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Abstract
In this paper, a new architecture comprising silicon nanoparticles inside a hole transport layer laid on a thin silicon layer is proposed to develop ultrathin film solar cells. Using generalized Mie theory, a fast analytical approach is developed to evaluate the optical absorption of the proposed structure for various geometries, polarizations and angles of incidence. The analytical results are verified through comparison with full-wave simulations, illustrating a reasonable agreement. The electrical performance of a distributed silicon nanoparticle solar cell is determined for selected configurations. To be able to predict the light-trapping in a solar cell comprising randomly distributed nanospheres, a new technique based on probability theory is developed and validated through comparison with the simulation results. Both analytical and numerical results show that the excited Mie resonant modes in the proposed structure lead to a significant enhancement in both absorption and the photo-generated current, in comparison to a conventional silicon solar cell with an equivalent volume of the active layer. In the case of random distributions, other advantages, including the simple fabrication process, indicate that the cell is a promising structure for ultrathin photovoltaics.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thin-film photovoltaics has emerged as a fascinating alternative to the conventional solar cells in generating electricity out of sunlight. These solar cells that possess multilayered structures containing an absorber material of micron thickness have been used in a wide range of applications including small-scale lighting and powering the electronic circuits [1]. This generation of solar cells addresses the need for photovoltaics with minimum material usage and potentially higher mechanically flexibility. However, the reduction of cell efficiency caused by less material consumption is a serious drawback - specially at higher wavelengths of the incident spectrum - to proceed ever thinner solar cells [2,3]. Due to that, attempts are focused to improve light absorption and increasing the photo-current generation via engineering the configuration of the cell [3–10].
Various techniques have been proposed to increase light trapping in thin-film cells, among them texturing the top and/or rear contacts and engineering the refractive index of the active layer have led to a number of configurations [9,11–13]. In terms of platforms used in thin-film cells, plasmonic gratings [14], metamaterial configurations [15] and nano-antennas [16] have shown to confine sunlight. Two-dimensional materials have also been introduced in the context of thin film photovoltaics; and interesting absorption at a fraction of nanometer thickness has been demonstrated [17,18]. The coupling between dielectric nanoparticles and quantum dots has been also used in [19,20] to achieve shown gained attentions toward light confinement beyond the diffraction limit, which can be used to enhance light absorption in solar cells. Beside these, nanowire solar cells have gained interests in photovoltaics due to their capacity to reduce reflection and to improve absorption by exciting the Fabry-Perot resonances [21,22]. The polyaniline nanowires doped with quantum dots, have also shown promising features in reducing the carrier recombination rates and improving the photo-generated carriers transport [23,24]. Despite all these efforts, drawbacks such as plasmonics loss [16,25], narrow bandwidth and dependency to the angle of incidence [16] and polarization [26,27], are serious challenges toward realizing an efficient thin-film solar cell.
The use of nanoparticles with the size of submicron has so far been mainly considered as a light harvesting scheme, or for enhancing the carrier transport from the cell absorber [28,29] in photovoltaics. In the case of light trapping, dielectric nanoparticles with spherical [30], spheroidal [31,32] have been used on the front side to focus light beneath the particle layer, or inside the active layer, to form localized absorbing centers [33]. In terms of the particle material, the use of silica nanoparticles has enabled absorption enhancement, which is the results of exciting the whispering-gallery modes [34,35]. Silicon particles have also been used for directional scattering into thin-film cells [36] and $\textrm {SiO}_2$ nanoparticles on the Ag back contact for the same purpose [37]. Plasmonic nanoparticles have also been proposed in the form of nanospheres [38–40], hemispheres [41] nano-disks [42]and nanovoids [43] to enhance light absorption in thin film solar cells. In these cases, cell structures were expected to work even without the contribution of these particles - despite achieving lower efficiencies. Using plasmon-mediated Whispering-Gallery-Modes realized by metallic nanospheres one can enhance the magnitude of the electric field in the active layer of the solar cell leading to the absorption improvement in the structure. Changing the size of the nanospheres, adding quantum dots, and using core-shell spheres are other approaches to tune the spectral range of the solar cell to absorb maximum solar energy [44,45]. Metallic (gold and silver) nanoparticles have also been investigated in polymer solar cells; Careful selection of size, shape and density of these particles has a direct influence on improving light absorption in these organic cells [46–48]. However, the potential use of nanoparticles as the active layer of a cell has not yet been fully addressed(See for instance [49]).
