Beyond 30% Conversion Efficiency in Silicon Solar Cells: A Numerical Demonstration

We demonstrate through precise numerical simulations the possibility of flexible, thin-film solar cells, consisting of crystalline silicon, to achieve power conversion efficiency of 31%. Our optimized photonic crystal architecture consists of a 15 μm thick cell patterned with inverted micro-pyramids with lattice spacing comparable to the wavelength of near-infrared light, enabling strong wave-interference based light trapping and absorption. Unlike previous photonic crystal designs, photogenerated charge carrier flow is guided to a grid of interdigitated back contacts with optimized geometry to minimize Auger recombination losses due to lateral current flow. Front and back surface fields provided by optimized Gaussian doping profiles are shown to play a vital role in enhancing surface passivation. We carefully delineate the drop in power conversion efficiency when surface recombination velocities exceed 100 cm/s and the doping profiles deviate from prescribed values. These results are obtained by exact numerical simulation of Maxwell’s wave equations for light propagation throughout the cell architecture and a state-of-the-art model for charge carrier transport and Auger recombination.

and 400 μm-thickness 31 . However, the feature-sizes of traditional inverted pyramid cells are typically 10 μm or more and light-absorption in such cells falls below the Lambertian ray-trapping limit. Traditional ray-trapping architectures require thick silicon (~160-400 μm) to achieve sufficient light absorption, with concomitant bulk carrier recombination that usually limits the conversion efficiency to below 27%. In contrast, our light-trapping geometry employs inverted pyramids with base-lengths ranging between 1.3-3.1 μm. This allows our cells to achieve beyond-Lambertian light-absorption through strong wave-interference effects. Using only 3-20 μm-thick silicon, resulting in low bulk-recombination loss, our silicon solar cells are projected to achieve up to 31% conversion efficiency, using realistic values of surface recombination, Auger recombination and overall carrier lifetime.
Although the surface of our silicon solar cell is patterned along a plane that is perpendicular to the incident light, the light-propagation characteristics are considerably richer than widely-studied grating-coupled waveguides. Our photonic crystal refracts and diffracts incoming light to numerous wave-vectors that are nearly parallel to the air-silicon interface. These wave-vectors couple to and experience the long-lifetime slow-light modes of the PhC. Vortex-like flow of the electromagnetic Poynting vector is evident in high density of optical resonances throughout the 800-1200 nm range. These modes are evidence of an enhancement of the overall electromagnetic density of states over this wavelength range and are characteristic of the higher bands of a photonic crystal. In contrast, the grating couplers exhibit a much narrower coupling band-width, typically about 10% of center frequency [33][34][35][36][37] . Figure 1 shows the schematic of our PhC-IBC cell. The front surface of the solar cell is textured with a square lattice of inverted micro-pyramids of lattice constant a. Such inverted pyramids are fabricated by KOH etching of the (100) surface of silicon, exposing the (111) surfaces and resulting in a pyramid side-wall angle of 54.7°2 1 . The cell has a dual-layer antireflection coating (ARC) of refractive indices n 1 and n 2 and thicknesses t 1 and t 2 , respectively. This ARC layer also acts as part of the front passivation of the cell. We consider c-Si cells with thickness (H) ranging over 3-20 μm. The p-type bulk is assumed to have a uniform doping concentration of 5 × 10 15 cm −3 . Both the front surface field (FSF) and the base consist of highly doped p-regions (denoted by p + ) with Gaussian doping profiles. Similarly, the emitter contains a highly n-doped region (n + ) with Gaussian doping profile. The peak doping concentrations of n + and p + regions are denoted by N n0 and N p0 , respectively. The corresponding Gaussian doping profiles are σ

Solar cell Geometry and Numerical Details
Here, z refers to the direction of the Gaussian variation and σ i denotes the depth of the doping profile. The widths of the base and emitter regions are assumed to be w pdop and w ndop . The separation between the edges of emitter and base is w pn . A rear passivation layer covers all back surfaces of the cell where the electrode fingers do not make direct contact with the n + and p + regions. The width of the base (emitter) contact, touching the p + (n + ) region, is denoted by w pcon (w ncon ). The emitter and base contacts extend below the rear passivation and acts as back-reflector for the cell.
