Enhancement of Plasmonic Photovoltaics with Pyramidal Nanoparticles

Light trapping as a result of embedding plasmonic Nano-Particles (NPs) into Photovoltaics (PVs) has been recently used to achieve better optical performance compared to conventional PVs. This light trapping technique enhances the eﬃciency of PVs by conﬁning the incident light into hot-spot ﬁeld regions around the NPs, which possess higher absorption, thus more enhancing of photocurrent. This research aims to study the impact of embedding metallic pyramidal-shaped NPs inside the PV’s active region for enhancing the eﬃciency of plasmonic silicon PVs. The optical properties of the pyramidal-shaped NPs in the visible and near-infrared spectrum have been investigated. The light absorption into silicon PV is signiﬁcantly enhanced by embedding periodic arrays of pyramidal NPs in the cell compared to the case of bare silicon PV. Furthermore, the eﬀects of varying the pyramidal-shaped NPs dimensions on the absorption enhancement are studied. In addition, a sensitivity analysis has been performed, which helps in identifying the allowed fabrication tolerance for each geometrical dimension. The performance of the proposed pyramidal NP is compared with other frequently used shapes, such as cylinder, cone and


Introduction
Due to the importance of solar energy as one kind of renewable energy, many studies have been performed to improve the efficiency of Photovoltaics (PVs) [1].PVs are used to directly convert optical energy to electricity through the absorption of the incident light, which results in a generation of electron-hole pairs [2].Silicon can be considered the most suitable material for PVs due to its economic availability, low cost, well-established technology, and non-toxicity [3].The main drawback of conventional thick PVs is their high cost, Thus, thin-film PVs of a few hundred-nanometer thicknesses are commonly used to overcome this problem [4].However, reducing the thickness results in lower absorption and lower cell efficiency.Therefore, light trapping techniques can be considered as an efficient way to improve the performance of thin-film PVs and bridge this gap.One of the promising light trapping techniques is using plasmonic metallic nanostructures or nanoparticles (NPs), which offer higher absorption because of the localized surface plasmon resonance (LSPR) phenomenon [5].Plasmons are the collective oscillation of the free electrons due to the excitation caused by the incident electromagnetic wave.Due to these excitations, at wavelengths near the plasmons' resonance, the metallic NPs are acting as nano-antennas in the receiving mode as they effectively confine the incident light in the form of hot spots around the NPs.If these hot-spot regions are well positioned within the PV, the efficiency of PV is enhanced.This integration of NPs and PVs is called Plasmonic PVs.The spectral response and optical characteristics of the NPs depend on the geometry, dimensions, periodicity, penetration depth within the PV, and the refractive index of the surrounding material.Thus, optimizing these parameters allows the engineering of the spectral response for the sake of maximizing the efficiency of the plasmonic PV.Due to promises of the plasmonic PVs, many studies have been recently conducted in literature using NPs with different geometrical shapes, such as spheres [6,7], pyramids [8], crescent [9], rods [10], gear [11,12], cones [8,13], and cylinders [8,13].
Silver is the most suitable plasmonic material for realizing plasmonic PVs because of its comparatively low losses in the visible and near-infrared spectra [14].
In [6], paired embedded spherical NPs are used for better absorption and more generation of photo-current in silicon photovoltaic compared to the case of using a single NPs.While in [8], clustering NPs is used as a technique for improving the optical absorption and the photo-current generation of silicon PV.The studied NPs were cylindrical, pyramidal, conical, and spherical NPs at the rear side of the PV.This clustering improves the performance significantly, compared to the single NP case.In [7], embedding arrays of spherical NPs within different configurations inside silicon PVs is investigated.Significant improvement of the optical absorption was achieved, resulting in an efficiency of 16.18% and an enhancement of 92.6% compared to bare silicon PVs.
In [3], a composite structure of different NPs is used to improve the absorption efficiency of amorphous silicon (a-Si) thin-film solar cells.The proposed NPs structure consists of silicon dioxide SiO 2 , nanospheres and silver hemispheres on the front and rear surfaces, respectively.A significant enhancement in short (below 600 nm) and long wavelengths (around 700 nm) due to the dielectric and metallic NPs, respectively, is achieved.In [15], clustering of four gold and silver cylindrical NPs of different sizes was performed.The NPs are distributed vertically above and inside the thin-film silicon solar cell.This improves the efficiency and photocurrent density due to confining and bouncing the incident light inside the cell and due to the different plasmons caused by each NP.In [16], a silver sphere-cube nano-dimer was placed on top of crystalline silicon, providing broadband absorption, specifically in the visible and infrared spectral bands.
In [23], higher-order multipole responses of different shaped silicon NPs, including pyramidal-shaped NPs, in a homogeneous medium are investigated using the multi-pole decomposition method.In [24], a model for the calculation of effective absorption coefficients is proposed and applied on PTQ: IDIC absorber with embedded silver NPs, that allows the fabrication of thinner PVs.
This paper investigates the improvement of the optical absorption inside silicon PVs as a result of embedding periodic arrays of pyramidal metallic nanoparticles.The organization of the presented work is as follows, First, the physical structure of the proposed NPs and the methodology used is introduced in section 2.Then, the modal electric field distribution around the NPs, showing the hot-spot regions of different modes of the proposed structure, is presented in section 3.In section 4, the advantage of the proposed plasmonic PV on the absorption improvement is demonstrated, along with studying the effects of varying the NP dimensions on the absorption enhancement.In section 5, the sensitivity of the response of the proposed structure to fabrication tolerance of the main design parameters is investigated.The proposed structure is compared with other commonly used structures of plasmonic NPs in section 6.The current density-voltage characteristics associated with the proposed structure are obtained in section 7. The paper is concluded in section 8.

