1. Introduction

Energy crisis in whole world has led many researchers to focus their efforts on developing new materials and methods to improve the performance of conventional solar cells. One of the main challenges to the performance of development of thin film solar cells is the creation of efficient light trapping methods [1–5]. Surface texturing, utilization of back reflectors, and various other approaches have been proposed for enhancing absorption in thin film solar cells [6,7]. Recently, plasmonics has been used for light trapping employing metal nanoparticles [1] and alternative plasmonic materials [8]. Surface plasmons are collective oscillations of the free charges at the metal surface. Metals sustain surface plasmons, either localized in the metal nanoparticles or propagating in the case of planar metal surfaces. By controlling the size and shape of the metallic nanostructures, and the polarization angle of the source, the surface plasmon resonance or plasmon propagation can be tuned. Since the surface plasmon resonance of metal occurs primarily in the visible or the infrared wavelength region, they are of particular interest for photovoltaic applications [9,10]. The development of glass plastic materials based thin film solar cells on the substrate could be an apace growing market [11]. However, these solar cells are affected by some common issues like low efficiency, intense degradation of recombination rate, and less stability. A convenient way to increase the effective path length into the cell is to use surface texturing, which causes the scattering of light into the solar cell over a large angular range [12–14]. The performance of these solar cells can be decreased by the larger surface area due to an increase in minority carrier recombination. Several solutions for more effective light concentration and trapping have been proposed in recent years to solve these issues, particularly using plasmonic structures made of thin metal films or nanoparticles [15]. They were quite successful at enhancing and redirecting the incident light beam. Plasmonic structures have diverse applications [16], especially in the field of photovoltaics [17–19] where many different technological approaches were proposed to exploit plasmon excitation and light localization for highly efficient solar cells. Many studies were conducted illustrating the coupling of plasmons to optical emitters, plasmon focusing, and hybridized plasmonic modes in nanoscale metal shells [20–23]. Despite all these instigating scopes, until recently little methodical thought has been given to the question of how whole EM spectra can be utilized in a solar cell using plasmon excitation and light localization.

To allow near-perfect light absorption and photo carrier collection conventionally, photovoltaic absorbers are required to be "optically thick". To improve the efficiency of solar cells the system must have minority carrier diffusion lengths several times the material thickness for all photo carriers to be collected [24]. There are several ways of reducing the physical thickness of the photovoltaic absorber layers. Among them, plasmonic structures can be utilized in at least three ways while keeping their optical thickness constant. First, metallic nanoparticles have opened the opportunity to be used as sub-wavelength scattering elements to couple and trap freely propagating plane waves from the sun into an absorbing semiconductor thin film, by folding the solar spectra into a thin absorber layer [25,26]. Second, metallic nanoparticles can be used as sub-wavelength antennas in which the plasmonic near-field is coupled to the semiconductor, increasing its effective absorption cross-section [27]. Third, a thin photovoltaic absorber layer can couple solar spectrum into surface plasmon polarization modes supported at the metal/semiconductor interface by a corrugated metallic film on the back surface [28]. Free electrons as plasma in plasmonic materials can support electronic or plasmon over a broad spectrum from infrared to ultra-violet light. As metals have high optical losses, alternative metals with the lowest ohmic losses can be preferable for plasmonic devices. The material library of nanophotonics for a long time has been limited to noble metals such as gold and silver. They are commonly used metals in plasmonic and optical metamaterial devices because of their small ohmic losses or high DC conductivity. However, loss arising from interband transitions occurs when an electron in the valence band in a metal absorbs a photon to jump to the Fermi surface or when an electron near the Fermi surface absorbs a photon to leap to the following unoccupied conduction band causing a high loss in conventional plasmonic materials. Again, the magnitude of the real part of the permittivity is very large in conventional metals. Most importantly, the optical properties of metals cannot be tuned or adjusted easily and they are not cost-effective.

