It has been shown by Cecilia Noguez Noguez
(2007) that nanoparticles with fewer faces and sharper vertices exhibit
resonances in a wider range of wavelengths. When a nanoparticle is truncated,
the main resonance is blue-shifted, and the surface plasmon resonances at
shorter wavelengths get closer to the dominant mode; so they can be hidden and
increase the full width at half-maximum. Kelly et al. have shown that sharp
edges increases the electric field enhancement. It has also been found that
the corners contain more surface plasmons in a wider range of energy Kelly
_et al._ (2003).

Since the shape and size of a metallic nanoparticle dictate the spectral
signature of its plasmon resonance, the ability to tune these parameters and
study the effect on the plasmonic solar cells is an important challenge.
Despite previous theoretical investigations on Al nanoparticles and their
plasmonic properties, the literature on the effect of different parameters of
these particles on the absorption in silicon-based thin-film solar cells is
limited. So, in this work, we aim to reveal the effect of different parameters
of aluminum particles such as their shape, height, and distance from the
substrate on the absorption in silicon-based thin-film solar cells. Our paper
is structured in the following manner. Details of the computational
methodology are described in Section 2. We discuss our results in Section 3.
Finally, conclusions and a summary of our results are given in Section 4.

##  II Computational methods and theory

Figure 2: Schematic of the FDTD simulation model, consisting of a-Si:H layer,
and aluminum (Al) nanoparticles thereon, exposed to the standard AM 1.5 solar
spectrum. The Perfectly Matched Layer (PML) boundary conditions are used for
upper and lower boundaries and Periodic Boundary Conditions (PBC) are used for
the side boundaries to model the periodic nature of the particles. side view
(a), top view (b), and considered shapes (c).

We have performed the finite-difference time-domain (FDTD) calculations
Sullivan (2013). Which is a method for solving Maxwell’s equations in complex
geometries, as implemented in the Lumerical FDTD Solutions FDT . The proposed
structure, shown in (Fig. 2), consists of a crystalline silicon thin-film
solar cell on which metal particles are distributed periodically. The
particles are exposed to the standard AM 1.5 solar spectrum. To model the
sunlight, a plane wave with a wavelength range from 400 nm to 1000 nm was
normally incident (shown as the yellow arrow indicated in Fig. 2-a). The
Perfectly Matched Layer (PML) boundary conditions are used for upper and lower
boundaries, and Periodic Boundary Conditions (PBC) are used for the side
boundaries to model the periodic nature of the particles. Symmetric and Anti-
Symmetric boundary conditions have been used to reduce the required memory
size and computation time. The thickness of Si is set to 800 nm, and the
diameter and the period of particles are denoted as D and P, respectively. The
refractive index of the surrounding medium is set to be 1. Experimentally, it
is very likely that a native oxide layer is present above the Si surface and
in contact with the Al nanoparticle. In the simulation, we have assumed that
the Al particle are surrounded by vacuum, and their lowest point is 2 nm above
the Si surface. It is well known that a thin oxide layer of about 2-3 nm
thickness forms on the Al nanoparticles surface upon exposure to air Pettit
_et al._ (1975); Gertsman and Kwok (2005). Since modeling a thin oxide layer
in FDTD calculations requires sub-nanometer grid sizes and impractically long
computing times, the natural oxide layer of Al is not taken into account.
However, it has been shown by Akimov et al. that due to its transparency,
$Al_{2}O_{3}$ (whose dielectric permittivity is too low) cannot cause a
significant change in the spectral absorption of Al nanoparticles Akimov and
Koh (2011), whereas Ag nanoparticles exhibit unstable enhancement and due to
their highly absorbing $Ag_{2}O$ shell, they even result in a negative effect
for certain ranges of oxide thicknesses Akimov and Koh (2010). The
experimental study on the Al nanoparticles on a Si substrate without and with
an oxide layer for silicon solar cell application performed by Piyush et al.
also suggests that with the native oxide shell (2-3 nm) on the Al
nanoparticles, the cell efficiency enhances compared to the cell without
nanoparticles Parashar  _et al._ (2016).