Physical Structure and Characteristic Figures
In this section, the absorption enhancement due to embedding periodic arrays of pyramidal NPs in plasmonic C-Si PV is studied.The power absorbed per unit volume in the PV P abs can be expressed using the Poynting theorem as [25] : where ω is the radial frequency of light, E is the electric field, and ε ′′ is the imaginary part of the medium's permittivity.
The quantum efficiency of the PV is expressed as [7]: where P in (λ) is the incident light power and P abs (λ) is the absorbed light power within the PV.Then, the integrated quantum efficiency IQE, which is defined by the number of photons that were absorbed by the PV divided by the total number of photons flowing into the PV [26], and be expressed as : where h is Plank's constant, c is the speed of light in the free space, and I AM 1.5 is the sun's solar spectrum for Air Mass (AM) 1.5.Absorption enhancement g(λ) and Figure of Merit F oM are used to characterize the importance of the embedded nanoparticles on the PVs absorption compared to bare PVs [27], and they are expressed as follows : In this paper, arrays of metallic pyramidal NPs are embedded inside a block of crystalline silicon (C-Si).The excitation source is a plane wave propagating parallel to the pyramid's axis, i.e. along -z-direction, and the incident electric field is polarized along the x-direction, as shown in Fig. 1(a).The spectrum under investigation is from 400 nm to 900 nm, corresponding to a frequency range from 333 THz to 750 THz.The simulations are performed using Lumerical FDTD [28], which is a trusted commercial software for plasmonic solar cells.The boundaries are asymmetric and symmetric along the xand y-direction, respectively.This represents periodic boundaries in the XY plane and along the perpendicular z-direction, PML boundaries are used.The base of the pyramid NP is square, whose side dimension is B, where the height of the pyramid is denoted by H and H/B is the aspect ratio of the NP, which is denoted by A. The material of the metallic NP is considered as silver where the pyramidal NPs are embedded at the centre of the C-Si substrate as shown in Fig. 1 (b).The separation between the centres of adjacent NPs along both x and y-direction is P , which is known as the periodicity of the array.The thickness of the C-Si substrate is denoted by T Si .