The numerous drawbacks of conventional plasmonic materials have motivated researchers to search for alternatives. There have been numerous reports of several alternative materials that can overcome one or more of the drawbacks mentioned above [29–31]. Metal nitrides such as titanium nitride (TiN), zirconium nitride (ZrN), and tantalum nitride (TaN) are notable for carrying metallic properties within the visible and infrared wavelengths [32–34]. These nitrides are non-stoichiometric, interstitial compounds with large free-carrier concentrations. These materials are refractory, stable, and hard. Amazingly, their optical properties are tunable by varying their composition. Moreover, they are compatible with silicon CMOS technology [20,35]. However, metal nitrides offer fabrication and integration advantages which could be useful to overcome the challenges of the metals.

In this work, we systematically studied scattering cross-section and absorption enhancement by nanoparticles consisted of alternative material, TiN using the finite-difference time-domain (FDTD) method. At first, we performed a comparative performance analysis of TiN with noble metals, Au and Ag to establish its functionality as a plasmonic material. Afterward, we designed and optimized a bowtie shaped plasmonic nanosystem, which consisted of two triangular nanoplates. Moreover, we determined the total scattering cross-section, the fraction of light scattered into the substrate, light scatter into the substrate, the absorption cross-section, and spatial mapping of the electric field in the plasmonic nanosystems. Additionally, we studied the polarization-sensitive performance of the bowtie nanosystem varying the polarization angle of the source. We conducted a shape-dependent comparative analysis of the nanoparticles. We examined the impact of the dielectric coating of the nanoparticles. Our study will be beneficial for utilizing TiN in a variety of plasmonic device applications.

2. Plasmonic nanostructure and simulation method

2.1 Structural design

Our proposed nanostructure was composed of alternative plasmonic material, TiN-based nanoparticles on a semi-infinite crystalline silicon substrate as can be seen in Fig. 1. TiN exhibits localized surface plasmon phenomenon and it is economically viable. Silicon, which is the most widely used semiconductor, was considered an absorber layer. The silicon was covered with a thin Si$_3$N$_4$ layer. The plasmonic particles were separated from the semi-infinite silicon absorber layer by this thin nitride layer. Such layers were present in solar cells for surface passivation. In this paper, we varied the shape of the particles and analyzed their properties. In addition to nanospheres, the shapes of nanoparticles considered were triangular and bowtie-shaped nanoplates. In both cases, nanoplates were equilateral. In Fig. 1, the thickness of the thin film and substrate were represented by t$_1$ and t$_2$, respectively, and the polarization angle of the source was represented by $\theta$. L denoted the side length, and h represented the thickness of the nanoparticle.

 

Fig. 1. Visual representations of (a) a bowtie-shaped TiN nanoparticle, (b) a bowtie-shaped TiN nanoparticle with a distance of 60 nm between the two triangular plates, and (c) a triangle-shaped TiN nanoparticle. These nanoparticles were placed on top of a thin Si$_3$N$_4$ layer on a Si substrate. (d) A cross-section of the simulation set up. Light from the source was incident on the particle in the z-direction.

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2.2 Methodology

We studied the mentioned nanosystems using the FDTD method where Maxwell’s equations were solved numerically and simultaneously in both 3D space and time. The FDTD computational space had characteristic dimensions 1.2 $\mu$m $\times$ 1.2 $\mu$m $\times$ 1.25 $\mu$m, and a perfectly matched layer (PML) boundary conditions were used in all directions during simulation. A non-uniform mesh type was utilized for surrounding conditions. In the case of the nanoparticle, the mesh size was 4 nm. A total-field scattered-field (TFSF) plane wave was incident on the particle in the z-axis. The TFSF source was polarized perpendicular to the surface normal on the particle from the air side as the nanoparticles will be at the front face of a solar cell. The incident source was a uniform wave with a wavelength ranging from 550 to 1100 nm, which contains the most effective irradiance of the solar spectrum (AM 1.5). To study the scattering properties of the nanoparticles, a TFSF source was very useful and it allowed us numerically to separate the incident field from the scattered field. The TFSF source volume comprised both incident field and scattered field, while outside the source volume, all the fields were detected as scattered light. The incident field inside these regions was subtracted from the total field to numerically give the value of scattered fields. We added electric field monitors outside the TFSF box to calculate the scattering spectra of the nanoparticles. The mapping of the near electric field intensity was calculated by positioning a monitor at the desired spatial point. The electric and magnetic fields around the particle were measured, afterward, we performed a Fourier transform of the time domain electrical field into a frequency domain. This transform is given by,