The ratio of the number of carriers collected by a solar cell to the number of
photons of a given wavelength incident on the solar cell is defined as the
quantum efficiency of a solar cell $QE(\lambda)$ Kalogirou (2017). Using the
quantum efficiency, integrated quantum efficiency , can be defined as Li  _et
al._ (2010); Singh and Verma (2018):

$IQE=\frac{\int\frac{\lambda}{hc}QE(\lambda)I_{AM1.5}(\lambda)d\lambda}{\int\frac{\lambda}{hc}I_{AM1.5}(\lambda)d\lambda}$
(1)

where $c$ is the speed of light in free space, $h$ is Plank’s constant, and
$I_{AM1.5}$ is AM 1.5 solar spectrum. In this equation, the numerator and
denominator mean the number of photons absorbed by the solar cell and that
falling into the solar cell. To see how metal particles can improve the
efficiency of the solar cell compared to a bare solar cell, we define the
following quantities,

$g(\lambda)=\frac{QE_{Particle}(\lambda)}{QE_{bare}(\lambda)}$ (2)

and,

$G=\frac{IQE_{Particle}}{IQE_{bare}}$ (3)

where $g(\lambda)$ and $G$ are enhancement in quantum efficiency and
integrated quantum efficiency, respectively. Enhancement factors larger than 1
mean that the number of photons absorbed in the Si is increased with the
nanoparticles compared to a bare Si surface.

The absorption per unit volume is obtained from the divergence of the Poynting
vector Wu and Wang (2014),

$P_{abs}=-\frac{1}{2}real(\vec{\triangledown}.\vec{P})$ (4)

the divergence of the Poynting vector can be defined as:

$\vec{\triangledown}.\vec{P}=-\left(\vec{J}.\vec{E}+\frac{\partial u}{\partial
t}\right)$ (5)

where J and u are charge density and energy density, respectively. Assuming
that there is no free charge density, equation (5) can be written as:

$\vec{\triangledown}.\vec{P}=-\frac{\partial u}{\partial t}$ (6)

and the energy density is given by:

$u=\frac{1}{2}\left(\vec{E}.\vec{D}\right)=\frac{1}{2}\varepsilon|E|^{2}$ (7)

where D is the electric displacement field. Hence,

$\vec{\triangledown}.\vec{P}=-\frac{1}{2}\frac{\partial}{\partial
t}\left(\vec{E}.\vec{D}\right)$ (8)

if: $\vec{E}\propto e^{-i\omega t}$

$\vec{\triangledown}.\vec{P}=-\frac{1\times 2}{2}\varepsilon i\omega
E^{2}=i\omega\varepsilon E^{2}$ (9)

with the substitution of equation (9) into the equation (4), following
equation yields Singh and Verma (2019):

$P_{abs}=-\frac{1}{2}real\left(i\omega\varepsilon
E^{2}\right)=-\frac{1}{2}\omega|E|^{2}imag(\varepsilon)$ (10)

Thus, by knowing the electric field intensity and the imaginary part of the
permittivity, we can calculate the absorption as a function of space and
frequency. Both of these quantities can be easily measured in an FDTD
simulation.

Figure 3: Absorption profile of x–z cross-section at wavelength of 500 nm in
solar cells with aluminum particles with period 400nm and different diameters.
From left to right: without particles, 50 nm, 100 nm, and 150 nm (a), and the
Absorption spectra (b). “NP” indicates the absorption without particles.

##  III Results and discussion

We start with the spherical particles. (Fig. 3-a) shows the absorption profile
of x–z cross-section in the silicon substrate and spherical Al particles with
different diameters at a wavelength of 500 nm and a period of 400 nm. From
left to right: without metal particles, particles with diameters of 100 nm,
150 nm, and 200 nm. The solar cell without nanoparticles is for comparative
purposes. The larger the particle is, the stronger the absorption
concentration and intensity becomes under the particle. The particle with a
diameter of 200 nm shows the highest absorption under itself and at distances
near to the surface of the substrate. Although the particle with a diameter of
150 nm shows less concentration of absorption under it, compared to a particle
of 200 nm, they have increased the absorption in a wider part of the
substrate. It can be concluded that the size of the deposited Al particles has
a large influence on the absorption rate of the solar cell. A more detailed
comparison is to see how the absorption increase at different wavelengths.
(Fig. 3-b) shows the absorption characteristics of solar cells with the
different diameters of particles in the 400 nm to 1000 nm wavelength range,
demonstrating that the absorption of solar cells with Al particles increases
significantly and that the peak value of absorption changes compared to that
without Al particles. “NP” indicates the absorption without particles. For
convenience, we use the normalized incident light power to unity. The distance
between the particles (P) here is the same as 400 nm. It is clear that the
aluminum particles increase absorption in the Si by the enhanced forward
scattering due to the surface plasmon resonance. This enhancement is mainly in
the wavelengths of 400 to 600 nm. Increasing the diameter from 100 to 150 nm
shows an increment in the substrate absorption, but with a further increase
from 150 to 200 nm, the absorption decreases. It is due to the more
concentration of absorption under the particle and its less dispersion to
other parts of the substrate. And confirmed that Al particles with a diameter
of 150 nm provide the strongest absorption enhancement among the 3 different
diameters of particles. This result indicates that the path length of incident
light in the active layer can be enhanced by scattering short-wavelength
electromagnetic waves by Al particles, thereby improving the optical
absorption rate inside the photoactive layer. It should be noted that the
calculation of the absorption spectra in FDTD is done by a more accurate
numerical method compared to the method of calculating the absorption profile.
And in these calculations, the error due to the discontinuity of the electric
field near the interfaces is less.