We here propose the concept of a particle-based ultrathin-film solar cell composed of doped silicon nanoparticles distributed on a very thin layer of n-type silicon. We describe the cell mechanism and explore the absorption and the photo-current generated via silicon nanoparticles when used as an active layer of the cell. In addition, the generalized Mie theory is applied to calculate the absorption of the cell; using this, the results obtained by full-wave analysis - obtained by numerically solving the electromagnetic wave equation - are validated and the absorption is obtained in a fast way. Different patterns of the nanoparticles are considered; in the random distribution, the possibility of coupling between two or more silicon spheres is higher than the periodic one, which results in a higher field confinement within the spheres and consequently a higher absorption for the solar cell. Finally, we demonstrate and validate a technique based on probability theory to estimate the achievable absorption in random distributions of the proposed cells.
2. Proposed solar cell with distributed silicon nanoparticles (DSN)
We begin with the three-dimensional sketch of our proposed cell as shown in Fig. 1(a). As can be seen, a single layer of p-type silicon nanoparticles, immersed in an HTL medium is laid on a thin layer of n-type silicon to form a PN junction. We assume that only a single layer of particles are deposited on the silicon layer. The nanoparticles are inside a Hole Transport Layer (HTL), which thickness is extended above the particles. Next, there is a very thin buffer layer to protect the HTL from parasitic absorption. This layer is in contact with the front electrode, which is assumed to be a Transparent Conductive Oxide (TCO) and helps in reducing the reflection from the cell surface. In order to introduce the critical dimensions, in general we consider Fig. 1(b) as a unit cell of the structure, in which a silicon nanoparticle with an small contact area is placed on a silicon layer. The overall thickness of the top layer, the buffer, the HTL on top of nanoparticles and the bottom silicon layer are denoted by $d_{\textrm {ARC}}$, $d_{\textrm {B}}$, $d_{\textrm {HTL}}$ and $d_{\textrm {n-Si}}$, respectively. The geometrical cross-section of the unit cell is a square with the side length of $W_{\textrm {cell}}$. In the next sections, we will discuss about the impact of nanoparticle shape on its electrical performance of the structure.
Fig. 1. (a) Three-dimensional schematic of a nanoparticle-based cell (b) A unit cell of the structure composed of silicon nanoparticle. The particle may have different shapes. In theses figures, they have a quasi-spherical shape with a small contact area.
In terms of typical materials, various HTLs can be conceived. Herein, and to study the electrical properties of a typical DSN cell, we use spiro-OMeTAD, which has a fairly high contrast refractive index in comparison to silicon, and also been used in silicon nanoparticles-based perovskite solar cells [50]. Note that, that spiro-OMeTAD is a material with a low carrier mobility, of the order of $10^{-3}\,\frac {\textrm {cm}^{2}}{\textrm {Vs}}$. Thus, the thickness $d_{\textrm {HTL}}$ has an upper constraint, proportional to the carrier lifetime in in the HTL. Furthermore, $\textrm {MoO}_3$ and ITO can be used as two thin layers on the front layer; $\textrm {MoO}_3$ is used as a buffer layer above the HTL and ITO is to act as an anti-reflection layer (Note that the surface of the silicon nanoparticles can also be weakly passivated by an ultrathin - of $\approx 2 \,\textrm {nm}$ - $\textrm {SiO}_2$). The likely procedure for the preparation of the cell absorbing medium is to deposit the N-type silicon nanoparticles through spin-coating to form a layer. Then, P-type silicon nanoparticles prepared in chlorobenzene solution are added to the weight of Spiro-OMeTAD and the resultant solution is again spin-coated on the silicon layer.