Here, $E(\omega$) is the electrical field as a function of angular frequency, $\omega$. $t=0$ is the starting time and $T_f$ is the final time. The radial Poynting vector, $S(\omega )$ is presented by,

Here, $H(\omega$) is the magnetic field. The scattered powers along +x and +y directions are defined as $P_1$ and $P_2$, respectively, which are given by,

Here, ${S_x(y,z,\omega )}$ is the radial Poynting vector in yz plane and ${S_y(x,z,\omega )}$ is the radial Poynting vector in xz plane. Similarly, we calculated $P_3$, $P_4$, $P_5$, and $P_6$ along +z, –x, –y, and –z directions. The sum of power detected along all six monitors in the scattered field region was calculated as the total power scattered by the nanoparticle, $P_{s}$ given by,

Total scattering cross-section, $Q_{T}(\omega )$ as a function of $\omega$ was calculated by,

Here, $I(\omega )$ is incident power intensity with unit W/m$^2$.

The absorption cross-section, $Q_{ab}$ is the total absorbed power divided by the power per unit area of the incident beam. The monitors located inside the TFSF source measured the net power flowing into the particle and the $Q_{ab}$ was calculated by normalizing it to the source intensity. Similarly, $Q_{T}$ was calculated from the power measured in the monitors outside the TFSF source. Here, $Q_{sc}$ is the cross-section of light scatter into the substrate. The fraction of light scattered into the substrate, $f_{sub}$ is defined as the ratio of power in the scattered field region inside the Si substrate to the power in the scattered field region in air and in Si. As single particle simulations provide a good means of predicting the experimental behavior of random particle arrays [9,36], we considered a single nanoparticle during our simulations. We systematically studied the effect of nanoparticles on $Q_{T}$, $Q_{sc}$, $f_{sub}$, and $Q_{ab}$ by sweeping their structural properties. And, we comprehensively studied the ability to enhance the absorption inside substrate due to the addition of nanoparticles. Moreover, the impact of the polarization angle of incident light was explored by varying incident light polarization from 0$^\circ$ to 90$^\circ$.

3. Results and discussion

At first, we considered the nanospheres and varied their material to comprehend the feasibility of TiN-based plasmonic structures. To comprehend the impact of nanoparticles and the polarization angle of the source, we explored nanostructures by varying shapes of nanoparticle structure and polarization of light in subsequent subsections. Moreover, a detailed study of the structural variation of nanoparticles embedded in thin films was studied to study their prospects in solar cells.

3.1 Absorption enhancement by TiN-based nanoparticle

We performed a comparative analysis between alternative plasmonic material, TiN [37], and conventional plasmonic materials, Ag and Au [38]. The spectra of $Q_{T}$, and $Q_{sc}$ for 100 nm Ag, Au, and TiN nanospheres can be seen in Fig. 2(a). From our calculation, it was apparent that TiN exhibited comparable $Q_{T}$, and $Q_{sc}$ values to those previously reported for noble metals [9]. Thus, using TiN as alternative plasmonic material instead of plasmonic material like Ag, and Au is efficient and cost-effective.

 

Fig. 2. (a) $Q_{T}$, $Q_{sc}$ and (b) Electric field for 100 nm nanosphere consisting of Ag, Au, and TiN on a 10 nm thin Si$_3$N$_4$ underlayer on Si substrate.

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The electric field enhancement at the surface of Ag, Au, and Tin nanospheres were calculated using the quasi-static dipole approximation as can be seen in Fig. 2(b). Localized surface plasmon resonance (LSPR) modes are utilized in many sensing applications [39]. TiN nanospheres had red-shifted LSPR compared to Ag and Au as can be seen in Fig. 2(b). As TiN has smaller absolute real permittivity, the magnitude of field enhancement in TiN nanospheres is slightly smaller than those of Ag and Au.