Figure 4: Enhancement of quantum efficiency $g(\lambda)$ of solar cells with
different diameters of Al spherical particles (a), and Sunlight spectrum (b).
Figure 5: Enhancement of integrated quantum efficiency (G) of solar cells with
different diameters and periods of Al spherical particles.

To further investigate the effect of increasing the size of the particles,
(Fig. 4) shows the increase in the quantum efficiency of the cell. All three
particles show values larger than 1 in increasing the quantum efficiency of
the cell, which means the particles increase the number of photons absorbed in
the silicon layer. Particles with a diameter of 150 nm show the highest peak.
And it shows a similar trend that we have already observed in increasing the
absorption with increasing particle diameter in the previous plot. That is, by
increasing the particle diameter from 100 to 150 nm, the peak of quantum
efficiency increases, and then by increasing it further from 150 to 200 nm,
the peak decrease remarkably. Particles with 150 nm in diameter show the most
increment for wavelengths around 400 nm to 600 nm. As the particle size
increase, the Al spectrum exhibits a redshift and broadening due to
retardation effects and the emergence of higher-order plasmon modes. In
addition, there is a trade-off between the absorptivity and the broadening of
the absorption spectra. The broadening of the spectral line is at the expense
of reducing the height of the peak.

Figure 6: Absorption profile (a), electric field intensity profile (b) of x–z
cross-section at wavelength of 500 nm for spherical particles with fixed
radius R (75 nm) and different semi-heights H, and the Absorption spectra (c).

In all calculations so far, the distance between the particles considered to
be 400 nm. For more investigation, we calculate the increase in the total
quantum efficiency of the solar cell as a function of the diameter and period
of aluminum particles (Fig. 5). Enhancement of integrated quantum efficiency
(G) takes minimum value when the period is equal to the diameter of the
particle. It strongly increases with increasing the period until it reaches
its maximum value. Adjacent particles exhibit strong coupling for periods
slightly larger than the diameter of the particle Rockstuhl  _et al._ (2004).
Further Increase in the period causes a decoupling of the metallic structures,
so the enhancement tends to 1 (no enhancement). This is because the surface of
the Si is strongly diluted that means the number of particles per Si surface
area decreases, so the particle enhancement effect can almost be disregard
Rockstuhl  _et al._ (2008). Particles with a diameter of 150 nm show the
largest increment in integrated quantum efficiency, and as the particle size
increases further, the integrated quantum efficiency decreases. For these
particles, periods between 300 and 400 nm have the largest increment in
integrated quantum efficiency. As we can see in (Fig. 5), for particles with a
diameter of 150 nm, there is a region where the conversion efficiency is
substantially more than one. The reason is that the wavelength peak of
g($\lambda$) for these particles is close to the sun spectrum peak, which is
around 500 nm (Fig. 4-b). From (Fig. 5), more than 30% absorption-enhancement
expected by using aluminum particles as a plasmonic solar cell compared to the
bare solar cell.

Figure 7: Absorption profile (a), electric field intensity profile (b) of x–z
cross-section at wavelength of 500 nm for particles with different shapes.
From left to right: Sphere, cylinder, hemisphere, and truncated pyramid, and
the Absorption spectra (c).