The operation of the DSN is based on excitation of Mie resonances in the presence of sunlight, which confines the light inside silicon nanoparticles resulting in enhanced light absorption. To investigate the optical absorption, we simulate electromagnetic response of the cell structure in CST Studio Suite, in which the Finite Element Method (FEM) was chosen as the solver [51]. We here assume that there is a dense periodic distribution of the nanoparticles such that in each unit cell $W = d_{\textrm {p-Si}}$ (see Fig. 1(b)). The dense distribution is also beneficial as it increases light absorption and hence improves the photo-generated current. By increasing the distance between the nanoparticles, the absorptance and accordingly the cell efficiency are reduced. This is mainly due to use of lower absorber in the cell. Each unit cell is composed of a single nanoparticle - as shown in Fig. 1(a); Periodic boundary conditions are applied on the cell walls and the incident radiation is considered to be normal to the cell front. As a reference system for the absorption comparison, we choose a conventional solar cell with an absorbing material equivalent to the nanoparticle cell structure (Consequently, $d_{\textrm {n-Si}} \neq d'_{\textrm {n-Si}}$, where $d'_{\textrm {n-Si}}$ is the equivalent layer thickness in a planar structure). Figure 2(a) shows the absorption enhancement for cells with $d_{\textrm {p-Si}} = 200, 400$ and $600$ nm, in which $d_{\textrm {ARC}} = 50\,\textrm {nm}$, $d_{\textrm {B}} = 5\,\textrm {nm}$, $d_{\textrm {HTL}} = 25\,\textrm {nm}$ and $d_{\textrm {n-Si}}= 30\,\textrm {nm}$. As can be seen, the absorption enhancement is significant at several wavelength intervals of the solar spectrum, particularly at higher wavelengths; For the case that $d_{\textrm {p-Si}} = 400$ nm at $\lambda = 1005$ nm, the absorption enhancement reaches $34.2$, that although below the Yablonovich limit (See [49]), shows a great enhancement for an ultrathin solar cell. For nanoparticles dimension of $600$ nm, the absorption enhancement reaches above four between $\lambda = 800$ nm and $\lambda = 1000$ nm in which silicon has intrinsically low absorption. As we will explain, by using nanoparticles of various sizes higher enhancement can be extended to a wider range of wavelengths. At lower wavelengths the enhancement reaches the order of a planar solar cell, however note that the optical path length is also reduced at this wavelengths interval, thus the absorption itself is naturally high for silicon. Figure 2(b) shows the absolute absorption $\mathcal {A}$ when $d_{\textrm {p-Si}} = 600$ nm, in comparison to the planar structure with equivalent volume. This is clear that through a wide range of solar spectrum, the absorption is improved. Particularly, this is the case at higher wavelengths where silicon shows intrinsically weak absorption. We use $\mathcal {A}$ and calculate the photo-current generated [52]:
where $c$ is the speed of light, $e$ is the electron charge, $h$ is Planck’s constant, $\eta _c$ is collection efficiency, $\mathcal {A}(\lambda )$ is the absorption of the solar cell, and $S(\lambda )$ is AM 1.5G solar spectrum [53]. Then, with the assumption that each absorbed photon releases an electron-hole pair (i.e. $\eta _c = 1$), the photo-generated current density is obtained via Eq. (1). The $J_{\textrm {ph}}$ for the two compared structures are shown in Fig. 2(b). To use Eq. (1), we first consider a unit cell composed of a single nanoparticle as shown in Fig. 2(b). Under normal illumination of a plane wave with TE polarization and normalized power density through the whole solar spectrum, the total absorbed power in the silicon areas of the cell is computed numerically.Fig. 2. (a) The absorption enhancement simulated in a DSN solar cell at $300\,\textrm {nm}<\lambda <1100\,\textrm {nm}$ for $d_{\textrm {p-Si}} = 200, 400$ and $600$ nm, in which $d_{\textrm {ARC}} = 50\,\textrm {nm}$, $d_{\textrm {B}} = 5\,\textrm {nm}$, $d_{\textrm {HTL}} = 25\,\textrm {nm}$ and $d_{\textrm {n-Si}}= 30\,\textrm {nm}$, as compared with an equivalent planar silicon solar cell. (b) The absolute absorptance $\mathcal {A}$ of a DSN solar cell with $d_{\textrm {p-Si}} = 600$ nm, compared with the absorption of equivalent planar silicon solar cell. (c) The absorption enhancement for $d_{\textrm {p-Si}} = 400\,\textrm {nm}$ and $d_{\textrm {n-Si}} = 30, 100, 300\,\textrm {nm}$, $d_{\textrm {ARC}} = 50\,\textrm {nm}$ and $d_{\textrm {HTL}} = 25\,\textrm {nm}$. The 3D simulations are obtained using CST.