To compare the performance of these plasmonic materials in a device, we calculated absorption enhancement, $g$, light absorption efficiency (LAE), and absorbed power. The external quantum efficiencies of a device by incorporating the plasmonic nanoparticles on top of a crystalline silicon ($EQE_{np}$) and a bare crystalline silicon ($EQE_{bs}$) were calculated [33]. The $g$ is a function of wavelength and given by,

The $g$ and $P_{abs}$ spectra are presented in Fig. 3. We simulated the spectra of $g$ for Ag, Au and TiN plasmonic nanospheres on a silicon for r = 50 nm, and 100 nm. For r = 50 nm, TiN showed effectively similar performance as those of Ag and Au for wavelengths longer than 780 nm wavelength. The peak value of $g$ for TiN nanoparticle was $\sim$1.2 and the peak value of Ag and Au of $g$ were $\sim$1.35 for r = 100 nm as can be seen in Fig. 3(a) (see Fig. S2 (a) of Supplement 1). The average enhancement, G for Tin, Au, and Ag were found to be 0.98859, 1.01428, and 1.05579 for r = 50 nm, respectively. It is quite clear that TiN can show comparable performance with conventional noble metals. As can be seen in Fig. 3(b), the $P_{abs}$ for TiN was higher than those of Ag and Au for 580 nm to 1100 nm wavelength and it was similar to Au for wavelengths 400 nm to 580 nm. It was evident that TiN efficiently absorbed light and consequently had comparable performance to Ag and Au. Furthermore, we calculated the LAE, which reveals how much of the incident light was absorbed by the TiN nanoparticle incorporated structure (for details see Section 2 of Supplement 1). The values of LAE for TiN nanosphere on a kesterite substrate with r = 15, 50, 75, and 100 nm were found to be 30.14%, 29.8%, 27.59%, and 22.2%, respectively.

 

Fig. 3. (a) Absorption enhancement, $g$ for plasmonic nanoparticles with different radii. (b) Absorption, $P_{abs}$ spectra for different plasmonic nanoparticles.

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3.2 Effect of the variation of nanoparticle thickness

We simulated bowtie-shaped nanoplates with various thicknesses and observed their scattering and absorbance behavior. We varied the thickness from h = 50 nm to 250 nm to determine an optimized structure for scattering cross-section. Here, t$_1$, t$_2$, and L were considered to be 10 nm, 250 nm, and 200 nm, respectively. When the thickness was 50 nm and 75 nm, $Q_{T}$ was highest for the wavelength 550 nm to 900 nm as can be seen in Fig. 4(b). As can be seen in Fig. 4(c)-(d), $Q_{sc}$ decreased and $Q_{ab}$ increased with the increase of thickness and as the thicknesses of 50 nm and 75 nm were comparatively negligible, hardly any light was absorbed and all the light was scattered. Hence, as a tradeoff, we considered an optimum thickness between 100 nm to 200 nm thickness for a great performance.

 

Fig. 4. (a) $f_{sub}$ , (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ as a function of wavelength for Tin bowtie shaped nanoplate with thickness, h= 50, 100, 150, 200, and 250 nm placed on top of a 10 nm Si$_3$N$_4$ on a Si substrate.

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Again, the most important factor for the representation of the path length enhancement of a scattering light-trapping structure is $f_{sub}$ [40]. As can be seen in Fig. 4(a), the value of $f_{sub}$ decreased significantly as the thickness was increased. For thicknesses 50 nm and 75 nm, $f_{sub}$ did not vary significantly. This allowed efficient coupling of the portion of the scattered light that was high in-plane wave vector and was evanescent in the air but propagated in silicon. As can be seen in Fig. 4(b), $Q_{T}$ decreased as the thickness increased from 50 nm to 150 nm. For h greater than 150 nm, it started to increase and the $Q_{T}$ became highest for 250 nm thickness for 1100 nm wavelength. The $Q_{sc}$ increased as the thickness decreased and $Q_{ab}$ increased as the thickness increased as can be seen from Figs. 4(c) and (d). As the thickness of 50 nm was quite small, most of the light was scattered and the TiN nanoplates did not absorb light which is evident from Figs. 4(b), (c), and (d). Hence, the optimum thickness should be considered for proper tunability of light.