A stable FDTD scheme, implemented using open source software package Electromagnetic Template Library (EMTL) 38 , is used to simulate Maxwell's equations and optimize the light-trapping performance of the c-Si solar cell. A unit cell of the c-Si inverted pyramid PhC is used for 3D FDTD computations. Perfectly matched layers (PML) are applied at the top and bottom boundary planes (normal to z-direction) of the computation domain. Periodic boundary conditions are assumed along the x and y− directions. The top of the inverted pyramids are coated with dual ARC layers. The bottom of the c-Si is coated with a 50 nm SiO 2 passivation layer (with refractive index 1.45), backed by a perfect electric conductor (PEC), acting as a back-reflector to the sunlight. The use of SiO 2 buffer layer reduces parasitic absorption losses in real-world back contact such as silver 39 and justifies the use of a PEC to simulate the back-reflector. Calculation of absorbed photon density in our PhC solar cell is a two-step Figure 1. Geometry of the proposed inverted-pyramid photonic crystal IBC solar cell. The front surface of the cell is textured with a square lattice of inverted pyramids and coated with dual-layer ARC with refractive indices n 1 and n 2 . The thickness of the silicon layer is given by H.The p-type bulk has a uniform doping concentration of 5 × 10 15 cm −3 . w pdop and w ndop denote the widths of the base and emitter dopings, respectively. The base and emitter contact widths are denoted by w pcon and w ncon , respectively. w pn represents the distance between the edges of the base and emitter regions of the cell.
process. In both steps, the cell is illuminated with a broadband plane wave, incident from +z-direction. In the first step, the incident wave has significant energy in the 300-1100 nm spectral range. In the second step, we accurately model solar absorption in the 1100-1200 nm range. This latter absorption in c-Si involves both electronic bandgap narrowing (BGN) 40 and phonon-assisted optical absorption comprising the Urbach edge [41][42][43][44] . As we show in sec. 3, the second effect is insignificant in conventional Lambertian light-trapping based solar cells but contributes significant sub-gap solar absorption in our PhC solar cell. In the second FDTD computation, we use an incident plane wave with significant energy in the 1100-1200 nm spectral range. A detailed model of the complex refractive index of c-Si in the 1100-1200 nm wavelength range appears in the "Methods" section.
Combining the results of the separate FDTD computations, we calculate the absorption coefficient of the c-Si over 300-1200 nm wavelength range as where R(λ) and T(λ) are the reflection and transmission coefficients of the structure. The maximum achievable photo current density (MAPD) of the cell under AM1.5G illumination is given by: MAPD nm nm 300 1200 Here, I(λ) is the intensity of the AM1.5G spectrum. We assume that each absorbed photon creates a single electron-hole pair. The short-circuit current (J SC ) of an ideal cell, without any surface and bulk recombination losses, coincides with J MAPD . The left panel of Fig. 2 shows a sample optical generation profile for a 10 μm-thick cell obtained through our FDTD calculation. The actual 3D profile has been integrated along the y-direction and converted into an equivalent 2D profile. This 2D profile is then repeated over many unit cells of the inverted pyramid PhC to cover the entire width of the 2D transport model of the IBC cell (shown in the right panel of Fig. 2). For the purpose of clarity, the carrier generation profile only over 3 unit cells is shown. The 2D carrier transport calculations are performed using Sentaurus 45 assuming a temperature of 25 °C. In all the calculations, the Shockley-Read-Hall (SRH) lifetime, τ SRH , is assumed to be 10 ms (except for the cases where we study the performance and optimum thicknesses of our solar cell as a function of τ SRH ) according to the experimental results obtained in 11 . The Auger recombination in our carrier-transport calculations is implemented using the state-of-the-art improved Auger model 11 . The surface recombination at the Si − SiO 2 interface is implemented using a microscopic, SRH recombination statistics-based model 46 (more details are given in the "Methods" section) that complies with the experimental data of 11,[47][48][49] . In all our computations involving inverted-pyramid PhC solar cells, the contact SRVs are chosen to be 10 cm/s. This low contact SRV allows us to compare the performance of our solar cell to the benchmarks that completely neglect surface recombination 10 . In addition, recent experimental developments suggest that passivated contacts allow realization of IBC cell with much lower contact SRVs than conventional contacts 50 . Nevertheless, we discuss the effect of higher contact SRVs on the performance of our cell in sec. 4.