Figure 2(c) shows the absorption enhancement when $d_{\textrm {p-Si}} = 400$ nm for various values of the silicon layer thickness. The figure shows that larger values of the absorption enhancement happen at higher wavelengths; It is also shown that even for ultrathin layers, there are wavelength intervals at which the absorption enhancement is higher than cells with thicker silicon layers.
3. Analysis of DSN solar cells using Mie theory
In the case of a very thin layer of silicon ($d_{\textrm {n-Si}} \ll d_{\textrm {p-Si}}$) we can derive analytic expressions for the absorption response of a cell. This feature is advantageous as it takes much less time to find the absorption of the cell, than computing with a commercial software like CST or COMSOL. In addition, this approach can be used to validate the result of simulations. For this, we begin by describing the electromagnetic (EM) fields in a dielectric medium composed of nanospheres with different size and optical properties distributed inside a homogeneous medium. To predict the behavior of the structure, we use the generalized Mie theory demonstrated in [54], to find the general solution for the EM scattering in this system. This model also accounts for the coupling between the spheres.
Figure 3 shows the schematic of this system, where $t^{\textrm {th}}$ sphere with the radius of $a^{(t)}$ is centered at $(X^{(t)},Y^{(t)},Z^{(t)})$ and light in the form of a plane wave with the wave number $k$ is incident to this medium. We first consider the $t^{\textrm {th}}$ nanosphere; the total electric field interacting with this sphere consists of three parts: the incident field - from the illuminated light - $\mathbf {E}_i$, the scattered field from the sphere $\mathbf {E}_s$, and the internal field of the sphere $\mathbf {E}_I$. Following the general framework for solving the vector wave equation, we define the two vector harmonics $\mathbf {M}$ and $\mathbf {N}$ as [55]
Fig. 3. A system of nanospheres each with the radius $a^{(t)}$, which are distributed arbitrarily in the space and illuminated by a plane wave with the wave number $k$.
After deriving the EM field coefficients we use Eq. (6) and calculate the total absorption of the distributed nanoparticles. For a normalized incident power, the total absorptance of nanoparticles can be expressed by
4. Effect of particle distributions
In this section we look at the impact of distribution as well as using nanoparticles of different size in the cell structure. For this, we simulate the absorptance for the two structures shown in Fig. 4. Assume that the structure is comprised of 21 nanospheres - including 9 spheres with the radius of 200 nm and 12 with the radius of 100 nm, which are distributed in a square unit cell with W = 1800 nm. The size and number of nanoparticles are selected based on parametric optimization, such that the absorption enhancement per silicon usage becomes maximum in the given unit cell. The corresponding absorptance for a normal incidence are depicted in Figs. 4(a) and (b) for the periodic and random distributions, respectively. In terms of the permittivity of silicon and silver, the experimental data in [57,58] are used in calculations. In addition, to verify the accuracy of the analytical method, we compute the absorption via full-wave analysis in CST. For this, structures shown in Fig. 4 are normally illuminated by a plane wave while the cell boundaries are set to be open, a standard boundary condition defined in CST that mimic an unbounded free space around the solar cell. The use of plane waves is due to the far distance of the Sun, as the light source, from the Earth, which makes the plane wave incidence a reasonable assumption as considered in other works in this area [30,59].
Fig. 4. The core of a solar cell based on silicon nanoparticles,(a) periodically and (c) randomly, inside the HTL layer (shown in green). (b) and (d) show the side views, in which the active medium is laid on a silver layer.