When a plane wave impinges on an object or a scatterer, the energy carried by the plane wave is deflected to other directions. The analysis of optical features such as the scattering cross-section, the electric field distribution as well as the E-plane and H-plane radiation patterns, provided a systematic near and far field characterization of the nanoparticles. The electric field map was observed in the xy and zx plane for the incident wavelength 1100 nm by varying the thickness of the TiN bowtie nanoplates which is presented in Fig. 5 and Fig. 6 respectively. These structures can create LSPR where free charges accumulate precisely at the edges of the nano-gap.

 

Fig. 5. Color map showing the xy spatial distribution of |E| for TiN bowtie-shaped nanoplate placed on top of a 10 nm thin Si$_3$N$_4$ on a Si substrate with varying thickness, h = (a) 50, (b) 75, (c) 100, (d) 150, (e) 200, and (f) 250 nm.

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Fig. 6. Color map showing the zx spatial distribution of |E| for TiN bowtie-shaped nanoplate placed on top of a 10 nm thin Si$_3$N$_4$ on a Si substrate with varying thickness, h = (a) 50, (b) 75, (c) 100, (d) 150, (e) 200, and (f) 250 nm.

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When h = 50 nm and 75 nm, free charge accumulation resulted in minimal E-field enhancement and the charge density scatter along the surface as can be seen in Figs. 5(a) and (b). Charge density accommodated strong surface plasmon coupling resulting in high charge distribution along the edges and the center of the bowtie nanoplate as can be seen in Fig. 5(d). Charge localization near the edges and the center of the metal surface fell rapidly for increasing the thickness of TiN metal film as presented in Figs. 5(e) and (f) which agreed well with the previous report [15] (see Subsection 1 of Supplement 1). For 50 nm thickness, the light was almost scattered and there was no absorption. Hence, the thickness increased charge accumulation and the far-field pattern increased.

Moreover, we investigated E-field intensity on the zx-plane as can be seen in Fig. 6. When h = 50 nm, 75 nm and 100 nm, a small amount of free charge accumulated and resulted in minimal E-field enhancement at the top and bottom edges of the material surface which was consistent with the xy and yz (see Fig. S1 of Supplement 1) planes. When h increased from 150 to 250 nm, charge density accommodated strong surface plasmon coupling resulting in high charge distribution along the bottom and top corners, and the center of the bowtie nanoplate as can be seen in Figs. 6(d)-(f).

3.3 Impact of light polarization

We varied the polarization angle of the light source and calculated the optical characteristics of the nanoparticles. For the structural design with the optimized scatterings and absorption, we considered a bowtie-shaped TiN nanoparticle and varied the polarization angle of the source, and detected the optimized polarization angle, $\theta$. Here, t$_1$, t$_2$, L, and h were considered to be 10 nm, 250 nm, 200 nm, and 100 nm, respectively. The $\theta$ was varied from 0$^\circ$ to 90$^\circ$. As the $\theta$ was increased, the $Q_{T}$, $Q_{sc}$ and $Q_{ab}$ increased and the peak wavelength of $Q_{T}$ was blue-shifted as can be seen in Fig. 7(b), (d) and Fig. S2 (b) of Supplement 1.

 

Fig. 7. (a) $f_{sub}$, (b) $Q_{T}$, (c) absorbed power spectra and (d) $Q_{ab}$ for various polarization angle of the source from $\theta$=15$^\circ$ to 90$^\circ$.