Here, we go further and study spherical particles with fixed radius R (75 nm)
and different semi-heights H (Fig. 6) by varying the aspect ratio H/R in the
range from 1/3 (oblate spheroids) to 4/3 (prolate spheroids). According to the
obtained results, by increasing the ratio H/R, absorption first increases and
then decreases, and it reaches its maximum when the ratio is equal to 1. For
the oblate spheroidal shape with H/R = 2/3, the absorption peak is almost the
same as particles with H/R = 1. Thus, for the particle size we have, oblate
(H/R=2/3) and spherical shapes (H/R=1) are more preferable for the better
performance of the cell. It worth mentioning that these results are in good
agreement with previous results for Ag particles by Akimov et al. Akimov and
Koh (2011).

Considering the diameter of 150 nm and the period of 400 nm as the optimal
diameter and period of the particles, we continue with other shapes. The
distance of particles from the substrate is set to be 2 nm. For cylindrical or
disk-shaped particles, the diameter and height of the particles are 150 nm.
Particles with a truncated pyramid shape also have a height of 150 nm and a
bottom and top x and y span of 150 nm and 75 nm, respectively. (Fig. 7-a)
shows the absorption profile for aluminum particles with different shapes. As
can be seen, spherical particles have the highest and hemispherical particles
the least increase of absorption in the substrate. (Fig. 7-b) shows the
intensity profile of the electric field for systems containing particles of
different shapes. As expected, the scattering of the electric field also
decreases similarly to the absorption. With the color bar limits that we have
defined here for the electric field intensity, we can’t see its dispersion
inside the silicon layer. (Fig. 7-c) shows the absorption spectra for the
solar cell in the presence of particles with different shapes. The highest
absorption in the substrate belongs to the spherical particles. Other shapes
of particles by shadowing have somewhat reduced the absorption in the
substrate compared to the solar cell with no particles. However, since the
particles in the form of cylinders or disks, after spherical particles, have
shown the highest amount of absorption near the surface (Fig. 7-a), we
continue with optimizing the effect of the height of these particles. But
before that, we present a comparison for two other parameters for different
shapes (Table. 1). The short circuit current density ($J_{SC}$) is the
electrical current generated by the solar cell when the voltage across the
solar cell is zero. The $J_{SC}$ can be calculated for both the bare Si
substrate (no particles) and the Si substrate with the particles. The
influence of different shapes of particles on the short circuit current was
calculated for the entire wavelength from 400 nm to 1000 nm. For this
analysis, the result shows a direct correlation to the absorption enhancement
analysis shown previously, the spherical Al particles are giving the highest
$J_{SC}$ value. As we can see from (Table. 1), the $J_{SC}$ increment for
spherical particles compared to that of the bare silicon substrate is almost
$34\%$. The other shapes show significantly lower $J_{SC}$ values, with the
hemisphere being the lowest, which its $J_{SC}$ is even lower than that of the
bare Si. The enhancement of integrated quantum efficiency (G) for the 400 nm
period also follows a similar trend, with spherical Al particles producing the
highest enhancement. This is in good agreement with the results of Choudhury
et al. They had found that spherical Ag particles significantly increase the
optical enhancement when compared to that of the other shapes Choudhury  _et
al._ (2017).

Table 1:  The short circuit current density ($J_{SC}$) and Enhancement of integrated quantum efficiency (G) for different shapes for the 400 nm period. Shape |  $J_{SC}$ ($mA/cm^{2}$) | G
---|---|---
Bare Silicon | 5.77 | 1
Sphere | 7.73 | 1.34
Cylinder | 5.89 | 1.02
Truncated pyramid | 5.96 | 1.03
Hemisphere | 5.64 | 0.98
Figure 8: Absorption profile (a), electric field intensity profile (b) of x–z
cross-section at wavelength of 500 nm for cylinder Al particles with different
heights. From left to right: 50 nm, 100 nm, 150 nm, and 200 nm, and the
Absorption spectra (c). Figure 9: Absorption profile (a), electric field
intensity profile (b) of x–z cross-section at wavelength of 500 nm for
cylinder Al particles of 150 nm diameter and 100 nm height, with different
distances from the substrate. From left to right: -100 nm, -50 nm, +2 nm, +50
nm, and +100, and the Absorption spectra (c). Negative values indicate the
part of cylinders lie within the silicon substrate.