One approach to achieve low SRVs is through passivated carrier-selective contacts using highly doped polycrystalline Si (poly-Si) thin films. Experimental study 51 has shown that a 20 nm-thick poly-Si front-contact leads to parasitic absorption loss of ~1.1 mA/cm 2 . In contrast, in our IBC design, poly-Si layers with similar thickness would be placed at the back of the cell. As shown by the integrated optical generation profile of Fig. 2, our photonic crystal architecture captures the vast majority of sunlight in the upper parts of the cell. Two orders of magnitude more photogenerated carrier density appears in the upper regions relative to the bottom of the cell. Accordingly, we expect that parasitic absorption in poly-Si bottom contacts is negligible in our PhC-IBC cell.
For all our charge-carrier transport calculations, we assume that the the contacts are ideal (i.e. zero resistivity). The resistive losses in the bulk and along the highly doped n + and p + regions depend on the doping concentrations and are automatically considered in our 2D drift-diffusion calculations. We account for the losses at the  Fig. 1. The optical generation, shown on the left side, is calculated using 3D FDTD computation that involves a unit cell of the inverted pyramid photonic crystal. This 3D generation profile is then integrated over y-direction and repeated over multiple photonic crystal unit cells to cover the entire width of the IBC cell. For illustration purpose, we have only shown the optical generation in a 10 μm-thick cell but each cell-thickness under consideration involves its own optical generation profile. Periodic boundary condition (PBC) is used along xdirection in our transport calculations. (2019) 9:12482 | https://doi.org/10.1038/s41598-019-48981-w www.nature.com/scientificreports www.nature.com/scientificreports/ semiconductor-dielectric and semiconductor-metal interfaces through SRH recombination statistics 46 and contact surface recombination velocities, respectively.

Light-Trapping Optimization
C-Si thin-films with low doping can provide solar cells with high open-circuit voltage due to reduced bulk recombination, but usually suffer from poor solar absorption. Maximization of light-trapping capability in c-Si thin-film is one of the most important aspects of high-efficiency, ultra-thin silicon solar cell design. We show below that 3-20 μm-thick c-Si inverted micro-pyramid PhCs are highly effective for wave-interference based light-trapping leading to solar absorption, comparable to (and in some cases more than) that of the 165-400 μm-thick conventional cells. Our simulations reveal that a dual-layer ARC with n 1 = 1.4, t 1 = 45 nm, n 2 = 2.6 and t 2 = 100 nm exhibits the best anti-reflection behavior, irrespective of the cell-thickness. Figure 3 shows the optimization results for the lattice constant, a, of c-Si inverted-pyramid PhCs with H = 3, 5, 7, 10, 15 μm for MAPDs over 300-1100 nm wavelength range. We also note from Fig. 3 that for a 3 μm-thick cell, the MAPD corresponding to a = 400 nm is 31.9 mA/cm 2 in comparison to the MAPD of 39.05 mA/cm 2 at the optimum lattice constant of 1300 nm. However, as the cell becomes thicker, this difference progressively decreases. For a 20 μm-thick cell, the difference between MAPD at a = 2900 nm and a = 400 nm is only 0.34 mA/cm 2 . For H = 15 μm, the MAPD shows a maximum variation of 0.25 mA/cm 2 over the 1700-3200 nm lattice constant range. The light-trapping performances of 15-20 μm-thick inverted PhC solar cells are extremely robust with respect to lattice constant variation. The total MAPD over the entire 300-1200 nm wavelength range, for the optimum cases of different cell-thickness, are shown in Table 1. The 1100-1200 nm absorption is calculated according to an accurate Lorentz model of experimental c-Si dispersion data as discussed in the "Methods" section. Table 1 also shows that wave-interference based light-trapping in our optimized thin-silicon inverted pyramid PhCs surpasses the ray-optics based Lambertian light-trapping limit. The MAPDs corresponding to the Lambertian limits of different cell-thicknesses are calculated using absorption coefficient from 9 and eq. 1. Here, a denotes the lattice constant of the photonic crystal. The details of the optimum lattice sizes are summarized in Table 1.  Fig. 4 over the 300-1200 nm wavelength range: a thin cell with H = 5 μm and a relatively thicker cell with H = 15 μm. These absorption spectra exhibit multiple resonance peaks and significant absorption in the 900-1200 nm wavelength range, whereas Lambertian cells and planar silicon are weak absorbers of sunlight. These peaks in the absorption spectra originate from purely wave-interference effects, absent in Lambertian light-trapping. To illustrate this point, we show a magnified view of the absorption spectrum of the 5 μm-thick, optimized inverted pyramid PhC cell over the 850-1200 nm wavelength range in Fig. 5(a). The red circles correspond to resonant absorption peaks located at λ = 1110, 1130 and 1176 nm. Figure 5(b-d) show the in-plane Poynting vector plots over the central xz-slice of the inverted pyramid PhC unit cell at these resonances. The energy flow-pattern reveals multiple regions with vortex-like flow and parallel to interface flow of light at these resonances leading to very long dwell-time of photons in the solar cell. On the other hand, Lambertian light trapping assumes that the distribution rays in the cell obeys a probability distribution f(θ) = 1/π cosθ, where θ is the angle that a ray within the cell makes with the cell-surface normal. According to this distribution, propagation of energy near θ = 90° (i.e. parallel to the interface) is insignificant. However, direct solutions of Maxwell's equations show that a significant amount of energy flows close to θ = 90° due to wave-interference based light-trapping in our PhC. Moreover, a ray-optics based picture cannot provide vortices in the power-flow pattern shown in Fig. 5(b-d).