From Fig. 5(a) and (b) the analytically calculated absorption has a reasonable agreement with computed results in CST, for both periodic and random DSN cells. it also verifies the accuracy of the analytical method. The difference between the two methods arises mainly from EM modeling of the non-homogeneous medium; while by using the Mie theory we consider that nanoparticles are inside a homogeneous medium, here the HTL thickness is finite. In addition, we have used image theory to consider the effect of the back contact. In this modeling, the back contact is removed and its effect is modeled by For this, the metal layer is approximated to be Perfect Electric Conductor (PEC) and is removed. Instead, image particles, which are identical particles to real nanoparticles and have the same distribution but are assumed to be located under the back contact [60]. The n-type silicon layer can also contribute in this discrepancy, although due to its rather trivial thickness - in comparison to the nanoparticles diameter - it causes a negligible effect (see [61]).
Fig. 5. Absorption of (a) periodic and (b) random DSN cell calculated using the generalized Mie theory (red) and full-wave(dashed blue).
Next, we study the effect of various random distributions to see how it affects the absorption of the structure. For this, we consider three different distributions of the nanospheres as shown in Fig. 6, and compute their absorption for two polarizations of TE and TM (see Fig. 7). Moreover, the results of these DSN cells are compared with the absorption obtained for a regular silicon solar cell with an equivalent volume of the active layer – with a thickness of 108 nm in our proposed architecture. We have also included the absorption of a periodic structure - with the same number of nanoparticles in the figure. As shown in Fig. 7, the absorption of random and periodic solar cells are much higher than that of a conventional structure; This demonstrates the superiority of the method proposed in this paper over simple thin-film solar cells. Although the same amount of crystalline silicon is used in all presented solar cells, the average absorption in random and periodic structures is much higher than the simple one (Note that, although the structures shown in Fig. 6 are chosen completely arbitrary, Figs. 7, 8 show that no remarkable discrepancy is observed in the results). In addition, this can be seen that periodicity does not make a serious change in the absorption. Considering the fact that fabrication of a solar cell with randomly distributed nanospheres is much easier than a cell with periodic distributions, a randomly DSN is an attractive design in terms of lower manufacturing costs.
Fig. 6. Schematic of three random DSN cells, each with 9 large silicon nanospheres with $a^{(t)} = 200\,{\textrm {nm} }$ together with 12 smaller ($a^{(t')} = 100\,{\textrm {nm} }$) ones.
Fig. 7. Absorption of the proposed solar cell with distributed silicon spheres, computed for (a) TE and (b) TM polarizations of incident light. The Random i (i = 1,2,3), refers to the three distributions shown in Fig. 6. The results are compared with the absorption of an equivalent conventional cell.
Fig. 8. The Photo-generated current versus different angles of incidence for the periodic and randomly DSN solar cells. i (i = 1,2,3), refers to the three distributions shown in Fig. 6. Results are compared with the equivalent conventional solar cell.
Using Eq. (1) the average $J_{\textrm {ph}}$ for solar cells with random spheres of Fig. 6 are $11.87\,{\textrm {mA}/\textrm {cm}^{2}}$ and $11.30\,{\textrm {mA}/\textrm {cm}^{2}}$ for TM and TE polarizations, respectively. In the case of a periodic solar cell and the equivalent conventional solar cell, $J_{\textrm {ph}}$ reaches $11.0 \,{\textrm {mA}/\textrm {cm}^{2}}$ and $4.63 \,{\textrm {mA}/\textrm {cm}^{2}}$, respectively ( for both polarizations). Next, we compute dependence of the proposed structures to the angle of the incident light. Figure 8 shows the $J_{\textrm {ph}}$ - averaged over the TE and TM modes - versus the angles of incidence in random (see Fig. 6) and equivalent periodic solar cells. As shown in this figure, the current reaches about $12\,\textrm {mA}/\textrm {cm}^{2}$ at normal incidence; This value shows a great enhancement as compared with a conventional solar cell with equivalent silicon.
The photo-generated current enhancement (with respect to an equivalent conventional thin-film cell) for the random solar cell is then 2.56 and 2.44 for TM and TE polarization, respectively and for a periodic DSN cell it reaches 2.37, for both polarizations. These results are comparable with those reported in the literature; for instance in [16,25,26,62] the enhancement factor is below 1.5 and in other notable reports, this value reaches 2.09 and 2.18 in [52] and [63], respectively.