Resonance Modes
In this section, the modal electric field distributions are presented for the different resonance wavelengths of the proposed pyramidal nano particle.The sum of the of absorption and scattering cross-sections is defined as the extinction cross-section.Figure 2 shows the extinction cross-section for the case of a single nano particle in free-space, where the base size of the pyramid is B = 200 nm, while its aspect ratio is A = 2.This means that the height of the pyramidal particle is 400 nm.Two resonance wavelengths appear clearly at 414 nm and 645 nm.
Figures 3 and 4 show the electric field intensity distribution on a logarithmic scale along eight different horizontal cut planes parallel to the xy-plane and perpendicular to the axis of the pyramid, i.e. z-axis.The spacing between two consecutive horizontal cut planes is 50 nm, while the spacing between the top (bottom)-most plane and the head (base) of the pyramid is 25 nm.It is clear that the hot-spot regions in each horizontal cut plane are surrounding the opposite edges of the pyramid's cross-section, such that the line connecting a pair of hot-spots is parallel to the electric field of the incident wave, i.e. parallel to the x-axis.These two hot-spots regions are associated with localized charges with opposite signs inside and near the surface of the pyramidal nano particle at both sides.The opposite signs of localized charges force the electric field to be in the same direction in the two hot-spot regions outside the pyramid.This pair of opposite localized charges form the conventional electric dipole, which acts as an equivalent nano dipole antenna in the receiving mode.The length of this equivalent dipole equals half of a guided wavelength at resonance.If this resonance condition is met, the field intensity at the hot-spot regions reaches its maximum value.The higher the intensity of the electric field in the hotspots region, the more generation of electron-hole pairs in the active region of the solar cell, which gives rise to more photo current generation.
It is clear that the field is more confined around the pyramidal nano particle at 414 nm if compared to 645 nm, as the fringing of the field lines is known to be more divergent with the increase in wavelength.This results in a decrease in the maximum field intensity from 10 2.1 (≊ 126) at 414 nm down to 10 1.6 (≊ 40) at 645 nm as a consequence of the more fringing of the electric field.At 414 nm, the field is more confined around the central region of the pyramid.On the hand, the concentration of the field is more towards the base of the nano particle at 645 nm.Consequently, the length of the equivalent dipole is relatively short (long) at 414 nm (645 nm), as expected.Now, a two-dimensional uniform array of pyramidal nano particles is formed.The same element's dimensions are used.The periodicity of the array in the two dimensions is P = 300 nm.The extinction cross-section versus wavelength for the array surrounded by free-space is plotted in Fig. 5.In addition to the previous two resonance wavelengths of the single element, an additional resonance wavelength appears in Fig. 5.The first two resonance wavelengths now appear at 375 nm and 503 nm instead of 414 nm and 645 nm before forming the array.This blue-shift of the resonance wavelengths after forming the array is expected as the effective refractive index is now bigger than before owing to the presence of many particles with high refractive index.The third resonance wavelength due to the array formation appears at 556 nm, as shown in Fig. 5.
Figure 6 shows the logarithmic distribution of the electric field intensity along different horizontal cuts across the pyramid nano particle in an array configuration surrounded by free-space at 556 nm.The high electric field intensity at the edges of the horizontal cuts near the central region of the pyramid indicates strong coupling between the adjacent elements of the array oriented  parallel to the incident electric field.The length of the equivalent dipole formed between the inner two central regions of adjacent elements is slightly longer than the equivalent dipoles formed within a single element.Consequently, the array resonance wavelength of 556 nm is higher than the resonance wavelengths of the isolated single element, 375 nm and 503 nm.The maximum electric field intensity of the array mode is about 10 1.6 (≊ 40), which is almost the same as that of the long equivalent dipole of the isolated single element.

Parametric Study
To study the impact of varying the dimensions of the array of pyramidal NPs on the absorption enhancement, a parametric study is performed by calculating the absorption enhancement at different values of periodicity, base size, and aspect ratio.

Variation of the Periodicity
Figure 7 demonstrates the impact of varying the periodicity P from 300 nm to 600 nm on both the quantum efficiency and the absorption enhancement, while the aspect ratio is fixed at 2 and the base size is kept at 200 nm.As shown in this figure, by increasing the periodicity, the absorption decreases, especially for wavelengths greater than 600 nm as increasing the periodicity decreases the strength of coupling between the adjacent NPs in the array, which deteriorates the role of the coupled modes and consequently decreases the absorption.

Variation of the Base Size
The effects of changing the base size B from 50 nm to 200 nm on both the quantum Efficiency and absorption enhancement are demonstrated in Fig. 8, where the aspect ratio is fixed at 2 and periodicity at 300 nm.The investigated range of the base size is from 50 to 200 nm to ensure enough spacing between the adjacent NPs in the array and sufficient volume of the absorbing material in between the adjacent cells.When the base size increases, the absorption enhancement increases.This can be attributed to the expansion of the hotspot regions around the NP as a consequence of increasing the base size B. As shown in this figure, it can be clearly noticed that the best performance can be reached when the base size reaches 200 nm.Compared with the bare silicon case, it is clear that the array of NPs significantly improves the absorption, especially in the visible and near-infrared ranges.