Electronic Optimization
Collection of the photo-generated carriers, before they recombine, is crucial for high power conversion efficiency in solar cells. Accordingly, the emitter, base and FSF regions of the IBC cell require higher doping levels in order to deflect minority carriers from contacts and other surfaces. However, high doping levels in these regions lead to high Auger recombination. Higher doping also reduces the open-circuit voltage due to larger BGN. Therefore, a careful balance between the peak doping concentration and depths of the Gaussian doping profiles is paramount to exploiting the full potential of wave-interference based light-trapping. For carrier-transport optimizations, we use a 2D model of the IBC cell (shown in Fig. 2). The design parameters such as the details of the Gaussian doping profiles and contact widths turn out to be independent of the PhC cell-thickness. For concreteness, we describe in detail the optimization process for a 10 μm thick cell. Figure 6 shows the optimization map for the Gaussian doping profile of the n + emitter region. For this optimization, the peak doping concentration (N p0 ) and doping depth (σ p ) of p + regions are kept fixed at 5 × 10 18 cm −3 and 100 nm (corresponding to a total p + region depth of 370 nm). The specific contact geometry and other parameters used for emitter optimization calculations are given in Table 2. Figure 6 shows that our 10 μm-thick IBC www.nature.com/scientificreports www.nature.com/scientificreports/ cell achieves a conversion efficiency of 30.74% for a peak emitter doping concentration N n0 = 2 × 10 18 cm −3 and emitter doping depth σ n = 220 nm, corresponding to a total emitter depth of 760 nm. This plot reveals that if we increase σ n to 300 nm (keeping N n0 fixed at 2 × 10 18 cm −3 which corresponds to an emitter depth of 1.038 μm), the cell efficiency exhibits a negligible drop of 0.01% (additive). Clearly, N n0 = 2 × 10 18 cm −3 allows a large tolerance toward emitter-fabrication.
For the optimization of p + regions (shown in Fig. 7), we choose N n0 = 2 × 10 18 cm −3 and σ n = 220 nm. For both p + base and FSF regions, we assume the same Gaussian doping profile, characterized by peak doping concentration N p0 and depth σ p . Other simulation parameters used in our base doping optimization are given in Table 2. Our cell achieves 30.75% conversion efficiency for N p0 = 4 × 10 18 cm −3 and σ p = 100 nm. www.nature.com/scientificreports www.nature.com/scientificreports/ The PERC cell described in 4 , has an emitter with N n0 = 3 × 10 18 cm −3 and a total depth of 730 nm. This is very similar to the optimized emitter of our IBC cell. However, the emitter optimization in 4 does not involve BGN and is a result of the balance between Auger recombination and sheet resistance. In contrast, the carriers in our IBC cells travel at most 70 μm lateral distance, unlike PERC cells where, the electrons inside the emitter region travel several hundreds of microns laterally before they reach front contacts. Consequently, sheet resistance is much less in our IBC cell. In addition to Auger recombination, we have included BGN in the present optimization study. This unavoidable effect limits N p0 and N n0 in a practical cell. The BGN-mediated drop in the open-circuit voltage reduces the power conversion efficiency for large N p0 and N n0 . Figure 8(a,b) show the optimization maps for p and n-contact widths with contact SRVs 10 cm/s and 100 cm/s, respectively. The IBC cell is assumed to be 10 μm thick with optimized p + and n + dopings. Table 3 shows the   Table 2. Parameters used in emitter and base doping optimization of IBC cell as described in Fig. 6. The Auger recombination of the carriers are described by improved Auger model of 11 . The BGN of Si and surface recombination at Si − SiO 2 interface are modeled according to the details illustrated in the "Methods" section. www.nature.com/scientificreports www.nature.com/scientificreports/ details of all the simulation parameters used in our contact optimization study. Figure 7(a) reveals