In the next case study, we cast a statistical view to parameters of random DSN cells. For this, we consider a cell with the same area as in the previous case studies, but with nine identical silicon nanospheres. For this study, we consider 15 random distributions of the silicon nanoparticles, wherein the particles locations are calculated using a computer code generating random numbers. First, we compare the average absorption of these structures with a periodic structure and an equivalent conventional solar cell. The results are calculated using the generalized Mie theory and shown in Fig. 9. As can be seen, the average absorption of a cell with random distributions is considerably increased for a wide range of solar spectrum, in comparison to the conventional solar cell. Moreover, random DSN cells have higher absorption than the periodic ones; The enhancement in light trapping in disordered cell arrangements has also been addressed in [64].
Fig. 9. The average absorption of 15 different random solar cells, compared with the absorption of a periodic and an equivalent conventional solar cell.
We then, compare photo-current generated by these solar cells in Table 1. The mean of $J_{\textrm {ph}}$ reaches $7.87\,{\textrm {mA}/\textrm {cm}^{2}}$, which experiences a $13\%$ enhancement in comparison to the periodic cell. This indicates the capacity of a random DSN cell to be used as an ultrathin-film solar cell.
Table 1. The photo-generated current of 15 different random solar cells, together with the current in the equivalent periodic structure. Each cell is composed of nine identical silicon nanospheres of radius 200 nm inside a HTL.
4.1 Discussion on the electrical performance
The electrical simulation of DSN cells is performed using finite element analysis in COMSOL. First, we perform the electromagnetic simulation of the structure using the RF Module, to compute the absorbed power $P_{\textrm {abs}}$- in $\textrm {Wm}^{-3}$ - inside the silicon nanoparticle and the silicon layer under it. Then, by dividing the computed photocurrent (see Eq. (1)) on surface-to-volume ratio of the cell absorbers, the carrier generation rate - in $\textrm {m}^{-3}s^{-1}$ – is obtained at each point of the active layer. Now, using the Semiconductor module of COMSOL, and having the carrier generation rate from previous steps, we derive the electrical response of the cell. The cell structure is assumed to be periodic with a unit cell shown in Fig. 1(b). We assume that nanoparticles form a dense layer such that $W_{\textrm {cell}} = 2R_{\textrm {p-Si}}+40\,[\textrm {nm}]$, where $R_{\textrm {p-Si}}$ denotes the radius of the nanosphere. Other geometrical parameters are assumed to be $d_{\textrm {ARC}} = 50\,\textrm {nm}$, $d_{\textrm {HTL}}= 50\,\textrm {nm}$, and $d_{\textrm {n-Si}}= 30\,\textrm {nm}$. With regard to the nanoparticle configuration, we choose it to have three different shapes of spherical, quasi-spherical and in the form of nano-disk. We note that despite their gripping resonant properties, exactly spherical nanoparticles are in practice hard to achieve; There are however techniques to fabricate them via Chemical Vapor Deposition(CVD) method [65]. In the case of (quasi-)spherical nanoparticles, $R_{\textrm {p-Si}} = 200\,{\textrm {and}}\,300\,\textrm {nm}$, and for the nano-disk we assumed that it has a circular cross-section with the radii of 200 nm and a volume equal to that of a spherical nanoparticle with $R_{\textrm {p-Si}} = 200\,\textrm {nm}$. In terms of materials, we assume that the HTL medium is Spiro-OMETAD, the buffer layer is $\textrm {MoO}_3$ and the top layer is ITO. In addition, an ultrathin oxide (with expected 2-4 nm thickness) surrounds the silicon nanoparticle. However, for the sake of simplicity, we do not include it in the simulations. Moreover, the refractive index and band gap properties of silicon and Spiro-OMETAD and ITO including are taken from the literature.