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For the wavelength ranging from 550 to 650 nm, the value of $f_{sub}$ decreased significantly as the $\theta$ was increased as apparent from Fig. 7(a) (see Fig. S2 (b) of Supplement 1). For the wavelength longer than 650 nm, $f_{sub}$ decreased significantly as $\theta$ was decreased. It can be seen from Fig. 7(b) that $Q_{T}$ increased significantly as the $\theta$ was increased. As can be seen from Fig. 7(d), $Q_{ab}$ increased as the angle increased for wavelength ranged from 550 to 780 nm. For wavelengths longer than 780 nm, $Q_{ab}$ decreased as the $\theta$ was increased. When the contributions from horizontal and vertical polarization are equal, resulting in a vanishing net polarization. The quantitative calculation of the scattering light of the polarized fraction has been carried out assuming that the scattering of each small volume of the nanoplate is isotropic in the bowtie nanoplate and scattering at the edges and corners was higher. By changing $\theta$, the degree of polarization which is the quantity of the portion of an electromagnetic wave that is polarized can be controlled.

Power absorption spectra can be seen in Fig. 7(c) for various $\theta$ from 0$^\circ$ to 90$^\circ$. The absorption power spectra increased with the increase of $\theta$ till around 700 nm wavelength and for the longer wavelengths, decreased as $\theta$ increased as can be seen in Fig. 7(c).

Figures 8(a)–(f) show E-field intensity on the xy-plane for 1100 nm wavelength for bowtie nanoplates with the variation of $\theta$. When $\theta$ was smaller, charge accumulation and E-field were enhanced at the centers as can be seen in Figs. 8(a)-(c). As $\theta$ increased, the charge accumulation increased for strong surface plasmon coupling resulting in high charge distribution at the corner of the edges of the bowtie nanoplate as can be seen in Figs. 8(d)-(f). Spatial distribution of E-field intensity on the zx plane (at y=0) can be seen from Figs. 9(a)-(f) for bowtie nanoplates for incident wavelength 1100 nm with the variation of $\theta$ (see Fig. S3 of Supplement 1 for E-field distribution on zx plane at y=100 nm ). When $\theta$ = 15$^\circ$, 30$^\circ$, and 45$^\circ$, a large amount of free charge accumulated resulting in E-field enhancement at the top and bottom edges of the material surface. At the corners that were along or nearby the E-field direction, charge accumulation was higher than other corners as is apparent in Figs. 9(a)-(c). When $\theta$ increased from 60$^\circ$ to 75$^\circ$ charge distribution of the corner mentioned earlier started to decrease and other corners started to increase as can be seen in Figs. 9(a)-(c). For $\theta$ = 90$^\circ$, charge density accommodated and surface plasmon coupling resulting in charge distribution equally along the corners and the center of the bowtie nanoplate as shown in Figs. 9(f). The E-field map on the yz plane is shown in S4 of Supplement 1.

 

Fig. 8. Color map showing the xy distribution of |E| for TiN bowtie-shaped nanoplate placed on top of a 10 nm thin Si$_3$N$_4$ on a Si substrate with varying the polarization angle of the source from $\theta$=15$^\circ$ to 90$^\circ$.

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Fig. 9. Color map showing the zx spatial distribution of |E| for TiN bowtie-shaped nanoplate placed on top of a 10 nm thin Si$_3$N$_4$ on a Si substrate with varying the polarization angle of the source from $\theta$=15$^\circ$ to 90$^\circ$.

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3.4 Effect of varying the shape and thickness of dielectric material

The shapes of TiN-based nanoparticles that we investigated were a bowtie shape, a bowtie shape consisting of two triangles with a 60 nm distance from each other, and a triangular shape. We considered the bowtie shape consisting of two triangles with a 60 nm distance from each other as a tradeoff when the distance was 60 nm the $f_{sub}$ and $Q_{ab}$ had the highest value and $Q_{T}$ and $Q_{sc}$ had comparable value to the distance of 10,30,50 and 70 nm (see Fig. S5 of Supplement 1). The thickness of dielectric material Si$_3$N$_4$ was varied to evaluate the optical performance. Here, t$_2$, L, and h were considered to be 250 nm, 200 nm, and 100 nm, respectively. We varied the thickness of dielectric thin film Si$_3$N$_4$ from 5 nm to 40 nm and observed the spectra of $f_{sub}$, $Q_{T}$, $Q_{sc}$ and $Q_{ab}$ for nanoparticles (see Fig. 10 and Fig. S7 of Supplement 1). The $f_{sub}$ remained unchanged for variation of the thin film. The $Q_{T}$ and $Q_{sc}$ increased as the thickness of the dielectric increased showing consistent results deduced from Figs. 11(b)-(c). As can be seen from Figs. 10(d), the nanoparticle had maximum absorption for 30 nm dielectric thickness. The optimum thickness of Si$_3$N$_4$ thin film is necessary as a large amount of hydrogen, originating from plasma gas dissociation, incorporated in the Si$_3$N$_4$ film is driven into silicon during the metallization step, leading to excellent bulk passivation for multi-crystalline silicon solar cells. The driving force was the surface passivation effect. Silicon nitride provides very low surface recombination velocities both on p-type and n-type cells. Moreover, the anti-reflective properties of the nitride layer reduce considerably the light reflection.