that for contact SRV = 10 cm/s and 10 μm emitter-contact width, the optimum value of base contact width, w pcon , is 140 μm. As w pcon increases from 10 μm to the optimum value of 140 μm, the cell efficiency increases by 0.2% (additive). In comparison to this, the variation of the emitter contact width (w ncon ) has even less influence on the power conversion efficiency of the cell. For w pcon = 140 μm, as w ncon increases from 10 μm to 100 μm, the power conversion efficiency of the photonic crystal IBC cell drops only by 0.08% (additive). As we increase the contact SRV to 100 cm/s, Fig. 8(b) shows that the relative influences of the variations in w pcon and w ncon on the cell efficiency remain approximately the same as for contact SRV 10 cm/s. However, the optimum base contact width and maximum power conversion efficiency now have lower values, 110 μm and 30.49%, respectively. Although the contact SRV increased by an order of magnitude, the cell efficiency changes only by 0.25% (additive).
Using the optimum contact widths and doping profiles obtained above, we now study the optimum thickness of our inverted pyramid PhC IBC cell. A thicker cell leads to a higher short-circuit current due to more light absorption but has a lower open-circuit voltage due to increased bulk recombination of photo-generated carriers. This trade-off leads to an optimum cell-thickness for each choice of τ SRH . The SRH lifetime, determined by bulk defects in the c-Si wafer, can vary widely depending upon the quality of the fabrication process. A higher SRH lifetime or lower bulk recombination allows larger solar absorption by using thicker c-Si layer without losing much photo-current in the recombination process. It follows that for a higher SRH lifetime, the optimum cell-thickness is larger and vice versa. In Fig. 9, we consider τ SRH = 0.1,0.5,1 and 10 ms to study the optimum cell-thickness (other simulation parameters appear in Table 4). Figure 9(b-d) show the variation of V OC , J SC and FF with cell-thickness. As the cell-thickness is increased from 3 to 20 μm, the FF of the IBC cell drops by 4% (additive) for τ SRH = 0.1 ms. As τ SRH increases, the drop in the FF becomes smaller for the same range of cell-thickness variation. For τ SRH = 10 ms, the FF becomes almost independent of cell-thickness.    Table 3. Parameters used in contact width optimization of IBC cell as described in Fig. 8. The Auger recombination of the carriers are described by improved Auger model of 11 . The BGN of Si and surface recombination at Si − SiO 2 interface are modeled according to the details illustrated in the "Methods" section.
It is instructive to compare the optimization of our realistic IBC cell with that of a hypothetical, ideal Lambertian cell. Here, we include the same surface recombination mechanism and SRH lifetime as used in our best IBC cell. The hypothetical Lambertian cell is also endowed with the same doping levels as the p-type bulk, n + and p + regions of our IBC cell. The optimum thickness of the Lambertian cell is found to be 90 μm with a maximum conversion efficiency of 28.37% (shown in Fig. 10). Thus, our thin-Si photonic crystal solar cell offers 2.7% (additive) higher conversion efficiency than the limiting efficiency of a Lambertian cell with practical doping configurations and loss mechanisms. Table 5 compares the performance of our inverted pyramid PhC IBC solar cell with the hypothetical Lambertian solar cell.
Lifetime measurements of well-passivated c-Si samples (Fig. 5 in 11 ) have shown that the effective lifetime (τ eff ) of p-type samples with a bulk doping concentration of 5 × 10 15 cm −3 is approximately 10 ms. Since, τ SRH > τ eff (as a result of the relation: τ eff = (1/τ SRH + 1/τ Aug ) −1 ), τ SRH = 10 ms is practically attainable. We now consider our 15 μm-thick IBC PhC cell, with optimized light-trapping, to study the degradation of performance with lower quality electronic parameters. We delineate below, how higher SRH lifetime, poor contact quality and non-optimized FSF, lower the efficiency of our PhC IBC cell.