Variation of the Aspect Ratio
Figure 9 shows the effect of varying the aspect ratio of the NP from 0.5 to 2 on both the quantum efficiency and the absorption enhancement, while the periodicity is kept at 300 nm, and the base size is kept at 200 nm as the upper limit of the aspect ratio is selected to ensure that the NPs are kept inside the absorbing material with a fixed practical thickness of 500 nm.It is clear from Fig. 9 that the increase in the aspect ratio leads to an increase in the absorption enhancement, as increasing the aspect ratio leads to an increase in the number of resonance wavelengths within the spectrum under investigation.Moreover, there is a redshift in the absorption spectrum as a consequence of increasing the aspect ratio, which is a direct application of the wavelength scaling law.
Table 1 lists the figure of merit (FoM) at different values of each design parameter keeping all other parameters fixed at their central values.It can be concluded that within the wavelength range under investigation, the best FoM of 1.39 is achieved if the base size = 200 nm, the periodicity = 300 nm, and the aspect ratio = 2. Further, optimization can be made to obtain even higher values of the FoM.

Sensitivity Analysis
In this section, the sensitivity of the array's response to the main design parameters is studied as a guide for the allowed fabrication tolerances and to illustrate there effects on the performance of the proposed plasmonic PV.Each parameter is perturbed with 5% and 10% of its optimal value.The FoM is studied in each case, where the design parameters are the aspect ratio, the depth of the NPs in silicon, and the volume while keeping the aspect ratio of 2. The sensitivity is calculated as follows [29]:  Figure 10 shows the sensitivity of the response functions to variations in each design parameter.For the case of periodicity = 300 nm, it is shown that the variation of the volume is the most effective parameter while the tolerance of depth is the least effective one.On the other hand, when the periodicity = 400 nm, tolerance of the aspect ratio is more effective than the tolerance of volume as the silicon around the NPs is increased, so the small change in the base size and aspect ratio will not be remarkable.It can be concluded that depth is the least effective design parameter for the sensitivity.In contrast, the volume is the most effective design parameter in the case of small periodicity, while the aspect ratio is the most effective one for larger periodicity.

Comparison with Alternative Nanoparticles
In this section, different NP structures are considered and compared with the proposed Pyramidal NP.The structures under investigation are cylindrical, conical and hemispherical NPs. Figure 11 shows the absorption enhancement for the different NP structures for the range of base size from 50 nm to 200 nm, where the periodicity is kept at 300 nm, and the aspect ratio is kept at 2. For comparison, the base size of the pyramidal NPs matches the diameter of the other structures, and the height is kept the same for all structures.From this figure, it is clear that at B = 50 nm, the cylindrical NP shows the best performance, while the conical NP is the best choice for B = 100 nm.Starting from B = 150 nm, the pyramidal NP is more efficient than other structures as it possesses the best absorption enhancement, especially in the visible and nearinfrared regions.Then, the proposed pyramidal NP is the best candidate at a relatively large base size.Table 2 summarizes the FoM of the structures under investigation.The table reveals the same conclusion, which is the superiority of the pyramidal NP at large base sizes starting from 150 nm.On the other hand, cylindrical and conical NPs are superior at smaller base sizes.

Current density-voltage characteristics
This section investigates the J-V characteristics of PIN-PV, where the intrinsic layer is embedded with the proposed pyramidal NPs and both N-layer and Player are heavily doped.The schematic structure of the PIN-PV is shown in Fig. 12.The absorption of the plasmonic PV, obtained in the previous sections, is used to calculate the photogeneration rate of the electron-hole pairs at wavelength range of [400−1000] nm.In these calculations, the absorption spectrum is scaled along with the studied wavelength range by taking the solar spectrum radiation of the sun for AM 1.5 into consideration.The generation rate ,G, is

Depletion region
Fig. 12: PIN junction structure defined as [12]: where p abs (λ) is the absorbed power in silicon, h is plank's constant, and c is the light speed in free-space.
Then both the Poisson's and the Carrier's continuity equations are solved along the structure [12,30].Poisson's equation is expressed as follows: where ψ is the potential, and ρ, the volumetric charge density that is expressed as ρ = q (p − n − N a + N d ), where n is the electrons concentration while p is the holes concentrations.N a is the ionized acceptors' concentration and N d is the donors' doping concentrations.The continuity equations for both electrons and holes are expressed by: where J n and J p are the electrons and holes current densities, respectively, which are defined as: and, where D n and D p are the diffusion constants for electrons and holes, respectively, R is the Shockley Read Hall recombination rate expressed as [31]: where the life time of electrons and holes are defined by τ n and τ p , respectively.Poisson's equation and continuity equations of the current density are both formulated in terms of potential and quasi-Fermi levels.Then, Poisson's equation and continuity equations of the current densities are solved using   3.The calculated J-V characteristics of the proposed plasmonic PV are compared in the case of various nanoparticle dimensions.Figure 13 shows the effect of changing the base size of the pyramidal NPs on the J-V characteristics of the proposed PV.These characteristics correspond to periodicity = 300 nm, aspect ratio = 2, and silicon thickness = 500 nm.As shown in this figure, increasing the base size of the NPs from 50 nm to 200 nm results in an increase in the generated current enhancement from 16.3% to 41.3 %, respectively, if compared to the base silicon case.This can be attributed to the improvement of the incident light absorption, as demonstrated by the calculated FoM.In Fig. 14, the effect of varying the aspect ratio on the current density is shown while keeping the periodicity at 300 nm, base size at 200 nm, and the silicon layer thickness at 500 nm.It is clear that the generated current enhancement increases from 20.6% to 41.3%, with respect to the bare silicon case, due to the increase in the aspect ratio from 0.5 to 2, respectively.The highest shortcircuit current density is 13 mA/cm 2 , which corresponds to B = 200 nm and aA = 2. Further rigorous optimization should be performed in order to achieve even higher values of the generated short-circuit current.