 

Fig. 10. Average (a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ for TiN bowtie-shaped nanoplate for various thickness of dielectric material, Si$_3$N$_4$ on a Si substrate.

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Fig. 11. (a) $f_{sub}$, (b) $Q_{T}$, (c) $Q_{sc}$, and (d) $Q_{ab}$ as a function of wavelength for TiN bowtie-shaped nanoparticles placed on top of a 30 thin Si$_3$N$_4$ on a Si substrate.

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We calculated cross-sections for TiN particles on 10 nm and 30 nm thick Si$_3$N$_4$ on semi-infinite silicon with triangular and, bowtie-shaped nanoplates (see Fig. S6 of Supplement 1 and Fig. 11 ). Structures with 30 nm thick Si$_3$N$_4$ performed better. For 30 nm Si$_3$N$_4$ film, $Q_{T}$ had the highest value of 11 Wm$^{-2}$ when the shape was triangular nanoplate and $Q_{ab}$ had the highest value of 4.76 Wm$^{-2}$ for bow tie-shaped nanoplate. The maximum $Q_{sc}$ of 4.58 Wm$^{-2}$ and 5.23 Wm$^{-2}$ were obtained for bowtie-shaped and triangular shaped nanoplate respectively. As can be seen in Fig. 11, for the entire wavelength range, the $f_{sub}$ was much higher for triangle and bowtie than that for bow tie with a distance of 60 nm. It can be seen from Fig. 11, $Q_{T}$ and $Q_{sc}$ had the highest value for triangular-shaped nanoplates, and $Q_{ab}$ had the highest value for bow tie-shaped nanoplate. Bowtie-shaped TiN nanoparticles had a peak value of 0.42. Hence, light trapping using a triangular shape or a bowtie shape nanoparticle will be efficient.

4. Conclusions

TiN is a promising alternative plasmonic material for plasmonic and metamaterial applications in visible and near-infrared wavelengths according to the results of our comprehensive study. The LAE of plasmonic nanospheres with radii of 15 nm and 50 nm were found to be 30.14${\%}$ and 29.80${\% }$ for TiN. The value of absorption enhancement was greater than 1 for the whole wavelength at r = 15 nm TiN nanosphere. The average enhancement, G for TiN, Au, and Ag were found to be 0. 98859, 1.01428, and 1.05579 for r = 50 nm, respectively. TiN, Au, and Ag had normalized absorption power 0.98, 0.96, and 0.68, respectively, at 450 nm wavelength and TiN had greater power spectra than Ag and Au. Comparative analysis of TiN and conventional plasmonic materials suggested that TiN offers comparable performance to Ag and Au for plasmonic applications. Moreover, high light absorption efficiency ($\sim$30%) was achieved by incorporating TiN nanoparticles in Si absorber layer. We showed that particle shape, and thickness were crucial parameters for designing plasmon-enhanced solar cells. TiN particles with a large fraction of their volume close to the substrate caused effective path length enhancement leading to enhanced near-field coupling. Bowtie-shaped TiN nanoparticle had peak $f_{sub}$ of 0.42. The $f_{sub}$ had no impact on the Si$_3$N$_4$ dielectric thickness variation. Hence, the scattering cross-section of the particles can be manipulated by varying the shape, and the thickness of the particles. The insight deduced from this work will be beneficial to design TiN-based plasmonic solar cells.