The effect of higher τ SRH is shown in Fig. 11. Other simulation parameters are same as given in Table 4. As shown in Fig. 11(d), J SC exhibits little variation with τ SRH (only ~0.22 mA/cm 2 over the entire range of 0.1 ms ≤ τ SRH ≤ 15 ms). In contrast, V OC and FF of the cell increase significantly as τ SRH changes from 0.1 ms to Figure 9. Thickness optimization of thin-silicon inverted pyramid PhC IBC solar cells with optimum lattice constants and dual-layer ARCs, given by Table 1. The cell-design parameters for transport computations are given in Table 4. For τ SRH = 0.1 and 0.5 ms, the optimum IBC cells are 7 and 12 μm thick, respectively. For both τ SRH = 1 ms and 10 ms, the optimum cell-thickness becomes 15 μm.  Table 4. Parameters used in thickness optimization of inverted pyramid photonic crystal IBC cell as described in Fig. 9. The Auger recombination of the carriers are described by improved Auger model of 11 . The BGN of Si and surface recombination at Si − SiO 2 interface are modeled according to the details illustrated in the "Methods" section.
1 ms. Within this range of τ SRH , V OC increases from 768 mV to 792 mV and FF increases from 79.95% to 86.24%. For τ SRH > 3 ms, V OC of our IBC cell falls very close to its saturation value of 794.1 mV. Similarly, FF of the cell almost reaches its saturation value of 88.2% as τ SRH becomes larger than 5 ms. Overall conversion efficiency of our 15 μm-thick IBC cell increases steeply from 27.12% at τ SRH = 0.1 ms to 30.3% at τ SRH = 1 ms. Power conversion efficiency of our cell crosses the 31% threshold for τ SRH > 5 ms. Clearly, τ SRH > 1 ms is a prerequisite for photonic crystal IBC cells to achieve efficiency beyond 30%. We now consider the effect of increased contact SRV and non-optimized FSF/BSF on our 15 μm-thick photonic crystal IBC cell. Apart from τ SRH = 10 ms and a variable contact SRV, all other simulation parameters are given by Table 4. Figure 12(a) shows that the power conversion efficiency of our IBC cell with optimized FSF and BSF (i.e. N p0 = 4 × 10 18 cm −3 and σ p = 100 nm) undergoes only 0.3% (additive) drop leading to 30.77% efficiency when the contact SRV is increased from 10 cm/s to 100 cm/s (red curves in Fig. 12). In contrast, the blue curve in Fig. 12(a) shows a 5% (additive) drop in the conversion efficiency for the same change in contact SRV, in a cell with inadequate FSF and BSF (N p0 = 1 × 10 17 cm −3 and σ p = 100 nm in this particular example). When the contact SRVs  Table 4). The Lambertian cells are assumed to have contact SRV = 10 cm/s and τ SRH = 10 ms. The Auger recombination is modeled using the improved Auger model of 11 . For BGN, we use the model illustrated in the "Methods" section. In comparison to a lossless, undoped Lambertian cell with maximum theoretical efficiency of 29.43% and optimum thickness 110 μm 10 , inclusion of practical doping profiles, bulk recombination and surface recombination reduces the maximum theoretical efficiency of the Lambertian cell to 28.37% with an optimum thickness of 90 μm. In contrast, our inverted pyramid PhC IBC solar cell with same design parameters achieves 31.07% conversion efficiency with an optimum thickness of 15 μm.

Conclusions
Through detailed and precise design optimization, we have identified a route to 31% power conversion efficiency in thin-film crystalline silicon solar cells. The architecture consists of a flexible 15 μm-thick c-Si sheet patterned as a square-lattice, inverted micro-pyramid photonic crystal with a grid of interdigitated back contacts. By choosing the micro-pyramid lattice spacing comparable to the wavelength of near-infrared light, it is possible to achieve  Table 4. The power conversion efficiency increases rapidly as τ SRH increases from 0.1 ms to 1 ms.  Table 4). The red curve corresponds to optimum FSF and BSF doping, showing a more gradual drop in the cell efficiency as contact SRV increases. In contrast, a rapid degradation in cell efficiency (blue curve) when FSF/BSF dopings are improperly chosen. (2019) 9:12482 | https://doi.org/10.1038/s41598-019-48981-w www.nature.com/scientificreports www.nature.com/scientificreports/ remarkable wave-interference-based light-trapping throughout the 800-1200 nm wavelength range. Together with an optimized anti-reflection coating, this leads to overall solar absorption in the 300-1200 nm range, well above the so called Lambertian limit. This unprecedented amount of light absorption in a thin-film, indirect band gap semiconductor, exploits the wave nature of light and suggests a paradigm shift in solar cell design. It leads to the remarkable conclusion that thin, flexible, silicon solar cells may outperform their traditional, thick, inflexible counterparts. Given the advanced technologies available for silicon surface passivation, it suggests that thin-film silicon may provide higher power conversion efficiency than any other single material of any thickness.