(Fig. 8-a) shows the absorption profile for cylindrical or disk-shaped
particles for different heights. As we can see in this figure, the absorption
in the substrate increases with increasing particle height and then decreases
with a further increase in height. The electric field profile (Fig. 8-b) shows
that increasing the height causes the concentration of the electric field to
decrease at the edges and to extend in the lateral circumference of the
cylinder. From (Fig. 8-c), we conclude that although increasing the height
from 50 nm to 100 nm, or in other words, changing the ratio of height to
diameter from $1/3$ to $2/3$, increases the absorption in the substrate, but
further increase in height reduces it. So that for the ratio of $4/3$, the
absorption has substantially decreased. Based on this, we choose the height of
100 nm for cylindrical particles with a diameter of 150 nm as the optimal
calculated height, and in the following, we study the effect of the distance
of the particles from the substrate.

Figure 10: Absorption profile (a), electric field intensity profile (b) of x–z
cross-section at wavelength of 500 nm for two cylinder Al particles of 150 nm
diameter and 100 nm height with +2 nm and +50 nm distances from the substrate.
Figure 11: Absorption profile (a), electric field intensity profile (b) of x–z
cross-section at the wavelength of 500 nm for pyramid Al particles of 150 nm
height, and different x and y span of the top area. From left to right: 0 nm,
50 nm, 75 nm, 100 nm, and 150 nm (inverted pyramid), and the Absorption
spectra (c).

Absorption profiles for particles with different locations are shown in (Fig.
9-a). Five locations of Al particles were designed. From left to right
include: placing the entire height of the particle inside the substrate, so
that top of the particle is at the same level as the substrate surface,
placing half of the height inside the substrate, placing the particles at a
distance of 2 nm from the substrate (accordance to previous calculations), the
particle at a distance of 50 nm, and at the distance of 100 nm from the
substrate. For particles that are located at a distance from the substrate, a
transparent layer that has a little absorption in the solar cell performance
region (i.e., the visible wavelength) can be used as the separating layer. As
we can see, when the whole particle height is placed inside the substrate, the
absorption in the substrate is low, and by separating the particle from the
substrate and placing it at 50 nm, the intensity and concentration of
absorption under the particle inside the substrate, increases. As the particle
gets further away from the substrate, absorption is still concentrated under
the particle but decreases in intensity. The cross-section of the electric
field intensity (Fig. 9-b) shows that as the distance from the substrate
increases, the concentration of the electric field changes from the edges to
the lateral circumference. And further increment shifts it to the lower edges
of the particles. Similar results have been observed by F. J. Beck et al. for
Ag nanodisks Beck  _et al._ (2011b). For a more detailed study, we plot the
absorption spectra in terms of wavelength for different distances of particles
from the substrate. (Fig. 9-c) shows that the absorption for the particles
with a 2 nm distance from the substrate is almost equal to the absorption for
the particles with a 50 nm distance from the substrate. This is because
although the particles at a distance of 50 nm from the substrate show a more
increment in absorption, compared to the particles that are located at a
distance of 2 nm from the substrate, most of this increase has concentrated
under the particle and the coupling between the adjacent particles has almost
gone (Fig. 10). Therefore, this makes the area where the absorption increases
in total to be a smaller part of the substrate. Increment in absorption under
the particles comes at the cost of reducing the area in which absorption has
increased. So, in (Fig. 9-c), we see that the absorption peaks of these two
are close to each other.

For the last part, we investigate pyramid Al particles with different x and y
span of its top area. As we can see in (Fig. 11), changing the x and y span of
the top area does not have much effect on absorption. However, as it expands
and changes to a more cubic shape, it shows a very little increase. Inverting
the pyramid causes the absorption to be more concentrated under the particle
and the coupling between the particles loss. In addition, it increases the
absorption of the particle itself.

##  IV Conclusions

In this paper, we demonstrated that particle shape is a crucial parameter for
designing plasmon enhanced solar cells. We investigated, through the time-
domain finite-difference (FDTD) method, a thin-film crystalline silicon solar
cell on which Al particles have distributed periodically. Various parameters,
including absorption profile and the electric field intensity profile of x–z
cross-section for different shapes, were studied. The enhancement of
integrated quantum efficiency was calculated as a function of particle
diameter and period. The structures were compared with bare thin-film Si solar
cells. Simulation results suggest that compared to a bare cell, more than
$30\%$ conversion efficiency for plasmonic solar cells can be achieved using
Al spherical particles. Spherical particles with a diameter of 150 nm and a
period of 400 nm show the greatest effect on the absorption. While in most of
the previous works, particles were treated as a single particle, so the
coupling between them was ignored, we considered the coupling between the
particles. So the situation can be more close to the experiments.

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