Conclusion
The light absorption enhancement in silicon PVs by embedding periodic arrays of square pyramidal-shaped nanoparticles is investigated in this paper.By studying the field mode distribution of the NP, it was demonstrated that the pyramidal NP can be considered a multi-resonator as there are separate hotspot regions along its axis, forming separate resonance dipoles, which confine the field around the NP, thus increasing the absorbed power.The embedding of the proposed NPs results in significant enhancement in the absorption compared to bare silicon PVs.This improvement appears significantly in the visible and near-infrared regions.The roles of the main design parameters of the pyramidal NPs in controlling the absorption enhancement and quantum efficiency have been investigated.It has been demonstrated that by increasing any of the design parameters, the resonance wavelengths shift to the right, and the absorption level increases slightly.Moreover, the redshift and the absorption level are mostly affected by the periodicity.The best performance has been achieved for the case of base size of 200 nm, aspect ratio of 2, and periodicity of 300 nm.The sensitivity analysis reveals that the response of the proposed plasmonic PV cell is most sensitive to volume for small periodicity, while it is most sensitive to aspect ratio for periodicity larger than 300 nm.Consequently, special care should be given to this dimension in the fabrication process.By comparing the proposed pyramidal NP with other commonly used shapes, it has been concluded that the proposed structure is the best candidate at relatively large base sizes starting from 150 nm.The current density-voltage characteristics have been calculated using the driftdiffusion model.An enhancement of 41% in the short-circuit generated current is achieved compared to the pure silicon PV cell with no nanoparticles.

Declarations
All data generated or analyzed during this study are included in this published article.
• Funding The authors declare that they did not receive any funding.
• Competing interests The authors declare that they have no competing interests.• Ethics approval Not applicable.
• Consent to participate Not applicable.
• Consent for publication Not applicable.
• Availability of data and materials Not applicable.
• Code availability The authors declare that they have made their custom code.• Authors' contributions In this manuscript, the efficiency of plasmonic silicon photovoltaics has been improved by embedding metallic pyramidal-shaped nanoparticles inside the photovoltaic's active region.This light trapping technique enhances the efficiency of photovoltaics by confining the incident light into hot-spot field regions around the nanoparticles, which possess higher absorption, thus more enhancing of photocurrent.The optical properties of the pyramidal-shaped NPs in the visible and near-infrared spectrum have been investigated.Sensitivity analysis and the effects of varying the pyramidal-shaped NPs dimensions on the absorption enhancement have been studied.The proposed structure has been compared with other frequently used shapes.
-Heba M. Yassin has performed the simulations of the proposed structure and writing the manuscript.

Fig. 3 :
Fig. 3: Electric field intensity distribution on loagarithmic scale, log 10 (E 2 ) , at λ = 414 nm along horizontal cut planes across a single pyramidal nano particle in free-space with B = 200 nm and A = 2.

Fig. 6 :
Fig. 6: Electric field intensity distribution on loagarithmic scale, log 10 (E 2 ) , at λ = 556 nm along horizontal cut planes across a pyramidal nano particle with B = 200 nm and A = 2 in an array configuration with P = 300 nm, in free-space.

Fig. 13 :
Fig. 13: JV-characteristics of the proposed PV while embedding pyramidal NPs at different base size P = 300 nm, A = 2, and T si = 500 nm

Table 1 :
FoM for different values of the design parameters of the array of pyramidal NPs

Table 2 :
FoM of various nanoparticle structures at different base sizes

Table 3 :
Simulation values of the PIN's model parameters