Our predictions remain robust over a reasonable range of photonic crystal structure parameters as well as a viable range of surface recombination velocities at the silicon-contact interfaces. The vital role of front and back surface fields obtained by realistic doping profiles was delineated. Major deviations from the prescribed profiles were shown to cause rapid degradation of solar cell performance. By elucidating the optimized photonic and electronic architecture, together with deviations from the optimum parameter choices, we provide a detailed roadmap for experimental efforts to realize power conversion efficiency beyond 30% in a thin-silicon solar cell.

Methods
Long wavelength absorption in crystalline silicon. Sub-gap absorption in c-Si arises from two distinct mechanisms. The first is electronic bandgap narrowing (BGN), denoted by ΔE g (in eV). This allows c-Si to absorb sub-gap photons with energies less than E g but above (E g − ΔE g ), where E g is the bandgap energy (in eV) of unperturbed c-Si. In Sentaurus, ΔE g is estimated using Schenk's model 40 and leads to a slight drop in V OC . The second mechanism allows c-Si to absorb photons with energies less than (E g − ΔE g ) and originates from the exponentially decaying Urbach tail below the continuum band edge 41,42 . For non-crystalline solids, static disorder contributes to an exponential band tail of localized states below the electronic band edge. In c-Si, a similar tail of phonon-assisted optical absorption gives rise to mobile electron-hole pairs 43,44 . The sub-gap absorption is characterized by an exponential of the form: α(ν) ~ exp[{hν − E G (T)/E 0 (T)}], where ν is the optical frequency, E G (T) is the downshift of the continuum band edge corresponding to BGN and E 0 (T) is the Urbach slope.
In order to model the sub-gap absorption, we take the frequency dependent dielectric constant of Si over the 1000-1200 nm wavelength range from 52  The fitting parameters ε ∞ , ω pj , Δε j and γ j , given in Table 6, are obtained using an open MATLAB program 53 . We compare the absorption length λ/4πk (where k is the imaginary part of the refractive index) calculated form our fit and that obtained from 52 in Fig. 13. The measured value of Urbach slope of c-Si at 300K is 8.5 ± 1.0 meV 54 . In comparison to this experimental data, microscopic modeling of the optical-absorption edge due to acoustic and optical phonons yields a slope of 8.6 meV 43,44 . The inset shows that the experimental data from 52 exhibits an Urbach slope of 8.6 meV over the 1160-1190 nm wavelength range.  Table 6. Fitting parameters for experimental Si dispersion data of 52 . Figure 13. Fitting of absorption length of c-Si in 1000-1200 nm wavelength range with experimental data obtained from 52 . Fitting parameters are given in Table 6. Inset: Urbach slope exhibited by the experimental absorption coefficient.
www.nature.com/scientificreports www.nature.com/scientificreports/ Surface recombination at insulator-silicon interface. For all our transport calculations, we use SRH statistics for the insulator-Si interface. According to this model, the recombination rate at the Si-insulator interface is given by 46  where, S j0 = v th,j σ j D interface with j = n,p (v th,j is the thermal velocity, σ j is the capture cross-section, D interface is the interface trap density at the oxide-semiconductor interface), n s and p s are electron and hole concentration at the Si surface and n N N E T k T exp( ( )/2 ) and T = 298K. The thermal velocity of holes is slightly lower due to higher effective mass . ⁎ m ( 1 5 ) e . We set D interface = 3 × 10 9 cm −2 according to the measured value of the near-midgap trap density at the Si-insulator interface in 47 . We take σ p = 6 × 10 −17 cm 2 for these traps from the measured data on capture cross-sections (Fig. 6 in 48 ). This figure also shows that the measured value of σ n varies over a large range. The choice of σ n = 6 × 10 −16 cm 2 results in a S n0 that closely approximates the effective SRV of the state of the art measurements in 11 . Accordingly, we choose S n0 ≈ 20.16 cm/s and S p0 ≈ 1.7 cm/s for all our transport calculations.