We first perform the electromagnetic simulation of the structure, to compute the absorbed power $P_{\textrm {abs}}$- in $\textrm {Wm}^{-3}$ - inside the nanoparticle and the silicon layer, under one sun illumination. Next, the $P_{\textrm {abs}}$ is used to compute the carriers generation rate - in $\textrm {m}^{-3}s^{-1}$ - at each point of the unit cell. Next, we perform electrical simulations. First, we consider that there exists a contact area between the nanoparticle and silicon layer, that facilitates carrier transport toward the electrodes. This contact area is shown in the lateral sketch of studied unit cells in Fig. 10(a) and (b). As can be seen in Fig. 10(a), in the case of spherical nanoparticles, they are required to be embedded in the silicon layer. We consider that the embedding thickness $t_{\textrm {EM}}$ to be 30 nm, 20 nm and 10 nm. As we will see, to achieve appropriate efficiency, it not necessary to have a deeply embedded nanoparticle and even a very small contact area is sufficient for proper carrier transport. In the case of quasi-spherical nanoparticle, we assume that a sphere is cropped to a value corresponding to the embedding situation $t_{\textrm {EM}} = 30$ nm. We choose the doping densities of the nanoparticle and the layer to be $10^{17}$ and $10^{19}\, \textrm {cm}^{-3}$, respectively. These values have been chosen based on the result of parametric optimization performed on the proposed unit cells. After performing the simulations, the electrical properties of the unit cells, including the short circuit current $J_{\textrm {sc}}$, open circuit voltage $V_{\textrm {oc}}$, Fill Factor $FF$, efficiency $\eta$ and the equivalent thickness of the cell Th. are tabulated in Table 2. Moreover, these results are compared with the data reported for very thin silicon solar cells.
Fig. 10. The lateral cross section of nanoparticles studied to obtained the electrical properties of the DSN solar cell. (a) Shows a unit cell composed of an embedded(quasi) spherical nanoparticle (b) shows a nano-disk with a volume to a spherical particle.(c) and (d) show the profile of Carrier Generation Rate (C. Gen. Rate) in $\textrm {m}^{-3}s^{-1}$ in a quasi-spherical nanoparticle of $d_{\textrm {p-Si}} = 400\,{\textrm {nm}}$ and a nano-disk of the same volume and identical cross-section at V = 0.55 V. Arrows show the distribution and strength of the total current densities inside the cross section.
Table 2. The electrical properties of DSN cells with various configuration of the nanoparticles
As can be seen, $\eta$ of a DSN cells reaches 9.8% for a cell comprising spherical particles with the dimension $d_{\textrm {p-Si}} =600\,\textrm {nm}$, which is embedded to the below layer with the thickness $t_{\textrm {EM}} = 10\,\textrm {nm}$. In the case of spherical embedded nanoparticles, the cell has fairly higher efficiency, which can be attributed to absorption of high number of resonant modes. We note that the absorption and accordingly the electrical performance of cell composed of nano-disks is a function of their diameter-to-length ratio; i.e. as this ratio approaches to unity (i.e. close to spherical shape), its absorption increases [66]. Despite this, a cell with non-spherical shapes - but with similar dimension - do not have a remarkably different efficiency. We believe that this is an interesting point as in practice the available nanoparticles, may have various shapes. The concentration of absorptance in the particle center leads to higher carrier generation rate in that area and thus, presence of the higher current density(see Fig. 10(c)). The generation rate is also depicted for the nano-disk in Fig. 10(d). As can be seen, In contrast to the distribution of the generation rate in nano-disk, which is much more consistent inside the particle, the generation rate in a quasi-spherical particle is highly concentrated in the core of the particle. In addition, the flow of carriers in the quasi-spherical case is remarkably limited to a small area on top of the particle. As a result, the impact of surface recombination on the particle is mainly limited to its upper surface. We also note that, these results are obtained for a DSN solar cell with an equivalent silicon layer of less than a micron. To compare the calculated efficiencies, the table shows results of relevant studies on ultrathin solar cells.
5. Fast estimating of absorption in a random DSN solar cell
We illustrated in the previous sections that the response of a random distribution of silicon nanospheres can be obtained via the generalized Mie theory. Moreover, the tolerance between the absorption as well as the photo-generated current of various samples - with similar particle densities - are small. Although these results are obtained for certain particle densities, they clearly indicate the possibility of taking a unique approach to estimate key parameters of a DSN solar cell with a various number of nanoparticles. This method is based on the probability of existing various groups of nanoparticles in a unit cell. With that, we only need to know about the average density of the distributed nanoparticles in the cell structure, which is practically known. For this, we define the filling ratio $F_{\textrm {R}}$ as
where $S^{\textrm {cell}}$ is the total area of the cell and $N'$ is the total number of silicon nanospheres (as noted in Eq. (12)), each with the cross-section $S^{\textrm {p}}$. We note that $F_{\textrm {R}}$ is large for dense distributions and therefore, the absorption normally increases in these structures.The nanospheres form clusters in a structure with random distribution. By a cluster, we mean that $t$ nanoparticles are close enough that there exists a strong coupling between them. With that, we assume that the distance $d_{j_0,j}$ between the centers of two adjacent nanoparticles is $d_{j_0,j} < 2a^{(j)}+d'$, where $d'$ depends on the size and material properties of nanoparticles. Figure 11 shows the profile of the electric fields in two clusters with two and three identical nanospheres of radius 200 nm. As can be seen, by sufficiently increasing the distance between the particles of a cluster, the contribution of their scattered fields on the internal fields becomes negligible. Using three-dimensional software simulation, we found that for nanospheres with $a^{(j)} \approx 200\,{\textrm {nm}}$ inside a medium with a refractive index of $\approx 1.8$ (such as spiro-OMETAD), if $d_{j_0,j} > 2a^{(j)}+100\,{\textrm {nm}}$, then the coupling between the adjacent nanoparticles becomes weak enough to be neglected.
Fig. 11. The profile of the electric field in different clusters of silicon nanospheres at $\lambda = 1000\,{\textrm {nm}}$, computed in CST. In (a) and (c) there are clusters with two and three silicon nanospheres. In (b) and (d) there is a distance of 200 nm between the nanoparticles.
In studying the EM response of randomly distributed nanoparticles, one way is to choose a computational domain size $S^{\textrm {unit}}$ proportional to the filling ratio with $F_{\textrm {R}}S^{\textrm {unit}} = N\times S^{\textrm {P}}$, in which $N$ is the number of nanoparticles within the considered unit cell. $S^{\textrm {unit}}$ should be sufficiently large to host at least one nanoparticle. We here assume that $S^{\textrm {unit}}$ is fixed to the area of a square with $W = 1800\,{\textrm {nm}}$ but $N$ can vary up to nine nanoparticles. Now, the total absorptance of $N$ nanoparticles distributed randomly in the form of several clusters within the unit cell can be expressed by
Table 3. Values of $g(N,t)$, for $\{N,t\} = \{1,2,\ldots ,9\}$.
Now, we evaluate the effectiveness of this method by computing the absorption of a randomly DSN cells. The absorption efficiencies of the likely cluster are computed separately. Then, by inserting to Eq. (15), we find the total efficiency. The results are shown in Fig. 12. To validate the obtained absorption, we compare the estimated results with those computed using the extended Mie theory. From the figure, there exists a very good agreement between the two graphs through the whole spectrum, which shows that the technique is applicable for the described DSN cell.
Fig. 12. Average absorption of 15 different random DSN cells each with 9 silicon nanospheres with the radius of 200 nm, calculated via extended Mie theory. These results are compared with the absorption calculated using the probability method.
6. Conclusion
In this paper, we presented an ultrathin-film solar cell with crystalline silicon spheres as active resonators. Solar cells with random or periodic distribution of silicon spheres showed great potential to trap sunlight due to the Mie resonances spanned in a wide frequency bandwidth, different angles of incident and polarizations. The proposed structures were analyzed with generalized Mie theory and validated with numerical full-wave simulations. The results showed that using DSN solar cells can significantly increase the absorption, in comparison to the conventional cells with the same amount of crystalline silicon. Also, random solar cells, which have an easier fabrication process, showed higher absorption than the periodic ones. The average of photo-current enhancement for random solar cells were 2.56 and 2.44 for TM and TE polarizations, respectively. This parameter reaches 2.37 for the periodic solar cell. In addition, we demonstrated in detail the simulation procedure to evaluate the absorptance together with the electrical performance of typical DSN solar cells. Finally, we proposed and validated a new method based on the probability of particle distributions to estimate the absorption of the random DSN solar cell.
Funding
University of Tehran Science and Technology Park (180/99772); Iran National Science Foundation.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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